Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics: Essays in Honour of the Philosophy of Décio Krause (Synthese Library, 476) 3031318390, 9783031318399

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Synthese Library 476 Studies in Epistemology, Logic, Methodology, and Philosophy of Science

Jonas R. B. Arenhart Raoni W. Arroyo   Editors

Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics Essays in Honour of the Philosophy of Décio Krause

Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 476

Editor-in-Chief Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, USA Editorial Board Members Berit Brogaard, University of Miami, Coral Gables, USA Steven French, University of Leeds, Leeds, UK Catarina Dutilh Novaes, VU Amsterdam, Amsterdam, The Netherlands Darrell P. Rowbottom, Department of Philosophy, Lingnan University, Tuen Mun, Hong Kong Emma Ruttkamp, Department of Philosophy, University of South Africa, Pretoria, South Africa Kristie Miller, Department of Philosophy, Centre for Time, University of Sydney, Sydney, Australia

The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology, all broadly understood. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines. Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. In addition to monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.

Jonas R. B. Arenhart • Raoni W. Arroyo Editors

Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics Essays in Honour of the Philosophy of Décio Krause

Editors Jonas R. B. Arenhart Department of Philosophy Federal University of Santa Catarina Florianópolis, Santa Catarina, Brazil

Raoni W. Arroyo Centre for Logic, Epistemology and the History of Science University of Campinas Campinas, Brazil Department of Philosophy, Communication and Performing Arts Roma Tre University Rome, Italy

This work was supported by grants #2021/11381-1 and #2022/15992-8, São Paulo Research Foundation (FAPESP)

ISSN 0166-6991 ISSN 2542-8292 (electronic) Synthese Library ISBN 978-3-031-31839-9 ISBN 978-3-031-31840-5 (eBook) https://doi.org/10.1007/978-3-031-31840-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To professor Décio Krause

Preface

This book is a collected volume of contributions by internationally recognized scholars aiming to discuss themes present in the work of the Brazilian philosopher Décio Krause on the occasion of his 70th birthday. In particular, all the invited authors interacted personally with Krause in the development of his philosophical ideas. Krause is well-known in philosophy of physics circles for his fierce defense of a metaphysics of non-individuals in quantum mechanics, and for his constant advocacy of a need to change classical logic for a non-reflexive logic in the context of quantum theory. This touches on many different topics that are mainstream philosophy today: the ontology of quantum mechanics, metaphysics and science, and the relation of logic with science and metaphysics. The wide spectrum of themes related reflects itself on the wide scope of the contributions present. The contributions collected in the volume are of two broad kinds: either directly related to themes currently explored by Décio, such as the metaphysics of quantum mechanics and the use of non-classical logic to deal with empirical science, or are more directly related to the major issue of philosophical foundations of quantum mechanics, dealing with the logical foundations and interpretations of the theory. Whichever way one goes, there are clear connections with themes that are present in Krause’s work. Besides, not only do the papers gathered here engage with themes of Krause’s work, they also present further developments, criticism, and connections not yet explored. Thus, the reader needs not to expect that this book presents a simple homage to Krause, but rather opens up new themes and new directions to be explored by researchers with interests related to the ones we find in Krause’s work. In brief, the contributions do point to extensions and possible improvements to be made in current research programs that are related to the metaphysics of quantum mechanics and the use of non-classical logics for empirical theories. Non-individuality, as a new metaphysical category, was thought to be strongly supported by quantum mechanics. No one did more to promote this idea than the Brazilian philosopher Décio Krause, whose works on the metaphysics and logic of non-individuality are now widely regarded as part of the consolidated literature on the subject. vii

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Preface

This volume brings together chapters elaborating on the ideas put forward and defended by Krause, developing them in many different directions, commenting on aspects not completely developed so far, and, more importantly, critically addressing their current formulations and defenses by Krause himself. Given that Krause’s ideas do connect directly and indirectly with a wide array of subjects, such as the philosophy of quantum mechanics, more broadly understood, the philosophy of logic and logical philosophy, non-classical logics, metaphysics, and ontology, the reader of this volume will most certainly find important material for the research on logic and foundations of science, broadly understood. All the invited contributors have already worked with the ideas developed by Décio (some of them still work with them), being also distinct authors and extremely relevant in their areas of expertise. We would like to thank all the authors for kindly and promptly accepting the invitation to this volume. Florianópolis, Brazil Rome, Italy June 2023

Jonas R. B. Arenhart Raoni W. Arroyo

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jonas R. B. Arenhart and Raoni W. Arroyo

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Part I Metaphysics and Ontology of Individuality 2

Quantum Individuality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dennis Dieks

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Quasi-structural Realism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steven French

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Not Individuals, Nor Even Objects: On the Ontological Nature of Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olimpia Lombardi

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The Roads to Non-individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jonas R. B. Arenhart and Raoni W. Arroyo

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Understanding Defective Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Moisés Macías-Bustos and María del Rosario Martínez-Ordaz

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A Phenomenology of Identity: QBism and Quantum (Non-)Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Michel Bitbol

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Open Problems in the Development of a Quantum Mereology . . . . . . . . 157 Federico Holik and Juan Pablo Jorge

Part II Logic and Formalism of Indiscernibility 9

Identity and Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Otávio Bueno

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On the Consistency of Quasi-Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Adonai S. Sant’Anna ix

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Contents

Troublesome Quasi-Cardinals and the Axiom of Choice. . . . . . . . . . . . . . . 203 Eliza Wajch

Part III Logic and Philosophy of Quantum Mechanics 12

Six Measurement Problems of Quantum Mechanics . . . . . . . . . . . . . . . . . . . 225 F. A. Muller

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Measuring Quantum Superpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Christian de Ronde

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Recognizing Concepts and Recognizing Musical Themes. . . . . . . . . . . . . . 297 Maria Luisa Dalla Chiara, Roberto Giuntini, Eleonora Negri, and Giuseppe Sergioli

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From Here to Eternity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Newton C. A. da Costa and Francisco Antonio Doria

Contributors

Jonas R. B. Arenhart Department of Philosophy, Federal University of Santa Catarina, Florianópolis, Brazil Graduate Program in Philosophy, Federal University of Maranhão, São Luís, Brazil Raoni W. Arroyo Centre For Logic, Epistemology and the History of Science, University of Campinas, Campinas, Brazil Department of Philosophy, Communication and Performing Arts, Roma Tre University, Rome, Italy Michel Bitbol Archives Husserl, Centre National de la Recherche Scientifique, Ecole Normale Supérieure, Paris, France Otávio Bueno Department of Philosophy, University of Miami, Coral Gables, FL, USA Maria Luisa Dalla Chiara Dipartimento di Lettere e Filosofia, Università di Firenze, Florence, Italy Newton C. A. da Costa Department of Philosophy, Federal University of Santa Catarina, Florianópolis, Brazil Christian de Ronde Philosophy Institute “Dr. A. Korn” Buenos Aires University, CONICET, Buenos Aires, Argentina Engineering Institute, National University Arturo Jauretche, Florencio Varela, Argentina Center Leo Apostel for Interdisciplinary Studies, Brussels Free University, Brussels, Belgium Dennis Dieks History and Philosophy of Science, Utrecht University, Utrecht, The Netherlands Francisco Antonio Doria Institute for Advanced Studies, University of São Paulo, São Paulo, SP, Brazil Steven French School of PRHS, University of Leeds, Leeds, UK xi

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Contributors

Roberto Giuntini Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Cagliari, Italy Federico Holik Instituto de Fsica (IFLP-CCT-CONICET), Universidad Nacional de La Plata, La Plata, Argentina Juan Pablo Jorge Facultad de Filosofía y Letras, Universidad de Buenos Aires, CABA, Argentina Instituto de Filosofía, Universidad Austral, Pilar, Argentina Olimpia Lombardi CONICET and University of Buenos Aires, Buenos Aires, Argentina Moisés Macías-Bustos Philosophy Department, University of MassachusettsAmherst, Amherst, MA, USA National Autonomous University of Mexico, Mexico City, Mexico María del Rosario Martínez-Ordaz Program of Postgraduate Studies in Philosophy, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil F. A. Muller Erasmus School of Philosophy (ESPhil), Erasmus University Rotterdam (EUR), Rotterdam, The Netherlands Eleonora Negri Scuola di Musica di Fiesole, Fiesole-Florence, Italy Adonai S. Sant’Anna Department of Mathematics, Federal University of Paraná, Curitiba, Brazil Giuseppe Sergioli Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Cagliari, Italy Eliza Wajch Institute of Mathematics, Siedlce University of Natural Sciences and Humanities, Siedlce, Poland

Chapter 1

Introduction Décio Krause, Quantum Mechanics, Non-individuality, and This Volume Jonas R. B. Arenhart and Raoni W. Arroyo

Abstract This chapter is an Introduction to this volume, presenting in brief each of the contributed chapters. We also indicate how the chapters relate to each other and how they connect to themes to be found in the philosophy of Décio Krause.

1.1 This Book This book is a collection of contributions by internationally recognized scholars aiming to discuss themes present in the work of the Brazilian philosopher Décio Krause on the occasion of his 70th birthday. In particular, all the invited authors interacted personally with Krause in the course of his career and during the development of his philosophical ideas. Krause is well-known in philosophy of physics circles for his fierce defense of a metaphysics of non-individuals in quantum mechanics, and for his constant advocacy of a need to change classical logic for a non-reflexive logic in the context of the discussions of the metaphysics of quantum theory. These topics touch on many different subjects that are mainstream philosophy today: the ontology of

J. R. B. Arenhart was partially funded by CNPq. R. W. Arroyo was supported by grants #2022/15992-8 and #2021/11381-1, São Paulo Research Foundation (FAPESP). J. R. B. Arenhart Department of Philosophy, Federal University of Santa Catarina, Florianópolis, SC, Brazil R. W. Arroyo () Centre For Logic, Epistemology and the History of Science, University of Campinas, Campinas, Brazil Department of Philosophy, Communication and Performing Arts, Roma Tre University, Rome, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_1

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quantum mechanics, the connection of metaphysics and science, and the relation of logic with science and metaphysics. Such a broad spectrum of related themes reflects the scope of Krause’s contributions and is reflected in the wide scope of the contributions present here. Perhaps we could summarise the connection of such themes in Krause’s work in the following terms. It all starts with quantum mechanics and the metaphysical understanding of the nature of quantum entities as non-individuals, as advanced by some of the founding fathers, such as Schrödinger. Non-individuality, as a new metaphysical category, was thought to be strongly supported by quantum mechanics. Krause pursued this line of thought with the rigor of logic, building logical foundations for the view that may be correctly qualified as the current standard approach to the view in the literature. It seems appropriate to say that the development of a theory of quasi-sets, an axiomatic set-theory which does away with identity right from the start, in the hands of Krause, contributed to rendering logically respectable the idea of non-individuality and the ‘loss of identity’ talk. Metaphysics, inspired by science and grounded in logic. That brief account clearly indicates that Krause’s ideas do connect directly and indirectly with a wide array of subjects, such as the philosophy of quantum mechanics, more broadly understood, the philosophy of logic and logical philosophy, non-classical logics, metaphysics, and ontology. It is no surprise, then, that the papers contributed to this volume all advance material connected to the research on logic and foundations of science, broadly understood (i.e., including the metaphysical foundations). All the invited contributors have already worked with the ideas developed by Krause (some of them still work with them), being also distinguished and extremely relevant authors in their areas of expertise. The contributions collected in the volume are of two broad kinds: they are either directly related to themes actively explored by Krause, such as the metaphysics of quantum mechanics and the use of non-classical systems of logic to deal with empirical science, or are more directly related to the broader issue of philosophical foundations of quantum mechanics, dealing with the logical foundations and interpretations of the theory. Whichever way one goes, there are clear connections with themes that are present in Krause’s work. But the current volume may also serve another purpose, besides paying tribute to Krause; because not only do the contributions gathered here engage with themes of Krause’s work, but they also present further developments, criticism, and connections not yet explored. In this sense, the present volume brings together chapters elaborating on the ideas put forward and defended by Krause, developing them in many different directions, commenting on aspects not completely developed so far, and, more importantly, critically addressing their current formulations and defenses as advanced by Krause himself. Thus, the reader needs not expect that this book presents a simple homage to Krause. Rather, it opens up new themes and new directions to be explored by researchers with interests related to the ones we find in Krause’s work. In brief, the contributions point to extensions and possible improvements to be made in current research programs related to the metaphysics

1 Introduction

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of quantum mechanics and the use of non-classical logics for empirical theories. Let us check.

1.2 Its Structure The book is structured in three Parts: the ontology and metaphysics of nonindividuality; the logical aspects of quasi-set theory; finally, the general philosophy of quantum mechanics. Here is a brief survey of the Chapters.

1.2.1 Part I Opening the first Part with a challenge to the non-individuals view, in Chap. 2 Dennis Dieks presents an account of the individuality debate that takes into account pragmatic aspects of the discussion and the metaphysical profile of individuality of quantum objects. As he argues, non-individuality (as per the Received View) is the result of an attempt to logically specify the metaphysics of quantum objects (e.g. through quasi-set theory, or Schrödinger logics). According to the pragmatic perspective, however, a metaphysics associated with individuality (what he calls the Alternative View) should be preferred from the point of view of working physicists. Chapter 3 is a piece by Steven French, a long-standing defender of structural realism, which suggests that quasi-set theory offers a formal framework for a new kind of structural realism. After carefully surveying different kinds of structural realism (viz. the moderate and the radical ones), French holds that quasi-set theory supports a moderate kind of structural realism in which quasi-sets play the relations between non-individual objects; hence the metaphysical “thisness” of quantum nonindividual objects is mitigated by the formalism of quasi-sets—thus reducing the strength of questions regarding the nature of such objects. In this sense, such kind of structuralism may be cashed out in terms of moderate structural realism in which the ontological category of objects is not eliminated—as in radical versions of structuralism (e.g. French’s own view)—but the metaphysical nature of such objects need not to be spelled out given their “thin” metaphysical nature. In Chap. 4, Olimpia Lombardi offers a thorough development of her ontology of properties, and how such an ontology is able to deal with some of the greatest difficulties present in the quantum foundations, namely: contextuality, non-separability, and indistinguishability. In this ontology, quantum objects are eliminated: there are no individuals (nor non-individuals) and not even objects; the fundamental ontology is one of properties that do not require objects to carry them at the fundamental level. In Chap. 5, Jonas R. Becker Arenhart and Raoni Arroyo present a methodological (or metametaphysical) discussion in which it is argued that the very meaning of the metaphysical concepts “individuality” and “non-individuality” does not come from physics or even from logic. Rather, they are described by such disciplines once

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they are available, but are not derived from them. In this sense, metaphysics, both for individuals and non-individuals, floats free from physics and from logics, and should be understood as an additional layer of explanation. Consequently, we should not expect such disciplines to determine this metaphysical profile of individuality: neither individuals nor non-individuals come from quantum mechanics nor quasi-set theory. Moisés Macías-Bustos and María Martínez-Ordaz present, in Chap. 6, an epistemological discussion on the metaphysics of non-individuality. The authors argue for a way to frame the debate concerning metaphysical underdetermination (e.g. whether quantum-mechanical objects are individuals or non-individuals) as a source of understanding defective theories. In Chap. 7, Michel Bitbol develops a meta-ontological interpretation in which ontological notions—such as “individual quantum particles” are understood within a phenomenological, specifically Husserlian, framework. To do so in the context of quantum ontology, Bitbol builds a bridge between Husserlian phenomenology and QBism on the basis that each of these views prioritizes how we expect to experience our experience. On the phenomenological side, this is done by the two-way relation between the meanings given by subject and the object poles of existence; on the QBist side, such a meaning is translated by expectation values and probabilities. As a result, Bitbol states that ontology should be understood as something that we, as humans endowed with perception, impose actively on the objects of our human experience, and not the joints of nature which we passively cut in doing science and philosophy. In particular, the non-individuality of the (Received View of) quantum objects is a metaphysical profile that depends on certain physical and philosophical assumptions, ones that determine the content of our metaphysics and ontology— rather than considering metaphysics and ontology to be a passive description of the outer world. Closing the first Part, Federico Holik and Juan Pablo Jorge advance in Chap. 8 some ontological perspectives for the development of a mereology for quantum objects, that is, the study of how discernible macroscopic objects can be formed by indiscernible quantum objects. Taking quasi-sets as a basic logical framework, Holik and Jorge point out the main difficulties for such a “quantum mereology”— the most challenging being the possibility of preparing superpositions of states with particles of different cardinalities.

1.2.2 Part II Opening Part II, Otávio Bueno articulates in Chap. 9 a concern according to which identity is a necessary prerequisite for quantification. Making sense of his long-time defence of the fundamentality of identity—not in a metaphysically robust sense, e.g. that identity is the ground of being, but in the sense of its being crucial for conceptual systems, as he is also a long-time defender of structural empiricism— Bueno’s chapter has far-reaching consequences for the non-reflexive family of non-

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classical logical systems, e.g. quasi-set theory, which do away with identity right from the start. As such, he offers a reductio for the very idea of the intelligibility of a quasi-set theory. Chapters 10 and 11 are complementary chapters, for both present a discussion on the role of the Axiom of Choice in the consistency of the quasi-set theory. In Chap. 10, Adonai Sant’Anna identifies severe difficulties in using the Axiom of Choice in quasi-set theory, pointing to the indiscernibility between current quasiset theory (with the Axiom of Choice) and the usual Zermelo–Fraenkel with Urelemente (ZFU) set theory. Sant’Anna indicates the following dilemma: it is necessary either to abandon or else to modify the Axiom of Choice in quasi-set theory. It is precisely the second horn of the dilemma that Eliza Wajch takes up in Chap. 11. Wajch modifies the axioms of quasi-set theory (.Q), proposing an alternative quasi-set theory (.Q∗ ) that does not have the Axiom of Choice as an axiom—thus, addressing the difficulties posed by Sant’Anna. Such an alternative quasi-set theory, however, is defined to serve as a model of ZFU, so it is an open problem whether .Q∗ responds to the problems posed by Sant’Anna to .Q, viz. the one of its being indiscernible from ZFU. We then close this volume with Part III, which is a collection of papers touching on the general philosophy of quantum mechanics (Chaps. 12 and 13) and are writings of personal friends of Décio Krause, who dedicated their contributions to him (Chaps. 14 and 15).

1.2.3 Part III Opening this last Part of the volume, Fred Muller presents in Chap. 12 the most complete survey so far of the measurement problem in quantum mechanics—arguably the most pressing issue in quantum foundations. Muller rigorously distinguishes between six measurement problems, ranging from the (mathematical) troublesome projection algebras to the (metaphysical) reality problem. In Chap. 13, Christian de Ronde criticises the measurement problem (the first one, according to Muller’s taxonomy offered in Chap. 12), identifying its presuppositions rooted on logical positivism. More precisely, de Ronde argues that the measurement problem as we know it is a version of the famous myth of the given in which the notion of “experience” thought of as given and unproblematic. Then, de Ronde offers a newfangled “realist” account of the quantum reality, inspired in the Einsteinian conception according to which the notion of “experience” is something to be derived from one’s conceptual framework—rather than be something presupposed by it. Chapter 14 is a kind of homage paid by Maria Luisa Dalla Chiara, Roberto Giuntini, Eleonora Negri, and Giuseppe Sergioli to Décio Krause, in which the authors identify several similarities between quantum machine learning process and musical patterns.

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Closing this book, Chap. 15 is also a kind homage from some of Krause’s dearest friends, Professor Newton da Costa and Francisco Doria. The authors conjecture the existence of inaccessible cardinals through fast-growing functions. In this short and speculative chapter on the axiomatic foundations of mathematics, da Costa and Doria explore the philosophical implications and the search for meaning for the existence of some mathematical entities implied by set theory. Acknowledgments We thank all the authors for kindly—and promptly—accepting the invitation to contribute to this volume.

The Selected Work of Décio Krause Arenhart, J. R. B., & Krause, D. (2014a). From primitive identity to the non-individuality of quantum objects. Studies in History and Philosophy of Modern Physics, 46, 273–282. Arenhart, J. R. B., & Krause, D. (2014b). Why non-individuality? A discussion on individuality, identity, and cardinality in the quantum context. Erkenntnis, 79, 1–18. Arenhart, J. R. B., & Krause, D. (2019a). Does identity hold a priori in standard quantum mechanics? In D. Aerts, et al. (Eds.), Probing the meaning of quantum mechanics. Singapore: World Scientific. Arenhart, J. R. B., & Krause, D. (2019b). Is Identity Really so Fundamental?. Foundations of Science, 24, 51–71. de Barros, J. A., Holik, F., & Krause, D. (2017). Contextuality and indistinguishability. Entropy, 19(9), 435. French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Oxford University Press. Krause, D. (1992). On a quasi-set theory. Notre Dame Journal of Formal Logic, 33(3), 402–411. Krause, D. (2000). Remarks On quantum ontology. Synthese, 125, 155–167. Krause, D. (2005). Structures and Structural Realism. Logic Journal of the IGPL, 13(1), 113–126. Krause, D. (2010). Logical aspects of quantum (non-)individuality. Foundations of Science, 15, 79–94. Krause, D. (2011). A calculus of non-individuals (ideas for a quantum mereology). In L. H. A. Dutra & A. M. Luz (Eds.), Rumos da Epistemologia: Linguagem, Ontologia e Ação (Vol. 10, pp. 92–106). Florianópolis: NEL/UFSC. Krause, D. (2017). Quantum mereology. In J. Seibt, G. Imaguire, & S. Gerogiorgakis (Eds.), Handbook of mereology. Berlin: Philosophia Verlag. Krause, D. (2019). Does Newtonian space provide identity to quantum systems?. Foundations of Science, 24, 197–215. Krause, D. (2022). Models and modeling in science: The role of metamathematics. Principia, 26(1), 39–54. Krause, D., & Arenhart, J. R. B. (2016). The logical foundations of scientific theories: Languages, structures, and models. Abingdon: Routledge. Krause, D., Arenhart, J. R. B., & Bueno, O. (2022). The non-individuals interpretation of quantum mechanics. In O. Freire, Jr. (Ed.), The Oxford handbook of the history of quantum interpretations (pp. 1135–1154). New York: Oxford University Press. Krause, D., & Coelho, A. M. N. (2005). Identity, indiscernibility, and philosophical claims. Axiomathes, 15, 191–210. Krause, D., & French, S. (1995). A formal framework for quantum non-individuality. Synthese, 102, 195–214. Krause, D., & French, S. (2007). Quantum sortal predicates. Synthese, 154, 417–430.

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Krause, D., Sant’ Anna, A. S., & Sartorelli, A. (2005). On the concept of identity in ZermeloFraenkel-like axioms and its relationship with quantum statistics. Logique et Analyse, 48(189/192), 231–260. Krause, D., Sant’ Anna, A. S., Volkov, A. G. (1999). Quasi-set theory for bosons and fermions: Quantum distributions. Foundations of Physics Letters, 125, 67–79.

Part I

Metaphysics and Ontology of Individuality

Chapter 2

Quantum Individuality Dennis Dieks

Abstract Décio Krause is one of the staunchest defenders of the “Received View” of “identical quantum particles”, i.e. quantum particles of the same kind. According to the Received View identical quantum particles do not possess individuating properties: they are entities without identity. Still, they are “different” from each other in the weak sense that there can be more than one of them. As Décio Krause has pointed out, such identity-less objects must be handled by a non-standard set theory—quasi-set theory, a subject to which he has made important contributions. In this Chapter we compare and contrast the ideas of the Received View with those of a rival interpretation that has recently started to attract attention. According to this Alternative View quantum particles are emerging entities—at a fundamental level, the world does not consist of particles. But once emerged, quantum particles of the same kind are distinguishable and possess identity, so that they can be dealt with by standard set theory and ordinary mathematics.

2.1 Introduction In a much-quoted article Schrödinger (1950) concluded that one of the most important differences between classical and quantum particles is that quantum particles fail to be individuals: they cannot be separately identified and lack “sameness” over time. In order to explain his point Schrödinger appealed to the difference between classical and quantum statistics, using a by now celebrated analogy. Three schoolboys, Tom, Dick, and Harry, deserve a reward. The teacher has two rewards to distribute among them. . . It is a statistical question to count the number of different distributions. The point is that the answer depends on the nature of the rewards. Three different kinds of reward will illustrate the three kinds of statistics.

D. Dieks () History and Philosophy of Science, Utrecht University, Utrecht, Netherlands e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_2

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D. Dieks (a) The two rewards are two memorial coins with portraits of Newton and Shakespeare respectively. The teacher may give Newton either to Tom or to Dick or to Harry, and Shakespeare either to Tom or to Dick or to Harry. Thus there are 3 times 3, that is 9, different distributions (classical statistics). (b) The two rewards are two shilling-pieces (which, for our purpose, we must regard as indivisible quantities) . They can be given to two different boys, the third going without. In addition to these three possibilities there are three more: either Tom or Dick or Harry receives 2 shillings. Thus there are six different distributions (Bose–Einstein statistics). (c) The two rewards are two vacancies in the football team that is to play for the school. In this case two boys can join the team, and one of the three is left out. Thus there are three different distributions (Fermi–Dirac statistics). Let me mention right away: the rewards represent the particles, two of the same kind in every case; the boys represent states the particle can assume. Thus, “Newton is given to Dick” means: the particle Newton takes on the state Dick.

Schrödinger continues by explaining that quantum particles of the same kind always follow either pattern (b), in which case they are called bosons, or pattern (c), fermions, and never classical (Maxwell–Boltzmann) statistics (a). The relevant difference between case (a), and cases (b) and (c) is that the rewards of (a) are distinguishable: it makes a difference whether Newton or Shakespeare is given to Tom. By contrast, it makes no difference which shilling Tom receives: in both cases he ends up possessing one shilling.1 So, the morale of the story is that applicability of one of the three types of statistics provides evidence concerning the identity of the objects we are dealing with: objects without identity, for which permutations make no sense, obey statistics of either type (b) or type (c), and not of type (a). Perhaps the most interesting case is that of fermions, since these are the building blocks of ordinary matter. In Schrödinger’s analogy fermions correspond to vacancies in a football team. This is meant to illustrate what it means for fermions to lack identity, namely that it makes no sense to inquire about differences resulting from switching fermions, in the same way as it is meaningless to discuss permutations of vacancies. The use of vacancies in a football team may be a bit misleading here, since such vacancies are different from each other if they are understood as corresponding to different field positions (goal keeper, etc.). Instead, one should think of club membership tout court, without any distinctions. Such vacancies constitute a pure absence of properties, so the analogy tells us that fermions are likewise without any distinguishing characteristics. A person can occupy only one membership vacancy; in the analogy this reflects Pauli’s exclusion principle according to which no two fermions can be in the same state (only two mutually orthogonal states can fill a two-fermion vacancy).

1 Schrödinger speaks about coins, which might create misunderstandings because actual coins will be distinguishable, for example by initial positions, trajectories and scratches on them. In randomized repeated distributions of the coins this would lead us back to the statistics of case (a). The intention of the simile becomes clearer when money in a bank account is considered— scriptural (deposit) money. If the teacher has an account containing two euros, and transfers exactly one euro to the account of one of the boys, it does not make sense to ask which of the teacher’s original two euros has been transferred.

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Although fermions represented by vacancies lack individuating properties, there can be different numbers of them. However, there can be no physical procedure that distinguishes them, assigns them individual names, puts them into one-to-one correspondence with the natural numbers, or traces them over time. In their seminal book on Identity in Physics (French & Krause, 2006) Steven French and Décio Krause call the view that quantum particles of the same sort cannot be distinguished from each other in the way explained by Schrödinger and lack identity, but can nevertheless present themselves in different numbers, the Received View. Décio Krause has made important contributions to the development of a formal scheme—quasi-set theory—able to handle the Received View’s identityless objects. Quasi-sets of such objects possess well-defined total numbers of members, but these members cannot be individually identified or numbered. By contrast, in standard set theory each element a of a set always has its own identifying property, defined by membership of the singleton set .{a}. This discrepancy between non-standard quasi-set theory applying to quantum particles and the standard set theory that lies at the basis of the theory’s mathematical framework has led Krause (2019) to suggest that there exists a tension in the framework of present-day mathematical physics, which it would be important to remove. In what follows we will argue, however, that the Received View’s central idea that identical quantum particles lack identity is not the only interpretative option. We will draw attention to an Alternative View according to which quantum particles possess an identity grounded in physical properties, like classical particles, so that they can be handled by standard set theory. Thinking of identical quantum particles as physically distinguishable entities is in line with physical practice, in which it is standard to refer to, e.g., electrons that are fired from a particular electron gun, are part of a certain experimental setup, travel from a particular source to a specific detector, and so on and so forth. One might even wonder whether the notion of a particle without identity is not a contradiction in terms: the very concept of a particle, and more generally an object, in daily life and in the practice of physics is bound up with identifiability. Conversely, in the absence of identity conditions speaking of separate objects seems esoteric. For example, it would be strange to insist that the balance of a banking account consists of separately existing “euros without identity”, instead of thinking of an undivided total sum of money (cf. Dieks, 2021).

2.2 Schrödinger’s Analogy and the Quantum Mechanical Formalism of Identical Particles Schrödinger’s parable has a counterpart in the quantum formalism of “identical particles”, i.e. particles of the same kind. It is a basic postulate of quantum mechanics that states of more than one identical particles must be either symmetric (bosons) or anti-symmetric (fermions). Here, we will focus on fermion states—

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fermions (protons, neutrons and electrons, for example) are the elementary building blocks of matter. A typical two-fermion state built up from the one-particle states .|ψ and .|φ is the following anti-symmetrized product state: 1 | = √ {|ψ1 |φ2 − |φ1 |ψ2 }. 2

.

(2.1)

This state is an element of the tensor product Hilbert space .H1 ⊗ H2 . The labels 1 and 2 in Eq. (2.1) refer to the one-particle Hilbert spaces (the “factor spaces”) .H1 and .H2 in this tensor product. Because the physical interpretation of these labels will turn out to be a bone of contention later, it should be stressed that the meaning of the labels within the mathematical formalism is not subject to doubt: the labels differentiate two isomorphic copies of the one-particle Hilbert space .H, namely the first and second as written down, from left to right, in the tensor product. In order to connect this formalism to Schrödinger’s story about Tom, Dick, Harry and the two vacant places in the football team, we have to imagine that there are three available states, .|φ, .|ψ and .|ζ , say, from which anti-symmetric two-particle states in .H1 ⊗ H2 are to be constructed. This gives us the following possibilities for the composite state: 1 | = √ {|ψ1 |φ2 − |φ1 |ψ2 }, . 2 1 | = √ {|ψ1 |ζ 2 − |ζ 1 |ψ2 }, . 2 1 | = √ {|ζ 1 |φ2 − |φ1 |ζ 2 }. 2

.

(2.2) (2.3) (2.4)

So, exactly as stated by Schrödinger, there are three possible cases, each corresponding to a different one-particle state that is left out. Experiment confirms that in scenarios in which there are no dynamical factors that favor one of the one-particle states .|φ, .|ψ and .|ζ  over the others, the states (2.2), (2.3) and (2.4) occur with equal probability .1/3. This shows that only the number of available oneparticle states plays a role in statistical considerations. The labels .1, 2 are irrelevant in the sense that their permutation does not yield a new physical state. More generally, if there are n mutually orthogonal states available, the  number of possible k-fermion states with an antisymmetrized product form is . nk , which leads to the prediction of Fermi–Dirac statistics; each of these states is completely determined by the specification of the one-particle states occurring in it. The counterparts of Schrödinger’s membership vacancies in this formalism are the factor spaces in the tensor product Hilbert space of the total system. The vacancies were “holders” of persons who, as explained by Schrödinger, are meant to stand for quantum states. The vacancies are property-less absences of persons, pure possibilities of occupation; similarly, the factor spaces represent possibilities of state realization. One might say that the factor spaces are waiting for the selection

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of a specific state from all possible states definable in them, just as Schrödinger’s vacancies were waiting for the selection of one specific person from the set of all possible candidates. The Hilbert spaces are labeled in the quantum formalism, but this labeling only serves the purpose of representing the spaces in linear order in the equations. It is a purely conventional device without physical meaning: permutations of the spaces or renumbering them makes no difference. Schrödinger’s vacancies could similarly be included in a numbered list, but as in the quantum formalism these numbers would only serve the purpose of creating an “administrative” spatial order on paper and would say nothing about the vacancies themselves. In principle, it would be better to speak about one two-person vacancy, instead of vacancy 1 and vacancy 2. Continuing Schrödinger’s analogy, the Hilbert space labels function as “bare particles”, waiting to be “dressed” by a state. There is a reason to be worried about such property-less bare particles, however. Schrödinger seems to take it for granted that there must be a kind of receptacle that can “hold” the particle state and that can exist independently of that state. But it seems against the spirit of modern physics to accept something like a property-less substratum as the basis of physical properties. Do we really want to say that an independently existing bare particle, represented by the mere possibility of taking on physical properties, waits to be invested with a specific particle state? Thinking this way seems an echo of a metaphysics that has long fallen into disrepute within physics.2

2.3 The Received View The core thought of the Received View is the interpretation of states of the form (2.2)–(2.4) as representing two particles, each associated with exactly one of the two labeled one-particle factor spaces—so that we may speak of particle 1 and particle 2. The symmetry of equations (2.2)–(2.4) implies that each of these two particles is in exactly the same state, a mixed state that can be obtained by the procedure of “partial tracing”. This sameness of physical states entails that switching the labels has no physical significance: all statistical predictions of quantum mechanics are invariant under permutations of particle labels. French and Krause (2006) comment (p. 143): from the point of view of the statistics, the particle labels are otiose. The implication, then, is that the particles can no longer be considered to be individuals, that they are, in some sense, ‘non-individuals’. This conclusion expresses what we have called the ‘Received View’: classical particles are individuals but quantum particles are not. . . . As we shall see in the rest of the book, one can in fact go beyond mere metaphor and underpin the Received View with an appropriate logico-mathematical framework.

2 Actually, substrata, bare particles, haecceities, etc. are explicitly rejected also by Décio Krause (2019).

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If the particle labels are otiose, as French and Krause conclude, the question naturally arises whether one should think of these labels as referring to particles at all—this is the route we are going to explore. But French and Krause, and the Received View, take a different turn. They accept that all quantum particles of the same sort are in exactly the same physical state and possess exactly the same physical properties. These particles can therefore not be distinguished and individuated by any physical procedure. Still, they can exist in different numbers, so they violate the principle that entities sharing all their physical properties must be one and the same. In chapter 7 of Identity in Physics (French & Krause, 2006) Décio Krause outlines a formal framework for dealing with these unusual entities without individual identities. This framework takes the form of “quasi-set theory”, a variation on ordinary set theory. Typically, elements of quasi-sets will lack the individuating property of self-identity. This contrasts with the situation in standard set theory, in which each element always has the individuating property of belonging to the singleton set of which it is the only member. Therefore, one of the important differences between ordinary (ZFC) set theory and quasi-set theory is that in quasiset theory the possibility of forming arbitrary singleton sets is excluded by axiom. Another striking difference with standard set theory is that quasi-sets with only identity-less members will have a “quasi-cardinality” indicating the total number of its elements, but will not have an ordinal number, since identity-less entities cannot be ordered and numbered. This axiomatic theory of quasi-sets is an admirably ingenious construction that provides the Received View with a fitting mathematical basis.

2.4 An Alternative View Nevertheless, the notion of a particle not possessing any individuating properties setting it off against other objects is difficult to reconcile with the original meaning of the word “particle”, a tiny part sitting at a definite place in a bigger whole. The application of the term particle in classical physics (and in daily life) rests precisely on the presence of this type of individuality: classical particles are impenetrable localized entities that travel along well-defined spatial trajectories. They can consequently be distinguished from each other at any instant of time (synchronically) and can also be followed over time (diachronically). This makes it possible to label classical particles in a physically meaningful way: such labels represent identifying physical properties, like the initial positions the particles came from. The use of the concept “particle” in the practice of quantum physics resembles the use in the classical context. Paradigm examples refer to such cases as paths that can be observed in bubble chambers, single electrons that can be trapped in a potential well and may be kept there for a long time, or an electron gun firing a single electron that a bit later is detected on a screen, after which a second electron is fired,

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and so on. In such situations the attribution of identity to particles is considered a matter of course, even in the case of identical quantum particles like electrons. There are also situations in which the use of the term particle seems inappropriate, and in which it appears better to think in terms of an undivided whole, or sometimes a wave. Think again of the Schrödinger-like analogy of bank money: if there is a sum of money in the account it does not make sense to conceive of this total buying power as made up of separately existing though indistinguishable and identity-less units. This seems analogous to what happens in certain boson systems, in particular Bose–Einstein condensates. In cases where particle talk is used in physical practice, these particles are not defined by reference to abstract labels, but they are directly associated with observable physical quantities. This is the cue that has led to the proposal of a rival to the Received View, which we will call the Alternative View. In this Alternative View the one-particle states occurring in composite states like (2.2)–(2.4), rather than the Hilbert space labels, serve to identify single particles in a many-particle system. Lubberdink (1998) was perhaps the first to suggest that it is consistent to interpret states of the form (2.1) as representing two individual particles, possessing well-defined individual properties, one characterized by the state .|φ and one characterized by the (orthogonal) state .|ψ. Ghirardi et al. (2002) devoted an extensive analysis to this interpretive option. They proved that anti-symmetrized product states like (2.1), although having the form of entangled states, nevertheless behave in many respects like simple product states representing individual particles. In the particle interpretation that ensues, the particles are represented by the orthogonal one-particle states occurring in the antisymmetric fermion state. These one-particle states correspond to individuating physical properties (via the observables of which they are eigenstates). This new way of interpreting many-particle states has been further developed by a number of authors, for example: Dieks and Lubberdink (2011), Caulton (2014), Friebe (2014), Bigaj (2015), Dieks (2020), Dieks and Lubberdink (2020), Bigaj (2022). We refer to this literature for many of the details. That this alternative approach is closer to the actual practice of physics than the Received View may be illustrated by considering an anti-symmetric state of the following form: 1 | = √ {|L1 |R2 − |R1 |L2 }, 2

.

(2.5)

where .|L and .|R represent narrow spatial wave packets on the far left hand side and the far right hand side of an experimental setup, respectively. According to the Alternative View this state represents two particles, one located on the left and one located on the right; the states .|L and .|R do the job of identifying the particles in this alternative approach. These two particles are clearly distinguishable; they possess synchronic identities based on their different locations. They may possess

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diachronic identity depending on circumstances relating to details concerning interactions, decoherence, etc.3 This is in accordance with how states of this kind are interpreted in physical practice. Of course, this practice is strongly suggested by the fact that measurements will only detect particle presence in the disjoint regions L and R. By contrast, according to the Received View the state (2.5) represents two particles with exactly the same location, “smeared out” evenly over L and R.4

2.5 Particles and Their Identity In the last three sections of his essay What is an elementary particle? Schrödinger (1950) elaborated on the characteristics of quantum particles. The particles defined earlier in his essay, represented by money units without individuating properties (bosons), or property-less club vacancies (fermions), do not possess an identity: permuting them does not even make sense as they cannot be individually addressed. It similarly does not make sense to speak about the identity over time of such property-less entities. However, Schrödinger acknowledges that in practice quantum particles are distinguished all the time, and seeks to explain this. As he points out, particle states are well-defined individuals and can be used to distinguish fermions-with-a-state, even though particles per se, without states, do not possess identity. Concerning diachronic identity Schrödinger notes that quantum states evolve continuously in time, which sometimes makes it possible to track a particlewith-a-state on the basis of a string of successively occupied states. As he puts it Such a string gives the impression of an identifiable individual, just as in the case of any object in our daily surrounding. It is in this way that we must look upon the tracks in the cloud chamber or in a photographic emulsion, and on the (practically) simultaneous discharges of Geiger counters set in a line, which discharges we say are caused by the same particle passing one counter after another. In such cases it would be extremely inconvenient to discard this terminology. There is, indeed, no reason to ban it, provided we are aware that, on sober experimental grounds, the sameness of a particle is not an absolute concept. It has only a restricted significance and breaks down completely in some cases.

For particle identity over time to be a valid concept, strings of successive events forming a trajectory must not be too close to each other, so that these trajectories do not intermingle inextricably. This requirement can be translated into a rough and ready criterion for the applicability of the notion of an identifiable particle. In order to distinguish a particle from its neighbors in space we must be able to locate it within a region . x that is much smaller than the average distance from other particles, denoted by L. But this entails an uncertainty in the momentum, .p = mv, with v the speed, of the order .h/x (because of the uncertainty relation).

3 We

will discuss diachronic identity more extensively in Sect. 2.5. to the mixed one-particle state .1/2(|LL| + |RR|).

4 Corresponding

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Because of this lack of sharpness of momentum and speed values, the vagueness of the particle’s position grows. If one requires that this lack of spatial determination remains well below L even after the particle has covered a distance of the order L, one arrives at the inequalities x  L,

.

v.L/v = (h/m.x).L/v  L, so that p.L  h.

.

(2.6)

So particles can be distinguished by their trajectories if their momenta are much larger than .h/L, a condition easy to satisfy for objects in everyday situations. Inequality (2.6) is in fact known as a criterion for the applicability of classical particle theory to the description of gases. Schrödinger’s focus on trajectories in three-dimensional space as a means for defining particle identity is reasonable, since the most prominent property of a particle in classical physics is its well-defined spatial position at any instant. One would expect quantum particles to mimic the behavior of classical particles in the classical limit of quantum theory, and Schrödinger’s argument goes some way to making it understandable how classical particles emerge from the quantum world. However, there is a mechanism, not yet well understood in Schrödinger’s days, that helps even better to understand why “position” has such a special role in classical physics, namely the process of decoherence. Decoherence may be described as the result of frequent interactions with the environment. Since interaction Hamiltonians are typically local, such interactions can be considered to be repeated position measurements by the environment, with the result that components of the measured quantum state associated with different spatial positions get correlated with (almost) orthogonal environment states. As a result, the state of the particle by itself (after tracing out environmental degrees of freedom) becomes a mixture of localized states. So, localization will be the most important identity-conferring particle property in many circumstances, in particular in the classical limit. Still, localization is not the only way of defining particle identity. In the case of fermions of the same kind the one-particle states that occur in an anti-symmetrized product state of the many-particle system are all mutually orthogonal and appear only once in the total state—the latter feature expresses Pauli’s exclusion principle. If we associate single particles with these orthogonal one-particle states, the resulting particles are completely distinguishable at any instant of time, since orthogonal states can always be told apart. These particles may also possess identity over time, diachronical identity or genidentity. This is the case, for example, when there is no interaction between the fermions described by an anti-symmetrized product state, so that each one-particle state evolves unitarily. Orthogonal states remain orthogonal to each

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other under unitary evolution, so that in this case an N-fermion system corresponds to N orthogonal one-particle states that trace out distinguishable paths in the total Hilbert space. However, as in the case of identification by trajectories, the existence of genidentity cannot be guaranteed. Wave packets of interacting particles may intermingle to such an extent that it becomes meaningless to ask which of the original particles is the same as any given post-collision particle. Moreover, it should be noted that antisymmetrized product states are special cases of many-particle fermion states, and that in general we will have to deal with superpositions of such product states. In this general case the strategy of defining distinguishable one-particle systems as building up the total system may well fail, which signals a breakdown of the applicability of the identifiable particle concept. It follows that particles, with both synchronic and diachronic identity, should not be considered as fundamental in quantum mechanics; rather, such particles are emergent entities (Dieks & Lubberdink, 2011, 2020; Dieks, 2020, 2023). Only if certain conditions are fulfilled, with inequality (2.6) as an example, does the notion of a particle become applicable. In his essay Schrödinger explains that it is exactly in situations where distinguishable properties can be assigned to physical systems that a particle picture arises. By contrast, situations in which distinguishing properties cannot be attributed to elementary physical systems are often better described as “wave-like”. With these observations Schrödinger comes close to our Alternative View. His remarks may be understood as a recommendation to reserve the term “particle” exactly for situation in which individuating states manifest themselves, and to drop the term “particle” in other cases. However, at the end of his essay Schrödinger falls back on the idea that particles are property-less bearers of states, like the membership vacancies representing fermions, and warns of the misconception that particles could possess a real identity. Schrödinger’s analysis is typical of the ambiguities besetting identical-particle discussions in the literature: on the one hand there is the practice of speaking about specific electrons entering a specific device and following a trajectory— here electrons are associated with distinguishing one-particle states. On the other hand there is the tradition, especially in literature of a more philosophical nature, of defining particles via identity-less bearers of properties—in Schrödinger’s membership analogy formless vacancies, in the Received View Hilbert space labels. These two ways of thinking about particles correspond to the Alternative View, and the Received View, respectively. In the Alternative View there are no particles without at least synchronic identity. The price to be paid for this introduction of identity is that so-called “many-particle systems” cannot always be interpreted as assemblies of single particles-withidentity. The particles of the Alternative View are emergent, which implies that the particle concept has a restricted validity according to this view.

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2.6 Décio Krause on Particle Identity Décio Krause is not only one of the originators of the term “Received View”, he is also one of the staunchest defenders of the View’s core ideas. Of course, he is aware of the fact that in the practice of physics elementary particles are often described as individuals, with properties that set them apart from other particles of the same kind. For example, in 1989 Hans Dehmelt, together with Wolfgang Paul, was awarded the Nobel prize in physics for isolating individual quantum particles by means of the so-called ion trap technique. Van Dyck et al. (1986) had even been able to keep a single positron in such a trap during three months; in a sense, Dehmelt had built up a relation with this specific positron and had called her “Priscilla”. In 2012 a Nobel Prize was awarded to Serge Haroche and David Wineland “for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems”. The Nobel citation stated (Royal Swedish Academy of Sciences, 2012) that “The Nobel Laureates have opened the door to a new era of experimentation with quantum physics by demonstrating the direct observation of individual quantum particles without destroying them.” Continuing Dehmelt’s program, Haroche and his research group have even been able to trap a single photon for a long time. The statements made by these physicists and by the Nobel committee contradict the doctrine that particles of the same kind cannot have distinguishing characteristics. Krause (2011, 2019) has taken up this challenge. He concludes that, contrary to appearances, such experiments with trapped particles, if properly analyzed, confirm that identical quantum particles do not possess identity. Comparing Krause’s Received View analysis with the one provided by the alternative interpretation we have outlined clarifies both interpretative options. An important element of Krause’s criticism of Dehmelt’s interpretation of the Priscilla experiment is that it is impossible to know which positron is located in the trap: it does not make any physical difference to the situation if Priscilla is replaced by an arbitrary other particle of the same kind—so who is Priscilla? Clearly, it is only spatio-temporal location that in Dehmelt’s account makes Priscilla stand out as an individual. Krause (2019) criticizes this spatio-temporal way of individuating a positron in the following way. He writes: suppose there are two quite similar traps in Dehmelt’s laboratory, both with a positron trapped in. Which one is Priscilla? Well, you can say: that one in the trap near the window. But suppose Dehmelt is absent from his laboratory for a moment, during which the students exchange the traps one another, and leave the light sufficiently weak so that Dehmelt doesn’t perceive the new configuration of his laboratory. Then the students ask him: where is Priscilla? For sure Dehmelt will point to the trap near the window and say: “She’s there!”. Without a careful analysis of the trap (not the positron), he cannot distinguish between the two positrons. Only by distinguishing between the traps, by noticing that they were changed, Dehmelt will be able to say that Priscilla is not that positron, but the another one.

This argument highlights several differences between the Received View and the Alternative View. When the students move the two traps with the two positrons inside, the Received View says that at each instant the two positrons are in exactly

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the same state, evenly smeared out over the two trap positions. The final situation after the swap is the same as the initial situation as far as the two positrons are concerned; only by investigating the traps, here to be taken as macroscopic and distinguishable devices, can it be shown that something has changed in the total configuration. This accords with Décio Krause’s description.5 By contrast, the Alternative View says that in the initial situation there is one positron with a position near to the window, and another positron farther away from the window. These two positrons are individuals, represented by two different and orthogonal states corresponding to narrow wave packets in the respective traps. These positrons remain trapped in their own traps while these are being switched by the students. In the final situation the positrons have been switched together with their traps, so it is an objective fact that the positron near the window is not Priscilla. It will not be easy for Dehmelt to find out whether the positrons have been switched or not, especially if the light is turned down, as stipulated in the quoted passage. But from this practical difficulty nothing follows about the identity of the positrons. Suppose that Dehmelt, smart as he is, mistrusts his students and has a camera installed in the lab. Looking at what has been recorded, it will be very easy for him to ascertain which positron is Priscilla. Indeed, the question being asked is about the relation between the initial and final stages of a swapping procedure, so it is a question about diachronic identity. It is not remarkable that considering the situation at only one instant of time may be insufficient to answer questions about diachronic identity; it is only to be expected that facts about what has happened in the past are needed. The final two positrons situation is the same as when everything started; but this does not demonstrate that it is without meaning to think that they have been exchanged. In the above story it is implied that the two trapped positrons followed continuous trajectories; this makes it possible to establish their points of origin. Consulting the camera footage will remove doubts about the location of Priscilla. Décio Krause asks the question “which positron is the one in Dehmelt’s trap?” This question, pertaining to synchronic identity, presupposes that one can speak of the identity of particles independently of their states (in this case the narrow wave packets inside the traps). This is an assumption that is typical of the Received View, as we have noted before: because the Received View starts by associating labels with particles, one may ask “Is it particle 1 or particle 2 that is located in the trap near the window?”. By contrast, in the Alternative View this question cannot be sensibly asked, because here the identity of particles is defined by their physical properties. So Priscilla is by definition the particle in Dehmelt’s trap, and it is without meaning to ask the further question “which one of all the positrons in the universe could this be?”.

5 The final sentence of the quoted passage seems to say that the positrons can be distinguished by the trap they are in, which agrees with the Alternative View but not with the Received View; but I take it that Krause is pointing out that Dehmelt must be arguing this way—which is mistaken according to the Received View (though common in practice).

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A similar analysis, devoted to a situation with two electrons trapped in two infinite potential wells at a distance from each other, 1 and 2, is presented in Krause (2019). The following anti-symmetrized product state is chosen to represent the situation:  1  ψ12 (a, b) = √ ψ1 (a)ψ2 (b) − ψ1 (b)ψ2 (a) , 2

.

(2.7)

defined in the Hilbert space .H1 ⊗ H2 where the sub-indexes name the wells and a and b ‘name’ the particles. Important to recall that the use of (anti-)symmetric vectors intends precisely to make these labels not conferring identity to the particles.

The idea is that .ψ1 and .ψ2 are wave functions that vanish everywhere except in trap 1 and trap 2, respectively. That a and b are “naming” the particles is typical of the Received View; a and b in Eq. (2.7) are the labels of the two factor Hilbert spaces in the tensor product space of the system (so they should equal the indices .1, 2 in .H1 ⊗ H2 ). A small calculation is sufficient to show that there is no interference between the wave packets in the two potential wells.6 However, according to the Received View this lack of interference and the linked distinguishability of the two wave packets is irrelevant for the distinguishability of the two particles. This contrasts with the Alternative View, which says that there are two independent particles, each particle in one potential well, associated with the two orthogonal states .ψ1 and .ψ2 , respectively. There is no need for independent names a and b in the Alternative View, the one-particle states do all the identification work. Décio Krause briefly considers this alternative way of thinking about particle identity when he comments: “The most we can do is to identify the particle (calling it a) with well 1”. He quickly follows this up with: “but it would be indifferent for all physical purposes if in well 1 the particle was b instead”. However, according to

6 The

interference term in the probability coming from .|ψ12 |2 is written as .ψ1 (a)|ψ1 (b)H

1

.ψ2 (b)|ψ2 (a)H2 = 0,

(2.8)

with the comment “it is supposed that .ψ1 (b) = ψ2 (a) = 0, for b is out of well 1 and a is out of well 2.” Strictly speaking, however, the expression on the left-hand side of Eq. (2.8) is not mathematically well-defined, because it involves inner products between states belonging to different Hilbert spaces. Thinking of a and b as names of particles given independently of the states has here led to the picture that .ψ1 , in well 1, assigns probability 0 to the presence of particle b, which is outside well 1. The correct counterpart to Eq. (2.8) reads .ψ1 (1)|ψ2 (1)H

1

.ψ2 (2)|ψ1 (2)H2 = 0,

(2.9)

which shows that it is the orthogonality of .ψ1 and .ψ2 that is responsible for the absence of interference. Thus, no arguments about particles existing independently of these states are needed.

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the Alternative View it is self-contradictory to think that the particle in trap 1 might in fact be the particle in trap 2. It may be added, more generally, that the weight of permutability arguments in order to cast doubt on the presence of individuals, depends on details of the case. Think of Max Black’s notorious pair of iron spheres, qualitatively the same and at some distance from each other in Newtonian space. It would not make any physical difference to the situation if, in thought, we instantaneously swapped the two spheres. The implication of this fact is simply that the spheres are qualitatively the same; they differ only in their locations. In this case it would be incorrect to go further and conclude that the spheres lack identity. The spheres can be distinguished synchronically by their different positions in Newtonian space,7 and if classical mechanics applies they have continuous trajectories, which can be used to trace them over time. In an actual physical swap, in which the spheres switch their positions by means of a dynamical process, they will surely retain their identity. For the purposes of physics in our world these common-sense remarks about synchronic and diachronic identity suffice, and there is no need to let our thoughts stray to other worlds and to ask questions about trans-world identity. According to the Alternative View the two positrons in Dehmelt’s lab are individuals with different positions, even though they are permutable like Black’s spheres. By contrast, according to the Received View all properties of the two positrons, including their positions, coincide, and only arbitrary labels without physical significance distinguish them “on paper”. Although Alternative-View particles resemble Black’s classical spheres in some respects, there remain essential differences. Generally, these quantum particles possess only a restricted diachronic identity, as already explained by Schrödinger. They may get annihilated or created, and if elementary quantum particles interact it may happen that after their interaction there is no fact of the matter concerning which post-collision particle is the same as one of the particles before the interaction. Compare and contrast this with what Krause (2019) writes about the meaning of identity: An individual, thus, is something that has identity, in the sense of possessing an identity card, . . . a document that enables us to identify the object as such in whatever situation or context. The particular object is the unique one with such an identity card, and every other object presents a distinction from it.[italics added]

This corresponds to unrestricted identity, something that is not subject to “decay”, extends to other worlds, and certainly is retained in collision processes. In using the notion of identity this way Krause follows (and quotes with approval) Toraldo di Francia, who disqualified restricted identity as “mock identity” (Toraldo di Francia, 1985; Dalla Chiara & Toraldo di Francia, 1993); see for more discussion concerning mock identity (Dieks, 2021). Thus, Krause (2019) writes, while con-

7 The situation is more complicated within a relationist scenario, in a Leibnizean space. In this case the identity of the spheres could be grounded with the help of “weak discernibility”, see, e.g., Dieks and Versteegh (2008).

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templating particles identified by the potential wells in which they are trapped (i.e., Alternative View particles): Although trapped in the infinite wells, they have only what Toraldo di Francia has termed mock individuality, an individuality (and, we could say, a ‘mock identity’) that is lost as soon as the wells are open or when another similar particle is added to the well (if this was possible). And this of course cannot be associated with the idea of identity. Truly, there is no identity card for quantum particles. They are not individuals, yet can be isolated by trapping them for some time.

It can only be admitted that quantum particles generally do not live forever, that processes of creation and annihilation may occur, and that in situations where there is synchronic distinguishability diachronic identity may still fail.8 In the Alternative View this is not problematic, because according to this view particles are not fundamental; they are emergent entities. That means that the particle concept is only applicable within a restricted range of situations. If the notion of identity starts failing to apply, the concept of a particle itself stops being valid. A final point to take into account is that according to a widely shared consensus in the physics community macroscopic objects are themselves basically quantum mechanical. It is true that in everyday situations quantum behavior of the objects surrounding us will not be noticeable, but this is only because of the enormously small probabilities of macroscopic quantum phenomena, which in turn is mainly due to the omnipresence of decoherence. But there is a growing list of recent laboratory results that confirm the existence, in principle, of macroscopic quantum effects. From a conceptual point of view the implication is that the identity of macroscopic objects, like tables and chairs, cannot have a character that is essentially different from that of the identity of elementary particles. The difference is one of degree, relating to time scales and measures of decoherence. Thus, even macroscopic objects cannot possess anything more than a “mock identity”.

2.7 Conclusion The Received View, according to which identical particles are identity-less objects, has been dominating theoretical and philosophical thinking about quantum mechanics for some time. With Décio Krause’s quasi-set theory this View has obtained a solid logico-mathematical underpinning. Still, there is a viable rival view, which we have called the Alternative View. The core thought of this Alternative View is that particles are emergent entities identified by distinguishable one-particle quantum states. Alternative View particles

8 Although the situation is not nearly as hopeless as suggested in the quotation: for example, opening of potential wells does not automatically lead to a loss of identity. As we noticed in Sect. 2.4, the states of non-interacting fermions trace distinguishable paths in Hilbert space, even if these fermions are not spatially confined in any way.

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are therefore distinguishable and possess identities; they are objects whose treatment does not require the introduction of non-standard mathematics. That these particles are emergent means that, according to the Alternative View, the particle concept cannot always be validly used in the description of physical processes. In some circumstances it is better to speak of wave-like phenomena, or of undivided wholes instead of “many-particle systems”. On a fundamental level a quantum field ontology (largely still to be developed) may be most suitable. In cases where particle talk is appropriate the Alternative View diminishes the distance with respect to physical practice. Moreover, it adds to conceptual clarity by avoiding the notion of identity-less objects.

References Bigaj, T. (2015). Exchanging quantum particles. Philosophia Scientiae, 19, 185–198. Bigaj, T. (2022). Identity and indiscernibility in quantum mechanics. Cham: Palgrave Macmillan. Caulton, A. (2014). Qualitative individuation in permutation-invariant quantum mechanics. arXiv:1409.0247v1. Dalla Chiara, M. L., & Toraldo di Francia, G. (1993). Individuals, kinds and names in physics. In G. Corsi, M. L. Dalla Chiara, & G. C. Ghirardi (Eds.), Bridging the gap: Philosophy, mathematics and physics (Vol. 140, pp. 261–283). Boston studies in the philosophy of science. Berlin: Springer Science + Business Media. Dehmelt, H. (1990). Less is more: Experiment with an individual atomic particle at rest in free space. American Journal of Physics, 58, 17–27. Dieks, D. (2020). Identical quantum particles, entanglement, and individuality. Entropy, 22, 134. https://doi.org/10.3390/e22020134 Dieks, D. (2021). Identical particles in quantum mechanics: Against the received view. arXiv:2102.02894v1 [quant-ph]. Dieks, D. (2023). Emergence and identity of quantum particles. Philosophical Transactions A. https//doi.org/10.1098/rsta.2022.0107 Dieks, D., & Lubberdink, A. (2011). How classical particles emerge from the quantum world. Foundations of Physics, 41, 1051–1064. Dieks, D., & Lubberdink, A. (2020). Identical quantum particles as distinguishable objects. Journal for General Philosophy of Science. https://doi.org/10.1007/s10838-020-09510-w Dieks, D., & Versteegh, M. A. M. (2008). Identical particles and weak discernibility. Foundations of Physics, 38, 923–934. French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Oxford University Press. Friebe, C. (2014). Individuality, distinguishability, and (non-)entanglement: A defense of Leibniz’s principle. Studies in History and Philosophy of Modern Physics, 48, 89–98. Ghirardi, G., Marinatto, L., & Weber, T. (2002). Entanglement and properties of composite quantum systems: A conceptual and mathematical analysis. Journal of Statistical Physics, 108, 49–122. Krause, D. (2011). Is Priscilla, the trapped positron, an individual? Quantum physics, the use of names, and individuation. Arbor, 187, 61–66.

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Krause, D. (2019). Does Newtonian space provide identity to quantum systems? Foundations of Science, 24, 197–215. Lubberdink, A. (1998). De individualiseerbaarheid van identieke deeltjes. (In Dutch. English title: The individualizability of identical particles.) Master Thesis, Utrecht University. http:// gradthesis.andrealubberdink.nl. An English summary is available at https://archive.org/details/ arxiv-0910.4642 Royal Swedish Academy of Sciences. (2012). Scientific background on the Nobel Prize in Physics 2012: Measuring and manipulating individual quantum systems. Press release. www. nobelprize.org/prizes/physics/2012/press-release/ Schrödinger, E. (1950). What is an elementary particle? Endeavour, 9, 109–116. Reprinted in: Schrödinger, E. (1952). Science and humanism. Cambridge: Cambridge University Press. Also In Castellani, E. (Ed.) (1998). Interpreting bodies: Classical and quantum objects in modern physics. Princeton: Princeton University Press. Toraldo di Francia, G. (1985). Connotation and denotation in microphysics. In P. Mittelstaedt & E. W. Stachow (Eds.), Recent developments in quantum logics (pp. 203–214). Bibliografisches Institut Manheim. Van Dyck, R. S., Schwinberg, P. B., & Dehmelt, H. G. (1986). The electron and positron geonium experiments. In R. S. Van Dyck & E. N. Fortson (Eds.), Proceedings, 9th International Conference on Atomic Physics: Seattle, USA, July 24–27, 1984 (pp. 53–74). Singapore: World Scientific.

Chapter 3

Quasi-structural Realism Steven French

Abstract Devising an appropriate formal framework for structural realism has long been an issue in the development of this position. Décio Krause has suggested that quasi-set theory might offer such a framework and here I explore that possibility in the context of so-called ‘moderate’ and ‘radical’ forms of Ontic Structural Realism (OSR). However, although the central claims of the former can indeed be captured by quasi-set theory, I argue that these claims cannot bear the metaphysical weight placed upon them and conclude that the search for an appropriate formal framework for OSR remains open.

3.1 Introduction Over the past 35 years or so, structural realism has become one of the dominant positions in the realism-antirealism debate. As is now well-known, it broadly divides into two versions: epistemic structural realism, which, also broadly, states that all that we can know, is structure (Worrall, 1989); and ontic structural realism (OSR), which insists that all that there is, is structure (Ladyman, 1998). One of the questions that is most often asked about this position (asked so often in fact that I’m not going to bother with any citations here!) is the following: What is this structure that we are supposed to be realists about? As I’ve pointed out in what might be seen as a companion piece to this paper, it is remarkable that critics of this view never seem to bother to ask the same question of their own, ‘object-oriented’ stance (French, forthcoming-a, hopefully). It is almost as if they think that the notion of ‘object’ is so metaphysically transparent that no such question needs to be asked of it (reader, it isn’t and it does!). In that other paper I adopted the ‘toolbox’ view of the relationship between science and metaphysics, according to which the latter can be thought of

S. French () School of Philosophy, Religion and History of Science, University of Leeds, Leeds, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_3

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as providing various devices and frameworks with which the theories of the former can be furnished (see French & McKenzie, 2012). From this perspective, there are a range of such devices that could be used to metaphysically ‘flesh out’ the notion of structure in different ways. So, for example, we could understand it as built up, in ‘bottom-up’ fashion, through the ‘fusion’ of properties in a certain sense, or approaching it from the ‘top-down’, as it were, view it in terms of the determinability-determinate relationship, with properties as the determinates of the relevant laws, taken as determinables; or when it comes to the modal force of structure, we could appropriate a recently developed notion of ‘potentiality’ for example, or we could adopt a Humean stance and deny that it has any such force at all (French, forthcoming-a). Indeed, there are various options, each of which could be conceived of as yielding a different answer to the question, what is structure? That might seem alarming and as a further example of metaphysical underdetermination (French, forthcoming-b) but, again, the same can be said of the notion of ‘object’, about which there have been centuries of reflection on the relevant possible options (Rettler & Bailey, 2017). However, it might be thought that the latter notion is comparatively less problematic precisely because it can be situated in a familiar formal framework, namely that of standard set theory. Here, to put it simply, one begins with a given set of objects and then defines relations between them, in the usual extensional manner, giving what is typically thought of as the associated ‘structure’. Indeed, Sider articulates his sense of structure in terms of first-order quantification plus (Zermelo–Frankel) set theory, with a dash of fundamental physics (Sider, 2011). But of course, this strategy retains at its very heart the object-oriented stance that the ontic structural realist seeks to do away with! Now, here we need to take a little more care, and within Ladyman’s general characterisation of structural realism as shifting focus away from objects (Ladyman, 2020), distinguish two broad forms of ontic structural realism (OSR). There is the more radical, ‘eliminativist’ version (EOSR) that seeks to remove the notion of ‘object’ from the picture completely and then there is the more moderate form (MOSR) that incorporates only a very ‘thin’ notion of object-hood. Let us consider each in turn and, in particular, reflect on what formal frameworks might be available in each case.

3.2 Eliminativist OSR It is perhaps worth recalling two of the central motivations for ontic structural realism: the first is to accommodate theory change in science without running afoul of the so-called ‘Pessimistic Meta-Induction’ (Worrall, 1989). The relevant structures are then seen as providing the necessary common elements that remain throughout such changes, thereby providing an appropriate target for the realist to latch onto. The second has to do with accommodating certain core features of modern physics, such as the role of symmetries for example (Ladyman, 1998).

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When it comes to quantum mechanics, the latter includes Permutation Symmetry, which, broadly speaking, states that permutations of particles of the same kind are not observable. At its very inception, this was taken to imply that quantum particles should not be regarded as individuals, which became the ‘Received View’ for many years, despite the absence of an appropriate set-theoretic framework. However, this symmetry principle can also be made coherent with an ontology in which the particles are still regarded as individuals, but with the caveat that the principle must be understood as a ‘super-selection rule’ that divides up the relevant Hilbert space into distinct sectors such that particles are constrained with regard to their behaviour in aggregate. As a result, then, we obtain a kind of metaphysical underdetermination generated by the role of Permutation Symmetry in quantum mechanics according to which we have two contrasting ontological packages: according to one, quantum particles can be understood as individual objects but subject to certain constraints on what states they can occupy; on the other, they can be understood as non-individual objects, where this can be formally accommodated via quasi-set theory (French & Krause, 2006). This underdetermination can be ‘cut off at the knees’, as it were, by eliminating from the ontological base the very notion of ‘object’ that it relies upon, and, thereby, adopting the eliminativist form of OSR (EOSR). Now, these two motivations – circumventing the Pessimistic Meta-Induction and accommodating the core features of modern physics – may not be equally well satisfied (French, 2006). Thus, it has been suggested that in the transition from classical to quantum mechanics, there are not the relevant structures that one can be realist about (although see Thébault, 2016). In that case it might be that we should focus on the second motivation and take OSR as the form of realism best suited to modern physics. That would mesh with certain recent moves in the realism debate that urge a shift in focus away from ‘global’ forms of the position that can handle the Pessmistic Meta-Induction and towards more local forms that are ‘bespoke’ in the sense of being tailored to the specific theory considered (Saatsi, 2017). Such a move would also obviate the concerns that have been expressed in the past about using category theory as a formal framework for OSR. So, category theory may seem to offer a way of representing the shift in focus from objects to structures that is central to OSR, because a category is characterised by its morphisms and not the relevant objects, with the latter regarded as secondary at best, or as definable in terms of, and consequently but more radically perhaps, reducible to, the former. However, although we could certainly characterise theories in category-theoretic terms (see da Costa & French, 2003, p. 26), it is not clear that the framework can perspicuously capture the kinds of inter-theory relationships that are presented as a response to the Pessimistic Meta-Induction. Of course, if we were to drop such a response as a motivation for OSR and shift to a local form of realism, as suggested above, then this concern would evaporate. That still leaves the second motivation, however, namely capturing the relevant structural features of our current most successful theories. Unfortunately, here too category theory may not be up to the job, as the relevant objects may not end up being entirely eliminated (Lal & Teh, 2017; Lam & Wuthrich, 2015). It turns out that the objects are eliminated from only some ‘models’ of the theory which is

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translated into category-theoretic terms and, even more problematically, relations may be eliminated as well. As a result, the category-theoretic representation of ‘structure’ we end up with is set at a formal distance from what we are trying to capture, since it is basically characterised in terms of category theoretic morphisms holding between category theoretic objects and these are not the same as relations between elements (of such category theoretic objects) which is what the structural realist is concerned with. All of which is to say that as a device for formally representing what the structural realist is concerned with, in a way that will help provide an answer to our initial question, category theory may not be that useful. An alternative option is take the notion of structure as primitive, and attempt a direct characterisation of it (Muller, 2010). So, just as the Zermelo–Frankel formalism takes the concept of set as a primitive, introduced via set variables, the notion of structure can be introduced as a primitive via ‘structure variables’, and not be reduced to either sets or category theoretic objects. The structural realist will take these variables to range over all the structures in physical reality, where it is science that tells us which of all the possible structures covered by our theory of structures are actually realized or instantiated. The structural realist can then assert that those predicates in the language of the theory of structures that single out these realized structures provide literal descriptions of them. This is a promising way forward but as I noted in (French, 2014), it remains just that. The final option is one that I have presented many times, so I will only sketch it here. I’ve called it Poincaré’s Manoeuvre (French, 2014, pp. 66–68) because it was used by Poincaré in his presentation of the group-theoretic approach to geometry (Poincaré, 1898). There he noted that we get to the group-theoretic characterization by starting with the standard geometric objects, considering the relevant transformations they can undergo and then once we have achieved that characterization, we can discard the objects themselves as simply a kind of ‘crutch’ used to enable us to get to the former. We can find a similar move deployed in the structuralist context by Eddington (1941) and, more recently, in support of nominalism, by Melia (2000). The advocate of EOSR can in turn appropriate it to capture the notion of structure they are focused on, while still using the familiar formal apparatus of standard set theory. So, the core idea is that we begin with our set of putative objects – quantum particles, say – denoted by A; we can then define a family of relations R over this set and form a set-theoretic structure .A, R in the usual way. However, from the structuralist perspective, this should be read, ontologically speaking, from right to left, so that it is the relations that have metaphysical priority, with the objects understood as constituted in terms of the former, so they can, in effect, be eliminated. The set A thus functions as a kind of formal ‘crutch’ enabling us to express the structure in set-theoretic terms but which can then be metaphysically discarded. This manoeuvre then meshes precisely with the iterative metaphysical stance adopted towards the particles themselves: we begin by regarding them as physical objects, between which various relations can be taken to hold, as expressed in the physics by the relevant laws and symmetries, but then the eliminativist shifts the focus around

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and takes the latter as ontologically prior, with the particles conceptualized in terms of them and hence as having lost their objecthood. This allows the ontic structural realist to assert the ontological primacy of the fundamental relations constituting the relevant structure, without having to come up with an entirely new formalism. However, it does suggest a view of relations that differs from the standard account. According to the latter, relations may be contrasted with properties in that ‘[ . . . ] relations aren’t relations of anything, but hold between things, or, alternatively, relations are borne by one thing to other things, or, another alternative paraphrase, relations have a subject of inherence whose relations they are and termini to which they relate the subject.’ (MacBride, 2020). Note the emphasis here on things – if there were no things, there could be no relations. Within that standard account, there are, of course, an array of different views of relations but even those that downplay the significance of things, such as primitivist or so-called positionalist views, retain this emphasis. So, take the latter for example: such views hold that relations should be understood as possessing argument positions, which can be filled in various ways by various things, thereby accommodating in a relative way certain higher-order features of relations, such as their possession of converses and so on (ibid.). But such positions are also generally regarded as entities themselves (Fine, 2000) so this sort of account appears to offer little comfort to the eliminativistically inclined ontic structuralist, although it does bear comparison to the ‘moderate’ stance to be discussed below. Having said that, so-called ‘relative positionalism’ may offer some basis for the more radical stance (Donnelly, 2016). According to this view, the positions are to be construed as relative properties, akin to the property of ‘being North of’, for example (ibid.). Crucially, this view ‘[ . . . ] construes positions as properties, not as extra relata of a relation. On this version of positionalism, positions are not independent items that need to be appropriately linked with relata in a given application of a relation. Rather, it is through having particular properties relative to one another that the relata are linked to one another in the way of a particular application of the relation’ (ibid., p. 98). Now of course, as is the case, still, with much of current metaphysics, the examples given of the relata in these discussions are typically ‘every-day’ examples, such as persons and buildings, and the relations considered are those such as ‘loves’ or ‘is North of’ and so it may seem that even on this view, we haven’t escaped the necessity of positing objects, since that is how the relata are understood in this particular context. But we have left that context far behind and when it comes to quantum mechanics, the ‘particular properties’ mentioned above are themselves to be reconceptualized in structuralist terms. So, first of all, Permutation Symmetry, expressed mathematically by the permutation group, yields the division of ‘elementary particles’ into the two fundamental kinds, namely fermions (to which electrons belong) and bosons (to which photons, for example, belong), which correspond to two of the irreducible representations of the group. In addition, if we encompass in our considerations the relativistic context of quantum field theory, we must also include the symmetries of Minkowski spacetime which are represented mathematically via the Poincaré group, the irreducible representations of which yield a classification of all elementary particles, with

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these representations indexed or characterised by mass and spin (the invariants of the group). Thus, the significance of these symmetries motivates a structuralist understanding of the so-called ‘intrinsic’ properties such as mass, charge and spin, where the relevant structure is represented group-theoretically (French, 2014). We can also extend this understanding to non-intrinsic properties, like energy or spin direction (as in spin ‘up’ or ‘down’), although here complications arise insofar as this understanding can be regarded as interpretation-dependent.1 Having said that, consider Mermin’s claim that ‘Correlations have physical reality; that which they correlate does not.’ (1998, p. 753). He takes this as ‘removing the mystery’ from quantum mechanics, in the same way as the statement ‘Fields in empty space have physical reality; the medium that supports them does not’ (ibid.) does for classical electrodynamics. And he goes on to argue that this is the outcome of the strategy of taking the formalism of quantum mechanics as given and inferring from the theory itself what it is trying to tell us about reality (ibid.).2 At the heart of the argument is the claim that it is the well-known no-hidden-variables theorems that ‘[ . . . ] deny physical reality to a complete collection of correlata underlying all these correlations’ (ibid., p. 760).3 Now, what he means by ‘correlata’ here are the particular values for the relevant observables (‘spin-up’ or ‘spin-down’, say) and it might be objected that the above claim does not amount to denying physical reality to the objects themselves. However, in that case the relationship between said objects and their properties needs carefully spelling out and, certainly, the standard ‘bundle’ account, according to which an object is nothing but a bundle of such properties, is going to run into difficulties. Of course, it could be maintained that an object simply is whatever it is that lies over and above such properties but then these objects, which are deemed by the opponent of EOSR to be conceptually necessary for an understanding of what structure is, are going to amount to nothing more than the bearers of such properties and relations. This is a ‘thin’ view of objecthood indeed, and one that I shall return to when I consider ‘moderate’ OSR below.4 What we get, then, is something like the following: the structure can be understood as relational, where the relations can be metaphysically analysed within this 1 It

is for this reason that I did not go into details in (French, 2014), leading to criticism by Esfeld in his review, for example (Esfeld, 2015), although in fact I think that most, if not all, current interpretations of quantum mechanics can be given a structuralist ‘spin’. 2 Interestingly this is the same strategy that was explicitly adopted by Everett and there are useful comparisons to be drawn between the latter’s ‘relative state’ interpretation and Mermin’s relational one. 3 Mermin also addresses the issue of ‘relations without relata’ and invokes consciousness in this context: ‘Consciousness enters into the interpretation of quantum mechanics because it and it alone underlies our conviction that a purely relational physics – a physics of correlations without correlata – has insufficient descriptive power’ (1998, p. 755). This is not a view that should be dismissed as lightly as it has been, although I won’t go into details here. 4 An interpretation similar to Mermin’s can be found in Rovelli’s ‘relational’ account (Laudisa & Rovelli, 2021) for which it has been argued, OSR provides an appropriate philosophical framework (Candiotto, 2017).

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framework of relative positionalism and, hence, as endowed with certain ‘positions’. These in turn should not be understood as ‘extra relata’ or ‘independent items’ but as relative properties supposedly possessed by the putative relata – conceived of prior to this metaphysical analysis as particles but subsequently, as a result of the analysis, regarded as devoid of any objecthood. This yields an account that bears considerable similarity to that of Mertz (2016), where, in effect, reality comes to be analysed as nothing but a network of relations – not quite structures all the way down but structures all across the board. Nevertheless, the ‘non-standard’ features associated with this account must be acknowledged and many contributors to the development of structural realism have come to prefer the non-eliminativist alternative, to which we shall now turn.

3.3 Moderate OSR According to Moderate Ontic Structural Realism (MOSR), on the one hand, the putative objects, as fundamental relata, are conceptually necessary and hence cannot be eliminated, but on the other, all there is to them are the relations that they bear (Esfeld & Lam, 2008, 2010). In particular, their intrinsic properties and identity are given entirely by these relations and thus by the structure. This latter feature is of course consonant with what I’ve said above, so MOSR shares with EOSR the capacity for accommodating the structuralist understanding of the role of group theory in modern physics. However, insisting that, nevertheless, we must retain the objects in order to pin down that structure, as it were, raises the following dilemma: that if all there is to objects are the relations in which they stand, then either they are ontologically so ‘thin’ that nothing can be said about them, qua objects, rendering them metaphysically mysterious; or it must be accepted that, indeed, there is nothing to them at all, in the sense that they ontologically evaporate, as it were, and the position collapses into eliminativist OSR (French, 2010; Chakravartty, 2012). The second horn is, I hope, straightforward. The claim is that if everything about an object, including even its identity, is cashed out in relational terms, then there is nothing there, metaphysically and the object can be dispensed with in favour of a wholly relational ontology. It is the first, perhaps, that needs a little further elaboration and this will indicate how the dilemma might be avoided. So, the advocate of MOSR wants to argue, as we have just noted, that both the intrinsic properties and the very identity of objects are given by certain relations but to make sense of that relational conception, and hence of the very notion of structure, we need to retain objects as the relata. The concern as expressed in the first horn of the dilemma is then that there appears to be nothing more to the objects, metaphysically speaking, other than to act as such relata. And as a result one has to wonder whether, from an ontological perspective, such ‘thin’ entities can carry the metaphysical ‘weight’ placed on them when there is apparently nothing we can say about their nature. I’m going to take this last claim as expressing the sense in which

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such objects are metaphysically mysterious. And my suggestion is that Krause’s quasi-set theory may go some way towards dispelling that sense of mystery.

3.4 Quasi-set Theoretic Structure In his (2005) Krause begins with the afore-mentioned criticism that no answer had – then, at least – been given to the question ‘what is a structure?’ and then suggests that the core principles of OSR indicate that ‘the concept of structure should not depend on the particular objects being structured’ (ibid., p. 113). What he then goes on to offer is ‘[ . . . ] a way of looking at relational structures where the involved relations do not depend on the particular objects being related but depend only on the ‘kind’ (or sort) they are’ (ibid., p. 115). The framework for this way of looking at structures is then provided by quasi-set theory, of course, which presents a relation in terms of a quasi-set of ordered pairs – and thus to this extent mirroring the standard account of relations – such that the elements of the quasi-set are indistinguishable in the sense that they may be exchanged with others of the same kind but the given relation will continue to hold, contrary to the usual conception (ibid., pp. 119–120). The formal details are by now well-known so I’ll just briefly recall them here (see Krause, 2005, pp. 123–125): the founding idea, as it were, is that a quasi-set is a collection of elements about which we cannot say either that they are identical or distinct, and this is not for epistemological reasons but because there are ontological grounds for not taking identity to apply to such elements. Quasi-set theory admits of two kinds of Urelemente: m-atoms, which represent the above elements and M-atoms, which represent entities for which (classical) identity holds. The theory is obtained by applying the standard ZFU (Zermelo– Frankel plus Urelemente) axioms to the domain consisting of m-atoms, M-atoms and aggregates thereof and one can define a translation from the language of ZFU into the language of the theory in such a way as to yield a copy of the former in the latter. This allows us to define the usual mathematical concepts, with ‘sets’ appearing as quasi-sets whose transitive closure does not contain m-atoms. The ‘quantity’ of elements in a quasi-set can be captured by a primitive notion of quasicardinality, which, unlike the classical notion, cannot be associated with ordinality in the usual way. ‘Pure’ quasi-sets contain only m-atoms and the axioms of the theory underpin the claim that there is nothing that distinguishes the elements from one another. Crucially, one such is the axiom of weak extensionality which states that quasisets that have the same quantity of elements of the same sort are themselves indistinguishable. The exchange of a given element with another that is indistinguishable from it yields a quasi-set that is indistinguishable from the original one in the sense of this axiom (Krause, 2005, p. 122). This allows for the accommodation within quasi-set theory of the unobservability of particle permutations, which is a fundamental feature of quantum statistics (not just Bose–Einstein but also Fermi–Dirac) and underpins one of the core symmetries incorporated within Ontic

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Structural Realism, namely Permutation Symmetry, as noted above. Indeed, given this result, it has been suggested that quantum statistics can be obtained within quasi-set theory without the need to postulate such a symmetry (ibid., p. 122; Krause et al., 1999). This framework then captures the idea that the relevant structure may be preserved across changes to the ‘material’ of which it is composed (ibid., p. 125; indeed, Krause takes it to capture Schrödinger’s comments about the ‘form’ or ‘shape’ of things, understood in terms of the relevant invariant properties). Of course, given OSR’s emphasis on such symmetries representing aspects of the fundamental structure of the world, the claim that quantum statistics may be obtained without the need to postulate Permutation Symmetry might be thought to stand in some tension with the aim of providing an appropriate formal framework for this position. However, this tension may be eased through reflection on the distinction between the physical ‘axioms’ of a theory and associated logical or settheoretical ones. According to Krause and Arenhart, the axiomatic basis of a given theory can be taken as encompassing three levels of postulates: the logical, where this may include non-classical logic; the ‘mathematical’ or set-theoretic, where, likewise, this may include non-standard set theories, such as quasi-set theory; and that of ‘specific’ postulates which depend on the field being axiomatized (Krause & Arenhart, 2017, p. 78). It is in the last that we might expect to find the postulate of Permutation Symmetry which is then reflected, as it were, in the axiom of Weak Extensionality at the set-theoretic level. Thus, although there is a sense in which the latter obviates the need to postulate the former, that is only the case if we remain at level 2 of our axiomatisation, which is not a pragmatically viable move to make when it comes to the actual practice of physics itself! Furthermore, the structure that is represented at the third level by such symmetry postulates (together with the relevant laws; see French, 2014), can be taken to be captured at level 2 in terms of the relevant quasi-set theoretic axioms and so the afore-mentioned tension dissipates. Now, of course, the relationships between these three levels are a matter for further discussion (see, again, Krause & Arenhart, 2017) but it must be allowed that there can be a certain degree of flexibility between them. An example of this has already been presented in the case of the ‘Poincaré Manoeuvre’ above but another can be found in the alternative package to that of quantum-particles-asnon-individuals, namely that of quantum-particles-as-individuals subject to certain accessibility constraints. Adopting the latter, quantum statistics can still be reproduced with a metaphysics of individuality but, again as already noted, with Permutation Symmetry understood as imposing certain constraints on Hilbert space which divide it up into different sectors, corresponding to the different particle ‘kinds’, such that any particle in such a sector must remain within it (see French & Krause, 2006, pp. 139–197). This, of course, represents the second ‘horn’ of the metaphysical underdetermination that is one of the motivations for the eliminativist form of OSR. With this package, one would retain the same postulates at ‘level 3’ of our axiomatisation scheme but subject to an alternative understanding, of course, and with standard set-theory at ‘level 2’ (and hence standard, ‘classical’ logic with

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identity at ‘level 1’, where this identity must be understood to be ‘veiled’, in some sense due to the constraints at ‘level 3’). There is now the ‘meta-level’ question of which set of relationships between these levels of axioms is to be preferred. One could adopt a ‘Principle of Smoothness’, according to which the three levels should be ‘smoothly’ inter-related in the sense that they should mesh with each other with as few metaphysical ‘add-ons’ as required. However, it is not clear how precise this notion of ‘meshing’ can be made and hence how much store can be set in such a principle. Certainly, a variety of factors must come into play. So, for example, the advocate of the particles-as-nonindividuals package might argue that the postulates at level 3 can only be made to mesh with standard set theory at level 2 by introducing a certain understanding of those postulates (in terms of constraints, say) which represents such an ‘add-on’. An advocate of the alternative might retort – as indeed they have – that such an apparent cost is more than outweighed by the widespread ‘meshing’ of standard set theory at level 2 with a much broader range of mathematical and scientific postulates. And so the debate would rage (and again, the eliminativist would urge that the entire discussion be cut off at the knees by abandoning such object-oriented stances to begin with!). As it turns out, this whole set of issues also arises in the context of the alternative approach to ‘moderate’ OSR that takes quantum entities to have a ‘contextual’ identity.

3.5 OSR and Contextual Identity As already noted, this form of OSR retains a notion of object, in some sense, thereby allowing it to be underpinned by standard set theory, but understands the identity of such objects to be ‘contextual’, in the sense that ‘[ . . . ] “physical identity’ is the qualitative identity that can be ascribed to physical objects on the basis of the physical description provided by the relevant physical theory’ (Lam, 2014, p. 1158). More specifically – and this is the structuralist aspect – this identity and indeed, the very existence of the objects, is given by the relevant physical relational structure. Crucially, this sense of identity ‘[ . . . ] has to be clearly distinguished from the purely formal (logical, set-theoretic) notion of identity, which the [ . . . ] physical descriptions make use of’ (ibid., p. 1158). This distinction enables the advocate of this view to evade the criticism, that it cannot accommodate there being a determinate number of objects associated with these physical structures. The argument boils down to the following: the standard notion of cardinality is articulated within standard set-theory and is dependent on the usual articulation of identity within the latter; MOSR rejects the latter; hence it cannot avail itself of the standard notion of cardinality and cannot accommodate numerical diversity (Jantzen, 2011; see Arenhart (2012) and Arenhart and Krause (2014) for a response in the quasi-set theoretic context).

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The response runs as follows: moderate OSR and the associated notion of contextual identity are about ‘physical ontology’, in the sense of certain claims about what there is in the physical world. It is such claims that must be kept in focus and ‘[c]onsiderations based on the logico-mathematical framework alone are [ . . . ] inept at producing well-grounded ontological claims about the physical world as described by fundamental physics’ (Lam, 2014, p. 1163). Such logico-mathematical considerations are taken to only acquire physical meaning within the framework afforded by the relevant physical description. Consequently, the assertion of a lack of physical identity at the level of physical theories – that is, at level 3 above – does not imply any lack of a notion of formal identity at the level of the formal representation of such theories – that is, at level 1. Given that the argument above is couched entirely in terms of the latter level, it fails to get a grip here. Nevertheless, the concern remains about how the determinate number of objects at the physical level might be accommodated. One response is to take numerical diversity to be a primitive notion (Lam, 2014, pp. 1165–1168). This might seem odd, not least because such primitiveness is typically associated with the notion of haecceity, which of course is problematic in the quantum context (see, again, French & Krause, 2006). However, here it must be understood as appropriately contextualised, in the sense that the diversity is that of ‘contextual objects’, whose identity and existence is dependent on the relevant structures. Insofar as haecceities are typically understood as independent of such structures, this notion is not implied by that of contextual numerical identity (Lam, 2014, p. 1166). Having said that, it has been argued that if we are to go down the ‘primitive’ route, we might as well opt for primitive intrinsic identity and abandon OSR altogether (Dorato & Morganti, 2013). This is because, first of all, if we accept that primitive intrinsic identity does not imply haecceistic differences either, then the ‘metaphysical cost’ of the former is no higher than that of primitive contextual identity. And secondly, it is claimed, this primitive intrinsic identity is effectively encoded within the formalism of quantum mechanics because a welldefined number of particles is always presupposed in the latter. Given that, it is concluded, it is more straightforward to accept this notion that its contextualised counterpart (ibid.). Clearly, what this amounts to is a form of meta-level claim that primitive intrinsic diversity with the standard set-theoretic underpinning, meshes more smoothly across the levels than primitive contextual diversity with the same underpinning. However, the argument assumes that the presupposition of a determinate number of particles requires a notion of primitive intrinsic identity and, of course, that can simply be denied – the particles can be taken to have a primitive contextual identity and still have a determinate number (Lam, 2014, pp. 1167–1168). Given that, the above claim of smooth meshing falls into doubt – if not implied by numerical diversity, primitive intrinsic identity must be regarded as a metaphysical ‘add-on’ and, it is claimed, one that, furthermore, does not fit so well with the features of the theory that the structuralist emphasises, such as Permutation Symmetry for example (ibid., p. 1168). Of course, this last claim can be disputed, on the grounds outlined above, namely that it is compatible with the theory to adopt the particles-as-

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individuals package but with Permutation Symmetry understood as an accessibility constraint. What we have then is that in addition to the metaphysical underdetermination between particles-as-individuals (where this individuality can be understood as primitive, or otherwise) and particles-as-non-individuals, we have a further underdetermination between different forms of ‘moderate’ ontic structural realism. In both cases we have structure sans individual objects, as usually understood. But in one, we have non-individual objects underpinned by quasi-set theory and in the other we have objects with a form of contextual identity, with this view underpinned by standard set theory. As already indicated, it is not clear what considerations can be drawn upon to determine which position should be preferred. On the one hand, if clear sense can be made of the claim that there should be a ‘smooth’ relationship between the physical postulates of a theory and the underlying set-theoretical axioms, then the non-individuals plus quasi-set theory form might be preferred. On the other, it can be argued that some flexibility in that relationship should be allowed, not least because the theory itself does not require non-individuality, and given that the standard set theoretic framework is applicable beyond the quantum domain in ways that quasi-set theory is not (or at least not so widely), there are strong grounds for adopting the alternative. However, both forms must still face the concern that by eliminating the notion of individuality entirely or just contextualising it, they have ‘trimmed down’ the notion of objecthood to the point where it is too ‘thin’ to carry the metaphysical weight imposed upon it. Indeed, one might worry that such a ‘thin’ notion amounts to no notion at all (French, 2014, pp. 178–179). If quantum particles-as-objects are conceived of as bare relation bearers with nothing to them, as it were, over and above the relevant relations (where these may yield the particles’ identity or not, depending on the form adopted), then the suspicion may arise that these relata are being posited on conceptual grounds only. Of course, you can posit whatever you like on conceptual grounds but for it to have any value in the context of developing a form of structural realism that is naturalistically acceptable, there needs to be an appropriate physical correlate and with all the relevant properties and even, in one case, the identity of the particles cashed out in structural terms, there is no physical correlate to this thin notion of objecthood. Putting it another way, if an object is nothing but what bears the relations, then, as Chakravartty notes, ‘[ . . . ] the question arises: “What is the what?” — that is to say, what is the thing that bears the relations?’ (2012, p. 204). As a result, it can fairly be suggested that we have lost our grip on what this ‘thin’ notion is, and how these views are really different from the supposedly more ‘radical’, eliminativist form of OSR. Indeed, Chakravartty has usefully explored the space of possible positions between a ‘thick’ conception of objects (such as that underpinned by a notion of substance, for example) and eliminativism and has concluded that there is simply no metaphysical room for a viable ‘thin’ conception (2012). Furthermore, referring back to Krause’s three levels of axiomatisation, what we have here is a further mismatch between levels 2 and 3: at the latter, modern physics presents us with a structural conception that we ‘naturally’ tend to understand in relational terms (naturally in that this is how we have understood

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structures previously). However, that very conception drains the putative objects of all their properties, and even their identity, but the standard understanding of relations in play at level 2 requires us to retain this ‘thin’ notion of object according to which it is nothing but the bearer of such relations. Better, insists the eliminativist structural realist, to get rid of such a notion entirely and restore a ‘smooth’ relationship between these levels by revisiting the understanding of relations and/or reconceptualising what is meant by structure.

3.6 Conclusion Quasi-set theory offers an appropriate formal framework for the view that quantum particles are non-individuals and also, then, for a form of Moderate Ontic Structural Realism constructed on this basis. The relevant sense of ‘structure’ would be one in which the relations hold between non-individual objects and the ‘thinness’ of such objects is mitigated somewhat by their being situated within such a formalism. However, with no individuality and all properties grounded in the relations, such objects do no metaphysical work except to ‘bear’ in some, perhaps primitive, sense, those relations. But of course, the above view is not the only one compatible with quantum statistics and one might prefer, for broadly metaphysical reasons, to retain the understanding of particles as individuals, where their individuality may be taken as primitive, or contextual or whatever. Although this then allows for the implementation, as formal underpinning, of standard set theory, additional constraints must be introduced so as to accommodate the physics. In either case, then, there is a cost and balancing these costs – both formal and metaphysical – is a tricky business. One can avoid paying them altogether by eliminating objects entirely from the structuralist metaphysical pantheon but such a move also has consequences of course, not least in how the notion of structure is to be conceived. Formally, one can adopt the Poincaré Manoeuvre and still appeal to the standard set-theoretic framework. Metaphysically, however, the notion of ‘relation’ then requires a more nuanced consideration. Thus, whichever view you prefer, there is further interesting work to be done!

References Arenhart, J. R. B. (2012). Many entities, no identity. Synthese, 187, 801–812. Arenhart, J. R. B., & Krause, D. (2014). Why non-individuality? A discussion on individuality, identity, and cardinality in the quantum context. Erkenntnis, 79, 1–18. Candiotto, L. (2017). The reality of relations. Giornale di Metafisica, 2, 537–551. Chakravartty, A. (2012). Ontological priority: The conceptual basis of non-eliminative, ontic structural realism. In E. Landry & D. Rickles (Eds.), Structure, object, and causality (Western Ontario series in philosophy of science) (pp. 187–206). Springer.

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da Costa, N. C. A., & French, S. (2003). Science and partial truth. Oxford University Press. Donnelly, M. (2016). Positionalism revisited. In A. Marmodoro & D. Yates (Eds.), The metaphysics of relations (pp. 80–99). Oxford University Press. Dorato, M., & Morganti, M. (2013). Grades of individuality. A pluralistic view of identity in quantum mechanics and in the sciences. Philosophical Studies, 163, 591–610. Eddington, A. S. (1941). Discussion: Group structure in physical science. Mind, 50, 268–279. Esfeld, M. (2015). Review of The structure of the world: Metaphysics and representation. Mind, 124, 334–338. Esfeld, M., & Lam, V. (2008). Moderate structural realism about space-time. Synthese, 160, 27–46. Esfeld, M., and Lam, V., (2010), ‘Ontic structural realism as a metaphysics of objects’, in Bokulich, A. and Bokulich, P., Scientific structuralism, Springer Fine, K. (2000). Neutral relations. Philosophical Review, 199, 1–33. French, S. (2006). Structure as a weapon of the realist. Proceedings of the Aristotelian Society, 106, 167–185. French, S. (2010). The interdependence of structures, objects and dependence. Synthese, 175, 89– 109. French, S. & McKenzie, K. (2012). Thinking Outside the (Tool)Box: Towards a more productive engagement between metaphysics and philosophy of physics. The European Journal of Analytic Philosophy, 8, 42–59. French, S. (2014). The structure of the world. Oxford University Press. French, S. (forthcoming-a). What is this thing called structure? Forthcoming in a volume edited by F. Muller. French, S. (forthcoming-b). Metaphysical underdetermination as a motivational device. French, S., & Krause, D. (2006). Identity in physics: A Historical, philosophical, and formal analysis. Oxford University Press. Jantzen, B. (2011). No two entities without identity. Synthese, 181, 433–450. Krause, D. (2005). Structures and structural realism. Logic Journal of the IGPL, 13, 113–126. Krause, D., & Arenhart, J. R. B. (2017). The logical foundations of scientific theories: Language, structures and models. Routledge. Krause, D., Sant’Anna, A. S., & Volkov, A. G. (1999). Quasi-set theory for bosons and fermions: Quantum distributions. Foundations of Physics Letters, 12, 51–66. Ladyman, J. (1998). What is structural realism? Studies in History and Philosophy of Science, 29, 409–424. Ladyman, J. (2020). Structural realism. In The Stanford encyclopedia of philosophy (Winter 2020 Edition) (E. N. Zalta, Ed.). https://plato.stanford.edu/archives/win2020/entries/structuralrealism/ Lal, R., & Teh, N. (2017). Categorical generalization and physical structuralism. British Journal for the Philosophy of Science, 68, 213–251. Lam, V. (2014). Entities without intrinsic physical identity. Erkenntnis, 79, 1157–1171. Lam, L., & Wuthrich, C. (2015). No categorial support for radical ontic structural realism. British Journal for the Philosophy of Science, 66, 605–634. Laudisa, F., & Rovelli, R. (2021). Relational quantum mechanics. In The Stanford encyclopedia of philosophy (Winter 2021 Edition) (E. N. Zalta, Ed.). https://plato.stanford.edu/archives/ win2021/entries/qm-relational/ MacBride, F. (2020). Relations. In The Stanford encyclopedia of philosophy (Winter 2020 Edition) (E. N. Zalta, Ed.). https://plato.stanford.edu/archives/win2020/entries/relations/ Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109, 455–479. Mermin, D. (1998). What is quantum mechanics trying to tell us? American Journal of Physics, 66, 753–767. Mertz, D. W. (2016). On the elements of ontology: Attribute instances and structure. De Gruyter. Muller, F. A. (2010). The characterisation of structure: Definition versus axiomatisation. In F. Stadler et al. (Eds.), The present situation in the philosophy of science. Springer. Poincaré, H. (1898). On the foundations of geometry. The Monist, 9, 1–43.

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Rettler, B., & Bailey, A. M. (2017). Object. In The Stanford encyclopedia of philosophy (Winter 2017 Edition) (E. N. Zalta, Ed.). https://plato.stanford.edu/archives/win2017/entries/object/ Saatsi, J. (2017). Replacing recipe realism. Synthese, 194, 3233–3244. Sider, T. (2011). Writing the book of the world. Oxford University Press. Thébault. (2016). Quantization as a guide to ontic structure. The British Journal for the Philosophy of Science, 67, 89–114. Worrall, J. (1989). Structural realism: The best of both worlds? Dialectica, 43, 99–124. Reprinted in Papineau, D. (ed.). The philosophy of science (pp. 139–165). Oxford University Press.

Chapter 4

Not Individuals, Nor Even Objects: On the Ontological Nature of Quantum Systems Olimpia Lombardi

Abstract To which ontological category do quantum systems belong? Although we usually speak of particles, it is well known that these peculiar items defy several traditional metaphysical principles. In the present chapter these challenges will be discussed in the light of certain distinctions usually not taken into account in the debate about the ontological nature of quantum systems. On this basis, it will be argued that an ontology of properties without individuals, framed in the algebraic formalism of quantum mechanics, provides adequate answers to the ontological challenges raised by the theory.

4.1 Introduction What kind of item is a quantum system? In the practice of physics it is common to speak of quantum particles, as if they were items of a similar nature to classical items, but obeying different laws of motion. However, as is well known, quantum systems have such peculiar features that they challenge certain ontological principles and categories as understood in traditional metaphysics. In general, these features are analyzed in the context of the so-called problem of indistinguishability, which is a consequence of the particular statistical behavior of quantum systems. But the fact that certain items are “indistinguishable” is not the only difficulty to be overcome in order to elucidate the ontological category of quantum systems. On the other hand, the issue about the nature of quantum systems usually revolves around the category of individual: the question is whether this category can be applied to understand the peculiarities of quantum behavior. However, some metaphysical subtleties are overlooked in this discussion. What is the difference between the notions of object and of individual? What metaphysical principles are quantum systems supposed to fulfill? What ontological kinds of properties are

O. Lombardi () University of Buenos Aires – CONICET, Buenos Aires, Argentina © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_4

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involved in the quantum domain? These questions should be taken into account when the issue at stake is the elucidation of the ontological nature of quantum systems and, more generally, of the structure of the quantum realm. The purpose of this chapter is twofold. On the one hand, different ontological distinctions that are not usually taken into account in the debate will be brought to the fore. In the light of these distinctions, the different ontological challenges that quantum mechanics poses to the ontological pictures presupposed by traditional metaphysics will be discussed. On the other hand, an ontology of properties, framed in the algebraic formalism of quantum mechanics, will be proposed. According to this view, quantum systems are not objects at all, but mere collections of properties that do not preserve their identity either synchronically or diachronically. The final aim of this work is to show how this proposal provides adequate answers to the ontological challenges raised by quantum mechanics to the categories coming from traditional metaphysics. On the basis of this purpose, the chapter is organized as follows. In Sect. 4.2, certain metaphysical topics, usually ignored in the discussion on the interpretation of quantum mechanics, will be introduced and clarified. This clarification will allow us to discuss, in Sect. 4.3, the main quantum ontological challenges in more precise terms. Section 4.4 will be devoted to present the proposed ontology of properties, in the context of which quantum systems turn out to be non-objectual bundles of properties. This ontology will show its advantages to face the quantum ontological challenges in Sect. 4.5. Section 4.6 will delve into the physical nature of quantum systems conceived as non-objectual bundles. Finally, Sect. 4.7 will introduce some concluding remarks.

4.2 Some Metaphysical Preliminaries 4.2.1 Ontological Categories In the discussions about the interpretation of quantum mechanics, it is commonly assumed that it introduces a deep break with respect to the classical view of reality. To recognize the extent of such a break, the first step is to recall what an ontological category is. A category is not a class defined by a concept, like “yellow” or “round”, which gathers certain objects together because they possess a certain property -or a cluster of properties-. A category is not a taxon, like “vegetal” or “mammal”, which classifies individuals into well-defined kinds. Categories are prior to any classification, since they are what endow reality with a certain structure. For this reason, they are conditions of possibility of any classification (Lewowicz, 2005). Ludwig Wittgenstein (1921), in his Tractatus Logico-Philosophicus, introduces the distinction between saying and showing. A state of affairs external to language itself is “sayable”, and is depicted by a proposition. A proposition has a content that is fully intelligible to a person who is fluent in the language, even if she

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does not know whether it is true or false. However, although propositions can depict the whole of reality, they cannot depict their own logical form, since this would require adopting a perspective outside of language itself. Thus, a proposition does not represent its logical form, but shows it in its own structure. In other words, the logical structure of language cannot be said; it can only be shown. As Wittgenstein states: “What can be shown, cannot be said.” (1921, Proposition 4.1212). These “unsayable things” are shown in the form of the propositions: they are there, in language, even though they cannot be said. In turn, the structure of language, shown by language itself, is also the structure of reality: “Propositions show the logical form of reality. They display it.” (Wittgenstein, 1921, Proposition 4.121). Therefore, the analysis of the logical structure of language allows us to understand the ontological structure of reality. A proposition as ‘the flower is pink’ says something, namely, it describes the fact that a certain flower is pink, but it cannot say that ‘flower’ is a noun representing an object and ‘pink’ is an adjective representing a property: the proposition can only show, through its structure, that it is speaking of a reality inhabited by objects and properties. Once this Wittgensteinian distinction is taken into account, it becomes clear that categories are said neither with nouns nor with predicates nor with any other kind of word. Categories are shown by language: each language manifests, in its own structure, the categories that inform and organize the ontology to which it refers. For example, the structure of language will tell us whether the ontology is inhabited by objects, properties, and relations, whether ontological items can be categorized as one or multiple, whether events are temporally ordered, whether there are causal links between them. In turn, one task of metaphysics is to cut reality into categories. From Aristotle through Kant, many authors proposed categories to structure reality. Here only those categories that play a relevant role in the ontological interpretation of quantum mechanics will be considered, in particular, the categories of object, property, and event. I will use the term ‘item’ to refer to anything that exists, regardless of the category to which it belongs: objects, properties, and events are ontological items. The ontological category of object is mirrored in language by the linguistic category of subject. As Ernst Tugendhat clearly explains: “There is a class of linguistic expressions which are used to stand for an object; and here we can only say: to stand for something. These are the expressions which can function as the sentence-subject in so-called singular predicative statements and which in logic have also been called singular terms” (Tugendhat, 1982: 23). This means that the category of object has its linguistic correlate in the so-called singular terms, which play the role of logical subjects of propositions and have singular references. Since the category of object is closely related to the idea of subject of predication, it is always complemented by the ontological category of property: properties are attributed to objects and correspond to language predicates. In other words, if an object is a subject of predication at the logical level, at the ontological level it is the bearer of the properties attributed to it (see Rettler & Bailey, 2022). It is relevant to emphasize that monadic properties, usually called ‘properties’, and nadic properties, usually called ‘relations’, both belong to the ontological category of

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“property” and are distinguished by their arity. Or, in the other conceptual direction, a monadic property can be conceived as a particular case of relation, corresponding to n = 1. On the basis of these distinctions, the most traditional metaphysical picture is that of an ontology of objects and properties. This picture is present, under different forms, in the Aristotelian dichotomy between primary substance and attributes and in Locke’s doctrine of the substance in general as the underlying substratum in which properties inhere, among many other cases in the history of philosophy. It is this kind of ontology that underlies Western ordinary languages and most systems of logic. In fact, in the propositions ‘Socrates is mortal’ and ‘water is liquid’, both of the form ‘S is P’, the linguistic distinction between the subject S and the predicate P expresses the ontological distinction between object and property. The same can be said in propositions with non-copulative verb, such as the case ‘The baby cries’: here too there is a subject, ‘the baby’, representing an object, and a predicate, ‘to be crying’, representing a property. Moreover, the predicate need not be monadic; in a relational proposition such as ‘Aristotle was the teacher of Alexander’, the names ‘Aristotle’ and ‘Alexander’ refer to objects, and the dyadic predicate ‘to be the teacher of’ denotes a relation. The application of a monadic property to one object or of a n-adic property to n objects leads to a fact, which is logically expressed by a true proposition (Armstrong, 1993). Another ontological category is that of event, which is endowed with a strong temporal connotation. Some philosophers consider that the link between events and facts is close enough to justify the assimilation of the two categories (Wilson, 1974; Tegtmeier, 2000). However, facts are commonly conceived as having a certain temporal stability: they last for a certain period of time, in general not too short. By contrast, events are thought of as occurrences that are instantaneous or last for a very short time. On the basis of this temporal connotation, if events are assumed as a primitive ontological category, then temporal instants or intervals can be obtained as derived items, for instance, as maximal sets of simultaneous or partially simultaneous events (Russell, 1914; Whitehead, 1929), or time itself can be constructed as a linear ordering of events, induced by the binary relation “x wholly precedes y” between them (Thomason, 1989).

4.2.2 Objects: Individuals Versus Stuff The traditional examples of objects are individual things, such as persons, tables, or flowers. However, if an object is understood strictly as a bearer of properties, the category of object is more general than that of individual: there are bearers of properties, such as water, soup, and jelly, which are not individuals but belong to a different subcategory. I will call it ‘stuff ’ in order to avoid the philosophical and physical connotations of terms like ‘matter’ and ‘mass’. An individual is an object that bears properties, but it needs additional features to be such. An individual is a whole unity in the sense that, as such an individual,

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it is indivisible. This means that either it cannot be divided (it is “atomic”) or, if it can be divided, the results of the division are individuals or parts of a different kind from the original one. In turn, an individual is subject to the Kantian category of quantity (unity-plurality): individuals are either one (each one of them) or many, that is, a plurality. In the plural case, individuals form aggregates, in which they can be counted. As Henry Laycock says, the key to the character of this general category “evidently rests in the notions of unity and singularity—and thereby perhaps, more generally, in the concepts of number and countability.” (Laycock, 2010: 8). In the Western philosophical tradition, the properties of an individual can be either (i) accidental, which are those that can change over time because the individual may or may not possess them, or (ii) essential, that is, those that the individual necessarily possesses and that in many cases allow the individual to be reidentified over time. In this sense, space-time properties always play a central role, either as essential properties of the individual or as the properties that confer individuality to the individual, in both cases under the assumption of impenetrability, which guarantees that two individuals cannot occupy the same place at the same time. Most systems of logic include individual constants and individual variables, which represent items belonging to the category of individual. For instance, in first order logic, a proposition ‘Pa’ says that the property referred to by the predicate ‘P’ applies to the individual denoted by the individual constant ‘a’; in turn, in the expressions ‘∀xPx’ and ‘∃xPx’, the range of the individual variable x is a domain of individuals. The presence of individual constants and variables is not specific of traditional logic: the vast majority of systems of logic, even extensions of the traditional logic and deviant logics (see Haack, 1974, 1978), all include symbols to represent individuals, so that an ontology inhabited by individuals is presupposed. In turn, in traditional set theory, the elements of a set are also individuals: when we say that ‘a∈A’, we mean that the individual denoted by ‘a’ belongs to the set represented by ‘A’, and this holds even in the case that the element denoted by ‘a’ is itself a set, since in this case the set behaves as an individual. Even if it may be difficult to define what an individual is, it seems quite clear that the ontology we usually talk about includes individuals, precisely because the symbols used to denote them are ubiquitous in our ordinary and formal languages. Perhaps for this reason, the idea of an ontology of individuals and properties has been the dominant view in Western philosophical thought. It has also shaped physics since Modern times, from the corpuscular philosophy of Galileo and Boyle, up to present-day physics, with the standard model of fundamental particles. On the contrary, the answer to the question ‘What is stuff?’ is far from easy, as there is no strong tradition to help us. Laycock introduces the issue with an everyday example: “Removing a fly from a bowl of soup inevitably involves removing some soup as well; but it seems grammatically inappropriate to say that in such a case, there is another thing which is removed, alongside the fly. [ . . . ] the spatio-temporal isolation of any such soup will be arbitrary or adventitious [ . . . ] that, in essence, is why soup must be served in discrete bowls.” (Laycock, 2010: 15). The key difference between the two cases is that a fly is an individual, whereas soup is stuff.

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As in the case of individuals, a stuff must have an identity principle, that is, a principle that distinguishes it from other stuffs of kinds of stuff. However, in this case such a principle has nothing to do with space and time: what distinguishes water from alcohol has no relation to spatio-temporal properties. Despite this, portions of stuff do exist in space and time: a portion of water can be located spatio-temporally, for example, now and here, in my glass. Unlike individuals, a portion of stuff can be further divided into portions of the same stuff, that is, it can be divided without losing its identity: if a piece of chalk is broken into smaller pieces, the resulting parts are also pieces of chalk. However, a stuff is not each of its portions: the meaning of ‘chalk’ cannot be established by pointing to one piece of chalk. But neither is a stuff the mere aggregate of its portions: the reference of the word ‘water’ is something beyond all the portions of water that exist in the universe. This means that a stuff embodies unity and multiplicity at the same time: it is one stuff, but it has multiple manifestations in its portions. Despite this, portions of stuff do not behave as individuals because, when they are put together in an aggregate, they cannot be counted: the aggregate of two portions of water is not “two waters” but “more water”. Moreover, whereas individuals preserve their identity in the aggregate, portions of stuff cannot be reidentified once they are put together: it cannot be said that “this” is one and “that” is the other of the original portions of, for instance, water or iron (see Lewowicz & Lombardi, 2013). Analytic philosophy has faced the problem of understanding the category of stuff from a linguistic perspective through the discussion of the so-called “problem of mass terms”. In the words of Donald Davidson (1967): the problem is to understand the difference between countable nouns and mass nouns or non-countable nouns (see Pelletier, 1979). In fact, a mass term refers to something that cannot be counted. It is in this sense that mass terms are said to have the semantic property of referring cumulatively: “any sum of parts which are water is water” (Quine, 1960: 91). Like plural nouns, mass terms are semantically non-singular, a fact reflected in their nonacceptance of singular determiners: one can speak of ‘all water’, ‘some water’ and ‘more water’, but not of ‘a water’, ‘each water’ or ‘one water’. As a consequence, mass terms do not denote individual portions of stuff. On the other hand, they have in common with singular nouns the distinction of being semantically non-plural: whereas we can say ‘all oranges are sweet’, we can only say ‘all water contains impurities’. This means that the reference of mass terms is neither singular nor plural, since they designate neither one nor many individual things: “we should not expect a successful reduction to singular reference and singular predication, something that the application of traditional first-order logic would require [ . . . ] when we say that water surrounds our island [ . . . ] our discourse is not singular discourse (about an individual) and is not plural discourse (about some individuals); we have no single individual or any identified individuals that we refer to when we use ‘water’” (McKay, 2008: 310–311). Summing up, when the category of object is in the spotlight, it is necessary to consider the distinction between individual and stuff, since the two subcategories lead to very different ontological pictures.

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4.2.3 Objects: Metaphysical Identity The question of identity is a traditional topic of metaphysics. A distinction is commonly made between qualitative identity and numerical identity (Noonan & Curtis, 2022). Qualitative identity is a relationship between two objects that have certain properties in common, and is a matter of degree. Numerical identity, on the other hand, is a relationship that an object has only with itself, and involves absolute qualitative identity. The present work is concerned exclusively with numerical identity, so the term ‘identity’ will always be used in the sense of numerical identity. It is in this context that different criteria of identity are proposed. A criterion of synchronic identity makes it possible to identify an object by distinguishing it from all others at a given instant of time. A criterion of diachronic identity criterion makes it possible to reidentify an object over time. Commonly, identity criteria are discussed in the case of individuals; here we will consider them in the more general case of objects. When the criterion of synchronic identity depends on an ontological category that transcends the properties of the object, a transcendental identity (transcendental individuality in terms of Post, 1963) is established. Duns Scotus’s haecceitas, a notion still present in contemporary metaphysics, is an example of this approach; substance in Locke’s sense is another typical example (Kaplan, 1975). By contrast, for the bundle theory, the criterion depends exclusively on the properties of the object (Armstrong, 1989), since the object is nothing but a bundle of properties. In this case, identity is defined as the equivalence relation (reflexive, symmetric and transitive) satisfying Leibniz’s law (conjunction between the principle of identity of indiscernibles and the principle of indiscernibility of identicals). The principle of indiscernibility of identicals states that if x and y are identical (they name the same object), then exactly the same properties apply to them (this is a case of absolute qualitative identity). Assuming that the identity relation satisfies this condition is not controversial. However, this principle alone cannot act as a criterion of identity, since the identity relation appears in its antecedent. The converse of this principle, the principle of identity of indiscernibles (PII), states that if exactly the same properties apply to x and y, then they are identical. The PII could work as an identity criterion, since the identity relation appears in its consequent. However, in the specification of the minimal subset of properties that suffices to obtain the identity, a new problem arises. In fact, three versions of the PII are distinguished on the basis of the subset of properties considered relevant (French & Krause, 2006): (i) PII(1), if two objects have all their monadic and relational properties in common, then they are identical, (ii) PII(2), if two objects have all their monadic and relational properties in common, except spatio-temporal properties, then they are identical, and (iii) PII(3), if two objects have all their monadic properties in common, then they are identical. These three versions are ordered from lowest to highest logical strength. PII(2) and PII(3), being stronger, have counterexamples in the classical (non-quantum) domain; the traditional one is the case of Black’s spheres (Black, 1952): two exactly equal spheres in an empty

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Newtonian space, sharing all their monadic and relational properties, except the spatio-temporal ones, and yet they are two objects. PII(1), being the weakest, is the only version that is a candidate for metaphysical necessity. Diachronic identity, although not as intensely discussed as synchronic identity, is not less relevant. What is it that makes an object the same at different times? Proponents of transcendental identity may claim that substance or haecceitas is what persists over time and thus reidentifies the object through change. Those who adopt the bundle theory are committed to selecting certain essential properties that do not change over time, while accidental properties may be replaced. In the case of the bundle conception of individuals, space-time properties are usually conceived as the essential properties that confer diachronic identity in terms of their continuity (see Gallois, 2016).

4.2.4 Properties: Universals Versus Tropes Properties are items that are present in almost any ontological picture, since they are necessary for classification: objects that are numerically different can, nevertheless, be similar in terms of a certain feature or characteristic and, as a consequence, can be grouped in the same class. However, despite their ubiquity, the metaphysical nature of properties has been the subject of controversy since the origin of philosophical thought. Since Plato, who used the term ‘ε῏ιδoς’ to designate them, properties have traditionally been conceived as universals, that is, items that can be shared by different objects, in contrast to particulars, which have a single existence. The peculiarity of universals is that they are “one-in-many”: a universal is one (e.g. redness), but it is instantiated in a multiplicity of cases (red in this case, red in that case). A universal is fully present in each of its instances, and the existence of a universal in one case is unrelated to its simultaneous existence in another case (see MacLeod & Rubenstein, 2006). The so-called “problem of universals” has permeated the history of philosophy from its beginnings, with questions about whether and how universals exist. There are three long-standing answers to these questions: realism, nominalism, and conceptualism. Whereas realists accept universals, conceptualists and nominalists refuse to accept their existence. Conceptualists explain the similarity between individuals by appealing to general concepts that exist only in minds. Nominalists, on the other hand, leave the relation of qualitative similarity as a brute and primitive fact. According to realists, by contrast, universals exist as mind-independent items. For transcendent realism, such as that proposed by Plato, they exist even though they are not instantiated, and they are thus “transcendent” or “ante res” (“before the things”). For immanent realism, such as that defended by Aristotle, universals are “immanent” or “in rebus” (“in things”), since they exist only if they are instantiated by objects. Whatever the differences between the two forms of realism, in both cases the instances of a universal property are many but absolutely indistinguishable: they are only numerically different.

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Since a universal has multiple instantiations, it is necessary to elucidate the concept of instantiation. Instantiation is traditionally conceived as the relationship that a universal property maintains with the substratum to which it is “attached” (see Orilia & Paoletti, 2022). However, from a bundle-theory viewpoint, there are no substances, substrata, or individuals in which the properties inhere. Therefore, in this case, instantiation is the relationship that a property maintains with its multiple manifestations, called ‘instances’. For example, the property “red” has many instances in different situations, that is, when it appears in different bundles. In the physical realm, energy and momentum are properties that are instantiated in their multiple empirical manifestations. In twentieth-century metaphysics, a new approach to properties entered the scene: properties as tropes, a stance that claims to occupy a middle position in between realism and nominalism with respect to universals. Tropes are particular properties, such as the particular shape, weight, and texture of an object (see Maurin, 2018). That two objects “share” a property (for example, a particular shade of redness) means that they each exemplify a redness-trope, where the two redness-tropes, though numerically distinct, nevertheless resemble each other exactly. Because tropes are particulars, trope theorists face the problem of providing a principle of individuation for tropes (see Schaffer, 2001). A natural answer is to individuate tropes by reference to the objects that instantiate them. However, this strategy does not work in the context of a bundle view: if objects are bundles of tropes, that principle of individuation becomes circular. A different approach appeals to space-time individuation, according to which two tropes are different when they are located at different space-time positions: the redness-trope now and here as different from the redness-trope then and there (see, e.g., Campbell, 1990). But this view needs to exclude space-time position as a property in the same sense as the rest of the trope-properties. For these and other reasons, many trope theorists have opted for a primitivist perspective, according to which the fact that two tropes are distinct is a primitive fact lacking any further metaphysical explanation (see, e.g., Keinänen & Hakkarinen, 2014).

4.2.5 Properties: Determinables and Determinates Another distinction regarding properties, which is often not sufficiently taken into account, is the traditional difference between determinables and determinates, that is, properties that stand in a distinctive specification relation; let us call it ‘determination’ (see Wilson, 2022). For example, color is a determinable having red, blue, and other specific shades of color as determinates; shape is a determinable having rectangular, oval, and other specific shapes (including many irregular ones) as determinates; mass is a determinable having specific mass values as determinates. The determination relation differs from other specification relations. Unlike the genus-species and conjunct-conjunction relations, in which the more specific property can be understood as a conjunction of the less specific property and

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some independent property or properties, a determinate cannot naturally be treated in conjunctive terms (whereas man can be conceived as the conjunction between animal and rational, red is not a conjunctive property having color and some other property or properties as conjuncts). And, unlike the disjunct-disjunction relation, in which disjuncts may be dissimilar and compatible (as in the case of ‘red or round’), the determinates of a determinable are both similar and incompatible (red and blue are similar in the sense that they are both colors, but nothing can be simultaneously and uniformly both red and blue). It is extremely important to emphasize that the determinable-determinate relationship should not be confused with the universal-instance relation. The latter is the relation between a universal property and its many instances: for example, the universal property “color” has countless instances of colored items. The first is the relation between a property and other more specific properties that are cases of it: for example, the determinable “color” has “red”, “green”, “yellow”, etc. as its determinables. The distinction between the two kinds of relationship is clear in a classical ontology. In fact, the position Q and the momentum P of a particular classical object, say, this billiard ball, are instances of the universal properties “position” and “momentum”, respectively. In turn, the position q1 = “10 cm from the corner of the table” and the momentum p1 = “20 gr cm/sec with respect to the table” (in both cases, the billiard table is taken as the reference frame) are determinates of the determinables Q = “position of the billiard ball” and P = “momentum of the billiard ball”, respectively. Furthermore, those determinates define the classical state s1 of the billiard ball, conceived as a classical system, at a given time t1 : s1 = (q1 , p1 ). The principle of omnimode determination is a principle that was generally accepted in Modern philosophy. For example, it already appears in the works of Wolff: “Apparet hinc, individuum esse ens omnimode determinatum” (“Hence it appears that an individual is a completely determined being”) (Wolff, 1728, p. 152). It can also be found in Bernoulli’s famous treatise on the calculus of probabilities: “Sed quicquam in se et sua natura tale esse [viz. incertum et indeterminatum], non magis a nobis posse concipi, quam concipi potest, inde simul ab Auctore naturæ creatum esse et non creatum” (“That anything is uncertain and indeterminate in itself and by its very nature is as inconceivable to us as it would be inconceivable for that thing both to have been created and not created by the Author of nature”) (Bernoulli, 1713, p. 227). It is also repeated several times by Kant in his lectures on metaphysics: “Alles, was existirt, ist durchgängig determinirt” (“Everything that exists is continuously determined”) (1902, AA 18:332, 5710; AA 18:346, 5759; see also LM XXVIII 554). The idea is that, in any object, all determinables are determinate: if the determinable “color” applies to an object, the object necessarily has some determinate color, say red, independently of its other determinate properties, and also independently of our knowledge about what that determinate color is. In other words, it is not possible for an object to have a determinable property that is not determinate: an object cannot be colored without being of some particular color, say, red, blue, white, etc.

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4.3 The Ontological Challenges of Quantum Mechanics The philosophical considerations introduced in the previous section are not a mere exercise in metaphysical thinking, but must be taken into account when specifying the structure of the quantum ontology. As will become clear in the present section, they are especially relevant for a clear understanding of the ontological challenges posed by quantum mechanics.

4.3.1 Contextuality One of the first reactions to the probabilistic character of quantum theory was the attempt to interpret it as a statistical theory, in the style of classical statistical mechanics, so that the probabilities could be explained as frequencies in ensembles of systems with definite but “hidden” values of their observables. The coup de grace for such classical-style statistical interpretations was the Kochen–Specker theorem (Kochen & Specker, 1967), which proves the impossibility of ascribing precise values to all the observables of a quantum system simultaneously while preserving the functional relations between commuting observables. It follows that the selection of observables to which precise values can be attributed must be contextual, i.e. situation-dependent (e.g., dependent on the measurement context). In the discussions on the interpretation of quantum mechanics, the distinction between determinable and determinate, which allows us to formulate the principle of omnimode determination, is almost never taken into account. This is surprising because quantum contextuality defies precisely that traditional and intuitive principle: while in all classical objects all determinables are determinate, in the quantum realm non-commuting observables correspond to determinables that are not simultaneously determinate. Furthermore, according to the Kochen– Specker theorem, a quantum system always have determinables (observables in the physical language) that are not determinate (that do not have precise values). It is interesting to note that the breach of the principle of omnimode determination is counterintuitive for any kind of object, not only for individuals, but also for stuff. In fact, one expects that not only in the case of a billiard ball but also in the case of soup, the determinable property, say, “color”, is determinate, say, “white”. Different approaches have been proposed to accommodate quantum contextuality. One of them is based on the adaptation of the logic used in the quantum framework: starting from the fact that contextuality is related to the non-Boolean structure of elementary quantum propositions, a non-classical propositional logic can be formulated in terms of the non-distributive, orthocomplemented lattice of the theory (see, e.g., Jauch & Piron, 1969; Piron, 1976; Beltrametti & Cassinelli, 1981). From a more physical perspective, other authors have dealt with quantum contextuality by selecting a context, via an interpretive assumption (see, e.g., Bub & Clifton, 1996; Dieks, 2005), or via a physical process such as decoherence (see,

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e.g., Zurek, 1982, 2003). However, the general problem of what a quantum system is and what structure the quantum ontology should have for contextuality to be natural has not been answered in a systematic way.

4.3.2 Non-locality and Non-separability Unlike the classical world, the quantum domain admits surprising correlations between the properties of distant non-interacting systems, such as those of the famous Einstein-Podolsky-Rosen experiment. Taken at face value, EPR-correlations strongly suggest non-locality, that is, non-local influences between spatially distant systems, i.e., systems between which no light signal can travel. However, since this idea is incompatible with special relativity, the exact nature of those quantum correlations is a subject of ongoing controversy (see Berkovitz, 2016). For instance, according to collapse interpretations, EPR-correlations imply a certain action at a distance which, nevertheless, does not allow sending information at a superluminal velocity. In the case of Bohmian mechanics, it is the quantum field that possesses the necessary non-local features to induce EPR-correlations. From another perspective, those correlations are consequences of the holistic nature of quantum systems, understanding holism as the opposite of separability. Separability implies that, if a physical object is constructed by assembling its physical parts, then its physical properties are completely determined by the properties of the parts and their relationships. Holism, by contrast, is the characteristic of some physical objects that are not composed of physical parts, but are indivisible wholes; so, EPR-correlations are correlations between properties of a single holistic object (see Healey & Gomes, 2022). From the viewpoint of the state of the composite system, correlations leading to non-locality and non-separability appear when the state is entangled. However, when discussing this issue the relativity of entanglement is rarely taken into account. As John Earman (2015) clearly stresses, a given state is entangled or not only in relation to the decomposition of the composite system into subsystems. In fact, a given state may be entangled with respect to a certain decomposition and nonentangled with respect to another. The typical case is that of the hydrogen atom, which can be decomposed into the proton-system and the electron-system, but also into the center of mass-system and the relative-system: the entanglement of the atom’s state is relative to the chosen decomposition (see, e.g., Harshman, 2012). It has even been proven that, given a state vector | in a finite-dimensional state space H with non-prime dimension d = m-n, there always exists a partition, expressed by a tensor product structure H = Hm ⊗ Hn , with respect to which | is factorizable and, therefore, non-entangled (Terra Cunha et al., 2007). And since “without further physical assumption, no partition has an ontologically superior status with respect to any other” (Zanardi, 2001: 4), there is no reason to privilege one claim about the entanglement of a quantum state over others. This relativity makes it difficult to conceive the correlations due to entanglement as consequence of non-local

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influences between subsystems of a composite system, since the very idea of welldefined subsystems goes into crisis. By contrast, entanglement-induced correlations favor a holistic view of the composite system, according to which correlations are internal to a single indivisible item.

4.3.3 Indistinguishability Most discussions about the ontological commitments of quantum mechanics focus on the challenge posed by the indistinguishability of so-called “identical particles” (particles of the same kind, that is, with the same state-independent properties) to the ontological category of individual. The usual story begins by counting how many distributions or complexions of two particles over two states are possible. The classical answer is given by the Maxwell–Boltzmann statistics, according to which there are four possible distributions of two individuals over two states. By contrast, in quantum statistics (Bose–Einstein and Fermi–Dirac), a permutation of the particles does not lead to a different complexion since particles are “indistinguishable”. Although the theory has formal resources to deal with quantum statistics, from a conceptual viewpoint the problem is to explain why a permutation of individual particles does not lead to a different complexion in the quantum case. Indistinguishability is often considered to be a feature that leads to the violation of the weakest version of the principle of identity of indiscernibles and, consequently, that challenges the category of individual. Already in the 1960s, Heinz Post (1963) argued that elementary particles cannot be regarded as individuals, but must be viewed as “non-individuals”: this led to the so-called “Received View” about quantum indistinguishability (see French, 2019). However, in general the Received View gives no metaphysical characterization of those items beyond their non-individuality: they are only negatively characterized. An exception is the case of Paul Teller (1998), who proposes an account of quantum indistinguishability in terms of stuff: if quantum objects are stuff, permutation invariance follows naturally, because permuting two portions of the same stuff over two states does not give rise to a different complexion. Several perspectives confront the Received View in order to restore the category of individual. For example, Bas van Fraassen (1985) recovers quantum statistics by renouncing the equiprobability of the different distributions of quantum particles in quantum states. Another position contrary to the Received View comes from Steven French (1989), who claims that states of indistinguishable particles that are not symmetric or antisymmetric are ontologically possible and only physically inaccessible: indistinguishability is a physical situation, not an ontological condition from which non-individuality can be inferred. Another alternative proposal to the Received View is based on the idea of weak discernibility (Saunders, 2003; Muller & Saunders, 2008): in the case of two fermions in a singlet state, the relation “having the opposite direction of each spin component with respect to...” that each fermion has with respect to the other is sufficient to establish numerical distinction between

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the objects, even if they are indistinguishable with respect to their monadic and relational properties (for an analogous argument for bosons, see Muller & Saunders, 2008). French and Décio Krause (2006) have rejected this strategy by claiming that it entails circularity: in order to appeal to such relations, one has had to discriminate beforehand between the related objects; therefore, the numerical difference between the objects has been presupposed by the relation itself. Although the problem of indistinguishability in quantum mechanics has generated extensive discussions, a common ground underlies all of them. In fact, despite the Received View and its critics differ with respect to conceiving quantum systems as individuals, they nevertheless agree with respect to subsuming them under the category of object.

4.4 Systems as Non-objectual Bundles of Properties Hardly anyone denies the underdetermination of metaphysics by physics: quantum mechanics is compatible with distinct metaphysical “packages”. In fact, the quantum domain has been conceived as structured on the basis of very different fundamental ontological categories. According to Bohmian Mechanics (see Dürr et al., 2013), the universe is a configuration of particles in precise positions relative to each another. Therefore, elementary quantum systems are individual particles with synchronic and diachronic individuality: they are discernible by their positions in the configuration, and they can be reidentified over time by the continuous trajectory traced by their motion. The difference with respect to the classical case is that the dynamics of quantum individuals is described by a law of motion known as “guiding equation”, which makes the evolution of each particle depend on the position of all the others, through the wave function. Proponents of Bohmian Mechanics believe that it provides the smallest deviation from the ontology of classical mechanics that is necessary to accommodate quantum phenomena (see, e.g., Esfeld, 2019). The Ghirardi-Rimini-Weber (GRW) collapse theory (Ghirardi et al., 1986) states that, in order to solve the quantum measurement problem, the dynamical equation of the standard theory must be modified by adding stochastic and nonlinear terms. This new equation describes the spontaneous jumps undergone by the wave function in configuration space at random times. This theory has been ontologically interpreted in two different ways: the matter-density ontology (GRWm) and the flash ontology (GRWf). In GRWm, the wave function describes a continuous matter density field (see Ghirardi et al., 1995), which varies at different points of the three-dimensional physical space and changes in time. This means that there is a single fundamental object in the universe, a stuff that “fills” the entire space. The defined outcomes of measurements are the result of the spontaneous contraction of the matter density field at certain points or regions of space: the collapse of the wave function represents such a contraction of the matter density field. Discrete individual objects are mere appearances that arise in certain regions of the physical space where

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the matter density is higher. According to GRWf, by contrast, each jump of the wave function in configuration space represents an event occurring at a point of physical space. These point-events are known as “flashes” (Tumulka, 2006). The time evolution of the wave function in configuration space represents the probability of the occurrence of future flashes, given an initial distribution of flashes. Therefore, physical space is not filled with objects, neither individuals nor stuff, but with a sparse distribution of discrete events. Objects are, from this view, nothing but clusters of a huge number of flashes. These cases show that the literature on the ontology of quantum mechanics has appealed to different fundamental ontological categories in order to design the structure of the quantum domain: individuals and properties in Bohmian Mechanics, stuff in the matter-density ontology, events in the flash ontology. However, the possibility of an ontology in which properties are the only fundamental items has scarcely been considered. In fact, the idea of bundle of properties has appeared only a few times in the literature on quantum physics. It has been proposed for quantum field theory in its algebraic version by Meinard Kuhlmann (2010), and suggested by Cord Friebe (2014) in his objections to Gian Carlo Ghirardi’s criterion for entanglement of indistinguishable particles. In this scarcity lies the novelty of the proposal for a ontology of properties, which was originally presented by Olimpia Lombardi and Mario Castagnino (2008) in a paper on the Modal-Hamiltonian Interpretation of quantum mechanics, and was subsequently developed in later works (da Costa et al., 2013; da Costa & Lombardi, 2014; Lombardi & Dieks 2016; Fortin & Lombardi, 2022). This section will briefly introduce the proposal, highlighting its ontological significance and conceptual connotations.

4.4.1 Formalism and Ontology At present it is clear that a formalism does not determine its interpretation: if a formal system has one interpretation, it may have an infinite number of interpretations. Nevertheless, this does not imply that formalisms are ontologically neutral: different formal systems, even if equivalent, may suggest different ontologies. A typical example is the theory of natural numbers, which can be formulated either on the basis of Peano’s axioms or in terms of Russell’s settheoretic construction: although mathematically equivalent, the two formulations have different ontological connotations. From Peano’s perspective, natural numbers admit a realist, Platonist interpretation, according to which they exist as abstract entities. Russell’s formulation, by contrast, is more favorable to a nominalist interpretation, according to which reality is populated by individuals and classes, but not by natural numbers. An example from physics is the case of Hamiltonian and Lagrangian classical mechanics, which are also mathematically equivalent: they lead to the same predictions but evoke different ontological pictures. Hamiltonian mechanics gives a dynamical picture: it describes time processes that, starting from an initial state, all evolve in the same temporal direction under the rule of a

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dynamical law. Lagrangian mechanics, by contrast, suggests a static picture, with a crystalized time, where there are two states at once, and the trajectory between them is given by a variational principle, in particular, the principle of least action, which selects one among all possible trajectories. Some prominent physicists have even endowed Lagrangian mechanics with a teleological meaning, regarding it as reintroducing final causes in physics (see discussion in Ben-Menahem, 2018, in particular, Planck quote in page 150). Analogously to the above cases, different formalisms for standard quantum mechanics, although mathematically equivalent, design different ontological pictures. In the Hilbert space formalism, a quantum system is represented by a Hilbert space, whose vectors represent the system’s states; observables are represented by operators acting on the Hilbert space. The mathematical priority of systems with their states over observables is easily reflected in an ontology of individuals, endowed with ontological priority over their properties. By contrast, in the algebraic formalism, a quantum system is represented by an algebra of observables, and states are functionals on that algebra. If this mathematical priority of observables over states is transferred to the ontological domain, the result is an ontology whose primary items are properties, and systems arise from the convergence of those properties. In the algebraic framework, a quantum system is represented by a *-algebra .A of observables .A ∈ A , closed under products, linear combinations, and involution. A state of the system is represented by a normed and positive expectation-value functional .ω : A → C belonging to the dual algebra .A  . A state ω is pure when it cannot be written as a non-trivial convex combination ω = λ1 ω1 + λ2 ω2 , with 0 < λ1 , λ2 < 1, λ1 + λ2 = 1, and .ω1 , ω2 ∈ A  ; otherwise ω is mixed. The Gelfand-Naimark-Segal (GNS) construction (Gelfand & Naimark, 1943, Segal, 1947) proves that, if .A is a C*-algebra, then it can be mathematically represented by a set .O of Hermitian operators O on a Hilbert space H , and states can be mathematically represented by normed trace (density) operators ρ on H . When the state ω, represented by the density operador ρ, is pure, then there is a vector | ∈ H such that ρ = | |. For different *-algebras, other representations of the algebra have been proved; for instance, a nuclear algebra can be represented by a rigged Hilbert space (see Iguri & Castagnino, 1999; for applications of rigged Hilbert spaces to quantum mechanics, see Bohm & Gadella, 1989). From now on, we will not distinguish between the abstract algebraic language and the language of the mathematical representation; then, we will say that a quantum system is represented by the algebra .O of observables .O ∈ O, and that the system’s states are expectation-value functionals ρ(O) = Tr(O ρ) = Oρ ∈ R, for all .O ∈ O. Moreover, given two component systems represented by the algebras of observables 1 2 1 2 .O and .O , the composite system is represented by .O ∨ O , that is, the minimal 1 2 algebra generated by .O and .O . In turn, we will also not distinguish between the physical language (e.g., observables, states) and the mathematical language (e.g., elements of an algebra, functionals on an algebra), under the assumption that the context clarifies the meaning of each term.

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It is not necessary to delve into the technical details of the formalism to emphasize that the algebraic approach brings to the fore the difference between observables and states, whose corresponding spaces may not even be the same; this is precisely the case in nuclear algebras, represented by rigged Hilbert spaces. In the algebraic theoretical framework, observables are the basic elements of the theory; states are secondary elements, defined in terms of the basic ones. To the extent that states are defined as expectation-value functionals on the algebra of observables, their “nature” is exhausted in accomplishing the task of computing the expectation values of the observables of the algebra. In other words, states are not to be confused with observables, they are not to be understood as any kind of property of the quantum system. As Earman (2015: 324) emphasizes, one should never forget “the mantra of the algebraic approach: a system state is an expectation value functional on the system algebra.”

4.4.2 The Structure of the Ontology Following the Wittgensteinian idea that the structure of language is also the structure of reality, the latter arises by establishing the ontological counterpart of the algebraic formalism, that is, by providing an interpretation for each physical/mathematical term. • The term ‘observable’ is used in quantum physics to refer to certain quantifiable magnitudes of physical relevance, which are mathematically represented by Hermitian operators. Ontologically, they correspond to items belonging to the category of property, in particular, determinable properties, which here will be referred to as ‘type-properties’. In addition, it is necessary to distinguish between universal type-properties (U-type-properties) and instances of universal typeproperties (I-type-properties). The ontological counterparts of general physical magnitudes are U-type-properties, and of observables are I-type-properties. We will symbolize an U-type-property as [A], and its I-type-properties as [Ai ]. An example of U-type-property is energy [H], which can be instantiated as the energy [H1 ] of this particular system. Let us stress that, although this talk suggests an ontology of objects, below we will define the concept of quantum system as a non-objectual ontological item. • Since a physical observable is a quantifiable magnitude, it has different possible values, which are mathematically represented by the eigenvalues of the corresponding Hermitian operator. Their ontological counterparts are determinate properties, which here will be referred to as ‘possible case-properies’ (P-caseproperties) of the corresponding I-type-property. Here the terms ‘type-properties’ and ‘case-properties’ stand for determinables and determinates, respectively; they are used just to emphasize that the corresponding items belong to the i ontological category of property. Given an I-type-property  ] of a U-type [A property [A], its P-case-properties will be symbolized as . aji . Following with

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  the above example, we can talk of the P-case-properties . ωj1 (the energy values that constitute the energy spectrum) of the energy [H1 ] of this particular system, where [H1 ] is an I-type-property of the U-type-property energy [H]. • In physics it is implicitly assumed that each observable, although having multiple possible values, cannot have more than one value at a time. The value actually acquired by an observable has no direct mathematical representation in the theory: there is no formal way to distinguish it from the remaining possible values. But, ontologically, it is essential to emphasize that, given an I-typeproperty [Ai ] of a U-type-property [A], no more than one of its P-case-properties   i . a becomes actual. That actual case-property (A-case property) will be j     symbolized as . a ik . In the above example, . ω1k is the actual value of the energy [H1 ] of this particular system. Notice that the clause “no more than one”, which corresponds to “exactly one” in the classical case, can be “zero” in the quantum case. In fact, as the Kochen–Specker theorem shows, not all the I-type-properties of a system have an A-case property. • Since in the algebraic formalism the physical concept of quantum system Si is mathematically represented by an  algebra its ontological   of  observables,  counterpart is a bundle .Bi = Ai , B i , C i , . . . of the I-type-properties [Ai ], [Bi ], [Ci ], . . . corresponding to the U-type-properties [A], [B], [C], . . . . The precise nature of these bundle-systems will be discussed in the following subsection. • The physical concept of state is mathematically represented by an expectationvalue functional over the space of observables. As mentioned above, in this interpretive framework states do not refer to properties but are endowed with a probabilistic nature. More precisely, the state of a system Si encodes the ontological propensities to actualization of all the P-case-properties of all the Itype-properties belonging to the bundle .Bi , which is the ontological counterpart of Si . In this ontological proposal, type-properties are conceived as universals. A legitimate question is why not to appeal to tropes. The answer is strongly linked to the problem of indistinguishability. Although they may be absolutely similar, tropes are neither absolutely indistinguishable nor only numerically different, precisely because they can be individuated and distinguished by their space-time position (redness here and now), by the object to which they apply (red of this individual balloon), or because the distinction between them is taken to be primitive (see Sect. 4.2.4). Since the elemental items of the quantum ontology should be adequate to provide the foundations of quantum indistinguishability, an ontology of tropes would face the same difficulties as an ontology of objects, since in both cases they are distinguishable items. By contrast, the instances of a universal are absolutely indistinguishable because they are manifestations of a same property: the roundness of a billiard ball and the roundness of a water drop are both instances of the universal roundness, and trying to distinguish them as different properties makes no sense. For this reason, an ontological approach based on universals and their

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instances paves the way towards an ontologically reasonable answer to the problem of indistinguishability. In Sect. 4.2.5, the importance of not confusing between the universal-instance relation and the determinable-determinate relation was emphasized by means of an example coming from classical mechanics: the position and the momentum of a particular billiard ball as determinable instances of the universal properties “position” and “momentum”, respectively, having determinate values with respect to a billiard table. The difference between the two relations is completely analogous in the quantum framework. The difference between the classical case and the quantum case lies not in those relationships, but in the role played by possibility and actualization. In the classical case, at a given time, a single possible determinate corresponds to each determinable and, as a consequence, such a determinate becomes actual. For example, at time t1 , the determinable Q = “position of the billiard ball” has a single possible determinate, say, q1 = “10 cm from corner of the table”, and this is the actual determinate position of the billiard ball at t1 . In the quantum case, by contrast, at a given time, determinables may have many different possible determinates, among which at most only one becomes actual. For example, at a given time, the determinable Sz = “spin in direction z of the quantum system S” has two possible determinates, Sz1 = “spin up in direction z” and Sz2 = “spin down in direction z”: it may be the case that one of them becomes actual; but, according to the Kochen–Specker theorem, it may also be the case that neither of them becomes actual. Finally, it is worth introducing some remarks about the concept of possibility, whose nature has been one of the most controversial issues in the history of philosophy. Two general ways of conceiving possibility can be distinguished (see Menzel, 2022). According to actualism, everything that exists, when analyzed in depth, turns out to be actual: the discourse on possibility can be reduced to a language that only refers to what actually exists; as a consequence, the predicate ‘actual’ is redundant. For possibilism, by contrast, possibility is an ontologically irreducible feature of reality: possible items need not become actual in order to be real. In Aristotelian terms, being can be said in different ways: as possible being or as actual being. Given the essential probabilistic nature of quantum phenomena, in the present proposal possibility is conceived in non-actualist terms. An I-typeproperty has—possible—P-case-properties, among which at most one becomes actual, and the state gives the measure of the corresponding possibilities, that is, the measure of the tendency to actualization of those P-case-properties. These facts have nothing to do with a limitation of our knowledge about an underlying actual state of affairs. Probabilities measure possibilities conceived as propensities to actualization, which are ontologically irreducible because the theory is irreducibly indeterministic (see Lombardi et al., 2022).

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4.4.3 Non-objectual Bundles The problem of the nature of objects remains one of the main areas of controversy in contemporary metaphysics: is an object a substratum supporting properties or a mere “bundle” of properties? (for a survey, see Loux, 1998). The conception of an object as a substratum acting as a carrier of properties has pervaded the history of philosophy. As already mentioned, it is present under different forms, for example, in Aristotle’s “primary substance” and in Locke’s “substance in general”. However, following Hume’s rejection of the idea of substance, many philosophers belonging to the empiricist tradition, such as Bertrand Russell, Alfred Ayer, and Nelson Goodman, have regarded the postulation of a characterless substratum as a metaphysical abuse, and have adopted some version of the bundle theory. According to this view, an object is nothing but a bundle of properties: properties have metaphysical priority over objects and are therefore the fundamental items of the ontology. In the literature, it has been argued that the difference between the substratum theory and the bundle theory is only verbal: in the bundle theory the object results from a “compresence relation” that serves the same purposes as the substance in the traditional substratum-plus-attributes theory. Therefore, the decision as to whether an object is a substratum supporting properties or simply a bundle of properties remains a matter of metaphysical taste (see Benovsky, 2008). While this may be the case in the classical domain, quantum mechanics calls this conclusion into question. Indeed, although in the present proposal quantum systems are ontologically characterized as bundles of properties, it is important to emphasize the peculiarity of this perspective. The first point to consider is related to the difference between determinables and determinates, which is rarely taken into account in discussions about the ontological interpretation of quantun mechanics (for exceptions, see Calosi & Wilson, 2019; Calosi & Mariani, 2021). According to the traditional versions of the bundle theory, an object is the convergence of certain determinate properties, under the assumption that the determinable properties are all determinate. For example, a billiard ball is the confluence of a definite value of position, say here, a definite shape, say round, a definite color, say white, etc. So, the problem is to decide whether this object is a substratum in which definite position, roundness and whiteness inhere, or is the mere bundle of those determinate properties. But in both cases the properties that compose the bundle are actual properties. In the quantum case, by contrast, not all the determinable properties of a system are determinate; as a consequence, the system cannot be identified with a bundle of determinate properties. For this reason, in the present proposal, a quantum system is conceived as a bundle of determinables, that is, type-properties (I-type-properties), each one of them with its possible caseproperties (P-case-properties). This is the first reason why this interpretation of the nature of quantum systems cannot be assimilated to the traditional notion of object. On the other hand, in its traditional versions, the bundle theory is a theory about particular objects, according to which objects are composed of items of

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a different ontological category (namely, properties). In other words, the bundle theory is designed to account for objects without appealing to a substratum on which properties inhere (see, e.g., O’Leary-Hawthorne, 1995; French, 2019). To this end, some properties must be selected to play the role of the principle that supplies synchronic and diachronic identity. The proposed quantum bundle view, by contrast, completely dispenses with the ontological category of object: bundles of properties do not behave as objects at all since they belong to a different ontological category. On this basis, when two bundle-systems combine, the composite system is also a bundle. And since bundles are not objects, there is no principle that preserves their identity in the composition: in the composite system the identity of the components is not preserved precisely because they are not objects at all. Also in this sense quantum systems are conceived as non-objectual bundles of properties. Precisely because of their non-objectual nature, bundle-systems require a different kind of logic. An ontological picture in which properties are the elementary items, and do not constitute objects, is not adequately captured by any formal theory whose elementary symbols are individual variables referring to classical objects. But, as remarked in Sect. 4.2.2, most systems of logic are designed to handle individual objects. A way out of this problem is to develop a “logics of predicates” in the spirit of the “calculus of relations” proposed by Tarski (1941), in which individual constants and variables are absent. A different strategy is to apply quasi-set theory (see, e.g., Krause, 1992; da Costa and Krause, 1999): although it was originally devised to provide a formalism for indiscernible quantum objects, it can be adapted to formally deal with aggregates of items that do not belong to the ontological category of object but to that of property, so that bundles turn out to be represented by quasi-sets of properties (Holik et al., 2022).

4.5 Revisiting the Ontological Challenges Let us insist again that metaphysics is underdetermined by physics; in particular, quantum mechanics is compatible with different ontological pictures. Thus, arguing in favor of a certain quantum ontology over others requires showing how fruitful it is in the task of offering reasonable solutions to interpretive problems. This section is devoted to show the advantages of the proposed picture for dealing with the ontological challenges of quantum mechanics: contextuality, non-locality, and indistinguishability.

4.5.1 Contextuality As explained in Sect. 4.3.1, the Kochen–Specker theorem proves the impossibility of ascribing precise values to all the observables of a quantum system simultaneously, while preserving the functional relations between commuting observables. For

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this reason, the contextuality resulting from the theorem defies the principle of omnimode determination, according to which all determinables are determinate in any object—individual or stuff. It is in this sense that quantum systems cannot be conceived of as classical objects: in any situation, they have determinable properties that are not determinate. In our ontological language, quantum contextuality is expressed by saying that, given a bundle, not all of its I-type-properties actualize, that is, acquire an (actual) Acase-property among all their (possible) P-case-properties: which I-type-properties acquire an A-case-property is a “contextual” fact. Of course, in each context one could insist on the classical idea of I-type-properties with their definite A-caseproperties with no contradiction. In other words, the picture of a bundle of actual case-properties that stands for a classical object could be retained in each context. But as soon as we try to extend this ontological picture to all the contexts by conceiving the object as a bundle of bundles, the Kochen–Specker theorem imposes an insurmountable barrier: in a quantum system, it is not possible to actually ascribe A-case-properties corresponding to all the I-type-properties in a non-contradictory manner. Therefore, the classical idea of a bundle of bundles of actual case-properties does not work in the quantum ontology. The Kochen–Specker theorem introduces a constraint with respect to A-caseproperties, more precisely, with respect to which P-case-properties of a bundle can enter actuality. But this restriction does not affect our concept of quantum system, because it is defined not as a bundle of actual case-properties, as in the traditional bundle theory, but as a bundle of I-type-properties, each with its corresponding possible P-case-properties. Precisely for this reason these quantum bundles do not constitute objects in the traditional sense: they are non-objectual bundles. Since the ontology is only populated by properties and bundles of properties, the principle of omnimode determination, valid for objects, is not false but does not apply. As a consequence, this ontology, devoid of objects, is immune to the challenge represented by the Kochen–Specker theorem.

4.5.2 Non-locality and Non-separability As recalled in Sect. 4.3.2, the quantum domain seems to admit correlations between the properties of distant non-interacting systems, strongly suggesting non-local influences between distant systems that are incompatible with special relativity. Despite disagreements about this particular feature of quantum mechanics, in general the arguments about non-locality are based on the assumption that quantum systems are individual objects, and subsystems are also individuals. Consequently, the problem is to explain how the properties of those individual subsystems are instantaneously correlated even though they are not in interaction and they are located in different spatial positions. However, from an ontology of properties, the problem appears in a new light.

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Recall that the ontological category of individual requires some “principle of individuality” that, regardless of its specific form, identifies a particular individual as distinct from another and as the same over time (see discussion in French & Krause, 2006). In turn, individuals can form aggregates, in which they can be counted (see Sect. 4.2.2): “countability” depends on the possibility of ontologically distinguishing each individual from the others. Therefore, if two individual systems interact as to yield a composite system in an entangled state, they should retain their identity as individual parts of the new whole. For this reason, EPR-correlations are conceived as the correlations linking the properties of those individual subsystems, and when they are distant in space and do not interact, such correlations become puzzling. By contrast, in our ontology of properties, quantum systems are not individuals, not even objects: they are non-objectual bundles of properties. Therefore, there is no principle of individuality that allows them to retain their individual identity when they merge into a new bundle-system. As a consequence, the issue of interpreting EPR-correlations acquires a new formulation from the outset. The problem is no longer to explain the correlations between the properties of distant non-interacting objects. Since the composite bundle is a single whole, non analyzable in component bundles, the EPR-correlations are correlations between properties of a single item. Thus, the mystery of the original formulation, which seems to require a certain unexplainable harmony between distant objects, vanishes, since correlations between properties of a single system are natural even in the classical ontological domain. For example, it is not surprising that the area of a table is correlated to its length, or that the kinetic energy of a car is correlated to its mass and to its velocity: there is no need of an enigmatic harmony to explain these correlations. In a sense, this view implies a kind of holism. However, in a traditional ontology of objects and properties, the indivisible whole is also an individual. But, according to the traditional view, an individual, if it is not “atomic”, can be split up giving rise to individual parts that are different from the original one (see Sect. 4.2.2). This implies that, from a holistic perspective framed in a traditional ontology, the challenge is to account for the fact that the individual composite system is a whole that cannot be decomposed into individual parts. This problem also disappears in an ontology that lacks the category of individual: since the holistic item is not an individual, the fact that it lacks individual parts turns out to be an expected consequence. An ontology of properties without objects also allows us to cope with the fact that any composite quantum system can be decomposed into subsystems in different ways, none of them privileged over the others. This fact makes entanglement essentially relative to the particular decomposition considered in each case. Therefore, if subsystems are conceived as individuals, it must be accepted that there may be multiple non-local entanglement-induced correlations between multiply defined individual subsystems, which must be accounted for. From the perspective of our ontology, by contrast, a quantum system is a single non-objectual bundle of properties. Thus, the relativity of entanglement with respect to the multiple partitions of the composite system is nothing but the manifestation of the multiple

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correlations between the properties of a single bundle, which have nothing to do with non-local influences between non-univocally defined individual subsystems. This way of conceiving quantum correlations can be expressed mathematically in the algebraic formalism of quantum mechanics. Although the relativity of entanglement is usually introduced in terms of different tensor product structures— the various ways in which a Hilbert space can be decomposed into a tensor product of Hilbert spaces—, some authors have diverted their attention from Hilbert spaces to algebras of observables. For instance, Paolo Zanardi and his collaborators have taken an algebra M on a finite Hilbert space H as the starting point, to prove that, given two independent subalgebras .A and B of M that satisfy (i) independence ( .[A , B] = 0, that is, [a, b] = 0 for all .a ∈ A and b ∈ B) and (ii) completeness ( .A ⊗ B ∼ = A ∨ B = M ), then .A and B induce a tensor product .HA ⊗ HB (Zanardi, 2001; Zanardi et al., 2004). The authors stress that, in this way, the partition of the algebra of observables and the resulting entanglement of the state of interest can be made to depend on the observables accessible in each situation. In turn, Nathan Harshman and Kedar Ranade (2011) provide an explicit constructive method for generating such subalgebras from a finite set of operators that, although may look arbitrary from the viewpoint of the unstructured Hilbert space, have the right properties to rigorously define locality, separability, and entanglement. This algebraic perspective on entanglement also dispenses with the concept of particle— individual—and places observables—properties—at the center of the scene.

4.5.3 Indistinguishability As already explained, the problem of indistinguishability arises in quantum statistics when the issue is to explain why a permutation of individual particles does not lead to a different complexion. Consequently, particles are considered indistinguishable, leading to the violation of the weakest version of the principle of identity of indiscernibles. As pointed out in Sect. 4.3.3, despite the Received View about indistinguishability and its critics differ with respect to conceiving quantum systems as individuals, they nevertheless agree with respect to subsuming them under the category of object. In this subsection it will be argued that, by dispensing with the category of object, the problem acquires a completely different formulation. According to the Received View, quantum systems are non-individuals; it has also been suggested that they are not even objects at all (Quine, 1976, 1990). But these views give no metaphysical characterization of those items beyond their non-individuality or non-objectuality: they are only negatively characterized. By contrast, in the proposed ontology of properties, non-objectual quantum systems are positively and precisely characterized in metaphysical terms as bundles of I-type-properties. Moreover, I-type-properties, whith their corresponding P-caseproperties, are ontological items metaphysically characterized in a clear way and physically/formally represented with precision by observables of an algebra. As

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will be argued below, this positive characterization makes it possible to draw many conclusions about the nature and the behavior of quantum systems. In the traditional treatment of the problem, indistinguishability is a relation between particles, that is, individuals: particles with the same state-independent properties are indistinguishable. By contrast, in an ontology of properties, indistinguishability is primarily a relation between two instances of a same universal type-property when they have the same case-properties: two I-type-properties [A1 ] and [A2 ] are indistinguishable when they are I-type-properties   of the  same  U-typeproperty [A] and they have the same P-case-properties, . aj1 = aj2 . From this primary meaning, indistinguishability acquires a derived meaning when applied to bundles: two bundle-systems are indistinguishable when their respective Itype-properties are indistinguishable. Both indistinguishable I-type-properties and indistinguishable bundle-systems are only numerically different. Nevertheless, this does not imply that the principle of identity of indiscernibles is false for them: whereas the principle refers to the identity of indiscernible objects, in our case indistinguihability is a relation between items belonging to the ontological category of property. It is precisely this positive characterization of quantum systems that makes the non-applicability of the principle to them conceptually meaningful. When indistinguishable bundles combine, it is natural to expect that the Itype-properties belonging to the new bundle do not distinguish between the original bundles. Simply phrased, when two indistinguishable bundles merge into a single whole, which component bundle is taken first and which second does not matter at all. Mathematically, this requires that the observables representing the I-type-properties belonging to the composite bundle-system be symmetric with respect to the permutation of the component bundles. In this way, the so-called “indistinguishability principle” is satisfied in a natural way. In fact, in the context of the traditional particle-view, the principle states that all quantum states that differ only by a permutation of indistinguishable particles are observationally indistinguishable, that is, they lead to the same expectation values for any observable of the system. This requirement can be satisfied by restricting states to be symmetric (bosonic) or anti-symmetric (fermionic), or by restricting observables to be symmetric (see Messiah & Greenberg, 1964). Both the assumptions that certain states are inaccessible and that certain observables are not allowed have a certain ad hoc flavor, since they are posited exclusively to satisfy the indistinguishability principle. By contrast, in the proposed ontology of properties, the restriction on observables is ontologically motivated. The observables of systems composed of indistinguishable subsystems are symmetric due to the very nature of the properties of the component bundle: they are indistinguishable and, then, the order in which they are incorporated into the composite bundle is absolutely irrelevant. Therefore, the indistinguishability principle need not be considered an ad hoc postulate of the theory, but turns out to be a consequence of the ontologically motivated symmetry of the observables of the composite system. As an additional advantage, dispensing with symmetrization and anti-symmetrization of states dissolves the traditional problems of defining entanglement in the case of indistinguishability (see Fortin & Lombardi, 2022).

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4.6 The Physical Nature of Non-objectual Bundles 4.6.1 Which Properties? Up to this point, the quantum ontology is described as a property-only realm, in which systems are non-objectual bundles of properties. However, this characterization remains formal, as it says nothing about which properties effectively inhabit the quantum domain. In fact, not any I-type-property can belong to a quantum bundlesystem. The properties involved in the quantum ontology are physical properties. As Leslie Ballentine (1998) points out, although the formal structure of quantum mechanics is a necessary basis for the formulation of the theory, it has by itself very little physical content. When concrete physical problems are to be solved, the relevant observables of the system, endowed with a clear physical meaning, must be identified. Those observables are closely related to space-time symmetry transformations. Let us begin by recalling that each physical theory has a corresponding group of symmetry transformations, in the sense that the dynamical law of the theory is covariant under the transformations of the group, that is, it preserves its form under these transformations. The group corresponding to quantum mechanics is the Galilei group. Since it is a Lie group, each Galilei transformation Tα can be represented by a unitary operator Uα , with the exponential parametrization .Uα = eiKα sα , where sα is a continuous parameter and Kα is a Hermitian operator independent of sα , called “generator” of the transformation Tα . So, the Galilei group is defined by ten symmetry generators associated to ten parameters: one time-displacement, three space-displacements, three space-rotations, and three boost-velocity components. Those symmetry generators represent the basic physical observables of the theory (strictly speaking, the generators are proportional to the corresponding observables with a factor 1/): the energy H (time-displacement), the momentum P = (Px , Py , Pz ) (space-displacement), the total angular momentum J = (Jx , Jy , Jz ) (space-rotation), and the position Q = (Qx , Qy , Qz ) (boost-transformation, whose generator is mQ, where m is the mass). It is worth noting that, if the Hamiltonian H is a function of time, in general it cannot be conceived of as the generator of time-displacements. This means that the time-independence of the Hamiltonian is what endows the Schrödinger equation with a clear physical meaning (precisely, that of expressing time-displacements) and, at the same time, what makes it strictly applicable to closed systems. This result implicitly supports the orthodox formulation of quantum mechanics, in which the quantum system is conceived as a closed, constant-energy system, which unitarily evolves according to the Schrödinger equation. The Hamiltonian (the energy) of the system only changes with time as the result of its interaction with other systems. Although in a—closed—quantum system the Hamiltonian H is time-independent and, then, invariant under time-displacements, it may or may not have the remaining space-time symmetries. When H is invariant under a certain continuous transformation, the generator of that transformation is a constant of motion of the system. In

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other words, each symmetry of the Hamiltonian defines a conserved quantity. For example, the invariance of H under space-displacements in any direction implies that the momentum P is a constant of motion; the invariance of H under spacerotations in any direction implies that the total angular momentum J is a constant of motion. If, on the contrary, H is invariant under space-displacements only in one direction, say x, only the component Px of P is a constant of motion. The central role played by the Hamiltonian in the dynamical law of the theory and in the definition of the constants of motion of the system has led some authors to consider it the touchstone of the interpretation of quantum mechanics. According to the ModalHamiltonian Interpretation (Lombardi & Castagnino, 2008; Ardenghi et al., 2009a, b; Lombardi et al., 2010; Fortin et al., 2018; Lombardi & Ardenghi, 2022), the Hamiltonian of the quantum system defines the preferred context, that is, the set of the observables—I-type-properties—that acquire an actual definite value—an Acase-property—among their possible values—P-case-properties. In summary, space-time symmetry transformations endow the formal skeleton of quantum mechanics with the physical flesh and blood that make it a well-specified physical theory. From the ontological viewpoint, they play a central role in the identification of the fundamental physical properties of the quantum realm.

4.6.2 What Holds Properties Together? According to the traditional bundle theory, objects are composed of items of a different category: they are bundles of properties. However, not just any collection of properties forms a bundle that is an object: following Russell (1940), properties must hold a relation that binds them together in order to constitute an object. Objects are either bundles of coinstantiated universals in the universals-view or bundles of compresent tropes in the tropes-view. Both coinstantiation and compresence, which tie properties together, are commonly regarded as primitive relations, serving the same purposes as substance or bare particulars in the traditional object-andproperties ontological view. In the quantum case, not every type-property of the bundle-system has an actual case-property. For this reason, in the present ontological proposal, a quantum system is not a bundle of actual case-properties but a bundle of I-type-properties, that is, instances of universal type-properties, and the bundle itself is formally represented by an algebra of observables. Bundling, in this case, does not require a coinstantiation or compresence relation that plays the role of the substance in equipping the bundle with a feature that distinguishes it from other bundles and reidentifies it over time. Such relations are not necessary precisely because bundles are not objects. However, does this mean that quantum bundles are mere collections or aggregates of properties, with nothing holding them together? The answer to this question is negative. Bundle-systems have a well-defined structure, given by the specific relations that link the I-type-properties of the bundle to each other. Those relations

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are formally represented by the commutation relations of the form [A, B] = C between the observables of the algebra. In other words, what “ties up” the Itype-properties of a bundle is the physically meaningful structure of relations accuratey represented by the mathematical structure of the corresponding algebra of observables. In turn, that structure of relations is neither vaguely defined nor arbitrary, but rather it also follows from the symmetry group of quantum mechanics. As a Lie group, the Galilei group is defined by the commutation relations between its generators, which, as explained in the previous subsection, represent the basic physical observables of the theory: (a) [Pi , Pj ] = 0 (b) [Gi , Gj ] = 0 (c) [Ji , Jj ] = iεijk Jk (d) [Ji , Pj ] = iεijk Pk (e) [Ji , Gj ] = iεijk Gk

(f) [Gi , Pj ] = iδ ij M (g) [Pi , H] = 0 (h) [Ji , H] = 0 (i) [Gi , H] = iPi

where  is taken as equal to one, and εijk is the Levi-Civita tensor, such that i = k, j = k, εijk = εjki = εkij = 1, εikj = εjik = εkji = − 1, and εijk = 0 if i = j. The rest of the physical magnitudes can be defined in terms of these basic ones: for instance, the three position components are Qi = Gi /m, the three orbital angular momentum components are Li = εijk Qj Pk , and the three spin components are Si = Ji − Li . In turn, the Galilei group has three Casimir operators which, as such, commute with all the generators of the group: they are the mass operator M, the spin-squared operator S2 , and the internal energy operator W = H − P2 /2m. The eigenvalues of the Casimir operators label the irreducible representations of the group; so, in each irreducible representation, the Casimir operators are multiples of the identity: M = mI, where m is the mass, S2 = s(s + 1)I, where s is the eigenvalue of the spin S, and W = wI, where w is the scalar internal energy. In his seminal paper, Eugene Wigner (1939) introduced the idea that kinds of elementary particles in a quantum theory are represented by the irreducible projective representations of the symmetry group of the underlying space-time corresponding to that theory. In that paper, he focused on quantum field theory in Minkowski space-time, claiming that elementary particles correspond to the irreducible projective representations of the Poincaré group, which is the symmetry group of the Minkowski space-time. But this idea can also be applied to nonrelativistic quantum mechanics, in such a way that, in this theoretical framework, each irreducible representation of the Galilei group represents a kind of elemental particle, characterized by its mass m, its spin s, and its internal energy w (see da Costa et al., 2013). This section has shown how the symmetry group of quantum mechanics endows the structure of the quantum ontology with a precise physical referent. In this sense, the Galilei group plays a dual role. On the one hand, it defines the physical content of the properties that make up the bundle-systems. On the other hand, it establishes the relations between these properties, which give cohesion to the bundle without

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turning it into an object of traditional metaphysics. From a more general point of view, this section shows that physics and metaphysics are not at odds at all. On the contrary, when metaphysical notions are elucidated from the outset, they acquire physical content in a natural way.

4.7 Final Remarks The interpretation of quantum mechanics has been discussed for over a hundred years because it challenges certain basic assumptions of traditional metaphysics. The usual strategy for dealing with this situation has been to focus on one of the various challenges posed by the theory and to devise an interpretation that solves it, leaving aside the remaining difficulties. However, one can aspire to formulate a “global” solution, according to which all the problems can be adequately addressed in terms of a single ontology. This was the aim of the present work. Here it was argued that the ontological problems are not a matter of kinds of objects, but of ontological categories: it is necessary to decide how the reality referred to by quantum mechanics is structured, which ontological categories underlie the quantum realm. In the light of this central goal, the interpretive obstacles were addressed from a radical position: there are no individuals, not even objects in the quantum ontology; the quantum world is populated by quantum properties that form bundles which, nevertheless, do not acquire the necessary features to be subsumed under the ontological category of object. First, it was shown how this ontological picture provides coherent and conceptually unified answers to the main quantum ontological challenges: contextuality, non-separability and indistinguishability. Second, it was emphasized that the structure of this ontology can be endowed with precise physical content on the basis of the symmetry group of the theory. Of course, what has been said in this chapter does not exhaust all the interpretative issues surrounding quantum mechanics. For example, one question that cannot be ignored concerns the nature of quantum possibility and, with it, the interpretation of probability. Another inescapable issue is that referred to how to talk about non-objectual quantum systems, given that our ordinary and formal languages are designed to describe an ontology of objects and properties. Connecting the two issues is the need for a logic that makes possible to speak not only of properties without objects, but particularly of possible properties, that is, a modal-property based logic. However, a detailed treatment of these matters is beyond the scope of the present chapter and will be addressed in future works. Acknowledgements This work was supported by grant PICT-04519 of the Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT) of Argentina.

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Chapter 5

The Roads to Non-individuals (And How Not to Read Their Maps) Jonas R. B. Arenhart and Raoni W. Arroyo

Abstract Ever since its beginnings, standard quantum mechanics has been associated with a metaphysical view according to which the theory deals with nonindividual objects, i.e., objects deprived of individuality in some sense of the term. We shall examine the grounds of the claim according to which quantum mechanics is so closely connected with a metaphysics of non-individuals. In particular, we discuss the attempts to learn the ‘metaphysical lessons’ required by quantum mechanics coming from four distinct roads: from the formalism of the theory, treating separately the case of the physics and the underlying logic; from the ontology of the theory, understood as the furniture of the world according to the theory; and, at last, we analyze whether a metaphysics of non-individuals is indispensable from a purely metaphysical point of view. We argue that neither nonindividuality nor individuality is to be found imposed on us in any of these levels so that it should be seen as a metaphysical addition to the theory, rather than as a lesson from it.

J. R. B. Arenhart was partially funded by CNPq. R. W. Arroyo was supported by grants #2022/15992-8 and #2021/11381-1, São Paulo Research Foundation (FAPESP). J. R. B. Arenhart Department of Philosophy, Federal University of Santa Catarina, Florianópolis, SC, Brazil R. W. Arroyo () Centre For Logic, Epistemology and the History of Science, University of Campinas, Campinas, Brazil Department of Philosophy, Communication and Performing Arts, Roma Tre University, Rome, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_5

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5.1 Introduction Ever since its beginnings, standard quantum mechanics has been associated with a metaphysical view according to which the theory deals with non-individual objects, i.e., objects deprived of individuality in some sense of the term. However, it is also a well-learned lesson of current metaphysical methodology that metaphysical terrain is slowly and hardly won, if ever. That is, one cannot claim to have learned metaphysical lessons so easily from quantum mechanics, because the theory does not wear its metaphysics on its sleeves. In this sense, if quantum mechanics could give us a helping hand, at least in determining a metaphysics of non-individuality to its entities, it would have been such a great methodological victory for metaphysics. But can we learn such lessons? In this paper, we investigate precisely this question and check for the source of non-individuality and its justification. In more precise terms, we inquire into what kind of justification could be attributed to the claim that quantum entities are non-individuals. Does this follow from quantum mechanics itself? Is it attributed to an external source other than physics? If so, does this source grant any kind of justification for such an attribution, or is it floating free from the relevant physics? The question of the justification for such attribution of metaphysics is important not only for epistemic reasons, related to the methodological question of the metaphysics of quantum mechanics, but also, we believe because it highlights important features of non-individuality that are not yet discussed in the literature, and which are related with the very characterization of non-individuality. To claim that quantum entities are non-individuals, notice, is to advance a characterization in negative terms: quantum entities have lost a feature that classical entities do have, individuality. This view is called ‘the Received View’ on quantum non-individuality, given that it is the view that we received as the standard account of quantum entities by the likes of Schrödinger and Weyl (see French & Krause, 2006, chap. 3). The former, for instance, suggested that quantum entities are radically different from their classical counterparts, and dedicated more than one occasion to emphasize this: This essay deals with the elementary particle, more particularly with a certain feature that this concept has acquired—or rather lost—in quantum mechanics. I mean this: that the elementary particle is not an individual; it cannot be identified, it lacks “sameness”. [. . . ] In technical language it is covered by saying that the particles “obey” a new-fangled statistics, either Bose–Einstein or Fermi–Dirac statistics. The implication, far from obvious, is that the unsuspected epithet “this” is not quite properly applicable to, say, an electron, except with caution, in a restricted sense, and sometimes not at all. (Schrödinger, 1998, p. 197)

Here we have the whole package associated with the Received View (the RV, for short, from now on) in a nutshell. It is claimed that quantum particles have lost their individuality (a metaphysical feature), and that the main reason for that is the presence of new statistics in quantum mechanics (a physical feature), and that the particles lack ‘sameness’ (if that is understood as a failure of the identity relation, then, this is a logical feature). There are distinct features that a non-individual have,

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and they are related somehow. As a result, we shall discuss whether these features help us characterize non-individuality and whether they do help us derive any kind of justification for non-individuality from quantum mechanics. Before we present the plan for this paper, however, let us advance a distinction that makes for the background of this paper. We are assuming a distinction between ontology and metaphysics, where the former is understood as a kind of subset of the latter; this is rather traditional but is not undisputed these days, so we state it here as a starting point, given that it is not our aim to defend the correction or fruitfulness of the distinction. In a nutshell, ‘ontology’ deals with questions of existence, and metaphysics with questions of nature (see Arenhart, 2012, 2019; Arenhart & Arroyo, 2021; Hofweber, 2016; Thomson-Jones, 2017). In this way, we understand that the ontological questions are: what exists, and how are those entities that exist? ‘Metaphysics’ would go further, speculating about the nature of what was obtained in ontology, that is, developing from the grounds provided by an ontology. The first stage of ontology, in the more naturalistic context we concentrate on here, would be to form a catalog about what exists according to a given scientific theory.1 To refer to the existents of a theory, we choose the word ‘entity’, because it seems to be more neutral than ‘thing’, ‘stuff’, or ‘object’.2 As our discussion will focus on quantum mechanics, it seems safe to say that in a straightforward reading, the theory tells us that there are electrons, protons, and other entities typically conceived of as particles. They are part of the furniture of the world according to quantum mechanics. Electrons exist according to the theory, but we still don’t know what kind of entities they are; at least, quantum mechanics doesn’t give us straightforwardly that kind of information. So, the second task of ontology would be to organize these entities that exist in more general categories, types of entities. In this paper, given our more specific interests, we will focus on the category of ‘objects’ (other categories include properties and relations, and events). Suppose electrons are objects: what does that say about the accompanying metaphysics? Well, we can understand them metaphysically at least in two different ways, e.g., as individuals or as non-individuals. But what are individuals? And what are nonindividuals? To answer these kinds of questions is to attribute what we call a ‘metaphysical profile’ to quantum entities, or, what we take to be equivalent here, to describe their ‘nature’.

1 This particular task of ontology is named in several other ways in the literature, many of them metaphorical, such as the establishment of an inventory of the furniture of the world according to the theory in question. Note that at this stage we are also assuming a metaontological posture inspired by the Quinean tradition, according to which we can speak about the ontological catalog associated with certain theories, although we do not wish to take sides here on questions of whether existence is better represented by quantification or predication, and the like. 2 This type of caution has gained attention; we see this, for example, in Meincke (2020, p. 2): “I deliberately speak of ‘entities’ rather than (as many scholars do) of ‘objects’ to leave open the possibility that also non-object-like entities, such as processes, events, states of affairs, structures etc”. We shall return on this matter, albeit briefly, in Sect. 5.4.

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Given this distinction, recall, our problem is to discuss the notion of nonindividuality and the kind of justification one may have to say that quantum mechanics provides for a metaphysics of non-individuals. We are concerned with the source of non-individuality, which would also confer to the notion a kind of legitimacy. As we have briefly discussed already, there are distinct features that are used to characterize a non-individual, metaphysical, logical, and physical ones. Could they shed any light on the justification for the attribution of non-individuality for quantum mechanics? Our goal is to investigate precisely that. In Sect. 5.2, we will try to answer this question by attributing this role to logic; in Sect. 5.3, by seeing physics as in charge for the answer; in Sect. 5.4, ontology; finally, in Sect. 5.5, we will look for the metaphysical profile of non-individuals in metaphysics. We will go through these different areas of knowledge with a Wanted sign, checking what kind of information on non-individuality one may obtain from them, and conclude in Sect. 5.6.

5.2 From Logic to Non-individuality We start with attempts to find non-individuality in logic. The plan, in a nutshell, is: if non-individuality could be codified logically, in terms of the underlying logic, then, perhaps, one could have some kind of justification for a clear notion of non-individual in a system of logic incorporating this notion. That is, perhaps nonindividuality is a matter of logic, and we could justify non-individuality if we could justify the adoption of such a logic. Recall now Schrödinger’s quote above. Part of the idea of non-individuality comes from the idea that quantum entities lack ‘sameness’. One typical way, which has been traditionally wedded to the Received View in such a way that it is currently conflated with the RV itself, is to follow one of the many indications by Schrödinger in the quote presented, that quantum particles lack “sameness” (see the discussion in Arenhart, 2017b). This has been taken literally as an inspiration for the development of non-classical logics codifying this suggestion, that is, systems of logic in which the idea of identity loses its meaning due to lack of application; these systems have been baptized non-reflexive logics, due to the failure of the reflexive law of identity (see French and Krause (2006, chap. 7–8) for the standard presentation of the systems).3 The intuition behind this formulation of the RV is that nonindividuals are entities to which the logical relation of identity does not apply. As non-reflexive logics capture this notion of non-individuals, the expectation is to find the metaphysical profile of non-individuals in non-reflexive logics. As French (2019a, p. 22) has put it, referring to quasi-set theory (a form non-reflexive set theory), “forms of non-standard set theory have been presented as a framework

3 However, non-reflexivity indicates only part of what is failing here; the system does not express any result concerning identity for the intended quantum entities, so every property of identity fails.

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in which to understand the claim that they are non-individuals.” Also, about the relation between non-reflexive logics and non-individuality: These developments [of non-reflexive logics] supply the beginnings of a categorial framework for quantum ‘non-individuality’ which, it is claimed, helps to articulate this notion and, bluntly, make it philosophically respectable. (French, 2019b, sect. 5)

So, the logical framework is supposed to make a substantial job in delivering at least the general contours of the metaphysics of non-individuals, its ‘articulation’. However, notice that a system of logic, per se, does not have any commitment to a metaphysical picture of reality unless we attribute it one (for the case of the RV and non-reflexive systems, see Arenhart (2018)). There must be something like an intended interpretation of the lack of identity as meaning specifically failure of individuality, otherwise, no lesson is learned from that system. Standard classical logic without identity could also be seen as a non-reflexive logic, although it is not typically seen as articulating a metaphysics of non-individuals. So, there must be more to it if logic is going to teach us something about non-individuality. Of course, one way to establish such an intended meaning, from which metaphysical lessons could be drawn, may be obtained if we add a metaphysical reading over the system, a reading that is not necessarily there but is the intended application of the system. Perhaps this is what is actually done. By recalling that individuality may be understood in terms of haecceities, non-qualitative properties that items have, and which intuitively mean ‘being identical to itself’,4 French and Krause (2006, pp. 13– 14) indicate that the metaphysical meaning of non-individuality may be understood in terms of the lack of a haecceity, and the formal expression of that is the failure of identity: [. . . ] the idea is apparently simple: regarded in haecceistic terms, “Transcendental Individuality” can be understood as the identity of an object with itself; that is, ‘.a = a’. We shall then defend the claim that the notion of non-individuality can be captured in the quantum context by formal systems in which self-identity is not always well-defined, so that the reflexive law of identity, namely, .∀x(x = x), is not valid in general. (French & Krause, 2006, pp. 13–14)

They recognize that by establishing such an association they “are supposing a strong relationship between individuality and identity [. . . ] for we have characterized ‘non-individuals’ as those entities for which the relation of self-identity .a = a does not make sense” (French & Krause, 2006, p. 248). This association, however, clearly does not come from logic, but from specific choices of a particular characterization of individuality that relates it to identity and, consequently, a relation of lack of identity to lack of individuality. So, what is really going on here is that we have a different attitude towards nonindividuality and its relation to the formalism of non-reflexive logics than the one we expected: it is not that the formal system teaches us about non-individuality, but rather, there is a previous notion of non-individuality that the system is supposed to

4 So,

Plato’s haecceity is the property ‘being identical to Plato’.

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capture. In this sense, the metaphysics of non-individuality comes from outside of the formal system and is neither derived from it nor articulated by it. The dignity of the Received View, in this case, or its philosophical respectability, does not rest on the success of the distinct systems of non-reflexive logics, but rather on the coherence of the idea that non-individuality may be reasonably couched in terms of the lack of haecceity. However, this is now a metaphysical view that is somehow described by nonreflexive logics, not extracted from such logics. There are no special lessons learned for this metaphysics that derive from the non-reflexive systems, rather, the expectation is that such systems must correctly capture the basic tenets of lack of haecceity. In this sense, the explanatory direction goes from metaphysics to logic, one must have the metaphysics clear beforehand, to develop the logic that will describe it and not the other way around. Things get more complicated for the non-reflexive approach to the Received View when one considers that such logic is not even necessary for the Received View. If one concedes that quantum entities are non-individuals but refuses to follow the suggestion of Schrödinger and understand it in terms of lack of identity, and more, refuses to follow French and Krause couching lack of individuality in terms of lack of haecceity, one can still adopt a form of non-individuality. Indeed, one may adopt another suggestion, briefly advanced but not developed by French and Krause (2006, chap. 4), for instance, and see individuality as conferred by the spatiotemporal location. In this case, given that quantum theory is typically seen in most of its formulations as suggesting that particles do not have a well-defined location all of the time, then, individuality fails. However, notice that this approach is not formulated in terms of the lack of identity. This illustrates the general thesis that, if non-individuality is to be understood in terms of a lack of individuality, then, our metaphysical framing of the notion will depend on a theory of what individuals are, to begin with, and distinct such theories will give rise to distinct theories of non-individuals (see Arenhart, 2017b). Most of them will not require that the logical relation of identity is abandoned, and that non-reflexive systems be adopted. Most metaphysics of non-individuals will be quite compatible with classical logic, and it would be incorrect to think that classical logic teaches us anything special about them. This corroborates the claim then, that logic, per se, does not teach us metaphysical lessons about non-individuality (and neither about individuality too, of course).

5.3 From Physics to Non-individuality Perhaps we need a more concrete jury, like an empirical science. As the quote in the introduction indicates, one of the main reasons that Schrödinger, among others, had to suggest that quantum mechanics deals with non-individuals comes from the new kinds of statistics that the theory obeys. So let’s see if a minimal portion of the

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formalism of quantum mechanics itself could succeed where logic failed to offer us a metaphysical profile for non-individuals.5 Although much discussion has already been made about the role of statistics in quantum mechanics in deriving metaphysical lessons, the issue that Schrödinger wanted to highlight may be put in quite simple terms, contrasting the new quantum statistics with the classical Maxwell–Boltzmann statistics. The idea that classical particles have individuality, even in the situations in which one considers that they do have all the same properties (that is, when they are indiscernible) is famously encapsulated in the way that Maxwell–Boltzmann’s statistics work. Let us illustrate it with the case in which two particles, labeled 1 and 2, must be distributed in two states, A and B, which are equiprobable. We have the following possibilities (where .A(1) means that particle 1 is in state A, and so on): 1. 2. 3. 4.

A(1)A(2); B(1)B(2); .A(1)B(2); .A(2)B(1). . .

All these possibilities are assigned the same weight, that is, . 14 , and even though the particles are being considered as indistinguishable by their properties (stateindependent and state-dependent alike), permutations of particles are counted as distinct possibilities in the steps 3 and 4; in fact, they differ only by a permutation of the labels of the particles. So, if the particles may be indistinguishable, what accounts for the difference in the two situations described in 3 and 4? Well, it seems that there must be something in the particles that explains the fact that exchanging 3 with 4 makes a difference, and this something is thought to be the particles’ individuality. The individuality of classical particles may be explained in a variety of ways (and the metaphysics of individuality for classical particles gets underdetermined by classical physics). It is possible to adopt the kind of individuality principles that add some metaphysical ingredient, the “Transcendental Individuality” approach, with such ingredient understood in terms of a substratum, or a haecceity6 or one may adopt a bundle theory, individuating the particles by granting that each individual has some unique set of properties that may be used to characterize it, and which accounts for its individuality (see French and Krause (2006, chap. 1) for the basic approaches to individuality). The latter can always obtain in classical mechanics by the adoption of spatial position as a property, and a Principle of Impenetrability, which holds in classical mechanics, and which says that no two particles occupy the same position. 5 Notice: the idea that logic failed to offer us a metaphysical profile means here that, from logic alone, one can neither derive the claim that quantum entities are non-individuals and nor obtain any specific information on the notion of non-individuality that could be used to describe the nature of quantum entities, were they non-individuals. 6 And they are distinct approaches, given that the substratum is a particular item that is added to the composition of the particular object, while haecceity is a non-qualitative property.

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In quantum statistics, on the other hand, permutations of indistinguishable particles are not counted as giving rise to a distinct state. This fact is related to the already mentioned Permutation Symmetry in quantum mechanics, and is behind the typical allegations of loss of identity and individuality; that is, quantum mechanics itself would be the source of the failure of individuality. This is illustrated as follows. For two systems, labeled 1 and 2, distributed in two possible states A and B, we can have the following possibilities: 1. .A(1)A(2); 2. .B(1)B(2); 3. . √1 (A(1)B(2) ± A(2)B(1)). 2

In fact, we have two different kinds of statistics here: Bose–Einstein (BE) for bosons and Fermi–Dirac (FD) for fermions. The difference comes in the third possibility, because bosons have the “.+” sign, and fermions have the “.−” sign. Also, for fermions only the third case obtains; they cannot be distributed according to the first two cases for they cannot be in the same state, due to the Pauli Exclusion Principle, which holds for fermions (but not for bosons). This third possibility indicates, briefly, that the description of a state in which the particles 1 and 2 are distributed over A and B must take into account, simultaneously, both cases of possible distributions; notice that swapping the labels of particles does not change the state for bosons, while it just changes the sign of the state for fermions (and does not change the probabilities associated with the state). Permutations of particles are not observable. Does this indicate anything special about metaphysics? As we have seen, Schrödinger, and many others, thought so (see French & Krause, 2006, chap. 3). The only explanation for this possibility of permuting the particles without resulting in any different state, according to such a view, is that there is nothing in the particles themselves that allow for such a difference that would be had in the states such as .A(1)B(2) or .A(2)B(1), which would distinguish the particles. So, the particles are not individuals, because they lack precisely that which makes the classical particles individuals. Furthermore, given that permutations are not observable, the idea seems to be that one cannot identify particles, they are not things that one can trace and find in distinct situations. So, these features all suggested to Schrödinger that identity, in general, fails to apply to such particles, giving rise to the now traditional association between non-individuality and lack of identity (see the discussion in Arenhart, 2017b). The problem with this view is that the association between the behavior of quantum particles as described by the statistics is not so straightforwardly connected with the failure of identity, and also, it is not so straightforwardly associated with non-individuality. Non-individuality, as a failure of identity, does not drop out of the quantum apparatus. Let us check briefly. We have already commented in the previous section that the identification of non-individuality and lack of identity is a kind of historical contingency. The lack of individuality, later it was discovered, could be described in many distinct ways (see,

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in particular, Arenhart, 2017b). Quantum mechanics does not explicitly indicate that it is the particular approach to non-individuality as a failure of identity that it endorses. The same could be said about non-individuality itself. As argued at length in French and Krause (2006, chap. 4), quantum theory is compatible with a metaphysics of individuals. It is enough that individuality is attributed to some principle that allows indiscernible objects, like substrata or haecceities. The fact that quantum particles cannot be distinguished by quantum mechanical means does not produce any difficulty for these approaches, whose main motivation is in fact to deal with scenarios in which objects are not discernible and, hence, not able to be individuated by their properties or relations. So, no wonder that quantum mechanics is compatible with a metaphysics of individuals too. The theory says nothing about its metaphysics of individuality and non-individuality. As French has put it, concerning classical mechanics (but the lesson applies to quantum mechanics too):7 [. . . ] that metaphysics does not supply the kinds of ‘objects’ over which appropriately formalised versions of scientific theories could be said to quantify (do first order formulations of classical statistical mechanics quantify over haecceities, for example?!). (French, 2018b, p. 216, original emphasis)

That is, metaphysics is not in physics. We cannot hope to get from the theory the metaphysical descriptions that would be required to confirm that the entities dealt with in quantum mechanics are non-individuals, and even for those that believe that quantum mechanics deals with non-individuals, the particular approach to nonindividuality is also not determined by the theory. Perhaps it is in another place that we should look for these metaphysical lessons.

5.4 From Ontology to Non-individuality In some sense, ontology is a naturalizable philosophical discipline, i.e., its major question can be answered by science—at least in part, as we shall see. In particular, by physics. So maybe ontology can tell us what non-individuals are. Let’s take a closer look at this claim. As Esfeld (2019, p. 222) claims, a physical theory is responsible for offering us a dynamic law and an ontology, i.e., to spell out “[w]hat is the law that describes the individual processes that occur in nature (dynamics) and what are the entities that make up these individual processes (ontology)”. In this sense, as Dürr and Lazarovici (2020, p. 2) recognize, “[t]he ontology of a physical theory specifies what the theory is about”. Furthermore, in quantum mechanics, as Ruetsche (2018, p. 298) acknowledges, this is largely dependent on the interpretation adopted: hence, “[. . . ] an interpretation of QM tells the realist 7 The

same point was made already in Arenhart (2012, pp. 344–345) in connection to quantum mechanics.

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about QM what she believes when she believes QM”. As this is pretty standard, we recall the current state of quantum foundations very briefly as follows. An interpretation of quantum mechanics is a solution to the measurement problem, whose standard way—i.e., adopting Maudlin’s (1995) taxonomy—is stated as a checklist of three simple questions about the quantum-mechanical (e.g. the wave function) description: (i) is it complete? (ii) is it linear? (iii) does it yield definite (unique) measurement outcomes?8 The most elementary definition of an interpretation of quantum mechanics, that is, of a solution to the aforementioned measurement problem, is by the negative answer to one of the questions, which poses a trilemma (see Dürr & Lazarovici, 2020, p. 47). To deny the first implies accepting theories of hidden variables, such as Bohmian mechanics; a negative answer to the second question implies the adoption of collapse theories, such as the standard quantum mechanics with collapse and the GRW theory; finally, a negative answer to the third question implies the acceptance of an Everettian quantum mechanics, such as the many-worlds interpretation. Concerning ontology, each of these interpretations populates the world with different kinds of entities: Bohmian mechanics postulates the existence of a ‘pilot wave’, which governs the behavior of quantum particles (which exist along with the wave); some collapse interpretations postulate the existence of a ‘consciousness’ endowed with causal powers, separated from matter; some Everettian quantum mechanics postulate the existence of a plurality of ‘worlds’, and so on. The debate over (non-)individuality is not about an interpretation of quantum mechanics in the sense of offering an answer to the measurement problem; however, it seems evident that it is about ‘interpreting’ quantum mechanics in some sense (see Krause et al., 2022). Therefore, a peculiarity of this debate is its dependence on the interpretation adopted. Traditionally, metaphysical underdetermination between individuals and non-individuals is discussed in interpretations that admit the dynamics of collapse (see French & Krause, 2006; Krause & Arenhart, 2016), but it might be also discussed concerning Bohmian mechanics (see Pylkkänen et al., 2016) and Everettian quantum mechanics (see Conroy, 2016). So far we are working with a very standard notion of the term ‘ontology’ which, according to the Quinean tradition, asks “[. . . ] what, according to that theory, there is” (Quine, 1951, p. 65). So, in the (fairly) uncontroversial part, quantum mechanics tells us that there are bosons and fermions, among other entities. It is in this sense that we say that ontology has been ‘naturalized’: after all, we can extract these entities from the theory, it commits itself ontologically to it (see also Arenhart & Arroyo, 2021). As we have seen, other entities can be extracted depending on the interpretation adopted. In this way, a catalog or inventory is formed about what exists according to each interpretation. This inventory does not answer as to whether the theory deals with non-individuals, and what they are. In order to properly address this question of non-individuality through ontology, it could be helpful to introduce another aspect of ontology. The aspect of ontology

8A

proof of inconsistency may be found elsewhere (see Maudlin, 1995; Esfeld, 2019).

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we have in mind is explicitly acknowledged, for instance, by Ney (2014, p. 30), when she describes ontology as dealing not only with the issue of the furniture of the world but also with “a particular theory about the types of entities there are”. That is, we need to establish an ontological catalog of reality, listing what things exist (tables, chairs, electrons, Everettian worlds, Bohmian pilot waves, etc.); but we also need to establish ontological categories, classifying into types the entities of the catalog (examples of types: objects, immanent powers, structures, wave functions, etc.). While interpretations are clearly concerned with the catalog, and one could, at least in principle, attempt to find in the theory along with an interpretation the answer to the catalog question, the same cannot be said about the types of entities. In fact, the theory does not say anything explicitly about the type of entities it deals with, and this shall cast doubt on the project to attempt to extract any information about non-individuality from the ontology of quantum mechanics. Let us check. One could hope that ontology and quantum theory have a close connection. It could go along the following lines: quantum theory commits us with objects (ontological type) behaving in many respects quite different from what individual objects typically do (the nature of such objects). These would be the first steps in framing a connection between ontology and the nature of non-individuals. It would enjoy the benefit of epistemic authority derived from quantum mechanics itself. It is common to find in the literature that the focus on ontological aspects is a metametaphysical attitude typical of metaphysical naturalism (see French, 2018a; Arroyo & Arenhart, 2019). That is, naturalists maintain ontology as the privileged part of metaphysics, often reducing it to one another. After all, ontology can be obtained from science. Thus, some naturalists, like Maudlin, conclude that: Metaphysics is ontology. Ontology is the most generic study of what exists. Evidence for what exists, at least in the physical world, is provided solely by empirical research. Hence the proper object of most metaphysics is the careful analysis of our best scientific theories (and especially of fundamental physical theories) with the goal of determining what they imply about the constitution of the physical world. (Maudlin, 2007, p. 104)

Realists of this type, according to Magnus’ (2012; see also French (2018a)) characterization, are ‘shallow’ scientific realists. That is, scientific realists who feel epistemically safe going only as far as science goes—no further steps. And science goes—they say—only to the shallow waters of ontology. Again, some suggest that ontological proposals are epistemically justified because they are somehow extracted from science. The problem is that people with very different ideas about “lessons learned from quantum mechanics” seem to have the same belief. As a sample, we can showcase three alternatives to object-based ontology: 1. Ontic structural realism (OSR): “structures” are the basic elements of ontology (see French, 2014); 2. The logos approach: “immanent powers” are the basic ontological stuff (see de Ronde & Massri, 2021); 3. The wave function realism: the “wave function” is the basic stuff (see Albert, 2013).

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The curious thing is: how can we ‘read off’ three ontologies so different from the same quantum theory? The lesson seems to be rather than quantum theory teaches us no lesson about it. Perhaps even the hope of extracting the ontology of types of entities, when we are restricted to a shallow realism, is too high. This seems to be impossible. A first argument is the finding that ontological results, of the types of ontology, are underdetermined by physics. But perhaps the most radical aspect is the following: the type-aspect of an ontology associated with interpretations of quantum mechanics, is placed on the catalog. Unlike the catalog-aspect, the typeaspect of ontology involves the creativity of the metaphysician of science, rather than extraction. Thus, this shallow option cannot even come close to answering the profile of non-individuals, given that it does not guarantee even an ontology of objects. One could argue that the shallow realist never set out to argue about the metaphysical profile of entities. However, if that is the answer to the concern with non-individuality, then, what we are indicating is that the shallow realist falls short of the task she proposes: that of extracting the ontology from the theory. An alternative that seems more appropriate would be to interpret the development of these ontologies of types of entities as an addition to what the theory says so that the process of ‘reading off’ the ontology would be better understood as a metaphor that indicates a better fit that each author sees, concerning their proposal, with what the theory provides. In this sense, we agree with Chakravartty’s diagnosis, when evaluating the OSR venture as offered by French, in these terms: When French speaks of reading ontology from fundamental physics, what he is doing, in fact, is implicitly appealing to some (one hopes) defensible criterion or criteria which he takes to point toward a preferred interpretation of the relevant physics—an interpretation which is (one hopes) demonstrably superior to others. (Chakravartty, 2019, p. 11, original emphasis)

To focus on the OSR only, the point is that, as Esfeld (2013, p. 29) states, “OSR is not an interpretation of QM in addition to many worlds-type interpretations, collapse-type interpretations, or hidden variable-type interpretations”, and this is so because, in addition to the postulate “radically different proposals for an ontology of QM”, interpretations of quantum mechanics also commit themselves to different dynamics. So even if one takes the ‘shallow’ road to realism, such a road is a slippery slope between (a) not having all ontology informed by science, and (b) not having an epistemic (objective) guarantee to adopt an alternative to the detriment of another. We started out wanting a metaphysical profile for non-individuals, and we saw that ‘shallow’ realism, as a naturalistic approach to ontology, is a road that didn’t lead us to that. It fails in two important directions: it does not concern itself explicitly with metaphysical profiles, and, worse yet, it does not grant epistemic authority from science to the types-ontology that is advanced in each case. In this sense, there is a failure in granting the required ontology of objects an appropriate justification, and one could see this as more trouble for the non-individuals interpretation. However, this more basic source of trouble is an issue we shall not discuss here.

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5.5 From Metaphysics to Non-individuality An alternative to ‘shallow’ realism, still within the scope of scientific realism, would be the ‘deep’ scientific realism. Here, metaphysicians deliberately venture into the metaphysical waters in search of a metaphysical profile for non-individuals. On the one hand, giving a metaphysical profile seems to be a necessary component of one’s scientific realism. After all, according to Chakravartty (2007, p. 26) “[o]ne cannot fully appreciate what it might mean to be a realist until one has a clear picture of what one is being invited to be a realist about” (i.e. Chakravartty’s Challenge). On the other hand, how to do that? Here’s French: A simple answer would be, through physics which gives us a certain picture of the world as including particles, for example. But is this clear enough? Consider the further, but apparently obvious, question, are these particles individual objects, like chairs, tables, or people are? In answering this question we need to supply, I maintain, or at least allude to, an appropriate metaphysics of individuality and this exemplifies the general claim that in order to obtain Chakravarttys clear picture and hence obtain an appropriate realist understanding we need to provide an appropriate metaphysics. (French, 2014, p. 48)

So, on what concerns metaphysical issues such as individuality (but not only individuality, of course), physics seems to be the source for the metaphysical profile. This is also indicated in the following claim by French and Krause, where it is thought that one expects that the ‘theoretical content’ of the theory should be articulated in metaphysical terms. That is, the theoretical content and the metaphysics in terms of which it is articulated come from the same source: [. . . ] if that theoretical content is taken to have a metaphysical component, in the sense that the realist’s commitment to a particular ontology needs to be articulated in metaphysical terms, and in particular with regard to the individuality or non-individuality of the particles, then the realist appears to face a situation in which there are two, metaphysically inequivalent, approaches between which no choice can be made based on the physics itself. [. . . ] The choice for the realist is stark: either fall into some form of antirealism or drop the aforementioned metaphysical component and adopt an ontologically less problematic position. (French & Krause, 2006, p. 244)

That is, in the face of metaphysical underdetermination between individuality and non-individuality, one solution is to change the ontological basis, with the hopes that a distinct approach to the general kind of entities that one is assuming the theory deals with could solve the problem of granting a metaphysical description along with the theoretical content (they suggest an ontology of relations in place of an ontology of objects). The great concern here is epistemic since in these are turbulent waters, which no longer find support in scientific theories: [. . . ] you get only as much metaphysics out of a physical theory as you put in and pulling metaphysical rabbits out of physical hats does indeed involve a certain amount of philosophical sleight of hand. (French, 1995, p. 466)

Hence, the metaphysical profile finds itself ‘floating free’ from physics. Methodologically speaking, we saw that we cannot extract the metaphysical profile from logic, physics, or ontology (see Arroyo & Arenhart, 2020); it remains to be seen

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whether we can extract the metaphysical profile.. . . from metaphysics! This is, roughly speaking, the proposal put forward by French and McKenzie, sometimes called the “Viking approach” (see French, 2014), and sometimes the “toolbox” approach (see French & McKenzie, 2012, 2015; French, 2018b). The general idea of this approach is as follows: recognizing the need for a metaphysical profile so that we can give content to ‘deep’ scientific realism, metaphysicians would be free to produce their metaphysical theories free from science. These theories, when necessary, can be used for interpretive purposes of science, that is, to provide an adequate metaphysical profile for scientific theories—thus enabling the adoption of a metaphysically informed (or ‘deep’) scientific realism. From ready-made metaphysical theories—as could be the case with substance dualism and some interpretations of collapse-based quantum mechanics (see Arroyo & Arenhart, 2019)—to methods and strategies available in the metaphysical literature—such as, for example, methodological lessons that we use to extract from Lewisian modal realism a metaphysical profile to certain interpretations of Everettian quantum mechanics (see Wilson, 2020). The major difficulty in attempting to pick metaphysical theories of individuality and non-individuality from metaphysics proper is that the variety of options is just too big, and one ends up losing sight of the epistemic anchorage that quantum mechanics could provide for any such metaphysical adventure. That is, the only restriction that one could find is a straight inconsistency between quantum theory and a given metaphysical package, but this is still not enough to avoid metaphysical underdetermination. Regarding the metaphysical layer, using the Viking approach, we can give a metaphysical body to the ontological bones of each of the interpretations. As there is no anchoring in physics, it is to be expected that there is a metaphysical underdetermination (cf. Arroyo & Arenhart, 2020). We are aware that many authors understand metaphysical underdetermination as a motivation for structural realism, and this is the classic case of the individuality of quantum objects (French & Krause, 2006): if our best scientific theories do not provide us with elements to decide between a metaphysics of individuals and a metaphysics of non-individuals, we should modify our ontological basis for structures, where this underdetermination of individuality does not occur (cf. Ladyman, 1998; French, 2014, 2020a). However, on the one hand, it is not clear what a structure is in metaphysical terms; furthermore, there is no guarantee that, if we have a precise definition of what a structure is in metaphysical terms (cf. Arenhart & Bueno, 2015), that definition is unique so that we don’t fall prey for the problem of metaphysical underdetermination (cf. French, 2020b). On the other hand, although structural realism can respond to metaphysical underdetermination, it certainly cannot respond to underdetermination of quantum mechanical interpretations (cf. Esfeld, 2013), so underdetermination is not a direct guide to the ontology and metaphysics of structures (cf. Ruetsche, 2018). As we saw in Sect. 5.2, being a non-individual requires something else than values of variables, as Ladyman (2016, pp. 186–187)

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also acknowledges; the issue becomes even more pressing as there is no consensus about what is this “something else”: • • • • •

persistence (French and Redhead, 1988) transworld identity countability and determinate identity (Lowe, 2003) laws of identity, perhaps including PII absolute discernibility (French & Krause, 2006; Muller & Saunders, 2008; Caulton & Butterfield, 2012) • possession of some form of transcendent individuality (Ladyman, 2016, pp. 186–187)

As it is to be expected, there is not a single clear picture concerning what individuality is; consequently, as the RV is the position according to which quantum objects lose their individuality, it is also unclear what non-individuality is. Worse yet: even if one adopts the idea, following Ladyman (2016, p. 187), that principles of individuality such as the principle of transcendent individuation are the most fundamental, there are also many options:9 • • • • • • •

haecceity primitive thisness (Adams, 1979) transcendental individuality (Post, 1963; French & Redhead, 1988) bare particulars individual substances substratum self-individuating elements (Lowe, 2003) (Ladyman, 2016, p. 187)

On what concerns individuals, as we have seen, the claim that quantum entities are individuals is not itself ruled out, to begin with. What is enough is that the principle of individuality is not based on any notion of qualitative discernibility (these principles would be, at least prima facie, incompatible with a quantum mechanical description of objects). One could, for instance, argue that quantum entities are individuated by haecceities, the non-qualitative property of being ‘identical to itself’, and which each individual allegedly bears to itself (see Arenhart, 2017b; French & Krause, 2006, chap. 1). Of course, one could complain that this inflates the metaphysics, but when we consider that the Viking merely requires that we look for the appropriate tools in the toolbox, what one finds is that this kind of approach is available, and does some extra job in the trans-world identity (on which we shall not comment). Another option would be to confer individuality as a primitive feature of particles. This is the approach taken by Morganti (2015), among others. The plan is not that individuality is conferred by any kind of intrinsic feature, but by a kind of thin notion of identity, which adds nothing else in the constitution of the entity, in metaphysical terms, although at the same time, it plays the role of granting

9 Although some of the terms of the list are sometimes taken as terminological variants of each other, given the lack of a unified definition, it is useful to keep the distinct terms, because they may reflect some differences of usage for some authors.

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individuality. Here, one may have to choose between the explanatory power of defining individuality in terms of haecceity (which in itself is a bit mysterious), or of conferring it primitively, having to pay the price of losing explanatory power (and this is a reason why French and Krause (2006, chap. 1) have restricted their discussion to principles defining individuality, not attributing it primitively). The dispute is common in metaphysics, and it is not something that one could address by appealing to quantum mechanics itself. The idea that quantum entities are non-individuals does not suffer a better fate when we consider the toolbox approach. Granted that one needs to provide for a metaphysical profile for non-individuality, that is, the absence of individuality, one could look in the toolbox for diverse principles of individuality and check how they fail in quantum mechanics. This certainly leaves one with plenty of tools available in the toolbox, as the previous discussion makes clear. At first sight, by adopting the association between self-identity and haecceity, one could follow the Received View and attempt to eliminate the latter by eliminating the former. Lack of individuality means lack of haecceity (see our previous discussion on this issue, and also Arenhart (2017b)). However, this is not the only option. As French and Krause (2006, chap. 4) have suggested, one could also adopt a view that the principle of individuality consists in the spatiotemporal position of an entity. Given the well-known restrictions imposed on the standard formulation of quantum mechanics, there is not always a well-determined position for a particle, and one could reasonably take it to mean that such particles are non-individuals, they lack the feature defining individuality (curiously, this same principle could be used in Bohmian mechanics as a principle of individuality, so that the toolbox has distinct roles, depending on the ontology, which here is seen as related to the interpretation too). These examples are enough to show that when one looks for a metaphysics of individuality and also for a metaphysics of non-individuality in the current literature of metaphysics, one does find the tools. The major obstacle is that there are just too many tools, and one cannot expect to get any help from quantum mechanics itself on these issues. The plan, of course, was that metaphysics, as a toolbox, could help us get a clearer picture of quantum theory. But in the measure that there are many different metaphysical pictures available, with no direct connection with quantum mechanics, one ends up with no clear picture, but with confusion. The situation improves, but not so much if we take a distinct approach to the relation between metaphysics and ontology. One could suggest that a metaphysical profile should be handcrafted, tailored for non-individuals. Doing so involves modifying the actual notions of non-individuality in order to make the metaphysical concepts somehow more responsive to what dictates the logic, physics, and ontology, i.e., one must acknowledge the theoretical requirements coming from science itself for such development. However, this also involves stepping out of such disciplines and having a certain autonomy: since neither logic, science nor ontology were able to determine the metaphysics of non-individuals, it seems that the only productive way to provide a metaphysical profile is the free development

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of a metaphysical profile for non-individuals, one that is tailored to fit quantum mechanics. French (2018b) argues that weak discernibility is a case in point. The argument is that Saunders (2003) used in a quantum-mechanical context the notion, originally developed by Quine (1976), of weak discernibility to interpret quantum objects. Leaving aside the questions of whether this determines a metaphysical profile (for a critical assessment of this claim, see Arenhart (2017a)), French’s (2018b, p. 223) point is this: “metaphysics may still be useful to the philosopher of physics even if it is grounded in considerations that appear to have nothing to do with modern physics”. In other words, metaphysics, even if it is not directly focused on specific scientific questions, can contribute to science if applied in science. This is made clear in passages such as the following: Quine himself was certainly not disinterested in science and although the notion of weak discernibility was clearly not motivated specifically by considerations of physics (quantum or classical), one could perhaps argue that the whole framework of his discussion of discernibility and identity may have been influenced by his reflections on science, even if only indirectly. (French, 2018b, p. 224)

We wish, however, to stress that it is not enough to transport the metaphysical theory to the scientific context, or merely apply it in the scientific context so that it can provide a “clear picture”, i.e. so that it can provide conditions to address Chakravartty’s Challenge. A tailoring work must be done, that is, the metaphysical profile must be developed to obey the logical, scientific, and ontological characteristics of the theory in question. Of course, many times this metaphysical theory tailored to physics will employ a familiar vocabulary in its development (French (2018b, p. 227) recognizes this), for reasons of intelligibility. But the point we would like to emphasize is this: no metaphysical theory that floats totally free from physics can serve as a metaphysical profile for physics; the tailor’s work is indispensable if we want to delve into the waters of ‘deep’ scientific realism and respond to Chakravartty’s Challenge. Nevertheless, while this metaphysics is inspired by scientific theory, it is not directly related to it, as it will need philosophical elements unrelated to it. Now, suppose more than one metaphysical profile can be tailored to fit the physics. This is always a possibility, and it highlights the fact that no metaphysical notion comes from quantum theory itself. There seem to be no objective criteria to help us when it comes to theory choice (see Benovsky, 2016; Chakravartty, 2017). Let us pause and reflect on what has been said so far. We argue that no metaphysics is guaranteed by anything other than metaphysics itself. So if we want to look for metaphysics, we shouldn’t place our expectations of finding them in logic, physics, or ontology—which is not to say that these disciplines are absolutely useless for metaphysical reflections (see Arenhart, 2012; Arroyo & Arenhart, 2019), only that these disciplines are not capable of determining metaphysics (see Arroyo & Arenhart, 2020). This puts the discipline in a fragile position. For, if metaphysics reappears as a necessary condition for scientific realism (in its so-called ‘deep’

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version), its floating-free aspect raises a problem against the very possibility of scientific realism, which is metaphysical underdetermination. Maybe things are not so bad. For example, we can quickly advance a metaphysical criterion coined by Benovsky (2016, pp. 82–84) as ‘widen the net’. The idea is simple: we should not look at metaphysical theories, when applied to science, in isolation; instead, we should ‘widen our net’ of investigation, and see how these theories relate not only to a scientific theory but to other neighboring theories. Here comes Krause’s (2005, 2019) suggestion: since physics is unable to decide between the metaphysics of individuality and non-individuality, perhaps chemistry can do it. As Krause (2019) suggests, take the chemical reaction of methane combustion for the sake of an example: .CH4 + 2O2 −→ CO2 + 2H2 O. There is no way to tell whether the Hydrogen atoms in the methane molecule are the same ones composing the water molecules: they are indiscernible. Other examples can be found in Krause (2005), but the point made is: chemistry doesn’t work without absolute indistinguishability; thus we should prefer non-individuality over individuality for this (pragmatic) reason. But which metaphysics of non-individuals should we adopt? There are several, or, as Arenhart (2017b, p. 1345) put it, the “‘nonindividuals horn’ produces its own underdetermination”. At this point, we are without the pragmatics on our side. Perhaps an alternative would stand out for being compatible with all disciplinary levels (logical, scientific, and ontological), which is precisely the tailor-made methodology for metaphysics that we had been proposing until then. But how to do it? Alas! This subject deserves the attention of an entire paper.10

5.6 Conclusion Let us take stock and check what has been achieved. The plan, recall, was to find some possible source for the nature of quantum objects as non-individuals, and that with some kind of good epistemic authority, which could in some sense be justified. Certainly, the idea of non-individuality arises with quantum mechanics, in its early days, as the founding fathers fought to grasp what kind of thing was it that the theory was attempting to describe; but in the face of the current multiplication of alternative approaches, many of them quite conservative in metaphysical terms, one may ask for the epistemic credentials of the view. After all, being the first to enter the stage in the metaphysics of quantum mechanics does not confer to it any kind of epistemic privilege. So, are we justified in believing that quantum entities are non-individuals, in any given sense? This may be asked both by realists, attempting to answer to Chakravartty’s Challenge, as well as by anti-realists willing to check how the world

10 What is worst: notice that appealing to indiscernibility in chemistry does not exclude the possibility of using some of the principles of individuality in the Transcendental Individuality family, which would just show that underdetermination is left precisely on the same point!

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could look like according to a specific reading of the theory; the latter could face it as an exploratory investigation, perhaps even to show underdetermination and throw it back on the realist. We have supposed that there is a legitimate interest in discovering some kind of connection between the idea of non-individuality and some privileged source that could help us shed some light on the notion. In this paper, we explored some of the options that are typically mixed in the literature as a source for the non-individuals character of quantum entities. Logic, physics, ontology, and metaphysics proper. All of them are found wanting, or, in other terms, not responsible for the metaphysical character or nature of nonindividuals attributed to quantum objects. Logic does not allow us to confer the required respectability on non-individuality, rather, it requires that such a notion be given beforehand. Physics, it is well known, cannot be held responsible for quantum non-individuality; indeed, there is nothing in physics that answers for metaphysics. The metaphysics must be put there from somewhere else, it seems. Ontology is not the source also, given the fact that an ontology of objects is not even the only possibility for quantum theory. It seems that disputes in ontology all seem to declare that their posits are a result of the needs of physics, but then, physics is certainly not sure of what it needs in ontological terms, anyway. Distinct options seem all to be vindicated by the theory, but on a closer view, what is obtained is at best a kind of compatibility with the theory. Finally, metaphysics itself cannot be responsible for non-individuality. Besides the fact that individuality is also compatible with physics, there are just too many ways to understand what non-individuals are, all of them equally good for physics. One looking for metaphysical tools in the field of traditional metaphysics just does not know which tool is the appropriate one, and metaphysicians can do no better than to open the gigantic toolbox and offer their best smile. The idea that quantum entities are individuals or not is clearly not justified by any kind of source that finds itself already involved in the debate. In this sense, we generalized the argument of metaphysical underdetermination: it is not just physics that does not determine its metaphysics. Logic does not determine its metaphysics nor ontology, neither ontology do so. In this sense, we should not place our hopes to justify metaphysics with the help of other areas of knowledge—recall that this is precisely the naturalist project—because the epistemic warrant is not something that metaphysics can inherit. Metaphysics as a discipline finds itself floating free not just from physics but pretty much from everything else. This is not a problem specific to non-individuality. The epistemic authority to be attributed to metaphysical claims is just as difficult to find for any other metaphysical theory that wants itself attached to a physical theory. Clearly, the source of the problem is to be found on the very idea of conferring epistemic authority to a metaphysical notion, in its relation to quantum mechanics. Perhaps we have been wrong in looking for some kind of privileged authority that could be attributed to metaphysical claims. Or, perhaps, we have been wrong in looking for a metaphysical complement to science that can still be somehow justified. But addressing these issues is more than we could do now.

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References Adams, R. (1979). Primitive thisness and primitive identity. Journal of Philosophy, 76, 5–26 Albert, D. Z. (2013). Wave function realism. In A. Ney & D. Z. Albert (Eds.), The wave function: Essays on the metaphysics of quantum mechanics (pp. 52–57). Oxford: Oxford University Press. Arenhart, J. R. B. (2012). Ontological frameworks for scientific theories. Foundations of Science, 17(4), 339–356. Arenhart, J. R. B. (2017a). Does weak discernibility determine metaphysics? Theoria, 32(1), 109– 125. Arenhart, J. R. B. (2017b). The received view on quantum non-individuality: Formal and metaphysical analysis. Synthese, 194(4), 1323–1347. Arenhart, J. R. B. (2018). New logics for quantum non-individuals? Logica Universalis, 12(3–4), 375–395. Arenhart, J. R. B. (2019). Bridging the gap between science and metaphysics, with a little help from quantum mechanics. In J. D. Dantas, E. Erickson, & S. Molick (Eds.), Proceedings of the 3rd Filomena Workshop (pp. 9–33). Natal: PPGFIL UFRN. Arenhart, J. R. B., & Arroyo, R. W. (2021). Back to the question of ontology (and metaphysics). Manuscrito, 44(2), 1–51. Arenhart, J. R. B., & Bueno, O. (2015). Structural realism and the nature of structure. European Journal for Philosophy of Science, 5(1), 111–139. Arroyo, R. W., & Arenhart, J. R. B. (2019). Between physics and metaphysics: A discussion of the status of mind in quantum mechanics. In J. A. de Barros & C. Montemayor (Eds.), Quanta and mind: Essays on the connection between quantum mechanics and the consciousness (pp. 31–42). Cham: Springer. Arroyo, R. W., & Arenhart, J. R. B. (2020). Floating free from physics: The metaphysics of quantum mechanics. Preprint. http://philsci-archive.pitt.edu/18477/ Benovsky, J. (2016). Meta-metaphysics: On metaphysical equivalence, primitiveness, and theory choice (Vol. 374). Synthese Library. Cham: Springer. Caulton, A., & Butterfield, J. (2012). On kinds of indiscernibility in logic and metaphysics. British Journal for the Philosophy of Science, 63, 27–84. Chakravartty, A. (2007). A metaphysics for scientific realism: Knowing the unobservable. Cambridge: Cambridge University Press. Chakravartty, A. (2017). Scientific ontology: Integrating naturalized metaphysics and voluntarist epistemology. New York: Oxford University Press. Chakravartty, A. (2019). Physics, metaphysics, dispositions, and symmetries – À la French. Studies in Hisory and Philosophy of Science, 74, 10–15. Conroy, C. (2016). Branch-relative identity. In A. Guay & T. Pradeu (Eds.), Individuals across the sciences (pp. 250–270). Oxford: Oxford University Press. de Ronde, C., & Massri, C. (2021). A new objective definition of quantum entanglement as potential coding of intensive and effective relations. Synthese, 198, 6661–6688. Dürr, D., & Lazarovici, D. (2020). Understanding quantum mechanics: The world according to modern quantum foundations. Cham: Springer. Esfeld, M. (2013). Ontic structural realism and the interpretation of quantum mechanics. European Journal for Philosophy of Science, 3(1), 19–32. Esfeld, M. (2019). Individuality and the account of nonlocality: The case for the particle ontology in quantum physics. In O. Lombardi, et al. (Eds.), Quantum worlds: Perspectives on the ontology of quantum mechanics (pp. 222–244). Cambridge: Cambridge University Press. French, S. (1995). Hacking away at the identity of indiscernibles: Possible worlds and Einstein’s principle of equivalence. The Journal of Philosophy, 92(9), 455–466. French, S. (2014). The structure of the world: Metaphysics and representation. Oxford: Oxford University Press.

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French, S. (2018a). Realism and metaphysics. In J. Saatsi (Ed.), The Routledge handbook of scientific realism (pp. 394–406). New York: Routledge. French, S. (2018b). Toying with the toolbox: How metaphysics can still make a contribution. Journal for General Philosophy of Science, 49, 211–230. French, S. (2019a). Defending eliminative structuralism and a whole lot more (or less). Studies in History and Philosophy of Science Part A, 74, 22–29. French, S. (2019b). Identity and individuality in quantum theory. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Winter 2019 Metaphysics Research Lab, Stanford University. French, S. (2020a). Metaphysical underdetermination as a motivational device. Preprint. http:// philsci-archive.pitt.edu/16922/ French, S. (2020b). What is this thing called structure? (Rummaging in the toolbox of metaphysics for an answer). Preprint http://philsci-archive.pitt.edu/16921/ French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Oxford University Press. French, S., & McKenzie, K. (2012). Thinking outside the toolbox: Towards a more productive engagement between metaphysics and philosophy of physics. European Journal of Analytic Philosophy, 8(1), 42–59. French, S., & McKenzie, K. (2015). Rethinking outside the toolbox: Reflecting again on the relationship between philosophy of science and metaphysics. In T. Bigaj & C. Wüthrich (Eds.), Metaphysics in contemporary physics (pp. 25–54). Leiden: Brill/Rodopi. French, S., & Redhead, M. (1988). Quantum physics and the identity of indiscernibles. British Journal for the Philosophy of Science, 39, 233–246. Hofweber, T. (2016). Carnap’s big idea. In S. Blatti & S. Lapointe (Eds.), Ontology after Carnap (pp. 13–30). Oxford: Oxford University Press. Krause, D. (2005). Structures and structural realism. Logic Journal of the IGPL, 13(1), 113–126. Krause, D. (2019). Does Newtonian space provide identity to quantum systems? Foundations of Science, 24, 197–215. Krause, D., & Arenhart, J. R. B. (2016). The logical foundations of scientific theories: Languages, structures, and models. Abingdon: Routledge. Krause, D., Arenhart, J. R. B., & Bueno, O. (2022). The non-individuals interpretation of quantum mechanics. In O. Freire, Jr. (Ed.), The Oxford handbook of the history of quantum interpretations (pp. 1135–1154). New York: Oxford University Press. Ladyman, J. (1998). What is structural realism?. Studies in History and Philosophy of Science, 29(3), 409–424. Ladyman, J. (2016). The foundations of structuralism and the metaphysics of relations. In A. Marmodoro & D. Yates (Eds.), The metaphysics of relations (pp. 177–197). Oxford: Oxford University Press. Lowe, E. J. (2003). Individuation. In M. Loux & D. Zimmerman (Eds.), The Oxford handbook of metaphysics. Oxford: Oxford University Press. Magnus, P. D. (2012). Scientific enquiry and natural kinds: From planets to mallards. New York: Palgrave-Macmillan. Maudlin, T. (1995). Three measurement problems. Topoi, 14(1), 7–15. Maudlin, T. (2007). The metaphysics within physics. Oxford: Oxford University Press on Demand. Meincke, A. S. (2020). Dispositionalism: Between metaphysics and the philosophy of science. In A. S. Meincke (Ed.), The wave function: Essays on the metaphysics of quantum mechanics (pp. 1–12). Cham: Synthese. Morganti, M. (2015). The metaphysics of individuality and the sciences. In A. Guay & T. Pradeu (Eds.), Individuals across the sciences (pp. 273–294). Oxford: Oxford University Press. Muller, F. A., & Saunders, S. (2008). Discerning fermions. British Journal for the Philosophy of Science, 59, 499–548. Ney, A. (2014). Metaphysics: An introduction. New York: Routledge. Pylkkänen, P., Hiley, B., & Pättiniemi, I. (2016). Bohm’s approach and individuality. In A. Guay & T. Pradeu (Eds.), Individuals across the sciences (pp. 226–249). Oxford: Oxford University Press.

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Post, H. (1963). Individuality and physics. The Listener, 10 October, 534–537. Quine, Wv. O. (1951). On Carnap’s views on ontology. Philosophical Studies, 2(5), 65–72. Quine, Wv. O. (1976). Grades of discriminability. The Journal of Philosophy, 73(5), 113–116. Ruetsche, L. (2018). Getting real about quantum mechanics. In J. Saatsi (Ed.), The Routledge handbook of scientific realism (pp. 291–303). New York: Routledge. Saunders, S. (2003). Physics and Leibniz’s principles. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 289–307). Cambridge: Cambridge University Press. Schrödinger, E. (1998). What is an elementary particle? In E. Castellani (Ed.), Interpreting bodies: Classical and quantum objects in modern physics (pp. 197–210). Princeton: Princeton University Press. Thomson-Jones, M. (2017). Against bracketing and complacency: Metaphysics and the methodology of the sciences. In M. H. Slater & Z. Yudell (Eds.), Metaphysics in the philosophy of science: New essays (pp. 229–250). Oxford: Oxford University Press. Wilson, A. (2020). The nature of contingency: Quantum physics as modal realism. Oxford: Oxford University Press.

Chapter 6

Understanding Defective Theories The Case of Quantum Mechanics and Non-individuality Moisés Macías-Bustos and María del Rosario Martínez-Ordaz

Abstract Here, we deal with the question of under which circumstances can scientists achieve a legitimate understanding of defective theories qua defective. We claim that scientists understand a theory if they can recognize the theory’s underlying inference pattern(s) and if they can reconstruct and explain what is going on in specific cases of defective theories as well as consider what the theory would do if non-defective—even before finding ways of fixing it. Furthermore, we discuss the implications of this approach to understanding the meta-metaphysics of Quantum Mechanics, specifically with regard to Quasi-set theory. We illustrate this by employing Quasi-set theory to structure a defective scientific theory and make possible the understanding of the theory. Keywords Scientific understanding · Structuralism · Ideology · Meta-metaphysics · Quasi-set theory · Non-individuality in quantum mechanics

6.1 Introduction If a sign of the maturity in a particular philosophical research program is that it begins to confront itself with some of the major topics in traditional or mainstream philosophy, in this paper, we would like to contribute to moving forward some of

María del Rosario Martínez-Ordaz’s research was supported by the Programa Nacional de PósDoutorado PNPD/CAPES (Brazil). M. Macías-Bustos Philosophy Department, University of Massachusetts-Amherst, Amherst, MA, USA National Autonomous University of Mexico, Mexico City, Mexico e-mail: [email protected] M. R. Martínez-Ordaz () Federal University of Rio de Janeiro, Rio de Janeiro, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_6

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Krause’s views in the philosophy of science and logic towards discussion of the implications of the tolerance of defective (partial, vague, conflicting, inconsistent, and false) information for broader concerns in the philosophy and the metaphysics of science. Specifically, we consider what role Quasi-set theory might play vis a vis rational agent’s understanding of the scientific and metaphysical elements of quantum mechanics. Broadly speaking, scientific understanding is considered to be knowledge of relations of dependence. When one understands a theory, one can build a comprehensive picture of that theory as well as of the relations that hold within it. Understanding a theory allows scientists to find new domains of application for it, and understanding an empirical domain makes it possible to build new theoretical approaches to that domain. Science is generally concerned with explanation, prediction, manipulation, and actual knowledge of what the world is like. This last factor is metaphysical in nature, for metaphysics is concerned ultimately with the question of how the world is fundamentally like.1 Therefore, it is undeniable that scientific understanding is a fundamental component of any successful scientific enterprise. So far, understanding has been considered to be factive and explanatory, meaning that its content should only include true propositions and that it should come only after the achievement of explanatory knowledge.2 Unfortunately, if this were the case, however, we wouldn’t be able to legitimately understand any theories, models, or phenomena that are formulated in a defective manner. At least we wouldn’t be able to do understand them qua defective—yet, if there was no need for understanding defective theories, this wouldn’t be a problem. However, many of our most successful scientific theories, at some point in their development, are or have been defective. Some of them, like Bohr’s model of the atom, have been, allegedly, inconsistent. Some others have conflicted significantly with observation, like Newtonian dynamics. And some others, like Quantum Mechanics, are conceptually vague and imprecise, as well as (depending on the philosophical reconstruction) inconsistent (Cf. Arenhart and Krause, 2014; da Costa & Krause, 2014). This shows that much scientific practice has used and uses defective theories and models. And even more importantly, these theories, even when defective, have grounded and shaped our current science. And yet, while philosophers of science scrutinized the rationality behind using defective theories, they have significantly struggled when explaining how, if possible, to achieve any legitimate understanding of them. Here, we deal with the question of under which circumstances can scientists achieve a legitimate understanding of defective theories qua defective. We claim that scientists understand a theory if they can recognize the theory’s underlying inference pattern(s) and if they can reconstruct and explain what is going on in specific cases of defective theories as well as consider what the theory would do if not defective—

1 We

are presupposing some form of scientific realism is true as part of their working assumptions. specifics about the components of scientific understanding are discussed in Sects. 6.2 and

2 More

6.3.

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even before finding ways of fixing it. Moreover, we claim that understanding the inferential structure of the theory involves understanding the structure of its domain. Furthermore, this understanding is modal in nature,3 in that the domain might not actually instantiate that structure, the structure need only be possible. This last point we illustrate with specific reference to quantum mechanics. In order to do so, we proceed in five steps. First, in Sect. 6.2, we introduce the generalities of scientific understanding and we use some illustrations from the philosophy of science and the history of analytic philosophy. Second, Sect. 6.3 is devoted to analyzing the challenges around the legitimate understanding of defective theories; here we also introduce our case study. In Sect. 6.4, we sketch a structuralist approach to understanding and furthermore elaborate on what sort of presuppositions from metaphysics and meta-metaphysics are required by this type of approach. In Sect. 6.5, we explain in which way the detection of specific inferential patterns and logical constraints allows for the promotion of scientific understanding in the case of the quantum theory with non-individuality (Cf Krause & French, 1995; Arenhart & Krause, 2014). Section 6.6 is devoted to drawing some conclusions.

6.2 The Generalities of Scientific Understanding In this section, we introduce some of the most common assumptions about scientific understanding and its role in scientific development.

6.2.1 Preliminaries There is a shared intuition of science as being mostly motivated by our need of making sense of the world. This idea of making sense goes beyond only attaining bits of disconnected knowledge; as a matter of fact, making sense of the world is not about discovering new facts alone, it is not even about providing specific explanations for these individual facts but it is about going to a broader more general level: understanding. Understanding has been traditionally considered to “consist of knowledge about relations of dependence. When one understands something, one can make all kinds of correct inferences about it” (Ylikoski, 2013: 100). Understanding a theory allows scientists to find new domains of application for it, and understanding an empirical domain makes it possible to build new theoretical approaches to that domain. The fact that science is significantly motivated by the possibility of satisfactorily connecting our different scientific beliefs about the world makes understanding a

3 It is important to notice that it’s no requirement of ours that this type of epistemic modality should be metaphysically primitive.

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fundamental component of any successful scientific enterprise. Scientific understanding is a relational phenomenon. It consists of combining doxastic bodies for the building of comprehensive pictures of a particular domain as well as for integrating theoretical frameworks that could, initially, look disconnected. Understanding then is a task of building networks that successfully connect our scientific beliefs about X and that allow us to get a better grasp of X. This notion of understanding is very modern but has predecessors in the historical records. To take an influential precursor, Russell (1913: “Logical Data”) claimed that understanding a proposition involved a subject’s acquaintance with the constituents of the proposition: objects, properties, and relations, together with their logical form and hence the ultimate role of the proposition and their objects in logical space. Furthermore, understanding for Russell precedes truth, for to understand a proposition only means to be acquainted with the logical form relating its objects, properties, and relations, and these need not be instantiated. A proposition is defined as true only if the object’s properties and relations are actually logically related in such a way. Clearly, this intuition can be extended to whole theoretical structures when considering the Ramsey sentence of a theory, i.e., the existential generalization of the conjunctive proposition specifying the theoretical predicates and relations realized by the objects in the theory’s domain (Cf. Ramsey, 1929; Lewis, 1970). Furthermore, such a notion of understanding involves theoretical elements inasmuch as it aims at characterizing the structure of the world according to our scientific theories, under the assumption that scientific success does involve at the very least a successful grasp of the world’s underlying structures and their corresponding logical space i.e., the underlying logic capturing valid inferences in the domain.

6.2.2 The Ten Elements of Understanding Scientific understanding has been seen as the combination of ten elements: (1) its structural character, (2) its standing state, (3) a subjective (psychological) element, (4) an objective element, (5) its coherence-constrain, (6) its order requirement, (7) the intelligibility component—that emerges from (5) and (6), (8) its epistemic robustness, (9) its gradability and (10) its praiseworthy character.4 First, the structural character of scientific understanding refers to the general idea of what it means to understand something. When one understands a particular phenomenon, one is capable of making sense of that phenomenon by connecting epistemic bodies (beliefs, knowledge, among others) in such a way that the inferences around the understood phenomenon allow us to portray it in the clearest possible way. Thus, understanding consists in selecting a particular way

4 For

a similar characterization of the components of understanding, see Bengson (2018: 19–20).

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to accommodate and relate epistemic bodies, this is, in building a structure that can connect satisfactorily sets of information around a specific object or phenomenon. The standing state of understanding refers to the fact that understanding seems to be of a different class of epistemic objects than belief and knowledge. Understanding can affect epistemic bodies by reinforcing them (or the relations between them) or by weakening them; when one understands something one can also make sense of how certain epistemic bodies hang together in order to build a more cohesive picture of what has been understood. This is, through understanding one can see how relevant or legitimate certain epistemic bodies are (or aren’t) with respect to what is understood. In this sense, understanding circulates between beliefs and different types of knowledge, and it provides and distributes the epistemic force within the bodies to which it relates. The subjective, psychological, component of understanding refers to the feeling of grasping what is being understood, and it is stronger presentation is often reported as the so-called “eureka effect”. This sensation reveals that we are aware of having acquired a new competence for putting together bits of information that make more sense when together than when separated. The objective component of scientific understanding results from considering that this feeling of grasping depends solely on the individual agent’s experiences and reports and that agents often get mistaken about how reliable these experiences are. Epistemologists have required that understanding requires also the grasping of a fragment of reality (Cf. Elgin, 2017: 35). The coherent constraint and the order requirement are direct results of the structural spirit of understanding. To understand something is to be able to order the components of what has been understood in a coherent way (Cf. Bengson, 2018: 19). The intelligibility component of the understanding can be generally characterized as the epistemic virtue that reflects a harmony between the content of an agent’s beliefs contained in the understanding of X and the agent’s actions around X (Cf. Chang, 2009). There is a sense in which intelligibility is the result of the combination of coherence and order; however, it also extends the scope of understanding into a performative area. The epistemic robustness of understanding consists “in a way that mere acquaintance with particular deeds or facts—even if intelligent and objective—is not” (Bengson, 2018: 19). This supports the idea that once legitimate understanding is achieved, it is very hard to find reasons to give it up.5 Nonetheless, while it is difficult to give up successful understanding already achieved, that is compatible with the possibility of improving or deepening our understanding; this feature has come to be known as the gradability of understanding. Finally, the praiseworthy character of understanding results from the combination of all of the above points; “attributing understanding to an individual is not merely to credit her with some kind of success (. . . ) but to compliment, or praise her for it” (Bengson, 2018: 19).

5 What happens in cases where agents might think they understanding, but they are wrong about that will be discussed in Sect. 6.3.

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While there are still important ongoing philosophical debates around ways to expand this view on scientific understanding, I believe that what has been said in this section reflects the current most common agreements about scientific understanding and will suffice for the purposes of the paper.

6.3 Defects, Theories, and Understanding Here, we address what defective theories are, why is it important to achieve a legitimate understanding of them, as well as which are the challenges that traditional views can face when trying to explain the achievement of scientific understanding of defective theories. The section is divided in two main parts: Sect. 6.3.1 explains very briefly what defective scientific theories are and what has been said in the literature about them. Section 6.3.2 introduces the case study of Quantum Mechanics being a defective theory and argues in favor of the need of it to be understood qua defective. Finally, Sect. 6.3.3, provides an overview of the challenges around the understanding of defective theories.

6.3.1 Defectiveness in Science Defective information is an umbrella term that covers cases of partial, vague, conflicting, inconsistent, and false information. These types of information carry a negative connotation given the fact that, when present, they make reasoning more challenging in different respects. For instance, when the information that we are reasoning from is defective, it is harder to determine which are the legitimate consequences of this information, as well as to estimate the reliability of the inferences carried out with such data. This makes the problem of the use of defective information inherently both epistemological and logical. Let empirical theories be formulated based on the following theoretical model: T.= D, R.ni  “where D is a particular domain (a set of objects to which the theory is supposed to apply) and R.i is a family of n-place relations holding between the elements of D” (Bueno, 1997: 588). While the domain could be selected and individuated depending on the methodological preference of the research program in which the theory is being used and vary from time to time; the set of relations, R.i , work in a very different way: first, they are what helps to order, classify, and evaluate the objects in the domain (and the propositions through which they are described). Second, they close under specific logical consequence relations the objects of D, allowing and forbidding certain interactions between them. And third,

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as they regulate the behavior of D, they will not necessarily change if D increases or decreases.6 Defective theories are theoretical constructs that operate on a defective basis— either assuming incompatible commitments, accepting defective procedures or characterizing defective entities, etc. When these theories are or have been successfully employed in scientific practice, this reveals that they can preserve and stress particular inference patterns between propositions—and it is expected that such patterns are what warrant the applicability of the theories in different contexts. A scientific theory (or model) can be defective in different ways: conceptually, meaning that some of the concepts at the core of the theory are defective; empirically, this is, the relations that hold between the theory and its intended domains of applications are defective—the most common cases of this are those of either descriptive gaps or conflicts between the theory’s predictions and relevant observational reports. Furthermore, a theory can be intertheoretically defective, this is, with respect to other relevant theories, by conflicting with them, or by being extremely partial even though complementarity between these theories was initially expected. Moreover, a theory can be metaphysically defective, inasmuch as it is not clear what picture of the world the theory paints either because it involves elements of indeterminacy, contradiction, incompatibility with other fundamental theories, lack of clarity with respect to its theoretical ontology and ideology and so forth.7 In recent decades, the rational use of defective theories has significantly captured the attention of epistemologists and logicians of science. First of all, it is a common intuition that scientific methods are a sort of extremely sophisticated epistemic filter that can and should guarantee the high quality of the information that is used and obtained through science. And when they do not succeed at doing so, we find ourselves puzzled by this—arguably, especially so in the so-called “hard” sciences. Nonetheless, if we pay close attention to the history of science, we would realize that scientific theories are and have been most of the time defective. Sometimes, observational reports conflict with one another, theories are conceptually vague or models are extremely partial; and in spite of this, science is still carried out as one of our most rational enterprises. More importantly, the defects of some theories are crucial components of either the theories themselves or the phenomenon that they are representing, and therefore an adequate understanding of them needs to grasp them qua defective.

6 We are aware of the fact that there is an ongoing philosophical debate about the status of the different characterizations of scientific theories (see Halvorson (2016) for a comprehensive revision of the different views on scientific theories); however, we think this will suffice for the purposes of the paper. 7 In the literature we can find examples of metaphysicians defending that possibility that reality itself is indeterminate, inconsistent, disjoint and so on (Cf. Cartwright, 1999; Priest, 1985; Torza, 2021). However, on the methodologically conservative picture reality is consistent or coherent; fully determinate and unified or integrated into a whole.

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6.3.2 A Case Study: Quantum Mechanics and Non-individuality Nowadays, Quantum Mechanics is considered to be our most successful and fundamental scientific framework. A significant amount of important physical phenomena have been explained by using the theory, it has provided us with valuable epistemic access to the nature of light, electricity, and elementary particles, among other objects of scientific study. The theory in general allows us to draw important connections between different areas of physics and other disciplines like (some subareas of) chemistry, in order to build an image of the world that is cohesively integrated and epistemically robust.8 Moreover, the theory has found numerous applications in engineering and the high-tech economy. And the combination of all of this can only speak about the richness, success, and trustworthiness of the theory. It seems even obvious that any scientist should aim at understanding the theory in order to later, understand the physical world through it. However, understanding might not be so straightforward in this case. In the vast literature concerning the foundational and conceptual issues of Quantum Mechanics, one of the most salient issues is the metaphysical status of the entities posed by the theory, in particular, quantum entities can be considered as individuals (Cf. Saunders, 2016; Krause & Arenhart, 2016). Regarding this point, three main options have been entertained: either they can and should be considered as individuals, they should be considered as non-individuals or one should neglect particular objects and endorse a kind of ontic structural realism. Now, this discussion is rooted in the origins of Quantum Mechanics, [H]istorically, the issue has been treated from a very naturalistic point of view. That is, the choice should be made bearing always in mind what quantum mechanics itself dictates us concerning those matters. In that case, the adoption of a metaphysics of non-individuals seems to have at least historical precedence over the other two options. Really, right from the beginning of the theory it was seen by some of the founding fathers of quantum mechanics that it dealt with items without identity, in the sense of having no individuality. That is, it seemed to follow from the strange statistical behavior of quantum particles that they had no individuality, no identity, and so were a very strange-behaved kind of thing. That view was called the Received View on quantum non-individuality. (Arenhart & Krause, 2014:2).

Individuals are taken to be the intuitive values of the bound variables of classical first-order logic. Furthermore, for first-order languages that have individual constants (as opposed to predicate constants), individuals will be those objects that will be unique referents of constants. As stressed by Krause (2012: 3) classical logic has Leibniz’s Identity of Indiscernibles as a theorem: II: ∀x, ∀y((x) iff (y) → x = y)

.

8 For a comprehensive critical analysis of the epistemic robustness of quantum mechanics see Hoefer (2020).

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Individuals which are qualitatively indiscernible are identical. In order to avoid trivializing the identity of indiscernibles the qualities under discussion should not include the property of being identical to some object o, for in such a case if both x and y are identical to o, then by the transitivity of identity they would be identical to each other. The identity of indiscernibles was used by Leibniz for a number of purposes, for example in the Newton-Clarke correspondence (Cf. AG, Philosophical Writings) Leibniz makes liberal use of the principle to prove that it is impossible to shift every material object in some direction in space; that is impossible to boost the whole material world such that the relative motion of every particle remains the same; that it is impossible for there to be absolute space (a substratum existing independently of material objects) and analogously that it is impossible for there to be a world temporally indiscernible to ours but that being existing either earlier or later. Famously Max Black (1952) argued that the identity of indiscernibles entailed the impossibility of what he took to be an obviously possible world, a world containing two indiscernible spheres differing solo numero—if the identity of indiscernibles is true, then there cannot be objects differing numerically without differing qualitatively. Therefore, the identity of indiscernibles must be false. We have to distinguish between properties in the world and predicates in our logical language. If our language lacks an identity two-place predicate it can in fact be introduced derivatively as shorthand for any two individuals satisfying the same predicates (as in Whitehead and Russell’s Principia Mathematica).9 Restricting ourselves to the first-order case, since predicates are sets in first-order model theory, by the extensionality axiom of ZFC (classical first-order Zermelo– Fraenkel set theory with choice) any two sets are identical whenever they have the same elements. So far so good, suppose we allow ourselves the hypothesis that there are individuals whose only particularizing property is being that unique individual, a haecceity, and in doing so we reject the identity of indiscernibles. We can always introduce names for these objects differing solo numero to distinguish them, even if in name only. We can thus define “indistinguishables” as those individuals which are indiscernible with respect to every non-haecceteistic property. Allowing ourselves, per impossible, the possibility of having names for every nonhaeccetistic property i.e., predicate letters standing for subsets of the domain and furthermore presupposing quantification over the all-inclusive domain, such that no larger structure can discern our objects, we can say that indiscernibles are those individuals which are invariant with respect to .n−place predicate permutations, such that they satisfy the same formulas of this ideal language.

9 Russell gives a metaphysical defense of the principle in his An Inquiry Into Meaning and Truth (1959). On the interpretation of Principia Mathematica’s metaphysical logic see Landini (1998), Linsky (1999), and Klement (2018). It is interesting, in this connection, to consider that philosophers such as Russell and Frege felt that there had to be a philosophical elucidation of the systems of higher-order logic they were working on in spite of them being systems of logic and not applied mathematics. We think this bolsters our view about the generality of understanding as an epistemic activity.

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As Krause (2012: 2) and Krause and Arenhart (2016: 2) point out however this framework is prima facie in tension with the standard interpretation of quantum mechanics inasmuch as standard Quantum Mechanics allows for the possibility of fundamental indistinguishables, but needn’t require that these be individuals. The basic idea is that quantum objects are fully invariant with respect to predicate permutations, so much so that we can intuitively regard them as the same object, while at the same time speaking of these plurally. According to Krause (2012) classical, intuitively macroscopic objects are indeed individuals for they cannot be substituted for each other with everything remaining qualitatively the same, but quantum objects can. An example cited in the literature is that of fermions and bosons with regard to certain states (Cf. Ladyman & Bigaj, 2010). In this context, the claim that quantum particles are indistinguishables boils down to the fact that if particles are in some state S then permuting those particles within S produces a state that is physically indiscernible from S. In standard Quantum Mechanics, physical systems are represented by vectors in a Hilbert space with specific states corresponding to vectors of length 1 in the space. Properties of such systems are represented by Hermitian operators on the vector space, mappings of the vector space onto itself (Cf. Okon, 2014) where the expectation value of a measurement for a property (e.g., spin, position) corresponds to the eigenvalue of the eigenvector of that Hermitian operator. As pointed out by Ladyman and Bigaj (2010) the expectation value of a Hermitian operator is the same for all indistinguishable particles, quantum states involving fermions and bosons are permutation invariant and this property is retained even if further particles are added to the system, unlike what happens in classical systems (e.g., enantiomorphs10 which are distinguished when further objects are added to the space). To take an example of what a fundamental physical theory should involve considering Maudlin’s (2018, p. 2) discussion of this issue. On his view a fundamental physical theory is a theory about “matter in motion”, hence it should involve: (1) local beables (matter); (2) non-local beables (if any), such as the quantum state; (3) a space-time structure and (4) the dynamical laws. Parts (1), (2), and (3) tell us what there is according to the theory, the ontology whereas part (4) tells us what it does. Standard quantum mechanics is muddled about the dynamics, by having two radically different types of evolution (deterministic Schrodinger evolution and non-deterministic collapse according to the Born rule). Standard quantum mechanics is defective inasmuch as it lacks clarity with respect to the status of the nature of the quantum state, and involves a notion of “measurement” which, though it doesn’t prevent the use of the theory in experimental settings, is mysterious, imprecise, vague and in no way allows us to understand how the world is supposed to be according to it, hence the “interpretations” of quantum mechanics, which really are ways of fleshing out the theory in a manner compatible with (1)–(4) above. In spite of its success as an instrument, there are

10 Such

as the left-hand and the right-hand.

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conceptual and inter-theoretic difficulties reconciling the theory with our experience of the world, with other fundamental physical theories, such as general relativity, as well (allegedly) as with the fundamental logical frameworks that underlie traditional mathematics, physics, and metaphysics, on account of the failure of the identity of indiscernibles for quantum objects: classical logic and ZFC set theory. We take this to have shown that there is an important sense in which Quantum Mechanics should be considered to be a successful, yet defective, theory. The question that we address in the following paragraphs is whether the achievement of scientific understanding of defective theories is really possible, and if so, under which conditions this can occur.

6.3.3 The Challenges for Understanding Defective Theories There are two broad types of scientific understanding: a theoretical one and a practical one. The former consists of making sense of either theories or phenomena (by using those theories); while the latter refers to having the ability to perform complex tasks in a systematically successful way. In sciences these two types of understanding are often seen as closely linked; the expert scientist is expected to understand the theories that she works with, as well as the phenomena that she studies by implementing such theories, and the procedures that are followed in her daily practice. (Theoretical) scientific understanding has been traditionally characterized as explanatory and factive. On the one hand, the explanatory requirement means that understanding comes only after having obtained causal explanatory knowledge (Cf. Grimm, 2006, 2014; Lawler, 2016, 2018). In this sense, understanding is the most demanding epistemic good that we can attain. On the other hand, the factivity requirement means that the content of understanding includes only true propositions. This is, we legitimately understand only propositions that we know are true and that adequately refer to facts of the world. The explanatory requirement has been justified by the (epistemological) grounding role that causal explanations seem to play in the sciences and the factivity requirement has been motivated by the aim of truth preservation. This gives the impression that understanding cannot be attained in absence of causal explanations and more importantly, that the content of understanding cannot include any defective (vague, incomplete, conflicting, inconsistent, impossible) data. In what follows, we focus only on the factivity condition of understanding and the challenges that it poses for the understanding of defective theories. The satisfaction of the factivity condition plays a crucial role in determining whether a case of alleged scientific understanding is legitimate. A case of understanding is legitimate when it is robust towards updates of information; while it can upgrade consistently, it should never decrease in quality—the content of understanding shouldn’t go from being consistent, complete, and precise, to being inconsistent, incomplete, and vague or partial. Having the impression of

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understanding something that is knowingly imprecise, incoherent, or false is called the illusion of depth of understanding (Cf. Ylikoski, 2013). The upshot of the factivity condition the factivity condition is that the reliability and the legitimacy of understanding depend on the truth of its content. The factivity requirement is false though, considering that even if some cases of legitimate understanding fully satisfy it, it is not clear that this should be a necessary condition for understanding. In particular, considering cases of theoretical understanding of superseded theories. If one takes seriously this requirement when a scientist reports having understood theory a from the past, which we already know is partially false, one should accept that this is a case of the mere illusion of depth of understanding—or at least that some of the main features of the theory, those that are false, cannot be understood. Nonetheless, systematically, scientists and philosophers more generally have a strong feeling of understanding abandoned theories. This rises the question of how can we explain cases of understanding theories that are vague, incomplete, conflicting, inconsistent, or even false consistently with having a normative epistemological approach to understanding. When trying to satisfactorily combine the normative elements of understanding with the actual cases that we find in scientific practice, epistemologists have adopted at least three different standpoints: factivism, quasi-factivism and non-factivism. Factivism. The content of understanding can only include true propositions (or at least approximations to the truth) that are known to be so. Factivism accounts for the clearest cases of epistemic success. Indirectly, it also encompasses the clearest cases of error; as understanding is extremely hard to achieve, in the majority of cases in which we thought we had understood something, we were very much mistaken, and we discover this only when faced with the falsehood of our beliefs. Unfortunately, this standpoint fails at addressing the gray area that exists between radical success and radical error. Quasi-factivism. The content of understanding might include elements that are known to be non-true, but these elements are to be located in the periphery of the content of understanding. Quasi-factivism addresses the cases in which understanding is only achievable thanks to certain epistemic tools, which might go from bits of logical rules to sophisticated idealizations, abstractions, and fictions, among others. Nonetheless, as the epistemic role of these non-true elements is only to ease the reasoning, they are part of the content of understanding only by being elements of its periphery, but they are not located at its core, this is, they are not part of what has been understood. Non-factivism. The content of understanding can include non-true propositions that are known to be so; and, when they are essential for the of understanding in virtue of being false, they are located in its core. This standpoint deals with the issue of the way in which non-true elements can be part of the content of understanding; but in doing so, it loses track of the warrant that truth provided the other two theses with. And for the supporter of non-factivism

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errors become a real challenge, there is no way to make a clear division between cases of error as cases of understanding in the gray area concerning the use of false statements.11 So far, these standpoints capture different features of real-life scientific understanding, the cases of impressive epistemic success, the cases in which scientists use tools to ease their grasping of a theory or a phenomenon, and those in which these tools are indispensable. Nonetheless, there is an important difference between saying that one can include some idealizations in the content of understanding and saying that one can legitimately understand a defective theory. Understanding a theory that is defective, especially if it is knowingly defective, requires paying attention not only to its successes but also to its defects. If one, for instance, tries to separate the phlogiston theory from the falsehoods that we now know were part of it, one would end up with a different theory and lacking understanding of the one that one was initially trying to grasp. Thus, for the case of the superseded theories, the inclusion of their defects in the content of our understanding is crucial for agents to be able to see the relations that hold between the elements of the theory, their connections with the domains of application as well as the reasons for which they were abandoned. Furthermore, for those theories that despite their defects are still in use, understanding them as defective allows scientists to interpret their defects in novel ways—not only to solve them but also to tolerate them or even accept them. And more importantly, for those theories that are defective because the phenomenon that they are representing is essentially defective, understanding them qua defective becomes a crucial task for explaining their success and furthering scientific development. Furthermore, when it comes to the commitments of the scientific realist vis a vis knowledge there is a metaphysical commitment to specifying what a possible world instantiating the structure of the theory would be like.12 Summing up, while epistemologists have shed light on the ways in which certain non-true propositions can be included in the content of understanding (either at its core or its periphery), they have failed to explain the ways in which agents can legitimately understand defective theories qua defective. In the following sections, we explain how this is possible and we illustrate this in more detail with the case of Quantum Mechanics and non-individuality.

11 It is important to notice that for the non-factivist, the non-true propositions that can be included into the content of understanding are exclusively those that lead to (empirical) success when being used. These propositions have been called felicitous falsehoods and are falsehoods that facilitate understanding by virtue of being the falsehoods they are and whose “divergence from truth or representational accuracy fosters their epistemic functioning” (Elgin, 2017: 1) . 12 We do not take a stance on the nature of possible worlds in this paper, we use the concept as shorthand for possibilities. For a contemporary sympathetic and systematic approach to possible world realism, however, see Bricker (2020).

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6.4 Understanding Defective Theories: A Structuralist Approach Here we provide the generalities of a structuralist approach to scientific understanding that, on the one hand, remains neutral with respect to the debates about the truth value of the content of understanding; and on the other hand, allows us to explain the legitimacy of some cases of understanding of defective theories. The section is divided in four parts: Sect. 6.4.1 is devoted to summarizing the defective-theories motivation behind this account. Section 6.4.2 addresses the structuralist roots of the proposal and Sect. 6.4.3 sketches its epistemological and metaphysical import.

6.4.1 The Motivations Defectiveness-Wise Scientific theories are epistemic vehicles that help scientists to filter, order and relate the varied information that they get about the world in order to provide accurate descriptions, predictions, and explanations of the domain that they are talking about. Broadly speaking, theories are clusters of information which are initially incomplete but that, in the long run, tend to incorporate new data in order to improve the picture of the world that they provide. In that sense, it is not surprising that theories are, at least initially, vague and incomplete—in some cases, this also causes the presence of contradictions. Much scientific practice makes use of defective theories; some of the most famous examples of this are: the early calculus, Bohr’s theory of the atom and Frege’s foundations of arithmetic, among others. And despite the fact that some of these theories are knowingly defective, scientists still report having ‘understood’ both the theories as well as the phenomena that they describe. Yet, according to traditional accounts of scientific understanding, these reports should be considered to be illusions. There are broadly two very general ways to go about explaining what is going on when we grasp or communicate some defective theory: either claim that in none of those cases do we understand the aforementioned theories, which is on its face exceedingly implausible, or to say that we do understand them and then offer an explanation about what understanding comes down to in those cases. In the rest of the paper, we adopt the latter. For doing this, we adopt a strategy similar to the one already employed by structural realists for salvaging the continuity and preservation of science in light of pessimistic meta-induction style arguments.

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6.4.2 The Proposal and Its Structuralist Roots First of all, we take the notion of structure in the sense of mathematical structures (Cf. Bricker, 1992/2020). As pointed out by Shapiro (2000) mathematical structures are often interpreted in terms of set theory, Frigg and Votsis (2011, pp. 229–230) illustrate this for the case of ontic and epistemic structural realism in their wideranging survey of that research program. However, there is underdetermination at the level of what set-theoretical structures or metaphysical structures (e.g., pluralities of tropes, universals, possibilia) should be posited (if any) as the ontic ground of these mathematical structures. We take no commitment here on this question.13 Part of the appeal of scientific realism is the claim that this view can explain why more mature scientific theories are more successful than their predecessors. However, on the one hand, there are substantial changes in the ontologies and explanatory relations between any pairs of predecessor-successor theories, even those very close in time. Among theories we now consider false, there are those which are strikingly successful, e.g. Newtonian mechanics. So it’s not clear why, assuming our newest theories are also successful, their success is explained via their truth, since the earlier cases it was not. The structural realist research program claims to be able to explain why reference is irrelevant for success, getting the right referents does not suffice for getting at the right structure. Specifically, structural realists claim that they can explain why a successful theory does not need a genuine reference: it does not matter if the relationship relating the terms is some specific relation R or if the property had by the terms related by R is some specific property P , the abstract description in terms of some set of objects in the domain (the objects having P ) and some relation relating these objects in the right way will suffice to deliver truths in as much as the domain instantiates this structure (Cf. Russell, 1927; Votsis, 2004, 2018). If structural realism is to deliver on these promises, it needs a more robust notion of structure than the set-theoretic one, as we will discuss below, but for all that, we believe the view is correct in its intuitive formulation: structure is what matters to science. Analogously, we want to say something similar when it comes to understanding. What is understood in cases of defective theories is, broadly speaking, that some structure is being posited of some objects in some domain for the purposes of saying explanatory things about them given the posited structure. This works in cases where the theories are contingently false since we can consider some possible structure instantiating the pattern with the ontology of the theory just so related and more importantly, where the theories are necessarily false: for example, whenever they are

13

Relatedly, the notion of “structure” plays an important role in debates about scientific realism, structural realism, and so on (Russell, 1927; Frigg and Votsis, 2011; French, 2014). We assume that the notion of a mathematical structure e.g., the natural number structure, the real number structure, is robust enough that there is no methodological need to dive further here given our aims.

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inconsistent and their underlying logic is classical i.e. they are explosive. For those cases, one can consider impossible structures: where the ontology of the theory would be related in some patterned way if it were not for the inconsistent elements. In a similar way, agents would be able to deal with theories that are either extremely partial, vague, or incomplete. When scientists report having understood a defective theory, even if clearly false or impossible, their claim might be legitimate. We argue that scientists understand a defective theory if they can recognize the theory’s underlying pattern(s) and if they can reconstruct and explain what is going on in specific cases of defective theories as well as consider what the theory would do if not defective—even before finding ways of fixing it. An important remark is that, while the purposes of this paper concern the accommodation of defects in the content of understanding, in general, our approach is neutral with respect to the corresponding debate.

6.4.3 The Epistemological and Metaphysical Value If what has been said here is along the right lines, one still might wonder what type of understanding is gained in cases of defective theories; this is, which is its epistemic status. This concern comes from the fact that the explanatory and factivity conditions of understanding are the result of aiming at a factual understanding of the empirical world. So when we decide to include non-true elements in the content of understanding, there is an important sense in which we might be driving away from that goal. Responding to this issue, we take the type of understanding that agents gain of defective theories qua defective is modal understanding. “One has some modal understanding of some phenomena if and only if one knows how to navigate some of the possibility space associated with the phenomena” (Le Bihan, 2017: 112. Our emphasis). The notion of possibility space is meant to be comprehensive. First, we consider the set S of possible worlds in which P, or some subset of P in the sense above, is the case. Next we consider the set of dependency structures that, when appropriately associated together, give rise to P, or to some subset of P, within S. The possibility space for P will be the set of dependency structures in those possible worlds that give rise to any subset of P and the relations between those structures. Note that the possibility space does not only include the set of possible dependency structures for P and the subsets of P: it also includes the relationships between these structures. (Le Bihan, 2017: 114)14

In the case of defective theories, to achieve modal understanding would be to determine the set of possible worlds that correspond to the generic structural features assumed by the theory, broadly speaking, as well as by its most salient models.

14 The

notion of modal understanding has been used by Le Bihan (2017) to address the way in which we understand theories and models that misrepresent the actual world by not being true. Here, we extend its scope in two directions: we cover other cases of defects, besides falsehood, and we explain its structuralist grounds.

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This is, if the theory and its components were to be true, which type of domain would they describe. But, what is the value of modal understanding? For the case of scientific theories, it grounds any further type of understanding that an agent would gain. Modal understanding reflects the expertise needed to identify and explain the multiple relations of dependence that hold within a theory and that a theory posits for its intended domain. As we have argued throughout this paper, understanding is non-factive in the sense that it does not presuppose truth. This is a subtle point, for we have also argued that understanding is structural and pattern-guided. In grasping the mathematical structures of theories we acquire understanding of their possible nature and their logical space, there is a factive element there involving this notion of structure. We wish to remain non-committal as to the nature of structure here, for there are many plausible candidates that are faithful to our core intuitions, however it is important to clarify two things. First, owing to model-theoretic considerations, any notion of structure must be distinguished structure, there will be variable elements but also fixed points, specific patterns or relations that are causally or naturally special in an objective way, this is a question of metametasemantics (Cf. Sider, 2011; Bricker, 2020). Second, the notion of structure pertaining to the ideological primitives of fundamental theories introduced by Sider (2011) is not the notion of mathematical structure but a more general notion involving the fundamental acceptance of the logical primitives of our most fundamental theories as part of ultimate reality.15 This notion of structure is relevant to the problem of understanding, not only for metaphysicians, those persecuted but noble beings, but also for philosophers of science, since scientific theorizing is greatly concerned with representation and inference and these involve ideological choices in the above sense. We take this section to have explained that when agents understand a defective theory qua defective, they only can do so by incorporating to the content of their understanding defective elements as well as the structural relations that allow them to remain well-behaved when leading to successful outputs (predictions, descriptions, explanations, among others). Here we have also explained that the type of understanding that is gained through doing so is modal understanding. For the purposes of this discussion, in the next sections we focus illustrating this with a case from Quantum Mechanics.

6.5 Understanding Via Quasi-Set Theory This section aims at two main things, first, introduce the technical and philosophical basics of Quasi-set theory, and second, to illustrate what has been said in the previous section considering the case study from Sect. 6.3.2.

15 This

point is discussed in more detail in Sect. 6.5.2.

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In order to do so, we proceed in three steps, first, we introduce the technical basics of a paraconsistent Quasi-set theory. Second, we discuss its metaphysical value, focusing on the issue of non-individuality. Third, we explain the way in which the use of Quasi-set theory can play a crucial role for the understanding of Quantum Mechanics as a defective theory.

6.5.1 The Basics of the (Paraconsistent) Quasi-Set Theory QP Quasi-set theories are mathematical systems that allow us to deal with indiscernible elements. As the reader might imagine, the main motivation for these theories is the presence of indiscernible entities in quantum physics. Here we describe the basics of one Quasi-set theory, .QP , which is paraconsistent; this, taking into account that in one of the most problematic scenarios, the non-individuality of some quantum objects can be understood as the root for inconsistency within the theory. Let .L be the language of .QP , the paraconsistent Quasi-set theory. .L includes the logical constants: negation .¬, conjunction .∧, disjunction .∨, and material implication .→, and the bi-conditional, .↔; all defined as usual. It also includes quantifiers, .∀ and .∃, and auxiliary symbols of punctuation. The specific symbols of .L are: ∗

.

∗

.

∗

.

four unary predicates: m, M, Z and C, two binary predicates: .≡ and .∈, a unary functional symbol qc.

“The terms of .L are the individual variables and the expressions of the form qc(x), where x is an individual variable” (Krause, 2012: 6); qc(x) indicates ‘the quasicardinal of x’. That said, .m(x) indicates that ‘x is a m-atom, a quantum object; .M(x) says that ‘x is a M-atom’, which acts as ZFU’s ur-elements. Furthermore, .Z(x) says that x is a set, and .x ≡ y that x is indistinguishable (or indiscernible) from y. Finally, .x ∈ y says that x is an element of y. Now, these are the crucial concepts around .QP are: 1. .α ◦ :.= .¬(α ∧ ¬α) We say that .α is well-behaved; otherwise, it is ill-behaved. 2. .¬ ∗ α:.= .¬α ∧ α ◦ This is the strong negation. It will have all the properties of standard negation. 3. .x ∗ = y:.= .[Q(x)∧Q(y)∧∀z(z ∈ x ↔ z ∈ y)] .∨ .[(M(x)∧M(y)∧∀Q z(z ∈ x ↔ z ∈ y)] This is the strong equality, or identity. It will have all the properties of classical equality. For simplicity, we shall write .x = y and read it “x is certainly identical to y”. 4. .x = y:.= .¬ ∗ (x = y) we read “x is certainly distinct from y”. 5. .Q(x):.= .¬m(x) ∧ ¬M(x) (x is quasi-set, or qset for short). 6. .E(x):.= .Q(x) ∧ ∀y(y ∈ x → Q(y)) (x is a qset whose elements are also qset, or x as no atoms as elements). 7. .x ⊆ y:.= .∀z(z ∈ x → z ∈ y) (subqset) Remark: since the notion of identity (.∗ = does not hold for m-atoms, in general we don’t have effective means to know either a certain m-atom belongs of does not belong to a certain qset. But the definition works in the conditional form.)

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8. .D(x):.= .M(x) ∨ Z(x) (x is a Ding, a “classical object” in the sense of Zermelo’s set theory, namely either a set or a macro-ur-element). (Krause, 2012: 6–7)

It is important to mention that the underlying logic of .QP is da Costa’s paraconsistent calculus .C1∗ . We think that this would suffice for the purposes of the paper but if interested in a comprehensive description of the theory, see Krause (2012).

6.5.2 Ideology, Ontology and Quantum Mechanics Metaphysics, broadly understood, is the philosophical inquiry into the ultimate structure of reality (van Inwagen, 1988: 11). This study crucially involves ontological questions such as “what exists according to our best metaphysical theories?”. Metaphysicians would like to find out what entities populate the world as part of this broader inquiry into the nature of reality, this is ontology. Tackling such questions requires in turn that we possess a reliable methodology for extracting ontological commitments from our best theories. In formulating theories about anything it is inevitable that we will presuppose primitive, undefined notions and assumptions which are not part of the ontology, in the sense of corresponding to objects, properties, or relations within the theory’s domain, but instead are required for even formulating it. Those primitive notions and assumptions in any theory we can characterize as the ideology of the theory (Cf. Quine, 1951; Cowling, 2013, 2018, 2020). Supposing we are metaphysical realists, that is, we believe the world has some structure that is mind-independent, then following Quine (1948, 1951) there are two interrelated questions we might ask from the standpoint of our theories: what is the theory’s ontology? and what is the theory’s ideology? These inquiries inevitably lead to metametaphysics, the philosophical study of the concepts, methods, and principles of metaphysics. Ontology is about what exists according to our theories, ideology is about the primitive concepts and notions, logical and non-logical, expressible within our theories that enable them to represent the domains they are about (Bricker, 2016). Since Quine (1948) the popular response to the question of ontology is inextricably linked to the metaphysical status of quantifiers, bits of logical ideology.16 Recently some realist metaphysicians have considered that we should go beyond the predicate (Sider, 2011), that is they have defended the claim that we need a distinction between distinguished structure and gerrymandered structure, where “structure” stands for the ideological primitives of our most fundamental metaphysical theory. This distinguished structure (Sider, 2011: 5) calls “metaphysical structure”. In his view, metaphysical structure: the operators, quantifiers, logical consequence relations, and logical connectives of our fundamental theory, are also metaphysically committing. More successful theories of fundamental science get at nature’s joints better than less successful ones. For Sider, this is evidence that the 16 For

a detailed discussion on this topic, see Macías-Bustos (2022a).

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primitive ideology of these theories is about the ultimate nature of reality. It is a realism that takes such primitive ideology as both irreducible and worldly. According to Sider, metaphysics is not only about ontology but about metaphysical structure, where “metaphysical structure” is to be read off from the primitive ideology of our most fundamental theory. Metaphysical structure, on this view, is not ontic structure but it is nevertheless about the world. Consider for example the difference between someone who believes in primitive modality and embraces the operators “possible” and “necessary” as distinguished (or joint-carving) as opposed to another philosopher who believes modality can be eliminated or reduced to facts about possible worlds and the objects within them. To be sure, modal operators are given an analysis in terms of possible worlds by some who embrace primitive modality, but they do not think this analysis gets the structure of reality right. Furthermore, some philosophers of primitive modality might simply refuse to give even an elucidatory analysis of their primitive modal ideology (Cf. Finocchiaro, 2021). On Sider’s (2011) view, the world will have individuals if the fundamental quantifiers of the fundamental theory range over elements of a domain.17 There will be a further question about anti-haeccetism and haeccetism, the views that all facts supervene on qualitative facts and its negation respectively. In his “Individuals”, Dasgupta (2009) further argued that individualistic facts were shown by physics to be redundant and empirically undetectable and proposed changing our fundamental logical ideology to that of Quine’s logic of functors, which has the same expressive power as classical first-order logic but avoids commitment to individual entity variables by using variable binding and instantiation operators for qualitative nplace predicates and relations. We want to highlight the compatibility of this research program with Krause (2012) and Krause and Arenhart (2016) motivation for introducing quasi-set theory. The ideology of quasi-set theory, its primitive predicates, logical constants, and logical consequence relation constitute a logical framework whose main aim is to allow for the representation of a different kind of ontic structure, quantum ontic structure. The classical ontic structure has been described by Turner (2010, 2011) in his influential papers on ontological nihilism and ontological pluralism as a pegboard structure. Think of reality, according to the ontological realist, as consisting of a pegboard with a series of pegs representing the variables quantifiers range over. We can attach a rubber band to the pegs, one rubber band for monadic predicates to an individual peg, one rubber band between two pegs for dyadic relations, three for triadic, and so on. To say the world has an ontological structure is to say it is structured like the pegboard and this structure is distinguished and mind-independent, the pegs are individuals.

17 For the sake of argument suppose you take all axioms of a plurality of fundamental theories as the axioms of the one fundamental metaphysical theory, as in the Best System Accounts of Laws according to which the laws of nature are the axioms of our best theory that best balance simplicity and strength.

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Seen in this light Krause’s (2012) Quasi-set theory is in the business of quantum metametaphysics in the sense of Torza (2021), i.e., it aims to determine the nature of quantum logical space. The motivation is thoroughly metaphysical in that it presupposes that quantum mechanics is broadly true about the world. If we follow these methodological guidelines and quantum mechanics is a fundamental theory, its acceptance should involve a reconceptualization of the structure of logical space and hence a transition from classical to quantum logical space. If this is right, Quasi-set theory is a contender for a metametaphysics of quantum mechanics taken seriously as a fundamental framework for objects in the most general possible sense, that of logic and set theory.18 In quasi-set theory there’s no individualistic presupposition at the level of the fundamental ontic structure unrestrictedly (as holding for all objects), but quasi-set theory can nevertheless recover the classical framework’s logical behavior for the objects of the domain under certain conditions: this is highlighted in the distinction between quantum objects which are indistinguishables and the M-objects, which are the classical emergent objects in this mathematical framework. In Krause’s quasi-set theory, the fundamental notion is that of the indistinguishable instead of that of the individual (Krause & Arenhart, 2016). Indeed, Krause’s quasi-set theory can recover ordinals and cardinals, the fundamental set-theoretic structures of order-types and sizes of sets respectively as special cases of quasi-set structures that hold for the classical part of the theory. Krause’s quasi-set theory and its metaphysical applications, as we’ve argued above, could be profitably investigated from Sider’s (2011) perspective. From this perspective, the world has more than ontic structure, it has metaphysical structure where which is a term of art introduced by him denoting the primitive ideology of our metaphysically fundamental theories: their primitive logical connectives, consequence relations, quantifiers, operators, notions of object, predicates and so on. For Sider, metaphysical structure is as much about the world as ontic structure, but it does not correspond to objects, our understanding of what those metaphysical structures are about results partly from adopting those ideological frameworks and using them for representation and inference. Successful scientific theories employ primitive ideology and hence scientific success is a guide to structural truth, both ontic and ideological, a guide to metaphysical structure.

6.5.3 Quantum Mechanics, Measurement and the Crisp Axiom All is not right with the world, however. There is a serious difficulty in Krause’s (2012) and Krause and Arenhart’s (2016) proposal for adopting quasi-set theory as our fundamental logical theory. To be clear, we have no objections to the

18 Quasi-set theory is also not a classical theory at the level of its logical consequence relation, which is paraconsistent i.e., it is inconsistency tolerant.

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formal details of the theory and it is certainly of great interest logically and mathematically, however its main motivation, as we’ve argued above, springs from the metametaphysics of quantum mechanics. It takes standard quantum mechanics (and its extension to quantum field theory) as a fundamental physical theory. This is captured in their system by the introduction of the Crisp axiom. Whenever an object is “crisp” then it has a classical structure, it becomes an M-object. Indeed, there is a “Crisp” predicate such that if a quantum object satisfies the predicate it becomes an M-object. Krause (2012) is explicit that intuition comes from the measurement postulate of Quantum Mechanics: upon measurement objects become Crisp. This is captured by the following axiom: (C) .∀x(m(x) → (C(x) → M(x))) The standard quantum mechanical formalism has two crucial postulates meant to guide scientists whenever they are applying them to some physical system: the Schrodinger equation and the Born rule (Albert, 1992). Physical systems however will evolve in vastly different ways depending on whether we take them to be evolving only according to the Schrodinger equation or the Schrodinger equation plus the Born rule at some time. More precisely, our formal descriptions of the evolution of the system will be radically different depending on whether we consider the system evolving only according to the Schrodinger equation or both the Schrodinger equation and the Born Rule for a given interval of time. Generally speaking, when considering the spin of a particle, the vector space associated with the spin-properties will be a two-dimensional complex vector space.19 Associated with the different spin properties there will be linear operators, mappings of the vector space onto itself which preserve the underlying structural properties of the space i.e., the ones specified by the axioms for vector spaces. We interpret the eigenvector-eigenvalue rule as telling us that the system is in a given state with some value for the property iff the vector associated with that state is an eigenvector of the operator associated with the property with a specific eigenvalue. The dynamical Schrodinger equation of standard quantum mechanics tells us that the system will evolve linearly between any two different times so long as no measurement of its physical properties is performed. Whenever we want to measure a property and the vector corresponding to the state of the system is not an eigenvector of the corresponding operator, we apply the Born Rule to the state vector of the system. It will tell us the probability that the system will be found to have some value or other. Schrödinger evolution and evolution that involves collapse upon measurement are the two types of dynamical evolutions at the root of standard quantum mechanics, however, these two types of evolution are completely at odds: deterministic and linear versus indeterministic and non-linear. It is not that there is some sort of formal inconsistency here (Cf. Albert, 1992; Maudlin, 2019; Okon, 2014; Okon

19 Infinite dimensional when we consider the position, even for a single particle as there will be an infinite number of mutually orthogonal eigenvectors associated with the position operator.

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and Sudarsky, 2014) rather it is extremely puzzling that the physical systems which are allegedly represented in the standard Quantum Mechanics formalism behave in ways that seem almost magical: evolving deterministically when not measured and collapsing nondeterministically when measured. But the measurement problem is not only that it is hard to make sense of the Quantum Mechanics formalism and its radically distinct laws of evolution as a physical theory: rather, it is that there is no physical theory specified here at all! Why? Because a crucial notion, that of “measurement” is completely vague. There is simply no formal counterpart in the theory for the act of “measuring”. Furthermore, there is no consensus in the community of physicists about when measurements take place or their nature. Fundamentally, it is useful to contrast the rest of the Quantum Mechanics formalism with the imprecise notion of measurement.20 The Quantum Mechanics formalism represents a mathematical structure: the structure of complex Hilbert spaces, a sort of generalization of Euclidean spaces. The mathematical objects that we can define there, such as operators or tensor products, behave in precise ways; it is possible to tell about the formal theory whether some proposition couched in its language is an axiom, a theorem, an application of an operation defined within the structure and so on. In contrast “measurements” aren’t specified in any sort of formally rigorous way; they are understood contextually and have no clear conditions of individuation or applicability. The crux of the issue however is this: since standard Quantum Mechanics can only be successfully applied if it combines the Schrödinger equation and the Born rule in ways that invoke the notion of “measurement” and since that notion is obscure in ways that go beyond lack of precision e.g. conditions of applicability, causal mechanisms, etc.; then it has to be said that standard Quantum Mechanics doesn’t in fact qualify as a theory of the physical world. The issue is not that standard Quantum Mechanics isn’t scientifically respectable as a predictive tool, its astounding success shows otherwise, rather it is that any theory of the physical world should give a clear and precise specification of what its objects are (its ontology) and how they behave (the dynamics): otherwise there is simply no fact of the matter as to how the theory says the world is. Quantum Mechanics is, metaphysically, a defective theory absent a solution to the measurement problem. Indeed some physicists (see for example, Hance and Hossenfelder, 2022) have wondered whether further progress in theoretical physics has been slowed down by these methodological considerations, if so then the measurement problem poses a difficulty that goes beyond understanding how quantum mechanics is meant to represent the world fundamentally, regardless of whatever instrumental and engineering successes it has had so far. Metametaphysical approaches such as Krause’s (2012) help in the project of understanding quantum mechanics from the logical side, but the defectiveness in this approach is a result of taking an already defective theory as metaphysically fundamental. It might be that we need to reconceptualize logical space if we take seriously the quantum theory

20 For

a more comprehensive discussion on this issue, see Macías-Bustos (2022b).

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as a theory about the world’s ultimate metaphysical structure, but arguably that will require a solution to the measurement problem and hence a reconceptualization of logical space that starts from a better foundation e.g., GRW, Bohmian Mechanics, Wave Function Realism, State Space Realism or Many Worlds. For these reasons, it would be a mistake to ignore the role that the identification and selection of specific structures play in enabling our understanding of scientific theories and their domains; especially when any of those are considered to be defective. As we hope to have shown for the case of Quantum Mechanics and nonindividuality, is the identification of specific inferential constraints and patterns that allows for the intelligibility of the theory qua defective in a non-problematic way.

6.6 Final Remarks Here, we addressed the question of under which circumstances can scientists achieve a legitimate understanding of defective theories qua defective. And as a response to it, we introduced a structuralist approach to scientific understanding according to which, scientists understand a theory if: – they can recognize the theory’s underlying inference pattern(s) and – if they can reconstruct and explain what is going on in specific cases of defective theories as well as consider what the theory would do if not defective—even before finding ways of fixing it. Understanding the inferential structure of the theory involves understanding the structure of its domain. Furthermore, this understanding is modal in nature, in that the domain might not actually instantiate that structure, the structure need only be possible. We illustrated the above with a case from Quantum Mechanics for which the theory is seen as defective, due to the non-individual metaphysical status of some of the entities of the theory. We contended that the identification of a structure that allows making the theory intelligible even if defective, but especially, qua defective, this structure should be included in the content of the understanding of the theory. For our case study, this structure was provided by the (paraconsistent) Quasi-set theory .QP (Cf. Krause, 2012). Acknowledgments The first author wants to thank Phil Bricker and Kevin Klement for valuable discussions on metaphysics and the philosophy of logic relevant to portions of this work. We thank the reviewer for the suggestions and critical comments. Also, thanks to Jonas Arenhart and Raoni Arroyo for their assistance during the process of getting the paper ready: they kindly dealt with our delays and other difficulties still inherent to editorial processes.

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Chapter 7

A Phenomenology of Identity: QBism and Quantum (Non-)Particles Michel Bitbol

Abstract Décio Krause has achieved a thorough reconstruction of logic and set theory, to account for the unusual objects or quasi-objects of quantum physics. How can one cope with the (partial) lack of criteria of individualization and reidentification of quantum objects, when the elementary operations of counting them, and constituting sets of them, are to be performed? Here, I advocate an alternative strategy, that consists in going below the level of logic and set theory to inquire how their categories are generated in the experience and activity of knowing subjects, and whether this mode of category generation is still relevant in the field of experimental quantum physics. This project of a “genealogy of logic” is borrowed from Husserl’s last treatise, entitled Experience and Judgment. It is transposed to the case of quantum physics by way of a QBist approach of Mott’s theorization of quasi-“trajectories” in Wilson cloud chambers. It is also shown that one of the most appropriate ontologies for quantum objects or quasi-objects involves reversing the (grammatical) roles of subject and predicate, as advocated by Japanese philosopher Nishida Kitarô in reasonable agreement with both Schrödinger’s and Krause’s approaches of the concept of “particle”.

7.1 Introduction Décio Krause is deservedly well-known for his edification and philosophical articulation of the quasi-set theory (Krause, 1992). The motivation for this remarkable work was to formulate a non-classical alternative to Zermelo–Frankel set theory that may be applied to the indiscernible particles of quantum mechanics and to the quasi-particles of quantum field theory (Krause et al., 2005). Indeed, addressing the

M. Bitbol () Archives Husserl, Centre National de la Recherche Scientifique, École Normale Supérieure, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_7

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status of sets of microphysical entities affords an appropriate logico-mathematical backbone to clarify ontological issues about the nature of the quantum particles. To understand this line of research, one must remember that at least four ontological options are (more or less elegantly) compatible with quantum statistics, namely with Bose–Einstein or Fermi–Dirac statistics. The first option is an ontology of particles devoid of individuality or selfidentity: this is the so-called “received view of quantum non-individuality” (French & Krause, 2006). The second ontology associates an N-times excited quantum field with a particlelike behavior of experimental events of detection (Teller, 1995, 1998; Sebens, 2022). By combining the views of non-individuality and quantum fields, some prominent authors of the past felt they have good reasons to doubt the very existence of particles. This is the case of Erwin Schrödinger (1952), who claimed that “(...) the particles, in the naive sense of the old days, do not exist” (See Bitbol, 1996). This is also the case of Willard Van Orman Quine (1976) who considered that standard conceptions of material objects or particles are dissolved straightaway in modern physics. The third ontology is both more conservative and less conventional than the two previous ones, since it does not even fit with Leibniz’s principle of identity of indiscernibles. This is an ontology of particles endowed with “transcendental individuality” or “haecceity” (Huggett, 1999), but with no observable feature that make them distinguishable (or discernible) from one another. The fourth ontology develops radical skepticism about the concept of monadic entities, especially monadic particles, but adopts a purely relational view instead. One may think of an ontology of pure structures (French & Ladyman, 2003; Ladyman & Bigaj, 2010), or an ontology of weakly, relationally, distinct particles (Saunders, 1994, 2003; Muller & Saunders, 2008; Rovelli, 2021). Several types of arguments (logical, physical, or metaphysical) have been presented in favor of each one of these variegated options, with the aim of showing that it alone is acceptable. But there are reasons to think that a form of ontological and representational underdetermination is bound to persist despite the strong constraints exerted by the structure of quantum physics (van Fraassen, 1991). The problem is that only antirealist philosophers of science can unreluctantly accept a persistent (possibly permanent) ontological underdetermination. Indeed, antirealists hold that a given ontology is to be construed as a mere heuristic pattern for research, among several others. Nothing then prevents them from adopting any one of the acceptable ontologies as a mental guide whenever this is appropriate, and to use another ontology (or none) in different circumstances. By contrast, from a realist standpoint, according to which scientific theories tend to provide us with a set of literally true statements about the world (or with structures isomorphic to the world), the plurality of ontologies and theories compatible with the available data is necessarily provisional (Earman, 1993). At the very least, the many available theories, ontologies, or representations, should agree on a unique ontological core (Alai, 2019). Efforts have therefore been made, in realist circles, to remove the

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ontological underdetermination of the quantum domain, and hopefully come back to traditional ontological patterns. One example of such effort, discussed by Arenhart and Krause (2014, 2019), was provided by Dorato and Morganti (2013). According to the latter authors, “to grant a well-defined cardinality for any plurality of items is itself the manifestation of individuality” (Arenhart & Krause, 2014). So, the fact that one can ascribe sets of particles a number, more specifically a cardinal, allegedly proves that particles are endowed with “transcendental individuality”. But, as Arenhart and Krause cogently argued, this argument is flawed. Ascribing a cardinal number to a set of items does not necessarily involve the standard procedure of counting, the imposition of an order between distinguishable items while they are being counted, and the correlative generation of an ordinal number. It just implies that some cardinal number is globally associated to the set. This point was illustrated qualitatively by Schrödinger (1950) while he was pondering on the meaning of quantum statistics. He noticed that, if we distribute a certain amount of money between several persons, the number of different distributions of those amounts is determined by the Bose–Einstein formula. Here, amounts of money are expressed in currency units, and they can be valued without such units being individuals at all. A cardinal but not an ordinal can be defined on such amounts of money; a procedure of counting and an ordinal could be defined only if the money were concretely represented by a classical set of individual coins. But the demonstration of Arenhart and Krause (2014) is even more general. They show that “a reasonable understanding of non-individuals may be consistent with the idea of a plurality of indistinguishable items with a well-defined cardinal”. This “reasonable understanding” is formalized by the axioms of a quasi-set theory “built with the specific purpose of encoding in a formal system the idea of . . . collections of non-individuals”. To sum up, what Arenhart and Krause (2014, 2019) have shown is that “one may embrace an ontology of non-individual objects, provided one accepts that some non-classical logic should be employed”. Under the assumption of a non-classical logic and set theory, the ontology of non-individual objects recovers its plausibility, and it may even be preferred for reasons of metaphysical parsimony or lack of surplus structure. A quasi-set approach of particle counting is a major step towards a proper understanding of the quantum ontology. However, this approach remains at an excessive level of abstraction. We would like to come closer to the concrete actions and experiences through which we, cognizant subjects, come to believe that there are individualized material bodies out there, and count them as if they were a multiplicity of objects independent of us. We would also like to examine to what extent this kind of belief and procedure of counting can (or cannot) be applied to microphysical particles. And we would finally like to articulate these concrete actions and experiences to the concrete predictive use of the quantum formalism. To get thus from the abstract realm of logic and set theory, to the concrete domain of acting, experiencing and predicting, three moves have to be performed. Our first move (Sect. 7.1.1) is to reach the level of the cognizant subject’s lived experience (Bitbol, 2020, 2021; Bitbol & De la Tremblaye, 2022). In the present

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case, we are interested in the experience of identification of bodies across spacetime, which was accurately documented by Edmund Husserl (1973). This is all the more significant since Husserl’s study of the experiential procedure of identifying bodies is associated to a retreat from the layer of logic and predicative judgment to the layer of pre-logical perception and pre-predicative experience. Phenomenology thus provides us with a tool to inquire into the pre-logical motivation for crossing the threshold from classical logic plus classical set theory, to non-classical logic plus non-classical set theory, as Décio Krause and other authors proposed. Our second move (Sect. 7.1.2) amounts to go beyond the mere assimilation of individuality to tautological self-identity (Krause et al., 2020). Instead, we choose Reichenbach’s (1965) principle of “genidentity” as an ultimate criterion of individuality: a principle of research of spatio-temporal continuity for sequences of local phenomena, according to a trajectory-like pattern. Our third move is to look for an equivalent of the two first moves at the level of quantum predictions. The connection between phenomenology and quantum mechanics will thus be analyzed (Sect. 7.1.3) by way of an interpretation of quantum mechanics that has a surprising amount of affinity with the phenomenological theory of perception: QBism or Quantum Bayesianism (Fuchs et al., 2014). The two themes, identity and quantum phenomenology, are then articulated by way of a QBist approach of particle tracks in Wilson or Bubble chambers (Sects. 7.1.4 and 7.1.5).

7.1.1 Husserl’s Prelogical Experience and the Constitution of Identity Any claim to knowledge can be doubted, but the precondition of the act of knowing, namely lived experience, is beyond any doubt; for doubt itself is given qua experience. Similar remarks were made thousands of years ago, most clearly by St Augustine:1 “Suppose they say to me ‘what if you were wrong, when you declare yourself certain that you are?’ I answer: if I am wrong, I am; what is not cannot even be wrong”. Such deep realization is the basis of René Descartes’ attempt to rebuild philosophy, and it also serves as an absolute beginning to Edmund Husserl (1929/1982) in his Cartesian Meditations. It is true that in situations called “normal science” by Thomas Kuhn, namely whenever general principles remain unquestioned and only special problems are at stake, returning to lived experience as the unshakable root of our quest for knowledge may seem superfluous. But when the foundations of science undergo a lasting crisis, when even our most entrenched ontological presuppositions are shaken, as it is the case in quantum physics, the motto can but be “back to experience”, for this is the only opportunity to make a fresh start from solid 1 St

Augustine, De Civitate Dei, XI, 26.

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ground. No wonder the lesson Bohr (1963, p. 10) drew from quantum physics was that “Physics is to be regarded not so much as the study of something a priori given, but rather as the development of methods of ordering and surveying human experience”. For, after the quantum revolution, no previously accepted entity was left unchallenged. In quantum physics, as in the general problem of knowledge, a tabula rasa of virtually every pre-“given” concept is then needed, and a careful survey of human experience is to be performed. The most radical tabula rasa was called “Epoché” and “transcendental reduction” by the founder of phenomenology. In Husserl’s approach, the two-step act of epochè and transcendental reduction is characterized as the gesture of (i) suspending (and withdrawing) one’s exclusive focus on the meant or intended objects, and (ii) coming back to the experience of meaning and intending (Husserl, 1929/1982, p. 56). In some extreme cases (Patoˇcka, 1995), this withdrawal is complete, thus opening a field of “blank”, uninterpreted, experience. But in other cases, the withdrawal is partial, just suspending our commitment to the scientific picture of the world, without calling into question the ordinary beliefs that pertain to our “lifeworld” (Husserl, 1936/1978, p. 167). What is more relevant to us, is the phenomenological description of the specific tabula rasa which must be performed for the sake of clarifying the foundations of logic (Husserl, 1938/1973). Husserl points out that, to uncover the ultimate source of the logical judgment and of its two archetypal components (subject and predicate), one must perform a “retrogression” (“Ruckgang” in German) towards the prelogical and even preverbal experience. “A retrogression in several stages is required in order to arrive at primal [ . . . ] self-evidences, which must then form the necessary point of departure for every elucidation of the origin of the judgment” (Husserl, 1938/1973, p. 22). The motivation for such retrogression towards the lower levels of the act of knowing is that “it is precisely in these lower levels that the concealed presuppositions are to be found, on the basis of which the meaning and legitimacy of the higher-level self-evidences of the logician are first and ultimately intelligible” (Husserl, 1938/1973, p. 13). As soon as the said “retrogression” is performed, one can observe the budding seeds of experiential evidence on the basis of which the logical judgment, together with its underlying ontology of substances and attributes, is edified. The concept of permanent, individual, self-identical objects, presented “out there” as if they were transcendent with respect to our experience, is the (substance-like) core of this ontology. But how do we come to believe in such objects? To start with, Husserl points out that, as soon as the epoché is performed, it becomes obvious that the so-called “transcendence” of the objects of our attention arises from the immanence of the consciousness that attends to them. “It is conscious life alone wherein everything transcendent becomes constituted, as something inseparable from consciousness” (Husserl, 1929/1982, p. 62). Even if one does not adhere to dogmatic idealism, this must be granted for a simple epistemological reason. This reason is that “Every rightness comes from evidence, therefore from our transcendental subjectivity itself; every imaginable adequation originates as our verification, is our synthesis, has in us its ultimate transcendental basis” (Husserl,

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1929/1982, p. 59). This being granted, one must inquire into the details of the process of “constituting” objects in (and from) the flux of consciousness. And one must also inquire into the root of the conscious feeling that these objects lie beyond consciousness, in a domain that “transcends” consciousness. Since the most accurate description of this process is given in Husserl’s Experience and Judgment (Husserl, 1938/1973), I’ll mainly follow this text. Special attention will be paid to the phenomenological analysis of object’s self-identity it contains. What is the very first structure of immediate experience, below the “garb of ideas thrown over the world of immediate intuition and experience” by elaborate logic and science (Husserl, 1938/1973, p. 44)? Such structure emerges from “the perceptive contemplation (“Betrachtung”) of the pregiven sensuous substrate” (Ibid. p. 59). It combines in one single “adumbration”: (1) the present sensory appearance of a certain “perspective”, and (2) a “horizon” of expected possible appearances of further perspectives. Husserl also insists that “this kind of evidence has an essential ‘one-sidedness’”. Indeed, at any single moment, each perspective shows only one side of a putative something, and nothing else. But even though no other side is visible, the still invisible is not deemed to be inexistent; instead, further sides are sketchily imagined and foreseen. Each perspective, each side, is surrounded by “[ . . . ] a multiform horizon of unfulfilled anticipations (which, however, are in need of fulfilment)” (Ibid. p. 61). This means that the one-sidedness of the present perspective is compensated by a blossoming of anticipated possible perspectives that are awaiting confirmation (fulfilment) in the next future. Crucially, despite the latent multiplicity of perspectives (the present perspective together with the anticipated perspectives), this basic structure of immediate experience involves a sense of unity: the unity of one (sometimes several) object(s) whose many aspects are successively manifested and anticipated. Indeed, if an appearance were not considered the presentation of some object having other aspects in store that could be shown later, there would be no reason to go beyond the present towards the expected. In immediate experience we are then bound to presuppose that a certain variety of perspectives (or aspects) pertain to one and the same object. “The naive consciousness, which, through all the perspectives, adumbrations, and so on in which the object of perception appears, is directed toward this object itself in its identity, has always in view only the result of this act: the object, which is explicated in perception as such and such.” (Ibid.) Instead of just attending to the indefinite unfolding of sensuous data, pre-predicative experience thus implies an ascription of identity, a unique nucleus taken as the permanent object of one’s intentional directedness. Pre-predicative experience thus associates a variety of presentations and sensuous contents whose only initial unity is established by their being recollected by the knowing ego, into a “noema” which imposes a pre-intellectual principle of unity of these presentations under the concept of a permanent object (Husserl, 1913/2018, §97). This being granted, the sequential appearance of various perspectives is taken as the progressive disclosure of the hidden (yet partly anticipated in a former horizon of adumbrations) faces, aspects, or profiles of such object. Husserl writes that the succession of perspectives is taken

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as an “explication” of the features of one and the same object in the course of its exploration. Let us recapitulate this phenomenological analysis of immediate (perceptive) experience, before any further elaboration. The strong association of presentations and expectations in one’s situated experience is the basic structure that gives rise to a belief in objects that can permanently afford more appearances, more information, and more opportunities to us. Such is the lived structure that supports our conviction that objects are “external” with respect to us, that they exist “in themselves” as an enduring repository of resources for us. One would not exceed a present profile by means of a ray of expectations without the motivation to discover the latent faces of some durably given and inherently existing object, namely to “explicate” its hitherto unseen aspects. “Every phase of perception is thus a radiating system of actual and potential intentions of anticipation” (Husserl, 1938/1973, p. 87). Conversely, the implicit belief in some inherently existing objects would not occur without being borne by a constant attempt to reveal new profiles under the presupposition of their belonging to one and the same entity. In other terms, the sense of transcendence of the object is provided by the indefinite openness of the process of “explication”. It arises from the feeling that the partly explicated object always promises something additional to explore; from the sense that accumulation of further evidence “about an object” is possible by way of a series of experiences that “are repeatable in infinitum” (Husserl, 1938/1973, p. 87). The lesson to be drawn from this study of the origin of our belief in objects external to us is momentous, according to Husserl. “Objectification is . . . always an active achievement of the ego, an active believing cognizance of that of which we are aware, this something being one and continuously the same through the continuous extension of consciousness in its duration” (Husserl, 1938/1973, p. 62). Husserl here points out that the “active achievement” of a pole of sameness, of an enduring self-identical entity beyond the flux of fleeting appearances, is based on a “synthesis” of such appearances by the (transcendental) ego. The presumption of an object’s self-identity relies on a mental act of the ego: the mental act of constantly aiming at the putative nucleus of an ongoing flow of changing phenomena. Only thus, by this active “retaining in grasp” of a putative “something” by an ego, can “it be apprehended in a simple perception as an enduring object, as one which not only is now but which was also the same just before and will be in the next now” (Husserl, 1938/1973, p. 109). Objects are constituted by a transcendental ego as a consequence of the latter’s tendency to presuppose (and retain in grasp) their selfidentity beyond apparent change. We have seen that this combination of (i) “explicating”, namely developing a stream of appearances across time, and (ii) “retaining in grasp” one and the same object, is called “explicative synthesis” by Husserl. Such is precisely the experiential pre-predicative origin of the couple of categories (i) “determination” and (ii) “substrate” that a predicative judgment articulates. But, of course, the scope of the explicative synthesis, and of the predicative judgment that arises from it, is not limited to the needs of a single subject at a certain moment of her conscious life. The stabilization of the formal concepts of “substrate” and “determination”,

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and their articulation into a logical judgment, is in fact our central resource for elaborating a body of intersubjective knowledge. One could even say that these formal concepts, and the resulting judgments, are the condition of possibility of an intersubjectively shared science. The explicative synthesis and its crystallization in predicative judgments indeed express “the fact that, beyond the momentary situation, we aspire to create a store of knowledge which is communicable and usable in the future” (Husserl, 1938/1973, p. 63). The explicative synthesis is what allows us to fill the gap between: (a) the immediacy of present-appearances-forourselves, and (b) the communication of a corpus of knowledge that holds for any future time and for anyone who will need it. “The act of perceiving and judging on the basis of perception is [ . . . ] that attitude which underlies theoretical science and makes possible a confirmation with the goal of objectivity, of validity ‘once and for all’ and ‘for everyone’.” (Husserl, 1938/1973, p. 65). This phenomenological analysis has given us access to the pre-logical ground of the logic of judgments and its classical set-theoretical counterpart. We have discovered that the pre-logical ground for logic is the knowing subject’s striving to unify her lived experience in an intersubjectively communicable way, beyond the ceaseless unfolding of a variety of situated appearances. Such a quest for unity beyond and above the unstable present involves the recollection of past events, the anticipation of a continuous series of future events, and their tentative ascription to one and the same object. Can we find an equivalent of this procedure in our ordinary language and in classical Physics? And, if so, how is it disrupted in Quantum Physics?

7.1.2 Identity and Trajectory The identity of a material body through time is presupposed by the speech acts that consist of referring to it. This point is made clear by several modern theories of reference. According to Searle (1969), referring to a certain thing implies that the speaker is capable both of identifying that thing in the present moment, and of re-identifying it whenever “it” will reappear in the future. As for Kripke (1980), he considers that naming a thing amounts to accepting either (prospectively) that one will be able to recognize again and again this thing which has just been “baptized”, or (retrospectively) that one has the capacity to link the thing whose name is currently used, back to a past baptism certificate. In physics, something similar occurs. Besides, the identity of a material body across time is presupposed by the tracking procedures of a region of space characterized by some unusual sensory or experimental feature. Such tracking procedures may involve either the oculomotor system, or some monitoring device that operates in the same mode as the eye at microscopic or macroscopic scales. In the absence of any other criterion of identity, it is then the connection of one region of space to another, ensured by a continuous procedure of tracking, which is taken as our only guarantee of the

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identity of something across time. In this latter case, one speaks of the genealogical identity, or the “genidentity” (Reichenbach, 1928/1965, p. 55), of material bodies. Genidentity is clearly distinct from other modalities of identification of bodies, made possible by their possession of discriminating individual features. Moreover, according to Reichenbach, genidentity is not just a criterion of recognition of some pre-given thing by way of its spatio-temporal track. It is a constitutive principle in a Kantian (and to some extent also Husserlian) sense. “When we speak of the path of an electron, we must think of the electron as a thing remaining identical with itself; that is, we must make use of the principle of genidentity as a constitutive category” (Ibid.). To understand properly this sentence of Hans Reichenbach, one should notice two points. Firstly, even though, in it, the electron is initially referred to as if it were a pre-given thing, this impression is corrected immediately afterwards. Reichenbach does not say that some electron is (or is not) a self-identical element; he rather says that we must think of it as selfidentical across time; he thus implies that self-identity is just a presupposition by which we put ourselves in a position to attribute a sequence of phenomena to one and the same “thing” which we call an “electron”. Secondly, this preliminary act of thinking turns out to be a “coordinating principle” (a presupposed coordination of phenomena), and a “constitutive category”: one that constitutes an electron qua object beyond its manifold appearance in space-time. In Reichenbach’s neo-Kantian approach of physics, the principles of coordination play a key role. They so to speak define objects of physical knowledge. They are constitutive of them, insofar as they prescribe the way in which phenomena must be articulated to one another so that they can indeed count as multiple manifestations of the properties of a single object. However, Reichenbach did not retain Kant’s construal of constitutive principles qua perennial norms of thought. He insisted that the constitutive principles of science are relative to each step of research; they are not fixed once and for all, but are just the enduring background presuppositions of each historical phase of science (Friedman, 2001). This is why scientific revolutions are not limited to better characterizations of pre-given objects; they rather imply a “change in the concept of object” triggered by a preliminary “change of the coordinating principles” (Reichenbach, 1928/1965, p. 94). The consequences of such radical change were evaluated by Schrödinger in the context of quantum physics (Bitbol, 1996, chap 4 and 5). According to Schrödinger, if two particles have all their properties in common, the only remaining possibility to distinguish them is to “genealogically” connect their present observed position to two distinct past positions, through two precise continuous trajectories. Usually, however, this does not work in the quantum domain. Indeed, the trajectories are submitted to Heisenberg’s “uncertainty principle” and therefore no pair of trajectories can be found without a mutual overlap. The standard, genidentical, constitutive principle of material particles thus falls apart. And the very “concept of object” is overturned, in agreement with Reichenbach’s lesson. This is why, in a letter to Henry Margenau of 1955, Schrödinger pointed out that “to me, giving up

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the path seems giving up the particle”. At the very least, giving up a spatiotemporal path of self-identification means giving up the individual particle. Schrödinger’s position about the “existence” of particles is probably too clearcut. Schrödinger indeed explicitly overlooked the fact that, in certain cases of low densities, the genidentity of particles may still be ascertained by means of their (almost) non-overlapping trajectories, thus restoring some operational sense to their individual permanent “existence”. Instead, he insisted that, however small, the non-zero probability of overlapping makes the concept of a permanent particle universally meaningless (Schrödinger, 1950). And he finally declared that, such ontology of spatiotemporal continuants having been completely shattered, what is left to physicists is only a cloud of isolated detection phenomena, to be organized under non-genidentical coordinating principles. At the end of the day, Schrödinger felt entitled to assert that particles are always non-individuals and therefore (according to him) absolutely non-existent as such. But, may be, even the binary alternatives of individuality and non-individuality, existence and nonexistence, are outdated in the field of quantum mechanics. May be the joint claim of (intrinsic) non-individuality and non-existence is just as unwarranted as the opposite claim of (intrinsic) individuality and existence of particles. These claims were scrutinized in recent studies (Goyal, 2019; Dieks, 2020; Jantzen, 2020) of a feature that underly them: persistence as a necessary but not sufficient condition of re-identifiability. Even though Goyal starts from Schrödinger’s devastating analysis, his philosophical outcome is much more nuanced. Goyal’s strategy is indeed overtly operationalist. He accepts from the outset that what is given to physicists is nothing else and nothing more than a cloud of scattered detection phenomena. He accepts from the outset that the only material out of which physicists try to elaborate a conception of particles is a set of “raw data (that) consist of identical localized events”. In the latter sentence, “identical” means “showing the same properties (such as mass or electric charge) in one location as in another location”. Then, instead of jumping immediately to an ontological interpretation of such raw data, Goyal wonders which model is more appropriate to account for various distributions of the manifest identical localized events. Is it a model of persistence (and space-time continuity) or a model of non-persistence (and space-time discontinuity)? In other terms, should we assume that localized events are produced by persistent situated entities, or that they manifest some nonpersistent distributed process? Clearly, the answer to the latter questions may vary according to the experimental context. For instance, the model of persistence will be more appropriate (yet not exclusive) in the context of a bubble chamber showing approximately aligned localized events (the tracks), whereas the model of nonpersistence will be more appropriate (yet not exclusive) in the context of a two-slit experiment yielding a bi-dimensional layer of localized events (the interference pattern). Let me emphasize at this point two innovative features of this philosophical approach. In it, ontological issues are both “en-theorized” (Fine, 1996) and “enexperimentalized”. What can be said about what there is (namely persistent particles or non-persistent phenomena) is relative: (i) to various theoretical models and (ii) to

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various experimental contexts. As a consequence, there is no absolute answer to the ontological questions “do particles exist or not?” and “are they individuals or nonindividuals?” Instead, just as in the earliest analysis of experimental contextuality in quantum mechanics, the two models are shown to be “complementary” in Bohr’s sense: although they are mutually exclusive, they are jointly indispensable to account for a host of weird features of quantum statistics (Englert, 1996; Goyal, 2019). The joint indispensability of the two models of persistence and nonpersistence is not just a matter of words. It has concrete consequences, since Goyal demonstrates that “Feynman’s and Dirac’s symmetrization procedures arise through a synthesis of a quantum treatment of persistence and non-persistence models”. These two “mutually exclusive” models are “jointly”, or synthetically, needed “for accounting in detail for the quantum rules employed in the treatment of identical particles”. It then turns out that ontology is not to be understood as something we receive ready-made from “external reality”. Ontology should rather be construed as a set of structures proposed by an experimenting and theorizing subject for the sake of accounting for phenomena. This displacement of the central axis of ontological thought from a putative “external reality” to a subject who can take epistemic initiatives, strongly evokes phenomenology. But whereas Goyal’s subject is a typical pragmatist acting and thinking being, the phenomenological subject is mostly a being capable of having experiences and of initiating gestures of attention or anticipation within her lived experience. To get even closer to a phenomenological approach of quantum physics, we must now turn to QBism.

7.1.3 QBism: A Physics in the First Person The word “QBism” originated as an acronym for “Quantum Bayesianism” (Caves et al., 2002a, b; Fuchs & Schack, 2013). But the proponents of QBism gradually changed their minds about the relevance of this reference to the Bayesian conception of probabilities. Aren’t there multiple versions of Bayesianism; do we not distinguish in particular an objectivist version from a subjectivist version of Bayesianism? To manifest a decision in favor of the latter, it was first proposed to make the “B” of QBism the initial of “Bettabilitarianism”: the thesis that quantum physics is equivalent to a betting system. But as this word in “ism” is too heavy, a halfserious (and only transient) suggestion was to consider that the letter B is the initial of Bruno, for Bruno De Finetti (Fuchs, 2016). After all, it is on the audacity of Bruno De Finetti (2008) that the audacity of the authors of QBism is based: if we want to make quantum theory intelligible, they claim, we must transform not only our interpretation of the theoretical symbols (by identifying them with pure probabilistic valuations), but also our conception of what probabilities do, and our idea of what physics is. In quantum theory, these transformations are stated as follows: quantum “states” look like Bayesian personalistic probabilities associated to bets (Fuchs, 2010). One could even say that

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quantum states are nothing beyond such personal probabilistic estimates. Quantum “states” do not denote the properties of some microscopic “system” (Fuchs & Schack, 2015), they do not represent waves, and even less do they “describe” particles; they express our propensity to bet about an experimental outcome. This position represents a complete overturn of our views about physics, and science in general, similar to Kant’s “Copernican revolution”. According to the standard view, a science must first offer a description of what there is, before predicting what will be seen, and before becoming interested in the very point of view from which everything is seen. The reverse is explicitly assumed in QBism. QBism prescribes that physics wonder about its own departure point in the lived life of physicists. QBism also prescribes that we suspend (at least temporarily) questions about what there is beyond the visible, in the post-physical or metaphysical margins of the inquiry, and rather focus on the visible things that can be anticipated probabilistically. But what are the visible things that turn out to be primarily relevant to physics? For Bohr and his successors, these are the dials of measuring devices, describable in everyday language supplemented by the concepts of classical physics. For QBists, instead, “seeing” is more immediate and more intimate; it directly concerns the lived experience of subjects, prior to any assignment of existence to macroscopic objects such as measuring devices. “In QBism the only phenomenon accessible to Alice which she does not model with quantum mechanics is her own direct internal awareness of her own private experience. This (and only this) plays the role of the ‘classical objects’ of Landau and Lifshitz for Alice (and only for Alice). Her awareness of her past experience forms the basis for the beliefs on which her state assignments rest. And her probability assignments express her expectations for her future experience” (Fuchs et al., 2014). All central concepts of physics are reconstructed by QBists out of the elementary material of present lived experience. Indeed, ascribing present experience the role of a radical origin of knowledge is crucial to the QBist dissolution of enigmas such as EPR correlations (Fuchs et al., 2014) or Wigner’s friend paradox (De Brota et al., 2020). As a consequence, just as in Kant’s and Husserl’s doctrines of the “constitution of objectivity”, the traditional hierarchy of knowledge is reversed in QBism (De la Tremblaye & Bitbol, 2022). The object to be known is no longer an entity given in advance, and subsequently studied by the knowing subject. The object is no longer posited before the subject, being thereby considered as what arouses her experience. Instead, it is the experience of the knowing subject which decides what counts as an object. Accordingly, a QBist thinker should not start from a preliminary notion of “system”,2 to then assign probabilities to the values of observables measured “on it”. On the contrary, she should come to speak of a “system” only if the order of the probabilities that she evaluates is in conformity with that which she would expect from an entity of this type. “Whereas the usual

2 Although QBists often speak, for the sake of communication, as if such “physical systems” were already given out there, ready to be probed.

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approach in physics would presuppose an objective world and explain the different perspectives by ‘carving out’ parts of it to serve as the perceiving subjects, QBism begins with a fundamental plurality of perspectives and seeks to derive some notion of objective concepts from them” (Pienaar, 2020). Even more directly perhaps, “any user’s own experience constitutes all of the raw material out of which she constructs her world” (Fuchs et al., 2014). QBism goes so far as to bring back all the epistemological dualities, that of subject and world, that of observers and environment, or that of agents and targets of their actions, to the unique field of lived experience wherein they take shape. “In QBism, an element of reality is an experience, which contains as a fundamental internal structure a pairing of an experiencing subject with an experienced object” (Pienaar, 2020). This is an accomplished form of Kant’s “Copernican revolution”, in which appearing is not only the starting point of knowledge, but the very fabric of what we call “reality”; and in which lived experience is not a place where objects allegedly external to it are given, but rather the original ground from which the subject-object duality arises. The final step of the probabilistic purification of quantum theory by QBism consists in realizing that “the things which can be seen” and which are foreseen (the quantum phenomena), are not obtained passively, but actively. QBism marks this recognition by replacing the word “observer” with the word “agent”. The agent does not just collect the information. She pre-orders information through the design of her measuring device. And she arouses information (she “creates” it) by the effective implementation of such device. In other terms, a phenomenon is co-generated by the agent equipped with some measuring device, and by the environment she explores. Thus, each experiment is a creative act. Each experiment gives birth to a pure novelty that manifests qua lived experience. “A measurement does not, as the term unfortunately suggests, reveal a pre-existing state of affairs. It is an action on the world by an agent that results in the creation of an outcome – a new experience for that agent” (Fuchs et al., 2014). To sum up, QBists hold that the proper foundation of quantum physics is the agent capable of experience. This view looks like a combination of pragmatism and phenomenology, or at least (to stick to the framework of William James philosophy) a combination of pragmatism and radical empiricism. But then, how do QBists carry out the task they have assigned to themselves, namely deriving objective concepts from agent’s perspectives? Even though such procedure is not described very explicitly in the QBist literature, several features suggest that it might be similar to that of the phenomenological constitution of objects. Let me first remind that, according to QBism, quantum knowledge associates (i) a system of probabilistic expectations of potential experimental outcomes, expressed by a state vector and an observable, with (ii) a prescription to use actual experimental outcomes as a basis for redefining probabilistic expectations (such prescription amounts to using the “projection postulate”, but without the reifying undertones that usually come with it). This couple of operations comes remarkably close to Husserl’s phenomenological theory of perception, with its horizons of expectations that can be fulfilled or disappointed by “intuitive contents” made of sensory experience,

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and that are then continuously redefined according to whether they are fulfilled or disappointed (De la Tremblaye, 2020). This being granted, the QBist procedure for constituting spatio-temporally persistent objects is bound to follow the additional rules indicated by Husserl (1973). In Husserl’s procedure, the expectation pattern for future appearances is informed: (i) by a synthesis of past and present appearances, and (ii) by the tacit presupposition that they do and will concern one and the same enduring object. Such twofold determination of the subject’s expectations is tantamount to “retaining (a selfsimilar enduring object) in grasp” beyond the changing past and future experiences. In the QBist interpretation of quantum physics, the contribution of the past is represented by taking into account earlier measurements of generic features such as mass and charge, but also by acknowledging the latest measurements of spatial, kinematical or any other coordinates. This allows the QBist agent to write down an initial quantum state that is likely to express the best basis for the probabilistic valuation of future measurements of those coordinates. Now, is there an equivalent of the “retaining in grasp” of one and the same enduring object in the QBist system of probabilistic valuations for future measurements? The answer to this question is positive; and it is not too difficult to find such equivalent. To calculate the probability of a future experimental outcome through the Born rule, any quantum physicist (and the QBist as well) performs the inner product of the eigenstate corresponding to this outcome by a certain quantum state. But which quantum state? Not the initial quantum state that immediately accounts for a set of past measurements, but another quantum state derived from the former one by having an evolution operator acting on it. This evolution operator is usually derived from the Schrödinger equation, and it crucially involves a Hamiltonian operator. The Hamiltonian operator includes a set of variables that express both the generic features, the (putative) spatio-temporal location, and the physical environment of a certain (putative) classical system of enduring self-identical particles. The latter feature of the procedure of probability valuation is highly significant. It means that a quantum agent builds her expectations under the open presupposition of the selfidentity and persistence of some “physical system” of particulate entities whose interaction with her experimental device is supposed to generate the relevant future phenomena. In the same way as the phenomenological subject, the QBist agent thereby projects a “horizon” of anticipations which is conditioned by her “retaining in grasp” some alleged enduring entity. In the same way as the expectations of the phenomenological subject, the expectations that the QBist agent formulates under the presupposition of the persistence of a “system” of bodily entities can either be fulfilled or disappointed. In the same way as in phenomenology, the enduring bodily entities are not pregiven items liable to description, but rather background assumptions that determine a structure of predictions. However, unlike the phenomenological subject, the QBist agent can make use of unconventional resources. The first resource is that the background (corpuscularian) presuppositions are not rigidly connected with the predictions, but only probabilistically, through a mathematical device involving a formalism of state vectors articulated with the Born

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rule. The second resource is that, in case the standard, unformal, semi-classical, unprobabilistic, expectations a QBist agent may formulate under the hypothesis of the persistence of some entity, are systematically disappointed, she is free to use the alternative presupposition (or model) of non-persistence, and combine it with the former “complementary” presupposition. For only combinations of the models of persistence and non-persistence can make the standard (unformal, semi-classical) expectations fit the probabilistic expectations afforded by the quantum formalism. An illustration of this procedure will now be given in the next section. It concerns the wave-mechanical account of tracks in a cloud chamber by Nevil Mott. Just as the QBist account, this one does not involve the description of a particle trajectory, but a method of probabilistic prediction of trajectory-like alignments of ionized droplets, under the open presupposition of the persistence of such particle. And just as in our reading of the QBist procedure, this presupposition for the sake of prediction is encoded in “the particle’s Hamiltonian”.

7.1.4 Particle-Track Prediction Without Pre-given Particles3 Nevill Mott’s paper entitled “The wave mechanics of α-ray tracks” (Mott, 1929), is deservedly well-known as a seminal work about certain fundamental aspects of the quantum theory of measurement. But here, we will downplay the relevance of Mott’s study for the measurement problem, and rather insist on its implications for the issue of the emergence of phenomena interpretable as manifestations of selfidentical particles (Figari & Teta, 2013; Ballesteros et al., 2021). As a preliminary, we must notice some differences in the way Mott and Heisenberg (1930) accounted for the phenomenon of tracks in Wilson’s cloud chambers exposed to radiation. Mott was clearly attempting to promote a purely wave-mechanical account of this phenomenon. Instead, while he mostly relied on Mott’s calculations, Heisenberg tried to enact Bohr’s concept of “complementarity”. Heisenberg indeed replaced the virtually exclusive wave-mechanical bias of Mott’s paper, by an alternation of wave and particle representations. He thus substantiated the common-sense interpretation of trails of water droplets in cloud chambers as the manifestation of particle trajectories. Under this interpretation, high-energy particles impose local ionization to water molecules in the cloud chamber along their path, and those ionized molecules in turn trigger condensation of other water molecules around them, thus forming a visible trail of water droplets. Let us begin with Mott. His starting point was the interpretation of Schrödinger’s wave mechanics expounded a few weeks earlier by Charles Galton Darwin (1929). But Mott made Darwin’s ideas more concise, more operational, and also perhaps

3 This section is partly derived from the (unpublished) text of a talk delivered at the workshop “History of Radioactivity”, organized by C. Blondel et P. Radvanyi as a satellite meeting of the annual conference of the French Society of Physics held in Paris, in 1997.

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less metaphysical. The initial motivation of both Darwin’s and Mott’s papers was Gamow’s (1928) theory of radioactive disintegration, which had also been formulated independently by Gurney and Condon (1928). In this theory, the emission of α-rays was explained wave-mechanically by means of the concept of potentialbarrier penetration. The amplitude of the wave function decreases exponentially with the thickness of the potential barrier; but it has a non-zero value outside, and this accounts for the non-zero probability of the leaking of α-rays out of the nucleus. Now, the problem is that, as soon as they have emerged, the α-rays appear to have essentially corpuscle-like properties, for they give rise to tracks in cloud chambers. Charles Galton Darwin’s project was then to restore a certain conceptual homogeneity between the explanation of the radioactive emission (which is based on pure wave mechanics) and the account of detection (which must apparently involve corpuscularian categories). He wished to make sense of the α-ray tracks without resorting to the process that consists in imagining that each time an observation is made, “the wave [turns] into a particle and then back again [into a wave]” (Darwin, 1929). He then wanted “to show how a discussion only involving the wave function ψ would give spontaneously the results which simple intuition would suggest could only be due to particles” (Ibid.). As for Mott (1929), he also insisted that “wave mechanics unaided ought to be able to predict the possible results of any observation that we could make on a system”. How could Darwin and Mott achieve this remarkable result? They did so first of all by ascribing a single multidimensional wave-function to the global system made of the α-particle plus every ionizable atom in the cloud chamber. “Before the very first collision, (the wave function) can be represented as the product of a spherical wave for the α particle, by a set of more or less stationary waves for the atoms. ... [The] first collision changes this product into a function in which the two types of coordinates are inextricably mixed” (Darwin, 1929). The latter sentence is an early statement of what we now call the entanglement of wave functions after Schrödinger (1935). Darwin (1929) further insisted that “the trouble ... in the quantum theory has only arisen through attempts to work with an incomplete ψ”, namely a ψfunction that does not include relevant elements of the measuring device. As for Mott (1929), he noticed that “... we are really dealing with wave functions in the multi-space formed by the coordinates both of the α-particle and of every atom in the Wilson chamber”. But what does this multidimensional wave-function represent? Can it be considered as a mathematical picture of some real process out there? By no means. According to Darwin and Mott, the quantum mechanical account, including in cases where this account uses entangled wave functions, does not provide the slightest element of description of the putative processes underlying the α-ray track phenomenon; it only enables us “to predict the possible results of any observation”. In other terms, “interpreting the wave function should give us simply the probability that such and such an atom is ionized”, and the probability that a series of such ionizations generate a sequence of more or less aligned droplets in the cloud chamber. This way of using the wave function is in perfect agreement with Schrödinger’s late conception of his wave mechanics. In a lecture given in the early 1950s,

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Schrödinger (1995, p. 50) insisted, almost in the same terms as Mott’s, that what he called the “interpretation” of a wave function is nothing more and nothing less than the (probabilistic) connection between the continuously changing overall wave functions and the observed outcomes of a measurement. This purely probabilistic interpretation of the wave function also fits neatly with QBism, which will allow us (in the next section) to base a QBist approach of particle’s self-identity on Mott’s wave-mechanical theory of α-ray tracks. Accordingly, Mott and Darwin insisted that, in a quantum approach of the αray tracks, the multidimensional wave-mechanical account must be pushed as far as possible, and that any reference to corpuscular or discontinuous pictures must be delayed as much as possible. This method is fully homogeneous and coherent, insofar as it consists in developing the continuous predictive formalism until the stage where a probabilistic prediction is required, rather than mixing up continuous predictive elements with unwarranted discontinuous descriptive stories. Mott (1929) thus recommended that “until this final [probabilistic] interpretation is made, no mention should be made of the α-ray being a particle at all”. As for Darwin, he took this delay as the pivotal concept of his interpretation of quantum mechanics, and as a sort of leitmotiv. I take it that the verb “to postpone” is the key-word in Darwin’s paper. Darwin’s major aim was indeed to show “how it is possible to postpone speaking of particles”; for according to him, “there is no need to invoke particle-like properties in the unobserved parts of any occurrence, since the wave function ψ will give all the necessary effects”. Each entangled wave function can be read as a disjunction of conditional statements, relating one ionization to a series of other expected ionizations approximately located on the straight line joining the radioactive nucleus and the first ionization. While the probability of the first ionization is evenly distributed, the conditional probability of obtaining an approximately straight track uniting the radioactive nucleus and this first ionization is very high. However, here again, says Darwin, “The decision as to the actual track can be postponed until the wave reaches the uncovered part where the observations are made”. Later on in his paper, Darwin went even further, in suggesting that it is only at the level of the brain that we are really compelled to stop the chain of entanglements, and that it is our consciousness that so to speak cuts sections of the overall wave-function when it becomes aware of the outcome of observations. He thus anticipated later (sometimes controversial) views of the measurement problem such as von Neumann’s, London’s and Bauer’s, or Wigner’s. But this urge to explain how it is that we finally see a single track, in spite of the multiple-track structure of the relevant overall wave function, is a clear sign that Darwin was still tempted with ascribing a partly descriptive status to wave mechanics, rather than a purely predictive one. Far cry from Darwin, Mott avoided such speculations straightaway; he contented himself with having proved that the probability of observing two (or more) ionized atoms in the cloud chamber vanishes unless the line that joins them passes near the radioactive atom. As I mentioned earlier, the interpretative strategy used by Heisenberg (1930) was quite different. Unlike Mott and Darwin (and owing to the influence Bohr exerted on him), Heisenberg had no reluctance to jump from the corpuscle representation to

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the wave representation and back again whenever it appeared convenient to do so. According to him, nothing prevents one from using the corpuscular picture (though submitted to the uncertainty relations) when the tracks in Wilson cloud chambers have to be accounted for quantum mechanically, notwithstanding the fact that a wave picture is more convenient to explain the radioactive emission. In the same spirit, he considered that including the α-particle and the ionizable hydrogen atoms of the cloud chamber within the same compound system, or taking the α-particle as the only system and the ionizable atoms as part of the observation device, is a matter of free choice. A cut has to be introduced somewhere between the system and the observation device, but, says Heisenberg after Bohr, the location of this cut is almost arbitrary; it only depends on pragmatic considerations. Accordingly, Heisenberg did his best to show that, in the problem of α-ray tracks, the method of successive reductions of a wave packet (in which the α-particle is the system) gives exactly the same predictions as the method of entangled wave functions (in which the system includes not only the α-particle but also the ionizable atoms in the cloud chamber). Now, what are we to think about this difference between Heisenberg on the one side and Mott and Darwin on the other? Heisenberg was certainly right to point out the strict predictive equivalence between wave packet reductions and entangled wave-functions. However, the two methods are not equivalent from a conceptual standpoint. The first method (favored by Heisenberg) is quite widespread, but it prevents us from appreciating the purely probabilistic status of the wave functions and state vectors of quantum mechanics. Instead, Darwin’s and Mott’s method is less popular, but it has the merit of putting probabilities to the fore, thereby connecting immediately with the remarkable capacity of QBism to dry up the source of the so-called “quantum paradoxes”. Besides, the method of successive wave-packet reductions is usually much simpler but more misleading. It consists in using the information afforded by each point-like observation to extract a new wave function for the α-particle alone, out of the compound wave function of the larger system consisting of the α-particle plus a certain ionized atom. But one usually forgets this process; one usually forgets that successive reductions are by no means changes of the initial wave function, but rather redefinitions of it for practical purposes. As a consequence of this forgetfulness, the discontinuous evolution of the wave function is mistaken for a sort of descriptive account of the real process that gives rise to the track, and this arouses spurious questions about the “physical mechanism” of the (allegedly physical) wave packet reductions. By contrast, the method of the entangled wave-functions has the merit of permanently maintaining a clear distinction between the continuous (wave-like) predictive model, and the series of predicted particle-like discontinuous phenomena. The only question which arises in this latter method concerns not the discontinuous collapse of the wave function, but the progressive shift from a wave-like to a classical probability theory when the system encompassed within the global wave-function grows bigger and bigger. In other words, the question here is that of a progressive loss of interference terms in the probabilistic formalism. As we now know, a plausible answer to that question, but not directly to the question about wave-function collapse, has been provided

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by decoherence theories. In terms borrowed from Bub and Pitowsky (2007), while decoherence solves the “small” measurement problem of probabilistic transition, the “big” measurement problem of how a superposed “state” collapses to some welldefined eigenstate of an observable, is a “pseudo-problem”. Despite some remarkable exceptions (Bell, 1987, p. 119), Mott’s approach of the track and self-identity of particles have usually been overlooked. It has also triggered many misunderstandings (Broyles, 1993; Marolf, 1994), which reveal how difficult it is to recognize the exclusively probabilistic status of the quantum formalism. Broyles and Marolf both wish to explain the track phenomenon in terms of wave mechanics, or pure unitary quantum mechanics; and they both acknowledge Mott’s pioneering work in this field. However, they also claim that something is missing in Mott’s approach. Broyles, for instance, considers that, in Mott’s approach, “the questions still remain how a wave function with a broad extent collapses to a track, and what causes the probability distribution of observed tracks to be proportional to the magnitude squared of the incident wave”. Yet, as it should be clear by now, Mott did not “fail” to answer these questions, for the simple reason that he did not treat them as questions at all. The first question, about the collapse, was made irrelevant by Mott’s choice to process an entangled wave function throughout, and by his using such overall wave function as a purely predictive device (for calculating the probabilities of the events), rather than a descriptive device (for giving a symbolic counterpart of the discontinuous events themselves). As for the second question, about the reason why there is a relationship between ψ-waves and probabilities, it did not even arise in Mott’s approach, for, according to the latter, there is not a real wave on the one side, and a probability assessment that has to be derived from it on the other side; there is a wave function whose only role is to afford probabilities, a wave function which is probabilistic from the outset. Broyles and Marolf are thus somehow closer to Darwin than to Mott. Indeed, Darwin was still trying to explain the selection of one term among many, within the overall superposed wave function, by a procedure that has been later compared with decoherence (Figari & Teta, 2013). Instead, Mott tended to dissolve the measurement problem altogether by means of a purely probabilistic interpretation of the wave function.

7.1.5 Track-Like Predictions and Presumed Self-Identity I will now propose a reading of Mott’s account of tracks in cloud chambers which entirely dispenses with the two concepts of extended wave and persistent particle. This reading combines the deeply first-personal probabilistic interpretation of quantum mechanics proposed by QBism with the time-oriented analysis of perception developed by phenomenology. It thereby offers an agent-centered conception of self-identity; a conception based on the structure of the agent’s expectations rather than on some intrinsic feature of pre-given particles. Along with this reading, the spherical -wave that emanates from the radioactive material does not “describe” the α-rays in any way. It rather expresses the agent’s

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anticipation of future phenomena, based on knowledge of her past choices and past observations. The spatial extension of the -wave thus expresses nothing else than the spatial distribution of the agent’s expectation of potential phenomena; in other (Kantian) terms, the spatial extension of the -wave is the a priori form of the agent’s experimental future-oriented receptivity. As for the local amplitude of the -wave, it expresses the intensity of the agent’s predisposition to bet in favor of the appearance of a detection phenomenon at the corresponding point, or in the corresponding volume. It is crucial to this reading that not only waves, but also particles, are not said to be described by the -functions. But then, what is the connection between this way of understanding the quantum formalism and the observed tracks, namely the observed particle-like appearances of water droplets trails in a cloud chamber? Since this connection is not one of description, it must be one of presupposition. Even though a -function does not describe the features of a “real” particle, its variables are the coordinates of a model-system previously composed (in the agent’s mind) of a model-particle plus a model-set of ionizable atoms. Inserting these coordinates in the argument of the -function associated with the cloud chamber, is a way to mentally presuppose that a persistent particle may ionize atoms there. And submitting this -function to evolution through a Schrödinger’s equation whose Hamiltonian depends on particles and fields variables, is a way to mentally presuppose that this particle moves in a certain energy (or potential) landscape. Then, the probabilities calculated by applying the Born rule to this time-evolved -function show the distribution of events that can be expected when this highly elaborated presupposition is made. At the end of the process, the observed phenomena are compared to such expectations, thus enabling one to test the presupposition. But what is remarkable in the quantum paradigm is that, since the agent’s expectations of particle-like behavior of finite sequences of phenomena are purely probabilistic, testing them in various experimental contexts may yield a continuous scale of agreement ranging from strict agreement to complete disagreement. Even in a particle-presuppositionfriendly experimental context like the cloud chamber, the outcome is not entirely in favor of the persistent particle model, since the sequence of ionized atoms and droplets is discontinuous, and these droplets are not perfectly aligned. This is why quantum phenomena can never be anticipated under the presupposition of a single model (such as the model of persistent particles), but only under the presupposition of a mix of mutually exclusive models (such as Goyal’s models of persistence and non-persistence). Mott’s wave-mechanical approach of the α-ray tracks in a cloud chamber is perfectly compatible with this purely presuppositional, rather than descriptive, status of the model of persistent particles. In this approach, each ionized atom, and each droplet, of the track is by no means treated as a direct manifestation of the sudden transformation of a wave into a particle, as it would be in Heisenberg’s multiple collapse approach. Instead, it is treated as the actualization of one of the possible phenomena that are probabilistically predicted by way of a -function whose structure and evolution convey the (mental) presupposition of a model of persistent particles. By extrapolation, a sequence of ionized atoms, and droplets,

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loosely distributed along a straight line is not treated as the direct manifestation of a “real” particle’s trajectory. Instead, it is treated as the actualization of a coherent set of consecutive phenomena, probabilistically predicted by way of a -function whose structure and evolution convey the (mental) presupposition of a combination of kinematical models (persistence and non-persistence, with a dominance of the persistence model). In every case, the model of persistent particles remains in the background (or in the backstage) of the probabilistic formalism of quantum mechanics, rather than being thrown onto the stage of some objective process which the theory allegedly represents; and it is denied any exclusivity, since it is combined, though to a small extent, with the model of non-persistence. As we have seen earlier, a consequence of the background presupposition of the predominance of a model of persistent particles is that the probability of observing a series of ionized atoms (and droplets) in the cloud chamber vanishes unless they are approximately aligned. But this global prediction, that bears on the whole track rather than on each of its points, does not require the assumption that a particle has materialized at each point by way of a series of real wave-packet reductions. This is what makes Mott’s purely probabilistic approach of the α-ray tracks, pushed to its ultimate consequences by the QBist interpretation of quantum mechanics, so closely akin to the phenomenological conception of self-identity. Here, as in Husserl’s analysis, self-identity is not found ready-made in nature. It is supposed in advance, and actively achieved, by the (highly elaborated) mental expectations of a subject, and by the coordinated experimental interventions of an agent. Here, as in Husserl’s analysis, achieving the self-identification of a putative entity requires the synthesis of a succession of appearances under its concept. In the phenomenological theory of perception, the appearances to be synthetized are the aspects or profiles tentatively ascribed to an extended body. Instead, in microphysics, the appearances to be synthetized are the ionized atoms and droplets (or bubbles) found in a cloud (or bubble) chamber and tentatively ascribed to a particle. Here, as in Husserl’s analysis, one tends to constantly aim at a single object for the sake of predicting “its” next manifestations. In the phenomenological theory of perception, such putative object is (mentally) “retained in grasp” while future aspects or profiles are anticipated in the horizon of present aspects of profiles, and later pop up one after another in experience. In microphysics, the (mental) model of some persistent particle is maintained as a component of each scientist’s background presuppositions. This mental model determines to a certain extent the probabilistic anticipations of future ionized atoms (and droplets), and it is repeatedly tested by the appearance of new ionized atoms (and droplets). In both cases, once again, the same process of “explicative synthesis” under the background presupposition of a selfidentical entity, is performed as the succession of phenomena (profiles or droplets) gradually unfolds. To sum up, in microphysics as in phenomenology, self-identity arises as the focus of a mental act of presupposition, synthesis, and explication of a single objectconcept maintained throughout the development of what is tentatively taken for “its” phenomenal manifestations. Self-identity is mentally posited (or withdrawn), rather than found ready-made out there.

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7.1.6 On Suspending the Presupposition of Self-Identity Since the model of self-identical and persistent particles looks so contrived, since it is based more on our intellectual inheritance than on a fully convincing body of evidence, is it possible to dispense with it altogether? Arenhart and Krause (2014, 2019) remind us that an alternative view has been available for some time now: the quantum field theory (QFT), with its Fock space formalism. Redhead and Teller (1992) thus “propose a shift to the Fock space formalism, in which one keeps only the number of particles in each situation, but no labels or counting would remain”. Among other significant features, the Fock space formalism involves creation and annihilation operators. However, such “creations” and “annihilations” do not concern persistent self-identical particles, but rather abstract units (quanta) to be added to (or subtracted from) the overall number of shadowy quasi-particle-like entities. How does the Fock space formalism work? What are its similarities and differences with the formalism of standard quantum mechanics (SQM)? Overall, the Fock space formalism works exactly in the same way as the formalism of standard quantum mechanics, namely as a structured system of probabilistic anticipations, or tendencies to bet, about phenomena. It bears another crucial similarity with standard quantum mechanics: the dependence of those phenomena to the experimental devices in the context of which they are observed; in short, their contextuality. The concept of quantum field is indeed extrapolated from a classical concept of field through the imposition of relations of commutation (or anti-commutation) to certain couples of field variables, which thereby become field “observables”. And these relations of commutation (or anti-commutation) translate the mutual exclusion of experimental contexts into a mutual exclusion of the values measured in such contexts. But besides these similarities, there are also differences that are highly significant for the issue of self-identity. In QFT, the probabilistic anticipations (and tendencies to bet) are based on the presupposition of a field that can be excited a whole number of times, whereas in SQM, they are based on the usual presupposition of an ensemble of self-identical (yet indiscernible) particles bearing various properties. The state vector in Fock space of QFT can thus be understood (in the spirit of QBism) as a bet on the number of detections to come, not as a bet on the number or properties of allegedly existent/(more or less) persistent particles. Unlike the literal existence of particles, the number of detections is highly contextual, as illustrated by the Unruh effect (Crispino et al., 2008). And unlike the number of particles which is pregiven, the number of detections is predicted by means of state vectors which can be superpositions of eigenstates corresponding to several eigenvalues of the observable “Number” (Dunningham et al., 2011). In such situation, the formal concept of particle is shattered (since it has lost the last remnants of its self-identity), and the word “particle” itself thus becomes little more than some verbal wreckage of an archaic conceptual system. As Teller (1995, p. 105) pointed out, “states [in Fock space] simply characterize propensities for what will be manifested with what probability under various activating conditions. Among

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the items for which there can be propensities for manifestation is the occurrence of various numbers of quanta[ . . . ]”. Accordingly, both concepts of “quantum state” and “observable” acquire a new meaning in quantum field theory. In QFT, a quantum state is a predictive structure derived from a presupposed quantum field model, with its potentially detectable excitations (quanta), instead of being a predictive structure derived from a presupposed particle model, with its potentially detectable properties. In QFT, a quantum state serves to evaluate the probabilities of events that are said to coemerge from an interaction between the presupposed quantum field and a device framed under the presupposition that its number of excitations can be measured. It no longer serves to evaluate the probabilities of events that are said to co-emerge from an interaction between the presupposed particles and a device framed under the presupposition that their “properties” can be measured. Also, in quantum field theory, a certain observable denotes the set of possible natural integers that can arise under appropriate detection and counting conditions. In more familiar terms, the observable “Number” denotes a potential number of future detections expressing the excitations (quanta) of a presupposed quantum field. By contrast, under the standard presupposition of a particle model, an observable denotes a set of possible values (be they real numbers or integers) that can arise under conditions that are deemed to allow the measurement of particles’ properties. From SQM to QFT, a complete turnabout of our ontological commitment has then occurred. What (in classical theories and SQM) was presupposed as substrates of properties, namely (the discernible or indiscernible) “particles”, has acquired the status of a collective quantitative property of specific quantum fields in QFT. From the standpoint of the history of philosophy, this ontological turnabout amounts to a shift from Aristotle’s metaphysics to a neo-Platonic view. The Aristotelian scheme of a plurality (individual) substances endowed with their essential or accidental properties is indeed overturned in favor of the neo-Platonic scheme of a unique universal instantiated several times (Alvarado, 2020). In the first case, individual substances are taken as the permanently existing entities that can bear variable properties, whereas in the second case what truly exists is a set of immutable universals (or “ideas”, or generic features) whose concretions in the manifest world take the form of individuals that “participate” in it. The quantum field theoretical scheme is even radicalized with respect to its neo-Platonic counterpart, since the instantiations (the quanta) of the underlying universal (the field) are a far cry from being engaged in a process of individualization. In the same way as the Mott-QBist analysis of identity through space-time is closely akin to Husserl’s phenomenological analysis of perceptive identityascription, the QFT inverted ontological scheme has an experiential counterpart. But this experiential counterpart was upheld by somehow marginal actors in the field of phenomenology: mostly Max Scheler (1923), Nishida Kitarô, respectively a German philosopher and a Japanese philosopher of the twentieth century. In Scheler’s phenomenological approach, the question to be resolved bears on how the feeling and concept of one’s individual consciousness crystallizes within “(...) the background of an omni-encompassing and ever more vague consciousness

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in which our own existence and the experience of others are in principle presented as included together”. To understand how the sense of individuality emerges out of a generic experiential background, Scheler provides us with a pictorial view of the most likely process of this emergence: “[ . . . ] What is given is a stream of immediate experience, undifferentiated into ‘mine’ and ‘thine’, which effectively contains our own experience and that of others in a state of indistinction and reciprocal fusion. Within this flow are gradually formed ever more stable vortices, which slowly draw additional elements into their circle and are thereby identified successively and very gradually with distinct individuals”. Vortices in an oceanic flow is Scheler’s metaphorical account of the process of formation of individuals selves out of a universal substrate which is nothing else than “(undifferentiated) immediate experience”. As for Nishida Kitarô, he bases his philosophy on the phenomenology of Zen meditative states which yield pre-personal forms of consciousness. And he then translates this phenomenological background into a new logic (the “logic of place”) that reverts the Aristotelian hierarchy of substance and predicate, by giving the predicate an ontological priority over the substance. As noticed by Masao (1995), “The logic of place is a predicative logic in the radical sense, not a logic of the grammatical subject. [ . . . ] It is not a logic about the act of seeing or of knowing nor is it a logic about that which is seen and known objectively in terms of the grammatical subject; rather, it is a logic of “place,” which is prior to, and the source of, both seeing and knowing and that which is seen and known. It is a subjective or existential logic prior to the opposition of subject and object”. From such anonymous “place”, namely from such a blank experience prior to space-time coordination and to the subject-object distinction, individuals may nucleate, and be thrown in space-time. From the pure predicate of being, (individual) beings may arise. To be sure, this parallel between the ontological turnabout of quantum field theory and the non-aristotelian phenomeno-Logic of Scheler and Nishida is prima facie little more than an analogy. By contrast, the comparison between the QBist probabilistic approach of quantum mechanics and Husserl’s phenomenological description of perception is not merely analogical. Indeed, the pattern of (quantum) probabilistic predictions of experimental phenomena turns out to be an extension and an amplification of the (phenomenological) pattern of expectations of object’s profiles that continuously unfold in perception. Taken together, anticipatory perception and formalized rules of betting represent two layers of a single cognitive operation of minimization of surprise (Friston, 2013): a biological/implicit layer and a rational/scientific layer. Yet, the phenomenology of pre-individual experience may have been not only an illustrative analogy, but also a productive analogy: a historical source of inspiration for the ontological turnabout of quantum field theory. Indeed, it is striking that the earliest ancestor of quantum field theory, namely the wave mechanical theory of quantum gases (Schrödinger, 1926), was indirectly suggested to Schrödinger by his almost contemporary reflection about the universal, pre-individual consciousness which is the central axis of the Indian Upanishads (Schrödinger, 2008).

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7.2 Conclusion In this paper, we have pursued a phenomenological inquiry about how the categories of persistent spatiotemporal continuants endowed with properties are generated within the experience and activity of knowing subjects. This inquiry was based on Husserl’s project of a thorough “genealogy of logic (and set theory)”. We have shown that, even though such phenomenological process of generation of standard ontological categories was successfully extrapolated in the formalism of classical physics, it fails to a very large extent in quantum physics. The crucial point, here, is expressed by the qualification “to a very large extent”. If the generation of standard ontological categories had failed entirely in the quantum domain, it would have been tempting to interpret this failure along with a “realist” philosophy of science. Would a complete failure not have been a proof that the standard entities of classical physics “do not exist”, while other non-standard entities (say fields) “exist” in their stead? By contrast, a partial failure of the phenomenological genealogy of logic and set theory tells us a very different story. It suggests that any ontology that may underly logic and set theory is nothing else than a mental scheme whose limited function is to provide us with a partial guide for betting about the outcome of our actions within an unfathomable (micro-)environment. Such a toy-ontology can be changed along the way without qualms, if needed. At the end of the day, an ontology (be it an ontology of individuals or an ontology of some pre-individual background “field”) is seen to be little more than the structure of a bundle of expectations that instructs in advance the physicist’s actions. More generally, an ontology does not represent the deepest layer of our knowledge about the world, but rather the underlying body of presuppositions with which we steer ourselves to explore more efficiently the unknown.

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Goyal, P. (2019). Persistence and nonpersistence as complementary models of identical quantum particles. New Journal of Physics, 21, 063031. Gurney, R. W., & Condon, E. U. (1928). Wave mechanics and radioactive disintegration. Nature, 122, 439–440. Heisenberg, W. (1930). The physical principles of the quantum theory. University of Chicago Press. Huggett, N. (1999). Atomic metaphysics. Journal of Philosophy, 96, 5–24. Husserl, E. (1973 [1938]). Experience and judgment: Investigations in a genealogy of logic. Routledge and Kegan Paul. Husserl, E. (1978 [1936]). La crise des sciences européennes et la phénoménologie transcendantale. Gallimard. Husserl, E. (1982 [1929]). Cartesian meditations. Martinus Nijhoff. Husserl, E. (2018 [1913]). Idées directrices pour une phénoménologie pure et une philosophie phénoménologique. Gallimard. Jantzen, B. (2020). Ad hoc identity, Goyal complementarity, and counting quantum phenomena. http://philsci-archive.pitt.edu/18487/ Krause, D. (1992). On a quasi-set theory. Notre Dame Journal of Formal Logic, 33, 402–411. Krause, D., Sant’Anna, A. S., & Sartorelli, A. (2005). On the concept of identity in Zermelo– Fraenkel-like axioms and its relationship with quantum statistics. Logique et Analyse, 48, 231– 260. Krause, D., Arenhart, J. R. B., & Bueno, O. (2020). The non-individuals interpretation of quantum mechanics. arXiv:2008.11550 [quant-ph]. Kripke, S. (1980). Naming and necessity. Basil Blackwell. Ladyman, J., & Bigaj, T. (2010). The principle of identity of indiscernibles and quantum mechanics. Philosophy of Science, 77, 117–136. Marolf, D. (1994). Models of particle detection in regions of space-time. Physical Review, A50, 939–946. Masao, A. (1995). The logic of absolute nothingness as expounded by Nishida Kitarô. The Eastern Buddhist (New Series), 28, 167–174. Mott, N. F. (1929). The wave mechanics of α-ray tracks. Proceedings of the Royal Society of London, A126, 79–84. Muller, F. A. & Saunders, S. (2008). Discerning fermions. British Journal for the Philosophy of Science, 59, 499–548. Patoˇcka, J. (1995). Papiers phénoménologiques. Jérôme Millon. Pienaar, J. (2020). Extending the agent in QBism. Foundations of Physics, 50, 1894–1920. Quine, W. V. O. (1976). Whither physical objects? In R. S. Cohen, P. K. Feyerabend, & M. W. Wartofsky (Eds.), Essays in memory of Imre Lakatos (Boston studies in the philosophy of science). Reidel. Redhead, M. L. G., & Teller, P. (1992). Quantum physics and the identity of indiscernibles. British Journal for the Philosophy of Science, 43, 201–218. Reichenbach, H. (1965 [1928]). The theory of relativity and a priori knowledge. University of California Press. Rovelli, C. (2021). Helgoland. Riverhead Books. Saunders, S. (1994). Time and quantum mechanics. In M. Bitbol & E. Ruhnau (Eds.), Now, time and quantum mechanics. Éditions Frontières. Saunders, S. (2003). Physics and Leibniz’s principles. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections. Cambridge University Press. Scheler, M. (1923). Wesen und Formen der Sympathie. Friedrich Cohen. Schrödinger, E. (1926). Zur Einsteinsche Gastheorie. Physikalische Zeitschrift, 27, 95–101. Schrödinger, E. (1935). Discussion of probability relations between separated systems. Proceedings of the Cambridge Philosophical Society, 31, 555–563. Schrödinger, E. (1950). What is an elementary particle? Endeavour, 9, 109–116. Schrödinger, E. (1952). L’image actuelle de la matière. In E. Schrödinger (Ed.), Gesammelte abhandlungen. Verlag der österreichischen Akademie der Wissenschaften, Friedrich Wievweg & Sohn, 1984, Volume 4.

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Schrödinger, E. (1995). The interpretation of quantum mechanics (M. Bitbol, Ed.). Ox Bow Press. Schrödinger, E. (2008). My view of the world. Cambridge University Press. Searle, J. (1969). Speech acts. Cambridge University Press. Sebens, C. T. (2022). The fundamentality of fields. Synthese, 200, 1–28. Teller, P. (1995). An interpretative introduction to quantum field theory. Princeton University Press. Teller, P. (1998). Quantum mechanics and haecceities. In E. Castellani (Ed.), Interpreting bodies: Classical and quantum objects in modern physics. Princeton University Press. van Fraassen, B. C. (1991). Quantum mechanics: An empiricist view. Oxford University Press.

Chapter 8

Open Problems in the Development of a Quantum Mereology And Their Ontological Implications Federico Holik and Juan Pablo Jorge

Abstract Mereology deals with the study of the relations between wholes and parts. In this work we will discuss different developments and open problems related to the formulation of a quantum mereology. In particular, we will discuss different advances in the development of formal systems aimed to describe the whole-parts relationship in the context of quantum theory. Keywords Quantum mereology · Quasiset theory · Quantum indistinguishability · Undefined particle number · Quantum non-separability

8.1 Introduction Mereology deals with the study of the relations between wholes and parts (see for example Tarski, 1969; Leonard & Goodman, 1940; Lesniewski, 1992). In this work we will discuss different developments and open problems related to the formulation of a quantum mereology (Krause, 2012, 2017). In particular, we will discuss different advances in the development of formal systems aimed to describe the whole-parts relationship in the context of quantum theory (Krause, 2017; da Costa & Holik, 2015; Holik et al., 2016; Obojska, 2019; Holik et al., 2012). Since macroscopic physical systems are assumed to be compounded by many quantum systems, understanding the challenges associated to the development of a quantum mereology is of relevance for the problem of explaining the emergence of a macroscopic classical reality out of a microscopic quantum realm. As our work will

F. Holik () Instituto de Física (IFLP-CCT-CONICET), Universidad Nacional de La Plata, La Plata, Argentina e-mail: [email protected] J. P. Jorge Facultad de Filosofía y Letras, Universidad de Buenos Aires, CABA, Argentina Instituto de Filosofía, Universidad Austral, Pilar, Argentina © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_8

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make clear, the notion of “component part” needs to be critically examined due to the peculiar features of quantum systems. This analysis has also implications for the discussion about whether it is possible to describe a macroscopical system—such as a measurement apparatus—as a simple collection of elementary particles, atoms, or molecules, interacting only through a unitary evolution. This point of view lies at the basis of the so-called measurement problem. Its very formulation implicitly assumes that quantum mechanics should be “universal” in that very specific sense and, as such, is an example of reduction. The assumption that everything can be reduced to a simple collection of interacting elementary entities, gathered as a simple whole, partes extra partes, is known to be very problematic. There is a lot of evidence that it is a naive assumption, even when trying to reduce thermodynamics to classical physics by appealing to statistical mechanics. As we will see below, the “partes extra partes” approach seems to be incompatible with the main features of quantum theory. Here is where the interaction between ontological problems and the concepts of practicing physicists play a key role, which cannot be disregarded. When quantum systems are considered in aggregates they can display several features that have no analog in classical physics. In this work we focus in three central characteristics of quantum mechanics which are, from a conceptual standpoint, different. The first one is entanglement: the information of the whole cannot be recovered in terms of that of its parts. This feature gave place to multiple developments that allow to characterize, from a logical point of view, the problem of quantum nonseparability (Holik et al., 2016, 2012). The second one, is that quantum systems of the same kind can be prepared in situations in which they become utterly indiscernible (see for example Omar, 2005; Holik et al., 2020). Quasiset theory is an example of a formal system that allows to capture the idea of collections of entities which are truly indiscernible (French & Krause, 2006). We will revisit it together with Quaset theory, another formal system aimed to deal with the properties of quantum entities. The third one is related to the fact that it is possible to prepare superpositions of states with a different particle number. This is the case, for example, of the coherent quantum states of the electromagnetic field, that have an undefined number of photons. This feature is particularly challenging for the development of a quantum mereology, since it is difficult to capture it formally. We will revisit previous approaches that deal with this challenge (da Costa & Holik, 2015; Domenech & Holik, 2007; Holik, 2014). It is also important to mention that it is also possible to represent an undefined number of components by appealing to the construction of a Fock-space using quasiset theory (Domenech et al., 2008, 2010). In this work we will discuss the above three features under the light of the formal system introduced in da Costa and Holik (2015) and the quantum logical approach presented in Holik et al. (2016, 2012), Domenech et al. (2010). The work is organized as follows. In Sect. 8.2 we analyze the main features of compound quantum systems. Next, in Sect. 8.3, we formally describe compound quantum systems using a quantum logical approach that is based on the convex subsets of the set of quantum states. In Sect. 8.4 we discuss examples of non-standard set-

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theoretical frameworks that are inspired in quantum theory. Even if they are not, strictly speaking, mereologies, it is instructive to discuss them here, because they can be used as a basis for the development of quantum mereologies in the future. In Sect. 8.5 we analyze a logical system that can formally represent quantum systems with an undefined number of components. Finally, in Sect. 8.6 we present some conclusions.

8.2 Compound Quantum Systems Mereology studies the relationships between the whole and its parts. There exist different formal systems for studying this problem in a formal way. In its essence, it can be formulated as a mathematical theory about the whole and the parts. Our problem here is: how to formulate a quantum mereology? In order to attack it, we start first by revisiting the main features of the physics of composite quantum systems. Next, we will review some formal systems developed with the aim of capturing those features in a rigorous way. This is relevant for the discussion about possible ontologies for quantum theory, given that the formal description provides a precise characterization of certain concepts which, in natural language, can only be formulated in an intuitive way.

8.2.1 Quantum States The (pure) state of a quantum system can be formally represented by a vector |ψ in a separable Hilbert space .H. As such, linear combinations of pure states— if properly normalized to unity—give place to new states. If a source produces quantum systems in the state

.

|ψ = α|a + β|b

.

(8.1)

then, if a measurement takes place, we would obtain the outcomes associated to |a and .|b with probabilities .|α|2 and .|β|2 , respectively. Equation 8.1 describes what is known as a superposition state. More generally, states can be given a matrix representation, using the outer product:

.

ρ = |ψψ|

.

(8.2)

In the matrix representation, the probabilities can be computed using the Born rule: .pa = tr(ρ|aa|) = |α|2 and .pb = tr(ρ|bb|) = |β|2 .

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The state represented by Eq. 8.1 (or, equivalently, Eq. 8.2) should not be confused with the one represented by the matrix: ρinc = |α|2 |aa| + |β|2 |bb|

.

(8.3)

Alike .ρ, the state .ρinc is an incoherent mixture between .|aa| and .|bb|. As such, it gives the same probabilities when we measure a and b, but will yield different probabilities if other experiments are performed. Thus, .ρ and .ρinc represent different physical states.

8.2.2 Entanglement In case we have two distinguishable quantum systems involved (such as a proton and an electron, which can be distinguished, for example, by their charges and rest masses), if the first system is prepared in state .|a and the other is prepared independently in state .|b, the state of the compound system can be represented by: |ψ = |a ⊗ |b ∈ H1 ⊗ H2

.

(8.4)

The fact that states of compound systems must be constructed using the tensor product, is an independent postulate in standard quantum mechanics. According to the rules of quantum theory, it is also possible to form a superposition |ψ = α|a ⊗ |b + β|a   ⊗ |b 

.

with a and b different from .a  and .b . Such states can be prepared using suitable interactions, as is the case when a pair of photons is emitted after a laser beam targets certain non-linear crystals. It turns out that such a superposition cannot written as a product state (i.e., a state of the form .|ψ1  ⊗ |ψ2 ). When this is the case, it is said that the system is entangled. Numerous experiments suggest that the information about an entangled state cannot be recovered out of the information contained in its parts. This fact gives place to the following slogan: Slogan 1 THE WHOLE IS NOT EQUAL TO THE SUM OF ITS PARTS. Most of the extant literature addressing the problem of developing a quantum mereology, focus solely in the study of the features of quantum theory related to entanglement.

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8.2.3 Indistinguishability But entanglement is no the end of the story. If the quantum systems considered are of the same class, as is the case, for example, when we have two electrons or two photons, we must use the symmetrization postulate, and the states must be symmetrized, according to whether the systems belong to two (and only two) classes: Bosons: .

1 |ψ = √ (|a ⊗ |b + |b ⊗ |a) 2

(8.5)

1 |ψ = √ (|a ⊗ |b − |b ⊗ |a) 2

(8.6)

Fermions: .

At the moment of writing this work, there are no other known types of quantum systems in nature. All empirical evidence suggests that there exist only two general classes of elementary quantum systems. The symmetrization postulate has a lot of consequences. Among them, the most known is, perhaps, the Pauli exclusion principle (Marton et al., 2013). But Bose Einstein condensates have become very relevant as well. The symmetrization postulate entails that Slogan 2 QUANTUM SYSTEMS OF A SAME KIND ARE INDISTINGUISHABLE. Of course, the slogan must be taken carefully. If an electron on earth is created in a completely independent way from an electron in a distant star (i.e., an electron coming from a space-like separated region), no non-trivial correlations can take place among them. There is no wave function overlap nor entanglement among them, and, for all practical purposes, they can be considered as distinguishable (Omar, 2005). In other words, considering them as distinguishable, works as a reasonable approximation, since all correlations originated in the symmetrization of the quantum state are negligible. But the “for all practical purposes” should not lead to confusion, due to the following fact: quantum indistinguishability is operating anyway, because, should the electrons undergo an interaction process (some causal event connecting them), a physical situation might take place in which there exists no operational way to determine which one is which. This is a purely quantum feature, in the sense that it has no classical analogue. Two classical particles can be, in principle, always distinguished by a suitably designed experiment. There will always exist an empirical procedure allowing us to tell which is which, even if, under certain circumstances, it may be extremely difficult to do so. On the contrary, quantum mechanics forbids the definition of an operationally well behaved way of distinguishing. In this sense, from the ontological standpoint, quantum systems cannot be considered as individuals in the usual sense. They lack “identity cards”,

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given that there exist situations in which there is no way to retrieve their identities. Taking this into account, one could say that any identity or labeling attributed to a quantum system is mock or fake. Of course, one can imagine that quantum systems have hidden identities, to which one cannot access (as is the case, for example, in some versions of Bohmian mechanics). But, if quantum theory is correct, there exist situations in which this assumption has no correlate with any empirical procedure. This is a distinctive feature of quantum theory, given that, even if nothing prevents us to assume that quantum systems are individuals, they can be prepared in states in which any labels attributed to them must be essentially hidden. Due to the above reasons, the standard formulation of quantum theory assumes that quantum systems are indistinguishable in an deep ontological sense. This is the intuition that guides working physicists when dealing with quantum systems of the same kind. As we will see below, this intuition can be formulated in a rigorous way using a non-standard set theoretical framework. Before we continue, it is very important to make the following remark: Remark 1 Entanglement and indistinguishability are not the same thing and should not be confused. Equations 8.5 and 8.6 look like entangled states. But no genuine entanglement can be present due solely to the symmetrization of a state. As an example, for Fermions, if we want two have non-null entanglement, we need to create a superposition of two Slater determinants (and Eq. 8.6 only contains one). The technicalities behind this fact are far beyond the scope of this article. For our purposes here though, it is important to keep in mind the following: Remark 2 Indistinguishability is, besides superposition and entanglement, an independent and crucial physical feature of quantum systems. The assumption that quantum systems can be in situations of utter indiscernibility, leads naturally to the derivation of the so called quantum statistics. It can be explained in terms of counting: if one has two Bosons and two states, if they were distinguishable, there would exist four possibilities. But, in the properly quantum regime, there is no sense in saying that Boson A is in state 1, and Boson B is in state 2. Since they are indistinguishable, those classical alternatives become one whenever the correlations originated in the symmetrization of the state cannot be neglected. Therefore, for two Bosons and two states, there are only three alternative configurations (and not four, as in the classical case). Similarly, due to the Pauli exclusion principle, one cannot place two Fermions in the same state. Therefore, for the case of two Fermions and two states, there is only one physically meaningful configuration in the quantum regime.

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8.2.4 Undefined Particle Number The reader who is not familiar with quantum theory might feel a little bit exhausted. Not only the whole is not equal to the sum of its parts but, besides that, some component parts can be indistinguishable in a deep ontological sense. Yet, there is still another crucial physical feature that plays its role when we deal with compound quantum systems: it is possible to prepare quantum systems in a superposition of different particle number states. If we take quantum physics literally, this implies that the number of components can be undefined in an ontological sense. To illustrate this, consider first a state of two Bosons: 1 |2 = √ (|a ⊗ |b + |b ⊗ |a) 2

.

(8.7)

(notice the symmetrization). Next, consider a state formed of three Bosons: 1 |3 = √ (|a ⊗ |b ⊗ |c + |b ⊗ |a ⊗ |c + |a ⊗ |c ⊗ |b+ 6

.

|c ⊗ |b ⊗ |a + |c ⊗ |a ⊗ |b + |b ⊗ |c ⊗ |a) Now, according to the rules of quantum theory, nothing prevents us of forming a coherent superposition between the above states: |ψ = α|2 + β|3

(8.8)

.

If, given a system prepared in the above state, we perform a particle number measurement, we will detect two particles with probability .|α|2 , and three particles with probability .|β|2 . Following the standard interpretation of quantum theory, a system prepared in a state such as the one given by Eq. 8.8, has no defined number of components. An important example is that of a coherent state: |α = exp

.

|α|2 2

∞  αn √ |n n! n=0

(8.9)

It is important to recall here the difference between a coherent state and a mixture: |αα| = exp

.

|α|2 2

∞  |α|2n n=0

n!

|nn|

(8.10)

Also, squeezed states (Walls, 1983) play a crucial role in quantum information tasks (see Weedbrook et al., 2012 for details): |S = √

1

∞ 

.

cosh r

n=0

√ tanh r

n

(2n)! |2n 2n n!

(8.11)

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The above states are prepared in nowadays laboratories on a daily basis. Out of the considerations of this section, we add a new slogan: Slogan 3 THE NUMBER OF COMPONENTS CAN BE UNDEFINED.

8.2.5 Conclusions of the First Part When quantum systems with multiple components are considered, there might be— at least—three crucial physical features at play: • Entanglement: the system contains more information than that of its parts. • Qantum systems of the same kind are indistinguishable (they can enter situations for which the notions of labeling, re-identification and individuation lack of any well defined physical meaning). • The number of components can be undefined. Most studies about quantum mereology focus in the phenomenon of entanglement. In some previous works, besides entanglement, indistinguishability and undefined particle number have also been considered. In what follows, we will review some of these developments, with the aim of illustrating the challenges that appear in the development of a quantum mereology.

8.3 A Formal Description of the Relation Between the Whole and Its Parts Let us first consider what happens with the propositions associated to classical systems in the compound case. For a classical system, if the subsystems are prepared in states .s1 = (p1 , q1 ) and .s2 = (p2 , q2 ), then, the state of the compound system can be described by the pair .(s1 , s2 ). In this sense, the information of the compound system is just the sum of that if its parts. Any testable proposition about a physical system has the form: “the value of the physical quantity A lies in the interval .”. It turns out that, for classical systems, the set of testable propositions forms a distributive lattice with regard to the classical logic connectives “.∧”, “.∨” and “.¬”. To fix ideas, one can represent those propositions as the measurable subsets of the phase space (according to the Lebesgue measure). In this representation, the connectives “.∧”, “.∨” and “.¬”, are represented by the set theoretical intersection “.∩”, the union “.∩”, and the set theoretical complement “.(. . .)c ”, respectively. The diagram displayed in Fig. 8.1 illustrates the relation between the lattice of propositions of the compound system and those of its subsystems. There exists a map—which is the canonical projection associated to the Cartesian product—that establishes a one to one correspondence between states.

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LCM1 × LCM2

Fig. 8.1 The different maps between .LCM1 , .LCM2 , .LCM1 × LCM2 , and .L. .π1 and .π2 are the canonical projections

@

π1

@ π2 @ @

@

LCM1

LCM2

But the testable propositions associated to quantum systems have a very different structure. To begin with, the standard way of describing quantum systems is essentially probabilistic. Accordingly, in most circumstances, information about a quantum system is expressed in probabilistic terms. Second, besides those properties which can be considered “classical” (such as charge, intrinsic spin and rest mass), we can also make assertions such as: “The value of the observable A lies in the interval .”. In the quantum formalism, these testable propositions are represented by closed subspaces .PA () (or, equivalently, by their associated orthogonal projections). The mathematical representatives of the propositions of quantum systems form an algebraic structure known as orthomodular lattice which, alike its classical counterpart, is not distributive. Here we call .Lv N to that lattice. In this case, the logical connectives “.∧”, “.∨” and “.¬” are represented by the closed linear subspaces intersection “.∩”, direct sum “.⊕”, and orthogonal complement “.(. . .)⊥ ”, respectively.

8.3.1 Improper Mixtures Can we find maps as in Fig. 8.1 for quantum systems? The situation is depicted in Fig. 8.2. Can we find maps .ξ1 and .ξ2 connecting states of the compound system with those of its subsystems? In order to answer that question, we notice that quantum systems can be prepared in mixed states. In order to illustrate that concept, let us consider a compound quantum system. To simplify the description, let us assume that its subsystems are distinguishable (i.e., they are not quantum systems of the LvN

Fig. 8.2 Is it possible to find maps connecting .Lv N , .Lv N1 and .Lv N2 ?

ξ1 ?

LvN1

@ @ ξ2 ? @ @ @

LvN2

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same kind). Assume that the compound system is in the entangled state 1 |ψ = √ (|+ ⊗ |− + |− ⊗ |+). 2

.

Its associated density operator is given by ρ = |ψψ| ∈ Lv N

.

Then, taking the partial trace over the second subsystem, we obtain that the state of the first subsystem is ρ1 = tr2 (ρ) =

.

1 (|++| + |−−|). 2

But .ρ1 is not a pure state, since it is written as a non-trivial convex combination of .|++| and .|−−|. Then, it is not possible to describe it as an element of .Lv N . This is a clear expression of the fact that the whole is not the sum of its 1 parts. It is therefore not possible to use partial traces to map the properties of the compound quantum system to the properties of its subsystems, given that states of the subsystems can be mixed, and therefore, they do not have representatives as elements of the lattice of properties. Let us dig a little bit on this feature of the quantum formalism. Actual properties are defined as those which are associated with testable propositions whose probability of occurrence equals one. If we restrict to pure states, the conjunction of all actual properties defines the state of the system (see for example Aerts, 1981) in the following sense: {s} =

.

 {X ∈ Lv N | X is actual}

(8.12)

The above Equation provides the connection between states and properties: a (pure) state can be considered as the collection of all properties that are true at a given time. But, since states of subsystems are, in most cases, unavoidably mixed, it is no longer possible to use Eq. 8.12 to establish a connection between states ad properties. This fact precludes the possibility of finding a correct way to connect the states of the whole with the states of its parts (at the level of the lattice of propositions associated to a physical system). A possible solution is to extend the notion of proposition associated to a quantum system to probabilistic assertions, such as: “The system has property E with probability p”. Thus, one can define something like: CE (p) = {ρ | tr(ρE) = p}

.

(8.13)

It is possible to show that if E is a quantum effect, .CE (p) is a convex subset of .C (Holik et al., 2012). Thus, consider the following set (that defines the collection of all possible density operators representing quantum states): C = {ρ|ρ = ρ † ; ρ ≥ 0; tr(ρ) = 1}.

.

(8.14)

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The faces of .C form a lattice (in a canonical way) which is isomorphic to the orthomodular lattice of projection operators (see for example Beltrametti et al., 1984). As an extension, in Holik et al. (2012), it was proposed to consider a lattice formed by convex subsets of .C. The operations are defined as follows. Definition 1 .LC := {C ⊆ C | is a convex subset of C} In order endow this set with a lattice structure, we introduce the following operations in .LC (where .conv(A) describes the convex hull associated to a given set A): Definition 2 For all .C, C1 , C2 ∈ LC ∧ ∨ .¬ .−→

C1 ∧ C2 := C1 ∩ C2 C1 ∨ C2 := conv(C1 , C2 ). It is convex and it is contained in .C. contained ⊥∩C .¬C := C .C1 −→ C2 := C1 ⊆ C2

.

.

.

.

The new lattice has the following properties: • Contraposition and non-contradiction hold: C1 −→ C2 ⇒ ¬C2 −→ ¬C1

.

C ∧ (¬C) = 0

.

• Double negation does no longer holds. Therefore, .LC is not an ortholattice. • .LC is a lattice that contains all convex subsets of the space of quantum states. • It includes a copy .Lv N (its elements represented as faces of the convex set of quantum states). • All possible quantum states are propositions of the form .{ρ} ∈ LC (because each point of .C is a quantum state). By appealing to the extended lattices described above, the relationship between a quantum whole and its parts can be expressed as described in Fig. 8.3. Using the extended lattices, it is possible to define different canonical maps between .LC1 , .LC2 , .LC × LC , and .LC . The impossibility of reducing the information of the whole to 1 2 that of its parts can be expressed in formal terms by the nonequivalence between the following functions (Domenech et al., 2010; Holik et al., 2012, 2013) (Fig. 8.3): ◦ τ = τ ◦

.

(8.15)

The above equation should be clear by itself: it means that going from the whole to the parts, and then going from the parts to the whole, will not have the same information as the reversed operation. In short: going down and going up, is not the same as going up and going down.

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Fig. 8.3 The different maps between .LC1 , .LC2 , .LC1 × LC2 , and .LC . .π1 and .π2 are the canonical projections

LC

τ1 π LC1  1

@ @ τ2 @ τ Λ @ @ ? π2 LC1 × LC2 LC2 6

The mathematical structures described in this section allow for a formal description of the connection between the properties of the whole and the properties of its parts. In the next section, we focus on the description of collections of quantum entities from the point of view of set theoretical structures.

8.4 Non-standard Set Theories When we deal with physical (or mathematical) objects, it is natural to describe them in terms of collections. Therefore, it is natural to assume that a mereology must have a built-in way of dealing with collections of entities or parts. But collections of quantum entities must be handled with care. To fix ideas, one can think in the collection of natural numbers, or the collection of planets in the solar system. In those examples, the components of the collection have well defined identities: each planet or number can be unambiguously distinguished from others. Also, the membership relationship is sharply defined: the number two is either even or not. Therefore, it belongs or not to the collection of even numbers. There is no third possibility. As explained above, quantum objects can enter situations in which they can be considered as truly indistinguishable. Therefore, the way in which electrons form a collection, should not be the same as in the examples of planets and numbers—at least, from a formal standpoint. Even the notion of membership has been questioned in the quantum domain: up to which point can we clearly define a membership relation for the collection of the electrons with spin up in a given material? The formal systems described below aim to capture these quantum features in a formal way, providing a rigorous formulation for the notion of a collection of quantum entities. Of course, these formal systems can be considered as the basis of a non-standard mathematics, inspired in quantum theory.

8.4.1 Quasisets and Quasets In order to formally capture the peculiar features of quantum systems, two different set theoretical frameworks were proposed: QST (Quaset theory) and .Q (Quasiset theory). Though both theories are inspired in quantum theory and they share some

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structural similarities, they are quite different with regard to the way in which they depart from .ZF . QST was developed by María Luisa Dalla Chiara and Giuliano Toraldo di Francia (Dalla Chiara & Toraldo di Francia, 1993). It is formulated in a first order language with identity and its main characteristic is that it relies on an additional primitive membership notion in its non-logical axiomatic layer: “. ∈”. The axioms are set in such a way that .x ∈ y can be interpreted as “x is not a member of y with certainty”. This new connective, differently from ZF, is independent of “.∈”. The following well formed formula of QST is not valid: .¬(x ∈ y) ←→ (x ∈ / y). The direct implication of this fact is the existence of undetermined membership relations. If a given quaset y admits elements whose membership status is undetermined, then, there exists an element x such that .[¬(x ∈ y) ∧ ¬(x ∈ y)]. This is the main feature that distinguishes QST from ZF. The proponents of this formal framework suggested that this feature could be related to the notions of quantum uncertainty and superposition states. On the other hand, Quasiset theory (.Q from now on) was proposed by Krause (1990, 1992), and subsequently developed by other authors (French & Krause, 2006; Krause et al., 1999; French & Krause, 2003). Differently from QST, its first order metalanguage does not includes identity as a primitive. It aims to describe collections of truly indiscernible objects, for which the notion of identity cannot be applied. Therefore, it is not possible to write formulas such as .x = y for some elements of the theory. For the so called m-atoms, we can only appeal to an indiscernibility relation “.≡”. Two m-atoms can be indiscernible, but still count as two.1 In this way, a formal description of the idea of collections of objects that are truly indiscernible (non-individuals) is obtained. It is a non-standard set theory (a variant of the Zermelo–Fraenkel system), but it is not a mereology strictu sensu. Notice that the membership relationship “.∈” has a different meaning than that of “being a part of. . . ” (which we will denote by “.” below). If .x, y represent quantum entities of the same kind (i.e., if they are m-atoms of the same kind), then, the formula “.x ≡ y” is interpreted as x is indiscernible from y. Thus, even if they are identical in all their attributes, they are not the same. In this sense, indiscernibility is weaker than identity, but the notions coincide when applied to entities that belong to a certain classical domain. Intuitively, if .x, y are variables meant to represent electrons, then, the formula .x = y is not well formed, while .x ≡ y expresses the fact that those entities are indiscernible. In this way, one can consider them as different “solo numero” (i.e., only numerically discernible). As a consequence, the Principle of Identity of Indiscernibles is challenged. .Q provides a formal framework to deal with collections of indiscernible items without the usual tricks of confining them into equivalence classes or non-rigid (deformable) structures: in quasi-set theory one can speak and treat those items without those standard tricks and with logical precision. In what follows, we describe the axioms of these non-standard set theories with some detail.

1 Notice

that, in a standard set theory, if .x = y, then, there are no two objects, but one.

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8.4.2 Brief Description of the Non-logical Layer of QST and Q In this section, we briefly describe the elementary basis of Quaset and Quasisets. The non-logical language of QST includes the following primitives: 1. A monadic predicate (ur-object or ur-element) “O”. 2. Three binary predicates: positive membership “.∈”, negative membership “. ∈”, and the inclusion relationship “.⊆”. 3. A unary functional symbol: the quasicardinal “qcard”. 4. A binary functional symbol: quasets intersection “.”. Definition 8.3 A quaset is something that is not a ur-object: Q(x) := ¬O(x)

.

Axiom 8.1 If something has an element, then, it is a quaset: ∀x

.

∀y (x ∈ y −→ Q(y))

Axiom 8.2 If we know with certainty that something does not belongs to a quaset, then, it is not true that it belongs with certainty to the given quaset. But the converse is not granted in general: ∀x ∀y (x ∈ y → ¬(x ∈ y))

.

The previous axiom implies that not all instances of the principle .((x ∈ y) ∨ (x ∈ y)) are true, and then, there exists the possibility of undetermined membership relations. On the other hand, the notion of “certainly knowing” can be interpreted as “belongs with certainty”—if one wishes to avoid an epistemic interpretation of the theory. As is well known, quantum mechanics admits both, ontological and epistemic interpretations of the uncertainty principle. Axiom 8.3 The inclusion relationship “.(⊆)” is a partial order (i.e., it is a reflexive, symmetric and transitive relationship). The symbol .⊆ has an intensional meaning but, in general, it lacks an extensional one. Also, “.x ⊆ y” can be understood as “concept x implies concept y”. Axiom 8.4 Quasets inclusion implies extensional inclusion. ∀x

.

∀y (x ⊆ y −→ ∀z ((z ∈ x −→ z ∈ y) ∧ (z ∈ y −→ z ∈ x)))

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In QST, conjunction (or intersection) is a primitive notion. This is denoted by the symbol . and satisfies: Axiom 8.5 “.” represents the weak conjunction of quasets. It coincides with the usual intersection when restricted to sets: ∀Q x ∀Q y ((x  y ⊆ x ∧ x  y ⊆ y) ∧ (Z(x) ∧ Z(y) → x  y = x ∩ y))

.

Notice that the axioms of quasets do not demand that there exist proper quasets (i.e., quasets which are not sets). From an intuitive point of view, the qextension of a proper quaset does not represents a suitable semantic counterpart for the usual notion of extension. As an example, we mention that the qextension of a quaset— even with a cuasicardinal greater than zero—could be empty. Let us now focus on .Q. The primitive non-logical symbols are a binary predicate ‘.≡’, which stands for ‘indiscernible’, or ‘indistinguishable’; three unary predicates ‘Z’, ‘m’ and ‘M’, which have the following intuitive interpretation: .Z(x) says that x is a set, a copy of an entity of ZFU; .m(x) says that x is a m-atom, which is intended to represent quantum systems (be they considered quantized particles or field excitations); .M(x) says that x is an atom of ZFU. The theory still has a derived notion of extensional identity, symbolized by ‘.=E ’, which is defined only for Matoms that belong to the same quasi-sets or quasi-sets having the same elements. Let us call .Q to a first order theory whose primitive vocabulary contains, beyond the vocabulary of standard first order logic without identity (propositional connectives, quantifiers, etc.), the following specific symbols: (1) three unary predicates m, M, Z, (2) two binary predicates .∈ and .≡, (3) one unary functional symbol qc. Notice again that identity is not part of the primitive vocabulary, and that the only terms in the language are variables and expressions of the form .qc(x), where x is an individual variable, and not a general term.2 The intuitive meaning of the primitive symbols is given as follows: (i) (ii) (iii) (iv) (v)

x ≡ y (x is indiscernible from y) m(x) (x is a “micro-object”, or an m-atom) .M(x) (x is a “macro-object” or an M-atom) .Z(x) (x is a “set”—a copy of a ZFU set) .qc(x) (the quasi-cardinal of x) . .

The underlying logic of .Q is a non-reflexive one, a logic where the standard theory of identity is limited in some way; in our case, by restricting the application of the very predicate of identity; see also French and Krause (2006), Arenhart (2014).

2 This

restriction prevents that, for example, .qc(qc(x)) end up being a term.

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8.5 The ZF  System and the Problem of Describing an Undefined Number of Components Both, QST and .Q, are set-theoretical frameworks. This means that they are meant to describe collections of certain entities that have quantum features. The formal analysis of collections is important for the goal of developing a quantum mereology, but it is not enough. QST and .Q lack of an explicit definition of one of the key features of any mereology, namely, the notion of part. A proper quantum mereology was introduced in da Costa and Holik (2015). The proposed system has several interesting properties: it can describe, at the same time, indistinguishability and an undefined number of components. As a by-product, it allows to describe superpositions as a form of logical undecidability. From now on, we refer to the system introduced in da Costa and Holik (2015) as .ZF ∗ . We give here a brief description focused in the ontological aspects of the problem, and refer the reader to da Costa and Holik (2015) for a detailed description of its axiomatics.

8.5.1 Undefined Number of Components as Logical Undecidability Let us first introduce the primitive symbols of .ZF ∗ : • • • •

First order logic symbols with “.∈” (membership). “.C(. . . )” (in such a way that “.C(X)” reads “X is a set”). “.”, in such a way that “.X  Y ” reads “X is a part of Y ”. The theory contains sets and physical things. The latter are not sets, that is, they are not collections. A definition of indistinguishability can be introduced as follows: α ≡ β := α  β ∧ β  α

.

(8.16)

Alternatively, it could be introduced following the same strategy as in .Q (i.e., based in a relationship such as “.≡”). Using the above, it is possible to give the following definitions. Given a well formed formula F , the notion of a collection of entities satisfying that formula can be defined as: ∃{x | F (x)} := (∃y)(∀x)(x ∈ y ←→ F (x))

.

(8.17)

We will also need the notion of a Cantorian entity: Cant (α) := ∃{β | β  α}

.

(8.18)

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The above equation says that a given entity is Cantorian whenever we can describe its parts as a well behaved collection. In case that a given entity is Cantorian, we can formally assign to it a number of components, as the cardinal associated to the collection of all its parts: (α) := ({β | β  α})

.

(8.19)

If .α satisfies that .¬Cant (α), then, there is no way to assign a cardinal to it. Therefore, as a consequence of the axioms of the theory, non-Cantorian entities might exist. For them, the number of components will be undefined. The reader can already notice that here, undefinitness, is associated to the logical impossibility of granting the existence of an object in a logical system. In a certain sense, one can represent something which is assumed to actually happen in the real world (i.e., the existence of physical systems for which the number of components is not defined, such as those described by Eqs. 8.9 and 8.11) as a limitation of a formal system.

8.5.2 Undefined Things As a byproduct of the above discussion, we can obtain a more general result. If .α is such that .¬Cant (α), given the formula .F (x), it is not possible—using the axioms of .ZF ∗ —to grant the existence of the set: αF = {β  α | F (β)}

.

(8.20)

As a consequence, we cannot, in general, describe the components that satisfy certain properties as a well behaved collection. This gives place to undefined properties in the following sense: if .α is non-Cantorian, we cannot assert that its parts have the property defined by .F (x), neither that it doesn’t has it. This type indeterminancy in the possession of a property can be used to describe the situation originated when a quantum system is prepared in a superposition state, such as: .

1 √ (| ↑ + | ↓) 2

(8.21)

According to the standard formulation of quantum theory, we cannot tell whether the system has spin up or down, because that property is undefined previous to measurement. The .ZF system can thus give a formal description of one of the most important concepts of quantum theory, namely, superposition states. How to understand undefined things? How to formalize the idea that something is undefined (in a strong ontological sense)? The message of .ZF ∗ can be summarized as follows: INDETERMINANCY ⇐⇒ UNDECIDABILITY

.

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We think that the above double implication opens a very interesting line of philosophical inquiry. Up to which point can we relate the notions of quantum indeterminancy and logical undecidability? The deepest intuitions that guide quantum physicists rely on the idea that there is an intrinsic randomness in nature. Notice that this guiding idea underlies, for example, the whole industry of building commercial quantum random number generators (i.e., of using quantum systems as sources of true randomness (Ma et al., 2016)). Due to the assumption of true randomness, it is not possible to ascribe results of observations to properties possessed by the system previous to measurement, because the results of future experiments are simply not defined before their concrete instantiation. In that sense, those properties are undefined. In this way, logical undecidability—or indefiniteness— could be used as a conceptual tool to give a formal representation of the intuitions of working physicists. The .ZF ∗ system is a first step in this direction. We end this section by noticing that this type of analysis is motivated by questions that go well beyond the philosophy of physics: a deeper understanding of the connections between randomness, undefinitness and logical undecidability, could have direct implications in physics and mathematics, in the fields of random numbers generation and randomness certification.

8.6 Conclusions As we have shown, the study of the relationship between the whole and its parts poses special challenges for the philosophy of quantum theory. When quantum systems are presented in aggregates, we can find non-separability, indistinguishability and an undefined number of components. While most studies focus in the problem of non-separability, we have discussed here different formal systems which allow to describe these notions in a way that help to overcome the ambiguities of natural language. It follows that it is possible to give a rigorous formal representation of many of the intuitions underlying quantum theory. The notions of indistinguishability, the impossibility of reducing the whole to its parts, and the possibility of having an undefined number of components, can be captured in a formal way. The formalization of the peculiar features that quantum systems display when considered in aggregates is relevant for a better understanding of the connections between the macroscopic and microscopic levels of reality. Our analysis suggests that the picture of a macroscopic object as a simple “partes extra partes” aggregate of small localized entities is highly misleading. Having this into account is crucial in the analysis of the so-called measurement problem, given that it assumes implicitly that a measurement apparatus can be described as a simple aggregate of elementary systems evolving unitarily. This “reductionist” point of view has been shown to be problematic in many other areas of research, and cannot be naively implemented in the quantum domain either. The analysis presented in this work reinforces the need

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of overcoming the measurement problem with a different non-reductionist strategy, adopting a critical re-examination of the notion of “part” in the quantum domain. We have also discussed the connection between quantum inteterminancy and logical undecidability. In other words, we have suggested that logical undecidability could be used as a tool to formalize many of the intuitions underlying quantum theory. In particular, we have pointed out the relevance of this ideas in the context quantum random number generation and certification. The formal treatment of the intuitions underlying quantum indeterminancy could be of help for a deeper understanding of the concept of true randomness.

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Krause, D. (2017). Quantum mereology. In H. Burkhard, J. Seibt, G. Imaguire, & S. Gerogiorgakis (Eds.), Handbook of mereology (pp. 469–472). Munchen: Philosophia Verlag. Krause, D., Sant’Anna, A. S., & Volkov, A. G. (1999). Quasi-set theory for bosons and fermions: Quantum distributions. Foundations of Physics Letters, 12, 67–79. Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. Journal of Symbolic Logic, 5(3), 113–114. Lesniewski, S. (1992). On the foundations of mathematics. In J. Surma, J. T. Srzednicki, D. I. Barnett, & V. F. Rickey (Eds.), Collected works (pp. 1927–1931). Dordrecht: Kluwer Academic Publishers. Ma, X., Yuan, X., Cao, Z., Qi, B., & Zhang, Z. (2016, June). Quantum random number generation. npj Quantum Information, 2(1), 16021. Marton, J., Bartalucci, S., Bertolucci, S., Berucci, C., Bragadireanu, M., Cargnelli, M., Curceanu, C., Di Matteo, S., Egger, J.-P., Guaraldo, C., Iliescu, M., Ishiwatari, T., Laubenstein, M., Milotti, E., Pietreanu, D., Piscicchia, K., Ponta, T., Romero Vidal, A., Scordo, A., ..., Zmeskal, J. (2013, July). Testing the Pauli exclusion principle for electrons. Journal of Physics: Conference Series, 447, 012070. Obojska, L. (2019, October). The parthood of indiscernibles. Axiomathes, 29(5), 427–439. Omar, Y. (2005, November). Indistinguishable particles in quantum mechanics: An introduction. Contemporary Physics, 46(6), 437–448. Tarski, A. (1969). Foundations of the geometry of solids. Oxford: Oxford University Press. Walls, D. F. (1983, November). Squeezed states of light. Nature, 306(5939), 141–146. Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N. J., Ralph, T. C., Shapiro, J. H., & Lloyd, S. (2012, May). Gaussian quantum information. Reviews of Modern Physics, 84, 621–669.

Part II

Logic and Formalism of Indiscernibility

Chapter 9

Identity and Quantification Otávio Bueno

For Décio Krause, who inspires all of us to look deeper and further.

Abstract This work examines a number of arguments to the effect that quantification requires identity of the objects that are quantified over; the arguments concern the domain of quantification, the range of quantifiers, the collapse of the existential and the universal quantifiers, and the intelligibility of quantification. The central role of identity in quantification is identified in each case. Also considered is quantification in non-classical contexts, and it is argued that even in logics and set theories that supposedly do not demand identity for quantification, identity is still presupposed. Along the way, some recent challenges to this overall approach are considered. Keywords Quantification · Identity · Indistinguishability · Quasi-set theory

9.1 Introduction Does quantification require the identity of the objects that are quantified over? Is quantification intelligible without such identity? In this paper, I examine a number of arguments in favor of identity’s fundamentality in the context of quantification, and argue that classical quantification (that is, quantification in classical logic and set theories) requires identity. (Fundamentality is not used here in any metaphysically robust sense.)

O. Bueno () Department of Philosophy, University of Miami, Coral Gables, FL, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_9

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In particular, I consider four arguments: (a) the arguments from the domain of quantification, according to which domains of quantification, whether set-theoretic or not, require the identity of the objects that are quantified over; (b) the argument from the range of quantifiers, according to which the specification of the range of quantifiers also requires the identity of those objects over which one quantifies; (c) the arguments from the collapse of the existential and the universal quantifiers, according to which in order to prevent that the universal and the existential quantifiers have the same inferential properties the identity of the objects that are quantified over is required. (d) the arguments from the intelligibility of quantification, according to which without objects that have well-specified identity conditions, quantification becomes incoherent. In each case, I identify the crucial role that the identity of the objects that are quantified over plays in quantification. I also consider quantification in non-classical contexts, and argue that even in logics and set theories that allegedly do not demand identity for quantification, identity is still required. Along the way, I respond to some recent challenges to this approach. In discussing identity, my primary concern is with the identity of the objects quantified over, and the way in which their identity dovetails with quantification. I argue that both in classical and in non-classical contexts, identity is required for quantification to be implemented. It is the identity of the objects that needs to be in place so that quantification can be applied. Before proceeding, I should note that by ‘identity’ I mean an equivalence relation for which substitutivity holds. I do not think that substantive metaphysical assumptions are required in order for this concept to be used—although it is common in the philosophical literature to add metaphysical principles to identity and then claim that they are part of the concept itself. Items such as criteria of identity, individuation conditions, and necessity of identity all come to mind in this connection. It seems to me that it is important to distinguish identity from these metaphysical additions. Identity can, and should, be understood in metaphysically more deflationary ways (see Bueno, 2014; Arenhart et al., 2019).

9.2 Quinean Roots In his review of Peter Geach’s Reference and Generality (Geach, 1962), W.V. Quine (1964) discusses Geach’s doctrine according to which identity is a relative notion: ‘x = y’ is meaningless except relatively to a parameter, that is, ‘x and y are the same F’. As Quine notes:

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This doctrine is antithetical to the very notion of quantification, the mainspring of modern logic. Quantification depends upon there being values of variables, same or different absolutely; grant quantification and there remains no choice about identity, not for variables. For a language with quantification in it there is but one legitimate version of ‘x = y’ (Quine, 1964, p. 101).

Rather than providing arguments for these claims, Quine simply states them. In what follows, I will offer a number of arguments to support the need for identity in quantification. To make sense of relative identity (‘x and y are the same F’ rather than just ‘x = y’), identity (‘the same F’) is presupposed—on pain of an unwanted regress. In ‘the same F’ either identity is relativized or it is not. If it is not relativized, then relative identity ultimately assumes un-relativized, absolute identity. If in the ‘the same F’ identity is relativized, relative to which property is identity supposedly restricted? Suppose that G is such a property. In that case, ‘F is the same G’ would, in turn, invoke identity. If identity, nevertheless, is relativized in this expression, we are off to a regress. In order to prevent it, un-relativized, absolute identity would be assumed, once again. Perhaps ‘x and y are the same F’ could be expressed in terms of predication as: (Fx ∧ Fy). However, this only states that x and y are Fs, not that they are the same F. Maybe the following characterization works better: .

(F x ∧ F y) ∧ ∀z (F z → (z = x ∨ z = y)) .

Unfortunately, it does not. First, this formula expresses that there are at most two Fs, not that they are the same F. Second, un-relativized, absolute identity is used on the second conjunct (on pain of a regress). Perhaps one could try with the following expression: .

(F x ∧ F y) ∧ ∀X ((Xx ∨ Xy) → X = F ) .

To a certain extent, this gets closer to expressing that ‘x and y are the same F’. But, once again, un-relativized, absolute identity is invoked in the second conjunct (otherwise, a regress is inevitable). Finally, maybe we could have simply: .

((F x ∧ F y) ∧ x = y) .

Nonetheless, in this case, un-relativized, absolute identity is explicitly invoked. The issue is not ultimately about identity’s absoluteness. It is whether the content of what needs to be expressed can be properly expressed without identity. In the end, it is just unclear that there is such a thing as relativized identity. There is identity, and it can be used to express identity claims that, as a matter of expressive choice, can be restricted. But this is not a restricted kind of identity, any more than a soccer player is a restricted kind of human. There are humans, and some of them are soccer players. There is identity, and it can be used restrictedly to certain properties. Thus,

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for the reasons just discussed, in ‘x and y are the same F’, the expression ‘the same F’ invokes identity, and not relativized identity. Given that identity is at issue, can it be dispensed with in the context of quantification? In what follows, I will consider a number of arguments to the effect that it cannot.

9.3 Arguments from the Domain I start by examining the connections between quantification and its domain, given the need to specify over which things quantifiers range. This generates two kinds of arguments from the domain, depending on whether set theory is assumed or not. As will become clear, the arguments go through in either case.

9.3.1 The Set-Theoretic Version Quantification requires a domain of things over which one quantifies. This domain is typically a set. In classical set theories, extensionality, of course, holds: x = y if, and only if, ∀z (z ∈ x ↔ z ∈ y) .

.

As a result, identity is assumed in order to characterize the domain of quantification as a set. The identity of a set that is formed from certain objects can only be specified given the identity of these objects. If members of a set A lacked identity, it would be indetermined whether they or something else also belonged to another set B, with the result that it becomes indetermined whether A and B are the same set or not. After all, by extensionality, two sets are the same just in case they have the same members. Since quantification is then implemented over a set (the domain of quantification), identity is assumed. Thus, there is no quantification without identity.

9.3.2 Non-Set-Theoretic Versions Quantification requires a domain of things over which one quantifies. This domain, however, need not—in fact, should not—be a set. Requiring a set as the domain of quantification moves the focus away from what matters: the things that are being quantified over rather than the set of these things. One needs things to be quantifed over, not a set of them. Having a set is just a convenient, but dispensable, device to specify the range of quantification. After all, this range can be identified by simply determining the things over which one quantifies. Instead of the set of Fs, that is, {x: x is F}, all that is needed to single out the domain of quantification are the Fs.

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Nonetheless, each F needs to be distinct from one another, and thus identity has to apply to them. Clearly, the last claim needs support, and it will be offered in terms of an explicit argument. I will discuss each premise in turn. (P1 ) Quantification requires counting. Only certain things can be quantified over (i.e., can be in the range of a quantifier, although they are not taken to be a set), namely, those that can be counted. In particular, it is inappropriate to use a quantifier over mass terms. It is not by chance that statements such as ‘Each water is liquid’ or ‘Every water is transparent’ are ungrammatical, since a count noun is required by each quantifier. If quantification did not demand counting, those statements would not have violated grammar. Note that by introducing a count noun over which the quantifier ranges, the statements become perfectly acceptable. Consider, for instance, ‘Each body of water is liquid’ or ‘Every body of water is transparent’. This results from the fact that quantification ranges over things that can be counted. Quantifiers, after all, deal with quantities of some sort: all, some, most, etc. (P2 ) Counting requires distinguishability (to prevent double counting). Suppose that things a and b can be counted. In this case, there is a one-to-one mapping from each of these things, a and b, to the first two natural numbers, 0 and 1. (This follows immediately from the assumption that these objects can be counted.) As a result, a and b can be distinguished via this mapping: one object is assigned to 0, while the other to 1. So, the assumption that certain things can be counted requires their distinguishability. (P3 ) Distinguishability requires identity. There are two ways of supporting this premise, depending on the use (or not) of Leibniz’s principles of identity. (i) The easy way: assume Leibniz’s principle of the indiscernibility of identicals: .

(x = y → ∀P (P x ↔ P y)) .

If things are distinguishable, they are not the same. For there is some property P that distinguishes (or can distinguish) them. (ii) The hard way (without assuming Leibniz’s principles of identity): If two things are distinguishable, then there is something that distinguishes them (or can distinguish them). Can these things still be the same (be identical to one another)? If they are the same, they will be distinct identical things—a contradiction. If they are not the same, they are just distinct things, yet identity is applicable to them, since they are not the same. In either case, identity is still required. It may be argued that, in some contexts, the question of the identity of distinguishable things cannot be raised, say, because (in)distinguishability is a primitive notion that does not presuppose identity (see French & Krause, 2006). In this case,

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an artificial restriction in the language in use has just been introduced, even if it has been motivated by the need to express the corresponding restriction as it is found in certain formulations of quantum mechanics. After all, prima facie, the question of the identity of any object can always be legitimately raised—which does not mean that we will always know the answer to it. Thus, we have the first (non-set-theoretic) version of the argument from the domain: (P0 ) Quantification requires a domain (the things that are quantified over, which are not taken to be members of a set). (P1 ) Given a domain, quantification requires counting (in the sense that a quantifier ranges over things that can be counted). (P2 ) Counting requires distinguishability. (P3 ) Distinguishability requires identity. Hence, quantification requires identity. Two conditions are needed to specify properly the range of quantifiers: Completeness: Quantifiers range over each object of the domain, so that all of the objects in the domain are quantified over. Adequacy: No object distinct from those in the domain is in the range of the quantifiers. The completeness condition guarantees that all objects in the domain are included in the range of the quantifiers. The adequacy condition assures that no object distinct from those in the domain is included in the quantifiers’ range. Nonetheless, both conditions require the identity of the objects in the domain of quantification. With regard to Completeness, quantification cannot be implemented over things to which identity does not apply, since as argued, quantifiers range over things that can be counted, which, in turn, requires identity. Moreover, Completeness requires that each object in the domain could, in principle, be quantified over. In order that quantifiers range over all such objects, no object distinct from those already quantified over can be excluded. It is not just that each object falls under the scope of the quantifiers; each object different from—that is, not identical to—those already quantified over needs to be included as well. Regarding Adequacy, in light of the demand that no object distinct from those in the domain be in the range of the quantifiers, identity is also required, given that, clearly, an object distinct from another is not identical to it. Someone may complain that the formulation of the adequacy condition above is inadequate. After all, adequacy can be expressed without identity: Adequacy*: No object that is not in the domain of quantification is in the range of the quantifiers; or, alternatively, only those objects in the domain are in the range of the quantifiers. As a result, the argument goes, quantification is enough to formulate the adequacy condition. In the end, identity is not required for that.

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Despite appearances to the contrary, Adequacy* ultimately presupposes identity. In full generality, for an object not to be in the domain of quantification, it needs to be different from those that are in the domain; otherwise, it would be just one of those objects. As a result, the specification of the range of quantifiers ends up requiring identity. Thus, we have the second (non-set-theoretic) version of the argument from the domain: (P0 ) Quantification requires a domain (the things that are quantified over, although not a set of them) and a range (the scope of the quantifiers). (P1 ) The range of quantifiers needs to be complete and adequate with respect to the domain of quantification. (P2 ) Completeness requires identity. (P3 ) Adequacy requires identity. Therefore, quantification requires identity. This argument explores the connection between the domain of quantification and the range of quantifiers, and the need to ensure that they are properly aligned to one another. If set theory is invoked, this need is addressed by identifying that the range of quantifiers with the set of objects that characterize the domain of quantification. Without recourse to set theory, however, Completeness and Adequacy need to be established—and this, in turn, for the reasons just provided, demands identity.

9.4 The Argument from the Range of Quantifiers Even in classical set theory, considerations of the range of quantifiers provide an additional reason in support of the close dependence between identity and quantification. Consider, for instance, the familiar device, in model-theoretic semantics, of invoking sequences of objects of the domain of quantification in the interpretation of the quantifiers (Mendelson, 1997, pp. 59–60). A sequence s of objects in the domain satisfies a formula ∀xFx as long as every sequence that differs from s in at most its i-th component satisfies F. (A corresponding formulation can be given for the existential quantifier with respect to some sequence that also differs from s in at most its i-th component.) As is well known, such sequences are functions from natural numbers to objects in the domain of quantification, and in order for two sequences to differ in their i-th component, these components need to correspond to different (that is, non-identical) objects in the domain. Clearly this requires the identity of the objects in question. This argument highlights that need for identity in the very formulation of the truth-conditions for quantifiers. The basic model-theoretic devices invoked to characterize such quantifiers and their range demand identity.

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9.5 Non-Individuals There are those who would challenge all of the arguments presented thus far, based on considerations from quantum mechanics (see French & Krause, 2006; Arenhart, 2012, 2014; Krause & Arenhart 2019; for some responses, see Bueno, 2016). On their view, the most natural way of making sense of certain interpretations of non-relativist quantum mechanics is by understanding quantum particles as nonindividuals, that is, things to which identity does not apply. Non-individuals are not blatantly inconsistent things that are non-self-identical, i.e., entities such that x = x. (It is unclear whether entities of this sort exist in the world at all.) Rather, non-individuals are things to which identity fails to apply. As French and Krause (2006) argue, a key motivation to introduce non-individuals emerges from the need to make sense of quantum statistics. In classical physics, given two objects (say, a and b) and two states, four combinations are possible: 1. 2. 3. 4.

[ab] and [ ]: both objects are in the first state; none is in the second. [ ] and [ab]: no objects are in the first state; both are in the second. [a] and [b]: object a is in the first state; object b is in the second. [b] and [a]: object b is in the first state; object a is in the second.

Assuming that each state is equally probable, the probability of each combination is then ¼. In quantum physics, the situation is different. In light of the indistinguishability of quantum particles (e.g., electrons), swapping them with one another does not change the quantum state that they are in. Given two quantum particles and two quantum states, the following combinations are possible: (1 ) (2 ) (3 ) (4 )

[xx] and [ ]: both objects are in the first state; none is in the second. [ ] and [xx]: no objects are in the first state; both are in the second. [x] and [x]: one object is in the first state; one object is in the second. [x] and [x]: one object is in the first state; one object is in the second.

Given the indistinguishability of quantum particles, states (3 ) and (4 ) are not distinct. Thus, assuming each state is equally probable, there are only three possible combinations, and the probability of each of them is 1/3. But how can one quantify over these particles given their lack of identity? How is quantification over nonindividuals possible? An important response consists in the development of a non-classical set theory in which such quantification is allowed for: quasi-set theory (French & Krause, 2006). This is a mathematical theory that does not require that identity be applied to all of its objects. Instead of identity (=), a weaker relation of indistinguishability (◦ ) is introduced: it is an equivalence relation. The first three axioms of quasi-set theory state just that (French & Krause, 2006, p. 277): (A1 ) ∀x (x ◦ x) (A2 ) ∀x ∀y (x ◦ y → y ◦ x) (A3 ) ∀x ∀y ∀z ((x ◦ y ∧ y ◦ z) → x ◦ z).

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The theory is then formulated so that it allows for the construction of collections of non-individuals, such as indistinguishable particles in non-relativist quantum mechanics, without the assumption, inherent in classical set theories, that their members have identity. However, in the axioms above, the variables ranging over quasi-sets ultimately require identity, since they play the same semantic role in each axiom. Reflexivity, symmetry and transitivity, as expressed in (A1 ) to (A3 ), cannot be properly characterized if the same object is not referred to by the variable x in each occurrence.

9.6 Arguments from the Collapse of the Existential and the Universal Quantifiers Suppose we are (somehow) quantifying over non-individuals, e.g., some quantum particles. Suppose one of them is F, say, Fa. Then, ∃x Fx. So, we have the inference: Fa, therefore ∃x Fx. But since non-individuals are indistinguishable, and identity doesn’t apply to them, if a non-individual a is F, then all non-individual indistinguishable from a are also F. In this case, we have the inference: Fa, therefore ∀x Fx. However, this inference is typical of the existential quantifier, not of the universal one. Similarly, if ∀x Fx holds (for non-individuals), then Fa. Thus, we have the inference: ∀x Fx, therefore Fa. Consider, now, that ∃x Fx holds (for some nonindividuals). Since non-individuals are indistinguishable, and identity does not apply to them, if ∃x Fx holds for some non-individuals, then all non-individual indistinguishable from them are also F. Thus, ∀x Fx, and hence, Fa. As a result, we have the inference: ∃x Fx, therefore Fa. Yet, this inference is typical of the universal quantifier, not the existential one. Given these arguments, the following inferences hold for non-individuals: Fa, therefore ∀x Fx ∃x Fx, therefore Fa. Hence, we have the collapse of the existential and the universal quantifiers. Note that this collapse only emerged due to the fact that identity was assumed not to hold among non-individuals. The validity of the inferences above is blocked in the presence of identity. Fa does not guarantee that ∀x Fx, since an object different from a may not be F. ∃x Fx does not guarantee that Fa, since a may be different from the object that guarantees that ∃x Fx, and a may not be F. In response, it may be objected that the collapse only holds because quantifiers are being restricted to non-individuals. In the case of non-individuals—since they are indistinguishable—what holds for one also holds for all, and this results in the apparent collapse of the quantifiers. Nevertheless, there is in fact no such collapse, any more than there would be one if we restricted quantification to domains with only one object, and concluded that if something is F, then everything is F! In

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particular, identity is not required to resist the alleged collapse of quantifiers. After all, Fa doesn’t guarantee that ∀x Fx since some object, say, b, may not be F. Moreover, ∃x Fx does not guarantee that Fa since a may not be F. In neither case is identity required. Quantification beyond non-individuals is enough to achieve these results. The analogy with single-object domains does not go through, though. The collapse takes place even if quantification is not restricted to non-individuals, but also includes individuals. It is the case that non-individuals behave in rather unusual ways, given their indistinguishability. Despite that, note that although nonindividuals of the same kind (say, electrons) are indistinguishable from one another, they are supposedly different from non-individuals of distinct kinds (say, protons). But to assert that electrons are different from protons, since they have different charges, requires that both such non-individuals have identity conditions, which is ruled out by their formulation. (The claim is that electrons and protons are different, not just that their equivalence classes are.) So, non-individuals would need to have identity conditions in the end. But, given their characterization, this claim is incoherent. Moreover, identity is indeed required to resist the quantifiers’ collapse. Suppose that the domain of quantification is not restricted to non-individuals, but includes both individuals and various kinds of non-individuals. To resist the collapse, it was argued that Fa does not guarantee that ∀x Fx, for some object b may not be F. But for the argument above to go through, b has to be different from a, otherwise one would end up with a contradiction (namely, that Fa and not-Fa). However, this requires that a non-individual a have identity conditions—so that it can be different from b—as opposed to what the characterization of a non-individual demands. Furthermore, to block the collapse of the quantifiers, it was also argued that ∃x Fx does not guarantee that Fa, since a may not be F. Suppose then that it is possible that a is not-F. If a were F, the inference would not be blocked, since in this case both the premise (∃x Fx) and the conclusion (Fa) would be satisfied. Thus, to block the inference and avoid the quantifiers’ collapse, the object that guarantees that ∃x Fx needs to be different from the object that is not-F. Nonetheless, once again, identity is required for that.

9.7 The Arguments from the Intelligibility of Quantification In light of the considerations advanced so far, is quantification without identity intelligible? It is not clear that it is. I will offer four arguments to this effect, building up from the points that have already been made. (i). Suppose that, in a non-set-theoretic setting, one is quantifying only over non-individuals—and, thus, identity does not apply to them. What is the range of the quantifiers in this case? The adequacy condition involved in the specification of the quantifiers’ range requires that no object distinct from

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those in the domain is in the range of the quantifiers. As it turns out, this is incoherent, given what non-individuals are. Identity, after all, does not apply to them. However, it is required to apply so that the range of quantification is suitably characterized. (ii). Perhaps the domain of quantification for non-individuals could be specified by a set of them. Clearly, such a set cannot be classical, since classical sets presuppose identity due to the extensionality axiom. But the domain of quantification for non-individuals could be a quasi-set (French & Krause, 2006). This means that the domain is specified by a quasi-set of indistinguishable non-individuals. Nevertheless, identity is ultimately assumed in the formulation of the axioms of quasi-set theory. Consider, for instance, the axiom of reflexivity: .

(A1 ) ∀x (x ◦ x) .

Since the variable x in this axiom needs to play the same semantic role, so that the same object is referred to in each side of the indistinguishability relation, identity is indeed required. Given that identity cannot apply to non-individuals, once again we have an incoherence. (iii). As noted, according to quantum mechanics, electrons and protons are different. In order to state this fact about quantum particles requires the application of identity to them. However, once again, this would be incoherent, given that they are non-individuals. (iv). Suppose there are three electrons in an apparatus. One cannot specify that one of them is the first, another is the second, and yet another is the third, for electrons are non-individuals and cannot be individuated. (So, the collection in question has a “cardinal” but no “ordinal”; see French & Krause, 2006; Arenhart, 2012, 2014) Nevertheless, it is unclear how to make sense of quantification over indistinguishable electrons without taking them to be distinct. To quantify over all indistinguishable non-individuals (of a certain kind), quantification over each of them is required (so that ∀x Fx entails Fa, for each a). But this, in turn, requires counting—since quantifiers range over things that can be counted—which ultimately, as argued, requires identity. The result, once again, is incoherence.

9.8 Conclusion For the reasons discussed above, quantification is closely tied to identity: (a) The domain of quantification requires identity in classical set theory, due to the extensionality axiom. (b) The domain of quantification also requires identity in nonclassical set theory, such as quasi-set theory, since the formulation of the axioms of quasi-set theory invokes variables that need to have the same semantic role. (c) The domain of quantification requires identity in order to specify the range of the

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quantifiers without set theory, so that distinct objects in the domain are covered. (d) Quantification requires identity of the objects that are quantified over, otherwise the existential and the universal quantifiers collapse into one another. Finally, (e) the intelligibility of quantification requires the identity of the objects that are quantified over. I started with Quine, and I end in a somewhat Quinean note with the slogan: No quantification without identity! Acknowledgements It is a pleasure to dedicate this paper to Décio Krause, who for years has patiently heard and carefully responded to my concerns about identity and quantification while we collaborated on a number of related projects. He has been throughout a model of intellectual honesty, curiosity and insight, insisting on the importance of questioning deeply held assumptions and not taking anything for granted. His work is endlessly stimulating. On so many occasions, I randomly opened a page of Identity in Physics (which he wrote with Steven French) and after reading just a few paragraphs had to stop to reflect on and engage with the abundance of fresh and perceptive ideas that his work is invariably so full of. A great friend and wonderful philosophical interlocutor, he inspired me and generations of philosophers and logicians to do better and go further—and I very much look forward to our many discussions, collaborations and exchanges ahead. For particularly helpful conversations on the issues examined in this paper, my thanks go to Ali Abasnezhad, Jonas Arenhart, Jody Azzouni, Newton da Costa, Steven French, Norbert Gratzl, Andreas Kapsner, Décio Krause, Chris Menzel, Lavinia Picollo, and David Ripley.

References Arenhart, J. R. B. (2012). Many entities, no identity. Synthese, 187, 801–812. Arenhart, J. R. B. (2014). Semantic analysis of non-reflexive logics. Logic Journal of the IGPL, 22, 565–584. Arenhart, J. R. B., Bueno, O., & Krause, D. (2019). Making sense of nonindividuals in quantum mechanics. In O. Lombardi, S. Fortin, C. López, & F. Holik (Eds.), Quantum worlds: Perspectives on the ontology of quantum mechanics (pp. 185–204). Cambridge University Press. Bueno, O. (2014). Why identity is fundamental. American Philosophical Quarterly, 51, 325–332. Bueno, O. (2016). Identity in physics and elsewhere. In Cadernos de História e Filosofia da Ciência 2/1, Série 4 (pp. 93–105). French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford University Press. Geach, P. (1962). Reference and generality: An examination of some medieval and modern theories. Cornell University Press. Krause, D., & Arenhart, J. R. B. (2019). Is identity really so fundamental? Foundations of Science, 24, 51–71. Mendelson, E. (1997). Introduction to mathematical logic (4th ed.). Chapman and Hall. Quine, W. V. O. (1964). Review of Peter Geach, reference and generality. Philosophical Review, 73, 100–104.

Chapter 10

On the Consistency of Quasi-Set Theory Adonai S. Sant’Anna

Abstract Quasi-set theory .Q is a first order theory which allows us to cope with certain collections of objects where the usual notion of identity is not applicable, in the sense that .x = x is not a formula, if x is an arbitrary term. The terms of .Q are either collections or atoms (empty terms who are not collections), in a precise sense. Within this context, .Q is supposed to be some sort of generalization of ZFU (ZF with atoms). Strangely enough, no one published any permutation model of quasi-set theory until now, although this theory is already 30 years old. In this paper I introduce a class of permutation models of .Q defined as .Q − {Axiom of Choice and all axioms involving quasi-cardinality}. I show, for example, that all permutation models of .Q are models of ZFU.−{Axiom of Choice}.

10.1 Introduction A comprehensive book about the so-called “problem of individuality in quantum physics” is Identity in Physics: A Historical, Philosophical, and Formal Analysis, by French and Krause (2006). So, if the reader needs any grasp on the vast literature about this subject, that book is arguably the best starting point, along with its list of references. In this paper I pay special attention to some of the mathematics in French and Krause’s book. In Chapter 7 of that book there is a detailed description of quasi-set theory, which is strongly based on Krause’s seminal paper (Krause, 1992). The main point of quasi-set theory .Q is the use of a first order language, without identity, who is able to replicate standard ZFU in the following sense: part of the universe of discourse of .Q is supposed to be translated as a universe of terms of ZFU (called the classical part of .Q), while part of that universe cannot (called the non-classical part of .Q). In order to accomplish that, two binary primitive

A. S. Sant’Anna () Department of Mathematics, Federal University of Paraná, Curitiba, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_10

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predicates .≡ (indistinguishability) and .∈ (membership) are the starting point for dealing with indiscernible terms (either atoms or collections known as quasi-sets) who are not necessarily identical. Besides, there are two kinds of atoms: microatoms and macro-atoms. Thanks to that, it is possible to define a binary predicate .=E termed extensional identity such that .=E has all the features of standard identity within ZFU. More precisely, if x and y are either both macro-atoms or both quasisets whose transitive closure have no micro-atoms, then .x ≡ y ⇒ x =E y. In other words, a new first order theory of collections (without identity) is able to reproduce a well known theory of collections (with identity). Nevertheless, the motivating aspect of quasi-set theory is the existence of certain terms who are indiscernible (indistinguishable) but not necessarily extensionally identical. There are quasi-cardinals in .Q, who are cardinal numbers defined in the classical part of the theory, and such that the quasi-cardinality of any term in the classical part coincides with the (classical) cardinality of that term. One of the astonishing aspects of the book, however, is that the corresponding Axiom of Choice of quasi-set theory is introduced 11 pages after the introduction of quasi-cardinalities. That is a major problem regarding quasi-set theory, at least from the point of view of its presentation in that book. The rationale used on page 285 of French and Krause (2006) is this: A translation from the language of ZFU to the language of .Q shows there is a ‘copy’ of ZFU within .Q; II Therefore (they say), usual concepts about cardinality in ZFU can be defined within this classical part of .Q. I

What French and Krause failed to see is that the translation mentioned above (without any previous discussion whatsoever about AC) shows there may exist (within .Q) a ‘copy’ of ZFU where the Axiom of Choice simply fails. In that case, there is the possibility that the axioms of .Q will not allow us to define cardinalities in any classical sense. That is why we need a better understanding about quasi-set theory before we intend to use it. Krause and colleagues claim that quasi-set theory provides the mathematics of non-individuality (observe the definite article). Actually, that is precisely the title of Chapter 7 of French and Krause (2006). That claim reveals an obvious intended interpretation of quasi-sets: they are supposed to cope with objects who are not individuals, in the sense that those objects are genuinely indiscernible but not necessarily identical. Actually, according to quasi-set theory, identity is not even applicable among certain terms. To reinforce that statement, on page 296 there is a theorem of unobservability of permutations. To understand its statement, it is important to know that .[z] corresponds to the quasi-set of all terms which are indistinguishable from z; and .z is a quasi-set whose quasi-cardinality is extensionally identical to 1 (the second finite ordinal) and such that .z ⊆ [z]. Thus, that theorem on page 296 states this: Let x be a finite quasi-set (i.e., its quasi-cardinality is a natural number) such that .x =E [z] and z is a micro-atom such that .z ∈ x. If .w ≡ z and .w ∈ x, then there exists .w  such that   .(x − z ) ∪ w ≡ x.

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Obviously, such a statement seems to suggest that .[z] may have a transfinite quasi-cardinality, something which is not said in any of the axioms presented in French and Krause (2006). But that is a minor issue if nothing is wrong about quasi-cardinalities. The major issue, notwithstanding, is the fact that the proof of that theorem relies on axioms about quasi-cardinalities. Therefore, we need a better understanding about the role of the Axiom of Choice and its possible relationship with micro-atoms. The point in French and Krause’s book is the unobservability of permutations of micro-atoms. But my point in this paper is that such an unobservability of permutations is exactly the main feature of any non-trivial permutation model of ZFU. Permutation models of ZFU allow us to define non-trivial .∈-automorphisms on a vast number of possible universes of ZFU. Due to the fact that the Axiom of Extensionality of ZFU does not provide any criterium for the identification of atoms, that is exactly the opportunity to grant the indiscernibility of atoms in ZFU, in a quite classical fashion. In other words, the axioms of ZFU do not distinguish between the atoms. Such a phenomenon of indiscernibility is precisely formulated within permutation models with identity. Thus, indiscernibility among atoms is not any surprise at all, regardless of the framework used, whether it is ZFU or quasiset theory. That indiscernibility is actually the feature which allows us to introduce permutation models where the Axiom of Choice fails and, consequently, it is not possible to define the cardinality of some sets in ZFU (if there is no AC). If there is any intended interpretation of quasi-sets regarding the evasive ‘nonindividuals’, that is something which is stated by French and Krause just at an intuitive level of thought, which is merely speculative but hardly philosophical (let alone mathematical). On the other hand, what can model theory say about quasisets? Did we really get rid of identity, as French and Krause claim in their book? Permutation models have been used since exactly 100 years ago (Fraenkel, 1967; Jech, 2003), to prove the independence of the Axiom of Choice in ZFU. Since quasiset theory states a specific form of AC as one of its postulates, that fact requires special attention. The class of permutation models I introduce here requires just some minor adjustment to the first Fraenkel Model (Jech, 1973), in order to cope with AC.

10.2 Permutation Models The standard references used in this Section are Jech (1973, 2003). Let A be an infinite denumerable (countable) set of atoms. Observe I am not suggesting that quasi-set-theoretical axioms entail there is a denumerable collection of atoms (whether micro-atoms or macro-atoms). I am just proposing a possible class of models of .Q. Since the axioms in Chapter 7 of French and Krause (2006) do not say anything about that, for all that matters, quasi-set theory postulates are consistent with either a proper class of all atoms or a set of all atoms. For the sake of simplification, I am first considering a class of models .Q where there is indeed a

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set A of atoms. At the end of this paper I discuss the possibility of a proper class of atoms, i.e., a collection of atoms which is not a set. If we employ a technique analogous to the construction of a von Neumann universe, we can denote by .P ∞ (A) the universe of sets of a transitive model .M of ZFU, where .P ∞ (A) denotes the hierarchical power of A for all ordinals. That means, for every set x, there is a least ordinal .α such that .x ∈ P α (A). In that case, .α is the rank of x. Besides, for every ordinal .α, we have .P α (A) ⊂ P ∞ (A). Since one of the axioms of quasi-set theory is Regularity, there is no problem adopting that technique for building .P ∞ (A). Our aim here is to build a specific class of submodels .Q of .M and show that every permutation model .Q of a variation .Q of quasi-set theory is a model of ZFU, provided we ignore the Axiom of Choice and all the postulates regarding quasicardinalities in French and Krause (2006). In other words, .Q is the theory .Q − {Axiom of Choice and all axioms involving quasi-cardinality}. The Axiom of Choice introduced in French and Krause (2006) says the choice quasi-set c exists for any quasi-set x whose elements are quasi-sets and such that x satisfies the usual conditions imposed to the standard Axiom of Choice in ZFC. Now, let’s start building our permutation models .Q. Every time we say ‘x is a set’, we mean ‘x is a term from .P ∞ (A)’. Every time we say ‘a is an atom’, we mean ‘a is an element from A’. Let .π be any permutation of A. As in standard literature, for any set x, we can define .π(x), by transfinite induction on the ordinals, as it follows: π(x) = {π(y) | y ∈ x}.

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As a consequence from that definition, we have that .x ∈ y iff .π(x) ∈ π(y). In other words, .π is an .∈-automorphism of the universe .P ∞ (A). Now, let .G be the group of all permutations of A (i.e., the symmetric group of A). Then we can introduce the next definition: Definition 1 A set .F is a normal filter on group .G iff I II III IV V VI

every element of .F is a subgroup of .G; G ∈ F; if .H ∈ F and K is a subgroup of .G such that .H ⊆ K, then .K ∈ F; if .H ∈ F and .K ∈ F, then .H ∩ K ∈ F; if .π ∈ G and .H ∈ F, then .π H π −1 ∈ F; for each atom .a ∈ A, .{π ∈ G | π a = a} ∈ F.

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Usually it is unnecessary to assume .G as the symmetric group of A. But that is useful for our purposes, since it allows us to talk about a vast number of subgroups of .G. Definition 2 Let x be a set from .P ∞ (A). Then, sym(x) = {π ∈ G | π x = x}.

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We say x is symmetric with respect to .G (or simply symmetric) iff .sym(x) ∈ F. Definition 3 A permutation quasi-model .V of .Q is the proper class V = {x ∈ P ∞ (A) | every z ∈ TC({x}) is symmetric},

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where TC denotes transitive closure. I am not using the terminology ‘quasi’, in ‘quasi-model’, in any sense analogous to quasi-set theory. A quasi-set is an ‘almost a set’, in the sense of not employing identity in any explicit way. On the other hand, a permutation quasi-model is an ‘almost a model’, in the sense we need something else to introduce our model .Q of .Q , besides last definition. After all, the framework we are using to build our proposal does not dispense identity. As it is well known, our quasi-model .V is a transitive model of ZFU (ZF with a set A of atoms) (Jech, 2003, 1973). Axiom Q5 of .Q (which is an axiom of .Q as well), on page 278 of French and Krause (2006), says no term is a micro-atom and a macro-atom. That is the main reason why we do not refer to .V is a model of .Q : simply because it is not! Somehow we need to discriminate between micro-atoms and macro-atoms. Our first step towards that is to introduce two new sets, namely, .Am and .AM , such that Am ∪ AM = A, Am ∩ AM = ∅, and III both .Am and .AM are infinite and denumerable. I

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II

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In other words, .{Am , AM } is a particular partition of A which obviously induces an equivalence relation over A. Next we choose a specific subgroup .Gq of .G. Definition 4 .Gq is a quasi-set-theoretical group of permutations iff Gq is a proper subgroup of the symmetric group .G; for every atom .a ∈ AM , and every permutation .π ∈ Gq , we have .π a = a; III for every atom .a ∈ Am , there is permutation .π ∈ Gq where .π a ∈ Am ∧ π a = a; IV for every atom .a ∈ Am , there is no permutation .π ∈ Gq such that .π a ∈ AM . I

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Thus, .π {Am , AM } = {Am , AM } for any .π belonging to .Gq . That means we have an invariance principle, namely, partition .{Am , AM } of A is invariant under the actions of .Gq . We did not need last definition, if we had followed the usual approach, by considering a specific normal filter generated by certain specific subgroups of .G. But we prefer last definition as a pedagogical choice for the sake of our purposes. Now we are able to define the concept of indistinguishability . within the context of our model .Q, which is still under construction.

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Definition 5 Let x and y be either sets from .P ∞ (A) or atoms from A. We say that x is indistinguishable from y, and denote this by .x  y, iff there is .π ∈ Gq such that .π x = y. Since .Gq is a group, obviously there is .π −1 ∈ Gq such that .π −1 y = x, if .π x = y. Some theorems are quite straightforward, as it follows: Theorem 1 If x and y belong to the kernel of .P ∞ (A), then .x  y ⇔ x = y; Theorem 2 If x and y belong to .P ∞ (A) such that no atom from .Am belongs neither to the transitive closure of x nor to the transitive closure of y, then .x  y ⇔ x = y; Theorem 3 If both a and b are atoms belonging to .AM , then .a  b ⇔ a = b; Theorem 4 If a is an atom and x is a set, then .¬(a  x); Theorem 5 If .a ∈ Am and .b ∈ AM , then .¬(a  b); Theorem 6 . is an equivalence relation. Theorem 5 says any element from .Am is distinguishable from any element from AM , where ‘distinguishable’ is the negation of ‘indistinguishable’. In particular, if both a and b are atoms belonging to .Am such that .a  b, then we may have either .a = b or .a = b. That depends on how we build our model .Q. Hence, the idea here is quite simple: the quasi-set-theoretical group .Gq has nontrivial actions only on those atoms from .Am and, consequently, on those sets x whose transitive closure have at least one atom from .Am as one of its elements. If .a ∈ AM we call it a macro-atom. If .a ∈ Am we call it a micro-atom. To complete all necessary ingredients to finish our model .Q, we need some final concepts. .

Definition 6 A family I of subsets of A is a normal ideal iff, for all subsets E and F of A, ∅ ∈ I; if .E ∈ I and .E ⊆ F , then .F ∈ I ; III if .E ∈ I and .F ∈ I , then .E ∪ F ∈ I ; IV if .π ∈ G and .E ∈ I , then .π E ∈ I ; V for each .a ∈ A, .{a} ∈ I . I

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For each set x, let fixGq (x) = {π ∈ Gq | πy = y for all y ∈ x}.

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Obviously, .fixGq (x) is a subgroup of .Gq . Let .Fq be the filter on .Gq generated by the subgroups .fixGq (E), where .E ∈ I . In that case, .Fq is a normal filter and, so, it defines a permutation model we refer to as .Q. Model .Q is our main concern here. So, VQ = {x ∈ P ∞ (A) | x is symmetric with respect to Gq }.

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Now a standard and crucial result. Theorem 7 VQ  (x can be well-ordered) iff fixGq (x) ∈ Fq .

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The proof of last theorem is analogous to what is done in Jech (2003) and sketched in Jech (1973). Proposition 1 .Q = A, P ∞ (A), AM , Am , Gq , Fq , VQ ,  is a model of .Q , i.e., quasi-set theory .Q without the Axiom of Choice Q28 from page 297 of French and Krause (2006) (which was informally stated somewhere above) and without axioms Q18.∼Q26 (all of them involving the concept of quasi-cardinality and presented on pages 285–291 of French and Krause (2006)). To prove that, all we have to do is to exhibit a translation .T from the language of quasi-set theory to the language of our model .Q such that, for any axiom F of .Q , defined as .Q − {Q28 ∧ (Q18 ∼ Q26)}, we have Q  F iff Q  T (F ).

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We read .Q  F as ‘axiom F is true in .Q’. Translation .T is quite natural here. Micro-atoms from .Q are translated as elements from .Am ; macro-atoms are translated as elements from .AM ; indistinguishability .≡ is translated as .; membership .∈ of .Q translates as .∈ in .Q; formula .Z(x) (x is a classical set) translates as x is a set whose transitive closure has no micro-atom as an element. Since in French and Krause (2006) there are sixteen axioms and two axiom schemes regarding .Q , and the available room here is limited, we omit the details of the proof. But that is obviously straightforward, since quasi-set theory is strongly committed to ZFU in the sense of somehow preserving it, at least without the Axiom of Choice and, consequently, without any non-trivial discussion about cardinalities. For example, the first three postulates of .Q say that .≡ is an equivalence relation, something granted by Theorem 6. The essential difference between ZFU and quasiset theory is that in the latter there are two kinds of atoms: micro and macro. In the case of macro-atoms, indistinguishability entails extensional identity (which, by the way, is a simple restriction of standard identity over the class of collections whose transitive closure have no micro-atoms). In the case of micro-atoms, however, non-trivial .∈-automorphisms can take place in any possible universe of discourse proposed as a model of .Q . Next we shall consider what happens with the Axiom of Choice in .Q and if it is possible to introduce a permutation model of quasi-set theory in full, i.e., with all its axioms.

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10.3 The Troublesome Axiom of Choice Following the same notation from last Section, suppose I is the set of all subsets of AM , all finite subsets of .Am , and any union of an arbitrary subset of .AM with an arbitrary finite subset of .Am . In other words, .x ∈ I iff

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x ⊆ AM or x ⊂ Am and x is finite, or III .x = y ∪ z, where .y ⊆ AM , .z ⊂ Am , and z is finite. I

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II

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Therefore, I is a normal ideal, according to Definition 6. Recalling .Am is infinite, that entails the subgroup .fixGq (Am ) does not belong to the filter generated by .{fixGq (E) | E is finite}. In particular, it does not belong to the filter .Fq on .Gq generated by the subgroups .fixGq (E), where .E ∈ I , for that particular normal ideal I just defined. Then, according to Theorem 7, .Am cannot be well-ordered. But, if .Am cannot be well-ordered, then the set x = {y ∈ P(Am ) | y is unitary}

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cannot be well-ordered as well. After all, within the context of our model .Q, there is a one-to-one correspondence .f : Am → x given by .f (a) = {a}. But since the Axiom of Choice in .Q (AC.Q ) grants the existence of a choice quasi-set c for any quasi-set x whose elements are non-empty and two by two disjoint quasi-sets, then such a formula is not true in this particular permutation model .Q (as introduced in Proposition 1). On the other hand, if J is the normal ideal of all subsets of A, the subgroup .fixGq (Am ) does belong to the filter .Fq on .Gq generated by the subgroups .fixGq (E), where .E ∈ J , for that particular normal ideal J . Thus, the Axiom of Choice AC.Q (postulate Q28 in French and Krause (2006)) is true in this new permutation model. Nevertheless, in that case, for any atom a, .π a = a. Besides, for any set x, .π x = x. So, there are no micro-atoms in this permutation model based on the normal ideal J , according to Definition 4. So, within the context of permutation models, quasi-set theory is either inconsistent (due to AC.Q ) or equivalent to ZFU .+ {Axiom of Choice} (provided there are no micro-atoms). In either case, the ‘theorem’ of unobservability of permutations on page 296 of French and Krause (2006) is not true in any permutation model of .Q − {Q28 ∧ (Q18 ∼ Q26)}. As long the proof of that ‘theorem’ depends on a clear concept of cardinalities (in the classical part of quasi-set theory), it seems that its statement is not a theorem at all. It is worth to remark that, in the permutation model produced from the normal ideal I , indistinguishability . shows its relevance (in the sense of non-trivial .∈automorphisms) only in the case where we are dealing with infinite sets of microatoms. That is consistent with an algorithm introduced in Sant’Anna (2005), where all micro-atoms belonging to a finite quasi-set are labeled by classical sets and, therefore, that labeling algorithm defines a well-order on x. Actually, that is not

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surprising at all. It is expected the Axiom of Choice to hold in any finite collection, whether it is a classical set or a quasi-set. If French and Krause wanted to grant cardinalities without the Axiom of Choice, they could try Scott’s trick (Scott, 1955). But that will work only on a standard transitive model based on the von Neumann universe, and heavily grounded on the Axiom of Regularity.

10.4 Extensional Identity In this Section I show there is no surprise at all to produce a permutation model of Q . In order to do that we need some pieces of information. In French and Krause (2006) it is introduced the concept of extensional identity .=E . We say x and y are extensionally identical, and denote this by .x =E y, iff either (i) x and y are both quasi-sets (i.e., terms who are not atoms of any kind) who share the same elements or (ii) x and y are both macro-atoms such that, for every quasi-set z, x belong to z iff y belongs to z. Besides, a given term x is a Dinge, and we denote this by .D(x), iff x is either a macro-atom or a quasi-set whose transitive closure has no micro-atom as an element. Moreover, axiom Q11 (page 279 of French and Krause (2006)) says any terms x and y who are indiscernible Dinge are extensionally identical. Finally, since indiscernibility is reflexive, transitive and symmetrical, and since substitutivity holds for extensional identity (axiom Q4 on page 277 of French and Krause (2006)), then extensional identity works as a restriction of usual identity .= in the next sense: if .U is one possible universe of discourse of .Q (such a universe is endowed with identity .=), then .x =E y iff .x = y and both x and y are Dinge belonging to .U . That is why the definition of extensional identity entails that, for every microatom a, we have .a =E a. Although it seems odd the existence of terms who are extensionally different from themselves, the fact is that extensional identity is just a way to ignore identity on the whole universe of quasi-set theory. But any permutation model is endowed with identity. Thus, although .a =E a in .Q , when a is a micro-atom, we have .a = a in any permutation model. In other words, French and Krause simply used a trick in their book: they introduced a predicate (extensional identity) who just ignores micro-atoms. Well, that can be done in any first-order theory with identity. .

10.5 Can Micro-atoms Be Non-individuals? Consider, for example, the next quasi-set x, where z is a micro-atom: x =E {t ∈ P([z]) | qc(t) = 1}.

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Separation Schema Q13, Power-qset Axiom Q15 and the axioms about quasicardinality in French and Krause (2006) allow us to introduce quasi-set x above, where .[z] is the quasi-set of all terms t such that .t ≡ z, .qc(t) = 1 says the quasi-cardinality of each t belonging to x is the ordinal number 1, and .=E is the extensional identity in .Q. In that case, extensional identity says every term who belongs to x belongs to .{t ∈ P([z]) | qc(t) = 1}, and every term who belongs to .{t ∈ P([z]) | qc(t) = 1} belongs to x. But postulate Q20 guarantees that .qc(x) =E 0. That means there is an ordinal .α = 0 such that .qc(x) =E α. In other words, it is possible to exhibit a model of quasi-set theory such that x is well-ordered, if we interpret .qc(x) as the cardinality of x. Therefore, there is a one-to-one quasi-function .f : x → α in such a model. That means f is a quasi-set whose elements are ordered pairs .t, β, where .β < α (in the sense that .β ∈ α). But the same axiom Q20 guarantees that .[z] itself has a nun-null quasi-cardinality .qc([z]). That fact by itself is enough to distinguish each and every micro-atom from .[z], where we label them with ordinals at least within our model. Since the remaining axioms about quasi-cardinalities preserve the standard cardinal arithmetic, that entails .qc([z]) = α. But if micro-atoms are indiscernible, how can we discern them and discern their correspondent singletons with quasi-cardinality 1 within a model endowed with identity? In other words, what is the real difference between quasi-set theory and ZFU? If quasi-set theory is supposed to grant the existence of non-individuals (as French and Krause claim), what kind of models are we supposed to look for? All of this points towards an idea: maybe a consistent theory of collections of indiscernible objects needs to avoid the Axiom of Choice, since that postulate allows us to define cardinalities. The fact that French and Krause talk about cardinalities before AC brings a lot of suspicion and doubts.

10.6 Final Remarks There are at least two possibilities to reformulate quasi-set theory, if we insist on using permutation models to evaluate some of its possible universes of discourse: I II

To abandon axiom AC.Q ; To rephrase AC.Q as holding just to quasi-sets whose transitive closure have no micro-atoms.

But those ideas would work only if the axioms of quasi-cardinality were completely deleted or, at least, significantly modified. Even if we do that, the mere existence of permutation models compromises the intended interpretation of quasi-set theory as a world who admits the existence of non-individuals. After all, relations of non-trivial indiscernibility are quite abundant among permutation models of ZFU without AC. Nothing new here.

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Another possibility is to try to keep quasi-set theory somehow similar to what is presented in French and Krause (2006) and to admit the possibility of the existence of models of .Q quite different from permutation models. But if that is possible, my question is this: why is there no version of quasi-sets without atoms? Was quasi-set theory conceived just because it is easy to admit the existence of indiscernible and not identical atoms? Regarding models of quasi-set theory with a proper class of atoms, remember it is possible to conceive a permutation group which is a proper class as well. After all, the use of such ‘huge structures’ is fairly common in the literature. Well known examples are .∈-automorphisms (Jech, 2003), non-trivial elementary embeddings of a universe into itself (Kanamori, 2003), and pairing functions on the whole Gödel’s Constructible Universe (Devlin, 1977). Nevertheless, any normal ideal defined from a proper class of atoms is supposed to have only sets as elements (although the normal ideal itself is a proper class). In this sense, there will be no choice function for proper classes of atoms. But the rest of the discussion is analogous to what has been done here for any infinite set of all atoms. In a situation like that, indiscernibility among micro-atoms will be relevant only for proper classes of micro-atoms. Issues about indistinguishability relations do not occur only in set theories with atoms, but also in infinitesimal analysis (Bell, 2008), with the huge advantage of the non-existence of atoms in infinitesimal analysis. That happens due to the intuitionistic logic used in that formal framework. Since intuitionistc logic has been used in quantum theories (Isham & Butterfield, 1998), why can’t we deal with quantum indiscernibility within such a framework? Quasi-set theory was strongly motivated by the problem of indistinguishability in quantum theories (Domenech et al., 2008, 2010; French & Krause, 2006; Krause, 2010; Krause et al., 1999, 2005; Krause & Arenhart, 2019; Sant’Anna & Santos, 2000). But even in quantum theories there are severe limitations of applicability of quasi-set theory (Sant’Anna, 2020) (even if we are able to guarantee some relative consistency of a variation of .Q). Besides, since indistinguishability among micro-atoms is relevant only to infinite collections of micro-atoms, that is hardly useful within the context of physical theories who deal only with finite ensembles of objects. So, why can’t we pursue another way, where we admit the possibility that a better way to cope with quantum indiscernibility is by abandoning classical logic and embracing intuitionistic logic? Acknowledgments I wish to express my gratitude to Décio Krause, Eliza Wajch, Asaf Karagila, Jonas Becker Arenhart, and Raoni Wohnrath Arroyo for criticisms on earlier versions of this work. Nevertheless, any shortcomings in this paper are the sole responsibility of its author.

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References Bell, J. L. (2008). A primer of infinitesimal analysis. Cambridge: Cambridge University Press. Devlin, K. J. (1977). Constructibility. In J. Barwise (Ed.), Handbook of mathematical logic (pp. 453–489). Amsterdam: North Holland. Domenech, G., Holik, F., & Krause, D. (2008). Q-spaces and the foundations of quantum mechanics. Foundations of Physics, 38, 969–994. Domenech, G., Holik, F., Kniznik, L., & Krause, D. (2010). No labeling quantum mechanics of indiscernible particles. International Journal of Theoretical Physics, 49, 3085–3091. Fraenkel, A. A. (1967). The notion “definite” and the independence of the axiom of choice. In J. van Heijenoort (Ed.), From Frege to Gödel (pp. 284–289). Cambridge: Harvard University Press. French, S., & Krause, D. (2006). Identity in physics: A formal, historical and philosophical approach. Oxford: Oxford University Press. Isham, C. J., & Butterfield, J. (1998). Topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations. International Journal of Theoretical Physics, 37, 2669–2733. Jech, T.(1973). The axiom of choice. New York: Dover. Jech, T. (2003). Set theory. Berlin: Springer. Kanamori, A. (2003). The higher infinite. Berlin: Springer. Krause, D. (1992). On a quasi-set theory. Notre Dame Journal of Formal Logic, 33, 402–411. Krause, D. (2010). Logical aspects of quantum (non-)individuality. Foundations of Science, 15, 79–94. Krause, D., & Arenhart, J. R. B. (2019). Is identity really so fundamental? Foundations of Science, 24, 51–71. Krause, D., Sant’Anna, A. S., & Volkov, A. G. (1999). Quasi-set theory for bosons and fermions: Quantum distributions. Foundations of Physics Letters, 12, 67–79. Krause, D., Sant’Anna, A. S., & Sartorelli, A. (2005). On the concept of identity in ZermeloFraenkel-like axioms and its relationship with quantum statistics. Logique et Analyse, 189–192, 231–260. Sant’Anna, A. S. (2005). Labels for non-individuals? Foundations of Physics Letters, 18, 519–533. Sant’Anna, A. S. (2020). Individuality, quasi-sets and the double-slit experiment. Quantum Studies: Mathematics and Foundations, 7, 179–193. Sant’Anna, A. S., & Santos, A. M. S. (2000). Quasi-set-theoretical foundations of statistical mechanics: A research program. Foundations of Physics, 30, 101–120. Scott, D. (1955). Definitions by abstraction in axiomatic set theory. Bulletin of the American Mathematical Society, 61, 442–442.

Chapter 11

Troublesome Quasi-Cardinals and the Axiom of Choice Eliza Wajch

Abstract The article concerns French–Krause quasi-set theory .Q, in particular, its very controversial axioms on quasi-cardinals, the Axiom of Choice and the Weak Extensionality Axiom. A modification .Q∗ of .Q is proposed. Similarly to what is going on in .Q, indiscernibility is a primitive concept of .Q∗ , objects of .Q∗ are either M-atoms or m-atoms, or quasi-classes. Some quasi-classes are quasi-sets. The ZFAkernel of .Q∗ is defined to serve as a model of Zermelo–Fraenkel set theory with atoms (usually denoted by ZFA or ZFU). The Axiom of Choice is not an axiom of .Q∗ . For a quasi-set x and a finite ordinal n, .qc(x, n) is the statement: “The quasi-set x has its quasi-cardinal equal to n”. Axioms on the statements .qc(x, n) are formulated in such a way that, for every .n ∈ ω, if x is in the ZFA-kernel of .Q∗ , then it holds in .Q∗ that .qc(x, n) is true if and only if x is a finite set equipotent to n; however, equipotence does not appear in the axioms concerning the statements .qc(x, n). Concepts of strongly finite, strongly infinite, strongly countable and strongly uncountable quasi-sets are proposed. A Proper Weak Extensionality Principle is introduced. One of the consequences of this principle is a theorem on the unobservability of permutations.

11.1 Introduction Nowadays, the set-theoretic framework for the work of most mathematicians is the Zermelo–Fraenkel system .ZFC in which there is a very controversial Axiom of Choice denoted by .AC. Still less popular than .ZFC system .ZF is obtained by

Dedicated to Décio Krause on his 70th birthday. E. Wajch () Institute of Mathematics, Faculty of Exact and Natural Sciences, Siedlce University of Natural Sciences and Humanities, Siedlce, Poland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_11

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deleting .AC from .ZFC, and leaving the remaining axioms of .ZFC unchanged (cf., e.g., Kunen, 2009, Chapter 1.2). If a system of axioms is not mentioned at all in a text containing mathematical theorems and their proofs, its reader must guess that the system .ZFC is assumed there because it is obvious that no proof concerning sets can be deduced without assumptions about sets and, moreover, it happens that, in such texts, the Axiom of Choice is carelessly involved in proofs even if .AC can be avoided by giving more subtle arguments. That .AC is overused in mathematics is inspiringly criticized, for instance, in Herrlich (2006). By choosing .ZF instead of .ZFC as the axiomatic foundation, much deeper insight into mathematics can be achieved (see, for instance, Herrlich, 2006; Howard & Rubin, 1998; Jech, 1973, 2002). Many theorems of .ZFC have already appeared independent of .ZF. Distinct models of .ZF or permutation models of .ZFA are applied to independence proofs and consistency results (see, e.g., Howard & Rubin, 1998; Jech, 1973). We recall that .ZFA (denoted by .ZF0 in Howard and Rubin (1998)) stands for Zermelo– Fraenkel set theory with atoms, as well as for its system of axioms. All mathematical objects of .ZF are sets. The objects of .ZFA are either sets or atoms. Atoms (called also urelements) in .ZFA are not sets and have no elements. In .ZFA, the axioms of extensionality and regularity are modifications of the axioms of, respectively, extensionality and regularity of .ZF (see Jech, 1973, Chapter 4.1; Howard & Rubin, 1998, p. 2). Usually, in .ZFA, it is assumed that the totality of atoms is an infinite set (see, e.g., Jech, 1973, Chapter 4.1). If the set of atoms of .ZFA (without .AC) were empty, .ZFA would be limited to .ZF. Some authors prefer to denote .ZFA by .ZFU (see, e.g., French & Krause, 2006; Krause, 1992); however, it seems that most set-theorists prefer the notation from Jech (1973) and Jech (2002), including .ZFA. Although many results in mathematical physics can be obtained within .ZF, there are problems of quantum physics that cannot have accurate descriptions in .ZF. One of the reasons is that, with the extensionality axiom of .ZF and the identity relation .= in hand, for arbitrary objects .x, y of .ZF, we know that either .x = y (in this case x is y) or .x = y (if x and y are not the same object). In this sense, every object of .ZF is an individual distinguishable from every other object of .ZF. The same happens in .ZFA. Many thanks to Stephen French and Décio Krause for having written the fascinating book French and Krause (2006) in which the concepts of individuality and indistinguishability are deeply discussed, a lot of convincing arguments are given that, in quantum physics, an elementary particle is an entity which is neither an individual nor indiscernible from other particles of the same sort and, therefore, axiomatic foundations of a universal theory of everything in mathematics and physics, with indiscernibility introduced “right from the start”, are desirable. Among the arguments supporting this point of view, very important, helpful quotations from some of the most influential physicists, mathematicians and philosophers (for instance, taken from Post (1963) and Schrödinger (1952)) are included in French and Krause (2006). Several attempts to create the desired axiomatic foundations are also discussed. Chapter 7 of French and Krause (2006), which is the basis of our present work, is devoted to the quasi-set theory .Q in which the notion of indistinguishability is a primitive concept. The axioms of .Q in French and Krause

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(2006) are taken from Krause (1992) or are modifications of some axioms given in Krause (1992). Our article consists of seven sections organized as follows. In Sect. 11.2, we introduce an altered quasi-set theory .Q∗cl by recalling in brief and adapting to our needs basic notation, terminology and only those axioms of .Q (taken from French and Krause (2006), French and Krause (2010) or Krause (1992)) which we are going to use here. We also add new axioms. Some of them concern the existence of a universe and proper quasi-classes. We define the classical, hierarchical and .ZF kernels of .Q∗cl . In Sect. 11.3, we clarify the concepts of weak singletons, weak pairs and strong singletons, given in French and Krause (2006), French and Krause (2010) and Krause (1992). We also introduce the concept of a strong ordered pair and prove several basic properties of strong singletons to apply them in Sects. 11.4–11.6. In Sect. 11.4, we add a binary functional symbol qc to the language of .Q∗cl . Informally, for any x and a non-negative integer n, .qc(x, n) is the sentence: “x is a quasi-set, and n is a quasi-cardinal of x”. We give axioms (.qc0)–(.qc4) concerning ∗ and the ∗ .qc(x, n). Our theory .Q is the theory whose axioms are all axioms of .Q cl five axioms on .qc(x, n). All theorems of Sects. 11.4–11.6 are theorems of .Q∗ . We define strongly finite quasi-sets and prove several basic properties of such quasi-sets. We show that if a quasi-set x is in the classical kernel of .Q∗ , then, for a non-negative integer n, .qc(x, n) is true if and only if x is an n-element set. For every pair x, y of quasi-sets such that x and y are either in the classical kernel of .Q∗ or every element of .x ∪ y is a strong singleton, we introduce the concept of the equipotence of x and y which extends the classical concept of equipotence. We introduce strongly infinite quasi-sets and prove that a quasi-set is strongly finite if and only if it is not strongly infinite. In Sect. 11.5, we define strongly countable, strongly denumerable, strongly uncountable, Dedekind strongly infinite quasi-sets and Dedekind strongly finite quasi-sets. In Sect. 11.6, we formulate the Axiom of Choice and some of its weaker forms in .Q∗ . We also modify the Weak Extensionality Axiom from French and Krause (2006) and French and Krause (2010), and introduce a new Proper Weak Extensionality Principle. We prove that it holds in .Q∗ that the Proper Weak Extensionality Principle implies our modification of the theorems on the unobservability of permutations from French and Krause (2006) and French and Krause (2010). Section 11.7 contains several conclusions from the content of the previous sections and shows desirable directions for possible future research on quasi-set theories.

11.2 Axiomatic Foundations of the Quasi-Class Theory Q∗cl Let us denote by .Q0 the quasi-set theory whose axioms are given in Krause (1992). Some of the axioms and definitions from Krause (1992) were changed in French and Krause (2006) and French and Krause (2010). Let .Q1 stand for the quasi-set

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theory whose axioms are given in French and Krause (2006, Chapter 7). Some of the concepts and axioms of .Q1 were modified in French and Krause (2010). Let .Q2 be the modified quasi-set theory described in French and Krause (2010). The most controversial axioms of .Q0 , .Q1 and .Q2 are the axioms concerning quasi-cardinals. During invited lectures at the 2nd International Conference on Physics in Brussels (Belgium) in 2017 (see Wajch, 2017), the 4th International Conference on Physics in Berlin (Germany) in 2018 (see Wajch, 2018) and the World-wide Congregation on Physics in Zurich (Switzerland) in 2019, E. Wajch gave several arguments that the axioms on quasi-cardinals can make the theories .Qi for .i ∈ {0, 1, 2} inconsistent. Other arguments that the axioms on quasi-cardinals from French and Krause (2006) were too imperfect to be left unchanged were given in French and Krause (2010) where a modification of this group of axioms was proposed. In this section, we aim to describe a system of axioms of a new theory .Q∗cl with proper qclasses, announced in Wajch (2017). Most axioms of .Q∗cl are based on the axioms of .Qi for .i ∈ {0, 1, 2} but with the axioms on quasi-cardinals deleted. We introduce new axioms concerning strongly finite qsets in Sect. 11.4. For simplicity, if this is not misleading, we denote by .Q any of the theories .Qi with .i ∈ {0, 1, 2}. Similarly to set theories in Ackermann (1956) and Mac Lane (1971) (see also Lévy, 1959; Muller, 2001), the language of .Q∗cl contains a constant .U which stands for the universe. The universe is a primitive concept that we use intuitively. Informally, the universe .U is the totality of all elements of .Q∗cl . Besides the constant .U, in the language of .Q∗cl , there are the connectives, quantifiers and so on, and, mimicking Krause (1992) and French and Krause (2006), there are binary predicate symbols: .∈ (membership) and .≡ (indistinguishability). There are three unary predicate symbols: .m, M and Z. Other unary symbols will be introduced in the sequel. For every .x ∈ U, we treat .m(x) as an abbreviation for the sentence “x is an m-atom”. The sentence “x is an M-atom” is abbreviated to .M(x), and the sentence “x is a (classical) set” is abbreviated to .Z(x). We can look at m, M and Z as formulae of one free variable. Definition 11.2.1 We define unary symbols .Qcl , Q, D, .Ecl and E as follows: Qcl (x) := ¬m(x) ∧ ¬M(x) ∧ (∀y)(y ∈ x → y ∈ U);

.

Q(x) := x ∈ U ∧ ¬(m(x) ∨ M(x));

.

D(x) := x ∈ U ∧ (M(x) ∨ Z(x));

.

Ecl (x) := Qcl (x) ∧ ∀y(y ∈ x → Q(y));

.

E(x) := Q(x) ∧ Ecl (x).

.

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Furthermore, for an arbitrary x, we say: (i) (ii) (iii) (iv) (v)

x x x x x

is a qclass (or, equivalently, a quasi-class) if .Qcl (x); is a qset (or, equivalently, a quasi-set) if .Q(x); is a ding if .D(x); is a proper qclass if .Qcl (x) ∧ ¬Q(x); is transitive if the following holds: Qcl (x) ∧ (∀y)(∀z)((Q(z) ∧ z ∈ x ∧ y ∈ z) → y ∈ x).

.

Definition 11.2.2 (i) We define the binary relation .⊆ (inclusion) as follows: (∀x)(∀y)(x ⊆ y ↔ (Qcl (x) ∧ Qcl (y) ∧ (∀t)(t ∈ x → t ∈ y)).

.

If .Qcl (x) and .Qcl (y) (respectively, .Q(x) and .Q(x)), and .y ⊆ x, we say that y is a subqclass (respectively, subqset) of x. (ii) We define the binary relation .=E (extensional equality) as follows: for all x and y, we write .x =E y and say that x is extensionally equal to y if one of the following statements is true: (.e1 ) .Qcl (x) ∧ Qcl (y) ∧ x ⊆ y ∧ y ⊆ x, (.e2 ) .M(x) ∧ M(y) ∧ (∀z)(Q(z) → (x ∈ z ↔ y ∈ z)). Now, we are in a position to formulate the axioms of .Q∗cl . The axioms of .Q∗cl are denoted by (.A0), (.A1) and so on. By adding to the language of .Q∗cl a functional symbol qc and the axioms concerning it which are formulated in Sect. 11.4, we obtain the axioms of .Q∗ . To some extent, our axioms of .Q∗cl are based on the axioms given in Krause (1992), French and Krause (2006), French and Krause (2010), Kunen (2009, Chapter I.2), Jech (1973, Chapter 4.1) and Ackermann (1956) (see also Lévy, 1959; Mac Lane, 1971, Chapter 1.6; Muller, 2001). (.A0) The universe .U exists, .U is transitive and .U ∈ / U. (.A1): .(∀x ∈ U)(x ≡ x). (.A2): .(∀x, y ∈ U)((x ≡ y) ↔ (y ≡ x)). (.A3): .(∀x, y, z ∈ U)(((x ≡ y) ∧ (y ≡ z)) → (x ≡ z)). (.A4): .(∀ψ(x, x))(∀x, y ∈ U)((x =E y) → (ψ(x, x) → ψ(x, y))) where .ψ(x, x) denotes any formula with free variable x, and .ψ(x, y) denotes the formula obtained from .ψ(x, x) by replacing in .ψ(x, x) some occurrences of x with y, with the restriction that y is free from x in .ψ(x, x). • (.A5): .(∀x ∈ U)¬(m(x) ∧ M(x)). • (.A6): .(∀x, y ∈ U)((x ∈ y) → Q(y)). • (.A7): .(∀x ∈ U)((Q(x) ∧ (∀y ∈ x)D(y)) ↔ Z(x)). • • • • •

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• (.A8): The conjunction of the following two statements: (∀x, y ∈ U)((m(x) ∧ x ≡ y) → m(y));

.

(∀x, y ∈ U)(((x =E y) ∧ Z(x)) → Z(y)).

.

• (.A9): The conjunction of the following two statements: (∀x, y ∈ U)((D(x) ∧ x ≡ y) → x =E y);

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(∀x, y ∈ U)((Q(x) ∧ x =E y) → x ≡ y).

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• (.A10)(Pairing): .(∀x, y ∈ U)(∃z ∈ U)(Q(z) ∧ x ∈ z ∧ y ∈ z). • (.A11)(Separation): For every well-formed formula .ψ(t) in which t appears free but z does not appear free, the following holds: (∀x)(Qcl (x) → (∃z)(Qcl (z) ∧ (∀t)(t ∈ z ↔ (t ∈ x ∧ ψ(t)))).

.

• (.A12): .(∀x)(∀y)((Q(x) ∧ y ⊆ x) → Q(y)). Although, we have not finished to formulate our axioms yet, let us observe that the following proposition follows from (.A7) and (.A12): Proposition 11.2.3 .(∀x)(∀y)((Z(x) ∧ y ⊆ x) → Z(y)). With (.A11) in hand, we can give the following definitions: Definition 11.2.4 Let .ψ be a well-formed formula in which t appears free but z does not. Let .x ⊆ U and .z ⊆ U. Suppose that the following condition is satisfied: (∀t)(t ∈ z ↔ (t ∈ x ∧ ψ(t))).

.

Then we write .z =E [t ∈ x : ψ(t)]. Furthermore, if .Z(x) holds, we write .z = {t ∈ x : ψ(t)}. Definition 11.2.5 Suppose that .u ⊆ U, .x ⊆ U and .y ⊆ U. (i) If .Ecl (u), then:  .

u := [t ∈ U : (∃z ∈ u)t ∈ z] and



u := [t ∈ U : (∀z ∈ u)t ∈ z];

  the qclasses . u and . u are called, respectively, the union and the intersection of u. (ii) .x ∪ y := [t ∈ U : t ∈ x ∨ t ∈ y] and .x ∩ y := [t ∈ U : t ∈ x ∧ t ∈ y]. The qclasses .x ∪ y and .x ∩ y are called, respectively, the union and the intersection of x and y. (iii) .x \ y := [t ∈ U : t ∈ x ∧ t ∈ / y].

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(iv) .P(u) := [t ∈ U : Q(t) ∧ t ⊆ u]. The qclass .P(u) is called the power qclass of u. Definition 11.2.6 The qclass .[t ∈ U : m(t) ∧ M(t)] is called the empty qclass and denoted by .∅. • (.A13): .∅ ∈ U. Remark 11.2.7 It follows from (.A7) and (.A13) that .Z(∅) holds. Hence, the empty qclass can be called the empty set. • (.A14)(Power qset): .(∀x)(Q(x) → P(x) ∈ U). Definition 11.2.8 If .x ∈ U and .Q(x) is true, then the qclass .P(x) is called the power qset of x. • (.A15)(Union): .(∀x ∈ U)(E(x) → • (.A16)(Infinity):



x ∈ U).

(∃x)(Z(x) ∧ E(x) ∧ ∅ ∈ x ∧ (∀y)(y ∈ x → y ∪ {y} ∈ x)).

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• (.A17)(Foundation): (∀x)((E(x) ∧ ¬(x =E ∅)) → (∃y)(y ∈ x ∧ y ∩ x =E ∅)).

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Remark 11.2.9 One can notice that (.A13) follows from (.A16); however, one can also want to investigate a finitistic theory whose axioms are all axioms of .Q∗cl except (.A16). This is partly why we have formulated (.A13). Definition 11.2.10 (a) Let .α ∈ U. Then .α is called an ordinal number if the following two conditions are satisfied: (i) .Z(α) ∧ E(α) ∧ (∀y)((y ∈ α) → y ⊆ α); (ii) .(∀x)(∀y)((x ∈ α ∧ y ∈ α) → (x ⊆ y ∨ y ⊆ x)). (b) The qclass On of all ordinals is defined as follows: On := [x ∈ U : x is an ordinal number ].

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Remark 11.2.11 (i) As in Krause (1992) and French and Krause (2006, Chapter 7), we notice that .=E restricted to the qclass .[x ∈ U : Z(x)] has all the required properties of identity .=. From now on, instead of .=E , we shall write .=. We remark that, in view of Definition 11.2.2, if .x, y ∈ U and either .m(x) or .m(y), then .x = y does not make any sense.

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(ii) Suppose that .x ∈ U is such that the following condition is satisfied: Z(x) ∧ E(x) ∧ ∅ ∈ x ∧ (∀y)(y ∈ x → y ∪ {y} ∈ x).

.

Let  .u(ω) := [z ∈ P(x) : E(z) ∧ ∅ ∈ z ∧ (∀y)(y ∈ z → y ∪ {y} ∈ z)], .ω := u(ω) and .N = ω \ {∅}. In the sequel, we shall use the standard terminology concerning ordinal numbers. In particular, an ordinal number .α is called a finite ordinal if .α ∈ ω. As usual, we put .0 = ∅ and, for every .n ∈ ω, .n + 1 = n ∪ {n}. We still need an axiom that will play the role of the Replacement Scheme. Let us mimic Krause (1992) and French and Krause (2006, Chapter 7.2.3) to formulate the following definition: Definition 11.2.12 Let .ψ(x, y) be a well-formed formula with free variables x and y. Let .u ⊆ U. We say that .ψ(x, y) is quasi-functional for x in u if the following two conditions are satisfied: (i) .(∀s ∈ u)(∃t ∈ U)ψ(s, t); (ii) (.∀s, s  ∈ u)(∀t, t  ∈ U)((s ≡ s  ∧ ψ(s, t) ∧ ψ(s  , t  )) → t ≡ t  ). • (.A18)(Replacement Scheme): For every .u ⊆ U and for every formula .ψ(x, y) such that .ψ(x, y) is quasi-functional for x in u, the following condition is satisfied: Q(u) → (∃v)(Q(v) ∧ (∀t ∈ v)(∃s ∈ u)ψ(s, t)).

.

One may prefer the following modification of (.A18) which looks stronger than (.A18): • (.A18∗ )(Replacement Scheme): For every .u ⊆ U and for every formula .ψ(x, y) such that .ψ(x, y) is quasi-functional for x in u, the following condition is satisfied: Q(u) → [t ∈ U : (∃s ∈ u)ψ(s, t)] ∈ U.

.

The following two axioms complete the list of axioms of .Q∗cl . • (.A19): .([x ∈ U : m(x)] ∈ U) ∧ ([x ∈ U : M(x)] ∈ U). • (.A20): .[x ∈ U : m(x)] =  ∅=  [x ∈ U : M(x)]. To summarize, the axioms (.A0)–(.A20) are axioms of .Q∗cl . The objects of .Q∗cl are elements of .U or qclasses contained in .U. Definition 11.2.13 The qset At of all atoms of .Q∗cl is defined as follows: At := [x ∈ U : m(x) ∨ M(x)].

.

An element .x ∈ U is called an atom if .x ∈ At.

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A motivation for (.A19) is that, in .Q, every entity which is neither an m-atom nor an M-atom is called a qset (see Krause, 1992; French & Krause, 2006, Chapter 7; French & Krause, 2010), so it seems that it is assumed in .Q that the totality of all atoms is a qset. In the following definition, for any .x ∈ U such that .Q(x) is true, we introduce a hierarchy in .U, analogous to that in Kunen (2009, Definition I.14.1), Jech (1973, Chapter 4.1) and Krause (1992). Definition 11.2.14 Let .x ∈ U be a qset. (i) .R0 (x) := x. (ii) For every .α ∈ On, .Rα+1 (x) := Rα (x)  ∪ P(Rα (x)). (iii) For every limit ordinal .α, .Rα (x) := Rγ (x). γ ∈α  (iv) .V(x) = Rα (x). α∈On

(v) (vi) (vii) (viii)

The qclass .V(∅) is called the .ZF-kernel of .Q∗cl . The qclass .V([t ∈ U : M(t)]) is called the .ZFA-kernel of .Q∗cl . The qclass .V(At) is called the hierarchical kernel of .Q∗cl . The qclass .[u ∈ U : Z(u) ∨ M(u)] is called the full .ZFA-kernel or, equivalently, the classical kernel of .Q∗cl .

At this moment, it does not seem possible to prove that .U = V(At), however, we can treat .V(At) as a model of .Q∗cl . We can also add the following axiom to the collection of axioms of .Q∗cl : • (.A21): .U = V(At). The theory whose axioms are (.A0)–(.A21) can be denoted by .Q∗cl + (A21). In the next section, we shall work in .Q∗cl . Although we cannot prove that the full ∗ is the .ZFA-kernel of .Q∗ , it is worth noticing that the following .ZFA-kernel of .Q cl cl proposition is true in .Q∗cl + (A21), and it shows that the .ZFA-kernel of .Q∗cl + (A21) is the full .ZFA-kernel of .Q∗cl + (A21). Proposition 11.2.15 In .Q∗cl + (A21), the following is true: (∀x ∈ U)((Z(x) ∨ M(x)) ↔ x ∈ V([t ∈ U : M(t)])).

.

11.3 Weak Singletons, Weak Pairs and Strong Singletons There is a problem with the definitions of a weak ordered pair given in Krause (1992), French and Krause (2006, Chapter 7) and French and Krause (2010). Namely, in Krause (1992) and French and Krause (2006), given entities x and y of .Q, the weak pair .[x, y] is defined as the qset of all t such that .t ≡ x or .t ≡ y, and the weak singleton .[x] of x is defined as the qset of all t such that .t ≡ x. The weak ordered pair .x, y of x and y is defined as .[[x], [x, y]]. This is criticized in

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French and Krause (2010) because .[x] and .[x, y] may be, in a sense, too large (in our terminology, they may be proper qclasses), so some limitations of the “size” of .[x, y] is desirable. Therefore, in French and Krause (2010), the axiom of weak pairing from Krause (1992) and French and Krause (2006) is replaced by the axiom of pairing, and a distinct definition of a weak pair is proposed in French and Krause (2010, p. 106). Let us reformulate it for .Q∗cl as follows: Definition 11.3.1 (Cf. French and Krause (2010, p. 106)) Let .x, y ∈ U and let z ∈ U be such that the following condition is satisfied: .Q(z) ∧ (x ∈ z ∧ y ∈ z).

.

(i) .[x, y]z := [t ∈ z : t ≡ x ∨ t ≡ y]. The qset .[x, y]z is called the weak pair of x and y relative to z. (ii) If .x ≡ y, then .[x]z := [x, x]z . The qset .[x]z is called the weak singleton of x relative to z. Remark 11.3.2 Let .x, y ∈ U and let .z ∈ U be such that the following condition is satisfied: .Q(z) ∧ Z(x) ∧ Z(y) ∧ (x ∈ z ∧ y ∈ z). Then .[x, y]z = [t ∈ z : t =E x ∨ t =E y]. In this case, we write .{x, y} to denote .[x, y]z ; moreover, if .x =E y, we write .{x} instead of .[x]z . For a qset z and .x, y ∈ z, the weak ordered pair (relative to z) .x, yz is defined in French and Krause (2010, Definition 3.2) as follows: .x, yz := [[x]z , [x, y]z ]z . Let us notice that this definition of a weak ordered pair (relative to z) has a serious defect. Namely, if z is a qset such that .x ∈ z and .y ∈ z, then the qsets .[x]z and .[y]z are subqsets of z but need not be elements of z. If .[x]z or .[x, y]z is not an element of z, .[[x]z , [x, y]z ]z does not make any sense. Therefore, let us propose new definitions of a weak singleton, a weak pair and a weak ordered pair in .Q∗cl below: Definition 11.3.3 Let .x ∈ U and .y ∈ U. (i) ⎧ ⎪ ⎪ ⎨[t ∈ U : m(t)] .H (x) := [t ∈ U : M(t)] ⎪ ⎪ ⎩P(x)

if m(x); if M(x); if Q(x).

(ii) .H (x, y) := H (x) ∪ H (y). (iii) Let .[x] := [x]H (x) . The qset .[x] will be called the weak singleton of x. (iv) Let .[x, y] := [x, y]H (x,y) . The qset .[x, y] will be called the weak pair of x and y. (v) The weak ordered pair .x, y with the first entry x, and the second entry y is defined as follows: x, y := [[x], [x, y]].

.

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In Krause (1992) and French and Krause (2006, Chapter 7.2.5), the concept of a strong singleton is introduced, but, unfortunately, by involving in it a not satisfactory definition of a weak singleton and the primitive concept of a quasi-cardinal of .Q. Therefore, we need to introduce strong singletons in .Q∗cl in a more appropriate way than in Krause (1992) and French and Krause (2006). Definition 11.3.4 Let .x ∈ U and .y ∈ U. H(x) :=[u ∈ P(H (x)) : x ∈ u]. [x]s := H(x). The qset .[x]s will be called the strong singleton of x. s s s .[x, y] := [x] ∪ [y] . s s s s s .x, y := [[x] , [x, y] ] . The qset .x, y will be called the strong ordered pair with the first entry x and the second entry  y. (v) .H∗ (x) := [u ∈ P([x]) : x ∈ u] and .x ∗ := H∗ (x).

(i) (ii) (iii) (iv)

. .

Definition 11.3.5 (i) Given .u ∈ U, we say that u is a strong singleton if .Q(u) and there exists .x ∈ U with .u = [x]s . (ii) .S[U] := [u ∈ U : u is a strong singleton]. (iii) Given .u ∈ U such that .Q(u), we put .S[u] := P(u) ∩ S[U]. Let us establish several basic properties of strong singletons in the following proposition: Proposition 11.3.6 Let .x ∈ U and .y ∈ U. The following conditions are all satisfied in .Q∗cl : (i) (ii) (iii) (iv) (v) (vi) (vii)

[x]s = x ∗ ⊆ [x]; ∗ .(∀w ∈ U)((Q(w) ∧ x ∈ w) → x ⊆ w); ∗ ∗ ∗ ∗ .x = y ↔ (x ∈ y ∧ y ∈ x ); ∗ ∗ ∗ .y ∈ x → y = x ; ∗ ∗ .y ∈ P(x ) → (y = ∅ ∨ y = x ); ∗ ∗ ∗ ∗ .x = y → x ∩ y = ∅;  .Q(x) → x = S[x]. .

Proof (i) Since .H∗ (x) ⊆ H(x), we have .[x]s ⊆ x ∗ . On the other hand, if .u ∈ H(x), then .[x]u ⊆ u and .[x]u ∈ H∗ (x), which implies that .x ∗ ⊆ [x]s . Hence ∗ s s ∗ .x = [x] . Now, it is obvious that .[x] ⊆ [x] because .x ⊆ [x]. ∗ (ii)–(iii) Let w be a qset such that .x ∈ w. Then .[x]∩w ∈ H (x), so .x ∗ ⊆ w. Hence (ii) holds. It follows from (ii) that (iii) holds. (iv) Suppose that .y ∈ x ∗ . It follows from (ii) that .y ∗ ⊆ x ∗ . Suppose that ∗ ∗ ∗ ⊆ x ∗ \ y ∗ which is impossible because .x ∈ x \ y . Then, by (ii), .x ∗ ∗ .y = ∅. The contradiction obtained shows that .x ∈ y . Hence (iii) implies ∗ ∗ that .x = y and, thus, (iv) holds.

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(v) Suppose that .y ∈ P(x ∗ ). For every .z ∈ x ∗ \ y, we have .z∗ ⊆ x ∗ \ y by (ii) and, by (iv), .z∗ = x ∗ . Hence, if .x ∗ \ y = ∅, then .y = ∅. This shows that (v) holds. That (vi) also holds, follows from (iv). We infer from (ii) that (vii) is true.   Remark 11.3.7 Proposition 11.3.6(ii) is relevant to French and Krause (2006, Lemma 25, p. 294). Item (v) of Proposition 11.3.6 is analogous to Domenech and Holik (2007, Proposition 4.4).

11.4 Strongly Finite Qsets and Their Qcardinals In this section, we enrich the theory .Q∗cl by adding to its language a binary functional symbol qc which stands for a qcardinal (equivalently, a quasi-cardinal), and adding several new axioms concerning qc. If .x ∈ U, .n ∈ ω and .Q(x), then .qc(x, n) can be read: x is a qset which has a quasi-cardinal, and n is a quasi-cardinal of x. Contrary to Krause (1992) and French and Krause (2006), we do not assume that every qset has a quasi-cardinal. Our axioms on qc are the following: • (.qc0): .(∀x ∈ U)(∀n ∈ ω)(qc(x, n) → Q(x)). • (.qc1): .(∀x ∈ U)(qc(x, 0) ↔ x = ∅). • (.qc2): .(∀x ∈ U)(∀y ∈ U)(∀n ∈ ω)(qc(x, n) ∧ y ∈ S[U] ∧ x ∩ y = ∅) → qc(x ∪ y, n + 1)). • (.qc3): .(∀x ∈ U)(qc(x, 1) → x ∈ S[U]). • (.qc4): .(∀x ∈ U)(∀n ∈ ω)(qc(x, n + 1) → (∃y ∈ P(x))(qc(y, n) ∧ x \ y ∈ S[U])). We denote by .Q∗ the theory whose language is the language of .Q∗cl enriched with qc, and axioms are all axioms of .Q∗cl , together with the new axioms (.qc0)–(.qc4). From now on, let us work in .Q∗ . All definitions in .Q∗cl introduced in the previous sections are also definitions of .Q∗ . In what follows, each time when .qc(x, n) appears but .Q(x) is not written, we apply (.qc0) as a piece of information telling that x is a qset. To show the role of (.qc0)–(.qc4), we are going to prove in .Q∗ several basic properties of qc by the mathematical induction (in abbreviation, .MI). In .Q∗ , .MI is the statement: (∀N ∈ P(ω))((∅ ∈ N ∧ (∀n ∈ ω)(n ∈ N → n + 1 ∈ N)) → N = ω).

.

Proposition 11.4.1 .(∀x ∈ U)(qc(x, 1) ↔ x ∈ S[U]). Proof This proposition follows directly from (.qc1), (.qc2) and (.q3).

 

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Theorem 11.4.2 (∀x, y ∈ U)(∀m, n ∈ ω)((qc(x, m) ∧ qc(y, n) ∧ x ∩ y = ∅) → qc(x ∪ y, m + n)).

.

Proof Let .m ∈ ω and let x be a qset such that .qc(x, m) holds. We shall apply .MI to the following set N: N = {n ∈ ω : (∀y ∈ U)((qc(y, n) ∧ x ∩ y = ∅) → qc(x ∪ y, m + n)}.

.

Clearly, .0 ∈ N. Suppose that .n ∈ N. Consider any qset y such that .x ∩ y = ∅ and .qc(y, n + 1). By (.qc4), there exists .z ∈ S[y] such that .qc(y \ z, n) holds. By our inductive assumption, .qc(x ∪ (y \ z), m + n) holds. It follows from (.qc2) that .qc(x ∪ y, m + n + 1) holds. Hence, by .MI, .N = ω.   Theorem 11.4.3 .(∀x ∈ U)(∀n ∈ ω)(qc(x, n) ↔ qc(S[x], n)). Proof Let .N0 = {n ∈ ω : (∀x ∈ U)(qc(x, n) → qc(S[x], n))}. To prove by .MI that .N0 = ω, we notice that it follows from (.qc1) that .0 ∈ N0 . Next, we assume that .n ∈ N0 . To prove that .n + 1 ∈ N0 , we consider any qset x such that .qc(x, n + 1) holds. By (.qc4), there exists .z ∈ S[x] such that .qc(x \ z, n) holds. Since .n ∈ N, ∗ .qc(S[x \ z], n) holds. We notice that .S[x] = S[x \ z] ∪ z . It follows from (.qc2) that .qc(S[x], n + 1) holds, so .n + 1 ∈ N0 . Hence .N0 = ω by .MI. Now, let .N1 = {n ∈ ω : (∀X ∈ U)((Q(X) ∧ qc(S[X], n)) → qc(X, n))}. Clearly, by (.qc1), .0 ∈ N1 . We assume that .n ∈ N1 , and prove that .n + 1 ∈ N1 . To this aim, we consider any qset X such that .qc(S[X], n + 1) holds. By (.qc4), there exists .Z ∈ S[S[X]] such that .qc(S[X] \ Z, n) holds. There exists .T ∈ S[X] such that .Z = T ∗ . We notice that, by Proposition 11.3.6, .S[X \ T ] = S[X] \ Z, so .qc(S[X \ T ], n) holds. Since .n ∈ N1 , we deduce that .qc(X \ T , n) holds. It follows from (.qc2) that, .qc(X, n + 1) holds. Hence .n + 1 ∈ N1 . All this taken together with .MI implies that .N1 = ω. The equalities .N0 = ω = N1 complete the proof.   Theorem 11.4.4 (∀x ∈ U)(∀n ∈ ω)(qc(x, n + 1) → (∀t ∈ S[x])qc(x \ t, n)).

.

Proof Let .N = {n ∈ ω : (∀x ∈ U)(qc(x, n + 1) → (∀t ∈ S[x])qc(x \ t, n))}. It follows from (.qc1) and (.qc3) that .0 ∈ N . Let us assume that .n ∈ N and prove that .n + 1 ∈ N. We consider any qset x such that .qc(x, n + 2) holds. By (.qc4), there exists .z ∈ S[x] such that .qc(x \ z, n + 1) holds. Let .t ∈ S[x]. Suppose that .t ∩ (x \ z) =  ∅. Then, by Proposition 11.3.6, .t ∈ S[x \ z], so .qc((x \ z) \ t, n) holds because .n ∈ N. Since .x \ t = ((x \ z) \ t) ∪ z, we infer from (.qc2) that .qc(x \ t, n + 1) holds.

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Now, suppose that .t ∩ (x \ z) = ∅. Then, by Proposition 11.3.6, .x \ z = x \ t, so .qc(x \ t, n + 1) holds. This completes the proof that if .n ∈ N , then .n + 1 ∈ N. Hence .N = ω by .MI.   Theorem 11.4.5 (∀x ∈ U)(∀n ∈ ω)(qc(x, n) → (∀y ∈ P(x))(∃k ∈ n + 1)qc(y, k)).

.

Proof Let M = {m ∈ ω : (∀z ∈ U)(qc(z, m) → (∀y ∈ P(z))(∃k ∈ m + 1)qc(y, k))}.

.

In the light of (.qc1), it is obvious that .0 ∈ M. Suppose that .m ∈ M and consider any qset z such that .qc(z, m+1) holds. Let .y ∈ P(z). By (.qc4), there exists .t ∈ S[z] such that .qc(z \ t, m) holds. If .y ⊆ z \ t, then, since .m ∈ M, there exists .k ∈ m + 1 such that .qc(y, k) holds. Now, suppose that y is not a subqset of .z\t. Then, since .t ∩y = ∅, it follows from Proposition 11.3.6 that .t ∈ S[y]. Furthermore, since .y \ t ⊆ z \ t and .qc(z \ t, m) holds, and .m ∈ M, we infer that there exists .k ∈ m + 1 such that .qc(y \ t, k) holds. Since .y = (y \ t) ∪ t, by (.qc2), .qc(y, k + 1) holds. This implies that .m + 1 ∈ M. By .MI, .M = ω, which completes the proof.   Theorem 11.4.6 .(∀x ∈ U)(∀m, n ∈ ω)((qc(x, m) ∧ qc(x, n)) → m = n). Proof Let N = {n ∈ ω : (∀x ∈ U)(qc(x, n) → (∀m ∈ ω)(qc(x, m) → m = n))}.

.

We prove by .MI that .N = ω. To this aim, we notice that, by (.qc1), .0 ∈ N. Now, suppose that .n ∈ ω is such that .n+1 ⊆ N . Consider any qset x such that .qc(x, n+1) holds. Suppose that .m ∈ ω is such that .qc(x, m) also holds. If .m ≤ n, then .m ∈ N , so .m = n + 1. Therefore, we may assume that .n + 1 ≤ m. By (.qc4), there exists .z ∈ S[x] such that .qc(x \ z, n) holds. By Theorem 11.4.4, .qc(x \ z, m − 1) also holds. Since .n ∈ N, we have .m − 1 = n. Hence .n + 1 = m, so .n + 1 ∈ N . This, together with .MI, completes the proof.   Definition 11.4.7 A qset x will be called: (i) strongly finite if there exists .n ∈ ω such that .qc(x, n) holds; (ii) strongly infinite if, for every .n ∈ ω, there exists .y ∈ P(x) such that .qc(y, n) holds. Theorem 11.4.6 makes it possible to introduce the following definition: Definition 11.4.8 If x is a strongly finite qset, then the unique .n ∈ ω for which qc(x, n) holds will be denoted by .qc(x) and called the qcardinal (equivalently, quasi-cardinal) of x.

.

Theorem 11.4.9 A qset is not strongly infinite if and only if it is strongly finite.

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Proof Suppose that x is a qset which is not strongly infinite. We define a set N as follows: .N := {n ∈ ω : (∀y ∈ P(x))¬qc(y, n)}. Since x is not strongly infinite, we infer that .N =  ∅. Therefore, we can define .n0 := min N. By (.qc1), .n0 > 0, so .n0 − 1 ∈ ω \ N. Hence, there exists .y ∈ P(x) such that .qc(y, n0 − 1) holds. If .y = x, then x is strongly finite. Suppose that .y = x. It follows from Proposition 11.3.6 that there exists .t ∈ S[x] such that .t ⊆ x \ y. Then .y ∪ t ∈ P(x). It follows from (.qc2) that .qc(y ∪ t, n0 ) holds, which implies that .n0 ∈ / N. The contradiction obtained shows that .y = x and, in consequence, x is strongly finite. On the other hand, if x is a strongly finite qset it follows from Theorems 11.4.5 and 11.4.6 that x is not strongly infinite   All standard concepts of .ZFA can be applied to members of the full .ZFA-kernel of .Q∗ . In particular, we have the following standard definition of a finite set: Definition 11.4.10 Let .x ∈ U be such that .Z(x) holds. (i) We say that x is a finite set if there exists .n ∈ ω such that x and n are equipotent. (ii) If .n ∈ ω is such that the sets x and n are equipotent, we write .| x |= n to say that n is the cardinal of x or, equivalently, x is an n-element set. Our next theorem shows that, in the full .ZFA-kernel of .Q∗ , the notions of a strongly finite set and a finite set are equivalent. Theorem 11.4.11 (i) .(∀n ∈ ω)qc(n, n). (ii) .(∀x ∈ U)(∀n ∈ ω)(Z(z) → (qc(x, n) ↔| x |= n)). Proof (i) Let .N0 = {n ∈ ω : qc(n, n)}. Then, by (.qc1), .0 ∈ N0 . It follows directly from (.qc2) that if .n ∈ N0 , then .n + 1 ∈ N0 . Hence (i) holds by .MI. (ii) Let .N1 = {n ∈ ω : (∀x ∈ U)((Z(x) ∧ qc(x, n)) →| x |= n)}. By (.qc1), .0 ∈ N1 . Suppose that .n ∈ N1 and consider any .x ∈ U such that both .Z(x) and .qc(x, n + 1) hold. It follows from (.qc4) that there exists .t ∈ x such that .qc(x \ {t}, n) holds. Since .n ∈ N1 , we infer that .| x \ {t} |= n. This implies that .| x |= n + 1, so .n + 1 ∈ N1 . Hence .N1 = ω by .MI. Now, let .N2 = {m ∈ ω : (∀y ∈ U)((Z(y)∧ | y |= m) → qc(y, m))}. It follows from (.qc1) that .0 ∈ N2 . Suppose that .m ∈ N2 and, to prove that .m + 1 ∈ N2 , consider any .y ∈ U such that both .Z(y) and .| y |= m + 1 hold. Fix any .z ∈ y. Then .| y \ {z} |= m. Since .m ∈ N2 , we deduce that .qc(y \ {z}, m) holds. By (.qc2), .qc(y, m + 1) holds. Hence .N2 = ω by .MI. Since .N1 = ω = N2 , we infer that (ii) holds.  

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Let us generalize the standard concepts of a function and equipotent sets in the following definition: Definition 11.4.12 Suppose that A and B are both elements of .[x ∈ U : Z(x)] ∪ P(S[U]). (a) .A ×s B := {a, bs : a ∈ A ∧ b ∈ B}. (b) A subqset f of .A ×s B is called a function defined on A and taking values in B if the following conditions are satisfied: (i) .(∀a ∈ A)(∃b ∈ B)a, bs ∈ f ; (ii) .(∀a ∈ A)(∀b, c ∈ B)((a, bs ∈ f ∧ a, cs ∈ f ) → b = c).

(c) (d) (e) (f)

(g)

That f is a function defined on A and taking values in B is denoted by .f : A → B. A function .f : A → B is a surjection if the following condition is satisfied: s .(∀b ∈ B)(∃a ∈ A)a, b ∈ f . A function .f : A → B is an injection if the following condition is satisfied: s s .(∀a, d ∈ A)(∀b ∈ B)((a, b ∈ f ∧ d, b ∈ f ) → a = d). .f : A → B is called a bijection if f is a surjection and an injection. The qset A is equipotent to B (or, equivalently, A and B are equipotent) if there exists a bijection .f : A → B. That A and B are equipotent is denoted by .| A |=| B |. If .n ∈ ω and A is equipotent to n, we write .| A |= n.

We remark that if .A = ∅ = B, then the unique bijection .f : A → B is .f = ∅. By applying Theorem 11.4.3 and arguing in much the same way, as in the proof of Theorem 11.4.11, one can show that the following theorem holds: Theorem 11.4.13 .(∀x ∈ U)(∀n ∈ ω)((Q(x)∧ | S[x] |= n) ↔ qc(x, n)). Remark 11.4.14 Suppose that x is a strongly finite qset and .qc(x) = n. One cannot deduce from Theorem 11.4.13 that there exists a bijection from n onto x if .x ∈ / [x ∈ U : Z(x)] ∪ P(S[U]). Similarly to the finite quasi-cardinals defined in Domenech and Holik (2007), the qcardinal .qc(x) does not tell us how many elements x has, but .qc(x) shows how many strong singletons are contained in x. Corollary 11.4.15 Let x and y be strongly finite qsets. Then .qc(x) = qc(y) if and only if the qsets .S[x] and .S[y] are equipotent.

11.5 Strongly Denumerable Qsets and Strongly Uncountable Qsets We recall that an ordinal number .κ is called an initial ordinal or, equivalently, a well-ordered cardinal if there does not exist .α ∈ κ such that .α and .κ are equipotent. With Definition 11.4.12 in hand, we can introduce the following concepts:

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Definition 11.5.1 Let .κ be a well-ordered cardinal and let x be a qset. (i) If .S[x] and .κ are equipotent, we write .qc(x) = κ, and call .κ the qcardinal of x. (ii) If .S[x] is equipotent to a subset of .ω, we say that x is strongly countable. (iii) If .S[x] and .ω are equipotent, we say that x is strongly denumerable. (iv) If x is not strongly countable, we say that x is strongly uncountable. (v) If x contains a strongly denumerable subqset, we say that x is Dedekind strongly infinite. Otherwise, we say that x is Dedekind strongly finite. The results of the previous section show that every strongly finite qset is Dedekind strongly finite, and every Dedekind strongly infinite qset is strongly infinite. However, since we have not equipped .Q∗ with any form of the Axiom of Choice, we cannot claim that every strongly infinite qset is Dedekind strongly infinite. We cannot claim that, for every qset x, there exists a well-ordered cardinal ∗ .κ such that .S[x] and .κ are equipotent. Therefore, we do not have in .Q any quasicardinal assignment in the sense used in both Krause (1992) and French and Krause (2006).

11.6 The Axiom of Choice and the Weak Extensionality Axiom in Q∗ In the following definition, we show how one can formulate variants of the Axiom of Choice in .Q∗ , but we are not going to define all possible choice principles in .Q∗ here. None of the choice principles defined below is an axiom of .Q∗ . Definition 11.6.1 (i) A qclass X will be called pairwise disjoint if the following condition is satisfied: Ecl (X) ∧ (∀x ∈ X)(∀y ∈ X)(x = y → x ∩ y = ∅).

.

(ii) Suppose that X is a non-empty pairwise disjoint qclass  such that every element of X is non-empty. Every qclass Z such that .Z ⊆ X and, for every .x ∈ X, .Z ∩ x ∈ S[U] will be called a choice qclass of X. Furthermore, if a choice qclass of X is a qset, we shall call it a choice qset of X. (ii) The Axiom of Choice for qclasses (in abbreviation, .ACqcl ) is the sentence: Every non-empty pairwise disjoint qclass whose all elements are non-empty qsets has a choice qclass. (iii) The Axiom of Choice for qsets (in abbreviation, .ACqs ) is the sentence: Every non-empty pairwise disjoint qset whose all elements are non-empty qsets has a choice qset.

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(iv) The Countable Axiom of Choice for qsets (in abbreviation, .CACqs ) is the sentence: Every non-empty pairwise disjoint countable qset whose all elements are non-empty qsets has a choice qset. Some other choice principles defined, for instance, in Howard and Rubin (1998), can be also adapted for use in .Q∗ . It would be beneficial to construct models of ∗ ∗ .Q showing that the above-mentioned choice principles are independent of .Q . Let ∗ constructions of models of .Q and independence proofs be topics of other research papers in the future. Let us finish this article with several comments concerning the Weak Extensionality Axiom, that is, the axiom .(Q26) in French and Krause (2006, pp. 290–291), and the axiom .≡5 in French and Krause (2010, p. 110). Contrary to French and Krause (2006) and French and Krause (2010), in our variant of the Weak Extensionality Axiom we cannot use quasi-cardinals because a qset in .Q∗ may fail to have a quasi-cardinal in a reasonable sense. Therefore, in the following definition, we must modify the concept of Q-similarity from French and Krause (2006, Definition 15(i), p.290). Definition 11.6.2 For .x ∈ U and .y ∈ U, .Qsim(x, y) is the following statement: Q(x) ∧ Q(y) ∧ x = ∅ = y∧ | S[x] |=| S[y] | ∧(∀t ∈ x)(∀z ∈ y)(t ≡ z). If .Qsim(x, y) is true, we say that x and y are Q-similar.

.

Definition 11.6.3 For a non-empty qset x, we define the qset .x/ ≡ as follows: x/≡ := [[t]x : t ∈ x].

.

It seems that the Weak Extensionality Principle, close to the axiom .(Q26) in French and Krause (2006, pp. 290–291) and the axiom .≡5 in French and Krause (2010, p. 110), can be defined in .Q∗ as follows: Definition 11.6.4 The Weak Extensionality Principle (in abbreviation .WEP) in .Q∗ is the following sentence: for every pair .x, y of non-empty qsets, if the following condition is satisfied: ((∀z ∈ x/≡ )(∃t ∈ y/≡ )Qsim(z, t)) ∧ ((∀t ∈ y/≡ )(∃z ∈ x/≡ )Qsim(t, z)),

.

then .x ≡ y. Unfortunately, Theorem 26 in French and Krause (2006, Chapter 7.2.6) and Theorem 3.1 in French and Krause (2010) concerning the unobservability of permutations are not clearly formulated, and their proofs given in French and Krause (2006) and French and Krause (2010) are too far from being perfect to be wellunderstood. It is difficult to guess what their authors had in mind. Therefore, let us define a new principle and deduce from it a new theorem on the unobservability of permutations.

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Definition 11.6.5 The Proper Weak Extensionality Principle (in abbreviation, PWEP) in .Q∗ is the following sentence: for every pair .x, y of non-empty qsets, if the following condition is satisfied:

.

.

| S[x] |=| S[y] | ∧((∀t ∈ x)(∃r ∈ y)(t ≡ r)) ∧ ((∀r ∈ y)(∃t ∈ x)(r ≡ t)),

then .x ≡ y. Theorem 11.6.6 (Unobservability of Permutations) Let x be a qset and let .z ∈ x. Suppose that .[z] \ x = ∅. Then it holds in .Q∗ that .PWEP implies that, if x is either strongly finite or Dedekind strongly infinite, then, for every .w ∈ [z] \ x, s s .(x \ [z] ) ∪ [w] ≡ x. Proof Assuming that x is either strongly finite or Dedekind strongly infinite, we consider any .w ∈ [z] \ x. Let .y = (x \ [z]s ) ∪ [w]s . Since x is strongly finite or Dedekind strongly infinite, and .[w]s ∩ (x \ [z]s ) = ∅, one can easily check that .| S[x] |=| S[y] |. It is obvious that the following statement is true: ((∀t ∈ x)(∃r ∈ y)(t ≡ r)) ∧ ((∀r ∈ y)(∃t ∈ x)(r ≡ t)).

.

Thus, it follows from .PWEP that .x ≡ y.

 

11.7 Conclusions We have outlined two new theories of qclasses and qsets, .Q∗cl and .Q∗ , by formulating their axioms. In particular, we have introduced a new system of axioms of .Q∗ concerning finite qcardinals assigned to strongly finite qsets. Our approach to finite qcardinals has been shown fruitful, leading to natural concepts of strongly infinite, strongly countable, strongly uncountable qsets. We have clarified the concepts of strong singletons, weak singletons, strong ordered pairs and weak ordered pairs. We have proposed a new Proper Weak Extensionality Principle to apply it to a theorem on the unobservability of permutations. We have also shown how to formulate some choice principles in .Q∗ but have not attempted to construct any models of our theories. Needless to say that such models are desirable for consistency and independence proofs. Therefore, we would like to encourage researchers to search for models of the theories presented in this article. Although we have not discussed possible applications of our theories to more accurate descriptions of what really happens in quantum physics, we hope that many researchers will find it beneficial to develop and apply to mathematics and physics the results obtained in this work. Acknowledgments The author is deeply grateful to Professor Décio Krause for a stimulating discussion while writing this article and for showing his unpublished note “Reformulation of the theory of quasi-sets (discussing the attribution of quasi-cardinals)” (2021) which has influenced our approach to .qc(x, n).

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References Ackermann, W. (1956). Zur Axiomatic der Mengenlehre. Mathematische Annalen, 131, 336–345. Domenech, G., & Holik, F. (2007). A discussion on a particle number and quantum indistinguishability. Foundations of Physics, 37(6), 855–878. French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Clarendon Press. French, S., & Krause, D. (2010). Remarks on the theory of quasi-sets. Studia Logica, 95(1–2), 101–124. Herrlich, H. (2006). Axiom of choice, Lecture notes in mathematics (Vol. 1876). Berlin–Heidelberg: Springer-Verlag. Howard, P., & Rubin, J. E. (1998). Consequences of the axiom of choice, Math. surveys and monographs 59. Providence, Rhode Island: American Mathematical Society. Jech, T. (1973). The axiom of choice, Studies in logic and the foundations of mathematics 75. Amsterdam: North-Holland. Jech, T. (2002). Set theory. The Third Millenium Edition, revised and expanded, Springer monographs in mathematics. Berlin: Springer. Krause, D. (1992). On a quasi-set theory. Notre Dame Journal of Formal Logic, 33(3), 402–411. Kunen, K. (2009). The foundation of mathematics, Studies in logic (Vol. 19) London: College Publications (2009) Lévy, A. (1959). On Ackermann’s set theory. The Journal of Symbolic Logic, 24(2), 154–166. Mac Lane, S. (1971). Categories for the working mathematician. New York: Springer-Verlag. Muller, F. A. (2001). Sets, classes and categories. The British Journal for the Philosophy of Science, 52, 539–573. Post, H. (1963). Individuality and physics. The Listener, 70, 534–537 (1963, October 10); reprinted in: Vedanta for East and West 132 (1973), 14–22. Schrödinger, E. (1952). Science and humanism. Cambridge: Cambridge University Press. Wajch, E. (2017). Computation within models of ZF minus the postulate of infinity. Journal of Physical Chemistry & Biophysics, 7(3), 52. Wajch, E. (2018). Problems on quasi-sets in quantum mechanics. Journal of Physical Chemistry & Biophysics, 8, 53.

Part III

Logic and Philosophy of Quantum Mechanics

Chapter 12

Six Measurement Problems of Quantum Mechanics F. A. Muller

Abstract The notorious ‘measurement problem’ has been roving around quantum mechanics for nearly a century since its inception, and has given rise to a variety of ‘interpretations’ of quantum mechanics, which are meant to evade it. We argue that no less than six problems need to be distinguished, and that several of them classify as different types of problems. One of them is what traditionally is called ‘the measurement problem’. Another of them has nothing to do with measurements but is a profound metaphysical problem. We also analyse critically Maudlin’s (Topoi 14:7–15, 1995) well-known statement of ‘three measurements problems’, and the clash of the views of Brown (Found Phys 16:857–870, 1986) and Stein (Maximal of an impossibility theorem concerning quantum measurement. In: R. S. Cohen et al. (Eds.), Potentiality, entanglement and passion-at-a-distance, 1997) on one of the six measurement problems. Finally, we summarise a solution to one measurement problem which has been largely ignored but tacitly if not explicitly acknowledged.

12.1 Exordium The other day, I wondered: who discovered ‘the measurement problem’ of Quantum Mechanics (QM), and who coined it? If the problem is that superpositions carry over from ‘microscopic’ physical systems (molecules, atoms, particles) to ‘macroscopic’ physical systems, then Einstein discovered it, when he wrote in 1935 to

F. A. Muller () Erasmus School of Philosophy, Erasmus University Rotterdam, Rotterdam, The Netherlands Descartes Centre for the History and Philosophy of Science, Faculty of Science, Utrecht University, Utrecht, The Netherlands e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_12

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Schrödinger about a bomb in an exploded and non-exploded state.1 As is wellknown, Schrödinger went public with this, but not before replacing the bomb with a cat in a superposition of states of the cat being alive and being death, making the cat neither dead nor alive. Recently Rovelli preferred to consider a cat in a friendlier superposition, of being wide awake and being purring asleep, and Norson a cat with a fat belly having drunk milk and a cat with an empty belly having drunk no milk.2 If ‘the measurement problem’ is however that when we describe the measurement interaction unitarily and end up with a measurement apparatus indicating no definite measurement outcome, then von Neumann discovered the problem. Von Neumann inaugurated quantum-measurement theory in his magisterial Mathematische Grundlagen der Quantenmechanik (1932), and introduced his notorious Projection Postulate in order to end up with a single definite measurement outcome; this is stricto sensu a solution to ‘the measurement problem’.3 Did Einstein obtain the explosive idea of the bomb from von Neumann’s measurement theory, both members of the Princeton Institute for Advanced Studies at the time, and Einstein being familiar with von Neumann’s Grundlagen? These questions have, I must sadly report, no definite answers. A quick survey in historical writings on QM came up empty, and posing the question on the e-mail list of the hopos community has taught me there are no definite answers to these questions. A few pertinent remarks merit mentioning, historical underdeterminacy notwithstanding. One remark is that since Einstein’s and von Neumann’s considerations saw the light of day earlier than Schrödinger’s, the Columbus Price for Landmark Discoveries does not go to this Austrian pussycat. Another remark is that the dawn of talk of ‘the measurement problem’ lies in the early 1960s, with Wigner (1961, 1963), which makes Wigner the undisputed prime candidate for ‘Eugene Paul (né Jenö Pál) the Baptist’ of ‘the problem of measurement’ of QM; in both mentioned papers, Wigner expounds ‘the measurement problem’ with crystal clarity—and then unhesingtately solves it by invoking the Projection Postulate. All expositions of QM, whether aimed at working physicists, mathematicians or students, then and even now, have included and do include the Projection Postulate, as being part and parcel of standard QM, whether their authors accept, doubt or reject it; and then there is no ‘measurement problem’. Unless ‘the measurement problem’ is how to get rid of the Projection Postulate. Enough history. Before we proceed to state no less than six measurement problems (in Sects. 12.3, 12.6, 12.7, and 12.8), we need to state some of the postulates of standard QM precisely, if only for the sake of reference. We compare the very first problem, the Reality Problem of Measurement Outcomes, critically with Maudlin’s well-known exposition of ‘three measurement problems’ (Sect. 12.4). ‘Insolubility

1 Fine

(1986, Ch. 5). (2021, Ch. 2), Norsen (2017, p. 73). 3 von Neumann (1932); for a thoroughly updated version, including a deposit story of results from mathematical physics about quantum theory achieved after 1932, see Landsman (2017). 2 Rovelli

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Theorems’ and the impossibility of describing measurement interactions unitarily are the topic of Sect. 12.5, as well as Brown’s (1986) and Stein’s (1997) clashing take on these theorems. Section 12.8 includes a summary of an explication of the concept of measurement, which is the most ignored measurement problem of the six. We recapitulate and draw some conclusions at the end (Sect. 12.9).

12.2 Some Postulates of Standard Quantum Mechanics The primitive physical concepts of the vocabulary of standard QM, i.e. the ones without definitions, are: physical system, subsystem, measurement cq. measurement apparatus, property, space, time, state, probability and physical magnitude (Dirac called physical magnitudes positivistically ‘observables’, a terminology that has stuck; von Neumann spoke of ‘physical quantities’, which terminology is also in use). Sometimes one speaks of ‘physical variables’, which is troublesome for certain reasons and not troublesome for different reasons (we park them). A subsystem of a physical system is a mereological part of it; the relation ‘is a subsystem of’ coincides with the part-whole relation in mereology and is governed by its axioms. The first two postulates tell us what the mathematical representatives are of the physical states and physical magnitudes.4  Pure State Postulate. Pure physical states of a physical system .S are represented by unit-vectors in some Hilbert-space (.H), up to a multiplicative complex constant of modulus equal to 1, a ‘phase factor’ .eiθ . Whenever physical system .S consists of N subsystems, its Hilbert-space .H consists of the N -fold directproduct Hilbert-space of the N subsystems: . H = H1 ⊗ H2 ⊗ . . . ⊗ HN .  Magnitude Postulate. Every physical magnitude pertaining to physical system .S is represented by some self-adjoint operator A (up to a real multiplicative constant) that acts on the Hilbert-space .H of .S. Needless to say that the restriction to self-adjoint operators can be loosened to e.g. normal operators (they commute with their adjoint) or to positive operators— one can prove a spectral theorem for normal operators, but not for positive operators, which makes positive operators mathematically minacious. Such loosenings will have no bearing on the problems (and their possible solutions) treated in this paper. Notice that the troublesome converse is not part of the Magnitude Postulate: not every self-adjoint operator needs to represent a physical magnitude.5 The following postulate connects magnitude operators to measurements.

square (.) marks a postulate of standard QM; a black box (.) marks a principle worth considering; these are both written in italics. A dark red triangle (▲) signals a problem: there are six of them: I–VI. 5 Recall Wigner’s famous question: which unmeasurable physical magnitude represents the selfadjoint operator .P + Q ? 4A

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 Spectrum Postulate. All measurement-outcomes of a physical magnitude belong to the spectrum of the representing self-adjoint operator. Spectral Theorems inform us that every self-adjoint (and every normal) operator A has a unique spectral family of projectors, . P A () : H → H, where .  ∈ B(R) is a Borel subset of .R. We denote by .H(A, ) the subspace of .H onto which A A .P () projects, and by .H(A, a) the subspace of .H onto which .P ({a}) projects. Further, we represent a determinate physical property by an ordered pair .A, a, where . a ∈ R is a member of the spectrum of A; to attribute .A, a to a physical system is the same as: assigning value a to physical magnitude (represented by operator) A. In consonance with current terminology in metaphysics, we shall call physical magnitude A that pertains to physical system .S a determinable physical property of .S. A postulate that has become known under the misnomer ‘the eigenstateeigenvalue link’ provides a criterion for the ascription of determinate properties to physical systems depending on their state:  Eigenlink. A physical system .S having pure physical state . |ψ ∈ H has determinate physical property .A, a iff . |ψ lies in the eigen-subspace of A that belongs to a, that is, . |ψ ∈ H(A, a). This Eigenlink is limited to operators with a discrete spectrum. A generalistion to all types of spectra, discrete, continuous and combinations thereof, is possible, provided one is prepared to attribute ‘vague’ properties, mathematically represented by .A, :  Generalised Eigenlink. A physical system .S having pure physical state . |ψ ∈ H has determinate physical property .A,  , where . ⊂ R is an interval, iff . |ψ lies in the eigen-subspace of A, that is, . |ψ ∈ H(A, ). The Eigenlink is the special case of the Generalised Eigenlink when the spectrum is discrete and . is the singleton-set of some single eigenvalue: . = {aj }. The Generalised Eigenlink is how von Neumann put it in Gundlagen, as a corollary of representing properties by projectors (1932, item .(β), p. 253); it is a straightforward generalisation of the Eigenlink. But one may frown about a property that is mathematically represented by .A, , due to . being a subset of .R and generically containing non-denumerably many values of A. From the attribution of .A,  to physical system .S, we are not supposed to infer, absurdly, that .S then jointly possesses property .A, a for every . a ∈  that lies in spectrum of A. Some have argued this is yet another quantum-mechanical novelty, not entirely unfamiliar to metaphysicists: a determinate vague property, with sharp boundaries, so not vague in the standard sense.6 If one rejects the idea of a vague property, perhaps because it is unfathomable, then one can remain aboard with the Eigenlink and its sharp determinate properties, and throw the Generalised Eigenlink overboard. 6 See

e.g. French and Krause (1995), and Bush et al. (1996, p. 127), who talk about “vague objectification”.

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Finally, a postulate that is not a postulate of standard QM, but is a postulate of nearly every other interpretations of QM, is the following one.  Universal Dynamics Postulate. Time is represented by the real continuum .(R). For every physical system .S, there is some connected continuous Lie-group of unitary operators . t → U (t) acting on .H such that the state at time t is . U (t)|ψ(0) = |ψ(t) when .|ψ(0) is the state at time .t = 0.

.

This Lie-group of time-translations is the solution of the Schrödinger equation for the self-adjoint Hamiltonian H , the operator that represents energy. Operator H characterises the physical system and determines how the state of the system changes over time via the Stone–von Neumann Theorem, which theorem associates such a Lie-group uniquely with every self-adjoint operator by means of the equation: . U (t) = exp[−iH t/]. The . Dynamics Postulate of standard QM says the same as the universal one, but conditional on that no measurements are performed (lege infra). For the sake of clarity, standard QM is the theory defined by the State, Magnitude, Spectrum, Probability, Dynamics, Projection, and Symmetry Postulate, and the Eigenlink.7

12.3 The Reality Problem of Measurement Outcomes What is generally known as ‘the measurement problem’, we shall call ‘the Reality Problem of Measurement Outcomes’. We state it as a logical incompatibility of five propositions, taking for granted the relevant parts of mathematics relied on. Before stating the problem, we need to express one more proposition:  Single Measurement Outcome Principle (SMOP). Every performed measurement has a single outcome, provided the pieces of measurement apparatus involved do not malfunction.

.

SMOP seems very much a universal empirical regularity: measurements obtained by properly functioning pieces of measurement apparatus always have single outcomes. In cases where there is no outcome, some involved piece of equipment malfunctioned. In case there is more than one outcome .. . . But that seems impossible. How can a LED or LCD display show more than one number? How can a pointer at any moment of time indicate more than one mark on a scale? Are such measurement events not simply physically impossible? Do we really need a principle (SMOP) to rule out what seem to be physically impossible? SMOP certainly seems a universal empirical regularity fully supported by the practice of performing measurements. But we need to state it nonetheless to make

7 Some of the postulates just mentioned have not been stated yet; they will be stated below, when they are needed.

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the proof below devoid of any logical gaps; and furthermore, we know that there are interpretations of QM that reject SMOP, e.g. the Everett and the Many Worlds Interpretation. We arrive at the first problem.8 This problem is what we call a polylemma: to reject at least one of any number of propositions because they are shown to be jointly inconsistent. ▲ I. Reality Problem of Measurement Outcomes. Granted the relevant background mathematics, and given the State, Magnitude and Spectrum Postulate; then the Universal Dynamics Postulate, the Property Revealing Condition (lege infra), the Single Measurement Outcome Principle (SMOP), and the Eigenlink are jointly inconsistent. Proof Consider the famous Stern–Gerlach experiment, where one performs measurements of the spin of a charged particle after it has passed the inhomogeneous magnetic field of a DuBois magnet. We are going to apply the mentioned postulates and principles to this experiment, which yields a QM-model of this experiment, and show how they clash logically. We have an electron (.e) and a piece of measurement apparatus (.M), with Hilbert-spaces . He = C2 and . HM = C3 , respectively, and Hilbert-space . He ⊗ HM = C6 for the composite system (.e M, State Postulate). Pauli-matrix .σz , an operator acting on .C2 , represents z-spin (Magnitude Postulate), which has two orthogonal eigenvectors in .C2 . The measurement-magnitude (pointer-magnitude, display-magnitude) we represent by operator M, which acts in .C3 ; M has by definition three orthogonal eigenstates, with the following associated determinate properties (Eigenlink): | ↑ : e has determinate property σz , ↑ ; | ↓ : e has determinate property σz , ↓ ; .

|+ : M has determinate property M, m↑  ;

(12.1)

|− : M has determinate property M, m↓  ; |m0  : M has determinate property M, m0  . In state .|m0 , the measurement device .M has been turned on and is ready to measure; M is prepared in this state before the measurement begins. Both .σz and M are selfadjoint operators.9 We are going to measure z-spin of the electron, a process that takes .τ seconds, say. The interaction between .e and .M, codified by the Hamiltonian, is supposed to be

.

8 Bush

et al. (1996, p. 91 ff.) call it, curiously, “the objectification problem”. numerical eigenvalues that .↑ and .↓ symbolise are .+/2 and .−/2, respectively. The values .m↑ , .m↓ and .m0 can be chosen arbitrarily, provided they are different, e.g. .m0 = 5, . m1 = m↑ = +1, .m2 = m↓ = −1; then . M|mj  = mj |mj  , for . j ∈ {0, 1, 2}. 9 The

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a measurement interaction; it determines . t → Um (Universal Dynamics Postulate). But what is a measurement interaction, an interaction between a physical system that is being measured and one that is doing the measuring? One straightforward necessary condition reads that if the measured system possesses some determinate property that is measured, then the measured apparatus must reveal it (an instance of the Property Revealing Condition, see remark 3.◦ below):   Um (τ ) | ↑ ⊗ |m0  = | ↑ ⊗ |+ and

.

  Um (τ ) | ↓ ⊗ |m0  = | ↓ ⊗ |− . (12.2)

Both initial and final states in (12.2) are eigenvectors of . σz ⊗ M. On the basis of the Eigenlink, we then can assign the correct determinate physical properties to .e and to .M. The Spectrum Postulate says that upon measurement of z-spin, we can only find, as measurement outcomes, the two eigenvalues of .σz : .↑ and .↓. Semantic convention has it that physical magnitude .σz having value .↑ or .↓ is the same as saying that .e has determinate property . σz , ↑ or . σz , ↓, respectively, and that .M indicating outcome .m↑ or .m↓ is the same as .M possessing determinate property .M, m↑  or .M, m↓ , respectively. Suppose that initially, at time .t = 0, the composite system is in the following state:   |ψ(0) = α| ↑ + β| ↓ ⊗ |m0  ,

.

(12.3)

with . α, β ∈ C being both non-zero, and . |α|2 + |β|2 = 1. At time . t = τ , the post-measurement state of the composite system is, due to the Universal Dynamics Postulate and requirement (12.2): |ψ(τ ) = Um (τ )|ψ(0) .

= Um (τ ) α| ↑ ⊗ |m0  + Um (τ ) β| ↓ ⊗ |m0 

(12.4)

= α| ↑ ⊗ |+ + β| ↓ ⊗ |− . By the Eigenlink, at time .t = τ , since the state of the composite system is not an eigenvector of operator . σz ⊗M, .e does neither have the determinate z-spin property up, .σz , ↑, nor down .σz , ↓, and .M does neither have the determinate property .M, m↑  nor .M, m↓ , and therefore does not indicate an outcome. This contradicts SMOP.

We end this section with a number of systematic remarks. 1.◦ . First of all, the terminology of ‘Reality’ in the ‘Reality Problem of Measurement Outcomes’ is inspired by the fact that if we describe the measurement interaction unitarily, as in the proof above, none of the possible measurement outcomes becomes real—in general, possession of a determinate property by .S and

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calling that property of .S real, or calling it actual, is saying exactly the same with different words.10 Then non-actual properties can be determinate, but are not real. 2.◦ . Notice that probability is not even mentioned in the Proof. Therefore, no matter how one interprets probability in QM (‘quantum probabilities’), this will never solve the Reality Problem of Measurement Outcomes. Also replacing commutative Kolmogorovian Probability Theory with non-commutative ‘Quantum Probability’ Theory is of no avail when it comes to the Reality Problem of Measurement Outcomes. 3.◦ . Requirement (12.2) on measurement interactions is an instance of the  Property Revealing Condition. If the pure state of the composite system . S M is . |a ⊗ |m0  ∈ HS ⊗ HM , where .|a is an eigenvector of measured magnitude A of physical system .S, so that .S has determinate property .A, a, and if the measurement device .M measures A by operator M, and the unitary measurement interaction is . t → Um (t), then after the measurement has ended, at time . t = τ , the state of . S M is such that .M reveals that property whilst .S may have lost it (the state is then an eigenvector of . 1 ⊗ M ); below . |φ ∈ HS is any state of .S and . |ma  ∈ HM is the state of .M that correlates with . |a ∈ HS :

.

  Um (τ ) |a ⊗ |m0  = |φ ⊗ |ma  .

.

(12.5)

When . |φ = |a, one speaks of an ideal measurement: the state .|a of .S is left undisturbed and .S still has the determinate property .A, a in the post-measurement state (12.5) that it had in the initial state. When . |φ =  |a, one speaks of a disturbance measurement: in the post-measurement state, .S will then have lost property .A, a, due to the measurement interaction. (You read your weight while standing on scales, leave the scales, and then have lost your weight—Quantum Weight Watching.) In the proof above, we assumed that the measurement interaction was ideal, leading to requirement (12.2). The proof remains intact when we consider disturbance measurements, as we shall point out next. Applied to the Stern–Gerlach experiment, we then have for the final states of .e M:     .Um (τ ) | ↑ ⊗ |m0  = |u ⊗ |+ and Um (τ ) | ↓ ⊗ |m0  = |v ⊗ |− , (12.6) where . |u, |v ∈ C2 can be any states of .e. Then the post-measurement state of . M e is not (12.4) but becomes: α|u ⊗ |+ + β|v ⊗ |− ,

.

(12.7)

call only possessed properties .A, a ‘determinate’ leaves one without terminology for such properties when they are not possessed. Not a good thing. We call them: mere possible but not actual determinate properties.

10 To

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which is neither an eigenstate of . σz ⊗ M , nor of . 1 ⊗ M, and therefore .M does not indicate an outcome. The logical clash with SMOP remains within deductive reach when we replace ideal with disturbance measurements. Since . |v, |u ∈ C2 , they are superpositions of .| ↑ and .| ↓ (standard basis of .C2 ). This implies that the disturbed post-measurement state (12.7) has terms . | ↑ ⊗ |− and . | ↓ ⊗ |+ , suggesting that the coefficients in front of these terms yield the probability of .M indicating the wrong outcome (‘false positives’ and ‘true negatives’). 4.◦ . If standard QM were to include all premises mentioned in the Reality Problem of Measurement Outcomes, then standard QM would be inconsistent. Standard QM escapes the inconsistency argument narrowly because it rejects the Universal Dynamics Postulate (which implies that measurement interactions are unitary), and replaces it with a conditional version:  Dynamics Postulate. Time is represented by the real continuum .(R). IF no measurements are performed on physical system .S during time interval . I ⊆ R, THEN there is some connected continuous Lie-group of unitary operators . t → U (t) acting on .H such that, when .|ψ(0) represents the state at time .t = 0, the state at every time . t ∈ I is:

.

U (t)|ψ(0) = |ψ(t) .

(12.8)

This raises, of course, the question what happens when a measurement is performed. For that case, von Neumann advanced another conditional postulate, such that the two postulates are mutually exclusive and jointly exhaustive:  Projection Postulate. IF one performs a measurement of physical magnitude represented by operator A on physical system .S, when .S has state . |ψ(t) ∈ H at the moment .t ∈ R of measurement, AND one finds spectrum value in interval .  ⊂ R as the outcome (. being the measurement accuracy), THEN immediately after this measurement outcome has been obtained, the postmeasurement state of the physical system is . P A ()|ψ(t), where .P A () is the projector that projects onto the eigen-subspace . H(A, ). The phrase ‘immediately after’ can be made mathematically precise in terms of upper and lower limits, but we gloss over this. Most interpretations of QM reject a different premise of the ones mentioned in the Reality Problem of Measurement Outcomes to avoid inconsistency. The Copenhagen Interpretation follows standard QM by adopting the Projection Postulate and amending the Universal Dynamics Postulate. Everett and Many Worlds reject SMOP. Rovelli’s Relational QM somehow amends the Eigenlink. Modal Interpretations reject (one conjunct of) the Eigenlink. Bohmian Mechanics adopts a stronger state postulate, an additional postulate for worldlines of particles, and an involved story about measurements (reducing them all to position-measurements); it escapes the contradiction by never having superpositions of worldlines. Spon-

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taneous collapse interpretations prevent repugnant superpositions of states of macroscopic physical systems to occur by replacing the Dynamics Postulate with a different one, positing some non-linear, and hence non-unitary change of state over time.

12.4 Comparison to Maudlin’s Three Measurement Problems We analyse Maudlin’s well-known three measurement problems. Problem 1 Maudlin (1995) discerned three measurement problems and one of them closely resembles the polylemma we have called ‘The Reality Problem of Measurement Outcomes’ (Maudlin: “Problem 1: the problem of outcomes”). Maudlin took the State, Magnitude and Spectrum Postulate for granted and did not even care to mention them; he showed the inconsistency between the following three “claims” (our italics): 1.A The wave-function of a system is complete, i.e. the wave-function specifies (directly or indirectly) all of the physical properties of a system. 1.B The wave-function always evolves in accord with a linear dynamical equation (e.g. the Schrödinger equation). 1.C Measurements of, e.g., the spin of an electron always (or at least usually) have determinate outcomes, i.e., at the end of the measurement the measuring device is either in a state which indicates spin up (and not down) or spin down (and not up).

Claim 1.A asserts that the state of every physical system .S must somehow yield all determinate properties of .S. We point out that the Eigenlink states a criterion that precisely achieves this. Hence Claim 1.A is more general: the Eigenlink implies Claim 1.A, but Claim 1.A does not imply the Eigenlink. The completeness of the state must be taken to imply that only the state and nothing else, autonomously determines what the possessed determinate properties are, notably not in combination with the measurement context. Claim 1.B follows from the Universal Dynamics Postulate, because unitary operators are linear mappings on Hilbert-space. Yet 1.B says more generally that the function . t →  |ψ(t) governing the change of state over time is ‘linear’. Claim 1.C equates (1) .M showing a determinate outcome to (2) .M being in a relevant eigenstate. This claim is a terse combination of SMOP, the Spectrum Postulate and the Eigenlink, which three distinct propositions ought to have to been unsnarled. Maudlin writes (1995, p. 8): So if 1.A and 1.B are correct, 1.C must be wrong. If 1.A and 1.B are correct, z-spin measurements carried out on electrons in x-spin eigenstates will simply fail to have determinate outcomes.

The post-measurement state . | (τ ) (12.4) is a superposition of . | ↑ ⊗ |+ and β| ↓ ⊗ |− . To deduce that neither the measured system has spin-z properties nor the measuring device displays the relevant outcomes in this state, one needs to

.

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assume that being in an eigenstate is necessary for the possession of these properties, which is ‘half’ of the Eigenlink. When starting to present his second measurement problem, Maudlin informs the reader that he has taken the reader for a ride when expounding Problem 1: The three propositions in the problem of outcomes are not strictu sensu [sic] incompatible. We used a symmetry argument to show that .S ∗ [our .|ψ(τ ) (12.4)] could not, if it is a complete physical description, represent a detector which is indicating ‘UP’ but not ‘DOWN’ or vice versa. But symmetry arguments are not a matter of logic. Since we have not discussed any constraints on how the wave-function represents physical states, we could adopt a purely brute force solution: simply stipulate that the state .S ∗ represents a detector indicating, say, ‘UP’. Then 1.A, 1.B and 1.C could all be simultaneously true.

Maudlin buried his Problem 1 right after having expounded it. A hidden premise in his argument was, peculiarly, some ‘symmetry assumption’, having to do with √ the equal coefficients .1/ 2 in the spin-singlet state. Since we did not need such an assumption at all in our proof of the Reality Problem of Measurement Outcomes, Maudlin’s proof cannot be the same as our proof. In the proof of inconsistency of Problem. 2 (another polylemma), this ‘symmetry assumption’ is relaxed, and hopefully we shall have a proof of inconsistency stricto sensu. Problem 2 Maudlin (1995, p. 11) carries on to present a resembling yet different threesome of claims, which are also mutually inconsistent (our italics): 2.A The wave-function of a system is complete, i.e. the wave-function specifies (directly or indirectly) all of the physical properties of a system. 2.B The wave-function always evolves in accord with a deterministic dynamical equation (e.g. the Schrödinger equation). 2.C Measurement situations which are described by identical initial wave-functions sometimes have different outcomes, and the probability of each possible outcome is given (at least approximately) by Born’s rule. Claim 2.A is identical to Claim 1.A. The difference between Claim 2.B and claim 1.B is that the ‘linear’ has been replaced with ‘deterministic’. Both claims 1.B and 2.B follow from the Universal Dynamics Postulate but sting at different properties of the unitary evolution. We can therefore restrict our attention to the only substantial difference between Maudlin’s Problem 1 and Problem 2, which is Claim 2.C. The first conjunct of Claim 2.C asserts that the relation between wave-functions and measurement outcomes is not a function, from .H to the spectrum of A, for every physical magnitude A: different measurement outcomes, different wavefunctions.11 Let’s call this a specification function, in consonance with the terminol-

11 More precisely: not related by a global phase factor, so stricto sensu a function from a partition of .H to the spectrum of A, with equivalence relation: . |ψ ∼ |φ iff there is some .α ∈ C such that . |φ = α|ψ.

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ogy of Claim 2.A (below .[H]∼ is the partition of rays, and Sp.(A) is the spectrum of A): fA : [H]∼ → Sp(A), [ψ] → fA (ψ) .

.

(12.9)

Of course there exists an infinitude of specification functions in the mathematical domain of discourse, but Claim 2.A asserts that one of these functions is somehow ‘realised in nature’; this function represents a relation in physical reality, just as each moment in time, one Hilbert-vector (better: one ray) of the non-denumerably many Hibert-vectors represents the state of a physical system, and all others do not represent the state at that moment. Claim 2.A essentially calls this the ‘completeness’ of the wave-function. The second conjunct of Claim 2.C involves the Probability Postulate of QM:  Probability Postulate. Suppose we perform a measurement on a physical system .S of physical magnitude (represented by self-adjoint operator) A at time . t ∈ R while the physical state of the system is represented by . |ψ(t) ∈ H. Then the probability of finding upon measurement some value in Borel set . ∈ B(R) is given by the Born probability measure: Pr|ψ(t) (A : ) = ψ(t)|P A ()|ψ(t) .

.

(12.10)

Problem 2 is indeed different from Problem 1, and different too from our Reality Problem of Measurement Outcomes, precisely because it involves probabilities. Maudlin (ibid.): The inconsistency of 2.A, 2.B and 2.C is patent: If the wave-function always evolves deterministically (2.B), then two systems which begin with identical wave-functions will end with identical wave-functions. But if the wave-function is complete (2.A), then systems with identical wave-functions are identical in all respects. In particular, they cannot contain detectors which are indicating different outcomes, contra 2.C.

If the state (wave-function) determines the determinate properties probabilistically, then the same state does not specify which determinate properties are possessed, and Maudlin’s argument collapses. The argument is valid if, and only if, the completeness of the state (2.A) is supposed to entail that the state specifies all possessed determinate properties non-probabilistically, or determines them, say, so that the same state always specifies the same possessed determinate properties, as Maudlin says (1995. p. 11), and as e.g. the Eigenlink ordains. The Eigenlink provides a specification function (12.9) for every A: fA (ψ)|ψ = A|ψ .

.

(12.11)

The inconsistency of Maudlin’s Problem 2 occurs already between the Probability Postulate (second conjunct of Claim 2.C), and the weaker claim that every

12 Six Measurement Problems of Quantum Mechanics

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state . |ψ ∈ H of system .S is measurement-complete, which is to say that .|ψ, in combination with an ideal measurement apparatus (.M) in some initial state, determines a unique measurement outcome .ma for every physical magnitude A, correlated to value a from the spectrum of A (call it claim 2.A. ). A relevant specification function .gA can be introduced that sends, for each A and A-measuring operator M of .M, rays in .HS ⊗ HM to ordered pairs of values from the spectra of A and M: gA : [HS ⊗ HM ]∼ → Sp(A) × Sp(M), [ ]∼ → gA ( ) = a, ma  .

.

(12.12)

Or perhaps only a specification function for M, similar but not identical to .fA (12.9): mA : [HS ⊗ HM ]∼ → Sp(M), [ ]∼ → mA ( ) = ma .

.

(12.13)

Consider a state that is a superposition in the measurement basis and we are done: the measurement outcome should always be the same due to assumed the measurement-completeness of the state, which is contradicted by the Probability Postulate because it generically gives non-zero probabilities for other measurement outcomes whilst the system is in the same state. The addition of the Universal Dynamics Postulate (which implies Claim 2.B) is logically superfluous. Claim 2.A, expressing the property-completeness of every state, implies 2.A. , which is only about measurement outcomes and not about possessed determinate properties. We can strengthen Maudlin’s Problem 2 as the ▲ II. State Completeness Problem. Given the State, Magnitude and Spectrum Postulate. Then the Probability Postulate is incompatible with the measurementcompleteness as well as the property-completeness of the state. The absence of the deterministic character of the change of state over time, as in the Universal Dynamics Postulate, and of any postulate about how states change over time for that matter, makes the State Completeness Problem irrelevant for the question whether or not measurement interactions are unitary or not. But the situation of Problem 2 is logically even worse—or better? Claim 2.A, stating the property-completeness of the state, by asserting the realisation in nature of some specification function .fA (12.9), almost contradicts the first conjunct of Claims 2.C, which denies that the relation between states and measurement outcomes is a function; this comes down to denying the realisation in nature of any specification function .gA (12.12) or .mA (12.13). Almost contradicts, we wrote, because to obtain a contradiction, we only have to add that the specification functions are consistent: if |  = |ψ ⊗ |φ , then gA ( ) = fA (ψ), mA (φ) .

.

(12.14)

Besides the Dynamics Postulate (Claim 2.B), even the Probability Postulate drops out now (second conjunct of Claim 2.C). Otiose. We shall not elevate this inconsistency to another polylemma ‘measurement problem’, forcing one to choose

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F. A. Muller

between the State, Magnitude and Spectrum Postulate, and the consistency of the specification functions (12.14), because it is silly to state the property-completeness of the state in one premise (Claim 2.A) and nearly to deny it in another premise (first conjunct of Claim 2.C). Since standard QM includes all of the four mentioned postulates, the State Completeness Problem implies that QM is measurement- as well as propertyincomplete, which points into the direction of indeterminism. This is similar to but not the same as the incompleteness conclusion of Einstein et al. (1935), but now reached without having to assume any locality condition or to employ entangled states of two particles, and, trotting in the footsteps of Fine (1986, Ch. 3), perhaps even closer to Einstein’s intentions.12 Not the same as, we wrote, because for EPR, completeness can only be established by first knowing which determinate properties are possessed by means of their reality condition (‘elements of physical reality’), and then inquiring into whether QM permits or forbids their possession. The measurement- and state-incompleteness is only a problem for determinists. Friends of standard QM, ready accept indeterminism governing physical reality at the scales of the tiny and the brief, will see nothing problematic about the ▲ State Completeness Problem; they will take it as an expression of the indeterministic character of QM. Problem 3 Maudlin’s third measurement problem (“the problem of effect”) concerns some subspecies of one species of interpretation of QM, namely the Modal Interpretation, and it concerns repeated measurements. In a nutshell, the problem is that the measurement outcome of one measurement, which reveals a possessed determinate property of the measured system according to Modal Interpreters, has no effect on subsequent measurement results, which also reveal possessed determinate properties. Maudlin does not deduce a contradiction from explicit claims, and therefore Problem 3 is not in the same logical category as his other two problems.13 To recapitulate, Maudlin buried Problem 1, but with some tweaking, and dispensing with his peculiar and surreptitious ‘symmetry assumption’, it becomes the Reality Problem of Measurement Outcomes (p. 230). We could improve on Maudlin’s Problem 2 by arriving at a contradiction between fewer premisses; in fact between, granted a few uncontroversial postulates of standard QM: the Probability Postulate and the claim that the state is measurement-complete, which is implied by being property-complete (State Completeness Problem). Both Problem 2 and Problem 1 are problems that compel one to reject at least one of a number of premises. Problem 1 is not a problem of standard QM due to its Projection Pos12 Traditionally, ‘the completeness problem’ is whether there is another theory, a ‘hidden-variables theory’, with additional degrees of freedom, that performs empirically just as good as standard QM but is not haunted by the problems presented in this paper. See Bub (1974, Ch. II). 13 G. Bacciagaluppi has suggested that the modal interpretation is a red herring in Problem 3, and that Problem 3 points at the general issue of repeated measurements, which could be elevated to a seventh measurement problem. Private communication, Utrecht, October 2022.

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tulate, and Problem 2 (State Completeness Problem) can be seen as a simple proof in QM of the measurement- and property-incompleteness of the state, expressing its indeterministic character—only a problem for determinists. Problem 3 is not a problem for standard QM, but for a subspecies of the Modal Interpretation of QM: ultimately it states the open problem of finding a dynamics of possessed properties, in light of the fact that measurement-outcomes seem to be irrelevant for subsequent property ascriptions in most modal interpretations. We move on to four other measurement problems.

12.5 The Probability Problem of Measurement Outcomes Introduction So-called ‘Insolubility Theorems’ suggest that the Reality Problem of Measurement Outcomes is insoluble. Wrong suggestion. These theorems are only about probability distributions of measurement outcomes when the measurement interaction is taken to be unitary, as implied by the Universal Dynamics Postulate. What the Reality Problem of Measurement Outcomes has in common with the Probability Problem of Measurement Outcomes (lege infra) is that both make trouble for taking measurement interactions to be unitary. A difference is that this new Probability Problem crucially involves mixed states and probability—both are absent from the Reality Problem of Measurement Outcomes. The core of the insolubility proof goes back straight to von Neumann’s discussion of the measuring process, in Section VI.3 of his Grundlagen (1932). We first expound this core as applied to the same Stern–Gerlach experiment we used in the proof of the Reality Problem of Measurement Outcomes. Then we ascend to levels of utmost generality. Core and Special Case The State Postulate needs to be extended from Hilbertvectors representing pure physical states to state operators, aka density operators, which are by definition self-adjoint, positive, trace 1 operators, collected in convex set S(H). On the boundary of this set, one finds 1-dimensional projectors, which are the pure states because they correspond one-one to (rays of) Hilbert-vectors: |φ and Pφ = |φφ|. The Probability Postulate generalises from (12.10) to von Neumann’s celebrated trace-formula, for state operator W ∈ S(H):   W A .Pr (A : ) = Tr W P () . (12.15) The Projection Postulate also generalises to mixed states, given by Lüders’ formula; but we shall not need it here and therefore gloss over it.14 We consider the Stern–Gerlach experiment again. Suppose the initial state of the electron (e) is a pure state: We (0) = P↑z ≡ | ↑↑ | .

.

14 Bush

et al. (1996, pp. 31, 40–41).

(12.16)

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F. A. Muller

Suppose the initial state of the measurement device (M) is mixed; we write it as a convex combination of orthogonal pure states, each of which projects on an eigenvector of the measuring operator M: WM (0) = w0 P0M + w1 P+M + w2 P−M .

.

(12.17)

where the real coefficients wj ∈ [0, 1] sum up to 1. Combination (12.17) is unique by the Spectral Theorem. The ideal measurement interaction Um (t) then leads to the following final state (at time t = τ ), using (12.16) and (12.17): W (τ ) = Um (τ )W (0)Um† (τ )   = Um (τ ) We (0) ⊗ WM (0) Um† (τ ) .     = w0 Um (τ ) P↑z ⊗ P0M Um† (τ ) + w1 Um (τ ) P↑z ⊗ P+M Um† (τ )   + w2 Um (τ ) P↑z ⊗ P−M Um† (τ ) (12.18) Being a projector is invariant under unitary transformations. Since P↑z ⊗ PkM are projectors on HS ⊗ HM , the final state is a convex combination of orthogonal pure states corresponding to vectors (cf. footnote 230, p. 230):   Um (τ ) | ↑ ⊗ |mj  .

.

(12.19)

Enter ensembles.15 Suppose we have large number of copies of composite systems e M, and we want to characterise ensembles by mixed state operators. Suppose further that every copy of e initially is the same pure z-spin-state, characterised by P↓z or by P↑z . Such an ensemble is called homogeneous, and is characterised by some pure state operator such as We (0) (12.16). Every copy of M in the ensemble is also in some pure state, but we assume not in the same one, and we do not know in which one. Such an ensemble is called heterogeneous, and characterised by WM (0) (12.17). The coefficient wj is the probability that a copy of M is in pure state PjM , in agreement with the trace-formula (12.15): 2    Tr WM (0)PjM = wk δkj = wj .

.

(12.20)

k=0

This is called the ignorance interpretation of mixtures. Can we now also interpret final state W (τ ) (12.18) in this fashion? That is, every copy of e M is in a pure state

15 von

Neumann (1927, 1932).

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241

P↑z ⊗PjM , and hence by the Eigenlink, e possesses determinate property σz , ↑, and M possesses accompanying determinate property M, mk  ? Our ignorance about the initial state of a copy of M in the ensemble is preserved as the same ignorance about the final state of that copy of M. Let us now consider a different initial pure state of e: |φ(0) = α| ↑ + β| ↓ ∈ He = C2 .

.

(12.21)

The initial state of the ensuing ensemble of copies of the composite system e M , where we retain the same mixed state WM (0) for M as before (12.17), then is: W (0) = |φ(0)φ(0)| ⊗ WM (0) .

.

(12.22)

The final, post-measurement state becomes: W (τ ) =

2 

.

  wj Um (τ ) |φ(0)φ(0)| ⊗ PjM Um† (τ ) ,

(12.23)

j =0

which is a heterogeneous mixture of three pure states:   Um (τ ) P|φ(0) ⊗ PjM Um† (τ ) .

.

(12.24)

Remarkably, the probability that measurement outcome mj obtains equals again wj , just as in the initial mixed state WM (0) (12.17), as expressed in (12.20). This probability does not depend on the initial state of e: the coefficients α, β ∈ C, characterising the initial pure state of e (12.21), are absent from (12.18), as also expressed in (12.20). This is in conflict with a second requirement on Um (t) to qualify as a measurement interaction, a condition that involves probabilities, which we did not need in the Reality Problem of Measurement Outcomes. We shall state it below in full generality.16 General Case First some general stage setting. We consider a physical system S, a measurement device M, their composite system S M, and their sets of mixed states S(HM ), S(HS ) and S(HS ⊗ HM ), respectively. Physical magnitude A of S we take to be self-adjoint. We subdivide the scale M of M, which is the spectrum of A-measurement operator M, in N intervals Ij ⊂ M , and mj being the midpoint of Ij ; the equal with of Ij coincides with the measurement accuracy. When we include the ready-to-measure pure state of M, then N + 1 orthogonal states of M suffice, which means that HM is finite-dimensional, with dimension N + 1. Suppose we can measure part A of the spectrum of A, perhaps even of the entire spectrum of A. A calibration function sends spectrum-values of A to measurement outcomes

16 Bush

et al. (1996, p. 29).

242

F. A. Muller

(spectrum-values of M): g : A → M , a → g(a)

.

(12.25)

One assumes g to be one-one and continuous, so that g correlates values in A to values in M perfectly. Just as the N intervals Ij partition measurement scale M , intervals g inv (Ij ) ⊂ A partition part A of the spectrum of A. The unitary measurement evolution t → Um (t) sends the initial mixed state W (0) to the mixed final state:   W (τ ) = Um (τ ) WS (0) ⊗ WM (0) Um† (τ ) .

.

(12.26)

Hence we arrive at the:  Probability Reproducibility Condition. The Born–von Neumann probability measure (12.15) for A in the initial state of S is the same as the probability measure of M for M in the final state of the composite system S M when unitarily evolved by Um (12.26):   PrWS (0) (A : j ) = PrW (τ ) 1 ⊗ M : g inv (j ) .

.

(12.27)

For the Stern–Gerlach case, we then must have, for arbitrary pure initial state |φ(0) of e (12.21), using (12.22) and (12.23): w1 = |α|2 ,

.

w2 = |β|2

and

w0 = 0 .

(12.28)

So for every initial mixed state WM (0) (12.17) with coefficients w1 and w2 different from |α|2 and |β|2 , respectively, we have a logical clash with the Probability Reproducibility Condition via Eqs. (12.28). This is essentially a proof of the core of the:17 ▲ III. Probability Problem of Measurement Outcomes. Granted the Mixed State Postulate and the Magnitude Postulate. Then the Probability Postulate, the Universal Dynamics Postulate, and the Probability Reproducibility Condition are jointly incompatible. A few supplementary remarks about this polylemma problem. (a) Notice that not among the six jointly inconsistent premises are: the Spectrum Postulate, the Eigenlink and SMOP, which are members of the inconsistent bouquet of the Reality Problem of Measurement Outcomes. (b) What von Neumann pointed out (lege supra) is the core of the proof. Wigner critically discussed von Neumann’s considerations and repeated the core (1963,

17 Cf.

Theorem 6.2.1 in Bush et al. (1996, p. 76).

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p. 12). Fine (1970) fashioned it into a strengthened theorem with a proof. Fine’s proof sadly was “seriously defective”, as Stein (1997, p. 233) would put it; Shimony (1974) performed a repair job. Bush and Shimony (1997) extended the Insolubility Theorem from self-adjoint operators to positive operators (‘unsharp’ physical magnitudes). Brown (1986, p. 862) claimed to provide “a simple and transparent proof”, but entangled it with the ignorance interpretation of mixtures and privileged convex expansions. Stein (1997, p. 240) flogged Brown for this: This simple proof in question is not a proof of the theorem I have presented here, or of the theorem demonstrated by Shimony; nor is it a proof of the theorem stated by Fine. What it establishes is something very much weaker — which, however, Brown maintains, is the only thing that genuinely bears on the problem of measurement.

Stein (1997, pp. 240–241) ends as follows: In other words, Brown’s proposal is the one already discussed in Section 1, above. Setting aside any questions about the viability of the notion of the “real mixture” — the notion, that is, that a quantum statistical state should be characterized by more than its assignment of probabilities to values of dynamical variables and, in particular, that such a state should be thought of as an assignment of something like probabilities to pure states — it has there been pointed out that with such a conception of the state it is trivial that appeal to the mixed initial state of the apparatus can contribute nothing to the measurement problem. It would hardly have been necessary for such a man as Wigner to undertake an examination of the question.

(c) The proposal Brown rules out with his ‘Insolubility Theorem’, and what Stein discusses and dismisses in his introductory Section 1, is the impossibility of a unitary measurement interaction such that the final, post-measurement state is a mixture of pure states of the measuring magnitude M to which an ‘ignorance interpretation of mixtures’ can be applied. Recall that the ignorance interpretation of mixtures is the idea that when we write W ∈ S(H) as some convex combination of pure states: W =

N 

.

wn Pn ,

(12.29)

n=0

we should think of W characterising an ensemble, each member of which is in a pure state Pn with probability wn . When we choose for Pn orthogonal members of the spectral resolution of magnitude A having a discrete spectrum, then the Eigenlink permits us to say that each member of the ensemble has determinate property A, an , where Pn then projects on eigensubspace H(A, an ). Call such a convex expansion an A-expansion. We can also choose a B-expansion for W such that B does not commute with A. Since non-commuting operators generically have no common eigenvectors, an ignorance interpretation of W in terms of pure eigenstates of both A and of B is impossible.

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F. A. Muller

To save the ignorance interpretation of mixtures, one can privilege certain convex expansions and permit an ignorance interpretation of only those privileged ones. This is essentially what Brown (1986) does. Then Brown further requires that if the expansion of the initial state of the composite system S M is M-privileged: W (0) =

N 

.

wk PjA ⊗ PkM ,

(12.30)

k=0

where every copy of S is taken to be in pure state PjA , and A is the magnitude of S that M is measuring, then the final state has the following privileged expansion: W (τ ) =

N 

.

  wk Um (τ ) PjA ⊗ PkM Um† (τ ) .

(12.31)

k=0

By considering empirically distinguishable initial pure states of S, and showing that a final state ensues with weights identical to weights of the initial mixed state of M, these final states are empirically indistinguishable, every term being an eigenstate of 1 ⊗ M. This contradicts the Probability Reproducibility Condition, but also the weaker condition that empirically distinguishable initial states of M ought to lead by Um to empirically distinguishable final states of M. Does the reliance of Brown’s proof on A-privileged convex expansions as well as on an ignorance interpretation of mixed states makes it diverge from the line of Insolubility Theorems that started with Fine and originated in von Neumann (1932)? We answer this question in the course of the next remark. (d) Somewhat remarkable is that all proofs of Insolubility Theorems start with a pure state for S and a mixed state for M. Brown (1997, p. 863) indeed wonders why one does not assume that M is prepared in a pure initial state, the ready-tomeasure state |m0 . Has the experimentator been drinking? Techo-House Party in the Laboratory with XTC? Is the pure ready-to-measure state |m0  the only proper initial state for M? No. Eigenvalue m0 can be zillion-fold degenerate, with zillion (Z) different joint states of the octillions of atoms that compose M. But then initial state WM (0) can also be a mixture of precisely these pure states, say Pn0 , n ∈ {1, 2, . . . , Z}. There will be fluke terms in the convex expansion of WM (0) in the M-basis, with epsilonic probability. One then has: 0 WM (0) = WM (0) +

N 

.

j =1

wk PjM =

Z  n=1

vn Pn0 +

N  j =1

wk PjM ,

(12.32)

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such that the sum of all wj for j  1 being equal to ε, where 0 < ε  1, and the sum of all vn being w0 . Then, when P0M projects onto the eigenspace HM (M, m0 ), we have: P0M =

Z 

.

Pn0

and

  0 Tr WM (0)P0M = 1 − ε ≈ 1 .

(12.33)

n=1

Of course, starting with this ‘realistic’ initial state of M does not makes one deviate from the collision course to the Probability Reproducibility Condition. Contrastively, why restrict the initial state of S to be pure? That seems an unnecessary restriction. Suppose we were to start with a mixed initial state of S, and pure initial state P0M ∈ S(HM ) of M: W (0) = WS (0) ⊗ WM (0) =

N 

.

pk PkA ⊗ P0M =

k=0

N 

pk PkA ⊗ |m0 m0 | .

k=0

(12.34) Then the final state would be: W (τ ) =

N 

.

  pk Um (τ ) PkA ⊗ P0M Um† (τ ) .

(12.35)

k=0

Suppose we further were to impose condition (12.2) of ideal measurements on Um (t), so that in terms of state operators:   Um (τ ) PkA ⊗ P0M Um† (τ ) = PkA ⊗ PkM .

.

(12.36)

Then the final state makes one deviate from a collision course to the Probability Reproducibility Condition: W (τ ) =

N 

.

pk PkA ⊗ PkM ,

(12.37)

k=0

because this final state depends on the initial state WS (0) of S. Then the probability for finding M indicating mk is:   PrW (τ ) (M : mk ) = Tr 1 ⊗ M)PkM = pk .

.

(12.38)

Safe at last! One can also start with both S and M in mixed initial states (the most general case conceivable), and obtain, in case of ideal measurements, a final state that has

246

F. A. Muller

a convex expansion in pure states PjA ⊗ PjM with coefficients pk wj . Then for various choices of these coefficients, one can collide again with the Probability Reproducibility Condition. The conclusion is that problems only arise when the initial state of M is mixed. We are now in a good position to answer the question posed at the end of the previous remark: Does the reliance of Brown’s proof on A-privileged convex expansions as well as on an ignorance interpretation of mixed states makes it diverge from the line of Insolubility Theorems that started with Fine and originated in von Neumann (1932)? As we have seen above in remark (b), Stein answered harshly in the affirmative and trashed Brown’s version. But this does not sit comfortably with Brown’s motivation, which crucially involves the ignorance interpretation of mixtures (vide supra, p. 240). This motivation is an attempt to interpret the statistical spread in measurement outcomes of magnitude A, say, when every copy of S is prepared identically, as ignorance about the pure state each member of the ensemble of copies of S M is in after the measurement, which in turn would reflect our ignorance about the pure initial state of M. This suggests that when the initial state of M of the ensemble is mixed, the initial state of the heterogeneous ensemble carries over the heterogeneity of its final state, explaining the spread in measurement outcomes. Every copy of the ensemble would then be in a pure state according to the ignorance interpretation, and would have a determinate property A, aj . We would be on our way to dissolve the Reality Problem of Measurement Outcomes. Brown’s (1986) is indeed titled ‘The Insolubility Proof of the Quantum Measurement Problem’, and his requirement ‘RUE’ (ibid., p. 860) then makes sense, which is that if the initial mixture is A-privileged (“the real mixture”), then the final, post-measurement mixture written in basis Um (τ )|aj aj | is the real one when the measurement interaction is unitary. Alas! The Probability Problem of Measurement Outcomes now teaches us that this way of interpreting the statistical spread in identically prepared physical systems runs afoul against the Probability Reproducibility Condition. Apparently Stein had little patience for this motivation: he considered the idea of an A-privileged expansion (“the real mixture” in Brown’s words) as a non-starter.18 (e) Bacciagaluppi (2014) has recently pointed out that an insolubility theorem follows from the No-Signalling Theorem of QM. Wut? The No-Signalling Theorem says that when two physical systems, S1 and S2 say, are not interacting, the probability measure for A of S1 (of outcomes) does not depend on which magnitude one measures on S2 (settings), and vice versa. This is also known as outcome-setting independence. In case of modelling measurement interactions unitarily, S and M however do interact, and one of them measures the other. So it seems we have two rather different situations on our hands, a non-interacting and an interacting composite system, giving rise to the same statement of independence. The reason for this independence then must be different in each case. And it is: the non-interaction 18 G.

Bacciagaluppi insisted on making this point (private communication, 7 October 2022).

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makes the relevant joint probability measures factorise versus the conditions on the unitary interaction to qualify as a measurement interaction; cf. Bacciagaluppi (2014). (f) Stein (1997) has claimed to provide “the maximal extension” of the Insolubility Theorem: the weakest assumptions and the largest reach. The issue Stein is concerned with is whether there always is a convex expansion of the final state in terms of pure states that are eigenstates of the measuring magnitude M whenever the initial state is thusly expanded. Since commuting operators share their eigenvectors (if they have any), the requirement on the unitary interaction Um (t) to qualify as a measurement interaction just mentioned is the same as the vanishing of the commutator of W (τ ) and M (mathematically more precise: of W (τ ) and 1 ⊗ M):    [W (τ ), 1 ⊗ M]− = Um (τ ) PjA ⊗ WM (0) Um† (τ ), 1 ⊗ M − = 0 ,

.

(12.39)

for every pure initial state PjA of system S. The Lemma that Stein proves is, put slightly more abstractly, as follows. Given bounded operators M, Q ∈ B(H2 ), bounded operator W ∈ B(H1 ⊗ H2 ), and projector P on H1 . If P ⊗ Q and W commute, then there is a unique bounded operator TQ on H2 such that the product of P ⊗ Q and W can be written as ⊗factorised operator P ⊗ TQ . As Stein (1997, p. 236) points out in supplementary remark 3, a consequence of the Lemma is that the existence of TQ is sufficient and necessary for the commutativity of P ⊗ Q and W : [P ⊗ Q, W ] = 0 ⇐⇒ ∃! TQ ∈ B(H2 ) : (P ⊗ Q)W = P ⊗ TQ , (12.40)

.

where TQ depends on Q (whence the subscript) but does not depend on P . The Insolubility Theorem is “an immediate and sweeping consequence” of this Lemma, as Stein (1997, p. 237) puts it. Let the initial state of S M be W (0) = PjA ⊗ WM (0) —system S is assumed to be in pure state PjA initially. The expectation-value for M at the end of the measurement, at time t = τ , is by the trace-formula (12.15):     1 ⊗ MW (τ ) = Tr Um (τ ) PjA ⊗ WM (0) Um† (τ ) (1 ⊗ M) .

.

(12.41)

The trace is invariant under cyclic permutation: 1 ⊗ MW (τ ) = Tr

.



  PjA ⊗ WM (0) Um† (τ ) (1 ⊗ M)Um (τ ) .

(12.42)

Since in general, operator X commutes with U Y U † iff Y commutes with , it follows from (12.39) that also PjA ⊗ WM (0) and Um† (τ ) (1 ⊗ M)Um (τ )

U † XU

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commute. According to the Lemma (12.40), when choosing PjA for P , WM (0) for Q, and W (τ ) for W (12.26), there is a unique bounded operator T ≡ TWM (0) on HM , which does not depend on PjA , and which is such that from (12.42) we obtain:     1 ⊗ MW (τ ) = Tr PjA ⊗ T = Tr PjA Tr(T ) = Tr(T ) .

.

(12.43)

Hence the expectation-value of the measurement operator M at time t = τ is independent of the initial state PjA of system S. In other words, the probability measure over measurement outcomes of A as determined by the initial state of S is not reproduced by the probability measure over values of the measurement operator M of M, thereby scandalising the Probability Reproducibility Condition. We have arrived at the Probability Problem of Measurement Outcomes (p. 242). Moral The moral of the Insolubility Theorem is that describing measurements unitarily (Universal Dynamics Postulate) in terms of mixed states, obeying the Probability Reproducibility Condition, breeds contradictions. (The absence of the Spectrum Postulate, the Eigenlink and SMOP among the premises leading to a contradiction we have already duly noted.) Some will draw the further moral that the Projection Postulate is inevitable. Standard and Copenhagen QM are off the hook. All interpretations of QM that reject the Projection Postulate and adopt a Universal Dynamics Postulate must face the Insolubility Theorem, which includes Modal Interpretations, Rovelli’s Relational Interpretation, and of course Everett and Many Worlds. Whereas rejecting SMOP makes Everett and Many Worlds escape the Reality Problem of Measurement Outcomes, this option is unavailable in the face of the Probability Problem of Measurement Outcomes—it may aggravate their ‘probability problem’. The only way to go, then, for adherents of the Universal Dynamics Postulate (only unitary measurements) seems to deny that measurement devices initially never are in mixed states, but always in the pure ready-to-measure state when the measurement begins. Then one is safe.

12.6 The Reality Problem of the Classical World The Reality Problem of Measurement Outcomes is a reality problem of properties of measurement devices when described unitarily, granted the Eigenlink and SMOP. We have seen how narrowly standard QM escapes the lethal inconsistency, due to its conditional Dynamics Postulate and its Projection Postulate. But there is another ‘reality problem’ about properties not restricted to measurement devices but about all actual physical systems that are not subjected to measurement, which is the overwhelming majority of physical systems in the universe—nearly all of them. When we talk about the world that surrounds us, the world we see, hear, smell, touch and feel, the observed observable world, the ‘manifest world’ (W.F. Sellars), we do this mostly in terms of spatially extended material objects that have properties and are interrelated, subjects and their capacities included. These properties and

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relations may or may not change due to the influence that objects exert on each other (by means of physical interactions). But at each moment in time, every object and every subject possess several properties and exhibits several relations to other subjects and objects. This is often called the Classical World, because the metaphysical picture just sketched, in terms of objects, subjects, possessed properties and exhibited relations, fits classical physical theories like a glove, as it does in fact all other scientific theories as well.19 Predicate Logic follows suit to cannonize the Classical World logically. In terms of QM, the states of every two physical systems that have interacted in the past, or are interacting in the present, will generically be superpositions due to the Dynamics Postulate in bases of eigenstates that we associate with properties that we observe: the state of their composite systems is entangled. But then the Eigenlink prohibits the attribution of these properties to the physical systems, and their interrelations when taken as properties of composite systems. We have arrived at the following profound metaphysical problem. ▲ IV. The Reality Problem of the Classical World. How is the Classical World, a world with physical systems possessing properties and exhibiting relations, compatible with QM, specifically in the light of its Eigenlink and the generically entangled states of physical systems? The Reality Problem of Measurement Problems (p. 230) can be seen as a very special case of the Reality Problem of the Classical World, where we consider two physical systems, a measurement device and a measured physical system, and let them interact unitarily, so that by vice of the Eigenlink we end up with two systems devoid of the properties we believe they must have when the measurement has ended. To repeat, the Projection Postulate saves the day for standard QM. But this leaves physical systems in the universe unmeasured by us without any properties and relations at all, which is, to repeat, nearly everything in the universe. The Classical World is lost. Interpretations of QM aim to regain the Classical World: it is the very reason of their existence. A different manner to express roughly the same problem are so-called problems of the classical limit: when processes become slow and physical systems macroscopic (constituted by very many particles), what QM then says about them must be approximately (‘in the limit’) the same as what the appropriate classical physical theories say about them, e.g. classical mechanics, classical electro-dynamics, thermo-dynamics, optics. This are inter-theoretical problems, about ‘limiting-relations’ between QM and other physical theories, predicated on the assumption that these other physical theories describe the macroworld correctly. These problems of the classical limit, and the reverse problem, of ‘quantisation’ (how to get from these theories to QM), has been and is an intense area of theoretical

19 Physical theories accepted from the Scientific Revolution in the seventeenth century onwards until 1900 have been baptized ‘classical’; the ones from 1900 onwards are then called ‘modern’.

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and mathematical research.20 Yet even if all the problems of classical limits have been solved, in the regimes where the mentioned theories fail and QM must take over, then in the world of the brief and the tiny, we still have neither properties nor relations. The Reality Problem of the Classical World then is at best partly solved, not completely.

12.7 The Measurement Explanation Problem The two postulates of standard QM that mutually exclude and jointly exhaust the change of state over time (Dynamics and Projection Postulate) evoke the question: why two, and why these two? More specifically, to measure something is to act, and to act is to do something with a purpose: gathering knowledge about a physical system in the case of measuring. To act is a manifestation of human agency. Indeed, human agency, because pickles, protons, peanuts, pandas and planets do not and cannot measure anything. Measurement is an anthropomorphic concept, and this concept occurs in both postulates governing the change of state of physical systems everywhere in the universe. This is without precedent in the history of physics, and perhaps of natural science. Von Neumann spoke of two types of processes in the universe, insipidly calling them Prozess 1 (measurement processes) and Prozess 2 (unitary processes, see Fig. 12.1b, p. 251). Measurement processes are indeterministic, discontinuous and non-linear, whereas unitary processes are deterministic, continuous and linear. Somewhat anachronistically, one could submit that Aristotle’s distinction between artificial and natural processes has been resurrected, like Lazarus from the dead. A why-question is a request for an explanation, so here we go: ▲ V. The Measurement Explanation Problem. Why is there an anthropomorphic concept of human agency, the concept of measurement, present in the postulates of a theory of inanimate matter (QM)? Why do physical interactions between physical systems obey anthropomorphic laws of nature as we use them in measurements? Of course, we have already four distinct fundamental physical interactions that obey distinct laws: electro-magnetic, nuclear (‘strong’), radio-active (‘weak’), and gravitational. Electro-magnetic interaction is a unification of electric and magnetic interaction. The Standard Model sort of unifies the electro-magnetic and the radioactive interaction in the electro-weak interaction, and hypothesizes that in the very early universe, the electro-weak and the nuclear force once were unified. The Standard Model is unification on crutches. The inclusion of gravity has become a head-ache dossier of theoretical physics: the Holy Grail of a theory of quantum gravity. So what’s the problem that we have a fifth type of interaction, the measurement interaction, governed by yet another distinct law of nature? A hand 20 E.g.

Ehrenfest et al. (1927), Landsman (1998), Bracken (2003).

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physical interactions measurementinteractions

physical interactions

Prozess 2

mental-physical interactions measurementinteractions Prozess 1

Fig. 12.1 (a) How today nearly everybody sees the relation between physical and measurement interactions: measurement interactions are physical interactions. (b) How von Neumann and Wigner saw measurement interactions: mental-physical interactions (consciousness causing collapse)

full of interactions, with one distinctively human finger. What’s the problem with that? Well, for starters, nearly all measurement interactions are electro-magnetic: this is simply how measurement devices work, ‘mechanical’ ones notably included, as a moment of thought will reveal. As soon as we baptise an electro-magnetic interaction between physical systems a measurement interaction, as soon as human agency is involved, it starts to obey a different law of nature. Why is that? Why does Mother Nature switch laws about the very same physical interaction as soon as we show our faces? Standard QM and Copenhagen QM face the Measurement Explanation Problem. Most interpretations adopt a Universal Dynamics Postulate and attempt to describe measurement interactions unitarily, and then face some of the reality problems expositioned above, but they do not face the Measurement Explanation Problem (Fig. 12.1a). A related but distinct problem is the final measurement problem, to which we turn next.

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12.8 The Measurement Meaning Problem Measurement, quantitative observation, qualifies as a species of knowledge acquisition. And all men desire to know, as Aristotle wrote in the opening sentence of his Metaphysics. To measure is, as mentioned in the previous section, a manifestation of human agency, a species of behaviour: moving the human body, or parts of the human body, with a purpose. To measure physical magnitudes teaches physicists what values these magnitudes have, if only at the point of measurement. In all non-quantum physical theories, and in all theories in other scientific praxes, to measure is to reveal what determinate property the measured object possesses. Not so in standard QM. The ascription of determinate properties to measured physical systems happens only at the time when the measurement result comes into being. The appearance of these determinate properties in the world seems to be an event of creatio ex nihilo, as if experimenters and observers are Wizards performing acts of Metaphysical Magic. When we put it in metaphysical vocabulary as transforming a determinable property into a determinate, we are at the level of being (ordo essendi) rather than at the level of knowing (ordo cognoscenti). The physics laboratory has become a place of Ontic Sorcery, rather than mere Epistemic Agency. Well, let’s not get carried away. Perhaps better to say prosaically that the determinate properties are ‘produced by’ the measurement interaction between measured physical system and measuring device. Let’s state the next and last measurement problem: ▲ VI. The Measurement Meaning Problem. What is a measurement? What makes an interaction between physical systems a measurement interaction? What makes a physical system a measurement device? One famous and often-quoted piece of ranting and raving about this problem is Bell’s (1990), his paper ‘Against Measurement!’ in Physics World: What exactly qualifies some physical systems to play the role of ‘measurer’? Was the wavefunction of the world waiting to jump for thousands of millions of years until a singlecelled living creature appeared? Or did it have to wait a little longer, for some better qualified system .. . . with a PhD? If the theory is to apply to anything but highly idealised laboratory operations, are we not obliged to admit that more or less ‘measurement-like’ processes are going on more or less all the time, more or less everywhere? Do we not have jumping then all the time? .(. . .) The first charge against ‘measurement’, in the fundamental axioms of quantum mechanics, is that it anchors there the shifty split of the world into ‘system’ and ‘apparatus’. A second charge is that the word comes loaded with meaning from everyday life, meaning which is entirely inappropriate in the quantum context. When it is said that something is ‘measured’ it is difficult not to think of the result as referring to some preexisting property of the object in question.

Bell (1990) looks in vain in classic texts expounding standard QM (Dirac, von Neumann, Landau & Lifshitz, Gottfried) for clarity and rigour about what a measurement is. Quantum-mechanical measurement theory (absent in the works mentioned by Bell) provides more detailed mathematical representations of measurement interactions, but it leaves the concept of measurement, remarkably, un-

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analysed. Take the authoritive monograph of Bush et al. (1996). In their introductory section on ‘The Notion of Measurement’, they write (ibid., p. 25, their emphasis, our symbols): The purpose of measurements is the determination of properties of the physical system under investigation. In this sense the general conception of measurement is that of an unambiguous comparison: the object system .S, prepared in a state W , is brought into a suitable contact — a measurement coupling — with another, independently prepared system, the measuring apparatus from which the result related to the measured observable A is determined by reading the value of the pointer observable M. It is the goal of the quantum theory of measurement to investigate whether measuring processes, being physical processes, are the subject of quantum mechanics. This question, ultimately, is the question of the universality of quantum mechanics.

The concept of a measurement is not analysed but taken for granted. In general, in philosophy, when faced with the problem of analysing a concept, we can walk two ways: Wittgenstein’s Way and Carnap’s Way. Let’s take a walk. Wittgenstein’s Way In the opening page of The Blue Book, Wittgenstein (1958, p. 1), advances that to answer the question ‘What is length?’, it helps to answer the question ‘How do we measure length?’, and draws the analogy that to answer the question ‘What is the meaning of a word?’, it is better to ask: ‘How is this word used?’ To ask what it means to measure the momentum of a scattered elementary particle in CERN is answered by an experimenter working in CERN explaining you how they do it. To ask what it means to measure the temperature of gas in a vessel, an engineer will show you a manometer and will explain how this instrument works. To ask what it means to measure an electric current in a circuit is answered by explaining how an ammeter works, which is made part of the circuit. And so forth. In general, what it means to measure physical magnitude A of some physical system can be explained by some relevant expert. Residues of unclarity will be cleared up by the expert whenever asked. When we have such explanations of every type of measurement performed by all relevant experts, then we are done. What more is there to explain? According to a use-conception of meaning, there is nothing more to explain. We may draw up lists of rules governing the use of the words ‘to measure’ and ‘measurement’. Is the unsatisfied philosopher not falling victim to the philosopher’s craving for generality and essence? We might seek something that all kinds of measurements have in common, which could then characterise what the concept of measurement is. This is presumably what Bell has been looking for, in vain. Bell craved for generality and essence, like a true philosopher. If there isn’t something that all types of measurement have in common, but every type of measurement has something in common with some other types, then measurement is what Wittgenstein baptised a family-resemblance concept. If satisfied with such a conclusion, the Measurement Meaning Problem evaporates, because it presupposes that all kinds of measurement in physics have something in common, which must be captured by an explication. Bell and most philosophers (of physics) will judge that to end the inquiry into measurement with this Wittgensteinian conclusion is a cop out. Wittgenstein’s Way is not most philosophers’ favourite way. They prefer Carnap’s Way.

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Carnap’s Way An explication of a concept is a criterion for that concept, which is a condition that is both sufficient and necessary. An explication must be an explicit logical combination of other concepts, and should besides being extensionally correct also be intensionally correct.21 Intensional correctness is that explicans and explicandum must be synonymous. Extensional correctness is that the same things fall under the extension of both explicans and explicandum. Inspection of how the concept of measurement is used when walking Carnap’s Way is as unavoidable as it is when walking Wittgenstein’s Way. In a Liber Amicorum for P.C. Suppes, yours truly took a stab at finding an explication of the concept of measurement; we end this section by summarising this explication, with slight improvements.22 We begin by recalling that a physical system .S is observable (to us, human beings) iff whenever an arbitrary healthy human being were in front of .S in broad daylight, and were looking at .S, she would see .S (cf. Muller (2005)). Next observation predicates. Criterion for an Observation Predicate A predicate F applied to observable physical system .S is an observation predicate iff whenever an arbitrary healthy human being were in front of .S in broad daylight, and were looking at .S, then she either would immediately judge that .F (S), or judge that . ¬F (S), relying only on looking at .S, not making any inferences or appealing to some theory. (Rather than in terms of judgement, one can phrase this criterion also in terms of immediately obtaining an occurrent perceptual belief.) Next a criterion for physical system .S being a piece of measurement apparatus. Criterion for an A-Measurement Apparatus. Physical system .M is an Ameasurement apparatus iff (M1) .M is observable; (M2) there is a correlation between observation predicates F of the type ‘.M displays value a’, and sets of values of A; and (M3) the correlation of (M2) is the result of the A-relevant physical interaction between .M and physical system .S, of which A is a determinable. Friends of causality can replace ‘is the result of’ in (M3) with: is caused by. A physical interaction between .S and .M is A-relevant iff the interaction is needed to explain why the correspondence in (M2) obtains. For friends of causality, this explanation will then be a causal explanation. The explanation of the Ontic Sorcery of determinable physical properties becoming determinate at the end of a measurement (granted the Projection Postulate) ought to be part of the explanation mentioned in the criterion of an A-relevant measurement (M3). Since Modal Interpreters take A-measurements to reveal possessed 21 And 22 See

must meet a few other conditions we gloss over. See Carnap (1950). Muller (2015) for elucidation of the various features of this explication.

12 Six Measurement Problems of Quantum Mechanics

light

human vision

belief/judgement

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observability

observation predicate

explanation

-relevant interaction

-measurement apparatus

-measurement

Fig. 12.2 Conceptual dependencies of the concept of measurement as explicated in the current paper, starting with the concepts of human vision, light, belief (or judgement), and explanation

determinate properties .A, aj , they will prefer a different explanation—there is no Ontic Sorcery going on in laboratories according to Modal Interpreters. We see that the explication of what an A-measurement apparatus is has a feature that depends on which interpretation of QM is at play; but only there, within the explanation in (M3) of the correlation in (M2). Finally, an explication of what it means to measure something by a human being (or by any other being in the universe that has comparable capacities): Criterion for Measurement Human beings measure physical magnitude A of physical system .S by means of A-measurement apparatus .M and obtain value a iff they make .S and .M physically interact A-relevantly, and this A-relevant interaction results in ascribing value a to A, which value .M registers or displays. The conceptual dependencies are depicted in Fig. 12.2 (p. 255). Some measurements in physics, e.g. time measurements, by means of clocks, do not seem to meet the criterion above—with what physical system does a clock interact? Yet the criterion does fit measurement theory of QM seamlessly, as expounded in e.g. Bush et al. (1996). Further, every interpretation of QM could adopt this explication of the concept of measurement; it is interpretation neutral, or so we claim.

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12.9 Recapitulation We have distinguished six distinct ‘measurement problems’ about QM, which are not all problems that standard QM must face. Three problems are polylemma problems: they present three bouquets of inconsistent propositions, and force one to choose which proposition of each bouquet to renounce. One problem is an how-to problem, and another is a why-problem and therefore is a request for an explanation. The sixth problem is a what-problem: a request for an explication, of the concept of measurement. The first problem is the Reality Problem of Measurement Outcomes (p. 230), which is a logical clash between three plausible propositions, granted the State, Magnitude and Spectrum Postulate (of standard QM): that all physical interactions are unitary (Universal Dynamics Postulate), that physical systems have properties iff their state is in the relevant eigenstate (Eigenlink), and that properly functioning measurement devices yield a single outcome upon measurement (SMOP). Standard QM escapes the contradiction by restricting unitary evolution to when no measurements are performed, and adopts the Projection Postulate for when measurements are performed. The second problem is the State Completeness Problem (p. 237), which states that standard QM is committed to the measurement- as well as the propertyincompleteness of the state. Friends of standard QM take this to express the indeterministic character of QM, and of microphysical reality. Not really a problem, unless one is a determined determinist. Then one is in trouble, big time. The third problem is the Probability Problem of Measurement Outcomes, which states, granted only the Mixed State and Magnitude Postulate: the Probability Postulate is incompatible with all physical interactions being unitary and obeying the Property Revealing Condition. Whereas the Reality Problem of Measurement Outcomes is about determinate properties and employs the Eigenlink and SMOP (but does not employ the Probability Postulate), the Probability Problem of Measurement Outcomes (p. 242) is about measurement outcomes and does employ neither the Eigenlink nor SMOP (but does employ the Probability Postulate). Both these problems create enormous problems for taking measurement interactions to be unitary. The first problem uses only the Property Revealing Condition (measurements reveal possessed properties) to deduce a contradiction, the second problem the Probability Reproducibility Condition (the probability distribution of measurement outcomes must reflect the initial probability distribution of the physical magnitude measured). Both conditions are necessary for unitary evolutions to qualify as measurement interactions, and they are jointly sufficient: t → U (t) measurement iff Prop. Rev. Cond. and Prob. Reprod. Cond. (12.44)

.

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Standard QM, which rejects measurement interactions to be unitary and thereby escapes the first two inconsistency problems, does not escape the fourth problem, the Reality Problem of the Classical World (p. 249), of how to reconcile the fact that physical systems we observe all around us with their properties generically are not in eigenstates. That such a general metaphysical conceptual framework could possibly clash with a scientific theory was inconceivable before the advent of QM. Yet this is how deep QM drills metaphysically. The fifth problem is a request for an explanation to friends of standard QM, when the Projection Postulate has been adopted and the Dynamics Postulate has been restricted: the Measurement Explanation Problem (p. 250). Why does a manifestation of human agency occur in laws of nature governing all matter in the universe? The sixth and final problem is a problem that has not been served with solutions over the past eighty years, say; it is a request for an explication of the concept of measurement (the Measurement Meaning Problem, p. 252), to answer the question what measurement is. We summarised an attempted solution to this Meaning Problem. Décio Krause is a Brazilian philosopher who appreciates clarity, precision, and rigour eminently, who is fascinated by QM, and who has payed attention to QM in several publications, notably about indiscernibles, vague objects and quantum logic.23 In spite of the fact that my contribution does not relate directly to any of the issues in QM Décio has addressed, I hope, and suspect that he will appreciate— eminently or not—the disentanglement of the six different problems that since the inception of QM have made, and are making, waves under the flag of ‘the measurement problem’. Arguably the Reality Problem of the Classical World is the flagship of these problems. But let’s not forget that the flagship heads a small fleet. Acknowledgments Thanks to Guido Bacciagaluppi, Maura Burke, Dennis Dieks, Menno Hellinga, Ronnie Hermens, Sam Rijken en Raoul Titulaer for comments, corrections, and suggestions.

References Bacciagaluppi, G. (2014). Insolubility from no-signalling. International Journal for Theoretical Physics, 53, 3465–3474. Bell, J. S. (1990). Against measurement. Physics World, 3, 33–40. Béziau, J.-Y., et al. (Eds.). (2015). Conceptual clarifications. Tributes to P.C. Suppes (1922–2014). College Press. Bracken, A. J. (2003). Quantum mechanics as an approximation to classical mechanics in Hilbert space. Journal of Physics A, 36(23), L329–L335.

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(2000, Krause and French (2007), Krause (2010)), French and Krause (1995, 2006), Krause and da Costa (1997).

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Brown, H. R. (1986). The insolubility proof of the quantum measurement problem. Foundations of Physics, 16, 857–870. Bub, J. (1974). The interpretation of quantum mechanics. Dordrecht: D. Reidel Publishing Company. Bush, P., & Shimony, A. (1997). Insolubility of the quantum measurement problem for unsharp observables. Studies in the History and Philosophy of Modern Physics, 27(4), 397–404. Bush, P., Lahti, P. K., & Mittelstaedt, P. (1996). The quantum theory of measurement (2nd Rev. ed.). Berlin: Springer-Verlag. Carnap, R. (1950). Logical foundations of probability. Chicago: University of Chicago Press. Dirac, P. A. M. (1928). The principles of quantum mechanics. Cambridge: Cambridge University Press. Einstein, A., Podolsky, B. Y., & Rosen, N. (1935). Can the quantum-mechanical description of physical reality be considered complete? Physical Review, 35, 777–781. Ehrenfest, P. (1927). Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Zeitschrift für Physik, 45(7), 455–457. Fine, A. (1970). Insolubility of the quantum measurement problem. Physical Review D, 2, 2783– 2787. Fine, A. (1986). The shaky game, einstein, realism and the quantum theory. Chicago: University of Chicago Press. French, S., & Krause, D. (1995). Vague identity and quantum non-individuality. Analysis, 55(1), 20–26. French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical and formal analysis. Oxford: Clarendon Press. Krause, D. (2000). Remarks on quantum ontology. Synthese, 125, 155–167. Krause, D. (2010). Logical aspects of quantum (non-)individuality. Foundations of Science, 15, 79–94. Krause, D., & da Costa, N. C. A. (1997). An intensional Schrödinger logic. Notre Dame Journal of Formal Logic, 38(2), 179–194. Krause, D., & French, S. (2007). Quantum sortal predicates. Synthese, 154, 417–430. Landsman, N. P. (1998). Topics between classical mechanics and quantum mechanics. Berlin: Springer-Verlag. Landsman, N. P. (2017). Foundations of quantum theory: From classical concepts to operator algebras. Berlin: Springer-Verlag. Maudlin, T. (1995). Three measurement problems. Topoi, 14, 7–15. Muller, F. A. (2005). The deep black sea: Observability and modality afloat. British Journal for the Philosophy of Science, 56, 61–99. Muller, F. A. (2015). Circumveiloped by Obscuritads. The nature of interpretation in quantum mechanics: hermeneutic circles and physical reality, with cameos of James Joyce and Jacques Derrida, in Béziau et al. (pp. 107–136). Norsen, T., Foundations of quantum mechanics. An exploration of the physical meaning of quantum theory. Cham: Springer. Rovelli, C. (2021). Helgoland: Making sense of the quantum revolution. London: Penguin. Shimony, A. (1974). Approximate measurements in quantum mechanics II’. Physical Review D, 9, 2321–2323. Stein, H. (1997). Maximal extension of an impossibility theorem concerning quantum measurement. In R. S. Cohen et al. (Eds.), Potentiality, entanglement and passion-at-a-distance. Dordrecht: Kluwer Academic Publishers. van Fraassen, B. C. (1991). Quantum mechanics: An empiricist view. Oxford: Clarendon Press. von Neumann, J. (1927). Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Göttinger Nachrichten, 1, 245–272.

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von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: SpringerVerlag. Votsis, I. (2015). Perception and Observation Unladed. Philosophical Studies 172, 563–585. Wigner, E. P. (1961). Remarks on the mind-body problem. In I. J. Good (Ed.), The scientist speculates (pp. 284–302). London: Heinemann. Wigner, E. P. (1963). The problem of measurement. American Journal of Physics, 31, 6–15. Wittgenstein, L. (1958). The blue and the brown book. Oxford: Blackwell Publishing.

Chapter 13

Measuring Quantum Superpositions (Or, “It Is Only the Theory Which Decides What Can Be Observed.”) Christian de Ronde

The highest would be: to realize that everything factual is already theory. Goethe

Abstract In this work we attempt to confront the orthodox widespread claim, present in the philosophical and foundational debates about Quantum Mechanics (QM), that ‘superpositions are never actually observed in the lab’. In order to do so, we begin by providing a critical analysis of the famous measurement problem which, we will argue, was originated as a consequence of the strict application of the empirical-positivist requirements to subsume the quantum formalism under their specific understanding of a physical ‘theory’. In particular, the ad hoc introduction of the projection postulate (or measurement rule) can be understood as the necessity of imposing a naive empiricist standpoint which presupposes that observations are self evident givens of “common sense” experience independent of metaphysical (or conceptual) presuppositions yet necessarily represented in binary terms. We then turn our attention to two “non-collapse” interpretations of QM—namely, modal and many worlds—which even though deny that the “collapse” is a real physical process retain anyhow the projection rule as a necessary axiom of the theory itself. In contraposition, following Einstein’s claim that “it is only the theory which decides what can be observed”, we propose a return to the realist representational understanding of physical theories in which ‘observation’ is not a “self evident”

Fellow Independent Researcher of the Consejo Nacional de Investigaciones Científicas y Técnicas. C. de Ronde () Philosophy Institute “Dr. A. Korn” Buenos Aires University, CONICET, Buenos Aires, Argentina Engineering Institute, National University Arturo Jauretche, Florencio Varela, Argentina Center Leo Apostel for Interdisciplinary Studies, Brussels Free University, Brussels, Belgium © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_13

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given of experience but something that can be only understood in the context of a theoretical (formal-conceptual) scheme. It is from this standpoint that we discuss a new non-classical conceptual representation which allows us to understand quantum phenomena in an intuitive (anschaulicht) manner. Leaving behind the projection postulate, we discuss the general physical conditions for measuring and observing quantum superpositions in the lab. Keywords Quantum superpositions · Measurement problem · Observation · Representation

13.1 Introduction The present work is greatly in debt to Décio Krause, one of the most influential contemporary figures in the philosophical and foundational debate about QM, not only in our continent, America, but worldwide. My personal contact with Décio has allowed me to observe his passion and openness to discuss about logical, physical and philosophical issues about the theory of quanta. Maybe, more importantly, is to stress his generosity, not only with his students but also with the younger generation of researchers—like myself. Without any doubt, Décio has been a key figure— together with Newton da Costa—in the construction of a school in philosophy of QM in Brazil which has greatly influenced the development of the field in the South of America. Through the years we have debated about the meaning of QM in general and, in particular, about quantum superpositions. This discussion has influenced my work deeply. The following work focusing the meaning of quantum superpositions is dedicated to Décio and hopes to become a new step further in our common attempt in trying to understand the theory of quanta. Within the philosophical and foundational literature about Quantum Mechanics (QM) there exists a widespread idea according to which superposed states are never observed in the lab. The orthodox claim is that we never find in our macroscopic world, after a quantum measurement has been performed, the pointer of an apparatus in a superposed state. Instead, it is argued that what we actually observe is a single outcome, a ‘click’ in a detector or a ‘spot’ in a photographic plate. Since according to the orthodox empirical-positivist understanding, theories must be able to account for what we actually observe, there seems to be something really wrong going on with QM: the theory simply does not seem to provide an account of the empirical observations it should be talking about. In order to fill this void, Paul Dirac and John von Neumann introduced during the early 1930s a “measurement rule” as an axiom of the theory itself that would allow them to turn any quantum superposition into only one of its terms—representing the actually observed (single) outcome. As it is well known, the addition of this axiom implied the introduction of a new “invisible process” that was not described by the theory and went explicitly against the linearity of its mathematical formalism. Suddenly, there were

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two different evolutions within QM: firstly, a deterministic evolution described by Schrödinger’s equation of motion which worked perfectly well when no one was looking; and secondly, a strange indeterministic evolution, a “collapse” of the superposition to only one of its terms, produced each time a measurement was actually performed. Very soon, the Dirac–von Neumann axiomatic presentation, in tune not only with Bohr’s ideas but also with the positivist Zeitgeist of the epoch, became standardized as the orthodox textbook formulation of QM. Strange as it might seem, while collapses became accepted as part of the theory, quantum superpositions—an essential part of the mathematical formalism and the main reason for introducing collapses—became to be regarded with great skepticism and even despise (see de Ronde, 2018a for a detailed analysis). One of the main reasons behind this general attitude present within the orthodox understanding of QM might be found in the difficulties for providing a conceptual representation that would match the mathematical features implied by such superposed states. Dirac (1974, p. 12) had already realized these difficulties pointing to the fact that: “The nature of the relationships which the superposition principle requires to exist between the states of any system is of a kind that cannot be explained in terms of familiar physical concepts.” This became even more explicit after the famous Gedankenexperiment proposed by Erwin Schrödinger in 1935 which—following the ideas of Einstein (Norsen, 2005)—exposed the fact that quantum superpositions could not be understood in terms of ‘classical entities’ when constituted categorically in terms of the principles of existence, non-contradiction and identity. Schrödinger used a ‘cat’ in what could be considered an ad absurdum proof of the impossibility to represent (mathematical) quantum superpositions in terms of the classical notion of object; i.e., a system with non-contradictory definite valued properties. Not only did he demonstrate that the properties of being ‘alive’ and ‘dead’ were quantified beyond the certainty of binary valuations, he also showed that, since an “ignorance interpretation” of the sates was precluded by the mathematical formalism itself, these contradicting properties had to be both considered as somehow truly existent. Indeed, as he (Schrödinger, 1950, p. 185) would later on remark: “Something that influences the physical behaviour of something else must not in any respect be called less real than the something it influences”. Unfortunately, the influence of the Bohrian-positivist (Faye & Jaksland, 2021) alliance followed by the U.S. instrumentalist development of post-war physics would preclude for many decades the possibilities of a deeper analysis. It was only during the first years of the 1980s, when Alain Aspect’s famous experiment testing Boole-Bell inequalities in an EPR situation showed that both quantum superpositions and entanglement— as discussed and critically analyzed by both Einstein and Schrödinger—could be used as a “resource” for information processing that things became to really change. As described by Jeffrey Bub (2017), “[. . . ] it was not until the 1980s that physicists, computer scientists, and cryptographers began to regard the non-local correlations of entangled quantum states as a new kind of non-classical resource that could be exploited, rather than an embarrassment to be explained away.”

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As he continues to explain: “Most physicists attributed the puzzling features of entangled quantum states to Einstein’s inappropriate ‘detached observer’ view of physical theory, and regarded Bohr’s reply to the EPR argument (Bohr, 1935) as vindicating the Copenhagen interpretation. This was unfortunate, because the study of entanglement was ignored for thirty years until John Bell’s reconsideration of the EPR argument (Bell, 1964).” In this new context of technical possibilities, the “shut up and calculate!” widespread instrumentalist attitude of post-war physicists had to unwillingly allow the reopening of foundational and philosophical debates about QM—which had been silenced for almost half a century. However, instead of focusing on the conceptualization of superpositions and entanglement, it was the justification of the measurement rule, known in the literature as ‘the measurement problem’, which became placed at the center of the philosophical stage. The fact that no one had ever been able to experimentally test in the lab the actual existence of such “collapses” did not seem very important for a physics community that had been trained to accept that QM could be used as a “tool” but could not be actually understood.1 The obvious conclusion that the collapse was an artificial fiction which led nowhere was never critically considered and Professors kept teaching young students in Universities all around the world about the existence of these strange unobservable and irrepresentable “quantum jumps” triggered by measurement observation. Confronting orthodoxy, in this work we attempt to provide a critical analysis of the general conditions for observing and measuring quantum superpositions. We begin, in Sect. 13.2, by analyzing the essential link between the ad hoc introduction of the projection postulate and the positivist reference to “common sense” observations in physical theories. In Sect. 13.3 we discuss the role of the measurement rule in both “collapse” and “non-collapse” interpretations of QM focusing our attention in the many worlds and modal interpretations. Section 13.4 discusses the missing link between the orthodox mathematical formalism of QM and its conceptual framework. In Sect. 13.5, we revisit what is actually observed in the lab and, in Sect. 13.6, we show how an objective conceptual account of QM can be derived from the operational-invariance already present within the mathematical formalism of the theory itself. Finally, from this standpoint, we discuss in Sect. 13.7 the general conditions for measuring and observing quantum superpositions in the lab.

1 As

Lee Smolin (2007, p. 312) would describe the post-war instrumentalist view of physics recalling his own experience as a student: “When I learned physics in the 1970s, it was almost as if we were being taught to look down on people who thought about foundational problems. When we asked about the foundational issues in quantum theory, we were told that no one fully understood them but that concern with them was no longer part of science. The job was to take quantum mechanics as given and apply it to new problems. The spirit was pragmatic; ‘Shut up and calculate’ was the mantra. People who couldn’t let go of their misgivings over the meaning of quantum theory were regarded as losers who couldn’t do the work.”

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13.2 Positivism, Quantum Theory and the Measurement Rule Positivist ideas had a completely different impact in QM, first helping to create the theory, and later on, during its progressive standardization. While in the second half of nineteenth century positivism played a subversive role fighting against the dogmatism imposed by classical Newtonian physics and its a priori Kantian metaphysical interpretation in terms of space-time atomism, during the early twentieth century positivism played an essential role helping the new generation of physicists to burry metaphysical debates while, at the same time, taking for granted the “commonsensical” atomist narrative that had been fought just a few decades before. While the result of the first impact allowed physicist to adventure themselves beyond the limits of the classical representation creating both relativity theory and QM, once QM had been finally developed by Werner Heisenberg as a closed mathematical formalism in 19252 positivism begun to play an oppressive role by limiting the possibilities of development and understanding of the same theory it had—undoubtedly—helped to create. It is this latter approach which David Deutsch (2004, p. 308)—pointing explicitly to empiricism, positivism, Bohr and instrumentalism3 —has characterized as ‘bad philosophy’, namely, “[a] philosophy that is not merely false, but actively prevents the growth of other knowledge.” Indeed, concomitant with positivism Niels Bohr would develop an interpretation which embraced the impossibility of a consistent theoretical representation, something he would soon become to regard as the very essence of his own quantum revolution. In this respect, his principles of complementarity and correspondence, together with several fictional images such as ‘quantum particles’, ‘quantum waves’ and ‘quantum jumps’, would allow him to support both a pragmatic understanding of the theory of quanta—in tune with the positivist principles—as well as an atomist metaphysical

2 In this respect, it is interesting to notice that matrix mechanics was developed by Heisenberg replacing Bohr’s correspondence principle and the problem of classical trajectories by Mach’s observability principle. 3 As Deutsch (2004, p. 312) remarks: “[. . . ] empiricism did begin to be taken literally, and so began to have increasingly harmful effects. For instance, the doctrine of positivism, developed during the nineteenth century, tried to eliminate from scientific theories everything that had not been ‘derived from observation’. Now, since nothing is ever derived from observation, what the positivists tried to eliminate depended entirely on their own whims and intuitions.” Regarding the Danish physicist, Deutsch (2004, p. 308) makes the point that: “The physicist Niels Bohr (another of the pioneers of quantum theory) then developed an ‘interpretation’ of the theory which later became known as the ‘Copenhagen interpretation’. It said that quantum theory, including the rule of thumb, was a complete description of reality. Bohr excused the various contradictions and gaps by using a combination of instrumentalism and studied ambiguity. He denied the ‘possibility of speaking of phenomena as existing objectively’—but said that only the outcomes of observations should count as phenomena. He also said that, although observation has no access to ‘the real essence of phenomena’, it does reveal relationships between them, and that, in addition, quantum theory blurs the distinction between observer and observed. As for what would happen if one observer performed a quantum-level observation on another, he avoided the issue.”

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narrative detached from theoretical representation and understanding (de Ronde, 2023a). Finally, in the year 1930, a young English engineer and mathematician named Paul Maurice Dirac was able to bring together both Bohrian and positivist ideas within a common scheme. In his book, The Principles of Quantum Mechanics (Dirac, 1974) Dirac presented an axiomatic vectorial formulation which claimed to provide a rigorous mathematical exposition of the theory of quanta. Of course, it did much more. Following both positivism and Bohr’s principles, Dirac begun by stressing that “[it is] important to remember that science is concerned only with observable things and that we can observe an object only by letting it interact with some outside influence. An act of observation is thus necessarily accompanied by some disturbance of the object observed.” Continuing with Bohr’s interpretation of Heisenberg’s inequalities, Dirac remarked that: “we have to assume that there is a limit to the finiteness of our powers of observation and the smallness of the accompanying disturbance—a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer.” Adopting the positivist standpoint Dirac claimed that in science we gain knowledge only through observation, but embracing Bohr’s ideas we must also accept that in QM we have reached a limit to the very possibility of observability and representation itself. Dirac’s scheme was then confronted to the mathematical formalism and the existence of quantum superpositions which he himself recognized as a kernel element of the theory itself. The nature of the relationships which the superposition principle requires to exist between the states of any system is of a kind that cannot be explained in terms of familiar physical concepts. One cannot in the classical sense picture a system being partly in each of two states and see the equivalence of this to the system being completely in some other state. There is an entirely new idea involved, to which one must get accustomed and in terms of which one must proceed to build up an exact mathematical theory, without having any detailed classical picture. (Dirac, 1974, p.12)

The fact that superpositions did not provide by themselves a consistent representation of what was going on did not seem to constitute a problem. After all, Bohr had not provided any consistent representation for his model of the atom nor the “quantum jumps” performed by electrons. As Dirac would famously argue: “it might be remarked that the main object of physical science is not the provision of pictures, but the formulation of laws governing phenomena and the application of these laws to the discovery of phenomena. If a picture exists, so much the better; but whether a picture exists of not is a matter of only secondary importance.” What remained a problem for Dirac’s positivist standpoint was the relation between these “weird” mathematical elements and the need to predict single measurement outcomes with binary certainty. Following the positivist understanding, he would explain that a formalism “becomes a precise physical theory when all the axioms and rules of manipulation governing the mathematical quantities are specified and when in addition certain laws are laid down connecting physical facts [i.e., binary observations] with the mathematical formalism”. It is at this point that, making use of an example of polarized photons, Dirac introduced today’s famous “collapse” of the quantum wave function:

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When we make the photon meet a tourmaline crystal, we are subjecting it to an observation. We are observing wither it is polarized parallel or perpendicular to the optic axis. The effect of making this observation is to force the photon entirely into the state of parallel or entirely into the state of perpendicular polarization. It has to make a sudden jump from being partly in each of these two states to being entirely in one or the other of them. Which of the two states it will jump cannot be predicted, but is governed only by probability laws. (Dirac, 1974, p. 9)

Two years later, in 1932, the Hungarian mathematician John von Neumann would turn Dirac’s “jump” into a measurement postulate of the theory itself; i.e., the famous Projection Postulate (von Neumann, 1955). With the help of Bohr, the books by Dirac and von Neumann were soon regarded by physicists as a sound and rigorous exposition of the now “standard” complementarity account of QM. It is in this way that both the projection postulate and the existence of “collapses” became accepted and taught to students though many different, though essentially equivalent, textbooks. Of course, from a realist perspective, the measurement rule as well as the existence of collapses were clearly problematic—to say the least. It is in this specific context that the so called “measurement problem of QM” became part of some—at the time—marginal foundational and philosophical debate: Quantum Measurement Problem (QMP) Given a specific basis (or context), QM describes mathematically a quantum state in terms of a superposition of, in general, multiple states. Since QM allows us to predict that the quantum system will get entangled with the apparatus and thus its pointer positions will also become a superposition,4 the question is why do we observe a single outcome instead of a superposition of them? In order to understand the centrality of the measurement problem in QM one needs to pay special attention to the twentieth century positivist re-foundation of physics which imposed within science a radically new meaning and understanding of physical theories—following Ernst Mach—as “economies of experience”. This radical shift took place through the introduction of a new scheme grounded on two main pillars: on the one hand, “common sense” observations, from which theories were derived; and on the other, mathematical formalisms which, in turn, allowed to predict future observations. Physical theories were not understood anymore as providing a formal-conceptual representation of states of affairs but instead as mathematical algorithms capable of predicting observations in the lab.5 In this  a quantum system represented by a superposition of more than one term, . ci |αi , when in contact with an apparatus ready to measure, .|R0 , QM predicts that system and apparatus will become “entangled” in such a way that the final ‘system + apparatus’ will be described by . ci |αi |Ri . Thus, as a consequence of the quantum evolution, the  pointers have also become—like the original quantum system—a superposition of pointers . ci |Ri . This is why the measurement problem can be stated as a problem only in the case the original quantum state is described by a superposition of more than one term. 5 Technically speaking, the distinction between empirical terms (i.e., the empirically “given”) and theoretical terms (i.e., their translation into simple statements) comprised this new understanding of theories. 4 Given

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context, the measurement problem appeared as the natural consequence of imposing the positivist requirement to predict single measurement outcomes—replacing the reference of Heisenberg’s matrix formulation to intensities. However, regardless of their aversion towards metaphysics, positivist philosophers as well as Bohr—who had become the main leader of the physics community—, were very careful not to loose all contact with conceptual (or metaphysical) narratives. In fact, as some kind of metaphysical ghost that would refuse to completely vanish, the reference to (microscopic) ‘particles’ would remain always present in the background of their discourse (de Ronde & Fernández Mouján, 2021). Furthermore, it became accepted that once the mathematics of an empirically adequate theory was constructed, it was possible—for those willing to do so—to also add an ‘interpretation’ that would explain how the world was according to the theory. In this way, positivism was able to include both empiricism and realism within its own—essentially anti-realist—scheme. But of course, since physical phenomena was now understood as a “self-evident” given—independent of physical concepts and metaphysical presuppositions—, the introduction of an ‘interpretation’ (of the mathematical formalism) was not really necessary. Any empirically adequate theory already did its job as a predictive device without the addition of a metaphysical narrative.6 This obviously implied that in the positivist scheme, empiricism was much more fundamental than the degraded form of (metaphysical) realism it—anyhow—tolerated. In a very astute manner empirical positivism had followed the recommendation by Michael Corleone in The Godfather: “Keep your friends close, and your enemies closer.” In order to retain a realist discourse, it was accepted by the new anti-realist trend of thought that an empirically adequate theory could have many different ‘interpretations’. Many different narratives could be added in order to explain what a theory was really talking about beyond the observable realm. Thus, realism became a kind of “dressing” for theories, something to be “added” by metaphysically inclined physicists and philosophers which continued to stubbornly wonder about a noumenic reality beyond phenomena. As remarked by van Fraassen (1980, pp. 202–203), the new orthodoxy considered that: “To develop an empiricist account of science is to depict it as involving a search for truth only about the empirical world, about what is actual and observable.” In this context, the question of reference was completely relativized (van Fraassen, 1991, p. 242): “However we may answer these questions [regarding the interpretation], believing in the theory being true or false is something of a different level.” After centuries of confrontations, antirealism had finally been able to confine realism into the prison of interpretations. Presenting realists as “naive” or even “fanatic” believers in ungrounded fictional narratives, anti-realists were finally able to diminish the fundamental role of reality (or physis) within physics to its minimum expression—as a stamp that could be used on absolutely anything (concepts, mathematics, observations, etc.) but meant

6 It is important to remark that the meaning of ‘metaphysics’ in the positivist context was understood as an ungrounded discourse about the un-observable.

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really nothing. The foundations of physics, namely, physis, had been finally erased. As Karl Popper would famously remark: The empirical basis of objective science has thus nothing ‘absolute’ about it. Science does not rest upon solid bedrock. The bold structure of its theories rises, as it were, above a swamp. It is like a building erected on piles. The piles are driven down from above into the swamp, but not down to any natural or ‘given’ base; and if we stop driving the piles deeper, it is not because we have reached firm ground. We simply stop when we are satisfied that the piles are firm enough to carry the structure, at least for the time being. (Popper, 1992, p. 111)

Positivism portrayed realism as a subjective belief in ‘pictures’ and ‘images’ created by people who preferred ungrounded metaphysical bla bla instead of down to earth perception. As David Deutsch makes the point: During the twentieth century, anti-realism became almost universal among philosophers, and common among scientists. Some denied that the physical world exists at all, and most felt obliged to admit that, even if it does, science has no access to it. For example, in Reflections on my Critics the philosopher Thomas Kuhn wrote: ‘There is [a step] which many philosophers of science wish to take and which I refuse. They wish, that is, to compare [scientific] theories as representations of nature, as statements about what is really out there’. (Deutsch, 2004, p. 313)

However, the main positivist claim according to which empirical terms (or observations) could be understood without making reference to conceptual (or metaphysical) presuppositions remained completely unjustified. This problem— which Kant had already discussed two Centuries before in his Critique of Pure Reason—was repeatedly addressed by the main figures of both positivism and postpositivism. Rudolph Carnap (1928), Ernst Nagel (1961), Popper (1992) and many others tried to present a solution without any success. And even though in the 1950s it was explicitly recognized by Norwood Hanson (1958)—and later on, during the 1960s, by Thomas Kuhn, Paul Feyerabend and many others—that observation in science could not be considered without a presupposed theoretical scheme (i.e., the theory-ladenness of observations), the general naive empiricist framework remained completely untouched. More importantly, the problems constructed in both physics and philosophy of physics continued to—either implicitly or explicitly—accept naive empiricism as a main standpoint of analysis. The failure of the whole empirical-positivist program which was implicitly recognized in another famous paper by Carl Hempel (1958), did not change anything. Regardless of the deep internal criticisms and the lack of answers, naive empiricism has continued to play an essential role in the discussions and debates within both physics and philosophy of physics. QM might be regarded as one of the most explicit exposures of the empirical-positivist influence within the physical sciences. This scheme of understanding of theories is not only responsible for having created the (in)famous measurement problem of QM, it is also guilty for maintaining it—still today—at the very center of foundational and philosophical debates. After the Second World War, the Bohrian-positivist alliance converged into a new pragmatic trend of thought even more radical than its predecessors. Instrumentalism, as part of the twentieth century anti-realist Zeitgeist, was presented by the U.S.

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philosopher John Dewey as the natural extension of both pragmatism and empirical positivism. While pragmatism sustained that the value of an idea is determined by its usefulness, instrumentalism, rejecting the need of any metaphysical fundament, was ready to take a step further and claim that the question regarding the reference of theories was simply meaningless. Scientific theories do not make reference to an underlying reality. Thus, there is no sense in which a theory can be said to be true or false (or better or worse) apart from the extent to which it is useful as a “tool” in solving scientific problems.7 Popper, a post-positivist himself, famously concluded at the beginning of the 1960s that instrumentalism had already conquered the whole of physics: Today the view of physical science founded by Osiander, Cardinal Bellarmino, and Bishop Berkeley, has won the battle without another shot being fired. Without any further debate over the philosophical issue, without producing any new argument, the instrumentalist view (as I shall call it) has become an accepted dogma. It may well now be called the ‘official view’ of physical theory since it is accepted by most of our leading theorists of physics (although neither by Einstein nor by Schrödinger). And it has become part of the current teaching of physics. (Popper, 1963)

This result could be already observed within the reproduction of textbooks exposing the “standard” formulation of QM which, regardless of its explicit reference to microscopic systems, was taught in Universities as a “recipe” to compute measurement outcomes. This orthodox account of QM has remained, even today, part of a unified vision in the field of physics. As recently described by Tim Maudlin: What is presented in the average physics textbook, what students learn and researchers use, turns out not to be a precise physical theory at all. It is rather a very effective and accurate recipe for making certain sorts of predictions. What physics students learn is how to use the recipe. For all practical purposes, when designing microchips and predicting the outcomes of experiments, this ability suffices. But if a physics student happens to be unsatisfied with just learning these mathematical techniques for making predictions and asks instead what the theory claims about the physical world, she or he is likely to be met with a canonical response: Shut up and calculate! (Maudlin, 2019, pp. 2–3)

Maybe as a kind of reaction to the “shut up and calculate!” instrumentalist program, during the 1980s a new field called “philosophy of QM” was established in order to, once again, debate about the reference of QM to the world and reality. In this way, those physicists with realist worries who had expelled from the filed

7

“In the US, which after the Second World War became the central stage of research in physics in the West, the discussions about the interpretation of quantum mechanics had never been very popular. A common academic policy was to gather theoreticians and experimentalists to gather in order to favour experiments and concrete applications, rather than abstract speculations. This practical attitude was further increased by the impressive development of physics between the 1930s and the 1950s, driven on the one hand by the need to apply the new quantum theory to a wide range of atomic and subatomic phenomena, and on the other hand by the pursuit of military goals. As pointed out by Kaiser, ‘the pedagogical requirements entailed by the sudden exponential growth in graduate student numbers during the cold war reinforced a particular instrumentalist approach to physics’.” (Osnaghi et al., 2009, pp. 2–3)

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after asking too many questions, were now able to find shelter in philosophy departments. However, the anti-realist philosophy had already limited their praxis to the creation of “interpretations”, namely, ungrounded metaphysical narratives that imagined what reality could be according to the theory. At the center of this—supposedly—“realist debate” stood of course the measurement problem of QM. How could the path from quantum superpositions to single measurement outcomes be explained in terms of something taking place in reality? It is in this context that “interpretations of QM” have reproduced themselves during the last decades at an exponential speed, leading today to a dead end; what Adán Cabello has characterized as an “interpretational map of madness” (Cabello, 2017).

13.3 Collapse or Non-collapse Interpretations? One of the main distinctions between interpretations of QM in the mainstream philosophical literature is that between “collapse” and “non-collapse” interpretations. While the first group accepts the idea that “collapses” do actually take place and require thus some kind of representation or account; the latter non-collapse group denies that the measurement postulate should require a “realist” interpretation. Instead, they argue, collapses should not be considered as a real process that takes place in nature when measuring quantum superpositions. This seems to create a tension between those who tend to believe in the real existence of measurement collapses and those who do not. However, as we shall see in the following, this distinction overlooks the fact that both collapse and non-collapse interpretations take for granted the need of the measurement postulate within the theory of quanta.

13.3.1 The Many Worlds Narrative: The Measurement Rule as a Branching Process Today, the existence of many parallel worlds, similar to ours, giving rise to a “multiverse” has become a popular idea not only in science fiction movies and TV series but in mainstream physics. Applied in String Theory, Cosmology and the Standard Model, the many worlds narrative has become during the last decade one of the most fashionable ideas in theoretical physics. Of course, this narrative has become also popular in the interpretational debate about QM. The so called Many Worlds Interpretation (MWI) of QM goes back to Everett’s relative state interpretation8 which might be considered as framing Bohr’s contextual relativism

8 What seems very paradoxical with respect to the present Oxfordian account of Everett’s ideas— mainly due to Deutsch, Wallace and Saunders—is the complete elimination of Everett’s positivist

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in the context of the positivist scheme (see de Ronde & Fernández Mouján, 2018; Osnaghi et al., 2009). In his PhD dissertation, taking as a standpoint Wigner’s friend paradox—which exposed the confronting representations of a superposition before and after an actual measurement was observed by an agent inside a lab and his friend outside—Hugh Everett attempted as a student to avoid the “collapse” of the quantum wave function and in this way escape the measurement problem right from the start. In short, he wanted to escape the problem that positivism had created— by introducing the collapse postulate within the theory—without abandoning the positivist understanding of physical theories. Everett strategy was to avoid any ontological reference and understand QM in purely epistemological terms. By accepting a relativist subjectivist account of observations Everett was making explicit the basic empirical-positivist presupposition according to which physics did not make reference to physis (or reality) and should instead be regarded in purely epistemological pragmatic terms: as a “tool” to be used by individual agents in order to predict observations. In his own words (Barrett & Byrne, 2012, p. 253): “To me, any physical theory is a logical construct (model), consisting of symbols and rules for their manipulation, some of whose elements are associated with elements of the perceived world.” Following the positivist program, Everett’s account of QM made only reference to the observation of ‘clicks’ relative to agents. In this respect, he stressed: “There can be no question of which theory is ‘true’ or ‘real’ —the best that one can do is reject those theories which are not isomorphic to sense experience.” Once QM was understood by Everett as making reference only to the relative observations made by agents, the ‘collapse’ seemed to have finally disappeared. And yet, the measurement rule (or projection postulate) had not. Instead, it had changed its reference from the strange “collapse” of the quantum wave function to an even stranger “branching process” also dependent on subjects. Alike the measurement axiom introduced by Dirac and von Neumann, this “branching” was not meant to be interpreted in realist or metaphysical terms. As explained by Jeffrey Barrett, the hole point of Everett’s (dis)solution of the measurement problem was that “the branching” had to be understood in purely operational terms (Barrett & Byrne, 2012). However, closing the Bohrian circle, and quite regardless of Everett’s intentions, during the early 1970s Bryce DeWitt’s and Neill Graham would conceive the branching in ontological terms. According to DeWitt and Graham, the quantum measurement interaction described by the branching process was actually responsible for creating many different real worlds—mathematically represented in terms of quantum superpositions. In this way, Everett’s relativist epistemological account of QM was turned completely upside-down. This explains why Jeff Barrett, who had access to Everett’s original notes, found written next to the passage

standpoint regarding observability, prediction and anti-metaphysical commitments (Sect. 13.3.1). Jefferey Babrret, who studied Everett’s original texts and was responsible for the edition of his complete works (Everett, 2012), has repeatedly remarked that Everett’s ideas have nothing to do with the present Oxfordian misuse of his name. In fact, it is very easy to see that Everett’s Relative State interpretation of QM is much closer to QBism or Rovelli’s Relational Interpretation than to the MWI.

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where DeWitt presented Graham’s many worlds clarification of Everett’s own ideas the word “bullshit”.9 In DeWitts’ and Graham’s many worlds reformulation the “branching” became once again—just like the “collapse”—a real physical process. A process which allowed a measurement in one—real—world to create a myriad of parallel—also real—worlds. Of course, nothing of this incredible “creationist process” was described by the theory.10 During the 1970s, DeWitt’s ideas were not taken very seriously by the physics community. A decade later, David Deutsch used the existence of many worlds in order to explain the efficiency of quantum computations (Deutsch, 1985). But still, the many worlds’ branching solution to the measurement problem remained as unclear as its hylomorphic collapse twin. The reason is quite simple, there is nothing in the mathematical formalism of QM which can be related to the measurement postulate on which both collapse and branching processes are explicitly grounded. There is simply no theoretical account which explains how or when any of these physical (or fictional?) processes really takes place. Since the mathematical formalism of the theory is linear, it becomes difficult to understand why Deutsch and Wallace have repeatedly claimed that the many worlds interpretation is a “literal” reading of QM. At most it seems just a very imaginative interpretation of the ad hoc measurement rule introduced by Dirac. Another troubling aspect of the many worlds proposal is its unclear representation of the state of affairs which seems to talk not only about particles but also about a possible future branching of worlds (see Sudbery, 2016) which are neither described by the theory. According to many worlds, QM is not describing an actual state of affairs, but instead a future state of affairs which will be only actualized after the branching has taken place. Thus, the quantum superposition seems to make reference to the many worlds created only after an actual measurement has been performed. This places the interpretation in a very difficult dilemma regarding the reference provided by the—present and future—representation of the theory. The many worlds interpretation seems to imply that QM makes reference only to a future state of affairs, but not to the present one. One might also wonder about the reference to ‘elementary particles’ which, before the measurement, also seems to be described by quantum superpositions. Quantum superpositions seem to make reference, before the measurement to microscopic particles, and after the measurement, to many macroscopic parallel worlds. But apart from the unclear reference of what the interpretation really talks about, there are many questions related to the branching process which do not seem to find a convincing answer. What about the interference of probabilities and superpositions? If quantum probability is understood as (epistemic) ‘degrees of belief’ (Deutsch, 1999), how can they (really) interact? And what about entanglement? Does it amount to an

9 See

Everett (2012, 364–366) for scans of Everett’s comments. this respect, it is very interesting to notice that we could think of the “branching process” as the mirror image of the “collapse process”—none of which is addressed nor explained by QM. While the collapse turns the superposition into only one if its terms, the branching goes from one single measurement into a superposition of parallel worlds. 10 In

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interaction of one superposition of many worlds with another superposition of many worlds. . . or is it elementary particles? Where does this interaction takes place? And when? And what would it mean that a world interferes or interacts with another world? How does one actual world affect another parallel one? How can the branching and interference of worlds be tested experimentally? And by the way, how many worlds exactly are created in each branching? How many of these worlds really exist in our multiverse?11 How is all this actually represented by the mathematical formalism of QM?

13.3.2 Dieks’ Modal Narrative: The Measurement Rule as a Semantic Rule Another realist “non-collapse” interpretation is Dennis Dieks’ modal version of Bas van Fraassen’s (anti-realist) modal interpretation. Contrary to van Fraassen, Dieks interpretation was conceived in order to make sense of QM in terms of ‘systems’ with definite valued ‘properties’—going in this way beyond observed actualities. According to him, the attempt was to do so staying close to the orthodox quantum formalism without adding anything “by hand” (for a detailed analysis see de Ronde, 2011; Vermaas, 1999). Something that, according to Dieks, implied leaving behind the existence of “collapses”. Collapses constitute [. . . ] a process of evolution that conflicts with the evolution governed by the Schrödinger equation. And this raises the question of exactly when during the measurement process such a collapse could take place or, in other words, of when the Schrödinger equation is suspended. This question has become very urgent in the last couple of decades, during which sophisticated experiments have clearly demonstrated that in interaction processes on the sub-microscopic, microscopic and mesoscopic scales collapses are never encountered. (Dieks, 2010, p. 120)

Dieks modal narrative follows van Fraassen’s semantic account of theories (Dieks, 1991) where “an uninterpreted theory is identified with the class of its models, in the sense of abstract model theory. [. . . ] To make an empirical theory of it, we have to indicate how empirical data can be embedded in the models.” In particular, in order “[t]o obtain quantum mechanics as an empirical theory, we have to specify the links with observation, by means of ‘interpretation rules’.” It is at this point that Dieks rephrases the measurement rule in his own terms:

11 In recent interviews the many worlds followers have been confronted to some of these questions.

Wallace’s (2017) answer is that: “It is hard to define exactly because this branching process is not 100 precise, but to put a number out of the air .1010 , so 10 to the number of particles in the Universe.” We might point out that the acknowledgment that “the branching process is not precise” in an interpretation which attempts to describe “literally” the quantum formalism seems, to say the least, very unsatisfactory. Another question which immediately pops up is how did Wallace compute the 100 number, .1010 ?

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Consider a state vector representing a composite system, consisting of an object system and the remainder of the total system. The total state vector will almost always have one unique bi-orthonormal decomposition: .|

=



ck |ψk |Rk 

where the .|ψk  refer to the object system and the .|Rk  to the rest of the system; .ψi |ψj  and .Ri |Rj  = δij . I now propose the following semantical rule: As soon as there is a unique decomposition of the form (2), the partial system represented by the .|ψk , taken by itself, can be described as possessing one of the values of the physical quantity corresponding to the set .{|ψk }. The probabilities for the various possibilities to be realized are given by .|ck |2 . (Dieks, 1989, p. 1406, emphasis added)

The attentive reader might have already realized that the semantic rule—also called by Dieks interpretational rule—is nothing essentially different from the measurement rule or projection postulate. Independently of the introduction of the Schmidt decomposition, Dieks’ rule ends up doing exactly the same job as its predecessors, namely, to magically “bridge the gap” between quantum superpositions and single measurement outcomes. The tension present in Dieks’ empiricist-realist program becomes then evident. At the end of the day, his narrative—in a completely Bohrian fashion—retains two contradictory claims. On the one hand, that QM should be understood as representing physical reality in terms of elementary quantum particles (or systems) with definite valued properties, and on the other, that QM should be used—through the application of his interpretational rule—as a “tool” in order to predict observations (or measurement outcomes). Even though Dieks begins by arguing that the mathematical formalism of the theory should be interpreted “in realist terms” as making reference to the microscopic world and ad hoc rules should be rejected, he immediately reintroduces the measurement postulate now renamed as the a “semantic” or “interpretational” rule. Even though Dieks denies the existence of “collapses”, he anyhow applies his semantic rule in order to justify the appearance of single ‘clicks’ in detectors later on interpreted as particles. In this way, the mathematical formalism is dissected in two levels, a supposedly realist level which is unclearly related to ‘systems’ with definite valued ‘properties’ (i.e., to quantum particles), and an anti-realist level which through the ad hoc introduction of the interpretational rule does the dirty job of providing an instrumentalist account of measurement outcomes. It is important to understand that the reduced states arising from the Schmidt decomposition—to which Dieks applies his interpretational (measurement) rule— are in fact improper mixtures which, just like in the case of quantum superpositions, cannot be interpreted in terms of ignorance (D’Espagnat, 1976, Chap. 6).12 Thus, absolutely nothing has been gained by shifting the reference of the orthodox quantum formalism from one single system to two correlated ones. Notwithstanding the fact that one might wonder about the meaning of a theory that would describe

12 This point is recognized explicitly by Dieks himself in many occasions. See, for example: Dieks (1989, p. 1407).

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‘composite systems’ but fails to describe ‘single systems’, the attempt to provide a consistent interpretation without the addition of ad hoc rules is given up right from the start. Following the same strategy, Vermaas and Dieks (1995) have argued that one needs to name the reduced state in two different ways. First, as a “mathematical state” which cannot be interpreted in terms of ignorance—because of the mathematical formalism—, and then as a “physical state” which can be interpreted in terms of ignorance—simply because it is now called “physical”. The solution is to shift the naming of the same state depending on the context of application.13 The fact that the properties of systems do not behave according to the classical intuition (Bacciagaluppi, 1995; Clifton, 1995; de Ronde et al., 2014; Vermaas, 1997) is then resolved in a completely Bohrian fashion: the modal interpretation does not provide the expected representation of systems simply because QM is not classical.14 It is obvious then that our proposed semantical rule, together with the formalism of quantum mechanics, can lead to consequences which are very much at variance with classical intuition. In classical physics the properties of all partial systems together always completely determine the properties of a total system. By contrast, quantum theory —with the interpretation as discussed above— sometimes associates with a composite system a description that contains more than just the properties of the partial systems by themselves. (Dieks, 1989, p. 1408)

To sum up. Just like in the Bohrian scheme analyzed in detail in Norsen (2005), there is in Dieks’ interpretation a dualistic (inconsistent) account of QM. On the one hand, there is an instrumentalist (anti-realist) account of the theory provided through rules which only make reference to measurement outcomes; and on the other, a metaphysical (realist) account which attempts to make reference to systems with definite properties. But while at the metaphysical level “collapses” are rejected by claiming that one should not accept additions made by hand (such as ad hoc measurement rules), at the empirical level the measurement postulate is accepted as part of the theory. Even worse, not only the orthodox mathematical formalism fails to provide a consistent account of ‘quantum systems’ as possessing definite valued

13 In fact, this linguistic duality had been already proposed by van Fraassen who distinguishes between dynamical states and value states. What van Fraassen does not do, is to provide a realist interpretation of such a distinct naming. 14 This argument was first used by Bohr who argued in 1927 that “quantum jumps” could not be represented. As recalled by Heisenberg (1971, p. 74), Bohr’s reply to Schrödinger’s criticisms related exclusively to the limits imposed by QM: “What you say is absolutely correct. But it does not prove that there are no quantum jumps. It only proves that we cannot imagine them, that the representational concepts with which we describe events in daily life and experiments in classical physics are inadequate when it comes to describing quantum jumps. Nor should we be surprised to find it so, seeing that the processes involved are not the objects of direct experience.” As Bohr (1935, p. 701) would argue: “The impossibility of a closer analysis of the reactions between the particle and the measuring instrument is indeed no peculiarity of the experimental procedure described, but is rather an essential property of any arrangement suited to the study of the phenomena of the type concerned, where we have to do with a feature of [quantum] individuality completely foreign to classical physics.” For a more detailed analysis see de Ronde (2023a).

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properties, the interpretation fails to produce any intuitive (anschaulicht) insight. The addition of quantum particles to the interpretation seems to play no other role than supporting a fictional narrative with no clear explanatory power nor contact with the mathematical formalism of the theory.

13.4 The Missing Link Between Mathematics and Physical Concepts The positivist scheme fails to provide any scientific methodology in order to address the link between the mathematical formalism and the conceptual framework of a given theory. Not only observations—which are considered as primary givens of experience—are independent of conceptual or categorical constraints, but also the mathematical formalism—which is developed in order to account for observations—remains completely independent of metaphysical concepts. According to orthodoxy, physicists must create mathematical models in order to account for what experimentalists observe in the lab. That’s it! Conceptual (or metaphysical) representation plays no role whatsoever within the construction of an empirical adequate theory. This becomes explicitly recognized by the role assigned to ‘interpretations’ as ‘fictional narratives’ or ‘stories’ which are introduced (by philosophers) a posteriori from the construction of theories (by physicists). In tune with the postmodern Zeitgeist of the twentieth century, according to mainstream physicists, such interpretations have no other fundament than the wishful hopes and beliefs of a metaphysically inclined community of armchair philosophers who still attempt to describe reality beyond what we actually observe in the lab. As remarked by David Deutsch: [Postmodernism itself] is a narrative that resists rational criticism or improvement, precisely because it rejects all criticism as mere narrative. Creating a successful postmodernist theory is indeed purely a matter of meeting the criteria of the postmodernist community —which have evolved to be complex, exclusive and authority-based. Nothing like that is true of rational ways of thinking: creating a good explanation is hard not because of what anyone has decided, but because there is an objective reality that does not meet anyone’s prior expectations, including those of authorities. The creators of bad explanations such as myths are indeed just making things up. But the method of seeking good explanations creates an engagement with reality, not only in science, but in good philosophy too —which is why it works, and why it is the antithesis of concocting stories to meet made-up criteria. Although there have been signs of improvement since the late twentieth century, one legacy of empiricism that continues to cause confusion, and has opened the door to a great deal of bad philosophy, is the idea that it is possible to split a scientific theory into its predictive rules of thumb on the one hand and its assertions about reality (sometimes known as its ‘interpretation’) on the other. This does not make sense, because —as with conjuring tricks— without an explanation it is impossible to recognize the circumstances under which a rule of thumb is supposed to apply. And it especially does not make sense in fundamental physics, because the predicted outcome of an observation is itself an unobserved physical process. (Deutsch, 2004, p. 1408)

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The problem present within the contemporary mainstream positivist scheme is that there is no methodology allowing to discuss in scientific terms—beyond wishful thinking and subjective preferences—if an interpretation provides (or not) a good explanation of the mathematical formalism (see Chakravartty, 2017; French, 2011). It is the complete lack of any objective criteria to distinguish between ‘good’ and ‘bad’ interpretations which, in turn, has allowed the wild introduction of fictional notions, principles, rules and pictures with no contact whatsoever with the mathematical formalism or experience. Let’s give a few examples. There is no mathematical element of the formalism describing a ‘particle’ in Bohmian mechanics. The same occurs in the many worlds interpretation where there is no description of what a ‘world’ is or how the branching process could be described or computed according to the theory. Just like in the case of ‘collapses’ and ‘quantum particles’ these notions help creating a story which does not relate to the mathematical formalism they—supposedly—attempt to represent. There is no experimental nor theoretical support of the fictions added to the models. And once the ontological picture finds difficulties philosophers are ready to re-turn to epistemological or pragmatic justifications. We find two paradigmatic examples in the just mentioned “non-collapse” solutions to the MP. While many worlds interpretations, when confronted to the problem of interpreting the Born rule, have ended up following the (anti-realist) Bayesian subjectivist interpretation of probability; when confronted to different no-go theorems which explicitly show that ‘quantum systems’ cannot be regarded as possessing definite valued properties, Dennis Dieks has turned his modal interpretation into an extreme form of antirealist perspectivalism (Bene & Dieks, 2002; Dieks, 2019). In order to address the basis problem both narratives have introduced the principle of decoherence which is an excellent example of the way in which the failure of ontological explanations are resolved in contemporary physics by simply shifting the justification in purely pragmatic terms, what is known as “FAPP”—short for For All Practical Purposes or “It works! So, shut up and calculate!”). What becomes clear is that, quite regardless of the ongoing battles between the many interpretations of QM, there is a common methodology which has allowed to construct all these fictional stories. This “Bohrian methodology”, as we might call it, works as follows. First dissect the theory in two parts. One which provides an epistemological pragmatic account of measurement outcomes, and another part which provides an ontological picture. In the latter part of the theory you are free to create a story with the notions of your preference: systems, worlds, particles, branchings, collapses, histories, flashes, propensities or whatever you add to your cart. This story will remain essentially unrelated to the mathematical formalism or experimental evidence (see Dorato 2006). Even if the narrative becomes unclear there are many tricks you can use to escape the need of explanation. If there is something that does not fit your story simply add something else which is less understood. For example, you can add ad hoc principles which do not necessarily explain anything but will either naturalize the problem the concepts were supposed to explain or shift the attention to another problem. This methodology is explicit in Dieks’ interpretational rule which has a purely operational reference but no

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link whatsoever to the way in which ‘quantum systems’ and ‘properties’ behave. And the same happens with the role played by quantum superpositions in the many worlds interpretation. While projection operators are interpreted as related to (real) ‘properties’ of elementary particles present in different worlds, the numbers which accompany the states are interpreted as (unreal) ‘degrees of beliefs’ of agents (or subjects). In order to make the narrative completely invulnerable to scientific criticism it is of outmost importance that—just like in the case of quantum jumps, collapses, branching, complementarity, correspondence, etc.—you do not relate the added “explanation” to an experimental procedure that could test the interpretation. There should exist no test in the lab allowing to conclude if what the interpretation says is tenable (or not) in a given situation. If you do this with “studied ambiguity”— as Deutsch has characterized it (Deutsch, 2004)—and vagueness the confusion will grow in such a way that the reader will not be able to follow you and at some point he will start doubting herself about her own capability of understanding. If nothing of this works, just blame the theory itself for its weirdness and failures. Just repeat the mantra: “It is quantum, beyond classical experience. That is why you cannot understand it!” In contraposition to the positivist anti-metaphysical scheme, we have argued repeatedly that metaphysics does play an essential role within scientific theories. But metaphysics is certainly not just a fictional story about the unobservable realm. Metaphysics involves nets of interrelated concepts which allows us to create in a systematic fashion objective-invariant moments of unity. Observability in physics has nothing to do with “common sense”, it is on the contrary a praxis derived from the formal-conceptual unity provided by the theory itself.15 As Tian Yu Cao makes the point: The old-fashioned (positivist or constructive empiricist) tradition to the distinction between observable and unobservable entities is obsolete. In the context of modern physics, the distinction that really matters is whether or not an entity is cognitively accessible by means of experimental equipment as well as conceptual, theoretical and mathematical apparatus. If a microscopic entity, such as a W-boson, is cognitively accessible, then it is not that different from a table or a chair. It is clear that the old constructive empiricist distinction between observables and nonobservables is simply impotent in addressing contemporary scientific endeavor, and thus carries no weight at all. If, however, some metaphysical category of microscopic entities is cognitively inaccessible in modern physics, then, no matter how basic it was in traditional metaphysics, it is irrelevant for modern metaphysics. (Cao, 2003, pp. 64–65)

It is only through the interrelation of formal-conceptual representations that theoretical experience becomes possible. As Einstein remarked to a young Heisenberg: “It is only the theory which decides what can be observed.” Some decades later Heisenberg (1973, p. 264) himself would state that: “The history of physics is not only a sequence of experimental discoveries and observations, followed by their mathematical description; it is also a history of concepts. For an understanding of

15 For a critical analysis of the role of observation in physical theories see Deutsch (2004), Maudlin

(2019).

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the phenomena the first condition is the introduction of adequate concepts. Only with the help of correct concepts can we really know what has been observed.” The creation of concepts and representations is thus essential to the very possibility of considering scientific experience. As Cao continues to explain: An important point here is that metaphysics, as reflections on physics rather than as a prescription for physics, cannot be detached from physics. It can help us to make physics intelligible by providing well-entrenched categories distilled from everyday life. But with the advancement of physics, it has to move forward and revise itself for new situations: old categories have to be discarded or transformed, new categories have to be introduced to accommodate new facts and new situations. (Cao, 2003, p. 65)

This is the essential point of disagreement with Bohr’s doctrine of classical concepts, positivism and contemporary post-positivism—or whatever you might like to call it. On this point, Einstein had already warned us about the dangers of dogmatism: Concepts that have proven useful in ordering things easily achieve such an authority over us that we forget their earthly origins and accept them as unalterable givens. Thus they come to be stamped as ‘necessities of thought,’ ‘a priori givens,’ etc. The path of scientific advance is often made impossible for a long time through such errors. For that reason, it is by no means an idle game if we become practiced in analyzing the long common place concepts and exhibiting those circumstances upon which their justification and usefulness depend, how they have grown up, individually, out of the givens of experience. By this means, their all-too-great authority will be broken. They will be removed if they cannot be properly legitimated, corrected if their correlation with given things be far too superfluous, replaced by others if a new system can be established that we prefer for whatever reason. (Howard, 2010)

It is from this realist understanding of physical theories, as providing many different representations of a state of affairs, that we have argued in de Ronde (2018a) that the (anti-realist) measurement problem should be replaced by what we have termed the superposition problem. Superposition Problem Given  a situation in which there is a quantum superposition of more than one term, . ci |αi , and given the fact that each one of the terms relates through the Born rule to a meaningful operational statement, the question is how do we conceptually represent this mathematical expression? What are the physical concepts that relate to each one of the terms in a quantum superposition? Just in the same way that the principles of existence, non-contradiction and identity have played a double role defining on the one hand the notion of ‘entity’ in the actual mode existence, and on the other, classical logic itself; quantum superpositions must also need to find a conceptual framework which unlocks its physical meaning. Heisenberg’s original mathematical formulation of QM was developed from empirical findings which escaped the classical representation. Thus, it is from the mathematical formalism and its operational content that we should derive a consistent and coherent conceptual scheme. Unfortunately, the notion of ‘elementary particle’ together with the notion of ‘certain event’ have acted as metaphysical and empirical obstructions, respectively, restricting the possibilities

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of a conceptual development. This was according to Wolfgang Pauli (Laurikainen, 1998, p. 193) “the most important and extremely difficult task of our time to work on the elaboration of a new idea of reality.” In this respect, Pauli also remarked that (Pauli, 1994, p. 126): “We [should] agree with P. Bernays in no longer regarding the special ideas, which Kant calls synthetic judgements a priori, generally as the pre-conditions of human understanding, but merely as the special pre-conditions of the exact science (and mathematics) of his age.” What we desperately need for QM is a new non-classical conceptual scheme which is consistently linked to the mathematical formalism and allow us to think—without ad hoc rules and principles—in a truly consistent manner about the states of affairs and experience described by the theory. In this respect, it is of outmost importance to go back to the start of the quantum voyage and recall the way in which the mathematical formalism of QM was actually developed in an invariant manner from a specific field of intensive experience.

13.5 What Do We Actually Observe in the Lab: A ‘Click’ or a ‘Pattern’? The idea that QM should be able to predict single ‘clicks’ in detectors is a naive empiricist prejudice that was established during the 1930s specially through the work of Dirac. Leaving behind the subversive spirit of the positivist principle which had allowed Heisenberg to escape classical atomism and develop matrix mechanics,16 Dirac would choose instead to dogmatically restrict the understanding of observability to binary certainty.17 We now make some assumptions for the physical interpretation of the theory. If the dynamical system is in an eigenstate of a real dynamical variable .ξ , belonging to the eigenvalue .ξ  , then a measurement off .ξ will certainly give as result the number .ξ  . Conversely, if the system is in a state such that a measurement of a real dynamical variable .ξ is certain to give one particular result (instead of giving one or other of several possible results according to a probability law, as is in general the case), then the state is an eigenstate of .ξ and the result of the measurement is the eigenvalue of .ξ to which this eigenstate belongs. (Dirac, 1974, p. 35) (emphasis in the original)

This distinction introduced within the theory—between certain and uncertain observations—was not to be found in the field of phenomena from which the original mathematical formalism had been actually developed. In fact, since its origin, the theory of quanta had been related to the intensity patterns appearing 16 As Heisenberg begun his foundational paper of 1925: “The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable. It was this same principle which was also kernel for Einstein’s development of special relativity and his criticism of the notion of simultaneity.” 17 As explained by Asher Peres See Peres (1993, p. 66): “The simplest observables are those for which all the coefficients .ar are either 0 or 1. These observables correspond to tests which ask yes-no questions (yes = 1, no = 0).”

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in radiation and frequency problems—in which single outcomes had been never considered. As described by Heisenberg: The origin of quantum theory is connected with a well-known phenomenon, which did not belong to the central parts of atomic physics. Any piece of matter when it is heated starts to glow, gets red hot and white hot at higher temperatures. The color does not depend much on the surface of the material, and for a black body it depends solely on the temperature. Therefore, the radiation emitted by such a black body at high temperatures is a suitable object for physical research; it is a simple phenomenon that should find a simple explanation in terms of the known laws for radiation and heat. (Heisenberg, 1958, p. 3)

Regardless of the simplicity of the problem, the difficulties were immense. It had become impossible to find—by following classical presuppositions—a suitable theoretical model that would account for the experience observed in the lab. This was until in the year 1900, Max Planck was finally able to operationally solve what was known as the “ultraviolet catastrophe” by introducing the famous quantum of action in the Rayleigh-Jeans law of intensive radiation for black bodies. It took 25 years for physicists to reach a closed mathematical formalism that would allow the theory of quanta to account in a consistent manner for what was actually observed in the lab. Heisenberg was able to develop matrix mechanics following two main guiding principles, first, to leave behind the classical (metaphysical) notion of particle-trajectory, and second, to take seriously Ernst Mach’s positivist rule according to which a theory should only make reference to what is actually observed in the lab. So what was observed? The answer is well known to any experimentalist: a spectrum of line intensities. This is what was described by the tables of data that Heisenberg had attempted to mathematically model in an invariant manner and finally led him—with the help of Max Born and Pascual Jordan—to the development of quantum mechanics. As he would himself recall: In the summer term of 1925, when I resumed my research work at the University of Göttingen —since July 1924 I had been Privatdozent at that university— I made a first attempt to guess what formulae would enable one to express the line intensities of the hydrogen spectrum, using more or less the same methods that had proved so fruitful in my work with Kramers in Copenhagen. This attempt lead me to a dead end —I found myself in an impenetrable morass of complicated mathematical equations, with no way out. But the work helped to convince me of one thing: that one ought to ignore the problem of electron orbits inside the atom, and treat the frequencies and amplitudes associated with the line intensities as perfectly good substitutes. In any case, these magnitudes could be observed directly, and as my friend Otto had pointed out when expounding on Einstein’s theory during our bicycle tour round Lake Walchensee, physicists must consider none but observable magnitudes when trying to solve the atomic puzzle. (Heisenberg, 1971, p. 60)

Of course, operational observations and their algorithmic prediction were not enough in order to constitute a unified, consistent and coherent account of phenomena. As Heisenberg (1973, p. 264) would make the point: “For an understanding of the phenomena the first condition is the introduction of adequate concepts. Only with the help of correct concepts can we really know what has been observed.” And this is the essential gap between the operational prediction of observations in the lab and the conceptual understanding of physical phenomena. As Heisenberg would recall in his autobiography the words of Pauli:

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‘Understanding’ probably means nothing more than having whatever ideas and concepts are needed to recognize that a great many different phenomena are part of coherent whole. Our mind becomes less puzzled once we have recognized that a special, apparently confused situation is merely a special case of something wider, that as a result it can be formulated much more simply. The reduction of a colorful variety of phenomena to a general and simple principle, or, as the Greeks would have put it, the reduction of the many to the one, is precisely what we mean by ‘understanding’. The ability to predict is often the consequence of understanding, of having the right concepts, but is not identical with ‘understanding’. (Heisenberg, 1971, p. 63)

Following the representational realist program (de Ronde, 2016b), a theory must be considered as a unity of meaning and sense, a wholeness constructed systematically in both mathematical and conceptual terms which allows us to represent in an immanent fashion a specific field of phenomena. It is exactly this consistent and coherent wholeness which provides the missing structural link between a mathematical formalism, a net of concepts and a field of phenomena. This unity, in order to be consistent, must also allow us to think beyond particular mathematical reference frames (i.e., invariance) and the specific viewpoint of empirical agents (i.e., objectivity). Thus, invariance and objectivity appear as necessary conditions for any consistent and coherent representation of physical reality which can be considered as detached from subjective viewpoints and reference frames. This invariant-objective requirement provides also a structural link between mathematics and concepts to which we now turn our attention.

13.6 From Operational Invariance to Conceptual Objectivity Taking as a standpoint that in physical theories concepts are essential for observation, that concepts are relational constructs which depend on their mutual interrelation and that this whole must be consistently related to mathematical formalisms, it is possible to provide a completely different approach to the orthodox empirical-positivist understanding of theories. This approach, which we have termed representational realism (de Ronde, 2016b), goes back not only to the writings of Einstein, Pauli and Heisenberg but also to the original ancient Greek meaning of physics. According to this understanding, physical theories provide a unified, consistent and coherent formal-conceptual invariant-objective representation of states of affairs and experience.18 In the case of QM, what we are missing is the conceptual framework in order to understand what we observe. As we have argued elsewhere (de Ronde & Massri, 2017), the key to develop such an objective conceptual representation can be found in the operational-invariant structure of the theory itself. As Max Born (1953) reflected: “the idea of invariant

18 It might be remarked that we consider the idea that such theoretical representations provide a description of reality-as-it is not only naïve but also extremely misleading for a proper understanding of the realist quest (de Ronde, 2016b).

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is the clue to a rational concept of reality, not only in physics but in every aspect of the world.” Indeed, in physics invariants—quantities having the same value for any reference frame—allow us to determine what can be considered—according to a mathematical formalism—the same independently of the particular choice of a reference frame. The transformations that allow us to consider the physical magnitudes from different frames of reference have the property of forming a group. While in the case of classical mechanics invariance is provided via the Galilei transformations and in relativity theory via the Lorentz transformations, in QM the invariant content of the theory is brought by Born’s famous rule. Born Rule Given a vector . in a Hilbert space, the following rule allows us to predict the average value of (any) observable P . |P | = P 

.

This prediction is independent of the choice of any particular basis. This rule, which gives the operational-invariant content to the theory, provides the guiding line to develop an objective representation without the need to refer to objects. Something that could be termed, objectivity without objects; i.e., a conceptual objectivity.19 While invariance allows to detach the mathematical representation from particular reference frames, it is objectivity—understood in abstract terms—which allows us to detach the conceptual representation from the particular observations made by (empirical) subjects. The detachedness of the subject is essential for physics to refer to reality. As Einstein remarked repeatedly, observation cannot change the representation of the state of affairs. The notions of invariance and objectivity are intrinsically related, the first being the mathematical counterpart of the second, and the latter being the conceptual counterpart of the first (see for a detailed analysis de Ronde & Massri, 2017). It is this interrelation between mathematics and concepts which allow us to create a formal-conceptual representational moment of unity which is able to subsume a multiplicity of phenomena independently of reference frames or—even—particular observations. Following this line of reasoning, it becomes natural to understand Born’s rule as providing objective intensive information of a (quantum) state of affairs instead of subjective information about measurement outcomes (see for a detailed analysis and discussion de Ronde, 2016a; de Ronde et al., 2019). It is in this way that the pieces of the puzzle begin to fall into place. Intensity becomes not only the phenomenological standpoint of development of the mathematical formalism of QM but also the objective support of a new conceptual scheme which must go beyond the classical binary representation of modern physics.

19 As

proposed in de Ronde (2016a) there is a natural extension of what can be considered to a Generalized Element of Physical Reality: If we can predict in any way (i.e., both probabilistically or with certainty) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.

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In several papers (de Ronde and Massri, 2019, 2022), we have presented a formal-conceptual representation which provides explicit relationships between Heisenberg’s mathematical formalism—without the measurement rule—and a specially suited set of (non-classical) concepts. While projection operators are related to the notion of intensive power, their potentia is computed via the Born rule in an invariant manner. This non-classical representation of reality implies a radical shift from a binary picture of ‘systems’ imposed by definite valued ‘properties’, to an intensive account of ‘powers’ with definite ‘potentia’. The Born rule is then understood as quantifying the mode of existence of powers (i.e., their potentia). Thus, unlike properties conceived in binary terms (as strictly related to the values 0 or 1), quantum powers extend the realm of existence to an intensive level in which any potentia pertaining to the interval [0,1] must be accepted as a certain intensive value. Thus, escaping from empirical and metaphysical prejudices, quantum certainty becomes detached from the (binary) actual realm. This means that the potentia .p = 0.764 is not the measure of the ignorance of an agent about the actualization of a measurement outcome but the computation of an intensive value which can be tested operationally in the lab. Of course, this move has deep consequences not only for the understanding of the formalism in purely intensive terms—beyond actualism—, but also for the experience discussed by the theory. The intensive valuation of powers has allowed us not only to escape the reference to measurement outcomes—and consequently, to avoid the measurement problem— but also to bypass Kochen–Specker contextuality. In this latter respect, we have derived a non-contextual intensive theorem (de Ronde & Massri, 2022, Theo. 4) which, through the provision of a global (intensive) valuation of all projection operators, allow us to understand contexts (or bases) in invariant-objective terms (de Ronde & Massri, 2017). From a metaphysical viewpoint, the existence of quantum powers implies a potential realm of existence. Consequently we obtain a shift from a (classicalbinary) representation in terms of an Actual State of Affairs (ASA) to a (quantumintensive) representation in terms of a Potential State of Affairs (PSA).20 As explained in detail in de Ronde and Massri (2022), intuitively, we can picture a PSA as a table,

 : G(H) → [0, 1],

.

⎧ P1 ⎪ ⎪ ⎪ ⎨ P2 : P 3 ⎪ ⎪ ⎪ ⎩

→ p1 → p2 → p3 .. .

Our objective representation in terms of a PSA constituted by powers with definite potentia (computed via the Born rule) allows us to bypass Kochen–Specker 20 It is important to stress that the potential mode of existence to which we refer is completely independent of the actual realm and should not be understood in teleological hylomorphic terms as referring to the future actualization of measurement outcomes (de Ronde, 2017).

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contextuality as an epistemic feature of the theory which provides a constrain to the simultaneous measurement of incompatible powers—avoiding relativist choices which explicitly change the representation of (quantum) reality. Let us remark that this same aspect of epistemic contextuality is also common to classical physics. Even though some powers are epistemically incompatible (i.e., they require mutually incompatible measurement set ups in order to be observed) they are never conceptually incompatible since they can be all considered to exist simultaneously through a Global Intensive Valuation determined via the Born rule. The PSA, constituted by the set of potentially existent quantum powers, is in this respect completely objective—i.e. detached from any particular basis or observation made by an agent. Notice that this is completely analogous to the way in which the notion of ASA, as constituted by sets of actually definite valued properties, is regarded as objective in the case of classical physics. Objective means in this case that there exists a coherent global representation of a state of affairs which is consistent with the multiple observations of phenomena and independent of the choice of the context (or basis). The main difference between an ASA and a PSA is determined by their distinct conditions of objectivity. While in the classical case an ASA is defined in terms of a set of systems with definite binary valued properties; in the quantum case the PSA is defined in terms of powers with definite (intensive) potentia. Our approach shows explicitly a path to derive a conceptual framework which matches consistently the mathematical formalism of the theory providing at the same time an intuitive grasp of the experience involved by it. In turn, this new (non-classical) conceptual framework has allowed us to reconsider some essential notions like quantum superpositions and entanglement from a completely new objective viewpoint (de Ronde & Massri, 2019, 2021, 2023).

13.7 Measuring Quantum Superpositions Given a representation of a state of affairs as described by a theory we can imagine the possible thought-experiences implied by it. This specific type of physical thinking is of course not restricted by what we have observed nor by our technical capabilities, it is only constrained by the mathematical and conceptual theoretical schemes of representation. A physicist can derive conclusions from a theory without ever observing or measuring anything. In fact, that is the true power of physics. One which allows us to advance into future developments through the abstract theoretical experience as provided by Gedankenexperiments. Einstein’s and Schödinger’s famous thought-experiences from 1935 are good examples of the way in which physicists can go far beyond the technical restrictions of their epoch and conclude the existence of fantastic—never before measured nor observed— phenomena like quantum entanglement. There are countless examples in the history of physics which show this same amazing theoretical capacity of predicting things that were never before observed nor even imagined. Of course, this does not mean that experimental testing should be left aside—as some contemporary influential

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physicists are arguing today.21 Experimental evidence is essential to physics in order to secure a consistent link between any theory and what actually happens in the here and now. Without empirical testing physical theories loose their compass, following mirages and lost and disoriented, become an easy pray for instrumentalism and dogmatic metaphysics—or a combination of both. As Einstein Dieks (1988, p. 175) would also make the point, even though measurement and observability cannot play an internal role within a theory, “the only decisive factor for the question whether or not to accept a particular physical theory is its empirical success.” It is only measurement which allows us to connect the detached experience coming from a theory with the hic et nunc phenomena observed in the lab, providing in this way the necessary link between (objective) theoretical representation and (subjective) empirical observability. While the field of thought-experience is strictly limited by the theory, observation is a purely subjective conscious action which cannot be theoretically represented. It is ‘measurement’ which stands just in the middle between the objective representation of physical experience and the subjective hic et nunc empirical observation itself. Measurements, at least for the realist, are conscious actions performed by human subjects which are able to select, reproduce and understand a specific type of phenomenon. This is of course a very complicated process created by humans which interrelates practical, technical and theoretical knowledge. Anyone attempting to perform a measurement must be able to think about a specific problem, she must be also able to construct a measurement arrangement, she must be capable to technically produce and analyze what might be going on within the process, and finally, she must be qualified to observe, interpret and understand the phenomenon that actually takes place, hic et nunc, when the measurement is actually performed. All these requirements imply human capacities and, in particular, consciousness. The table supporting the measurement set-up does not understand what complicated process is taking place above itself. Tables and chairs cannot construct a measuring set-up. The chair that stands just beside the table cannot observe a measurement result, and the light entering the lab through the window cannot interpret what is going on. Tables, chairs and dogs do not and cannot perform measurements. It is only a conscious (empirical) subject (or agent) who is capable of performing a measurement. And as any physicist who has been trained in a lab knows very well, these actions have nothing to do with “common sense”; in fact, they imply a very specific theoretical and technical knowledge.

21 Gerard t’ Hooft (2001) has argued that: “Working with long chains of arguments linking theories

to experiment, we must be able to rely on logical precision when and where experimental checks cannot be provided.” Following the same line of reasoning Steven Weinberg (2003) has gone even further claiming that: “I think 100 years from now this particular period will be remembered as a heroic age when theorists cut themselves temporarily free from their experimental underpinnings and tried and succeeded through pure theoretical reasoning to develop a unified theory of all the phenomena of nature.” More recently, Richard Dawid has also argued in favor of considering nonempirical arguments in order to justify mathematical theories (Dawid, 2013).

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Our theoretical standpoint allows us to reject right from the start not only the (dogmatic metaphysical) existence of quantum “collapses” and “jumps” but also the (naive empirical) requirement—imposed by the twentieth century positivist understanding of theories—to introduce the measurement rule (or projection postulate) within the theory of quanta. It is only adequate concepts which can allow us to understand an observation for it is only the theory which decides what can be observed. That is the reason why, when learning physics as a graduate student, you first learn the theory, and only then you go to the lab—not the other way around. It makes no sense to try to measure an ‘electromagnetic field’ if you have not yet learned what is the mathematical and conceptual representation of a ‘field’. But even having learned the theoretical definition of a ‘field’, theories do not come with a user’s manual telling us how to measure them. Every physicist has experienced as a student the abysm of entering a lab after finishing a theoretical course in classical mechanics or electromagnetism. As a student who has just learned a theory you simply have no clue what to do in a lab in order to test the theory you just learned in the classroom. It requires a lot of technical skills and knowledge—that you need to learn in a lab—in order to be capable to test something the theory implies. Of course, the theory must provide the conditions for restricting the possibilities of what can be measured. And in this respect, the most important qualitative link is provided by physical concepts. A physical concept must be able to account for the operational conditions under which it becomes possible to test experimentally its own consistency. As famously remarked by Einstein when discussing about the definition of the concept of simultaneity: The concept does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case. We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.) (Einstein, 1920, p. 26)

A physical concept must be designed in order to bring into unity the multiplicity found within a specific field of experience. In order to do so, there are two main conditions which are essential for the construction of any consistent and coherent physical concept that attempts to represent reality, namely, operationalrepeatability and operational-invariance. First, operational-repeatability points to the fact that a physical concept must be able to bring into unity the multiplicity of physical phenomena observed in different subsequent tests. If every time we observe something it refers to something different, it becomes then impossible to keep track of anything. This is a problem which is well known since Heraclitus’ theory of becoming. If there is no sense of repeatability, the reference of different experiences is precluded right from the start and just like in the famous story by Jorge Luis Borges, Funes the Memorious, the necessary link between the observation of ‘the dog at three-fourteen’ and ‘the dog at three fifteen’ is completely lost (see for a detailed discussion Norsen (2005)).

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Definition 13.7.1 (Operational Repeatability) A physical concept must be able to bring into unity the multiplicity of physical phenomena observed in different subsequent tests. Second, operational-invariance points to the fact that a physical concept must be also able to provide a consistent ground for the operational testability with respect to different frames of reference (or perspectives). Observers from different perspectives should agree about what they observe. The ‘dog observed form a profile’ should be considered as the same ‘dog observed from the front’. In mathematical terms, this means there must exist an operational-invariant formalism which allows us to discuss what is the same independently of the reference frame. Definition 13.7.2 (Operational Invariance) A physical concept must be also able to provide a consistent unified account of the operational testability considered with respect to different frames of reference (or bases). Without these two conditions, allowing us to discuss about experience not only from different instants of time but also from different perspectives, the possibility to produce a coherent and consistent linguistic discourse about reality becomes precluded right from the start. In classical physics, the concepts that secure the tenability of these conditions are the physical concepts of ‘particle’ and ‘wave’ supplemented by the Galilean transformations. We might let Einstein conclude that: By the aid of speech different individuals can, to a certain extent, compare their experiences. In this way it is shown that certain sense perceptions of different individuals correspond to each other, while for other sense perceptions no such correspondence can be established. We are accustomed to regard as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perceptions. (Einstein, 1922, p. 1)

Unfortunately, in the context of QM, these conditions have been wiped out from the theory. While it has been argued that operational-repeatability is unattainable due to the fact ‘quantum particles’ are destroyed within each measurement, operational-invariance has been simply replaced by Bohr’s famous principle of complementarity. On the one hand, the idea that quantum particles are destroyed each time they are measured not only precludes the possibility to consider a repeatable analysis of the object under study, it also gives an alibi to justify why we have no clue of what quantum particles really are. On the other, complementarity has allowed us to naturalize not only the inconsistent representations of a state of affairs in terms of ‘waves’ and ‘particles’—none of which is linked to the mathematical formalism—, but also to build a new “contextual common sense” which precludes the possibility to describe the properties of quantum systems from different frames of reference (or bases). Lacking both operational-repeatability and operational-invariance the orthodox discourse of QM which makes reference to “quantum particles” has become nothing more than a fictional story with no link to the mathematical formalism nor to the experience we observe in the lab. In this

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context, both physicists and philosophers acknowledge the discourse about quantum particles should be considered as “just a way of talking” (de Ronde & Fernández Mouján, 2021). Since the tragic slaughter of operational-repeatability and operational-invariance in QM, the theory has been accused of multiple crimes. Apart from its incapacity to represent “quantum particles” convincingly, one of the most famous has been its inability to provide certain predictions. So they say, QM is intrinsically “random”, an “unpredictable” theory which, due to the existence of quantum superpositions, is incapable to predict—apart from very special cases—the result of a measurement outcome produced by an “elementary particle”. This has created an unspeakable aversion against superpositions within a community which silently agrees that these quantum weirdos should be somehow removed from the theory (de Ronde, 2018a). All these difficulties arise in the theory given we accept dogmatically that QM should make reference to single measurement outcomes which are consequence of quantum particles. On the contrary, if we simply recognize the fact that QM talks about intensive (non-binary) patterns all difficulties linked to contextuality suddenly disappear and it becomes quite easy to produce a mathematical invariant representation of what the theory talks about. From this standpoint, it becomes completely natural to argue that the Born rule should be understood as making reference to intensive patterns represented by values pertaining to the interval .[0, 1]—instead of making reference to single ‘clicks’ with possible values .{0, 1}. As a matter of fact, Heisenberg’s matrix formalism already provide a consistent link between the mathematical formalism and experience, one which has been repeatedly tested in the lab through the statistical analysis of mean values—which is what QM actually predicts. Since Heisenberg created matrix mechanics following Mach’s observability principle, QM has provided the operational conditions required to discuss measurements in a consistent manner right from the start. If we avoid the temptation to talk about ‘elementary particles’—i.e., falling pray of dogmatic metaphysics—or about single ‘events’—i.e., falling pray of naive empiricism—, the theory of quanta can be naturally restored as a theoretical (formal-conceptual) representation which is both operationally-repeatable and operationally-invariant. The opinion that an intensive representation of reality is untenable just because it escapes our classical binary representation in terms of an Actual State of Affairs22 —common to all classical physics, including relativity—or the conviction that observations should be understood as referring to single outcomes—common to the positivist contemporary understanding of theories—, is a belief supported only by dogma.

22 As

discussed in detail in de Ronde and Massri (2022), an Actual State of Affairs (ASA) can be defined as a closed system considered in terms of a set of actual (definite valued) properties which can be thought as a map from the set of properties to the .{0, 1}. Specifically, an ASA is a function . : G → {0, 1} from the set of properties to .{0, 1} satisfying certain compatibility conditions. We say that the property .P ∈ G is true if .(P ) = 1 and .P ∈ G is false if .(P ) = 0. The evolution of an ASA is formalized by the fact that the morphism f satisfies .f = .

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What is needed today, is to begin to accept that QM does not make reference to a binary experience consequence of a microscopic realm constituted by unobservable particles. The Born rule does not need to be understood as making reference to the position of single ‘particles’ and ‘clicks’, instead it can be naturally read as providing invariant intensive information of a state of affairs—yet to be objectively described (i.e., conceptually). In fact, as we have shown explicitly in de Ronde and Massri (2022), if we choose to stay close to the quantum formalism and make reference to the intensive data computed by the theory (i.e., mean values), invariance can be easily restored. Since we already have a consistent invariant mathematical formalism, the problem remains to derive an objective physical concept which can be quantified in terms of a number pertaining to the closed interval .[0, 1]. In de Ronde (2017), de Ronde (2018b), de Ronde and Massri (2022), de Ronde and Massri (2019), de Ronde and Massri (2021) we have presented a new (nonclassical) conceptual architectonic which through the notion of intensive power allows us to provide an anschaulich objective-invariant content to the theory of quanta. According to this representation, QM talks about a Potential State of Affairs constituted by powers with definite potentia. A quantum power can be understood in terms of a potential action which— even though has an expression in actuality—does not require its actualization in order to exist and interact with other powers. Powers exist in a potential mode of existence which is not reducible to a teleological reference to actuality. It is the widespread confusion between metaphysical actuality—which makes reference to a categorical representation of existence—and empirical actuality—which makes reference to here and now observability—which has allowed to mix up the notion of object with the observation of its profile; i.e., the obvious fact that the adumbration of an object is not the observation of the object itself (i.e., the totality of its profiles). the concept of ‘object’ is a metaphysical machinery capable of unifying multiple experiences, it is not a ‘thing’ in the ontological sense- We never observe an object in its theoretical totality. Obviously, observing the side of a table does not imply the existence of the side which remains invisible (see for a detailed analysis de Ronde, 2023b). Only through multiple observations in the here and now, it is possible to grasp a (metaphysical) object of experience. Indeed, a physical concept has the goal of bringing into unity distinct experiences, it is built in order to see what is the same within difference. The intensive powers that we find in QM are, in this particular respect, not very different from the objects we observe in classical mechanics. Just like it is only through multiple adumbrations of an object that we gain knowledge about its constitution, it is only through multiple observations that we can measure the intensive quantification of a power. Thus, the shift required by QM is twofold. While in the mathematical level we must go from a binary representation to an intensive one, in the conceptual level we must move from the notion of binary actual object to that of intensive potential power. The latter switch also implies a metaphysical shift in the consideration of the mode of existence which must now turn from an Actual State of Affairs to a Potential State of Affairs. A Potential State of Affairs can be understood as the set of powers with potentia represented formally by an abstract matrix, .ρ, and conceptually as all

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 the experimental possible actions of a lab. A quantum superposition, . ci |αi , or a density operator in a specific basis .ρB , can be then regarded as a specific experimental arrangement in that lab; i.e., a quantum perspective specified by a basis (or reference frame) in which a particular subset of powers can become actualized (de Ronde & Massri, 2019). The important point here is that once we accept an intensive form of quantification, all quantum powers can be considered as existent beings independently of the observation of their adumbration exposed in a measurement outcome. Powers interact between each other—in a potential realm— as described by the mathematical formalism. Consequently, one must distinguish between potential effectuations related to the process of entanglement and actual effectuations related to measurement outcomes (de Ronde & Massri, 2021). Powers are not binary, they are intensive in nature. Thus, their understanding requires always a statistical level of analysis which captures the possibility of repeatable measurements right from the start. Unlike the case of elementary particles which cannot be measured repeatedly, an intensive power remains the same independently of its particular observations (operational-repeatability) or the basis chosen to measure it (operational-invariance). And since the potential state of affairs does not change due to measurements, it makes perfect sense to claim that a set of powers are objectively real. All this can be consistently tested in the lab. In order to measure different powers we need to observe their actualizations from different quantum perspectives. Of course, in order to measure the potentia of each power we require many repeated measurements. Due to our global intensive representation the context we chose to measure a power has no influence whatsoever in the results we will obtain. This goes in line with the already mentioned operational conditions presented by the Born rule itself. Our definitions are completely objective, in the sense that the potential state of affairs remains always completely independent of the choice of a specific measurement set up or any observed measurement outcome. Unlike the orthodox contextual understanding of QM, our approach restores the necessary global consistency in order to produce an objective representation of a (potential) state of affairs. Grounded on an empiricist understanding of observation Dennis Dieks has argued in Dieks (2010, p. 133) against the real power of conceptuality: “I think it is unclear how a realist interpretation of p as some kind of ontologically objective chance can help our understanding of what is going on in nature. Clearly, such an interpretation cannot change the empirical content and predictive power of the theory that is involved.” On the contrary, we have shown in our referred works how the conceptual understanding of modality as intensive objective information about powers does provide not only an increase in the explanatory capacity of the theory of quanta, but also a radical change in the way in which its empirical content must be considered and analyzed in the context of experimental testing in the lab. Some explicit examples of the manner in which the conceptual representation has inevitable consequences for the analysis of data have been already provided in the context of quantum computation (de Ronde et al., 2019) and quantum entanglement (de Ronde & Massri, 2021, 2023). We expect that our research proposal will provide

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a new path to continue investigating what really needs to be measured according to the theory of quanta. Acknowledgments This work was partially supported by the following grants: the Project PIO CONICET-UNAJ (15520150100008CO) “Quantum Superpositions in Quantum Information Processing”, UNAJ INVESTIGA 80020170100058UJ.

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Dorato, M. (2006). Properties and dispositions: Some metaphysical remarks on quantum ontology. Proceedings of the AIP, 844, 139–157. Einstein, A. (1920). Relativity. The special and general theory. New York: Henry Holt & Company. Einstein, A. (1922). The meaning of relativity. New Jersey: Princeton University Press. Faye, J., & Jaksland, R. (2021). What Bohr wanted Carnap to learn from quantum mechanics. Studies in History and Philosophy of Science, 88, 110–119. French, S. (2011). Metaphysical underdetermination: Why worry? Synthese, 180, 205–221. Everett, H. (2012). In J. A. Barrett & P. Byrne (Eds.), The Everett interpretation of quantum mechanics: Collected works 1955–1980 with commentary. Princeton: Princeton University Press. Hanson, R. (1958). Patterns of discovery: An inquiry into the conceptual foundations of science. Cambridge: Cambridge University Press. Heisenberg, W. (1958). Physics and philosophy. London: World perspectives, George Allen and Unwin Ltd. Heisenberg, W. (1971). Physics and beyond. New York: Harper & Row. Heisenberg, W. (1973). Development of concepts in the history of quantum theory. In J. Mehra (Ed.), The physicist’s conception of nature (pp. 264–275). Dordrecht: Reidel. Hempel, C. G. (1958). The theoretician’s dilemma: A study in the logic of theory. Minnesota Studies in the Philosophy of Science, 2, 173–226. Howard, D. (2010). Einstein’s philosophy of science. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Summer 2010 Edition). http://plato.stanford.edu/archives/sum2010/entries/ einstein-philscience/ Laurikainen, K. V. (1998). The message of the atoms, essays on Wolfgang Pauli and the unspeakable. Berlin: Springer. Maudlin, T. (2019). Philosophy of physics. Quantum theory. Princeton: Princeton University Press. Nagel, E. (1961). The structure of science: problems in the logic of scientific explanation. New York: Hackett Publishing Company. Norsen, T. (2005). Einstein’s boxes, American Journal of Physics, 73, 164–176. Osnaghi, S., Freitas, F., & Freire, O. (2009). The origin of the Everettian heresy. Studies in History and Philosophy of Modern Physics, 40, 97–123. Pauli, W. (1994). In C. Enz & K. von Meyenn (Eds.), Writings on physics and philosophy. Berlin: Springer-Verlag. Peres, A. (1993). Quantum theory: Concepts and methods. Dordrecht: Kluwer Academic Publishers. Popper, K. R. (1963). Conjectures and refutations: The growth of scientific knowledge. London: Routledge. Popper, K. (1992). The logic of scientific discovery. New York: Routledge. Schrödinger, E. (1935), The present situation in quantum mechanics, Naturwiss, 23. In J. A. Wheeler & W. H. Zurek (Eds.), Translated to English in Quantum Theory and Measurement, 1983 (pp. 807–812). Princeton: Princeton University Press. Schrödinger, E. (1950). What is an elementary particle? Endeavor, IX(35), 109. Smolin, L. (2007). The trouble with physics. The rise of string theory, the fall of a science, and what comes next. New York: Mariner Books. Sudbery, T. (2016). Time, chance and quantum theory. In D. Aerts, C. de Ronde, H. Freytes, & R. Giuntini (Eds.), Probing the meaning and structure of quantum mechanics: Superpositions, semantics, dynamics and identity (pp. 324–339). Singapore: World Scientific. ‘T Hooft, G. (2001). Can there be physics without experiments? Challenges and pitfalls. International Journal of Modern Physics A, 16, 2895–2908. van Fraassen, B. C. (1980). The scientific image. Oxford: Clarendon. van Fraassen, B. C. (1991). Quantum mechanics: An empiricist view. Oxford: Clarendon. Vermaas, P. E. (1997). A No-Go theorem for joint property ascriptions in modal interpretation. Physics Review Letters, 78, 2033–2037. Vermaas, P. E. (1999). A philosophers understanding of quantum mechanics. Cambridge: Cambridge University Press.

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Vermaas, P. E., & Dieks, D. (1995). The modal interpretation of quantum mechanics and its generalization to density operators. Foundations of Physics, 25, 145–158. von Neumann, J. (1955). Mathematical foundations of quantum mechanics. Princeton: Princeton University Press. Wallace, D. (2017). Interview closer to truth. https://www.youtube.com/watch?v=Rql9tHE3pGA Weinberg, S. (2003). Viewpoints on string theory, published by NOVA online at www.pbs.org/ wgbh/nova/elegant/view-weinberg.html

Chapter 14

Recognizing Concepts and Recognizing Musical Themes A Quantum Semantic Analysis Maria Luisa Dalla Chiara, Roberto Giuntini, Eleonora Negri, and Giuseppe Sergioli

Abstract How are abstract concepts and musical themes recognized on the basis of some previous experience? It is interesting to compare the different behaviors of human and of artificial intelligences with respect to this problem. Generally, a human mind that abstracts a concept (say, table) from a given set of known examples creates a table-Gestalt: a kind of vague and out of focus image that does not fully correspond to a particular table with well determined features. A similar situation arises in the case of musical themes. Can the construction of a gestaltic pattern, which is so natural for human minds, be taught to an intelligent machine? This problem can be successfully discussed in the framework of a quantum approach to pattern recognition and to machine learning. The basic idea is replacing classical data sets with quantum data sets, where either objects or musical themes can be formally represented as pieces of quantum information, involving the uncertainties and the ambiguities that characterize the quantum world. In this framework, the intuitive concept of Gestalt can be simulated by the mathematical concept of positive centroid of a given quantum data set. Accordingly, the crucial problem “how can we classify a new object or a new musical theme (we have listened to) on the basis of a previous experience?” can be dealt with in terms of some special quantum similarity-relations. Although recognition procedures are different for human and for artificial intelligences, there is a common method of “facing the problems” that seems to work in both cases.

M. L. Dalla Chiara () Dipartimento di Lettere e Filosofia, Università di Firenze, Florence, Italy R. Giuntini · G. Sergioli Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Cagliari, Italy E. Negri Scuola di Musica di Fiesole, Fiesole-Florence, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_14

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14.1 Introduction There is a story about a king who was able to distinguish between two different pieces of music only: the “Royal March” and the “Non-Royal March”. One is dealing with an extreme case of a “fully non-musical personality”. Normally people behave differently; and children often show an early capacity of recognizing and repeating some simple songs they have listened to. Of course, such capacity generally depends on what is usually called the “musical talent” of each particular child. And, in the case of adult persons, one shall distinguish the behavior of professional musicians from the behavior of generic music-listeners who may have different degrees of musical culture.1 Recognizing a musical theme is a cognitive operation that is very similar to what happens when we recognize an abstract concept that may refer either to concrete or to ideal objects (say, table, star, triangle,. . . ). We know how quickly children learn to abstract concepts from the concrete objects they have met in their brief experience. On the basis of a small number of examples that have appeared in the environment where they are living, they easily recognize and use (mostly in a correct way) general concepts like table, house, toy. How are abstract concepts formed and recognized on the basis of some previous experience? What happens in the human brain when we recognize either an abstract concept or a musical theme? These questions have been intensively investigated, with different methods, by psychologists, neuroscientists, artificial intelligence researchers, logicians, philosophers, musicians and musicologists. Some important researches in the field of neurosciences (which have used sophisticated brainimaging techniques) have provided some partial answers. Interestingly enough, neuroscientific investigations have recently interacted with an important approach to psychology: the Gestalt-theory that had been proposed by Wertheimer, Kofka and Köhler in the early twentieth century.2 As is well known, a basic idea of Gestalt-psychology is that human perception and knowledge of objects is essentially connected with our capacity of realizing a Gestalt (a form) of the objects in question: a holistic image that cannot be identified with the set of its component elements. The cognitive procedure goes from the whole to the parts, and not the other way around! These general ideas can be naturally applied to investigate the question “how are abstract concepts formed and recognized?” A human mind that abstracts the concept table from a given set of concrete examples, generally creates a tableGestalt, a kind of vague and out of focus image that does not fully correspond to a particular table, with well determined features. When we ask different people the question: what do you see in your mind when you hear the word “table”?, we may receive different answers. An interesting answer that has been given by a person submitted to a psychological test is the following: “I see a table, with an indefinite

1 See, 2 See,

for instance, Honing (2009). for instance, Ehrenstein et al. (2003).

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color, floating in an indefinite space”. In this case the vagueness of the table-Gestalt has been expressively described by the metaphorical image of a “floating object”. Creating a Gestalt associated to a given concept is a cognitive operation that is quite natural for human intelligences. But what happens in the case of artificial intelligences? Is it possible to find a mathematical definition for a Gestalt-like concept that could be taught to an intelligent machine? We will see how this intriguing question can be successfully investigated in the framework of a quantum inspired approach to pattern recognition and to machine learning. From a logical point of view the relationship that connects gestaltic patterns with particular concrete or ideal objects can be analyzed by using the concept of similarity. Consider a child (let us call her Alice) who has recognized as a table a new object that has appeared in her environment. Apparently, her recognition is essentially based on a quick and probably unconscious comparison between the main features of the new object and the ideal table-Gestalt that Alice had previously stored in her memory. Generally, any comparison involves the use of some similarity-relations, weak examples of relations that are • reflexive: any object a is similar to itself; • symmetric: if a is similar to b, then b is similar to a; • generally non-transitive: if a is similar to b and b is similar to c, then a is not necessarily similar to c. Just the failure of the transitive property is one of the reasons why similarityrelations play an important role in many semantic and cognitive phenomena. A significant example is represented by metaphorical arguments, which frequently occur in natural languages as well in the languages of art. Metaphorical correlations generally involve some allusions that are based on particular similarity-relations. Ideas that are currently used as possible metaphors are often associated to concrete and visual features. Let us think, for instance, of a visual idea that is often used as a metaphor: the image of the sea, correlated to the concepts of immensity, of infinity, of pleasure or fear, of places where we may get lost and die. In the tradition of scientific thought metaphorical arguments have often been regarded as “fallacious”. There is a deep logical reason that justifies such suspicion. Metaphors, based on particular similarity-relations, do not generally preserve the properties of the objects under consideration: if Alice is similar to Beatrix and Alice is clever, then Beatrix is not necessarily clever! Wrong extrapolations of properties from some objects to other similar objects are often used in rhetoric contexts, in order to obtain a kind of captatio benevolantiae. We need only think of the soccermetaphors that are so frequently used by many politicians! In spite of their possible “dangers”, metaphors have sometimes played an important role even in exact sciences. An interesting example in logic is the current use of the metaphor of possible world, based on a general idea that had been deeply investigated by Leibniz. In some situations possible worlds, that correspond to special examples of semantic models, can be imagined as a kind of “ideal scenes”, where abstract objects behave as if they were playing a theatrical play. And a

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“theatrical imagination” has sometimes represented an important tool for scientific creativity, also in the search of solutions for logical puzzles and paradoxes. The classical concept of possible world is characterized by a strong logical determinism: due to the semantic excluded middle principle, any sentence that refers to a given possible world shall be either true or false. Quantum information and quantum computation theories have recently inspired the development of a new form of quantum-logical semantics: classical possible worlds have been replaced by vague possible worlds, where events are generally uncertain and ambiguous, as happens in the case of microobjects.3 In the next sections we will see how recognition-processes that may concern either abstract concepts or musical themes can be naturally investigated in the framework of this quantum-semantic approach.

14.2 A Quantum Semantics Inspired by Quantum Information Theory For the readers who are not familiar with quantum mechanics it may be useful to recall some basic concepts of the theory that play an important semantic role. Suppose a physicist is studying a quantum physical system .S (say, an electron) at a given time. His (her) information about .S can be identified with a particular mathematical object that represents the state of .S at that time. In the happiest situations our physicist might have about .S a maximal information that cannot be consistently extended to a richer knowledge. In such a case, the information in question is called a pure state of the system. An observer who has assigned a pure state to a given system knows about this system all that even a hypothetical omniscient mind would know. Following a happy notation introduced by Paul Dirac, quantum pure states are usually denoted by the expressions .|ψ, |ϕ, . . . (where .|. . . represent the so called “ket-brackets”). Mathematically, any pure state .|ψ is a vector (with length 1) living in a special abstract space, called a Hilbert space (usually indicated by the symbol .H).4 A strange logical feature of the quantum formalism is the following: although representing a maximal piece of information, a pure state .|ψ cannot decide all physical properties that may hold for a quantum system described by .|ψ. Due to the celebrated Heisenberg’s uncertainty principle some basic properties (that may concern, for instance, either the position or the velocity) turn out to be essentially indeterminate.

3 See,

for instance, Dalla Chiara et al. (2018). spaces are special examples of vector spaces that represent generalizations of geometric Euclidean spaces. A simple example of a Hilbert space is the geometric plane, whose set of points corresponds to the set of all possible ordered pairs of real numbers. 4 Hilbert

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Generally, a quantum pure state .|ψ can be represented as a superposition (a vector-sum) of other pure states .|ψi : |ψ =



.

ci |ψi ,

i

where .ci are complex numbers (called amplitudes). The physical interpretation of this formal representation is the following: a quantum system whose state is .|ψ might verify the properties that are certain for a system in state .|ψi  with a probability-value that depends on the number .ci .5 From an intuitive point of view one can say that a superposition-state seems to describe a kind of ambiguous cloud of possibilities: a set of potential properties that are, in a sense, all co-existent for a given quantum object. Special examples of superpositions that play an important role in quantum information are represented by qubits. As is well known, in classical information theory information is measured in terms of bits. One bit represents the informationquantity that is transmitted (or received) when one answers either “Yes” or “No” to a given question. The two bits are usually indicated by the natural numbers 0 (corresponding to the answer “No”) and 1 (corresponding to the answer “Yes”). On this basis, complex pieces of information are represented by sequences of many bits, called registers. For instance, a register consisting of 8 bits represents one byte. In quantum information theory the quantum counterpart of the classical notion of bit is the concept of qubit. In this framework, the two classical bits still exist and are represented as two particular pure states (usually indicated by .|0 and .|1) that live in a special two-dimensional Hilbert space, based on the set of all ordered pairs of complex numbers.6 On this basis the concept of qubit is then defined as any pure state .|ψ (living in this space) that is a possible superposition of the two bits .|0 and .|1. Thus, the typical form of a qubit is the following: |ψ = c0 |0 + c1 |1.

.

From an intuitive point of view, the qubit .|ψ can be interpreted as a probabilistic information: the answer (to a given question) might be “No” with a probabilityvalue that depends on the number .c0 and might be “Yes” with a probability value that depends on the number .c1 .7 Not all states of quantum systems are pure. More generally, a piece of quantum information may correspond to a non-maximal knowledge: a mixed state (or

2 precisely, this probability-value is represented  by the real number .|ci | (the squared modulus of .ci ). Since the length of .|ψ is 1, we have: . i |ci |2 = 1. 6 In this space (usually indicated by the symbol .C2 ) the two classical bits .|0 and .|1 are identified with the two number-pairs .(1, 0) and .(0, 1), respectively. 7 More precisely, the probability of the answer “No” is the number .|c |2 , while the probability of 0 the answer “Yes” is the number .|c1 |2 . 5 More

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mixture), that is mathematically represented as a special Hilbert-space operator called density operator. Quantum mixed states give rise to a kind of second degree of ambiguity: while any pure state verifies with certainty some specific quantum properties, a mixed state may leave indeterminate all non-trivial quantum properties. At the same time, all pure states correspond to special examples of density operators.8 Complex pieces of quantum information (which may involve many qubits) are supposed to be stored by composite quantum systems (say, systems of many electrons). Thus, the quantum theoretic representation of composite systems comes into play, giving rise to one of the most mysterious features of the quantum world: entanglement, a phenomenon that had been considered “potentially paradoxical” by some of the founding fathers of quantum theory (for instance, by Einstein and by Schrödinger). Consider a quantum composite system S = S1 + S2

.

(say, a system consisting of two electrons). Any state of .S shall live in a particular Hilbert space .H that is a special product (called tensor product) of the two spaces .H1 and .H2 , associated to the subsystems .S1 and .S2 , respectively. It is customary to write: H = H1 ⊗ H2 ,

.

where .⊗ indicates the tensor product. Unlike the case of classical physics, quantum composite systems (say, a system .S = S1 + S2 ) have a peculiar holistic behavior: the state of the global system (.S) determines the states of its component parts (.S1 , .S2 ), and generally not vice versa. Thus, the procedure goes from the whole to the parts, and not the other way around. Entanglement-phenomena arise in the case of particular examples of composite systems that are characterized by the following properties: • the state of the composite system is a pure state .|ψ (a maximal information); • this state determines the states of the parts, which (owing to the peculiar mathematical form of .|ψ) cannot be pure. One is dealing with mixed states that might be indistinguishable from one another.

8 Any

density operator .ρ of a given Hilbert space can be represented (ina non-unique way) as a weighted sum of some projection-operators, having the form: .ρ = i wi P|ψi  , where the  weights .wi are positive real numbers such that . i wi = 1, while each .P|ψi  is the projection operator that projects over the closed subspace determined by the vector .|ψi . Thus, any pure state .|ψ corresponds to a special example of a density operator: the projection .P|ψ . The physical  interpretation of a mixed state .ρ = i wi P|ψi  is the following: a quantum system in state .ρ might be in the pure state .P|ψi  with probability-value .wi .

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Thus, the information about the whole turns out to be more precise than the information about the parts. Consequently, against the classical compositionalityprinciple, the information about the whole cannot be determined as a function of the pieces of information about the parts. Metaphorically, we might think of a strange puzzle that, once broken into its component pieces, cannot be reconstructed again, recreating its original image. Let us now briefly recall the basic ideas of the semantics that has been suggested by quantum information theory.9 In this semantics (which is often called quantum computational semantics) linguistic expressions (sentences, predicates, individual names,. . . .) are supposed to denote pieces of quantum information: possible pure or mixed states of quantum systems that are storing the information in question. At the same time, logical connectives are interpreted as quantum logical gates: special operators that transform the pieces of quantum information under consideration in a reversible way. Consequently, logical connectives acquire a dynamic character, representing possible computation-actions. Any semantic model of a quantum computational language assigns to any sentence a meaning that lives in a Hilbert space whose dimension depends on the linguistic complexity of the sentence in question. In this way, meanings turn out to preserve, at least to a certain extent, the “memory” of the logical complexity of the sentences under consideration. In accordance with the quantum-theoretic formalism, quantum computational models are holistic: generally, the meaning of a compound expression (say, a sentence) determines the contextual meanings of its well-formed parts. Thus, the procedure goes from the whole to the parts, and not the other way around, against the compositionality-principle, that had represented a basic assumption of classical semantics (strongly defended by Frege). In some interesting situations it may happen that the meaning of a sentence (say, Bob loves Alice) is an entangled pure state, while the contextual meanings of the component expressions (the names Bob, Alice and the predicate loves) are proper mixtures. In such a case the parts of our sentence turn out to be more vague and ambiguous than the sentence itself. One is dealing with a semantic situation that often occurs in the case of natural languages as well in the languages of art. This is one of the reasons why quantum computational semantics gives rise to some natural and interesting applications to fields that are far apart from microphysics.

14.3 A Quantum Approach to Pattern Recognition and to Machine Learning How are abstract concepts formed and recognized on the basis of some previous experience? This question can be successfully investigated in the framework of a

9 Technical

details can be found in Dalla Chiara et al. (2018).

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quantum inspired approach to pattern recognition and to machine learning, which has been intensively developed in recent years.10 Consider an agent (let us call her Alice) who is interested in a given concept .C that may refer either to concrete or to abstract objects. The name Alice may denote either a human or an artificial intelligence. We will use .AliceH for a human mind and .AliceM for an intelligent machine. Alice will then correspond either to .AliceH or to .AliceM . We suppose that Alice (on the basis of her previous experience) has already recognized and classified a given set of objects for which the question “does the object under consideration verify the concept .C?” can be reasonably asked. And we assume that the possible answers to this question are: • “YES!” • “NO!” • “PERHAPS!” As an example, Alice might be a child who has already recognized in the environment where she is living: • the objects that are tables; • the objects that are not tables. At the same time, she might have been doubtful about the right classification of some particular objects. For instance, she might have answered: “PERHAPS!” to the question “is this food-trolley a table?”. While .AliceH may have seen the objects under consideration, seeing is of course more problematic for .AliceM . Thus, generally, one shall make recourse to a theoretic representation that faithfully describes the objects in question. As happens in physics, one can use some convenient mathematical objects that represent objectstates. In the classical approach to pattern recognition and to machine learning an object-state is usually represented as a vector − → x = (x1 , . . . , xd ),

.

that belongs to the real space .Rd (the set of all ordered sequences consisting of d → x is supposed to real numbers, where .d ≥ 1). Every component .xi of the vector .− correspond to a possible value of an observable quantity (briefly, observable) that is considered relevant for recognizing the concept .C; while d represents the number of the relevant observables that are taken into consideration. Each number .xi is usually → x . As an example, suppose called a feature of the object represented by the vector .− we are referring to a class of flowers and let .C correspond to a particular kind of flower (say, the rose). We can assume that each flower-instance is characterized by

10 See, for instance, Schuld and Petruccione (2018), Sergioli et al. (2017) and Sergioli et al. (2019).

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two features: the petal length and the petal width. In such a case, any object-state → x = (x1 , x2 ) that belongs to the space .R2 . will be a vector .− The basic idea of a quantum approach to pattern recognition can be sketched as follows: replacing classical object-states with pieces of quantum information, possible states of quantum systems that are storing the information in question. From a semantic point of view, these quantum object-states can be regarded as possible meanings of individual names of a convenient quantum computational language. In some pattern-recognition situations it may be useful to start with a classical → x . Then, the transition to a quantum information represented by an object-state .− pure state .|ψ can be realized by adopting an encoding procedure: − → → x ⇒ |ψ− x,

.

→ where .|ψ− x represents the quantum pure state into which the encoding procedure → x . An example of a “natural” encoding has transformed the classical object-state .− is the so called amplitude encoding, which is defined as follows.

Definition 14.1 (Amplitude Encoding) Consider an object-state − → x = (x1 , . . . , xd ) ∈ Rd .

.

→ x is the following unit vector that lives in the The quantum-amplitude encoding of .− (d+1) (real) Hilbert space .R : → |ψ− x =

.

(x1 , . . . , xd , 1) (x1 , . . . , xd , 1)

(where . (x1 , . . . , xd , 1) is the length of the vector .(x1 , . . . , xd , 1)). → Thus, .|ψ− x is a quantum pure state that preserves all features described by the → x. classical object-state .− Of course, one could also directly “reason” in a quantum-theoretic framework, → x . In such a avoiding any reference to a (previously known) classical object-state .− case, one will assume, right from the outset, that an object-state is represented by a quantum pure state .|ψ living in a given Hilbert space. We can now discuss in a quantum framework the problem: how is a concept .C recognized on the basis of a previous experience? Suppose that (at a given time .t0 ) Alice is interested in the concept .C. Her previous experience concerning .C can be described by the formal notion of quantum .C- data set according to the following definition.

Definition 14.2 (Quantum .C-Data Set) A quantum .C-data set is a sequence C DS = (C H, C St, C St + , C St − , C St ? ),

.

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where: 1. .C H is a finite-dimensional Hilbert space associated to .C. 2. .C St is a finite set of pure states .|ψ of .C H for which the question “does the object described by .|ψ verify the concept .C?” can be reasonably asked. 3. .C St + is a subset of .C St, consisting of all states that have been positively classified with respect to the concept .C. The elements of this set are called the positive instances of the concept .C. 4. .C St − is a subset of .C St, consisting of all states that have been negatively classified with respect to the concept .C. The elements of this set are called the negative instances of the concept .C. C 5. . St ? is a (possibly empty) subset of .C St, consisting of all states that have been considered problematic with respect to .C. The elements of this set are called the indeterminate instances of the concept .C. 6. The three sets .C St + , .C St − , .C St ? are pairwise disjoint. Furthermore, .C St + ∪ C St − ∪ C St ? = C St. We will indicate by .n+ , n− , n? the cardinal numbers of the sets .C St + , .C St − , C ? . St , respectively. Suppose now that at a later time (.t1 ) Alice “meets” a new object described by the object-state .|ϕ. Alice shall find a rule that allows her to answer the question “does the object described by.|ϕ verify the concept .C?” And this answer shall be based on her previous knowledge that is represented by the quantum .C-data set C DS = (C H, C St, C St + , C St − , C St ? ).

.

A winning strategy is based on the use of two special concepts: the quantum positive centroid and the quantum negative centroid of a quantum .C-data set. Definition 14.3 (Positive and Negative Centroids) Consider a quantum .C-data set C C C St, C St + , C St − , C St ? ). . DS = ( H, 1. The quantum positive centroid of .C DS is the following density operator of the space .C H: ρ+ =

.

  1 C St + . P : |ψ  ∈ |ψi  i n+ i

2. The quantum negative centroid of .C DB is the following density operator of the space .C H:   1 − C . P|ψi  : |ψi  ∈ St .ρ = n− −

i

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The concept of quantum positive centroid seems to represent a “good” mathematical simulation for the intuitive idea of Gestalt. Both the quantum positive centroid and the intuitive idea of Gestalt describe an imaginary object, representing a vague, ambiguous idea that Alice has obtained as an abstraction from the “real” examples she had met in her previous experience. As happens in the case of the intuitive idea of Gestalt, the quantum positive centroid, represented by the   1 C St + , ambiguously alludes density operator .ρ + = P : |ψ  ∈ i i n+ |ψi  to the concrete positive instances that Alice had previously met (which are mathematically represented by the pure states .|ψi  ∈ C St + ).11 It is worth-while noticing that the characteristic ambiguity of quantum positive centroids is not shared by the notion of positive centroid that is defined in many classical approaches to pattern recognition. In the classical case, a positive centroid represents an exact object-state, that is obtained by calculating the average values of the values that all positive instances assign to the observables under consideration. Thus, unlike the quantum case, classical positive centroids turn out to describe imaginary objects that are characterized by precise features, without any “cloud” of ambiguity. As we have noticed, human recognitions and classifications are usually performed by means of a quick and mostly unconscious comparison between the main features of some new objects we have met and a gestaltic pattern that we had previously constructed in our mind. We also know that any comparison generally involves the use of some similarity-relations that are mostly grasped in a vague and intuitive way by human intelligences. Similarity-relations play a relevant role in the quantum theoretic formalism. Important examples of quantum similarities can be defined in terms of a special function, called fidelity. In the case of pure states this function is defined as follows. Definition 14.4 (Fidelity) Consider a Hilbert space .H. The fidelity-function on .H is the function F that assigns to any pair .|ψ and .|ϕ of pure states of .H the real number F (|ψ, |ϕ) = |ψ | ϕ|2

.

(where .ψ | ϕ is the inner product of .|ψ and .|ϕ). From an intuitive point if view, the number .F (|ψ, |ϕ) can be interpreted as a measure of the degree of closeness between the two states .|ψ and .|ϕ.

11 We

recall that .P|ψi  indicates the projection operator that projects over the closed subspace determined by the vector .|ψi : a special example of a density operator that corresponds to the pure state represented by the vector .|ψi . According to the canonical physical interpretation of mixtures, + .ρ represents a state that ambiguously describes a quantum system that might be in the pure state 1 .|ψi  with probability-value . + . n

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The definition of fidelity can be easily generalized to the case of density operators, which may represent either pure or mixed states. Thus, in the general case we will write: .F (ρ, σ ). It is interesting to recall the main properties of this function, which play an important role in many applications: 1. 2. 3. 4.

F (ρ, σ ) ∈ [0, 1]. F (ρ, σ ) = F (σ, ρ). .F (ρ, σ ) = 0 iff .ρσ is the null operator. .F (ρ, σ ) = 1 iff .ρ = σ . . .

From a physical point of view, the fidelity-function can be regarded as a form of symmetric conditional probability: .F (ρ, σ ) represents the probability that a quantum system in state .ρ can be transformed into a system in state .σ , and vice versa. The concept of fidelity allows us to define in any Hilbert space .H a special class of similarity-relations, called r-similarities, where r is any real number in the interval .[0, 1]. Definition 14.5 (r-similarity) Let .ρ and .σ be two density operators of a Hilbert space .H and let .r ∈ [0, 1]. The state .ρ is called r- similar to the state .σ (briefly, .ρ ⊥r σ ) iff .r ≤ F (ρ, σ ). One can easily check that (owing to the main properties of the fidelity function) this relation is reflexive, symmetric and generally non-transitive. Now .AliceM has at her disposal the mathematical tools that allow her to face the classification-problem. Suppose that .AliceM ’s information about a concept .C is the quantum .C-data set C DS = (C H, C St, C St + , C St − , C St ? ),

.

whose positive and negative centroids are the states .ρ + and .ρ − , respectively. And let .r ∗ be a threshold-value in the interval .( 12 , 1], that is considered relevant for .C DS. The main goal is defining a classifier function, that assigns to every state .σ (which describes an object that .AliceM may meet) • either the value .+ (corresponding to the answer “YES!”); • or the value .− (corresponding to the answer “NO!”); • or the value .? (corresponding to the answer “PERHAPS!)”. Definition 14.6 (Classifier Function) Let .C DS = (C H,C St,C St + ,C St − ,C St ? ) be a quantum .C-data set and let .r ∗ be a threshold-value for .C DS. The classifier function determined by .C DS and .r ∗ is the function .Cl C that satisfies the [ DS,r ∗ ] following condition for any state .σ of the space .H:

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⎧ ⎪ +, if σ ⊥r ∗ ρ + and not σ ⊥r ∗ ρ − . ⎪ ⎪ ⎪ ⎪ ⎪ In other words, σ is “sufficiently similar” to the positive ⎪ ⎪ ⎪ ⎪ ⎪ centroid and is not “sufficiently similar” to the ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨negative centroid.

Cl C (σ ) = −, if σ ⊥r ∗ ρ − and not σ ⊥r ∗ ρ + . [ DS,r ∗ ] ⎪ ⎪ ⎪ ⎪ In other words, σ is “sufficiently similar” to the negative ⎪ ⎪ ⎪ ⎪ ⎪ ⎪centroid and is not “sufficiently similar” to the ⎪ ⎪ ⎪ ⎪positive centroid. ⎪ ⎪ ⎪ ⎩ ?, otherwise.

.

This definition of the classifier function turns out be quite suitable for .AliceM , who can now perform a simple computation in order to decide whether a new object submitted to her verifies a given concept. Apparently, .AliceM ’s algorithmic procedure corresponds to what is intuitively grasped by .AliceH , when she quickly compares her description of a new object with a gestaltic pattern stored in her memory. Even if recognition-procedures are different for human and for artificial intelligences, there is a common method of “facing the problems” that seems to work in both cases.

14.4 Musical Themes and Musical Similarities Although music and quantum theory belong to two far apart worlds, the semantics suggested by quantum information theory can be successfully applied to a formal analysis of music.12 We will see how this special form of quantum musical semantics represents a useful tool for investigating the intriguing question of musical recognitions. Any musical composition (a sonata, a symphony, a lyric opera,. . . ) is generally determined by three basic elements: 1. a score; 2. a set of performances; 3. a set of musical ideas (or musical thoughts), which represent possible meanings for systems of musical phrases written in the score. Scores represent the syntactical component of musical compositions: systems of signs that are, in a sense, similar to the formal systems of scientific theories. Performances are, instead, physical events, that occur in space and time. As is well known, not all pieces of music are associated with a score. We need only think of

12 See

Dalla Chiara et al. (2012).

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folk songs or of jazz music. However, in classical Western music compositions are usually equipped with a score that has been written by a composer. Musical ideas represent a more mysterious element. One could ask: is it reasonable to assume the existence of such ideal objects that are similar to the intensional meanings investigated by logic? We give a positive answer to this question. In fact, a musical composition cannot be simply reduced to a score and to a system of sound-events. Between a score and the sound-events created by a performance there is something intermediate: the world of the musical thoughts that underlie the different performances. This is the ideal world where normally live composers and conductors, who are often accustomed to study scores without any help of a material instrument. The basic principle of quantum musical semantics is that musical ideas can be formally represented as special cases of pieces of quantum information, which may have the characteristic form of quantum superpositions. Accordingly, we can conventionally write: |μ =



.

ci |μi ,

i

where: • .|μ is an abstract object representing a musical idea that alludes to other ideas .|μi  that are all co-existent: • the number .ci measures the “importance” of the component .|μi  in the context .|μ. The use of the superposition-formalism is a powerful abstract tool that allows us to represent, in a natural way, the allusions and the ambiguities that play an essential role in music. And in some special cases musical ideas can be even represented as peculiar mixtures, that are characterized by a deeper degree of ambiguity. In accordance with the quantum-theoretic formalism we will use the symbols .|μ, .|μ1 , .|μ2 , . . . . for pure musical ideas that behave as quantum pure states. At the same time, generic musical ideas that may behave either as pure or as mixed states will be indicated by the symbols .μ, .μ1 , .μ2 , . . . . As is well known, an important feature of music is the capacity of evoking some extra-musical meanings: subjective feelings, situations that are vaguely imagined by the composer or by the interpreter or by the listener, real or virtual theatrical scenes (which play an important role in the case of lyric operas and of Lieder). The interplay between musical ideas and extra-musical meanings can be naturally represented in the framework of our quantum musical semantics: extra-musical meanings can be dealt with as examples of vague possible worlds, where events are generally ambiguous, as happens in the quantum world. Musical scores are characteristic two-dimensional syntactical objects that can be formally represented as special kinds of matrices (with rows and columns). Each column contains symbols for notes or pauses that shall be performed at the same time; while each row is a sequence of symbols corresponding to notes or pauses

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Fig. 14.1 The incipit of Beethoven’s Fifth Symphony

that shall be performed in succession. In this way, chords can be represented as fragments of columns, while melodies can be represented as fragments of rows. The two-dimensional configuration of scores clearly reflects, in the musical notation, the role played by parallelism in music. As an example we can refer to the celebrated incipit of Beethoven’s Fifth Symphony (Fig. 14.1). Any score is subdivided in complex systems of musical phrases. Generally, a phrase may be either a monodic phrase, represented by a one-dimensional horizontal fragment of the score; or a polyphonic phrase (with horizontal and vertical components), represented by a two-dimensional fragment of the score. Unlike the notion of sentence of a formal scientific language, the concept of musical phrase does not generally represent a rigid notion. The subdivision of a score in musical phrases may also depend on the interpreter’s choices. And it is not by chance that one often speaks of the “phrasing” that characterizes the interpretation of a given performer. Interpreting a given score (say, Beethoven’s Fifth Symphony) means assigning to every system of musical phrases written in the score a convenient musical idea that evolves in time. And, as happens in the quantum computational semantics, musical meanings have a characteristic holistic behavior: generally, the meaning of a global phrase-system determines the contextual meanings of all its parts (and not the other way around).

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Let us consider again the incipit of Beethoven’s Fifth Symphony. And let us briefly indicate this first phrase of the symphony by .PhrFifth1 . Every interpretation of .PhrFifth1 realizes a particular musical idea. As is well known, different conductors have proposed different interpretations of this famous musical phrase; and every interpretation may depend on a particular choice of the dynamics or of the tempo. In the framework of our quantum musical semantics a musical idea that represents a possible interpretation of the phrase .PhrFifth1 can be conventionally indicated as follows: |μF if th1 .

.

One could ask: what kind of abstract object is a musical meaning? Does the ket-notation, used in our quantum musical semantics, simply play a “metaphorical” role”? In fact, we could be technically more precise, representing a musical meaning (say, .|μF if th1 ) as a piece of quantum information that, in principle, could be stored by a quantum composite system .S. In this representation, each column of the score should be associated to a particular subsystem of .S. However, as we can imagine, the details of such a technical representation would not be interesting for the aims of a musical analysis. What is important is regarding musical meanings as special examples of intensional meanings that behave according to the general rules of the quantum computational semantics. A critical question concerns the relationship between musical ideas and musical themes. But what exactly are musical themes? The term “theme” has been used for the first time in a musical sense by Gioseffo Zarlino, in his Le istitutioni harmoniche (1558), as a melody that is repeated and varied in the course of a musical work. In the framework of our semantics the concept of musical theme cannot be simply identified with a musical idea that represents a possible interpretation of a particular musical phrase (written in a given score). We have seen how any interpretation of the Fifth Symphony associates to the first phrase of the symphony a musical idea: |μF if th1 .

.

At the same time, what is usually called the main theme of the Fifth Symphony’s first movement is something more abstract that neglects a number of musical parameters, which may concern, for instance, the pitch or the timbre. Musically cultivated people generally recognize this famous Beethoven’s theme when it is played by different instruments, in different (low or high) registers and in different (minor) tonalities. Which are the characteristic features of this theme that represent some invariant parameters, that cannot be neglected? First of all, a particular sequence of melodic intervals and pauses. Then, a particular meter (. 24 ) and a peculiar rhythmic structure, which is independent of the notes that shall be played. Thus, generally, a theme can be regarded as a highly abstract musical idea that is determined by a sequence of melodic intervals and pauses, embedded in a given rhythmic structure. This suggests to consider an abstraction from a given phrase written in a score. We can introduce

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Fig. 14.2 A notation for the abstract main theme of the Fifth Symphony’s first movement

the concept of abstract musical theme, which represents an invariant with respect to possible timbre and pitch-transformations. As an example, let us refer again to the first phrase of the Fifth Symphony. We will briefly represent the abstract theme associated to this phrase by the notation described in Fig. 14.2. The symbolic convention assumed in this notation is the following: • the squared brackets mean that we are abstracting from the “real notes” written in the score-fragment inside the brackets. What we are referring to is a particular sequence of melodic intervals and pauses, embedded in a rhythmic structure (which is determined by the score-fragment under consideration).13 • The ket-brackets mean that we are representing our abstract theme as a particular example of a pure musical idea .|μ dealt with as a meaning in the framework of the quantum musical semantics.14 We can assume that just this abstract theme, briefly indicated by .|[μF if th1 ], can formally represent what is usually called the main theme of the Fifth Symphony’s first movement. Of course, abstract themes represent special examples of musical ideas. On this basis we can say that: the musical idea |μF if th1 

.

that represents a possible interpretation of the phrase .PhrFifth1 expresses the abstract theme |[μF if th1 ].

.

As expected, one could also directly associate abstract themes to phrases, asserting that a given phrase expresses a corresponding abstract theme. 13 The use of the squared brackets is suggested by a notation often used in mathematics, where operations involving an abstraction are frequently indicated by the brackets .[. . . . . .]. 14 Of course, we might use a “more mathematical” notation, indicating all melodic intervals by convenient arithmetical expressions. However, this kind of notation (which plays an important role in the framework of computer music) would be too heavy and hardly interesting for the aims of our semantic approach.

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So far, we have considered examples of monodic abstract themes, which correspond to fragments of rows in the formal representation of a given score. However, in some cases it may be interesting to consider also examples of polyphonic abstract themes, that correspond to polyphonic phrases of the score in question. For instance, we might consider the complex abstract theme that corresponds to the global first phrase of the Fifth Symphony (represented in Fig. 14.1). In such a case, the harmonic relationships between the different parts of our theme would come into play. Notice that what is usually called “The Theme” of a composition having the form Theme and Variations is generally represented by a polyphonic musical idea that, in turn, can include a particular monodic abstract theme. And just this monodic theme represents the “main character” which all Variations allude to, according to modalities that may appear more or less hidden. Let us now turn to the intriguing question that concerns musical recognitions. What does recognizing a melody or a musical theme mean? Recognition processes in music seem to be quite similar to what happens when we recognize an abstract concept, which may refer either to concrete or to ideal objects. In the case of music, the role played by abstract concepts can be naturally replaced by musical themes. We suppose that Alice is interested in a particular musical theme (say, the theme expressed by the incipit of Beethoven’s Fifth Symphony). The musical idea that Alice’s mind associates to this theme may be quite vague (depending, of course, on Alice’s musical knowledge). What is important is that Alice can use a label (a name) that, in principle, can refer to a particular abstract musical theme; in this case, the theme that we have previously indicated by the notation .|[μF if th1 ]. Accordingly, in our musical applications of pattern recognition-methods we will write .T (theme), instead of .C (concept). As happens in the case of concepts, we suppose that, at a given time .t0 , Alice has already classified a (finite) set of pure musical ideas MI d = {|μ1 , . . . , |μn }

.

with respect to the theme .T. In other words, for every musical idea .|μi  in the set MI d, Alice has answered either “YES!” or “NO!” or “PERHAPS!” to the question “does the musical idea .|μi  express the theme .T?” As expected, the elements of the set .MI d (which are formally dealt with as pieces of quantum information) represent musical thoughts (stored in Alice’s memory) corresponding to pieces of music that Alice might have either listened to or performed as a musician. On this basis, we can naturally introduce the notion of quantum musical .T-data set, identified with a system T MDS = (MI d, MI d + , MI d − , MI d ? ),

.

where: 1. MI d is a finite set of pure musical ideas .|μi  for which the question “does the musical idea .|μi  express the theme .T?” can be reasonably asked.

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2. .MI d + , .MI d − , .MI d ? represent, respectively, the sets of the positive, of the negative and of the indeterminate instances for the theme .T. We assume that any quantum musical .T-data set satisfies the same conditions that we have required in the case of quantum .C-data sets (Definition 14.2). The concepts of positive centroid and of negative centroid of a quantum musical .T-data set can be now naturally defined as we have done in the case of quantum .Cdata sets. The positive centroid is determined as a musical idea .κ + that is a mixture + + T of all positive instances .|μ+ 1 , |μ2 , . . . , |μn+  of . MDS. Each weight occurring in .κ + is identified with the number . n1+ (where .n+ is the number of all positive instances). Thus, from an intuitive point of view, .κ + represents an ambiguous musical idea that vaguely alludes to the pure musical ideas + + |μ+ 1 , |μ2 , . . . , |μn+ .

.

In a symmetric way, one can define the negative centroid of .T MDS as the musical idea .κ − that is a mixture of all negative instances of .T MDS. Suppose now that at a later time .t1 Alices listens to a new musical phrase, say the incipit of Beethoven’s Fifth Symphony, which might be performed by a regular orchestra or by a piano or even simply sung by someone. And suppose that Alice asks herself “is this piece of music the main theme of the Fifth Symphony’s first movement?” Of course, Alice’s answer should be based on her previous knowledge, which is formally represented by the quantum musical .T-data set .T MDS. In such situation, our natural wish would be trying and applying the same classifier function that we have successfully used for concept-recognitions. But is this procedure possible in the case of music? We have seen how, in the case of concepts, our definition of the classifier function has been essentially based on the notion of r-similarity, which admits a precise mathematical definition in the framework of quantum information theory. To what extent can this strategy be reasonably extended to musical recognitions? It is well known that similarity-relations play a very important role in the structure of music. Musical themes are normally transformed in different ways in the framework of a given composition. And all variation-phenomena (which some authors have described as the Urprinzip of music) are characterized by the occurrence of some similarity-relations. We may only think of the structure of Fugues, of the Sonata form and of the Theme and Variations-form, where abstract themes often appear as a kind of “ghosts” in a somewhat mysterious way. It is customary to distinguish different kinds of musical similarities: melodic, rhythmic, harmonic, timbric,. . . . And different musicians have sometimes attributed a privileged role to one or to another form of similarity. For instance, according to Arnold Schönberg rhythm represents the most important feature that allows us

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Fig. 14.3 Beethoven, Sonata op.10, n.1. The incipit of the primary theme

Fig. 14.4 A major transformation of the primary theme

to recognize musical motifs.15 Other authors, like Heinrich Schenker, have instead focused on the role played by specific intervallic relationships.16 In the case of tonal music some important similarity-relations are often connected with a modetransformation: from a major tonality to a minor tonality or vice versa. As an example, let us refer to the first movement of Beethoven’s piano sonata op.10 n.1, in C minor (the same tonality of the Fifth Symphony). The primary theme of the movement is proposed at the very beginning (Fig. 14.3). One is dealing with an ascending phrase, based on the three elements of the triad of the C minor key (C, E flat, G). The dynamic indication (forte) as well as the peculiar rhythmic structure (a dotted rhythm) seem to suggest a strong statement (a kind of “act of will”). Soon after, this theme is suddenly transformed into a major mode (the C major key) (Fig. 14.4), restating (in a major version) the same strong idea that had been asserted before. 15 See

Schönberg (1995). for instance, Schenker (1935).

16 See,

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Fig. 14.5 Beethoven, the Fifth Symphony’s first movement. A “virtual” major transformation of the main theme

Fig. 14.6 Beethoven, the Fifth Symphony’s first movement. A major variant of the main theme

Unlike the case of the Sonata Op.10 n.1, the main theme of the Fifth Symphony’s first movement is never directly transformed into a major version. The phrase represented in Fig. 14.5 does not belong to the symphony’s score. There is, however, a new theme that appears very soon (at bars 59–62): a phrase played fortissimo by the horns (in the B flat major tonality) that is naturally perceived as very close to the idea expressed by the main theme (Fig. 14.6). In the framework of our quantum musical semantics it is interesting to study how Alice lets different forms of musical similarities interact. For the sake of simplicity, we will now restrict our attention to two particular examples of similarity (which play an essential role in the structure of music): melodic and rhythmic similarities. Let .μ1 and .μ2 represent two musical ideas. When .μ1 is considered melodically similar to .μ2 , we will briefly write: μ1 SimM μ2 .

.

And we will write: μ1 SimR μ2 ,

.

when .μ1 is considered rhythmically similar to .μ2 . Melodic and rhythmic similarities can be logically combined in different ways. In some cases it may be interesting to consider the conjunction between a melodic similarity .SimM and a rhythmic similarity .SimR . This gives rise to a new relation, that can be called strong similarity. In some other cases it may be interesting to consider the disjunction of the two relations .SimM and .SimR . This gives rise to a different relation that represents a weaker form of similarity. As we have done in the case of concepts, we can distinguish different degrees of musical similarity. Although musicologists do not normally speak of r-similarity

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relations, musical analyses often use some locutions like “highly similar”, “somewhat similar”, “slightly similar”, that can be conventionally associated to some particular numerical values (in the interval .[0, 1]). Let Sim represent any form of musical similarity. When a musical idea .μ1 is considered r-similar to a musical idea .μ2 , we will briefly write: μ1 Simr μ2 .

.

And as happens in the case of concepts, there are musical situations, where it may be useful to choose a particular threshold-value .r ∗ (in the interval .( 12 , 1]), that is considered relevant for the musical context under consideration. Suppose, for instance, that: .μ1 Simr ∗ μ2 , while .r ∗ is “very close” to 1 (say, .r ∗ = 0.9). In such a case, it seems reasonable to conclude that: μ1 and μ2 are highly similar.

.

As an example, let us refer again to the first movement of Beethoven’s Sonata op.10 n.1. Suppose that .μ1 is a musical idea that corresponds to the movement’s primary theme (Fig. 14.3), while .μ2 corresponds to its major transformation (Fig. 14.4). Apparently, the primary theme and its major transformation have exactly the same rhythmic structure. Thus, it seems reasonable to conclude that: μ1 SimR 1 μ2 .

.

In other words, our two musical ideas are maximally similar from the rhythmic point of view. The situation changes if we refer to melodic similarity. Clearly, the primary theme and its major transformation do not have the same melodic structure, since in the major variant the minor triad (C, E flat, G) has been replaced by the major triad (C, E, G). However, a musical .AliceH , who is familiar with tonal music, “perceives” the two musical ideas .μ1 and .μ2 as “very close” to each other. Hence, by a convenient choice of the threshold-value .r ∗ (for instance, by choosing .r ∗ = 0, 9), it seems reasonable to conclude that: μ1 SimM r ∗ μ2 .

.

In other words, .μ1 and .μ2 are melodically very similar. A quite different situation arises if we consider two musical ideas .μ1 and .μ2 that correspond to the main theme of the first movement of Beethoven’s Fifth Symphony (Fig. 14.2) and to its major variant (Fig. 14.6), respectively. In such a case, both the melodic structure and the rhythmic structure of .μ1 and .μ2 are different. In spite of this, we perceive a deep relationship that connects the musical thoughts expressed by .μ1 and .μ2 . The major variant seems to restate, in a more incisive and permanent way, the strong assertion that had been proposed by the main theme (in a minor tonality).

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Musical similarities are often grasped in an intuitive and rapid way by human listeners who are familiar with classical Western music. At the same time, trying to analyze, by abstract methods, the relationships between different forms and different degrees of similarity is not an easy task. We have seen that, in the case of concepts, r-similarity relations (defined in terms of the fidelity-function) allow us to define a classifier function that has an objective and universal behavior. This precise mathematical situation can be hardly reproduced in the case of music, where any particular choice of a similarity-relation seems to depend, at least to a certain extent, on subjective preferences. What we can do is referring to a class of possible musical similarity-relations, admitting that, in different contexts, we have the freedom of choosing some special elements in this class. Once chosen a particular similarity-relation Sim, associated to a threshold-value ∗ .r , it will be possible to apply the same method used for recognizing concepts. Suppose that Alice’s information about a given abstract theme .T is represented by the quantum musical .T-data set T MDS = (MI d, MI d + , MI d − , MI d ? ),

.

associated to a threshold value .r ∗ , and let .κ + and .κ − be, respectively, the positive centroid and the negative centroid. As happens in the case of concepts, a musical classification function MCl (based on .T MDS and on .r ∗ ) shall assign to any musical idea .ν (which may represent a new musical example that Alice has listened to) either the value .+ or the value .− or the value .?. On this basis, we can assume by definition that: 1. .MCl(ν) = +, if ν Simr ∗ κ + and not ν Simr ∗ κ − .

.

In other words, .ν is sufficiently similar to the positive centroid .κ + and is not sufficiently similar to the negative centroid .κ − . 2. .MCl(ν) = −, if ν Simr ∗ κ − and not ν Simr ∗ κ +

.

In other words, .ν is sufficiently similar to the negative centroid .κ − and is not sufficiently similar to the positive centroid .κ + . 3. .MCl(ν) = ?, otherwise. The application of pattern-recognition methods to musical problems has confirmed the interest of investigating by abstract tools the intriguing concept of musical similarity, which has been analyzed, with different perspectives and methods, by musicians, musicologists as well by researchers in the field of musical informatics. Finally, we would like to conclude with the following general remark: adopting a quantum approach to pattern recognition has allowed us to obtain some natural

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simulations for artificial intelligences of the intrinsic vagueness and ambiguity that characterize many human cognitive behaviors. And, interestingly enough, one has shown that a systematic use of quantum uncertainties gives rise to significant improvements of the accuracy and of the algorithmic efficiency of some machinelearning procedures.17 These results seem to confirm that a quantum inspired investigation of ambiguity-phenomena can represent a powerful resource both for theoretic and for technological achievements.

References Dalla Chiara, M. L., Giuntini, R., Luciani, R., & Negri, E. (2012). From quantum information to musical semantics. London: College Publications. Dalla Chiara, M. L., Giuntini, R., Leporini, R., & Sergioli, S. (2018). Quantum computation and logic. How quantum computers have inspired logical investigations. Berlin: Springer. Ehrenstein, W. H., Spillmann, L., & Sarris, W. (2003). Gestalt issues in modern neuroscience. Axiomathes, 13, 433–458. Honing, H. (2009). Musical cognition. A science of listening. London: Transaction Publishers. Schuld, M., & Petruccione, F. (2018). Supervised learning with quantum computers. Berlin: Springer. Sergioli, G., Bosyk, G. M., Santucci, E., & Giuntini, R. (2017). A quantum-inspired version of the classification problem. International Journal of Theoretical Physics, 56, 3880–3888. Sergioli, G., Russo, G., Santucci, E., Stefano, A., Torrisi, S. E., Palmucci, S., Vancheri, C., & Giuntini, R. (2018). Quantum-inspired minimum distance classification in a biomedical context. International Journal of Quantum Information, 16, 1–15. Sergioli, G., Giuntini, R., & Freytes, G. (2019). A new quantum approach to binary classification. PLoS One, 14(5), e0216224. Sergioli, G., Militello, C., Rundo, L., Minafra, L., Torrisi, F., Russo, G., Chow, K. L., & Giuntini, R. (2021). A quantum-inspired classifier for clonogenic assay evaluations. Scientific Reports, 11(1), 2830. Schenker, H. (1935). Der freie Satz. Universal Edition A.G. Wien. Schönberg, A. (1995). The musical idea and the logic, technique, and art of its presentation. New York: Columbia University Press.

17 See,

for instance, Sergioli et al. (2018) and Sergioli et al. (2021).

Chapter 15

From Here to Eternity Newton C. A. da Costa and Francisco Antonio Doria

Abstract We conjecture that the existence of some fast-growing functions implies in a simple way the existence of some inaccessible cardinals. This note expands some previous work by the second author.

15.1 Introduction This note reviews a possible relation between two weird animals that come up in foundational discussions: fast-growing intuitively total recursive functions and inaccessible cardinals. Such a relation may also bear on practical problems, such as problems whose formal treatment leads to major constructions in complexity theory. The first discussion by the present authors of the central idea in this paper, namely the relation between fast-growing total recursive functions and inaccessible cardinals appears in Doria (2017).The exposition in this paper sort of better organizes the field (at least we hope so). In a nutshell: G. Kreisel contacted us on computer complexity issues, and mentioned that—among several other poorly known and poorly explored facts—the counterexample function to .[P = N P ] grows horrendously fast (we found that it grows in its peaks at least as fast as the Busy Beaver function).1 The axiomatic background where we do our exploration is supposed to be ZFC plus a few simple extensions which are made explicit whenever needed; actually

1 For

notation and basic facts, see da Costa and Doria (2017).

N. C. A. da Costa Department of Philosophy, Federal University of Santa Catarina, Florianópolis, Brazil e-mail: [email protected] F. A. Doria () Institute for Advanced Studies, University of São Paulo, São Paulo, SP, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. R. B. Arenhart, R. W. Arroyo (eds.), Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics, Synthese Library 476, https://doi.org/10.1007/978-3-031-31840-5_15

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we require rather standard mathematics, as we need a landscape which is quite commonplace. And we find those mathematical beasties which are supposed to be conceptually rich enough to describe their main properties, yet even if we have a very rich conceptual background we still have undecidable properties that won’t go away.

15.1.1 Summary of the Main Ideas We start from a specific question, at first not clearly related to our main queries: Let .f be a fast-growing, intuitively total, recursive function. We wonder about the meaning of .f(ℵ0 ): is it an inaccessible cardinal? Does it belong to a specific class of large cardinals?

We think it may be related to some class of large cardinals, granted that .f cannot be proved or disproved total in our axiomatic framework.

15.1.2 A Wild Idea: Inaccessible Cardinals Out of Fast-Growing Recursive Functions ? The ideas which we present here developed out of exchanges with friends and colleagues in the early 1990s; we explicitly mention our good friend the late M. Guillaume, who acted as a strong sparring to guide us in an unfamiliar terrain (Guillaume, 2000), and G. Kreisel, who introduced us to the counterexample function to .[P = NP ], which—he told us without proof—is fast-growing in its peaks, and appeared to be related to an inaccessible cardinal. M. Guillaume was an old friend, who offered to criticize our preliminary work on the counterexample function to .[P = NP ], which we had from Kreisel (1992) that it was twice intractable: • Noncomputable; and • Fast-growing. Moreover, we soon found out that the counterexample function grows at least as fast as the Busy Beaver function. Our preliminary work was presented at the EBL 2000, Juquehy, S. Paulo, Brazil. We managed to prove Kreisel’s main contentions, the counterexample function was a strongly growing object, and it was at least as fast-growing as the Busy Beaver function.

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15.1.3 An Example A fast-growing total recursive function .f(n) is a recursive sequence of natural numbers that satisfies f(n) < f(n + 1)

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and which grows in some unbounded way. Consider a very simple example: Example 15.1.1 Let us be given the sequence 1, 2, 22 , 23 , . . . , 2n , . . .

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of exponentials. Now we can give a clearcut meaning to: 2ℵ0 ?

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As we well know, that weird exponential of a transfinite cardinal represents the cardinal of the continuum. Can we extend the construction behind .2ℵ0 so that we   give some analogous meaning to .f(ℵ0 ) ? We stress it: Which “natural” meaning, if any, can be given to .f(ℵ0 ), where .f is an increasing, fastgrowing intuitively total recursive function ?

In order to try to answer that question—which is the “natural” meaning, if any, of .f(ℵ0 ) ?—we will make a few conjectures, and argue out of them.

15.2 1 -Soundness and F The next results show that .F sort of codes, or represents, deep facts about the structure of the axiomatic system where it is defined. Namely, if function F is proved total within theory T , then the axiomatic system T w.r.t. which .F is defined, is consistent. We also show that F is total if and only if T is .1 sound. We start from two apparently unrelated objects, function .F and some “principles of reflection.” Let us give a more rigorous definition for .F: : • For each n, .F(n) = maxk≤n ({e}(k)) + 1, that is to say, it is the sup of those .{e}(k) so that: 1. .k ≤ n. 2. .PrS (∀x∃zT (e, x, z) ≤ n.

 

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For our discussion of the relation between reflection principles and fast-growing functions we use the argument in Carnielli and Doria (2008)—it is ours, anyway. A principle of reflection added to theory T sort of reinforces the validity of arguments and proofs in the formal theory T . And .F is a function which is intuitively total, but cannot be proved so in our theory.

15.3 Reflection Principles and Theory T Remark 15.3.1 The local reflection principle Rfn(T ) for theory T is: PrT (φ ) → φ,

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(that is, if there is a proof for φ, we actually find it in T ); and the (first) uniform reflection principle, RFN(T ) is: [∀x PrT (φ(x) )] ˙ → [∀x φ(x)],

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all φ with only x free, and x˙ standing for the x that can be represented by actual constants in T . This means that once one can list instances φ(0), φ(1), . . . (which are derivable due to the first supposition in the Reflection Principle) for all nameable n, then a restricted application of the ω-rule leads to the principle. For 1 -soundness one restricts φ to all ∃x φ(x), φ primitive recursive; the corresponding restricted reflection principle is noted RFN1 (T ).   For T as PA,ZFC, and first-order extensions as considered here: Corollary 15.3.2 T  RFN1 (T ) → Consis(T ). Proof • Suppose T  ¬(0 = 0). • Therefore, T  PrT [¬(0 = 0)]. • Given 1 -soundness, as this is a trivial 1 sentence, PrT [¬(0 = 0)] → [¬(0 = 0)].

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• By contraposition, {¬[¬(0 = 0)]} → ¬PrT [¬(0 = 0)].

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• Or, (0 = 0) → ¬PrT [¬(0 = 0)]. • However, T  (0 = 0). Then follows: • ¬PrT [¬(0 = 0)]. A contradiction. (This proof is due to N. C. A. da Costa.)

 

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15.3.1 1 -Soundness Is Equivalent to [F Is Total] The next theorem is a folklore theorem; it appears in the present form in Carnielli and Doria (2008). We argue here for ZFC, since we want to have as much “elbow room” as possible. However we could have argued for any theory like ZFC; the present version is due to F. A. Doria.2 If we allow for the abuse of language that subsumes the infinitely many sentences of the Reflection Principle in our formulation as a single one in the statement of our result: Lemma 15.3.3 ZFC.[F is total].↔ [ZFC is .1 -sound]. Proof Recall that [F is total].↔ ∀x ∃z T (eF , x, z), where—here—T is Kleene’s T predicate and .eF is a Gödel number for .F. .(⇐). We first prove: assuming .RF N 1 (ZFC), then .∀x ∃z T (eF , x, z). Given the (recursively enumerable) infinite set of conditions .RFN1 (ZF C), for T we get: [∀x PrZFC (∃z T (eF , x, ˙ z) )] → [∀x ∃z T (eF , x, z)].

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Now, for each ZFC constant .n˙ we have that: ZF C  [∃z T (eF , n, ˙ z)].

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Then there are proofs of each of these sentences, for each .n. ˙ Therefore we conclude: .[∀x PrZF C (∃z T (eF , x, ˙ z) )], as .x˙ only ranges over the constants. From the corresponding restriction of .RFN1 (ZF C) we conclude that: [∀x ∃z T (eF , x, z)]

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holds, by modus ponens. (⇒). For the converse: given .∀x ∃z T (eF , x, z), we deduce .RFN1 (ZF C). (We will have to deduce each instance of the Reflection Principle, for each 1-variable .∃z ψ(z, x), .ψ primitive recursive—for the meaning of .z see below.) Recall that given each e we can explicitly construct a Diophantine polynomial .

pe (x, y, z1 , z2 , . . . , zm )

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2 This is a much quoted result which has never appeared in print despite its obvious importance. There are proofs in particular cases. We decided to offer this pedestrian but general proof which has been checked by M. Guillaume. The proof we exhibit is quite general and intuitive; it has originally appeared in Carnielli and Doria (2008). To avoid misunderstandings we give it verbatim—we are the authors, anyway.

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so that: [∀x∃z T (e, x, z)] ↔ [∀x∃y, z1 , . . . , zm pe (x, yz1 , . . . , zm ) = 0].

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(Of course .[∃z1 , . . . zm pe (x, yz1 , . . . , zm ) = 0] ↔ [y = {e}(x)]. Since we aren’t using an universal equation, m may depend on e.) We will abbreviate the .z1 , . . . , zm by .z. 1. Now, if [.F is total], then, for each .n ∈ ω we have that .F(n) is the sup of all .{e}(k) so that .k ≤ n and: PrZFC (∀x ∃z T (e, x, z) ) ≤ n.

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2. Since n is explicitly given, it is a bound on the Gödel number of the proof. Therefore we can also obtain a .n > n so that: PrZFC (∃z T (e, x, z) ) ≤ n .

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3. Or, for another .n , .PrZFC (∃y, z pe (x, y, z) = 0 ) ≤ n . This follows from: • Every recursive function .{e}(n) = m can be represented by a predicate .Fe (n, m). (The algorithm to produce .Fe given e is in Machtey and Young (1979, p. 126 ff).) • Given .Fe (n, m) we can use the procedure described in Davis’ paper on Hilbert’s 10th Problem (Davis, 1973) to get a polynomial .pe out of .Fe . 4. Since .n is explicitly given, we can then recover proofs in ZFC of: [∃y, z pe (x, y, z) = 0],

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all e under the specified conditions. 5. We then establish that ZFC proves, for all such e, PrZFC (∃y, z pe (x, y, z) = 0 ) → [∃y, z pe (x, yz) = 0].

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6. We now add the universal quantifier for x, and as it distributes over .→, [∀x PrZFC (∃y, z pe (x, y, z) = 0 )] → [∀x ∃y, z pe (x, yz) = 0].

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7. This will of course also hold for (due to the implication’s properties): [∀x PrZFC (∃y, z pe (x, ˙ y, z) = 0 )] → [∀x ∃y, z pe (x, yz) = 0].

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8. Finally the .∃wpe provide an enumeration of all .1 relations in ZFC of interest for .1 -soundness. Notice that in ZFC, .f and .F recursive, [f is total] ↔ {[F is total] → [F dominates f ]}.

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9. To show that the enumeration is exhaustive: suppose that for some p.r. .ψ one has: ∀x PrZFC (∃y ψ(x, ˙ y)),

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and that moreover the following sentence is proved: [∀x PrZFC (∃y ψ(x, ˙ y))] → ∀x ∃y ψ(x, y).

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10. Put .fψ (x) = miny ψ(x, y). Then: [∀x ∃y (fψ (x) = y)] ↔ [∀x ∃y ψ(x, y)].

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11. That is, .fψ is ZFC-provably total recursive, and therefore falls into the preceding case.   Remark 15.3.4 The preceding result shows the essential inner connections between the inability of an axiomatic theory like S to “see” the totality of functions like .F and beyond, and the usual formal sentences that assert the consistency of S itself. That is to say, the obstructions we have to face when trying to prove the totality of .F in S have to do with the impossibility of proving the consistency of S itself with its own tools. The proof of the preceding result for PA can be found in the original Paris– Harrington paper; ours is more general. The construction we present in this paper allows us to relate inaccessible cardinals to reflection principles: reflection principles .→ function .F .→ inaccessible cardinal.  

15.4 Inaccessible λ; Plus f and F We require a lot of hand waving in the discussion that follows. Recall that a strongly inaccessible cardinal is defined as follows: Definition 15.4.1 .λ is strongly inaccessible if it satisfies: 1. .λ > ω. 2. If .α is a cardinal and .α < λ, then .2α < λ. 3. For every family of cardinals .βi , i ∈ ι, ι < λ, and for each .i, βi < λ, then .supi (βi ) < λ.  

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Let .Consis(T ) be the (usual) sentence that asserts that theory T is consistent. Let .Card(λ) mean that .λ is a cardinal, and let .SInacT (λ) mean that .λ is strongly inaccessible for theory T . .f is as above, and let .F be the fast-growing, partial recursive function that appears in what we’ve called the exotic formulation (see da Costa & Doria, 2022, p. 166ff). Always waving hands we wonder if we have something like λF = F(ℵ0 ).

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With extra detail, [T proves .FT is total] if and only if .λFT = F(ℵ0 ) is inaccessible in T . Thus here .λF depends on .F. This series of equivalences is completed by our main folklore theorem: Proposition 15.4.2 T.[T is .1 -sound] .↔ [F is total].

 

Here .F is .FT , of course. Given those hypotheses and conjectures: Conjecture 15.4.3 There are .λF so that: (ZFC + [Fistotal ])  Card(λ) ∧ SInacZFC λF ).

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Remark 15.4.4 We suppose that the consistency of ZFC holds, and that we have the axiom of choice also holds where required.   Tentative sketch of proof, to be further developed : 1. As we suppose that .Consis(ZFC + [F is total]) holds, then it has a model .M. 2. Now, .ZFC + [F is total ]  Consis (ZFC). 3. It is a theorem of ZFC that: Consis(ZFC) ↔ ∃x[x | ZFC].

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(We can also begin our argument here and take this as a definition for Consis(T ).) 4. Given that: .

ZFC + [F is total ]  Consis (ZFC),

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there is a set .x ∈ M that is a model for .ZFC. 5. Write .Vλ = M − x. 6. Since .Vλ is nonempty and as the axiom of choice holds, there are ordinals in it. 7. Therefore there is at least a cardinal in .Vλ .

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8. Pick up the smallest of such cardinals; note it .λ: (a) One easily sees that for each cardinal .α ∈ V , .λ is different from .2α . (b) Also for each sequence .βi etc., .λ is different from .supi βi . (Both conditions hold because if not, .λ would be in V .) 9. Finally for all cardinals .α ∈ V , .λ > α. For if not, there would be a .β ∈ V , and .λ < β, and .λ would be in V . This also means that .V is in fact a set, V .  

15.5 Tentative Conclusion This argument only shows (or purports to show) that our extended theory .ZFC + [F is total] implies the existence of an inaccessible cardinal. One must now show that .F(ℵ0 ) can be interpreted as an inaccessible cardinal w.r.t. ZFC. We would then have .λ ≤ F(ℵ0 ), all .λ as above.

15.5.1 Extra Detail The finite .2n , and .2ℵ0 , can be pictured as binary trees. We can extend that picture to .f and .F. Now why is the totality of .F an undecidable property within our axiomatic framework? .F is total in the standard model for arithmetic, and in the corresponding segment of the model for the theory where .F is interpreted. .F is partial in models with nonstandard arithmetic. The fact that from within our axiomatic background we cannot decide most properties of .F can be read as inaccessibility. Briefly, that’s what is going on here.   With some extra detail: • We start from the construction of Kleene’s function .F.3 • We reach the first surprise: theory T proves the theorem [.F is total] if and only if [T is .1 -sound]. It is a surprise: .1 soundness is a kind of syntactic property that affects lots of terms in our theory T , while .F looks like a very restricted function, so it isn’t obvious why a general syntactic property should mirror a very specific property in an algorithmic function (namely F). We rest our case. • The next step—the fact that our function F appears related to an inaccessible cardinal—is only known in particular cases, to our knowledge. We feel that our argument isn’t enough, but the overall picture seems to be reasonably plausible.

3 Kleene

made .F into the centralpiece of his incompleteness phenomenon, see S. C. Kleene, “General recursive functions of the natural numbers” (Kleene, 1936).

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15.5.2 Caveat The proofs given in this paper are the sole responsibility of the authors; we give them in full detail and mention where they have appeared in print. However the backbone and general perspective which have guided us has never been published. Now: we would like to obtain a similar result out of the Halting Function .θ (da Costa & Doria, 2022): we conjecture that .λf (θ ) is inaccessible, for an adequate relation expressed by .λf . Is there any meaning in that? Acknowledgments It is a pleasure to dedicate these sketchy doodles to our good friend and colleague Décio Krause. We hope it contains the hidden pearl we think we have perceived in those ideas. The construction we present here originated in an e-mail exchange between the authors and G. Kreisel on the fast-growth properties of the so-called counterexample function to .P = N P , however the quasi-trivial machines are the authors’ own responsibility. We also discussed it with M. Guillaume who helped us with his usual acumen. And of course the title refers to the 1953 movie featuring Burt Lancaster, Montgomery Clift and Deborah Kerr—and to Ian Stewart’s book From Here to Infinity (Stewart, 2009).

References Carnielli, W. A., & Doria, F. A. (2008). Are the foundations of computer science logic-dependent? In C. Degrémont, et al. (Eds.), Essays in honor of S. Rahman. London: College Publications. da Costa, N. C. A., & Doria, F. A. (2017). Variations on a complex theme. Preprint. da Costa, N. C. A., & Doria, F. A. (2022). On Hilbert’s sixth problem. Berlin: Springer. Davis, M. (1973). Hilbert’s 10th problem is unsolvable. American Mathematical Monthly, 80, 233– 243. Doria, F. A. (Ed.). (2017). The limits of mathematical modeling in the social sciences. Singapore: World Scientific. Guillaume, M. Private correspondence with the authors, c. 2000–2003. Kleene, S. C. (1936). General recursive functions of the natural numbers. Mathematische Annalen, 112, 721–731. Kreisel, G. Exchange with da Costa and Doria (c. 1992–2000). Machtey, M., & Young, P. (1979). Introduction to the general theory of algorithms. New York: North-Holland. Stewart, I. (2009). From here to infinity. Oxford: Oxford University Press.