On Gödel and the Nonexistence of Time - Gödel und die Nichtexistenz der Zeit: Kurt Gödel essay competition 2021 - Kurt-Gödel-Preis 2021 3662670445, 9783662670446, 9783662670453

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Oliver Passon · Christoph Benzmüller · Brigitte Falkenburg Hrsg.

On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit Kurt Gödel essay competition 2021 – Kurt-Gödel-Preis 2021

On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit

Oliver Passon · Christoph Benzmüller · Brigitte Falkenburg (Hrsg.)

On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit Kurt Gödel essay competition 2021 – Kurt-Gödel-Preis 2021

Hrsg. Oliver Passon Bergische Universität Wuppertal Wuppertal, Deutschland

Christoph Benzmüller   Universität Bamberg Bamberg, Deutschland

Brigitte Falkenburg Technische Universität Dortmund Dortmund, Deutschland

ISBN 978-3-662-67044-6 ISBN 978-3-662-67045-3  (eBook) https://doi.org/10.1007/978-3-662-67045-3 Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. © Der/die Herausgeber bzw. der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung, die nicht ausdrücklich vom Urheberrechtsgesetz zugelassen ist, bedarf der vorherigen Zustimmung des Verlags. Das gilt insbesondere für Vervielfältigungen, Bearbeitungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Die Wiedergabe von allgemein beschreibenden Bezeichnungen, Marken, Unternehmensnamen etc. in diesem Werk bedeutet nicht, dass diese frei durch jedermann benutzt werden dürfen. Die Berechtigung zur Benutzung unterliegt, auch ohne gesonderten Hinweis hierzu, den Regeln des Markenrechts. Die Rechte des jeweiligen Zeicheninhabers sind zu beachten. Der Verlag, die Autoren und die Herausgeber gehen davon aus, dass die Angaben und Informationen in diesem Werk zum Zeitpunkt der Veröffentlichung vollständig und korrekt sind. Weder der Verlag, noch die Autoren oder die Herausgeber übernehmen, ausdrücklich oder implizit, Gewähr für den Inhalt des Werkes, etwaige Fehler oder Äußerungen. Der Verlag bleibt im Hinblick auf geografische Zuordnungen und Gebietsbezeichnungen in veröffentlichten Karten und Institutionsadressen neutral. Planung/Lektorat: Caroline Strunz Springer Spektrum ist ein Imprint der eingetragenen Gesellschaft Springer-Verlag GmbH, DE und ist ein Teil von Springer Nature. Die Anschrift der Gesellschaft ist: Heidelberger Platz 3, 14197 Berlin, Germany

Preface

In 2021, René Talbot and Hans Schwarzlow from the Kurt Gödel Freundeskreis (Kurt Gödel circle of friends) in Berlin announced the second edition of the Essay competition for the Kurt Gödel Award.1 The prize was endowed with 15.000 EUR and awarded with assistance of the University of Wuppertal (Germany). This time, the prize question was “What does it mean for our world view if, according to Gödel, we assume the nonexistence of time?” Famously, Kurt Gödel was not just a brilliant logician and mathematician but also contributed important results to the general theory of relativity. In the 1940s, he found a solution to Einstein’s field equations that allows for so-called “closed time-like curves”. However, what does this mean in everyday language? Vividly explained, Gödel found a universe in which it is possible to time travel! Now, travelling through time—in particular into one's own past—certainly compromises our very notion of time in the first place. Gödel suggested, therefore, that already the theoretical possibility of such a universe questions the existence of time.2 All this is the background to the prize question on the implications of such a nonexistence of time for our world view. A multidisciplinary jury, consisting of Brigitte Falkenburg, Christoph Benzmüller and Oliver Passon had to sift the anonymised submissions and finally decided to award the first prize to logician and philosopher Reinhard Kahle from the University of Tübingen (Germany) for his essay entitled “The Philosophical Meaning of the Gödel Universe” (Chap. 3). The second prize was shared among five contributions, which span a wide range of approaches towards this issue. This is also reflected in the authors' diverse professional backgrounds in philosophy (Michał Pawłowski and Bartosz Wesół), physics (Claus Kiefer), theology (Thorben Alles) or computer science (Tim Lethen). This, in case readers are wondering, is also the order of the contributions in the book. However, Kahle’s contribution did what good philosophy is supposed to do: It immediately triggered a controversy. The recognised philosopher and Gödel

1

See https://kurtgoedel.de for further details and information on the previous edition of the prize in 2019. 2

In addition, Gödel’s solution describes a rotating universe. This is also relevant to the issue of time; see, e.g. Chap. 2. V

VI

Preface

scholar Palle Yourgrau (Brandeis University) took umbrage at some of Kahle’s arguments, and thus we invited him to contribute a reply to Kahle’s paper, in which he elaborates in particular Gödel’s temporal idealism (see Chap. 5). Reinhard Kahle feels honoured that Prof. Yourgrau took the time to discuss his contribution in this article. In his opinion, the philosophical differences might be much fewer than it seems, as Kahle does not intend to questions Gödel’s temporal idealism, but rather the role of the Gödel universe for this position. He will reply in more detail at some other occasion. He would only like to point out that two of the issues raised are based on ambiguities in the translation of the original paper, which should have been clarified earlier (see Footnotes 10 and 19 in Chap. 5). Moreover, there is more material that we would like to provide for a proper contextualisation of the essays. In order to keep the volume self-contained we decided to reprint the Gödel essay Remark About the Relationship Between Relativity Theory and Idealistic Philosophy from 1949 (Chap. 1). Furthermore we have included some relevant material of the chapter “Time Travel and Time Machines” by Chris Smeenk and Christian Wüthrich (originally published in 2011 in The Oxford Handbook of Philosophy of Time, Craig Callender (Ed.) Oxford University Press) to provide some technical background (Chap. 2). Finally, another editorial decision concerned the language of publication. The essay competition invited contributions in English and in German. We decided to translate the German contributions to achieve a book for an international readership (while at the same time including the German versions; this concerns the essays from Kahle, Kiefer and Alles). However, there is one exception here. Guido Stemme’s contribution to the essay competition has a special artistic touch to it, which renders a proper translation challenging. While we did not award a prize to this essay, we decided to enrich the present publication by including it in the original (i.e. German) version (Chap. 13). On the whole, we hope to have achieved an enjoyable and thought-provoking text that provides a multi-faceted approach to a perennial issue: time. Wuppertal & Berlin Dezember 2022

Oliver Passon Christoph Benzmüller Brigitte Falkenburg

Inhaltsverzeichnis

Teil I  Historical and technical context 1

Remark About the Relationship Between Relativity Theory and Idealistic Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Kurt Gödel

2

Excerpts from: Time Travel and Time Machines. . . . . . . . . . . . . . . . . 9 Chris Smeenk and Christian Wüthrich 2.1 Implications of Time Travel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Teil II  The contributions 3

The Philosophical Meaning of the Gödel Universe . . . . . . . . . . . . . . . 19 Reinhard Kahle 3.1 Gödel as a Logician. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Einstein’s Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 The Gödel Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Is Time Travel Possible? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 The Philosophical Challenge: Physical Principles That Exclude the Gödel Universe . . . . . . . . . . . . . . . . . . . . . . . . 23 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4

Die philosophische Bedeutung des Gödel-Universums. . . . . . . . . . . . 27 Reinhard Kahle 4.1 Gödel als Logiker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Einsteins Relativitätstheorie. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Das Gödel-Universum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 Sind Zeitreisen möglich?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.5 Die philosophische Herausforderung: Physikalische Prinzipien, die das Gödel-Universum ausschließen. . . . . . . . . . . 31 Literatur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5

Gödel’s Temporal Idealism: A Reply to Prof. Kahle. . . . . . . . . . . . . . 37 Palle Yourgrau References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 VII

VIII

Inhaltsverzeichnis

6

Self or the World. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Michał Pawłowski 6.1 No Order in the World. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 No Self. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7

Without Time, the World Becomes Leibnizian. . . . . . . . . . . . . . . . . . . 59 Bartosz Wesół 7.1 Preliminary Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2 Nonexistence of Time and Metaphysics . . . . . . . . . . . . . . . . . . . 62 7.3 Answer: Leibniz’s Monadology. . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8

What Does it Mean for Our World-View If We Assume with Gödel the Nonexistence of Time? . . . . . . . . . . . . . . . . . . . . . . . . . 69 Claus Kiefer 8.1 Time and the Gödel Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.2 Time and Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.3 Time and Quantum Gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.4 Where Is the Path Going? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

9

Was bedeutet es für unser Weltbild, wenn wir mit Gödel die Nichtexistenz der Zeit annehmen? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Claus Kiefer 9.1 Zeit und Gödel-Kosmos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.2 Zeit und Wirklichkeit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9.3 Zeit und Quantengravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 9.4 Wohin geht der Weg?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Literatur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10 Consequences from the Impossibility of Objectively Identifying Change. Philosophical Considerations Following Kurt Gödel. . . . . . 91 Thorben Alles 10.1 Gödel’s Argument for the Non-Objectivity of Change and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 10.2 Philosophical Interpretations of Gödel’s Argument. . . . . . . . . . . 93 10.3 Consequences of a Differentiation of the Concept of Time. . . . . 96 10.4 Outlook: The Problem of Intersubjectivity . . . . . . . . . . . . . . . . . 99 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Inhaltsverzeichnis

IX

11 Konsequenzen aus der Unmöglichkeit einer objektiven Bestimmung von Veränderung. Philosophische Überlegungen im Anschluss an Kurt Gödel. . . . . . . . . . . . . . . . . . . . . 103 Thorben Alles 11.1 Gödels Argument für die Nicht-Objektivität von Veränderung und Zeit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 11.2 Philosophische Anschlussmöglichkeiten an Gödels Argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.3 Konsequenzen aus einer Ausdifferenzierung des Zeitbegriffs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 11.4 Ausblick: Das Problem der Intersubjektivität . . . . . . . . . . . . . . . 112 Literatur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 12 How much Time Does a Logical Inference Take? . . . . . . . . . . . . . . . . 117 Tim Lethen 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 12.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 12.3 Time and the Kripke Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 12.4 Time and Algorithmic Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . 125 12.5 Time and the Evolution of Knowledge . . . . . . . . . . . . . . . . . . . . 128 12.6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 13 Was bedeutet es für unser Weltbild, wenn wir mit Kurt Gödel die Nichtexistenz der Zeit annehmen?. . . . . . . . . . . . . . . 139 Guido Stemme

Herausgeber- und Autorenverzeichnis

Über die Herausgeber Oliver Passon is a private lecturer in physics education at the University of a Wuppertal (Germany). He has studied physics, mathematics, educational science and philosophy in Wuppertal and obtained his PhD in Particle Physics. His areas of work and interest cover physics education of quantum theory, history and philosophy of science, phenomenological optics and Goethe’s theory of colour. Together with Tim Lethen he has edited Kurt Gödel’s notebooks on quantum mechanics (Kurt Gödels Notizen zur Quantenmechanik, Springer 2021). Other recent publications include (together with C. Friebe et al.) The Philosophy of Quantum Physics (Springer, 2018) and (together with C. Benzmüller) Wider den Reduktionismus (Springer 2021). Christoph Benzmüller  is Professor of AI Systems Engineering at the University of Bamberg. As an Adjunct Professor (apl.); he is also affiliated with the Department of Mathematics and Computer Science at Freie Universität Berlin (where he was the first UNA-Europe Visiting Professor) and he maintains a close research collaboration with the University of Luxembourg. In addition, he advises AI startup companies in Germany and abroad. Benzmüller’s research addresses topics such as the automation of rational argumentation and normative reasoning (e.g. to control intelligent systems), universal logic and universal reasoning, computational metaphysics and the mechanization of foundational theories in mathematics and philosophy. Benzmüller has conducted research as a Visiting Professor/Research Fellow at numerous prestigious universities, including Stanford University (USA), Cambridge University (UK) and Carnegie Mellon University (USA). Brigitte Falkenburg is Professor of Philosophy (retired) in the Department of Philosophy and Political Science at the Technische Universität Dortmund. She holds a diploma in physics, a PhD in Philosophy and a second PhD in Physics. Her areas of specialization are philosophy of science (philosophy of physics, philosophy of neuroscience), history of philosophy (Kant, Hegel, neo-Kantianism), philosophy of technology. She is a member of the Académie Internationale de Philosophie des Sciences. Selected publications as author or editor include Kant’s Cosmology (Springer 2020), Mechanistic Explanations in Physics and Beyond (Springer 2019, XI

XII

Herausgeber- und Autorenverzeichnis

with Gregor Schiemann), Why More Is Different (Springer 2015, with Margaret Morrison), From Ultrarays to Astroparticles (Springer 2012, with Wolfgang Rhode) and Mythos Determinismus: Wieviel erklärt uns die Hirnforschung? (Springer 2012).

Autorenverzeichnis Thorben Alles  University of Bonn, Bonn, Germany Reinhard Kahle  Universität Tübingen, Tübingen, Germany Claus Kiefer  Institut für Theoretische Physik, Universität zu Köln, Köln, Deutschland Tim Lethen  Department of Philosophy, University of Helsinki, Helsinki, Finland Michał Pawłowski Department of Economics, European University Institute, Florence, Italy Chris(topher) Smeenk  Department of Philosophy, University of Western Ontario, Ontario, Canada Guido Stemme  Mainz, Deutschland Bartosz Wesół  University of Warsaw, Warsaw, Poland Christian Wüthrich  Department of Philosophy, University of Western Ontario, Ontario, Canada Palle Yourgrau  Department of Philosophy, Brandeis University, Waltham, MA, USA

Teil I

Historical and technical context

1

Remark About the Relationship Between Relativity Theory and Idealistic Philosophy Kurt Gödel

One of the most interesting aspects of relativity theory for the philosophically minded consists in the fact that it gave new and surprising insights into the nature of time, of that mysterious and seemingly self-contradictory1 being, which, on the other hand, seems to form the basis of the world’s and our own existence. The very starting point

1 Cf.,

e.g. J.M.E. McTaggart, “The Unreality of Time”. Mind, 17, 1908.

2 At least if it is required that any two point events are either simultaneous or one succeeds the other,

i.e. that temporal succession defines a complete linear ordering of all point events. There exists an absolute partial ordering. Reprinted with kind permission of the Shelby White and Leon Levy Archives Center of the Institute for Advanced Study from Kurt Gödel: Collected Works: Volume II (Publications 1938–1974). Edited by Solomon Feferman, John W. Dawson, Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort (1990, pp. 202–207). This work was first published in 1949 for a collection intended to honour and discuss Einstein’s work (Schilpp, P. A. (Ed.) The Library of Living Philosophers, Volume 7. Albert Einstein: Philosopher-Scientist. Open Court 1949, pp. 557–562). For the German edition (Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der idealistischen Philosophie. In: Albert Einstein als Naturforscher und Philosoph, P.A. Schilpp (Ed.) Kohlhammer 1955, pp. 406–412) Gödel added two remarks in the footnotes. The present version follows the original text of 1949 but includes these additions from the German version in translation. Note that these additions apparently make some difference – Kahle’s contribution (Sect. 3 and 4) refers to them. Kurt Gödel ist verstorben. K. Gödel (B) New Jersey, USA

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_1

3

4

K. Gödel

of special relativity theory consists in the discovery of a new and very astonishing property of time, namely the relativity of simultaneity, which to a large extent implies2 that of succession. The assertion that the events A and B are simultaneous (and, for a large class of pairs of events, also the assertion that A happened before B) loses its objective meaning, in so far as another observer, with the same claim to correctness, can assert that A and B are not simultaneous (or that B happened before A). Following up the consequences of this strange state of affairs, one is led to conclusions about the nature of time, which are very far reaching indeed. In short, it seems that one obtains an unequivocal proof for the view of those philosophers who, like Parmenides, Kant, and the modern idealists, deny the objectivity of change and consider change as an illusion or an appearance due to our special mode of perception.3 The argument runs as follows. Change becomes possible only through the lapse of time. The existence of an objective lapse of time,4 however, means (or, at least, is equivalent to the fact) that reality consists of an infinity of layers of “now”, which come into existence successively. However, if simultaneity is something relative in the sense just explained, reality cannot be split up into such layers in an objectively determined way. Each observer has his own set of “nows”, and none of these various systems of layers can claim the prerogative of representing the objective lapse of time.5

3 Kant,

in the Critique of pure reason, 2nd ed. (1787, p. 54), expresses this view in the following words: “Those affections which we represent to ourselves as changes, in beings with other forms of cognition, would give rise to a perception in which the idea of time, and therefore also of change, would not occur at all”. This formulation agrees so well with the situation subsisting in relativity theory that one is almost tempted to add: such as, e.g. a perception of the inclination relative to each other of the world lines of matter in Minkowski space. 4 One may take the standpoint that the idea of an objective lapse of time (whose essence is that only the present really exists) is meaningless. This is no way out of the dilemma; for by this very opinion, one would take the idealistic viewpoint as to the idea of change, exactly as those philosophers who consider it as self-contradictory. For in both views one denies that an objective lapse of time is a possible state of affairs, a fortiori, that it exists in reality, and it makes very little difference in this context, whether our idea of it is regarded as meaningless or as self-contradictory. Of course, for those who take either one of these two viewpoints the argument from relativity theory given below is unnecessary, but even for them it should be of interest that perhaps there exists a second proof for the unreality of change based on entirely different grounds, especially in view of the fact that the assertion to be proved runs so completely counter to common sense. A particularly clear discussion of the subject independent of relativity theory is to be found in Paul Mongré, Das Chaos in kosmischer Auslese, 1898. 5 It may be objected that this argument only shows that the lapse of time is something relative, which does not exclude that it is something objective, whereas idealists maintain that it is something merely imagined. A relative lapse of time, however, if any meaning at all can be given to this phrase, would certainly be something entirely different from the lapse of time in the ordinary sense, which means a change in the existing. The concept of existence, however, cannot be relativized without destroying its meaning completely. It may furthermore be objected that the argument under consideration only shows that time lapses in different ways for different observers, whereas the lapse of time itself may nevertheless be an intrinsic (absolute) property of time or of reality. A lapse of time, however, which is not a lapse in some definite way seems to me as absurd as a coloured object that has

1

Remark About the Relationship Between Relativity Theory …

5

This inference has been pointed out by some, although by surprisingly few, philosophical writers, but it has not remained unchallenged. Actually to the argument in the form just presented it can be objected that the complete equivalence of all observers moving with different (but uniform) velocities, which is the essential point in it, subsists only in the abstract space-time scheme of special relativity theory and in certain empty worlds of general relativity theory. The existence of matter, however, as well as the particular kind of curvature of space-time produced by it, largely destroys the equivalence of different observers6 and distinguishes some of them conspicuously from the rest, namely, those that follow in their motion the mean motion of matter.7 Now in all cosmological solutions of the gravitational equations (i.e. in all possible universes) known at present the local times of all these observers fit together into one world time, so that apparently it becomes possible to consider this time as the “true” one, which lapses objectively, whereas the discrepancies of the measuring results of other observers from this time may be conceived as due to the influence that a motion relative to the mean state of motion of matter has on the measuring processes and physical processes in general. From this state of affairs, in view of the fact that some of the known cosmological solutions seem to represent our world correctly. James Jeans has concluded8 that there is no reason to abandon the intuitive idea of an absolute time lapsing objectively. I do not think that the situation justifies this conclusion and am basing my opinion chiefly9 on the following facts and considerations:

no definite colours. However, even if such a thing were conceivable, it would again be something totally different from the intuitive idea of the lapse of time to which the idealistic assertion refers. 6 Of course, according to relativity theory all observers are equivalent in so far as the laws of motion and interaction for matter and field are the same for all of them. However, this does not exclude that the structure of the world (i.e. the actual arrangement of matter, motion, and field) may offer quite different aspects to different observers, and that it may offer a more “natural” aspect to some of them and a distorted one to others. The observer, incidentally, plays no essential role in these considerations. The main point, of course, is that the [four-dimensional] world itself has certain distinguished directions, which directly define certain distinguished local times. 7 The value of the mean motion of matter may depend essentially on the size of the regions over which the mean is taken. What may be called the “true mean motion” is obtained by taking regions so large that a further increase in their size does not any longer change essentially the value obtained. In our world, this is the case for regions including many galactic systems. Of course, a true mean motion in this sense need not necessarily exist. 8 Cf. Man and the Universe, Sir Halley Stewart Lecture (1935), 22–23. 9 Another circumstance invalidating Jeans’ argument is that the procedure described above gives only an approximate definition of an absolute time. No doubt, it is possible to refine the procedure so as to obtain a precise definition, but perhaps only by introducing more or less arbitrary elements (such as, e.g. the size of the regions or the weight function to be used in the computation of the mean motion of matter). It is doubtful whether there exists a precise definition that has so great merits that there would be sufficient reason to consider exactly the time thus obtained as the true one.

6

K. Gödel

There exist cosmological solutions of another kind10 than those known at present, to which the aforementioned procedure of defining an absolute time is not applicable, because the local times of the special observers used above cannot be fitted together into one world time. Nor can any other procedure that would accomplish this purpose exist for them; i.e. these worlds possess such properties of symmetry that for each possible concept of simultaneity and succession there exist others that cannot be distinguished from it by any intrinsic properties, but only by reference to individual objects, such as, e.g. a particular galactic system. Consequently, the inference drawn above as to the non-objectivity of change doubtless applies at least in these worlds. Moreover it turns out that temporal conditions in these universes (at least in those referred to in the end of footnote 10) show other surprising features, strengthening further the idealistic viewpoint. Namely, by making a round trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present, and future, and back again, exactly as it is possible in other worlds to travel to distant parts of space. This state of affairs seems to imply an absurdity. For it enables one, e.g. to travel into the near past of those places where he has himself lived. There he would find a person who would be himself at some earlier period of his life. Now he could do something to this person that, by his memory, he knows has not happened to him. This and similar contradictions, however, in order to prove the impossibility of the worlds under consideration, presuppose the actual feasibility of the journey into one’s own past. However, the velocities that would be necessary in order to complete the voyage in a reasonable length of time11 are far beyond everything that can be expected ever to become a practical possibility. Therefore, it cannot be excluded a

10 The most conspicuous physical property distinguishing these solutions from those known at present is that the compass of inertia in them rotates everywhere [in the same direction] relative to matter, which in our world would mean that it rotates relative to the totality of galactic systems. These worlds, therefore, can fittingly be called “rotating universes”. In the subsequent considerations I have in mind a particular kind of rotating universes, which have the additional properties of being static and spatially homogeneous, and a cosmological constant < 0. For the mathematical representation of these solutions, cf. my forthcoming 1949 [and, for a general discussion of rotating universes, my 1952 “Rotating universes in general relativity theory”. In: Proceedings of the International Congress of Mathematicians. Edited by L.M. Graves et al. Vol. 1, p. 175–181]. 11 Basing the calculation on a mean density of matter equal to that observed in our world, and assuming one were able to transform matter completely into energy, the weight of the “fuel” of the rocket ship, in order to complete the voyage in t years (as measured by the traveller), would have to be of the order of magnitude of 1022 /t 2 times the weight of the ship (if stopping, too, is effected by recoil). This estimate applies to t  1011 . Irrespective of the value of t, the velocity of the ship must √ be at least 1/ 2 of the velocity of light. [Translation of the author’s addition to the German edition (1955): A second reason for excluding a priori the universes mentioned above could be found in the possibility of “telegraphing a message into one’s own past”. But the practical difficulties in doing so would hardly seem to be trifling. Moreover, the boundary between difficulties in practice and difficulties in principle is not at all fixed. What was earlier a practical difficulty in atomic physics has today become an impossibility in principle, in consequence of the uncertainty principle: and the same could one day happen also for those difficulties that reside not in the domain of the “too small”, but of the “too large”.]

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Remark About the Relationship Between Relativity Theory …

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priori, on the ground of the argument given, that the space-time structure of the real world is of the type described. As to the conclusions that could be drawn from the state of affairs explained for the question being considered in this paper, the decisive point is this: that for every possible definition of a world time one could travel into regions of the universe that are past according to that definition.12 This again shows that to assume an objective lapse of time would lose every justification in these worlds. For, in whatever way one may assume time to be lapsing, there will always exist possible observers to whose experienced lapse of time no objective lapse corresponds (in particular also possible observers whose whole existence objectively would be simultaneous). However, if the experience of the lapse of time can exist without an objective lapse of time, no reason can be given why an objective lapse of time should be assumed at all. It might, however, be asked: Of what use is it if such conditions prevail in certain possible worlds? Does that mean anything for the question interesting us whether in our world there | exists an objective lapse of time? I think it does. For: (1) Our world, it is true, can hardly be represented by the particular kind of rotating solutions referred to above (because these solutions are static and, therefore, yield no red-shift for distant objects); there exist however also expanding rotating solutions. In such universes, an absolute time also might fail to exist,13 and it is not impossible that our world is a universe of this kind. (2) The mere compatibility with the laws of nature14 of worlds in which there is no distinguished absolute time, and [in which], therefore, no objective lapse of time can exist, throws some light on the meaning of time also in those worlds in which an absolute time can be defined. For, if someone asserts that this absolute time is lapsing, he accepts as a consequence that whether or not an objective lapse of time exists (i.e. whether or not a time in the ordinary sense of the word exists) depends on the particular way in which matter and its motion are arranged in the world. This is not a straightforward contradiction; nevertheless, a philosophical view leading to such consequences can hardly be considered as satisfactory. Kurt Gödel was born 1906 in Brno (German Brünn; today Czech Republic, then the Austro-Hungarian Empire) and died at Princeton (NJ, USA) in 1978. He studied mathematics and physics in Vienna and was probably the most important logician of the twentieth century. In 1931

12 For this purpose, incomparably smaller velocities would be sufficient. Under the assumptions made in footnote 11, the weight of the fuel would have to be at most of the same order of magnitude as the weight of the ship. 13 At least if it required that successive experiences of one observer should never be simultaneous in the absolute time or (which is equivalent) that the absolute time should agree in direction with the times of all possible observers. Without this requirement an absolute time always exists in an expanding (arid homogeneous) world. Whenever I speak of an “absolute” time, this, of course, is to be understood with the restriction explained in footnote 9, which also applies to other possible definitions of an absolute time. [Translation of the author’s addition to the German edition (1955): By an “absolute time” I understand a world time that can be defined without reference to particular objects and that satisfies the requirement formulated at the beginning of this footnote. More precisely, this should be called a “possible absolute time”, since several can exist within one world, even though that is only exceptionally the case in spatially homogeneous universes.]

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he proved the incompleteness of axioms for arithmetic (his most famous result). Other important results include the relative consistency of the axiom of choice and continuum hypothesis with the other axioms of set theory. His interests covered a wide range, including modern physics, philosophy and theology. Due to Austria becoming part of Nazi-Germany in 1938, he and his wife Adele left Vienna for Princeton in 1940 were he accepted a position at the Institute for Advanced Study (IAS). Given his reluctance to publish his findings the rich and only partly edited Nachlass of Gödel is expected to be good for many surprises still.

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Excerpts from:Time Travel and Time Machines Chris Smeenk and Christian Wüthrich

2.1

Implications of Time Travel

Given that time travel cannot be straightforwardly ruled out as incoherent or logically impossible, we now face the following difficult questions. In what sense is time travel physically possible, and what does this imply regarding the nature of time? More precisely, what are the novel consequences of time travel, that is, ones that do not follow already from more familiar aspects of special or general relativity? As a first step towards answering these questions, we will consider Kurt Gödel’s (in)famous argument for the ideality of time.1

1 The following papers, which we draw on below, discuss aspects of Gödel’s argument: Stein (1970),

D. B. Malament (1984), Savitt (1994), Earman (1995), Dorato (2002), Belot (2005). Ellis (1996) discusses the impact of Gödel’s paper. Reprinted with kind permission from the The Oxford Handbook of Philosophy of Time, Craig Callender (Ed.) Oxford University Press 2011, Chapter 20, Time Travel and Time Machines (pp. 577–630). While we highly recommend reading the full article, some parts of Section 4 (pp. 586ff) contain particularly relevant material on Gödel’s cosmological solution, closed time-like curves (CTC) and the role of rotation in general relativity. We have included this part (keeping the original footnote numbering) in order to make our presentation more self-contained. C. Smeenk (B) · C. Wüthrich Department of Philosophy, University of Western Ontario, Ontario, Canada e-mail: [email protected] C. Wüthrich e-mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_2

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Gödel (1949b) was the first to clearly describe a relativistic spacetime with closed CTCs.2 Gödel’s stated aim in discovering this spacetime was to rehabilitate an argument for the ideality of time from special relativity within the context of GR. In special relativity, Gödel asserts that the ideality of time follows directly from the relativity of simultaneity. He takes as a necessary condition for the existence of an objective lapse of time the possibility of decomposing spacetime into of a sequence of “nows”—namely, that it has the structure R×, where R corresponds to “time”, and  are “instants”, three-dimensional collections of simultaneous events. However, in special relativity, the decomposition of the spacetime into “instants” is relative to an inertial observer rather than absolute; as Gödel puts it, “Each observer has his own set of ‘nows’, and none of these various systems of layers can claim the prerogative of representing the objective lapse of time” (Gödel, 1949a, p. 558). This conclusion does not straightforwardly carry over to GR, because there is a natural way to privilege one set of “nows” in a cosmological setting. The privilege can be conferred on a sequence of “nows” defined with respect to the worldlines of galaxies or other large-scale structures. It is natural to require the surfaces of simultaneity to be orthogonal to the worldlines of the objects taken to define the “cosmologically preferred frame”. The question is then whether one can extend local surfaces of simultaneity satisfying this requirement to a global foliation for a given set of curves. For the cosmological models usually taken to be the best approximation to the large-scale structure of spacetime, the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, the answer is yes. These models have a natural foliation, a unique way of globally decomposing spacetime into a one-dimensional “cosmic time” and three-dimensional surfaces  representing “instants”, orthogonal to the worldlines of freely falling bodies (cosmic time in this case would correspond to the proper time measured by an observer at rest with respect to this privileged frame.). Thus, Gödel’s necessary condition for an objective lapse of time is satisfied in the FLRW cosmological models, and in this sense, the pre-relativistic concept of absolute time can be recovered. However, in Gödel’s spacetime one cannot introduce such a foliation. The spacetime represents a “rotating universe”, in which matter is in a state of uniform rigid rotation.3 Due to this rotation it is not possible to define a privileged frame with global “instants” similar to the frame in the FLRW models.4 An analogy due to D. Malament (1995) illustrates the reason for this. One can slice through a collection of

2 Although

Stockum (1937) discovered a solution describing an infinite rotating cylinder that also contains CTCs through every point, this feature of the solution was not discussed in print, to the best of our knowledge, prior to Tipler (1974). Gödel does not cite von Stockum’s work. Others had noted the possibility of the existence of CTCs without finding an exact solution exemplifying the property (see, e.g. Weyl, 1921, p. 249). 3 More precisely, in Gödel’s universe a congruence of time-like geodesics has non-zero twist and vanishing shear. Defining rotation for extended bodies in general relativity turns out to be a surprisingly delicate matter (see, especially, D. B. Malament, 2002). 4 As John Earman pointed out to us, Gödel does not seem to have noted the stronger result that Gödel spacetime does not admit of any foliation into global time slices.

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parallel fibres with a single plane that is orthogonal to them all, but if the fibres are twisted into a rope, there is no way to cut through the rope while remaining orthogonal to each fibre (The “twist” of the fibres is analogous to the rotation of worldlines in Gödel’s model.). The construction of global “instants” described above can be carried out if and only if there is no “twist” (or rotation) of the worldlines used to define the cosmologically privileged frame. Demonstrating that such rotating models exist by finding an explicit spacetime model solving Einstein’s field equations was clearly Gödel’s main aim. However, the welcome discovery that in his rotating universe there is a CTC passing through every point further bolstered his argument for the ideality of time.5 It is noteworthy that many chronology-violating spacetimes resemble Gödel’s solution in the following sense: they contain rotating masses and CTCs wind around the masses against the orientation of the rotation.6 What, then, is Gödel’s argument? The crucial problem is how to get from discoveries regarding the nature of time in this specific spacetime to a conclusion about the nature of time in general. Gödel could avoid this problem if his spacetime, or a spacetime with similar features, were a viable candidate for representing the structure of the observed universe. Then his results would obviously have a bearing on the nature of time in our universe. Gödel apparently took this possibility quite seriously, and subsequently discovered a class of rotating models that incorporate the observed expansion of the universe (Gödel, 1952) In these models, one can construct suitable “instants” as long as the rate of rotation is sufficiently low, and recent empirical work places quite low upper limits on the rate of cosmic rotation.7 Gödel goes on to argue that even if his model (or models with similar features) fails to represent the actual universe, its mere existence has general implications (p. 562):8

5 Malament

observed that the existence of CTCs is not mentioned in three of the five preparatory manuscripts for Gödel (1949b), and it appears that Gödel discovered this feature in the course of studying the solution. In addition, in lecture notes on rotating universes (from 1949), Gödel emphasises that he initially focused on rotation and its connection to the existence of global time slices in discovering the solution. See D. Malament (1995) and Stein (1995, pp. 227–229). 6 Cf. Andréka, Németi and Wüthrich (2008). That rotation may be responsible for the formation of CTCs is also suggested by Bonnor’s (2001) result that stationary axially symmetric solutions of Einstein’s field equations describing two spinning massive bodies under certain circumstances include a non-vanishing region containing CTCs. 7 These instants are not surfaces orthogonal to time-like geodesics, as there is still rotation present, but Gödel (1952) establishes that surfaces of constant matter density can be used to define a foliation that satisfies his requirements for an objective lapse of time. For recent empirical limits on global rotation based on the cosmic microwave background radiation, see, for example, Kogut, Hinshaw and Banday (1997). 8 As Sheldon Smith pointed out to us, if this is taken to be Gödel’s main argument, then it is not clear why the mere existence of Minkowski spacetime, regarded as a vacuum solution of the field equations, does not suffice. Why did Gödel need to go to the effort of discovering the rotating model granted that there is no distinguished absolute time in Minkowski spacetime? Although we do not find a clear answer to this in Gödel (1949a), we offer two tentative remarks. First, Gödel may have objected to classifying Minkowski spacetime as physically reasonable because it is a vacuum spacetime. Second, and more importantly, Gödel took the prospect of discovering a rotating and expanding model consistent with observations more seriously than most commentators allow for.

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C. Smeenk and C.Wüthrich The mere compatibility with the laws of nature of worlds in which there is no distinguished absolute time, and, therefore, no objective lapse of time can exist, throws some light on the meaning of time also in those worlds in which an absolute time can be defined. For, if someone asserts that this absolute time is lapsing, he accepts as a consequence that, whether or not an objective lapse of time exists ... depends on the particular way in which matter and its motion are arranged in the world. This is not a straightforward contradiction; nevertheless, a philosophical view leading to such consequences can hardly be considered as satisfactory.

Despite disagreement among recent commentators regarding exactly how to read Gödel’s argument, there is consensus that even this modest conclusion is not warranted. The dynamical connection between spacetime geometry and the distribution of matter encoded in Einstein’s field equations insures that, in some sense, many claims regarding spacetime geometry depend on “how matter and its motion are arranged”. Nearly any discussion of the FLRW models highlights several questions regarding the overall shape of spacetime—e.g. whether time is bounded or unbounded and what is the appropriate spatial geometry for “instants”—that depend on apparently contingent properties such as the value of the average matter density. What exactly is unsatisfactory about this? What does the mere possibility of spacetimes with different geometries imply regarding geometrical structure in general? Earman (1995, Appendix to Chapter 6) challenges the implicit modal step in Gödel’s argument. How can we justify this step on Gödel’s behalf and elucidate what is unsatisfactory about objective time lapse in general, without lapsing back into pre-GR intuitions? Perhaps the argument relies on an implicit modal assumption that lapsing, in the sense described above, must be an essential property of time. Then (given that  (¬P) ↔ ¬ (P)), the demonstration that  (¬P), where P is the existence of an objective lapse of time, via finding the Gödel spacetime would be decisive. However, what is the basis for this claim about the essential nature of time, and how can it be defended without relying on pre-relativistic intuitions? Earman (1995) considers this and several replies that might be offered on Gödel’s behalf, only to reject each one. Steve Savitt (1994) defends a line of thought (cf. Yourgrau, 1991) that is more of a variation on Gödelian themes than a textual exegesis. On Savitt’s line, Gödel’s argument rests not on essentialist claims regarding the nature of time, but instead on a claim of local indistinguishability. Suppose that it is physically possible for beings like us to exist in a Gödel spacetime, and (1) that it is possible for these denizens to have the “same experience of time” as we do. Assume further that (2) the only basis for our claim that objective time exists in our universe is the direct experience of time. Then the existence of the Gödel universe is a defeater for our claim to have established objective time lapse on the basis of our experience, because (for all we know) we could be in the indistinguishable situation—inhabiting a Gödel universe in which there is no such lapse. While this variation does not require a modal step

This suggests that the argument in the quoted passage is a fall-back position, and that Gödel put more weight on the claim that he had discovered a viable model for the observed universe that lacks an objective lapse of time.

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as suspect as the original version, neither (1) nor (2) are obviously true—and it is unclear how they can be established without begging the question.9 One response to the challenge is simply to abandon Gödel’s modal argument and formulate a different argument to the same effect. Consider an alternative argument that adopts a divide and conquer strategy rather than relying on a shaky modal step (suggested to us by John Earman). Divide the solutions of Einstein’s field equations into (1) those that, like Gödel spacetime, lack a well-defined cosmic time, and (2) solutions that do admit a cosmic time.10 The considerations above show that the spacetimes of type (1) lack an objective lapse of time in Gödel’s sense. The spacetimes of type (2) have, by contrast, an embarrassment of riches; there are many well-defined time functions, and in general no way to single out one as representing the objective lapse of time. The definition of the cosmologically preferred reference frame in the FLRW models takes advantage of their maximal symmetry. Thus, we seem to have an argument, without a mysterious modal step, that generic solutions of the field equations lack an objective lapse of time. A different approach spelled out by Gordon Belot (2005) offers a methodological rather than metaphysical response to Earman’s challenge. Belot concedes to Earman’s challenge given a “natural-historical” construal of Gödel’s argument, according to which the nature of time can be established based on empirical study of “how matter and its motion are arranged”. On this reading, time in our universe is characterised by the appropriate spacetime of GR that is the best model for observations—and the mere existence of alternative spacetimes is irrelevant. However, on a “law-structural” construal, questions regarding the nature of time focus on the laws of nature rather than on contingent features of a particular solution. Belot makes a case that a lawstructural construal of the question is more progressive methodologically, in that it fosters deeper insights into our theories and aids in the development of new theories.11 If we grant that understanding the laws may require study of bizarre cases such as Gödel’s spacetime alongside more realistic solutions, then we have the start of a response to Earman’s challenge. It is only a start, because this suggested reading remains somewhat sketchy without an account of “laws of nature”, which is needed to delineate the two construals more sharply. Even if we had a generally accepted account of the laws of nature, the application of “laws” to cosmology is controversial; how can we distinguish nomic necessities from contingencies in this context, granting the uniqueness of the universe? Setting this issue aside, Earman’s challenge can be reiterated by asking which spacetimes should be taken as revealing important properties of the laws. Why should Gödel spacetime, in particular, be taken to reveal something about the nature of time encoded in the laws of GR? Suppose we expect that only a subset of the

9 See

Belot (2005) and Dorato (2002) for further discussion. terms of the causality conditions in §3, a global time function exists for “stably causal” spacetimes—a condition slightly weaker than global hyperbolicity. 11 Belot finds inspiration for this position in several brief remarks regarding the nature of scientific progress in manuscript precursors to Gödel (1949b); however, he does not take these considerations to be decisive (see p. 275, fn. 52). 10 In

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spacetimes deemed physically possible within classical GR will also be physically possible according to the as-yet-undiscovered theory of quantum gravity. How would we argue that Gödel spacetime should fall within that subset, and that it should be taken to reveal a fundamental feature of the laws of GR that will carry over to quantum gravity? The features Gödel used to establish the lack of absolute time in his model are often taken to support a negative answer to this question that does not appear to be ad hoc. Many approaches to quantum gravity simply rule out spacetimes with CTCs ab initio based on the technical framework adopted.12 As we will discuss below, much of the physics literature on spacetimes with CTCs seeks clear physical grounds to rule them beyond the pale; insight into the laws of a future theory of quantum gravity would come from showing why the laws do not allow CTCs. Yet we agree with Belot that what is more unsatisfying regarding Gödel’s argument, even on the “law-structural” construal, is that an argument by counter-example does little to illuminate deeper connections between the nature of time and the laws of the theory. Assessing the implications of Gödel’s spacetime clearly turns on rather delicate issues regarding modality and the laws of nature. Perhaps our failure to articulate a clear Gödelian argument indicates that the properties of such bizarre spacetimes can be safely ignored when we investigate the nature of time in GR. Tim Maudlin (2007) advocates a dismissive response to CTCs, which would otherwise pose a threat to his metaphysical account of the passage of time: “It is notable in this case that the equations [Einstein’s field equations] do not force the existence of CTCs in this sense: for any initial conditions one can specify, there is a global solution for that initial condition that does not have CTCs”. He anticipates a critic’s response that his metaphysical account of passage boldly stipulates that the nature of time is not compatible with the existence of CTCs, and replies: “...But is it not equally bold to claim insight into the nature of time that shows time travel to be possible if we grant that it is not actual and also that the laws of physics, operating from conditions that we take to be possible, do not require it” (Maudlin, 2007, p. 190). These assertions would follow from the proof of the following form: CTCs do not arise from “physically possible” initial states under dynamical evolution according to Einstein’s equations. Below we will consider a more precise formulation of this “chronology protection conjecture” (in §6). However, at this point, we wish to emphasise that this is still a conjecture, and that there are a number of subtleties that come into play in even formulating a clear statement amenable to proof or disproof.13 Perhaps a claim like Maudlin’s, suitably disambiguated, will prove to be correct, but part of the interest of the question is precisely due to the intriguing technical questions that remain open.14 12 Gödel’s solution might be ruled out due to the symmetries of the solution, as Belot notes: symme-

tric solutions pose technical obstacles to some approaches to quantisation, and it seems precarious to base assertions regarding features of quantum gravity on properties of special, symmetric solutions. However, this argument seems too strong, in that it would also rule out the FLRW models, which are currently accepted as the best classical descriptions of the large-scale structure of the universe. 13 We should note that absent from some further qualifications, Maudlin’s first claim is false—as established by Manchak’s theorem discussed in the next section. 14 Our treatment here is influenced by the clear discussion of these issues in Stein 1970.

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References Andréka, H., Németi, I., & Wüthrich, C. (2008). A twist in the geometry of rotating black holes: seeking the cause of acausality. General Relativity and Gravitation, 40, 1809–1823. Belot, G. (2005). Dust, Time, and Symmetry. The British Journal for the Philosophy of Science, 56, 255–291. Bonnor, W. B. (2001). “The Interactions Between Two Classical Spinning Particles”. Classical and Quantum Gravity, 18(7), 1381–1388. https://doi.org/10.1088/0264-9381/18/7/318. Dorato, M. (2002). On Becoming, Cosmic Time, and Rotating Universes. Royal Institute of Philosophy Supplement, 50, 253–276. https://doi.org/10.1017/S1358246100010596. Earman, J. (1995). Bangs, Crunches, Whimpers, and Shrieks – Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press. Ellis, G. (1996). “Contributions of K. Gödel to Relativity and Cosmology”. In: P. Hájek (Ed.), Gödel ’96: Logical Foundations of Mathematics, Computer Science and Physics – Kurt Gödel’s Legacy (pp. 34–49). Springer-Verlag. https://doi.org/10.1017/9781316716939.004. Gödel, K. (1949). A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-Scientist (pp. 557–562). Open. Gödel, K. (1949b). “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation”. In: Reviews of Modern Physics, 21(3), 447–450. https://doi.org/10. 1103/RevModPhys.21.447. Gödel, K. (1952). “Rotating universes in general relativity theory”. In: Proceedings of the International Congress of Mathematicians. American Mathematical Society, pp. 175–181. Kogut, A., G., Hinshaw, and A. J. Banday (1997). “Limits to global rotation and shear from the COBE DMR four-year sky maps”. Physical Review D, 55(4), 1901–1905. https://doi.org/10. 1103/PhysRevD.55.1901. Malament, D. (1995). “Introductory Note to *1949b”. In: S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons, & R. Solovay (Eds.),kurt Gödel: Collected Works: Volume III: Unpublished Essays and Lectures. (Vol. 3, pp. 261–269). Oxford University Press on Demand. Malament, D. B. (1984). “Time Travel” in the Gödel Universe“. In: PSA 1984 2. Ed. by P. D. Asquith and P. Kitcher, pp. 91–100. Malament, D. (2002). A No-Go Theorem about Rotation in Relativity Theory. In D. B. Malament (Ed.), Reading Natural Philosophy (Essays Dedicated to Howard Stein on His 70th Birthday) (pp. 267–293). Press: Open Court. Maudlin, T. (2007). The Metaphysics Within Physics. Oxford University Press. https://doi.org/10. 1093/acprof:oso/9780199218219.001.0001. Savitt, S. F. (1994). “The replacement of time”. Australasian Journal of Philosophy, 72(4), 463–474. https://doi.org/10.1080/00048409412346261. Stein, H. (1995). “Introductory Note to *1946/9”. In: S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: Collected Works: Volume III: Unpublished Essays and Lectures, (Vol. 3, pp. 202–229). Oxford University Press on Demand. Stein, H. (1970). “On the Paradoxical Time-Structures of Gödel”. Philosophy of Science, 37(4), 589–601. van Stockum, W. J. (1937). The Gravitational Field of a Distribution of Particles Rotating About an Axis of Symmetry. Proceedings of the Royal Society of Edinburgh, 57, 135–154. Tipler, F. J. (1974). “Rotating cylinders and the possibility of global causality violation”. Physical Review D, 9(8), 2203–2206. https://doi.org/10.1103/PhysRevD.9.2203. Weyl, H. (1921). Space-Time-Matter. translated by S. Brose. Methue Yourgrau, P. (1991). The Disappearance of Time: Kurt Gödel and the Idealistic Tradition in Philosophy. Cambridge University Press.

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Chris(topher) Smeenk is Professor in the Department of Philosophy at the University of Western Ontario (Canada) and Director of the Rotman Institute of Philosophy. He received a BA degree in Physics and Philosophy from Yale University in 1995 and pursued graduate studies at the University of Pittsburgh leading to a PhD in History and Philosophy of Science in 2003. His main research interests are in history and philosophy of physics, the foundations of physics (especially spacetime physics and cosmology), and general topics in philosophy of science, and seventeenthcentury natural philosophy. Christian Wüthrich is Associate Professor of Philosophy at the University of Geneva (Switzerland). He works primarily in philosophy of physics, philosophy of science and metaphysics. He also has interests in the philosophy of mathematics, epistemology, and the philosophy of mind. Wüthrich has studied physics, mathematics, philosophy, and history and philosophy of science at the Universities of Bern, Cambridge, and Pittsburgh, where he received his PhD in 2006.

Teil II

The contributions

3

The Philosophical Meaning of the Gödel Universe Reinhard Kahle

During his time at the Institute for Advanced Study, Kurt Gödel became a close friend of Albert Einstein, and in particular studied the theory of relativity. One result of this study was the discovery of the so-called Gödel universe1, a model of Einstein’s field equations of general relativity in which no absolute time can be defined. Theoretically, time travel would be possible in such a universe. Sometimes, far-reaching philosophical consequences are attributed to this result. Palle Yourgrau, for example, argues that Gödel concluded that there is no time even in our world.2 In fact, only much more cautious conclusions can be found in Gödel’s writings. In this paper, we will first reconstruct the “discovery” of Gödel’s universe in the context of his work as a logician. On the basis of this reconstruction, the specifically philosophical challenge that Gödel saw can then be discerned: our cosmological theories should be capable of being strengthened by means of physical principles, so that the nonexistence of Gödel universes—and thus the existence of time—does not depend “on the particular way in which matter and its motion are arranged in the world”.3

1 Strictly

speaking, it is about a whole class of universes; the singular form stands here as a collective noun. 2 Can we really infer the nonexistence of time in this world from its absence from a merely possible universe? In a word, yes. Or so Gödel argues” (Yourgrau, 2005, p. 130). 3 Compare Gödel (1956, p. 412). R. Kahle (B) Universität Tübingen, Tübingen, Germany e-mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_3

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R. Kahle

Gödel as a Logician

The most outstanding result of Kurt Gödel in the list of his scientific achievements is without doubt the incompleteness theorems (Gödel, 1931). Before that he had already shown the (semantic) completeness of the given axiomatisation of predicate logic (Gödel, 1930) and he had introduced recursive functions as a technical tool to prove his incompleteness theorems. Later on, he succeeded to prove the consistency of the axiom of choice and the continuum hypothesis with the axioms of set theory of Zermelo and Fraenkel (Gödel, 1938). In addition there are many “smaller” results, like a form of the double negation interpretation for Peano arithmetic in Heyting arithmetic (Gödel, 1933) or the definition of his system of functionals of higher type T , which allows a non-finitist consistency proof of arithmetic (Gödel, 1958). All these works belong to mathematical logic, a discipline that has obtained its modern development not least by Gödel’s contributions. It is known that Einstein said towards the end of his life that he only went to the Institute at Princeton to have discussions with Gödel.4 . The subject of these discussions will certainly have been relativity and in particular Gödel’s famous Universum, which seems to call our concept of time into question. In the following reflection on the philosophical consequences that Gödel drew from the discovery of his universe, however, it is important to keep in mind that this discovery, too, should be considered against the background of Gödel’s logical competence.

3.2

Einstein’s Theory of Relativity

Albert Einstein had first developed the special theory of relativity (Einstein, 1905). It is true that Einstein himself was not a logician. But we can obtain the special theory of relativity absolutely with the help of certain logical guidelines. The central idea is: the knowledge that there is an absolute speed of light c, which is at the same time the physically highest speed, can be set up as a kind of axiom, so that all further theory components of a physical description of our universe must be compatible with this axiom. This can be illustrated by the addition of velocities. According to classical physics velocities can be added arbitrarily. For example, let a train with the speed v be on the way. If a person in the train goes forward with velocity v, the velocity considered from the outside is u + v. Now a passenger holds a flashlight in the direction of travel. The light of this flashlight would then, viewed from outside the train, travel at the speed u + c, which would be greater than c, according to the classical view. However, if c is to be an absolute maximum speed, the addition must be calculated

4 Oskar

Morgenstern in a letter to Bruno Kriesky, the Austrian Federal Minister of Foreign Affairs, Oct. 25, 1965, (Morgenstern, 2002): “Einstein often told me that in the last years of his life he always sought Gödel’s company in order to be able to discuss with him. Once he said to me that his own work no longer meant much, that he merely came to the Institute building to have ‘the privilege of walking home with Gödel’”

3 The Philosophical Meaning of the Gödel Universe

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in a different way. It now requires elaborate mathematical considerations,5 in order to arrive at the “correct” addition formula of Einstein: u+v . 1 + uc·2v

(3.1)

In a simplified description one arrives at the general theory of relativity (Einstein, 1916) from the special theory of relativity, if one still considers the further “axiom” that there is no distinguished frame of reference. For this one must rework the theory with still far more difficult mathematical tools in such a way that also this axiom is respected in all conclusions. Viewed through logical glasses, one could say that Einstein developed both the special and the general theory of relativity by thinking through new axioms. He reworked the existing theories to the point where they were consistent with these axioms. This, of course, is not to claim an adequate rendering of Einstein’s historical approach. It is only about pointing out the specific status of the new assumptions by which one finally obtains a mathematical theory. With Hilbert such mathematical theories can always be grasped axiomatically (Hilbert, 1918).

3.3

The Gödel Universe

Gödel (1949b) could show for the field equations of general relativity that there is a model in which—theoretically—time travel is possible (see Fig. 3.1). Even if extensive and complicated mathematical calculations were necessary to prove the existence of this universe, the qualitative conception is a purely logical one; the axiomatic version of the theory of relativity in its field equations admits non-standard models; i.e. in particular, that it is formally incomplete. One could characterise Einstein’s procedure to the effect that he investigated what follows from the constancy of the speed of light. Gödel then examined in his turn what is compatible with the result of Einstein’s investigations. The incompleteness discovered thereby has no specific relation to the incompleteness of arithmetic theories, shown by Gödel in his first incompleteness theorem. This was obtained syntactically and has a generic character, i.e. it carries over to (recursive) extensions of the given theory. Here, we are rather dealing with the phenomenon known from absolute geometry (i.e. Euclidean geometry without the parallel axiom). This is incomplete, precisely because the parallel axiom can neither be proved nor disproved, and this has been shown model-theoretically, i.e. by the

5 Note

Minkowski’s astonishment at Einstein’s mathematical achievements, handed down by Max Born: Oh, Einstein, he was always skipping lectures—I wouldn’t have believed it of him. From Constance Reid comes the following quote of Minkowski (Reid, 1970, p. 112): Einstein’s presentation of his deep theory is mathematically awkward—I can say that because he got his mathematical education in Zurich from me.

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Fig. 3.1 Graphical illustration of a Gödel universe with “time travel” drawn in, taken from Németi et al. (2010)

construction of the non-Euclidean geometries. Correspondingly, the casually formulated sentence: “There is no time travel” has been proved to be independent of the theory of relativity by the construction of the Gödel universe.

3.4

Is Time Travel Possible?

The question whether time travel is possible with the discovery of the Gödel universe is badly posed in itself. The modal concept of possibility first requires a definition of the “possibility space”, i.e. of those properties and facts that shall be allowed to be changed. This question concerns us in the next section. However, even without a determination of the possibility space Gödel could already deny the posed question. He refers to the time and energy requirement for a time travel that is only theoretically possible in his universe; practically the spacetime ship would be too heavy, and a journey during which one could visit oneself in the own past would take much too long (respectively, would require acceleration that would hardly be acceptable for the human body). However, Gödel’s main argument against time travel is another one. Even if time travel should be possible in “his universe”, this does not mean, of course, that “our universe”, i.e. the one in which we live—and in which also Gödel lived—is such a universe. Of course, the theory also allows that it is exactly as we and Einstein have imagined it “actually”. The Gödel universe possesses a space (time) curvature that would be measurable. In our universe, however, the corresponding red shift cannot

3 The Philosophical Meaning of the Gödel Universe

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be observed. Therefore, we live—fortunately—in a “standard universe”, in which also the concept of time does not collapse.

3.5

The Philosophical Challenge: Physical Principles That Exclude the Gödel Universe

For Gödel, the actual philosophical challenge lies in the following dilemma. It may be true that “our universe” does not allow time travel, but the fact that the physical framework allows other universes in which there is no meaningful concept of time is philosophically more than unsatisfactory. In fact, the existence of time would then depend only “on the particular way in which matter and its motion are arranged in the world” (Gödel, 1956, p. 412). These are contingent properties of our world, which do not already necessarily follow from the physical framework. Thus, if we take up the question of possibility space raised above, for Gödel the physical principles incorporated in the general theory of relativity established by Einstein—which are expressed, in particular, in the field equations—are firmly given as necessary boundary conditions. These allow models (in the logical sense) in which “there can be no objective course of time”6. If these models can be excluded only by contingent properties—namely the matter distribution and its motion—this can “hardly be regarded as satisfactory” from the philosophical point of view7. The task, which Gödel discerns here, consists exactly in finding further physical principles, from which the non-existence of his universe would already follow, without the necessity to use contingent properties of our universe.8 We have to admit that it is not determined from the beginning what should be considered as a physical principle.9 The constancy of the speed of light, as well as the absence of an distinguished reference frame are certainly such principles; matter distribution and its motion in a concrete universe, however, are not. A trivial solution would be to put the existence of an objective course of time as a principle at the beginning. Such a petitio principii, of course, cannot satisfy.. Nevertheless, exactly this was tried by Stephen Hawking with his “Chronology Protection Conjecture” (Hawking, 1992), but just only as a conjecture and not as an axiom.

6 Gödel

(1956, p. 412). (1956, p. 412). 8 With this evaluation we are in contrast to Yourgrau, who writes: “First, he remarks that in an attempt to ‘specify’ the above definition of ‘cosmic’ or ‘absolute’ time, arbitrary elements come into play which can never be fully eliminated” (Yourgrau, 2002, p. 276, our emphasis). Here, we refer to footnote 9 in Gödel (1949a). There, however, Gödel only opposes a precise definition of absolute time in the sense of Jeans, and instead of “never” Gödel carefully speaks of “perhaps”, even if he subsequently considers the existence of such a precise definition as doubtful. Regardless of any precise definition of absolute time that Jeans may have had in mind, one certainly cannot imply to Gödel that all forms of physical completion of relativity could be accomplished only by arbitrary elements. This follows already from his own considerations, which will be briefly addressed below. 9 Gödel says this explicitly at the end of the quotation given in footnote 12. 7 Gödel

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Thus, one must credit Hawking that this conjecture is still to be proved by other principles.10 Until today there has been no satisfactory answer to the question of such other physical principles that exclude the Gödel universe. Gödel himself, for example, has considered the entropy theorem as a possible candidate. However, he himself has stated that this is also compatible with his universe (Gödel, 1956, p. 411).11 Gödel has also thrown a particularly interesting spotlight on the question raised about the distinction between physical principles and contingent properties:12 What used to be a practical difficulty in microphysics has now, as a result of the indeterminacy relation, become an impossibility in principle; and the same may one day occur for the difficulties based not on a “too small” but on a “too large”.

Thus, if quantum mechanics invalidates the (by now) known laws of relativity at (too) small scales, Gödel speculates on the possibility of a “cosmological physics”, which, for its part, would revise the known laws at (too) large scales.13 To our knowledge, however, this idea has not been pursued further to this day. Finally, we want to emphasise that Gödel carries out the philosophical speculation about the existence of time in a specific context, namely in what he calls idealist philosophy.14 This is prominently referred to in the title of the work Gödel (1956), and he writes, with regard to the problems to which the nature of time is exposed by relativity: “It seems […] that one obtains a clear proof of the view of those philosophers who, like Parmenides, Kant, and the modern idealists, deny the objectivity of change and regard it as an illusion or as an appearance which we owe to our particular mode of perception”. On closer examination of Gödel’s argumentation in this article, one can see that he does not primarily support the (only apparent?) “clear proof”. He merely argues against the fact that such a proof can already be brought down by the fact that one can save the existence of time by recourse to the contingent properties of our universe.15 Although he does not comment on whether he still considers a

10 From

this perspective, Yourgrau’s harsh criticism of Hawking (Yourgrau, 2005, pp. 8 and 136) is both incomprehensible and unjustified. 11 This remark can be found as an “addition by the author in the German translation to footnote 14” only in the German edition of the article. 12 This can be found again in an “addition of the author to footnote 11 of the German edition”, which is introduced as follows (Gödel, 1956, p. 411): A second reason to exclude the above universes a priori could be found in the possibility of a “telegraphing into one’s own past”. However, the practical difficulties arising in this case are hardly likely to be less [than in the case of the previously discussed time travel]. By the way, the border between practical and principal difficulties is by no means immovable. 13 Here, one may well feel reminded of the Aristotelian distinction between sublunary physics and celestial mechanics. 14 The terminology used by Gödel is not unproblematic. In which concrete sense the presented position should correctly be called idealistic may be discussed. The tension is increased by the fact that Gödel himself is usually regarded as a realist; see, e.g. Yourgrau (2005, p. 171 f.). 15 He explicitly attributes such a rescue attempt to James Jeans (1935) (Gödel, 1956, p. 408).

3 The Philosophical Meaning of the Gödel Universe

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non-idealistic philosophy to be viable as an alternative;16 he clearly looks after the possibility of guaranteeing the existence of time through deeper physical principles. To find such principles is the philosophical challenge that Gödel has left us.

References Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Annalen der Physik und Chemie, 17, 891–921. Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354(7), 769–822. Gödel, K. (1930). Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik, 37, 349–360. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik, 38, 173–198. Gödel, K. (1933). Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34–38. Gödel, K. (1938). The consistency of the axiom of choice and of the generalized continuumhypothesis. Proceedings of the National Academy of Sciences, 24, 556–557. Gödel, K. (1949a). A remark about the relationship between relativity theory and idealistic philosophy. In P. A. Schilpp (Ed.), Albert Einstein, Philosopher-Scientist (pp. 555–562). Harper & Row. Gödel, K. (1949b). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 21(3), 447– 450. Gödel, K. (1956). Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der indealistischen Philosophie. In P. A. Schlipp (Ed.), Albert Einstein als Philosoph und Naturforscher (pp. 406–412). Kohlhammer (Translation of Gödel, 1949a with important additions in two footnotes). Gödel, K. (1958). Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, 12, 280–287. Gödel, K. (1995). Some observations about the relationship between theory of relativity and Kantian philosophy. In S. Feferman, et al. (Eds.), Kurt Gödel—Collected works (Vol. III, pp. 230–259). Oxford University Press (Drafts from Gödel’s Nachlass for the article Gödel, 1949a). Hawking, S. W. (1992). Chronology protection conjecture. Physical Review D, 46(2), 603–611. Hilbert, D. (1918). Axiomatisches Denken. In Mathematische Annalen 78.3/4 (pp. 405– 415). (Talk delivered at the Swiss Mathematical Society on September 11, 1917). Jeans, J. (1935). Man and the universe (pp. 22–23). Sir Halley Steward Lecture. Morgenstern, O. (2002). Brief an Bruno Kreisky. In E. Köhler, et al. (Eds.), Kurt Gödel – Wahrheit und Beweisbarkeit (Vol. I: Dokumente und historische Analysen, pp. 23–24). öbv & hpt. Németi, I., Madarász, J., Andréka, H., & Andai, A. (2010). Visualizing some ideas about Gödel-type rotating universes. In M. Scherfner & M. Plaue (Eds.), Gödel-type spacetimes: History and new developments (Vol. X. Collegium Logicum, pp. 77–127). Kurt Gödel Society. Reid, C. (1970). Hilbert. Springer. Stein, H. (1990). Introduction note to 1949a. In S. Feferman, et al. (Eds.), Kurt Gödel—Collected works (Vol. II. “1949a” ist der Artikel Gödel, 1949a, pp. 199–201). Oxford University Press.

16 Howard

Stein (1990), in his introduction to the reprint of the article Gödel (1949a), explicitly regrets the lack of more material that would shed light on Gödel’s philosophical position, especially vis-á-vis Kant. We at least have more detailed drafts of this article, in which “idealistic philosophy” was replaced by “Kantian philosophy” in the title, and which are dated 1946–49 (Gödel, 1995).

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Yourgrau, P. (2002). Philosophische Betrachtungen zu Gödels Kosmologie. In B. Buldt, et al. (Eds.), Kurt Gödel – Wahrheit und Beweisbarkeit (Vol. II: Kompendium zum Werk, pp. 269–288). öbv & hpt. Yourgrau, P. (2005). A world without time. Basic Books. Reinhard Kahle holds the Carl Friedrich von Weizsäcker chair for theory and history of science at the University of Tübingen. Previously he was Professor of Mathematics in Coimbra and at the Universidade Nova de Lisboa. He is a member of the Académie Internationale de Philosophie des Sciences. His research interests cover proof theory and the history and philosophy of modern logic, especially of the Hilbert school and its periphery. Together with colleagues he has edited numerous books and special issues, among others: Gentzen’s Centenary: The quest for consistency (Springer, 2015) and The Legacy of Kurt Schütte (Springer, 2020), both together with Michael Rathjen.

4

Die philosophische Bedeutung des Gödel-Universums Reinhard Kahle

In seiner Zeit am Institute of Advanced Study wurde Kurt Gödel ein enger Freund von Albert Einstein und hat sich insbesondere mit der Relativitätstheorie beschäftigt. Ein Ergebnis dieser Untersuchung war die Entdeckung des sogenannten GödelUniversums1, einem Modell der Einsteinschen Feldgleichungen der allgemeinen Relativitätstheorie, in dem sich keine absolute Zeit definieren lässt – theoretisch wären in einem solchen Universum Zeitreisen möglich. Diesem Ergebnis werden z. T. weitreichende philosophische Konsequenzen zugeschrieben. Palle Yourgrau argumentiert etwa, dass Gödel geschlossen habe, dass es selbst in unserer Welt keine Zeit gäbe.2 Tatsächlich finden sich in Gödels Schriften nur sehr viel vorsichtigere Konklusionen. In dieser Arbeit wollen wir zuerst die „Entdeckung“ des Gödelschen Universums im Kontext seiner Arbeit als Logiker rekonstruieren. Auf der Grundlage dieser Rekonstruktion lässt sich anschließend die spezifisch philosophische Herausforderung erkennen, die Gödel gesehen hat: Unsere kosmologischen Theorien sollten mithilfe physikalischer Prinzipien zu verstärken sein, sodass die Nichtexistenz von Gödel-Universen – und damit die Existenz der Zeit – nicht „von der besonderen Weise abhängt, in der die Materie und ihre Bewegung in der Welt angeordnet sind.“ 3

1 Genau

genommen handelt es sich um eine ganze Klasse von Universen; die Singularform steht hier als Kollektivum. 2 „Können wir wirklich daraus, dass die Zeit in einem nur möglichen Universum nicht existiert, ableiten, dass sie auch in unserer Welt nicht existiert? Kurz gesagt, ja, das können wir! So zumindest argumentiert Gödel.“ (Yourgrau, 2005, S. 135). 3 Vergleiche Gödel (1956, S. 412). R. Kahle (B) Universität Tübingen, Tübingen, Deutschland E-Mail: [email protected] © Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_4

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4.1

R. Kahle

Gödel als Logiker

Das herausragenste Resultat von Kurt Gödel in der Liste seiner wissenschaftlichen Leistungen sind ohne Zweifel die Unvollständigkeitssätze (Gödel, 1931). Davor hatte er schon die (semantische) Vollständigkeit der vorliegenden Axiomatisierung der Prädikatenlogik gezeigt (Gödel, 1930), und für die technische Etablierung seiner Unvollständigkeitssätze die rekursiven Funktionen eingeführt. Anschließend gelang ihm der Beweis der Verträglichkeit des Auswahlaxioms und der Kontinuumshypothese mit der Axiomatisierung der Mengenlehre nach Zermelo und Fraenkel (Gödel, 1938). Dazu gesellen sich viele „kleinere“ Resultate, wie eine Form der Doppelnegationsinterpretation für die Peano-Arithmetik in der Heyting-Arithmetik (Gödel, 1933) oder die Definition seines System von Funktionalen höheren Types T , das einen nicht-finiten Widerspruchsfreiheitsbeweis der Arithmetik erlaubt (Gödel, 1958). Alle diese Arbeiten sind der mathematischen Logik zuzurechnen, einer Disziplin, die nicht zuletzt durch Gödels Beiträge ihre moderne Ausgestaltung erfahren hat. Es ist bekannt, dass Einstein gegen Ende seines Lebens gesagt hat, er gehe nur noch zum Institut in Princeton, um mit Gödel zu diskutieren.4 Gegenstand der Gespräche wird sicherlicher auch die Relativitätstheorie gewesen sein und insbesondere Gödels berühmtes Universum, das unseren Zeitbegriff infrage zu stellen scheint. In der folgenden Reflexion der philosophischen Konsequenzen, die Gödel aus der Entdeckung seines Universums zog, ist es aber wichtig, sich vor Augen zu halten, dass auch diese Entdeckung vor dem Hintergrund der logischen Kompetenz von Gödel betrachtet werden sollte.

4.2

Einsteins Relativitätstheorie

Albert Einstein hatte zuerst die heute sogenannte spezielle Relavitätstheorie aufgestellt (Einstein, 1905). Zwar war Einstein selbst kein Logiker. Aber wir können die spezielle Relativitätstheorie durchaus mithilfe gewisser logischer Vorgaben gewinnen. Die zentrale Idee dabei ist: Man kann die Erkenntnis, dass es eine absolute Lichtgeschwindigkeit c gibt, die zugleich die physikalisch höchste Geschwindigkeit darstellt, als eine Art Axiom aufstellen, sodass alle weiteren Theoriebestandteile einer physikalischen Beschreibung unseres Universums mit diesem Axiom verträglich sein müssen.

4 Oskar

Morgenstern in einem Brief an Bruno Kriesky, den österreichischen Bundesminister für Auswärtige Angelegenheiten, vom 25. Okt. 1965, (Morgenstern, 2002): Einstein hat mir öfters gesagt, dass er in den letzten Jahres seines Lebens immer wieder Gödels Gesellschaft gesucht hat, um mit ihm diskutieren zu können. Einmal sagte er zu mir, dass seine eigene Arbeit nicht mehr viel bedeute, dass er lediglich ins Institutsgebäude käme, „um das Privileg zu haben, mit Gödel zu Fuß nach Hause gehen zu dürfen.“

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Dies lässt sich an der Addition von Geschwindigkeiten veranschaulichen. Nach der klassischen Physik lassen sich Geschwindigkeiten beliebig addieren. Sei dafür etwa ein Zug mit der Geschwindigkeit u unterwegs. Wenn eine Person in dem Zug mit Geschwindigkeit v nach vorne geht, beträgt ihre von außen betrachtete Geschwindigkeit u + v. Jetzt halte ein Passagier eine Taschenlampe in Fahrrichtung. Das Licht dieser Taschenlampe würde sich dann, von außerhalb des Zuges betrachtet, nach klassischer Auffassung mit der Geschwindigkeit u + c fortbewegen, was aber größer als c wäre. Wenn c aber eine absolute Höchstgeschwindigkeit sein soll, muss die Addition in gewisser Weise anders berechnet werden. Es bedarf jetzt aufwendiger mathematischer Überlegungen,5 um mit Einstein auf die „richtige“ Additionsformel zu kommen: u+v . (4.1) 1 + uc·2v In einer vereinfachten Beschreibung kommt man von der speziellen Relativitätstheorie zur allgemeinen Relativitätstheorie (Einstein, 1916), wenn man noch das weitergehende „Axiom“ berücksichtigt, dass es kein ausgezeichnetes Bezugssystem gibt. Dazu muss man die Theorie mit noch weit schwierigerem mathematischem Rüstzeug so umarbeiten, dass auch dieses Axiom in allen Folgerungen respektiert wird. Durch eine logische Brille betrachtet könnte man sagen, dass Einstein sowohl die spezielle als auch die allgemeine Relativitätstheorie dadurch entwickelt hat, dass er neue Axiome „zu Ende gedacht hat“: Er hat die bestehenden Theorien soweit umgearbeitet, dass sie mit diesen Axiomen in Einklang stehen. Damit soll natürlich kein Anspruch auf eine adäquate Wiedergabe von Einsteins historischem Vorgehen verbunden sein. Es geht nur darum, den spezifischen Status der neuen Annahmen herauszustellen, durch den man letztlich eine mathematische Theorie erhält. Und seit Hilbert lassen sich solche mathematischen Theorien auch immer axiomatisch fassen (Hilbert, 1918).

4.3

Das Gödel-Universum

Gödel (1949b) konnte für die Feldgleichungen der allgemeinen Relativitätstheorie zeigen, dass es ein Modell gibt, in dem – theoretisch – Zeitreisen möglich sind (siehe Abb. 4.1). Auch wenn zum Nachweis der Existenz dieses Universums umfangreiche und komplizierte mathematische Rechnungen nötig waren, ist die qualitative Konzeption

5 Man beachte dazu Minkowskis Erstaunen über Einsteins mathematische Leistungen, das von Max

Born überliefert wurde: Ach der Einstein, der schwänzte immer die Vorlesungen – dem hätte ich das gar nicht zugetraut. Von Constance Reid stammt das nur auf englisch erhaltene Zitat (Reid, 1970, S. 112): Einstein’s presentation of his deep theory is mathematically awkward – I can say that because he got his mathematical education in Zürich from me.

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Abb. 4.1 Graphische Illustration eines Gödel-Universums mit eingezeichneter „Zeitreise“, entnommen aus Németi et al. (2010)

eine rein logische: Die axiomatische Fassung der Relatitätstheorie in ihren Feldgleichungen lässt Nicht-Standard-Modelle zu; d. h. insbesondere, dass sie formal unvollständig ist. Man könnte Einsteins Vorgehen dahingehend charakterisieren, dass er untersucht hat, was aus der Konstanz der Lichtgeschwindigkeit folgt. Gödel hat dann seinerseits untersucht, was mit dem Ergebnis der Einsteinschen Untersuchungen verträglich ist. Die dabei entdeckte Unvollständigkeit hat keine spezfische Beziehung zur Unvollständigkeit arithmetischer Theorien, die Gödel in seinem ersten Unvollständigkeitssatz gezeigt hatte. Dieser wurde syntaktisch gewonnen und hat einen generischen Charakter, d. h., er überträgt sich auf (rekursive) Erweiterungen der gegebenen Theorie. Hier haben wir es eher mit dem Phänomen zu tun, das aus der absoluten Geometrie (d. i. die euklidische Geometrie ohne das Parallelenaxiom) bekannt ist. Diese ist unvollständig, eben weil sich das Parallelenaxiom weder beweisen noch widerlegen lässt, und das wurde modelltheoretisch, d. h. durch die Konstruktion der nichteuklidischen Geometrien, gezeigt. Entsprechend ist der salopp formulierte Satz: „Es gibt keine Zeitreisen.“ durch die Angabe des Gödel-Universums als unabhängig von der Relativitätstheorie erwiesen worden.

4.4

Sind Zeitreisen möglich?

Die Frage, ob mit der Entdeckung des Gödel-Universums Zeitreisen möglich seien, ist für sich genommen schlecht gestellt. Der modale Begriff der Möglichkeit erfordert

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zuerst eine Festlegung des „Möglichkeitsraumes“, d. h. derjenigen Eigenschaften und Sachverhalte, die zu verändern erlaubt sein soll. Diese Frage wird uns im nächsten Abschnitt beschäftigen. Doch auch ohne Festlegung des Möglichkeitsraumes konnte Gödel die gestellte Frage bereits verneinen. Er verweist auf den Zeit- und Energiebedarf für eine in seinem Universum eben nur theoretisch mögliche Zeitreise; praktisch wäre das Raumzeitschiff zu schwer und eine Reise, bei der man sich selbst in der eigenen Vergangenheit besuchen könnte, würde viel zu lange dauern (beziehungsweise eine Geschwindigkeit erfordern, die dem menschlichen Körper kaum zuzumuten wäre). Gödels Hauptargument gegen Zeitreisen ist allerdings ein anderes: Auch wenn in „seinem Universum“ Zeitreisen möglich sein sollten, heißt das natürlich noch nicht, dass „unser Universum“, d. h. das, in dem wir leben – und in dem auch Gödel lebte – ein solches Universum ist. Die Theorie lässt natürlich auch zu, dass es genau so ist, wie wir und Einstein uns das „eigentlich“ vorgestellt haben. Das Gödel-Universum besitzt eine Raum(zeit)krümmung, die messbar wäre. In unserem Univerums lässt sich die entsprechende Rotverschiebung aber nicht nachweisen. Demnach leben wir also doch – glücklicherweise – in einem „Standarduniversum“, in dem auch der Zeitbegriff nicht kollabiert.

4.5

Die philosophische Herausforderung: Physikalische Prinzipien, die das Gödel-Universum ausschließen

Die eigentliche philosophische Herausforderung liegt für Gödel in dem folgenden Dilemma: Es mag zwar richtig sein, dass „unser Universum“ keine Zeitreisen zulässt, aber der Umstand, dass die physikalischen Rahmenbedingungen andere Universen zulassen, in denen es keinen sinnvollen Zeitbegriff gibt, ist philosophisch mehr als unbefriedigend. Tatsächlich würde dann die Existenz der Zeit nur „von der besonderen Weise abhäng[en], in der die Materie und ihre Bewegung in der Welt angeordnet sind“ (Gödel, 1956, S. 412). Das sind aber kontingente Eigenschaften unserer Welt, die sich nicht schon notwendig aus den physikalischen Rahmenbedingungen ergeben. Wenn wir also die oben aufgeworfene Frage nach dem Möglichkeitsraum aufgreifen, sind für Gödel die in der von Einstein aufgestellten allgemeinen Relativitätstheorie inkorporierten physikalischen Prinzipien – die sich nicht zuletzt in den Feldgleichungen äußern – als notwendige Randbedingungen fest gegeben. Diese lassen aber Modelle (im logischen Sinne) zu, in denen es „keinen objektiven Zeitverlauf geben kann“ 6. Wenn sich diese Modelle nur durch kontingente Eigenschaften – nämlich die Materieverteilung und ihre Bewegung – ausschließen lassen, kann das aus philosophischer Sicht „kaum als befriedigend betrachtet werden“ 7.

6 Gödel 7 Gödel

(1956, S. 412). (1956, S. 412).

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Die Aufgabe, die Gödel hier sieht, besteht also gerade darin, weitere physikalische Prinzipien zu finden, aus denen die Nichtexistenz seines Universums bereits folgen würde, ohne dass man kontingente Eigenschaften unseres Universums heranziehen müßte.8 Wir müssen dabei zugeben, dass es nicht von vornherein festgelegt ist, was als physikalisches Prinzip gelten sollte.9 Die Konstanz der Lichtgeschwindigkeit als auch das Fehlen eines ausgezeichneten Bezugssystems sind sicherlich solche Prinzipien; die Materieverteilung und ihre Bewegung in einem konkreten Universum dagegen nicht. Eine banale Lösung wäre, die Existenz eines objektiven Zeitverlaufs als Prinzip an den Anfang zu stellen. Eine solche Petitio Principii kann natürlich nicht befriedigen. Dennoch wurde genau dies von Stephen Hawking mit seiner „Cronology Protection Conjecture“ versucht (Hawking, 1992), aber eben nur als Vermutung und nicht als Axiom. Damit muss man Hawking zugutehalten, dass diese Vermutung erst noch durch andere Prinzipien zu erweisen ist.10 Bis heute gibt es allerdings noch keine befriedigende Antwort auf die Frage nach solchen anderen physikalischen Prinzipien, die das Gödel-Universum ausschließen. Gödel selbst hat z. B. den Entropiesatz als möglichen Kandidaten ins Auge gefasst. Er hat aber selbst festgestellt, dass dieser auch mit seinem Universum verträglich ist (Gödel, 1956, S. 411).11 Ein besonders interessantes Schlaglicht hat Gödel auch auf die angesprochene Frage nach der Unterscheidung von physikalischen Prinzipien und kontingenten Eigenschaften geworfen:12

8 Mit

dieser Bewertung stehen wir im Gegensatz zu Yourgrau, der schreibt: „Zunächst bemerkt er, dass bei einem Versuch, die obige Definition der kosmischen‘ oder absoluten‘ Zeit zu präzisieren, ’ ’ willkürliche Elemente ins Spiel kommen, welche niemals vollkommmen beseitig werden können“ (Yourgrau, 2002, S. 276, unsere Hervorhebung). Dabei wird Bezug genommen auf die Fußnote 9 in Gödel (1949a). Dort wendet sich Gödel aber lediglich gegen eine precise definition der absoluten Zeit im Sinne Jeans’, und statt „niemals“ spricht Gödel vorsichtig von „perhaps“, auch wenn er anschließend die Existenz einer solchen präzisen Definition für „doubtful“ hält. Unabhängig von einer präzisen Definition der absoluten Zeit, die Jeans im Sinn gehabt haben mag, kann man Gödel sicher nicht unterstellen, dass alle Formen einer physikalischen Vervollständigung der Relativitätstheorie nur durch willkürliche Elemente erfolgen könnten. Das folgt allein schon aus seinen eigenen Versuchen, die im Folgenden kurz angesprochen werden. 9 Gödel sagt das explizit am Ende des in Fußnote 12 gegebenen Zitats. 10 Aus dieser Perspektive ist die harsche Kritik Yourgraus an Hawking (Yourgrau, 2005, S. 15 und 160) gleichermaßen unverständlich wie auch ungerechtfertigt. 11 Diese Bemerkung findet sich als „Zusatz des Autors bei der deutschen Übersetzung zu Fußnote 14“ nur in der deutschen Ausgabe des Artikels. 12 Dies findet sich wieder in einem „Zusatz des Autors zu Fußnote 11 der deutschen Auflage“, der wie folgt eingeleitet wird (Gödel, 1956, S. 411): Ein zweiter Grund, die obigen Universen a priori auszuschließen, könnte in der Möglichkeit eines „Telegraphierens in die eigene Vergangenheit“ gefunden werden. Doch dürften die dabei auftretenden praktischen Schwierigkeiten kaum geringer sein [als bei den zuvor diskutierten Zeitreisen]. Die Grenze zwischen praktischen und prinzipiellen Schwierigkeiten ist übrigens durchaus nicht unverrückbar.

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33

Was früher in der Mikrophysik eine praktische Schwierigkeit war, ist heute, infolge der Unbestimmtheitsrelation, eine prinzipielle Unmöglichkeit geworden; und dasselbe könnte eines Tages auch für die Schwierigkeiten eintreten, die nicht auf einem „zu klein“, sondern auf einem „zu groß“ beruhen.

Wenn also die Quantenmechanik die (mittlerweile) bekannten Gesetze der Relativitätstheorie im (zu) Kleinen außer Kraft setzt, spekuliert Gödel hier über die Möglichkeite einer „kosmologischen Physik“ die ihrerseits die bekannten Gesetze im (zu) Großen revidieren würde.13 Diese Idee ist nach unserer Kenntnis aber bis heute nicht weiter verfolgt worden. Abschließend wollen wir noch hervorheben, dass Gödel die philosophische Spekulation über die Existenz der Zeit in einem spezifischen Kontext vollzieht, nämlich in dem der von ihm so bezeichneten idealistischen Philosophie.14 Darauf wird prominent im Titel der Arbeit Gödel (1956) Bezug genommen, und es heißt im Hinblick auf die Probleme, denen das Wesen der Zeit durch die Relativitätstheorie ausgesetzt ist: „Es scheint […], dass man einen eindeutigen Beweis für die Ansicht jener Philosophen erhält, die, wie Parmenides, Kant und die modernen Idealisten, die Objektivität des Wechsels leugnen und diesen als eine Illusion oder als eine Erscheinung betrachten, die wir unserer besonderen Art der Wahrnehmung verdanken.“ Bei genauer Betrachtung der Argumentation Gödels in diesem Artikel kann man feststellen, dass er nicht vorrangig den (nur scheinbaren?) „eindeutigen Beweis“ unterstützt, sondern lediglich dagegen argumentiert, dass ein solcher Beweis schon dadurch zu Fall gebracht werden kann, dass man mit Rückgriff auf die kontingenten Eigenschaften unseres Universums die Existenz der Zeit retten könne.15 Zwar äußert er sich nicht dazu, ob er eine nicht-idealistische Philosophie als Alternative noch für tragfähig hält,16 aber er sieht klar die Möglichkeit, die Existenz der Zeit durch tieferliegende physikalische Prinzipien zu garantieren. Solche Prinzipien aufzusuchen ist die philosophische Herausforderung, die uns Gödel hinterlassen hat.

13 Hier darf man sich durchaus an die Aristotelische Unterscheidung von sublunarer Physik und Himmelsmechanik erinnert fühlen. 14 Die von Gödel verwendete Begrifflichkeit ist nicht unproblematisch. In welchem konkreten Sinne die dargestellte Position zu recht idealistisch genannt werden sollte, darf diskutiert werden. Das Spannungsfeld erhöht sich noch dadurch, dass Gödel selbst in der Regel als Realist betrachtet wird; siehe z. B. Yourgrau (2005, S. 126 f.). 15 Einen solchen Rettungversuch schreibt er explizit James Jeans (1935) zu (Gödel, 1956, S. 408). 16 Howard Stein (1990) bedauert in seiner Einleitung zum Nachdruch des Artikels Gödel (1949a) ausdrücklich das Fehlen von mehr Material, das Aufklärung über die philosophische Position Gödels, gerade gegenüber Kant, liefern würde. Wir haben wenigstens ausführlichere Entwürfe zu diesem Artikel, bei denen im Titel „idealistic philosophy“ durch „Kantian philosophy“ ersetzt war und die auf 1946–49 datiert werden (Gödel, 1995).

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Literatur Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Annalen der Physik und Chemie, 17, 891–921. Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354(7), 769–822. Gödel, K. (1930). Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik, 37, 349–360. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik, 38, 173–198. Gödel, K. (1933). Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34–38. Gödel, K. (1938). The consistency of the axiom of choice and of the generalized continuumhypothesis. Proceedings of the National Academy of Sciences, 24, 556–557. Gödel, K. (1949a). A remark about the relationship between relativity theory and idealistic philosophy. In P. A. Schilpp (Hrsg.), Albert Einstein, Philosopher-Scientist (S. 555–562). Harper & Row. Gödel, K. (1949b). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 21(3), 447–450. Gödel, K. (1956). Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der idealistischen Philosophie. In P. A. Schlipp (Hrsg.), Albert Einstein als Philosoph und Naturforscher (S. 406–412). Kohlhammer (Übersetzung von Gödel, 1949a mit wichtigen Zusätzen in zwei Fußnoten). Gödel, K. (1958). Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, 12, 280–287. Gödel, K. (1995). Some observations about the relationship between theory of relativity and Kantian philosophy. In S. Feferman, et al. (Hrsg.), Kurt Gödel – Collected works (Bd. III, S. 230–259). Oxford University Press (Entwürfe aus Gödels Nachlaß für den Artikel Gödel, 1949a). Hawking, S. W. (1992). Chronology protection conjecture. Physical Review D, 46(2), 603–611. Hilbert, D. (1918). Axiomatisches Denken. In Mathematische Annalen 78.3/4 (S. 405–415). Vortrag vom 11. September 1917 gehalten vor der Schweizerischen Mathematischen Gesellschaft. Jeans, J. (1935). Man and the universe (S. 22–23). Sir Halley Steward Lecture. Morgenstern, O. (2002). Brief an Bruno Kreisky. In E. Köhler, et al. (Hrsg.), Kurt Gödel – Wahrheit und Beweisbarkeit (Bd. I: Dokumente und historische Analysen, S. 23–24). öbv & hpt. Németi, I., Madarász, J., Andréka, H., & Andai, A. (2010). Visualizing some ideas about Gödel-type rotating universes. In M. Scherfner & M. Plaue (Hrsg.), Gödel-type spacetimes: History and new developments (Bd. X. Collegium Logicum, S. 77–127). Kurt Gödel Society. Reid, C. (1970). Hilbert. Springer. Stein, H. (1990). Introduction note to 1949a. In S. Feferman, et al. (Hrsg.), Kurt Gödel – Collected works (Bd. II. „1949a“ ist der Artikel Gödel, 1949a, S. 199–201). Oxford University Press. Yourgrau, P. (2002). Philosophische Betrachtungen zu Gödels Kosmologie. In B. Buldt, et al. (Hrsg.), Kurt Gödel – Wahrheit und Beweisbarkeit (Bd. II: Kompendium zum Werk, S. 269– 288). öbv & hpt. Yourgrau, P. (2005). Gödel, Einstein und die Folgen. Beck.

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Reinhard Kahle ist Inhaber des Carl Friedrich von Weizsäcker-Lehrstuhls für Wissenschaftstheorie und Wissenschaftsgeschichte an der Universität Tübingen. Zuvor war er Professor für Mathematik in Coimbra und an der Universidade Nova de Lisboa. Er ist Mitglied der Acad´emie Internationale de Philosophie des Sciences. Seine Forschungsinteressen umfassen die Beweistheorie sowie die Geschichte und Philosophie der modernen Logik, insbesondere der Hilbert-Schule und ihres Umfelds. Zusammen mit Kollegen hat er zahlreiche Bücher und Sonderhefte herausgegeben, unter anderem: Gentzen’s Centenary: The quest for consistency (Springer, 2015) und The Legacy of Kurt Schütte (Springer, 2020), beide zusammen mit Michael Rathjen.

5

Gödel’s Temporal Idealism: A Reply to Prof. Kahle Palle Yourgrau

In his Kurt Gödel Award 2021 essay, “The Philosophical Meaning of the Gödel Universe,” Prof. Kahle takes a fresh look at the philosophical ramifications of Gödel’s cosmology and helps clarify what Gödel’s intentions were and what the significance of his arguments is. I have several reservations, however, concerning Kahle’s discussion. To begin with a minor point, he states in his opening that, During his time at the Institute for Advanced Study, Kurt Gödel became a close friend of Albert Einstein, and in particular studied the theory of relativity.

This gives the misleading impression that it was due to his friendship with Einstein that Gödel studied relativity, although it is well known that Gödel’s initial focus in his university studies was on physics. How deep his knowledge of physics was can be gleaned from the fact that he suggested in a letter to John von Neumann in 1934, “with regard to his stay at the IAS […] that he would gain a great deal from the opportunity to hear von Neumann lecture or direct a seminar on quantum mechanics, a subject in which he had a “lively interest”…To Veblen, too, Gödel expressed his interest in the seminar …”1 This was long before he became a friend of Einstein. Further, Gödel himself said that the inspiration for his cosmology was not his friendship with Einstein, but rather his philosophical interest in (the reality of) time, in particular, with respect to Kant’s philosophy: “My work on rotating universes,” said Gödel, “was not stimulated by my close association with Einstein. It came from my interest

1 John

Dawson (1997, 92).

P. Yourgrau (B) Department of Philosophy, Brandeis University, Waltham, MA, USA e-mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_5

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in Kant’s views. In what was said about Kant and relativity theory, one only saw the difference, nobody saw the agreement of the two. What is more important is the nature of time. In relativity there is no passage of time, it is coordinated with space.”2 Solomon Feferman notes, further, that, at that time, “Einstein himself was preoccupied, as he had been for a long time, with constructing a unified field theory, a project about which Gödel was skeptical.”3 There is more about Kahle’s take on the relationship between Einstein and Gödel that I find potentially misleading. The standard view, I take it, is that in his writings on relativity, Gödel the logician turned into a part time physicist (which took his colleagues at the IAS by surprise). By contrast, Kahle chooses to view the relationship “through logical glasses,” and concludes that it was not Gödel who became a part time physicist, but rather Einstein, although not a logician, who nevertheless acted like one. Thus, Kahle writes: It is true that Einstein himself was not a logician. But we can obtain the special theory of relativity absolutely with the help of certain logical guidelines. The central idea is: the knowledge that there is an absolute speed of light c, which is at the same time the physically highest speed, can be set up as a kind of axiom, so that all further theory components of a physical description of our universe must be compatible with this axiom.

Kahle says similarly that Einstein obtained the general theory of relativity by deducing, logically, consequences that follow from another basic axiom: Viewed through logical glasses, one could say that Einstein developed both the special and the general theory of relativity by thinking through new axioms: He reworked the existing theories to the point where they were consistent with these axioms.

As for the Gödel universes, Kahle says that, [T]he qualitative conception is a purely logical one: the axiomatic version of the theory of relativity in its field equations admits non-standard models; i.e., in particular, that it is formally incomplete. One could characterize Einstein’s procedure to the effect that he investigated what follows from the constancy of the speed of light. Gödel then examined in his turn what is compatible with the result of Einstein’s investigations.

There is something to this way of viewing things, to be sure, but one must be careful not to take things too literally. To begin with, Kahle neglects other principles Einstein took to be basic in developing the special theory of relativity (or STR), including a kind of verificationism that banished the aether hypothesis and the idea that there are no privileged reference or inertial frames. Moreover, Einstein’s verificationism was a philosophical assumption. As the noted physicist John S. Bell pointed out, “since it is experimentally impossible to say which of two uniformly moving systems is really at rest, Einstein declares the notion ‘really resting’ and ‘really moving’ as

2 Hao

Wang (1996, 88). Feferman (1986, 13 f.).

3 Solomon

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meaningless […] [whereas] Lorentz […] preferred the view that there is indeed a state of real rest, defined by the ‘aether’, even though the laws of physics conspire to prevent us identifying it experimentally. The facts of physics do not oblige us to accept one philosophy rather than the other.”4 Indeed, Hao Wang told me that, in conversation, Gödel expressed great interest in the Lorentz vs. Einstein interpretation of the formalism of STR. Unsurprisingly, given that he made much the same comment we have just seen Bell make: “[I]n perfect conformity with Kant, the observational results by themselves [on which STR rests] really do not force us to abandon Newtonian time and space as objective realities, but only the observational results together with certain principles, e.g., the principle that two states of affairs which cannot be distinguished by observation are also objectively equal.”5 Naturally, it was not Einstein’s competence as a “logician” that inspired him to take those principles as basic, as “axioms”, when formulating STR, but rather his physical (and philosophical) intuition. Moreover, in constructing the general theory of relativity (or GTR), his physical (and philosophical) intuition had modified. More generally, of course, logical competence concerns derivation from axioms, not the adoption of the axioms themselves; and even “derivation” from physical axioms is not a purely logical affair. Einstein’s former student, Hans Reichenbach, the well-known logical positivist, wrote a book entitled, Axiomatization of the Theory of Relativity.6 Einstein, himself, of course, was not operating with such an axiomatization from which he was merely deriving “theorems.” Indeed, Gödel noted “the lack of interest of physicists in the axiomatic method […].”7 Similarly, Gödel’s construction of the Gödel universes involved a great deal more than logical competence. In addition to the obvious fact that Gödel’s construction depended on an advanced knowledge of physics, there is the fact that, as Feferman said, “in this work, Gödel brought to bear mathematical techniques and physical intuitions that one who was familiar only with his papers in logic would not have expected. The mathematics, however, harks back to his brief contribution to differential geometry in the 1930’s, as well as his studies of analysis with Hahn in Menger’s colloquium.”8 In sum, it was not primarily Gödel’s prowess as a logician that enabled him to construct the Gödel universes, but rather his surprising abilities as a physicist and pure mathematician. These reservations, however, are relatively minor concerns. More serious is the question of what Gödel believed about the reality of time, i.e., the question of Gödel’s temporal idealism. With respect to this question, Kahle states in the opening of his essay that, Palle Yourgrau (2005), for example, argues that Gödel concluded that there is no time even in our world. In fact, only much more cautious conclusions can be found in Gödel’s writings.

4 John

S. Bell (1989, 77). (1946/9-B2, 245–246); brackets added. 6 Hans Reichenbach (1969). 7 Wang (1996, 307). 8 Feferman, op. cit., 14. 5 Gödel

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This is a peculiar way of putting things. I am far from alone in making this suggestion. Moreover, Kahle is not alone in rejecting it. John Byron Manchak, for example, also suggests that although Gödel’s reasoning “is sometimes interpreted to be an argument that time in our universe is ideal […] this reading seems to be a bit strong.”9 In any case, Kahle does provide textual evidence that at least in Kurt Gödel (1949), Gödel was indeed rather cautious in his formulations, and that is a fact worth bringing out. To be sure, Kahle acknowledges that Gödel says early in Gödel (1949) that from special relativity (to quote Gödel) “it seems that one obtains an unequivocal proof for the view of those philosophers, who, like Parmenides, Kant and the modern idealists, deny the objectivity of change and consider change as an illusion”, but he adds (to my mind, without any basis) the words “(only apparent)” before “an unequivocal proof”, thus changing Gödel’s claim. What Gödel is saying here, surely, is that on the basis of special relativity alone, one obtains an unequivocal proof (not an “only apparent” unequivocal proof) that change (hence time) is an illusion. Gödel goes on to add, however, in Gödel (1949), that of course, one needs also to examine the general theory of relativity before reaching a final conclusion about the objective existence of time. No doubt, that explains Gödel’s use of the words, “it seems”.10 It is also important, here, to recall the fact that Gödel tended in much of this writing (formal and informal) to be extremely, perhaps, at times, excessively cautious. Witness, the fact, for example, that though his incompleteness theorem is known for its profound results concerning the relationship between proof and truth, when he published his theorem in 1931, he avoided mention of the controversial notion of (mathematical) truth. As Solomon Feferman remarks, “[T]he concept of truth in arithmetic was for Gödel a definite objective notion and […] he had arrived at the undefinability of that notion in arithmetic by 1931. On the other hand, he did not state this as a result […] and he took pains to eliminate the concept of truth from the main results of 1931.”11 Gödel himself, in correspondence with Hao Wang, said that, “the heuristic principle of my construction of undecidable number theoretical propositions in the formal systems of mathematics is the highly transfinite concept of ‘objective mathematical truth’, as opposed to that of ‘demonstrability,’ […] with which it was generally confused before my own and Tarski’s work. […] [F]ormalists consider formal demonstrability to be an analysis of the concept of mathematical truth, and therefore were of course not in a position to distinguish the two.”12 Much the same can be said about the failure of many physicists and philosophers— including, it seems, Kahle—to distinguish the intuitive concept of time from the formal/relativistic concept,13 which is as essential to a proper understanding of the

9 Manchak

(2016, 1053).

10 Comment by the editors: Reinhard Kahle acknowledges this point and suggests that “seemingly”

expresses the intended meaning of the German “scheinbar” better than the expression “apparently”. He also points out that his text contains a question mark here (“only apparent?”). 11 Feferman (1998, 159). 12 Hao Wang (1974, 9 f.). 13 Indeed, arguably, even Einstein failed to make that distinction. He remarked, as Gödel did, that in STR, there is no such thing as a nonrelative, global notion of simultaneity, but insisted, nevertheless

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significance of Gödel’s cosmology as the distinction between truth and proof is to a proper understanding of Gödel’s incompleteness theorem. (I go into this in great detail in Yourgrau 1999, 2005.) Gödel’s extreme caution to the side, then, most serious commentators on Gödel’s essay have drawn the parallel conclusion about Gödel’s temporal idealism, which contrasts the “formal”/relativistic conception of time with the intuitive. Gödel’s temporal idealism is directed only at the latter, and on that, commentators widely agree. The noted philosopher of science Karl Popper, for example, recalls a discussion he had with Einstein about Gödel 1949. “When I visited Einstein,” Popper wrote, “Schilpp’s Einstein volume […] had just been published; this volume contained the now famous contribution of Gödel’s which employed, against the reality of time and change, arguments from Einstein’s two relativity theories. […] The reality of time and change seemed to me the crux of realism. [Einstein] clearly disagreed with Gödel’s idealism. […] Einstein did not want to give up realism […] though I think he was ready to admit, as I was, that we might be forced one day to give it up if very powerful arguments (of Gödel’s type, say) were to be brought against it.”14 If Kahle is to be believed, both Einstein and Popper were confused about Gödel’s temporal idealism. As we have seen, Gödel (1949) is not the only place where Gödel indicated his thoughts about the reality of time. In conversations he had about that with Hao Wang, for example, he said: “As we present time to ourselves it simply does not agree with fact. To call time subjective is just a euphemism for this failure.” Again: “Time is no specific character of being. In relativity theory the temporal relation is like far and near in space. I do not believe in the objectivity of time. The concept of Now never occurs in science itself, and science is supposed to be concerned with the objective.”15 Wang himself comments that “on the whole, Gödel seems to favor the fundamental perspective of seeing objective reality, both the physical and the conceptual, as eternal, timeless, and fixed.” (op. cit., 322) (I will have more to say about Gödel’s conceptual realism further on.) Again, in Gödel (1946/9-B2), a (previously) unpublished draft of Gödel (1949), “Some observations about the relationship between theory of relativity and Kantian philosophy”, Gödel is less cautious than he is in Gödel (1949), stating outright at the beginning of his essay that relativity theory confirms Kant: “It is a remarkable fact, to which, however, very little attention is being paid in current philosophical discussions, that at least in one point relativity theory has furnished a very striking confirmation of Kantian doctrines.”16 He affirms, however, that “I am not an adherent of Kantian philosophy in general,” and notes that “unfortunately, whenever this fruitful viewpoint [of Kant’s] of a distinction between subjective and objective elements in our knowledge (which is so impressively suggested by Kant’s comparison with the Copernican system) […] appears in the history of science, there is at once a tendency

(contra Gödel), that “the concepts of happening and becoming are indeed not completely suspended, but yet complicated.” (Einstein 1961, 150) 14 Karl Popper (1976, 129–130). 15 Wang (1996, 320). 16 Op. cit., 230.

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to exaggerate it into a boundless subjectivism […]. Kant’s thesis of the knowability of the things in themselves is an example.”17 The specific Kantian doctrines he has in mind that he believes relate to relativity theory are that “time is neither ‘something existing in itself’ […] nor ‘a characteristic ordering inherent in the objects’, but only exists in a relative sense […].”18 I discuss in detail this (previously) unpublished essay on Kant—including drafts of it that have not yet been published—in Chap. 5 of my more strictly academic book, Yourgrau (1999). Unfortunately, Kahle seems to be unaware of that book, hence unaware of my extended discussion there of Gödel on Kant. As indicated above, it was in his attempt to find in relativity theory a confirmation of Kant on time that Gödel arrived at his discovery of the Gödel universes, in which, provably, the physical conditions necessary for the existence of time (in anything like the ordinary sense) are lacking. Yet can the possibility that the actual world is such a universe be ruled out by physical considerations? Perhaps, but only by adducing contingent features of our world. Gödel, however, finds it philosophically unsatisfactory that only contingent features of the actual universe concerning the global distribution of matter and motion rule out the possibility that the actual world is a Gödel universe. The existence of time, he believes, must be an essential feature of the fundamental laws of nature. In Gödel’s words, “if someone asserts that this absolute time is lapsing, he accepts as a consequence that, whether or not an objective lapse of time exists (i.e., whether or not time in the ordinary sense of the word exists), depends on the particular way in which matter and its motion are arranged in the world. This is not a straightforward contradiction; nevertheless, a philosophical view leading to such consequences can hardly be considered satisfactory.” Kahle seems to believe that by these words Gödel laid down a challenge to see if one can find a way to rule out the possibility of a Gödel universe based, rather, on basic physical principles. I say Kahle “seems to believe”, since I find his final sentence unclear: “To find such principles is the philosophical challenge that Gödel has left us.” It is unclear whether Kahle is suggesting that Gödel was challenging us to find such principles, or only that, given Gödel’s results, we, on our own, should challenge ourselves to find such principles. From what Kahle also says, however, the former appears to be the case: The task, which Gödel discerns here, consists exactly in finding further physical principles, from which the non-existence of his universe would already follow, without the necessity to use contingent properties of our universe.

Certainly, Gödel’s final words in Gödel (1949) as we have just given them do not suggest that Gödel was laying down a challenge or “discerning a task” that needs to be fulfilled. On the contrary, Gödel states outright what he sees as a consequence for someone who still wishes to maintain that objective time exists in our world. After

17 Op. 18 Op.

cit., 230; 257–258. cit. 230.

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all, it was Einstein, in his reply to Gödel 1949 in the Schilpp volume, not Gödel, who ended his comments by saying that “[i]t will be interesting to weight whether these [i.e., Gödel’s cosmological solutions] are not to be excluded on physical grounds.” (Brackets added) Further, it was Stephen Hawking, not Gödel, who proposed a conjecture that would rule out their physical possibility, which he dubbed (no doubt, with tongue in cheek), the Chronology Protection Conjecture. Kahle discusses Hawking’s conjecture, but I found his discussion confusing: A trivial solution would be to put the existence of an objective course of time as a principle at the beginning. Such a petitio principii, of course, cannot satisfy. Nevertheless exactly this was tried by Stephen Hawking with his “Chronology Protection Conjectur” (Hawking, 1992), but just only as a conjecture and not as an axiom. Thus one must credit Hawking that this conjecture is still to be proved by other principles.

To what does “exactly this” refer? Clearly to “a trivial solution”, which Kahle describes as “a petitio pricipii, [which] of course, cannot satisfy.” A harsh criticism, indeed. Yet Kahle adds that Hawking advanced his conjecture “just only [as] a conjecture and not as an axiom. Thus one must credit Hawking that this conjecture is still to be proved by other principles.” This added comment absolves Hawking from the harsh criticism Kahle just advanced against him. So, what exactly is Kahle saying? I think the latter, but the passage is clearly poorly written and confusing—since the latter claim contradicts the former. Further confusing is what Kahle said in footnote 10, that, “From this perspective, Yourgrau’s harsh criticism of Hawking (Yourgrau, 2005, pp. 8 and 136) is both incomprehensible and unjustified.” For my part, I found that footnote itself “incomprehensible.” It was not clear to me at first how my harsh criticism of Hawking differed from Kahle’s. I revisited, however, what I said about Hawking in my book, and it finally became clear what Kahle was trying to say in his criticism of me, and that in fact, he was correct. I had mistakenly taken Hawking’s Chronology Protection Conjecture as a proposed new axiom, not as a principle to be derived from more fundamental axioms, and thus considered it to be ad hoc, or in Kahle’s words, as “a trivial solution [that] would put the existence of an objective course of time as a principle at the beginning …; a petition pricipii, [that] of course, cannot satisfy.” Given what Kahle himself asserted about the conjecture (albeit, I have argued, in a confusing manner), it should have been obvious to him what my mistake was. It is incomprehensible to me, therefore, why Kahle described what I said as “both incomprehensible and unjustified.” Unjustified, yes; but incomprehensible?19 There is still more to this story, however. Kahle fails to pose an obvious next question: suppose Hawking’s conjecture proves false, in the sense that it proves incapable (in whatever sense one can make of this supposition) of being supported

19 Comment by the editors: Reinhard Kahle acknowledges this point. In fact, the English translation

might be ambiguous here and he rather meant to express that the argument seems unfounded to him.

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from more basic physical principles. What then? Should one propose a new axiom that would rule out cosmologies like Gödel’s? Kahle draws attention to “additions by the author” to footnotes to the German edition of Gödel 1949. He says that in those additions, Gödel speculates on the possibility of a “cosmological physics” which, for its part, would revise the known laws at (too) large scales.

If the idea here is that at some point cosmologists may revise the known laws of physics with respect to cosmological scales, then at this point, the problem of the petitio pricipii Kahle mentions does raise its head. Would it be an acceptable reason for proposing such revisions, or for adopting new fundamental physical principles, that they would rule out the possibility of the Gödel universes? This question was in the back of my mind when I wrote those passages in Yourgrau (2005). Note, further, that it was surely a priori, philosophical intuitions about the nature of time that led Hawking to make his conjecture in the first place, to look into whether one could deduce from more basic physical principles that the Gödel universes are a physical possibility. The situation, of course, was symmetric with regard to Gödel. He says explicitly that the reason he looked into the question of whether there are possible universes where it would be demonstrable that objective, intuitive time fails to exist is because of his philosophical intuitions about the nature of time, shared by idealists like Kant, that time is merely ideal. What if, however, Hawking’s Conjecture proves right, or if the discovery or postulation of deeper physical principles rules out the physical possibility of the Gödel universes? Kahle’s answer is clear: [H]e [Gödel] clearly looks after the possibility of guaranteeing the existence of time through deeper physical principles. (brackets and emphasis added)

Kahle here fails to distinguish between necessary and sufficient conditions for the existence of time. The significance Gödel attributed to his discovery of the Gödel universes was that in them it was provable on physical grounds that time in the intuitive sense fails to exist, i.e., their possibility was a sufficient condition for the nonexistence time in those worlds, and (via his modal argument), also in the actual world. It does not follow that the possibility of the Gödel universes is necessary for the nonexistence of time. Simply put, according to Gödel, if the Gödel universes are possible, time doesnot exist. It does not follow that if the Gödel universes are not possible, time does exist. And yet, as we have just seen from the quotation from Kahle, he seems to be making just this elementary logical mistake when he asserts that Gödel was looking after the possibility of guaranteeing the existence of time through deeper physical principles that would rule out the possibility of the Gödel universes. “There’s no such thing as time travel, therefore time exists” is hardly a valid inference. By contrast, Gödel’s inference, “there is such a thing as time travel, therefore time fails to exist,” has considerable force, which explains his concern to establish the premise.

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One reason Kahle may have made this mistake is his failure throughout his essay to distinguish the intuitive from the “formal”/relativistic concept of time, an astonishing failure, since, as pointed out above, Gödel’s entire philosophical discussion of time presupposes this distinction, much as his philosophical assessment of his incompleteness theorems brings out the analogous distinction between the intuitive and the formal, between truth and proof. (Indeed, Wang describes Gödel’s methodology as “a dialectic of the formal and the intuitive.”) Hawking’s Chronology Protection Conjecture, even if confirmed, would not, after all, protect intuitive time, but only formal/relativistic “time.” As mentioned above, I discuss the analogy between intuitive mathematical truth and intuitive, “Kantian” time at length in both of my books on Gödel and Einstein, yet Kahle, somehow, fails to note it. I describe Gödel’s characterization of intuitive time in detail on p. 74 of Yourgrau (1999). Gödel calls the concept of intuitive time the ordinary, Kantian, pre-relativistic concept, according to which time objectively lapses, which results in a change in the existing, and presupposes an objective, world-wide present or Now. (Recall what Gödel said to Wang: “I do not believe in the objectivity of time. The concept of Now never occurs in science itself, and science is supposed to be concerned with the objective.”) One can ask, however, whether the advancement of science provides deeper insight into a concept or rather replaces an intuitive concept with increasingly formal simulacra. Thus, Carlo Rovelli has described ten distinct notions of time in physics. “[A] single, pure and sacred notion of ‘Time’,” he says, “does not exist in physics.”20 He traces a sequence of different stages in the notion of time appropriate to stages in the advancement of science. “[T]he development of theoretical physics,” he says, “has modified substantially the ‘natural’ notion of time. A first modification was introduced by special relativity. After Einstein’s analysis of simultaneity, we know the notion of time is observer dependent.”21 The modification of the concept of time becomes increasingly extreme. For example, “[q]uantum fluctuations of physical clocks, and quantum superpositions of different metric structures make the very notion of time fuzzy at the Planck scale.”22 First time becomes relative, later it becomes fuzzy. “We have discussed attributes of time that progressively disappear,” he writes, “in going toward more ‘fundamental’ physical theories.”23 And so on. In response, it should be said that Gödel does not deny that a formal simulacrum of time remains in relativity (special and general). Indeed, he speaks often of “what remains of time” in relativity theory. What is left of intuitive time, minus the idea of objective becoming, a change in the existing, however, according to Gödel, “hardly deserves the name of time.” “[A]n entity corresponding in all essentials to the intuitive idea of time,” Gödel said, “[…] in relativity theory […] exists only in our imagination.”24

20 Carlo

Rovelli (1995, 82). cit., 84. 22 Op. cit., 86. 23 Op. cit., 86. 24 Gö95, 258, footnote 28. 21 Op.

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Even if it turns out, then, that Hawking’s Chronology Protection Conjecture was right and the Gödel universes are not physically possible, it does not follow that in the actual world time-intuitive time—lapses and that there is a change of the existing over time. As Steven Savitt puts it in discussing what he calls “Yourgrau’s epistemic argument”:25 [W]e presumably have in our universe (and the Gödelians lack in theirs) a global time function and the global time slices, but these structures, we recall, are only necessary conditions for objectively lapsing time. The other conditions, the successive coming into existence, is precisely what is at issue.

Similarly, Manchak comments that, “following the literature […] we take the existence of a global time function to be a necessary (but not sufficient) condition for the objective lapse of time”.26 Gödel’s excessive caution may, in fact, have caused him to overstate the importance of deriving his new cosmological models in order to argue for temporal idealism. It was typical of him to seek, cautiously, the most unassailable arguments for his philosophical views—in the present case, arguing from his new discoveries about relativity theory—and then to be even more cautious about the lack of decisiveness about even those arguments. Consider, for example, what the philosopher of science Lawrence Sklar has written with regard to Gödel’s cosmology and the intuitive concept of time: “I don’t think Gödel really needed such a cosmology to ‘refute’ Jeans, since conceptual reflection shows us, I think, that even if such a cosmological time exists, it is hardly a restoration of an ‘absolute time’ in anything like the pre-relativistic sense of that notion.”27 Indeed, one can go even further than this. As we have seen, Gödel concluded that it was philosophically unsatisfactory to maintain that the existence of time in our world depends on the particular distribution of matter and motion, and not on a fundamental law of nature. John Earman disagrees: “Why […] is there a lurking contradiction,” he says, “or something philosophically unsatisfactory in saying: ‘Time in our universe elapses, but if the distribution and motion of matter were different, there would be no consistent time order and so time would not elapse’?”28 Similarly, Manchak asks “[if] it is philosophically unsatisfying (although not a contradiction) for one to assert that time is objectively lapsing in one’s universe when from the perspective of the observer making the assertion there remains the nomological possibility that, after the time of the assertion, matter and its motions might be smoothly (re)arranged in such a way so as to prohibit an objective time lapse.”29 The essential point, however, I believe, does not concern the question of the relevance of a particular distribution of matter and motion. The determining factor, surely, is that if there is any motion

25 Steven

Savitt (1994, 468). (2016, 1052). 27 Lawrence Sklar (1984, 106). 28 John Earman (1995,198). 29 Manchak (2016, 1054). 26 Manchak

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at all in the universe, no matter the distribution, there must be change, and therefore time. In a word: no time, no motion. How, then, can the existence of time depend on a particular distribution of motion and matter? Sklar was right to note that “even if such a cosmological time exists, it is hardly a restoration of an ‘absolute time’ in anything like the pre-relativistic sense of that notion.” As to the question of the relevance of epistemologically remote cosmological facts to questions about the existence and nature of time, what Sklar has written about the question of the so-called direction of time applies as well to our present discussion:30 Surely, there is something implausible about a philosophical analysis that makes the existence and knowability of an obviously present feature of the world hinge upon the existence of other features that may, in fact, not exist, and that, even if they do exist, have observational consequences only in the most unusual situations.

If this is so, Gödel’s reasoning about the nonexistence of intuitive time with respect to special relativity—if it is sound31 —is already decisive. Moreover, based on formal results he claims to have established concerning expanding Gödel universes and what he calls “weakly observationally indistinguishable” space-times, Manchak argues that “it remains an epistemic possibility, just as Gödel claimed it was, that we inhabit a world that has no objective time lapse.”32 If it is indeed epistemically possible that in our world there is no objective time lapse, then clearly even our direct experience (as of) the lapse of time is not decisive. To borrow Gödel’s words, “this is not a straightforward contradiction; nevertheless, a philosophical view leading to such consequences can hardly be considered satisfactory.” In the end, how shall we describe Gödel’s view about the existence or reality of intuitive time, bearing in mind Kahle’s emphasis on how cautious Gödel was in Gödel (1949b)? The answer should be clear by now. Gödel, although a mathematical, space-time33 , and conceptual realist, was a temporal idealist. Kahle, however, refers to Howard Stein’s Introduction to Gödel (1949b), where he, in Kahle’s words,

30 Quoted

in David Malament (1976, 322). needs to add this caveat, since it remains a debated issue whether the becoming or lapsing of time is consistent with—even, for some, is a direct consequence of—the STR. 32 Manchak, op. cit., 1053. Manchak reminds us that “even late in his life, Gödel had still not given up on the possibility that we inhabit a Gödel-type model. Indeed, he would remain intensely interested in the collection of all astronomical data relevant to this possibility …” (1052) Just how interested can be gleaned from the fact that Gödel’s biographer, John Dawson, was astonished to find in Gödel’s Nachlass two notebooks containing detailed calculations concerning the angular orientation of galaxies. 33 We, in turn, must be cautious here. In the context of his writings on relativity, Gödel adopted a realist perspective with regard to space-time. Fundamentally, however, we know that his primary allegiance was to Leibniz, who was an idealist with regard to space and time. Interestingly, however, Gödel’s writings on Kant far exceed his discussions of Leibniz. Indeed, “I have never attained anything definite,” Gödel said, “on the basis of reading Leibniz. Some theological and philosophical results have just been suggested [by his work]. One example is my ontological proof [of the existence of God].” (Wang 1996, 87) 31 One

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“explicitly regrets the lack of more material that would shed light on Gödel’s philosophical position, especially vis-á-vis Kant”, although Kahle does mention that “at least” we have Gödel’s Kant paper. I hope, however, to have made it clear, briefly, above, and at length in Chap. 5 of Yourgrau (1999), that Gödel was a temporal idealist in precisely Kant’s sense, although he was not, in general, an adherent of Kant’s philosophy. Kahle only confuses the issue when he remarks that “Gödel himself is usually regarded as a realist; see, e.g., [Yourgrau 2005, p. 171 f.].” What I say there is only that Gödel is a conceptual realist. “I have been a conceptual and mathematical realist,” Gödel said, “since about 1925.”34 I emphasize, by contrast, in both Yourgrau (1999, 2005) that Gödel was, at the same time, a temporal idealist.35 As he made clear in Gödel (1946/9-B2) Gödel believed relativity theory confirmed Kant’s thesis that time is a mere appearance or illusion. Of course, Kant, unlike Gödel, was not in addition a conceptual realist. Plato, by contrast, was, like Gödel, both a temporal idealist and a conceptual realist (if we take the Platonic Idea as the ancestor of the modern notion of a concept). “I am for the Platonic view,” said Gödel. “If there is nothing precise to begin with, it is unintelligible to say that we somehow arrive at a precise concept. […] It was the anti-Platonic prejudice [with respect to mathematics] which prevented people from getting my results. This fact is a clear proof that the prejudice is a mistake.”36 In holding with Kant that (intuitive) time is a mere appearance or illusion, Gödel was, of course, denying that time exists, yet, curiously, this has been denied. “What advantage,” writes Savitt, “could there be to maintaining, as I insist, that Time does not exist rather than that Time is an illusion […] All over Vancouver […] people arrive at meetings […] buses arrive, all more-or-less on schedule. Could an illusion coordinate this intricate choreography? […] [C]locks don’t measure Time; they measure spacetime.”37 Savitt forgets here what has just been emphasized, that while Gödel is a temporal idealist, he is a space-time realist. In saying that (intuitive) time is an illusion, Gödel is not saying, perversely, that the world is haunted by this illusory phenomenon. Clocks cannot measure illusions. In maintaining that time is an illusion, Gödel is saying that we have the illusion that time exits. If I say to you that witches are a figment of your imagination, I am not claiming that imaginary witches are flying around the world. Savitt is not delivering the news to Gödel that clocks measure space-time, not (intuitive) time. One final point. The Kurt Gödel Award 2021 question was: “What does it mean for our world view if, according to Gödel, we also assume the non-existence of time? It is not about a confirmation or refutation of Gödel’s reasoning; instead for the contest this is assumed to be correct.” It is not clear to me how, exactly, Kahle

34 Wang

(1996, 235); 6; brackets added.

35 A subsection of Chap. 5 in Yourgrau (1999) is entitled, “Gödel’s Theorem and Gödel’s Cosmology:

Mathematical Realism versus Temporal Idealism” and Chap. 7 in Yourgrau (2005) is entitled, “The Scandal of Big ‘T’ and Little ‘t’,” where big T is truth, with respect to Gödel’s incompleteness theorem, and little t is time, with respect to Gödel’s results on relativity. 36 Wang (1996, 83); brackets added. 37 Savitt, op. cit., 469–470.

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has answered this question. His conclusion seems to be that Gödel left us a task he himself was engaging in, to find fundamental physical principles that would rule out the possibility of the Gödel universes and thus “guarantee” the existence of time. Thus, although his essay can stand on its own, Kahle appears to have ignored the requirement that “we also assume the non-existence of time,” and that we do not attempt to “confirm or refute Gödel’s reasoning.”

References Bell, J. S. (1989). How to teach special relativity. In J. S. Bell (Ed.), Speakable and unspeakable in quantum mechanics (pp. 67–80). Cambridge University Press. Dawson, J. (1997). Logical dilemmas: The life and work of Kurt Gödel. A K Peters. Earman, J. (1995). Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic spacetimes. Oxford University Press. Einstein, A. (1961). Relativity: The special and general theory. Crown (Transl. by R. W. Lawson). Feferman, S. (1986). Gödel’s life and work. In S. Feferman, J. W. Dawson Jr., S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: Collected works: Volume I: Publications 1929–1936 (Vol. 1, pp. 1–36). Oxford University Press. Feferman, S. (1998). In the light of logic (pp. 150–164). Oxford University Press. Gödel, K. (1946/9-B2). Some observations about the relationship between theory of relativity and Kantian philosophy. In S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: Collected works: Volume III: Unpublished essays and lectures (Vol. 3, pp. 230–259). Oxford University Press (1995). Gödel, K. (1949). A remark about the relationship between relativity theory and idealistic philosophy. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-scientist (pp. 555–562). Harper & Row. Hawking, S. W. (1992). Chronology protection conjecture. Physical Review D, 46(2), 603–611. https://doi.org/10.1103/PhysRevD.46.603. Malament, D. (1976). Review: L. Sklar, space, time, and space-time. The Journal of Philosophy, 73(11), 306–323. https://doi.org/10.2307/2025892. Manchak, J. B. (2016). On Gödel and the ideality of time. Philosophy of Science, 85(5), 1050–1058. https://doi.org/10.1086/687937. Popper, K. (1976). Unended quest: An intellectual autobiography. Fontana books 4416. Open Court. Reichenbach, H. (1969). Axiomatization of the theory of relativity. University of California Press (Transl. by M. Reichenbach). Rovelli, C. (1995). Analysis of the distinct meanings of the notion of “time” in different physical theories. Il Nuovo Cimento B (1971–1996), 110(1), 81–93. Savitt, S. F. (1994). The replacement of time. Australasian Journal of Philosophy, 72(4), 463–474. https://doi.org/10.1080/00048409412346261. Sklar, L. (1984). Comments on Malament’s “‘time travel” in the Gödel universe’. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 106–110. Wang, H. (1974). From mathematics to philosophy. Library of philosophy and scientific method. Routledge and Kegan Paul. Wang, H. (1996). A logical journey: From Gödel to philosophy. MIT Press. Yourgrau, P. (1999). Gödel meets Einstein: Time travel in the Gödel universe. Open Court. Yourgrau, P. (2005). A world without time: The forgotten legacy of Gödel and Einstein. Basic Books.

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Palle Yourgrau is the Harry A. Wolfson Professor of Philosophy at Brandeis University (MA, USA). He earned his Ph.D. at the University of California at Los Angeles. His research and publication interests include the metaphysics of death, the philosophy of mathematics, the philosophy of language, and the philosophy of space and time. The philosophers who are of special interest to him are Gottlob Frege, Kurt Gödel, Simone Weil, and Plato.

6

Self or the World Michał Pawłowski

‘Wake up’—with the sound of these words, uttered by his girlfriend, starts Nyles’ morning in the film Palm Springs, released in 2021. That and every following morning, to be precise, as the film protagonist has been trapped in a time-loop on a Californian wedding. Each time he falls asleep, no matter when and where, he wakes up in the same bed in the same morning, accompanied by the same girl. Terrifying as it may seem, the situation tends to be not as depressive for Nyles. The never-ending repetition of the wedding day in the town Palm Springs gives him an opportunity to test (social) reality around him to its maximal limits. He can repeat interactions with the same people and check their responses in various scenarios, ruin the party with inappropriate jokes without consequence (as long as he manages to get away by the end of the day), or simply enjoy each morning drinking beer in a pool. However, as the plot unfolds, serious doubts arise, not only because Sarah, a female protagonist who joins Nyles in the time-loop, confronts him with existential questions. Unlike Sarah and Nyles, who are aware of the situation they have fallen into and keep memories of previous ‘days’, people around them perpetually repeat the same words, gestures, actions, without any knowledge of the state of affairs. They resemble robots or animated puppets, programmed to perform a limited set of activities. A viewer might then ask about their status—are they real? Do they have real memories? If they do not live in the time-loop, what does their next day look like? Which version of the wedding party (devoid of any follow-up and ever-repeating for Nyles and Sarah) does the next day follow? Which future (of all possible ones) would enter Nyles and Sarah if they managed to escape the time-loop? What speaks in favor of the interpretation in line with which others are no real persons is the high repeatability of their behavior, the lack of memories concerning

M. Pawłowski (B) Department of Economics, European University Institute, Florence, Italy

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_6

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previous versions of the day, and the following manipulability of their actions. They are not only deluded as to the condition they are in (as one could try to reply with such a Cartesian scenario) but also devoid of persons’ typical features such as (at least partially) free agency and reflectivity, which is shown by the fact that interactions with them can be endlessly repeated and checked like in an algorithm, without their knowledge. However, what if everybody at the wedding has fallen into the timeloop? What—on the contrary—if nobody, in fact, has? Would this day still endlessly repeat, yet this time without any mind to observe next iterations? Could the skepticism about others’ personhood be maintained? What if, in fact, time does repeat? Are we conscious subjects? Thus, the status of cognizing minds—all minds—seems the first concern resulting from the possibility of time travel. The second problem (expressed above in questions about potential futures after the time-trapped wedding day) is the character of nonconscious reality. How can the apparently persistent sequence of events ordered in time be sustained when time travel is possible and, in fact, there is no time?1 Both subjects and objects seem endangered by its nonexistence. Let me discuss it briefly as two hypotheses in the following paragraphs. I will start with objects (and the world).

6.1

No Order in the World

The heart of Kurt Gödel’s argumentation for the possibility of time travel and, in consequence, against the reality of time, as presented in Gödel (1949) and Yourgrau (2005, pp. 152–154), is the refutation of the so-called A-series, just like in the presumably most famous argument against time by McTaggart (1908). While the alternative B-series of time (in McTaggart’s view logically dependent on A-series as the condition of its temporality) is based on fixed relations of being earlier or later between events, the A-series’ character is revealed in constant passing from future— through present—to past (Loux, 2006). ‘Now’ as a temporal point is continuously in motion. Where for McTaggart the A-series entails attribution of mutually exclusive predicates to the same event in time and, thus, entails contradiction, Gödel argues that the special relativity theory being true rules out the existence of a privileged (and independent of a frame of reference) time point, the universal ‘now’. Intuitive time of everyday existence disappears, a spectacular instance of which is the possibility of time travel (Gödel, 1949). Leaving aside a more comprehensive presentation of Gödel’s argument as laying outside the scope of these considerations (the con-

1 Of course, the possibility of time travel and the nonexistence of time can be prima facie treated as two separate hypotheses. Here, however, in accordance with the contest’s question, they are taken as logically interrelated.

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test question required assuming that time does not exist), I would like to focus on phenomenological aspects of the nonexistence of time.2 First, following the Kantian tradition of thinking about time as a precondition or form of a phenomenal world, it is worth looking closer at the role it plays in our perceptions. Time, as the inner sense (next to the outer sense, space), allows for organizing experience of outside objects and inner intuitions, so that they can be perceived by a cognizing subject (Kant, 1998, pp. 174–177 A22/B36–A26/B42; pp. 180–181 A33/B49–A34/B51). However, it must be noted that it does not need to entail ascribing to Kant a form of phenomenalism understood as constructing a world based entirely on sense impressions—as a matter of fact leaving the door open to a realist interpretation of the first Critique (Sellars, 1968, pp. 48–50; Heidemann, 2021). In accordance with Kant’s own understanding of his doctrine, one may add, since he openly identifies transcendental idealism with so-called empirical realism (Kant, 1998, pp. 426–427 A369–A372). At the same time, the role of time and space (together with categories introduced in transcendental analytics) is not solely reduced to conditioning the actual experience. What they facilitate, as Kant repeats many times, is all possible experience. It is then fully legitimate to call time and space the forms of the world (as representation), not only of cognition (Pore˛ba, 2014, pp. 101–102).3 If time does not exist, then, apparently, the world seems to lose its form. What does this mean? A hint might be provided by the words of Kant: If cinnabar were now red, now black, now light, now heavy, if a human being were now changed into this animal shape, now into that one, if on the longest day the land were covered now with fruits, now with ice and snow, then my empirical imagination would never even get the opportunity to think of heavy cinnabar on the occasion of the representation of the color red; or if a certain word were attributed now to this thing, now to that, or if one and the same thing were sometimes called this, sometimes that, without the governance of a certain rule to which the appearances are already subjected in themselves, then no empirical synthesis of reproduction could take place. (Kant, 1998, p. 229 A100–A101)

The world in this vision is no longer a structured whole, where objects (as phenomena) exist in time and space, interact with each other due to, among others, causality, and thus as res extensa (to use the Cartesian term) compose a calculable and predictable unity. Without time, through which objects, thoughts, events are organized as subsequent or simultaneous, they cease to relate to each other, that is, they cease to

2 According to Gödel’s reasoning, the possibility of time travel suffices to demonstrate the nonexis-

tence of time. It should be emphasized that his claim aims at proving time’s unreality, not simply its nonstandard structure. Thus, the film plot discussed above suffices to cast doubt on the existence of time as well. 3 In Marcin Pore˛ba’s words, who reads the first Critique in such a realistic spirit, ‘(...) space and time in Kant’s understanding are not merely forms of perception in the local sense of the word, i.e., forms that organize, as it were, individual appearances or scenes. For these forms are meant to apply to all objects of possible experience (phenomena), i.e., not only those that we actually perceive at a given moment, but also those that we could perceive (...).’ (Pore˛ba, 2014, p. 101)

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be one reality. That this problem is not an internal difficulty of Kant’s theory4 (which understands time as a form of pure intuition determining the structure of empirical reality via the schematism of the categories) can be seen in another famous paragraph from the Critique of Pure Reason, Refutation of Idealism, as well as in some other claims made by Kant. Conclusions he reaches there are to a big extent independent of his earlier assumptions. In short, the Refutation of Idealism presents a transcendental argument for the existence of outside world by showing that it is necessarily presupposed by the (expectation of reliable) measurements of time, otherwise no frame of reference for such a procedure would be provided (cf. Stern, 2021 for a very clear description of that reasoning). Apart from that, however, Kant seems to be also arguing that valid measurements of space require time: The measurement of a space (as apprehension) is at the same time the description of it, thus an objective movement in the imagination and a progression; by contrast, the comprehension of multiplicity in the unity not of thought but of intuition, hence the comprehension in one moment of that which is successively apprehended, is a regression, which in turn cancels the time-condition in the progression of the imagination and makes simultaneity intuitable’ (Kant, 2002, p. 142, AA 5: 258–259, italics mine).

Further, ‘That time is expressed by a line (which is however space) and space through a time (the distance traveled in an hour) is a schematism of the concept of the understanding.’ (Kant, 2005, p. 395, AA 18: 687). All spatial scales can be expressed with a reference to different metrics, at the end of the day; however, the search for more primitive terms describing these very measures demands employing temporal intervals (such as expressing distance by the time needed to walk it). As Pore˛ba notes, contemporary physics provides a very good candidate for such a temporal benchmark—the speed of light speed. ‘It allows a universal expression of time by space (for example, one light-meter is the time it takes light to travel one meter) and space by time (one light-second is the distance light travels in one second).’ (Pore˛ba, 2014, p. 112). Consequently, both space is said to determine time, and the other way round—time can be said to determine space. The resulting interrelation between time and space implies that disappearance of any of them will suffice for world’s (understood as structured whole of things or events) dissolution. Synthesis of possible experience then proves impossible. This, however, does not have to mean that Kant’s transcendental argument, according to which the empirical reality of a coherent world presupposes (as its necessary conditions) the validity of forms of experience or categories, holds irrefutable. Quentin Meillassoux offers an interesting reply to it. He argues that while the answer given by Kant to Hume’s problem of causation remains within the same realm as the question

4 One

could obviously posit another theory of time. However, for the two other most well-known conceptions, the one proposed by Newton and the one by Leibniz (where, respectively, time and space are both a kind of χωρα ´ independent of objects or are made up by the relations between them), the consequence of the nonexistence of time for the world appears only more burning.

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formulated by the Scottish philosopher (unlike Popper’s solution, which in Meillassoux’s view confuses the ontological order with the epistemological), it is based on an unjustified premise. According to Meillassoux, Kant illegitimately assumes that the conditions of regular appearances in nature (and thus, science) and a perceiving, empirical consciousness are the same (Meillassoux, 2015, pp. 24–31). As Strawson puts it: ‘He (Kant—MP) is concerned (...) with the general conditions of the employment of concepts, of the recognition of the particular contents of experience as having some general character; and he regards these conditions as being at the same time the fundamental conditions of the possibility of ordinary or empirical self-consciousness’ (Strawson, 1966, p. 26). The crucial category here, causation, is for Kant at the same time a precondition for the objective temporal order of appearances (which cannot be based only on subjective order of representations), according to the argumentation from the Second Analogy. In this sense, time is epistemically determined by causal laws (Guyer, 1987, pp. 242–248).5 Thus, the Kantian scenario, as Meillassoux sees it, implies that a causally and temporally unstructured, unsynthesized world would fall into imperceptible chaos, incognizable for the empirical self, being itself its element: But since every form of temporal continuity would come to be broken, I myself could not subsist, in the form of a self-consciousness capable of witnessing the spectacle of this frightening desolation, for my own memory would disappear in its turn as soon as it emerged. Everything would be reduced to the punctual and perpetually amnesiac intuition of a point of chaos without density and without relation to its past.’ (Meillassoux, 2015, p. 30)

However, as one could reply, the subjective notion of time essentially employing our memory (grasped in the form of McTaggart’s A-series) has still not been fully reduced to the objective time stemming from the application of the principle of causality to the outer appearances (cf. Falkenburg & Schiemann, 2016). Indeed, Meillassoux seems to be following that proviso when he argues that the above described scenario is not the only possible. In the two other types of ‘chaotic’ worlds that he describes, the irregularities would not be big enough to rule out the existence of consciousness. We could freely imagine universes where the nonexistence of laws of nature would cause some disturbances to science, limiting its applicability and explanatory/predictive powers (Type-1 worlds, as he calls them) or even completely precluding its existence (Type-2), yet human subjectivity would survive (Meillassoux, 2015, pp. 31–41). This suggests that without time (understood as a factor organizing objects of possible experience—and the world) a conscious self could possibly exist, even though the

5 ‘So the determination of the temporal order of the represented states must be grounded on something

other than either the order of the representations (...) or the order of the objective states themselves. (...) This Kant takes to imply that, since necessity is required, there must be a pure concept of understanding, and that it must be that of “the relation of cause and effect” (B 234)’ (Guyer, 1987, pp. 244, 246).

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world would not be fully structured and could not be described as an orderly arranged whole. Having this in mind, the first consequence of the nonexistence of time can be formulated. The world without time (intuitive flux of ‘now’) could possibly still be in existence, yet deprived of order. At the same time, the (possible) presence of a conscious self would be maintained. The alternative consequence looks quite different.

6.2

No Self

If we look closer at the thesis formulated by Gödel, affirming the possibility of time travel, which necessarily leads to the nonexistence of time, one should ask: who travels? First, let us imagine that I (or any other individual) can travel in time. Possibly, I could participate in events from the distant past or future, but what if I tried to re-experience events from my own life? This does not appear problematic as long as I do not try to imagine my presence at events such as my own birth. Would I reappear in the world, or simply exist in two ‘copies’? The former would contradict the reality of time travel. The latter, in turn, would cancel the reality of self. If in time travel I can meet myself and look upon (or even interact with) myself from the outside, maybe it is not me who travels. Who is it, then? Again in reference to the transcendental tradition, the traveling self could be called a transcendental ego. This time, however, more in the Husserlian sense than in the Kantian (designating the transcendental unity of apperception): ‘The “I” which transforms the world into mere phenomenon is, in so doing, aware of itself as transforming the world and cannot be subjected to the same transformation. (...) this “I” is completely devoid of any content which could be studied or explicated. It is completely indescribable, being no more than a pure ego’ (Schmitt, 1959, p. 240).

By very definition, such a transcendental ego would be something different than an empirical self, instantiated in our everyday experience. It would constitute conditions of the phenomenal world’s reality, not being itself its part. Moreover, because time, at least as long as being perceived (in case someone defended its strictly substantialist, Newtonian vision—or similarly posited empty time (cf. Allison, 2004, pp. 372–373)) arranges the succession of objects or events in the world, let them be physical or mental, the latter’s (viz., world’s) conditions of validity appear to stay beyond its domain. Otherwise, to argue again in a Kantian manner, we would need a time of a higher order (cf. Kant, 1998, pp. 508–509 A487/B515). Consequently, the transcendental ego’s perspective would contain an element of timelessness, rendering it close to how St. Augustine imagined the divine, allencompassing standpoint. For if all experience can (at least locally) be ever-repeated and tested towards its consequences, so that nothing really new occurs, it ceases to provide a truly fresh insight into empirical reality, constituting an outside, frozen bird’s eye instead. The quasi-divine character of that perspective would not have to

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mean that it is God who travels in time. Moreover, in a sense, there would be no travel at all. The transcendental ego would be reducible to the structure of the world, being a kind of its scaffolding. As Pore˛ba emphasizes, from the truly transcendental perspective there is neither any relativity nor a contingently distinguished frame of reference. Such a Parmenidean world would be untranslatable to our worldview (Pore˛ba, 2014, pp. 56–57).6 This world would have nothing chaotic at all. It would be a timeless vision, lacking any relativity or individual perspective. Thus, also lacking a conscious subject, as incapable of a fully objective, external perspective ‘from nowhere.’ Here, in this proposition, the world is saved, yet at the cost of the empirical subject’s disappearance. The remaining transcendental ego is rather the world (its structure) than the self. What we are left with is a timeless world without a conscious empirical subject, unlike in the first scenario, where the nonexistence of time entails the inevitability of chaos after the collapse of the time-based synthesis of experience. There, however, the self survives. It is beyond the scope of this short sketch to affirm one of the alternatives and dismiss the other. Whether we see the timeless Gödelian universe as an unstructured chaos perceived by a time-traveling individual or as a static and perfectly nonrelative whole without any subjectivity might depend on some deeper philosophical intuitions. Their investigation, however, must be carried on elsewhere (not to say elsewhen). As a concluding remark, let me say that in a timeless universe either the very world itself or a conscious subject can be maintained. To save them both, Nyles and Sarah have nothing else but to leave the wedding in Palm Springs as quickly as they can. Acknowledgements I would like to thank Prof. Marcin Pore˛ba and Maciej Głowacki for constructive comments to early versions of this paper as well as for fruitful discussions of some problems investigated in it. I am also grateful to the editors, especially Prof. Brigitte Falkenburg, whose remarks enabled me to expand the awarded essay to the form presented in this monograph.

References Allison, H. (2004). Kant’s transcendental idealism. Revised and enlarged edition. Yale University Press. Falkenburg, B., & Schiemann, G. (2016). Too many conceptions of time? Mc-Taggart’s views revisited. In S. Gerogiorgakis (Ed.), Time and tense. Unifying the old and the new (pp. 353–383). Philosophia. Gödel, K. (1949). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 21(3), 447–450. https://doi.org/10.1103/RevModPhys. 21.447. Guyer, P. (1987). Kant and the claims of knowledge. Cambridge University Press.

6 ‘From

the divine point of view, even if it takes the “secularized” form of the transcendental attitude, variability becomes pure extension, so to speak: different states, different variants of a certain structure or property can be seen from it as equally given, actualized, if not as a matter of fact, then at least in some broader space of possibilities’ (Pore˛ba, 2014, pp. 56–57).

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Heidemann, D. (2021). Kant and the forms of realism. Synthese, 198, 3231–3252. https://doi.org/ 10.1007/s11229-019-02502-4. Kant, I. (1998). Critique of pure reason. Cambridge University Press (trans. P. Guyer). Kant, I. (2002). Critique of the power of judgment, ed. P. Guyer, trans. P. Guyer, E. Matthews. Cambridge University Press. Kant, I. (2005). Notes and fragments, ed. P. Guyer, trans. C. Bowman, P. Guyer, F. Rauscher. Cambridge University Press. Loux, M. (2006). The nature of time. In M. Loux (Ed.), Metaphysics: A contemporary introduction (pp. 205–229). Routledge. McTaggart, J. E. (1908). The unreality of time. Mind, 17(68), 457–474. Meillassoux, Q. (2015). ‘Science-fiction and extro-science fiction’ in Q. Meillassoux, science-fiction and extro-science fiction followed by ‘The Billiard Ball’ by Isaac Asimov, trans. A. Edlebi. Univocal Publishing. Pore˛ba, M. (2014). Granice wzgle˛dno´sci. Opis metafizyczny. PWN SA, Fundacja na Rzecz My´slenia. Schmitt, R. (1959). Husserl’s transcendental-phenomenological reduction. Philosophy and Phenomenological Research, 20(2), 238–245. https://doi.org/10.2307/2104360. Sellars, W. (1968). Science and metaphysics. Variations on Kantian themes. Routledge & Kegan Paul, and Humanities Press. Stern, R. (2021). Transcendental arguments. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Fall 2021. Metaphysics Research Lab, Stanford University. Strawson, P. (1966). The bounds of sense. An essay on Kant’s ‘critique of pure reason’. Routledge Taylor & Francis Group. Yourgrau, P. (2005). A world without time: The forgotten legacy of Gödel and Einstein. Basic Books. Michał Pawłowski was born in 1996 in Warsaw. He graduated from the University of Warsaw with Master’s degrees in Philosophy and Econometrics and authored the book “O granicy poznania. Metafizycznie i einzcyzifatem” (On Limits of Knowledge. Metaphysically and yllacisyhpatem), Warsaw 2021. Currently, he holds a research position at the Department of Economics of the European University Institute in Florence. His primary philosophical interests include Kant, Wittgenstein, the philosophy of science, as well as self-referential argumentations.

7

Without Time, the World Becomes Leibnizian Bartosz Wesół

There is no general agreement on whether Gödel’s argument for the nonexistence of time is valid. Nonetheless, if there is even a small chance that this could be the case it is surely worth considering how the nonexistence of time would change our worldview. In this essay, I want to focus on the question: how would it affect our metaphysics, especially the notion of causality, if we assume that objective time does not exist? I also argue that the best candidate for the metaphysics of the world without time is Leibniz’s monadology.

7.1

Preliminary Remarks

Time is ‘directional’ Let me start with a couple of remarks concerning the notion of time. I want to stick to the Kantian conceptual framework, which will serve as a background for further reflections.1 According to Kant, one of the essential features of time is that ‘[t]ime is in itself a series (and the formal condition of all series)’ (Kant, 1998, B 438–4392 ). Accordingly, time ‘has only one dimension: different times are not simultaneous, but successive (just as different spaces are not successive, but simultaneous)’ (B 47).

1 Considering

issues discussed here, I believe such an approach is well motivated. Gödel as well as Einstein were ‘raised’ in the Kantian tradition, which, surely, dominated the German-speaking world of their times. 2 When referring to Kant’s Critique of pure reason (1998), I give the pagination of the second original edition (B) (i.e. Kritik der reinen Vernunft, Riga 1787). B. Wesół (B) University of Warsaw, Warsaw, Poland e-mail: [email protected] © Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_7

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Conversely, in space ‘there is no difference between progress and regress, because it constitutes an aggregate, but not a series, since all its parts exist simultaneously’ (B 439). From these passages, I want to extract the simple intuition that lays behind our common-sense notion of time, namely that time is ‘directional’. It simply means that time passes in a certain direction: from past to future and not in any different direction.3 In other words—to use the term used by Gödel (1946/9-B2, p. 235)—‘the flowing’ of time has a certain, fixed direction. On the other hand, things are different when it comes to space. As Kant puts it, space is ‘an aggregate’, not a series, and there are no differentiated directions in space.4 It is also worth noting that these considerations accord with Leibniz’s view on space and time as specific kinds of relations between substances and not substances themselves. Interestingly, one can also draw a connection between the ‘directionality’ of time and the classical Aristotelian notion of time. As Aristotle states: ‘[t]ime is a measure of motion’ (Aristotle, 2004, p. 61). If time depends on motion in a strong sense then it seems reasonable to assume that if motion always has a direction, then its measure, i.e., time, should also have a direction, ‘corresponding’ to the direction of this motion. Temporal series and causality According to what was said earlier, it seems that for Kant the causal relation between two objects is based on the idea of the ‘temporal series.’ The concept of time (as it is the ‘formal condition of all series’) should then be regarded as a fundament of the concept of causation. However, this is only partly true.5 In his ‘Second Analogy of Experience’ (see B 232–257), Kant inverts the relationship between time and causation saying that ‘[a]ll alterations occur in accordance with the law of the connection of cause and effect’ (B 232) (to be precise: ‘all alterations’ meaning ‘all change in time’). How can we make sense of this? Is Kant contradicting himself? In the first case, Kant speaks of subjective time—a transcendental form of the perceiving subject. Subjective time is a precondition of the scheme of causal relationships between appearances in the inner intuition, and in this case, time is more fundamental than the category of causality. However, the second case concerns objective time of the outer world where the concept of causality is a necessary condition for establishing the objective temporal order (cf. Kant’s ‘Refutation of Idealism’ in his B 274–294). Here, when considering the objective world, causality comes first, before temporal order. Gödel’s view accords with the second presumption. Wang in his Time in philosophy and in physics: From Kant and Einstein to Gödel (1995) presents some of Gödel’s remarks made by him in their private conversations. According to (Q6), Gödel said:

3 If

we assume that time is one-dimensional (which also seems to be the essential feature of the intuitive notion of time) it is enough to say that it does not go in the opposite direction, i.e., from future to past. 4 I am considering here an intuitive, physical space, i.e., the three-dimensional Euclidean space. 5 I want to thank Prof. Dr. Dr. Brigitte Falkenburg for drawing my attention to this matter.

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The real idea behind time is causation: the time structure of the world is just its causal structure. Causation in mathematics, in the sense of, say, a fundamental theorem causing its consequences, is not in time, but we take it as a scheme in time. (Wang, 1995, p. 229)

In general, I think we can call Gödel’s approach to philosophy more ‘objective’ than Kant’s transcendental one. Gödel used to insist that our human concepts need to be derived from something objective—in this case, that the concept of time should come from objective causation, which, according to Gödel, is the structure of the world in itself. Intuitive and metaphysical causation Let me here draw a terminological distinction that I regard as crucial in the issues discussed. To put things in order, I want to sketch the distinction between two opposing notions of causation. The first one is intuitive causation and the second is metaphysical causation. To put it pictorially, intuitive causation is the ‘causation of pool balls’.6 It is a causal relation between objects—in most cases, physical objects like pool balls— which directly interact with each other. It can also be a causal relation between two events (like the explosion of the bomb causing the collapse of the bridge) that involve some physical objects (in this case: the bomb and the bridge). It can be a relation between an agent and an object or an event as well.7 I believe that the roots of this notion lie in Newtonian physics, and when Kant spoke of ‘causality’ he probably had this notion of causation in mind. There are two distinctive features of intuitive causation that I would like to emphasize. First, usually in the case of intuitive causal relations, we can easily distinguish cause from effect, and what follows is that we can easily put them in a temporal order. Second, this ‘direct interaction’ is always local. The notion of metaphysical causation can be understood in various ways, depending on the given metaphysical theory, but the key idea behind it is that there exists a specific structure—namely the causal structure—in the world in itself (the world of objects, events or substances that in their existence are independent of any cognitive subject). This kind of causal structure can be broader and more general than the intuitive causal structure. Especially, it is not limited to physical objects or agents. Let us come back here to the passage cited from Gödel, where he speaks of the mathematical theorem causing its consequences (see Wang, 1995, p. 229). At first glance, this concept seems counterintuitive, for in this example, there is no interaction between any objects—we do not see any ‘action’ here and hence any ‘causation’ (in the intuitive sense). There is rather a logical relation between propositions. However, when we go deeper, we can see that there is a strong analogy between the relation of

6 Famously

brought up by Hume (1964, 164) in his Treatise on Human Nature, where he discusses the nature of causality. See the section ‘Of the idea of necessary connexion’ (Hume, 1964, 155–172). 7 In this case, causal relation can also involve nonphysical entities like mental states (e.g., my desire to know Gödel’s theorem better can be a cause of me grabbing the handbook of mathematical logic). Anyway, I would still claim that we should call the ‘agentic causation’ an intuitive causation.

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logical consequence and the intuitive relation between cause and effect. Both have the structure of the series, both have fixed directions, and both provide a specific type of ordering, which we can call metaphysical causation. The essential difference between these two types of causation is that intuitive causation (i.e., the direct interaction action between objects) always takes place in time. On the contrary, metaphysical causation is not so limited. According to the ‘classical view’,8 in the world of mathematical objects (or mathematical truths), there is no change and, thus, no need for time. However, and this is a crucial insight, there can still be a time-like (or series-like) ordering in this world, and that is exactly what ‘metaphysical causality’ stands for.

7.2

Nonexistence of Time and Metaphysics

Having set the grounds, let me consider now ‘the question’ of this essay: What does it mean for our worldview if, according to Gödel, we also assume the nonexistence of time? I want to focus here on the matter of how the nonexistence of time would affect our metaphysics,9 especially in one of the most fundamental of its aspects, namely: in the matter of causation. It is because I believe that causality plays a crucial role in our worldview (let alone the role it plays in the scientific worldview!) as it expresses one of the most basic aspects of the way we think about how things are related to each other. Furthermore, the concept of cause in the essence is attached to the category of explanation, since to explain something—to answer the question why?—is to indicate the cause.10 Moreover, a much deeper connection between causality and metaphysics can be drawn. When speaking of causal law as the condition of possibility of experience, Kant explicitly says: ‘This causality leads to the concept of action, this to the concept of force, and thereby to the concept of substance’ (B 249). If Kant is right here, the concept of causality constitutes the concept of substance—arguably the most important concept in the history of metaphysics. One can also see the ‘Newtonian theme’ in this quotation, namely, the crucial role of the concept of force—absolutely central in the whole of physics. No time, no intuitive causation So, if we assume that there is no objective time, how does it affect the concept of the world’s causal structure? As was pointed out earlier, time is directional and so is the intuitive causality. Furthermore, in our intuitive worldview, the direction of causal chains perfectly corresponds to the direction of the flow of time. Hence, without

8 Also

Gödel’s view which sometimes he would call “Platonism” (see Gödel, 1951, p. 311). this essay, I perpetually use the term ‘metaphysics’, but I suppose that in most cases it can be exchanged for the term ‘ontology’. 10 This is actually the traditional, Aristotelian view of what the notion of cause stands for. See, e.g., Aristotle (2009, p. 4). 9 In

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objective time, there would be no objective ordering of events, and no objective direction of causality. Let us consider a simple example: the explosion of the bomb (let us call this event ‘A’) causing the collapse of the bridge (event ‘B’). From the intuitive perspective, it is absolutely obvious that A can cause B and that B cannot cause A. Saying that the collapse of the bridge can cause the explosion of the bomb seems ridiculous. Moreover, what is far more significant here, it seems pointless to use the category of ‘cause’ (in the intuitive sense) in such a case. However, if we consider the special case within Gödel’s rotating universe with closed temporal loops (see Yourgrau, 1991, pp. 129–133), in principle it is possible that the time order of two events can get inverted: if we imagine that A and B are connected by the time-like loop, it is true to say both: that A happened before B, and that B happened before A. Problems occur if we still claim that events A and B are causally related. In an intuitive sense, it means that the causal order corresponds to the temporal order. In this example, it would follow that it is true that A caused B, and it is true that B caused A. However, ‘B caused A’ means that, for example, ‘the collapse of the bridge caused the explosion of the bomb’ (let us call this proposition ‘p’). Again, if we use the term ‘caused’ in the intuitive sense, this seems ridiculous, and I believe everyone would agree that (p) is just false and, moreover, that it cannot be true. Let me summarize this argumentation. We assumed that: 1. In Gödel’s universe with time loops it is possible that for some pairs of events A and B, A happened before B and B happened before A. 2. According to the intuitive view, there is a strong correspondence between temporal and causal order. So, if two events A and B are causally related and A happened before B then it follows that A caused B. 3. According to the intuitive view, if A caused B, then B could not have caused A and vice versa.11 4. A and B are causally related. If we then combine these four assumptions we conclude that: A caused B, and B caused A. However, if we consider our bomb-bridge example ‘B causes A’ is false. This example is aimed to show that the intuitive concept of causality does not apply to such cases in Gödel’s universe. Furthermore, it seems that it does not apply to the ‘world without objective time’. In such a world, there would be no objective flow of time and, hence, no objective causation (in intuitive sense).

11 For the matter of simplicity, I leave aside the problem of ‘reflective causation’, i.e., the question whether given event A can be the cause of itself. Logically speaking, if A = B, and we assume the possibility of reflective causation, then it is possible that A caused B and B caused A (which contradicts (3)).

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Need for a different notion of causality However, maybe if we broaden the concept of causality and make it independent from any temporal order, causation in the world without time can be maintained? First, I want to start with aforementioned Kant’s insight: if we were to change the concept of causation, then we would be forced to change our concept of interaction between objects, therefore also the concept of the object itself, and, eventually, the concept of substance. Hence, we need to change our metaphysics. Let me say something about the metaphysics of intuitive causation, i.e., the ‘metaphysics of the pool balls’—the ‘everyday-life metaphysics’ of macroscopic objects, which, I believe, lies behind classical, Newtonian physics. For the time being, we can stick to the image of pool balls, which serves as a paradigmatic example of objects that interact with each other causally. These kinds of objects have welldefined space-time positions and momenta and interact with each other directly—by contact. Through this interaction, they pass some of their properties to each other— like velocity and kinematic energy. It is very easy to apply the concept of causality in the case of pool balls as we can trace their interaction moment by moment and distinguish the series of causes and effects. However, one can easily see that this intuitive metaphysics cannot be applied so efficiently to other kinds of interactions (other than these basic ‘kinematic’ ones). Let us consider Newtonian gravity. It is still an interaction between two macroscopic objects, but the nature of this interaction is significantly different to the one described above. First, the interaction is not direct as there is no need for any contact, for there to be a gravitational interaction. Second, it is not so easy to apply the concept of causality to this interaction. Let us consider the Sun and the Earth going around it. Maybe, it seems right to say that the force of gravity makes the Earth follow the orbit (and not escape it), and so this force is the cause of this particular form of motion of the Earth. However, keeping in mind the view shared by Kant and Gödel, i.e., that causality has the time-like structure of a series, we can see that this notion of a cause is in this respect different to in the ‘pool balls causation’. It is very hard to impose any kind of ‘serial causation’ on the gravitational interaction. There is no straightforward causal chain that we can trace back in order to find out why the Earth is now in this particular position in reference to the position of the Sun. A few years after Einstein’s Special Relativity was published, Bertrand Russell wrote: All philosophers, of every school, imagine that causation is one of the fundamental axioms or postulates of science, oddly enough, in advanced sciences such as gravitational astronomy, the word ‘cause’ never occurs. (Russell, 1912, p. 1) (…) In the motions of mutually gravitating bodies, there is nothing that can be called a cause, and nothing that can be called an effect. (Russell, 1912, p. 14)

Russell concludes that the notion of ‘cause’ does not fully apply to the world as described by the science of his time. Fully geometrized general relativity as well as quantum mechanics with its indeterministic processes in a way have even worsened the situation for the use of the intuitive notion of causality in contemporary physics.

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So the question is even more imperative: if there is no time, how do we need to alter our notion of causation in order to maintain the basic idea laying behind it, namely that there is an order in the objective world or, in other words, that the objects are related to each other in an ordered way?

7.3

Answer: Leibniz’s Monadology

As we have seen, in a world without time there would be no room for intuitive causation and, therefore, no room for intuitive interaction between objects (i.e., substances). Therefore, considering the nonexistence of time, we are looking for a metaphysical system in which there are no interacting substances and no intuitive causation. However, as stated at the end of the previous section, it should be a system that allows us to preserve the idea of ordering in the world. Surprisingly, it is Leibniz’s monadology (one of the most ‘exotic’ metaphysical systems!) that comes with help. Let me then present some of the fundamental tenets of this system that answer our ‘metaphysical needs’. First, in Leibnizian metaphysics, there is no interaction between substances—no monad can cause any change in the other. ‘Monads have no windows (…) the natural changes in monads come from an internal principle, since an external cause could not influence [influer dans] their interior’ (Leibniz, 1989, pp. 643–644). Moreover, there is no genuine physical interaction between them as monads are simple spiritual, hence not material, substances (see Leibniz, 1989, pp. 643–644). Any change (essentially associated with time) of a monad’s attributes comes from its ‘interior’. Thus, Leibniz’s metaphysics is not limited to the idea of temporal ordering of causal relations (which intuitively refers to the interaction between objects).12 Second, there is a specific ordering in the world of monads—specific ‘harmony’ as it is in Leibniz’s concept of ‘pre-established harmony’ (Leibniz, 1989, p. 651). This harmony sets relations between substances, but these are not causal relations (in the intuitive sense). Even though substances do not interact with each other, changes of their attributes are perfectly ‘harmonized’. Moreover, this harmony is established independently from the perspectives of individual substances.

12 Furthermore,

it is also not limited to locality: ‘It follows that this communication extends to any distance whatever. As a result, every body responds to everything which happens in the universe, so that he who sees all could read in each everything that happens everywhere, and, indeed, even what has happened and will happen, observing in the present all that is removed from it, whether in space or in time’ (Leibniz, 1989, p. 649). Presumably, it can also help to set the metaphysical background for nonlocal interactions in quantum mechanics. If that would be the case, it would also show that Leibniz’s monadology not only serves as a good ground for the theory of relativity but also for quantum mechanics—two most important paradigms in contemporary physics. This issue has been addressed in detail by a Polish philosopher of physics Marek Woszczek in his ‘Serie Leibniza i problem dynamiki w kwantowaniu grawitacji’ (2011) (Eng. ‘Leibniz’s Series and the Problem of Dynamics in the Quantization of Gravity’).

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Third, Leibniz speaks of causes (and describes several types of them: efficient, final, and formal), but he uses this term in a different sense than we usually do. In simple words, I would say that for him the notion of cause has a more formal character. With his rather bizarre metaphysics, he is not limited to the physical causes and local interaction. From the historical point of view, I believe that he inherited the notion of cause from the scholastic tradition, whose roots go back to Aristotle himself (who introduced four types of causes: the same as Leibniz, plus the material cause13 ). Moreover, his philosophy is considered to be deterministic. However, it is crucial to see that this is a significantly different determinism than, e.g., Laplace’s determinism based on classical physics. In monadology, the determination does not come from causal interactions between substances that follow some necessary laws. It is established by universal harmony and, hence, is independent of causality and the ‘time parameter’ present in the dynamical laws of classical physics. Fourth, finally, let us consider the Leibnizian notion of time. By my lights, Leibniz’s metaphysical system is absolutely consistent with the nonexistence of the ‘cosmic time’ as it is described by Gödel.14 According to Leibniz, the world in itself is the world as it is perceived by God. However, from God’s perspective, there is no change and, hence, no flow of time. In monadology, time is only a phenomenon of finite substances and their ‘subjective’ perspectives, which fits surprisingly well with the conceptions of the ‘relative time’ of Einstein’s theory and the ‘subjective time’ in Kantian philosophy.15 In the case of Einstein, it is not of coincidence as is argued in Agassi (1969), where the author writes that ‘Einstein professed himself a Leibnizian and declared (…) the superiority of Leibnizianism [over Newtonianism]’ (Agassi, 1969, p. 331). Furthermore, we can see a deep connection between Leibniz’s view on the world in itself and Gödel’s idea of a world without time. It is worth noting here what Wang writes about Gödel’s own view on his philosophy: “On several occasions Gödel said that his philosophy is, in its general outline, like the monadology of Leibniz.” (Wang, 1995, p. 233).16 Reading Wang’s memories from conversations with Gödel, one can get the impression that Gödel would insist on the irrelevance of time in our perception of the world. For instance, when speaking of Hegel, Gödel was supposed to say that Hegel ‘[took] time too seriously’ (Wang, 1995, p. 229).

13 See

Aristotle (2004, pp. 19–21). term ‘cosmic time’ is used by Gödel in order to refer to the ‘objective’ or ‘real’ time in models of general relativity (see Gödel, 1946/9-B2). This ‘cosmic time’ is, in principle, supposed to be independent form the ‘subjective’ time of any particular observer. For Gödel it is the time of the world in itself. 15 About the connection between conceptions of time in theory of relativity and Kantian philosophy, see Gödel (1946/9-B2). 16 For the expansion of this thought, see Lethen (2021), especially the second section ‘Monads and Types’. 14 The

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Summary

I have tried to show that in the world without time there would be no causality in the intuitive sense. However, it does not have to mean that we need to abandon this fundamental notion. It rather forces us to consider the broader concept of causality that can be found in Gödel’s as well as Leibniz’s philosophy. The disappearance of intuitive causality leads us to the counterintuitive metaphysics of noninteracting substances, which are the base of Leibniz’s monadology.

References Agassi, J. (1969). Leibniz’s place in the history of physics. Journal of the History of Ideas, 30(3), 331–344. Aristotle. (2004). Physics. NuVision (Translated by R. P. Hardie and R. K. Gaye). Aristotle. (2009). Metaphysics. (Translated by W.D Ross). http://classics.mit.edu/Aristotle/ metaphysics.html. Accessed 30 Sept 2021. Gödel, K. (1946/9-B2). Some observations about the relationship between theory of relativity and Kantian philosophy. In S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: Collected works: Volume III: Unpublished essays and lectures (1995) (pp. 230–246). Oxford University Press. Gödel, K. (1951). Some basic theorems on the foundations of mathematics and their implications. In S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: Collected works: Volume III: Unpublished essays and lectures (1995) (pp. 304–321). Oxford University Press. Hume, D. (1964). A Treatise on Human Nature. (Edited by L.A. Selby-Bigge). Oxford: Clarendon Press. Kant, I. (1998). Critique of pure reason. Cambridge University Press (trans. P. Guyer). Leibniz, G. W. (1989). The monadology. In L. E. Loemker (Ed.), Philosophical papers and letters. The new synthese historical library. (Vol. 2, pp. 643–653). Springer. Lethen, T. (2021). Monads, types, and branching time-Kurt Gödel’s approach towards a theory of the soul. In O. Passon & C. Benzmüller(Eds.), Wider den Reduktionismus: Ausgewählte Beiträge zum Kurt Gödel Preis 2019 (pp. 13–24). Springer. https://doi.org/10.1007/978-3-662-631874_3. Russell, B. (1912). On the notion of cause. Proceedings of the Aristotelian Society, 13, 1–26. Wang, H. (1995). Time in philosophy and in physics: From Kant and Einstein to Gödel. Synthese, 102(2), 215–234. Woszczek, M. (2011). Serie Leibniza I Problem Dynamiki W Kwantowaniu Grawitacji. Filozofia Nauki, 19(2). Yourgrau, P. (1991). The disappearance of time: Kurt Gödel and the idealistic tradition in philosophy. Cambridge University Press. Bartosz Wesół is a graduate Master’s student in philosophy. He received his Bachelor degree in Philosophy within the Inter-faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw (within which he also studied physics) and received his Master’s degree at the Faculty of Philosophy, University of Warsaw. He is the president and cofounder of the Student Association of Logic and Philosophy of Religion, University of Warsaw. His academic interests include Kantian philosophy, philosophy of physics, logic and philosophy of religion.

8

What Does it Mean for Our World-View If We Assume with Gödel the Nonexistence of Time? Claus Kiefer

Certainty can only be reached in mathematics. But unfortunately it only touches the skirt of things. Who has ever felt a thorough astonishment about the world needs more. He philosophises … Wilhelm Busch in a letter to Maria Anderson, 2 May 1875

8.1

Time and the Gödel Universe

From 30 May to 1 June Cornell University hosted a small conference on the nature of time, organised by the originally Viennese astro-physicists Hermann Bondi und Thomas Gold. This meeting gathered an illustrious circle of physicists, mathematicians and philosophers. Among them the latter Nobel laureates of physics Subrahmanyan Chandrasekhar, Roger Penrose and Richard Feynman.1 Kurt Gödel did not attend, although roughly 15 years earlier the famous mathematician had made a significant contribution to this field. He had given an exact solution of Einstein’s field equations, which allowed for a rotating universe with closed time-like world lines.2 Especially the latter caused some excitement, since the existence of such world lines implies that such a universe allows for observers that may travel via their future into their own past—with all the consequences involved.

1 In

the Proceedings, the latter was denoted as “Mr. X” only; cf. Gold (1967).

2 The original publication is Gödel (1949); an expanded exposition is Gödel (1952). Both pieces are

(introduced by Stephen Hawking) reprinted in Gödel (1990). C. Kiefer (B) Institut für Theoretische Physik, Universität zu Köln, Cologne, Germany e-mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_8

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Why did Gödel no attend this meeting? A possible explanation is indicated in the brief presentation of Chandrasekhar; together with the subsequent discussion it is contained in the Proceedings (Gold, 1967, p. 68). The talk is entitled Geodesics in Gödel’s Universe and, thus, refers explicitly to Gödel’s cosmological solution. Chandrasekhar first reported at a seminar given by Gödel roughly 15 years earlier at Princeton.3 Subsequently, he raises criticism that he had already published previously. Chandrasekhar shows that such an observer moving on a geodesics (i.e. on a force-free curve of “free fall”) cannot reach his own past. That is indeed so, however, this claim was never raised by Gödel. His closed time-like curves, along which one can travel into one’s own past, are precisely no geodesics but describe an accelerated motion. One may expect that Gödel would have been given the opportunity to reply to Chandrasekhar’s criticism. However, this did not happen, and it can be suspected that Chandrasekhar’s criticism has contributed to the longtime lack of interest in Gödel’s solution.4 The discussion that followed this talk centred mainly on the possibility in principle of such closed time-like world lines within Einstein’s theory. As emphasised by the physicist Wolfgang Rindler there are also other solutions with this property. One is the so-called Anti-De Sitter space, which has currently gained popularity due to string theory.5 The Proceedings also contain a contribution by John Wheeler—like Gödel and Einstein based at Princeton, although not at the Institute for Advanced Study but the university. Wheeler comments only briefly on Gödel, and his concern is not the cosmological solution but his philosophical views, especially the relation between the subject and the object.6 We will see later that also these issues are related to Gödel’s interest in cosmology. How should we evaluate Gödel’s cosmological solution from today’s perspective? The “Gödel-universe” as presented in Gödel (1949) has not only closed time-like world lines but describes in particular a rotating universe. In contrast to the universe we live in, it is not expanding. This expansion of our universe was known already in 1949, and presumably this is one reason why Gödel introduced a rotating solution with expansion in Gödel (1952). However, this solution does not allow us to travel into the past. The Gödel-universe is matter-dominated with positive pressure and a negative cosmological constant, i.e. negative vacuum energy. The latter property is shared by the already mentioned Anti-De Sitter space, which is, however, matter-free and admittedly only of questionable empirical importance. The relevant literature on relativity contains Gödel’s solution; see, for example, the standard work by Hawking and Ellis (1973, pp. 168–170). In the absence of any absolute reference-frame (and this is a key element of Einstein’s theory) it is at first not obvious what rotation means. One may ask: rotation with respect to what? What is meant is that a local inertial system (e.g. realised by a free-falling gyroscope) rotates with respect to some far away object (e.g. a quasar).

3 This

talk was given on 7 May 1949; the handwritten notes were found in Gödels’s remains and published posthumously in Gödel (1995, 261 ff.). 4 See Dawson (1997) for a detailed presentation on the history of how Gödel’s work was received. 5 However, closed time-like curves may be avoided there by moving to the universal covering space. 6 See Gold (1967, p. 91).

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Or as Wheeler and Ford put it vividly: “In Gödel’s theory, individual galaxies rotate more in one direction than another—just as the hands of clocks on a wall rotate more in one direction than another”.7 The relevance of Gödel’s solution is due to its impact on the physical notion of time. For one thing, there is the possibility to have world lines that allow travel into one’s own past. In addition, the presence of rotation implies that even the illusion of an approximate global time can no longer be entertained. Admittedly, that Einstein’s theories compromise the notion of time is one of their fundamental properties anyway. However, one may shirk away from this consequence by adopting the view that only cosmological solutions with expansion and without rotation are relevant. Under this assumption there is a distinguished time t, which holds for all co-moving observers and which allows for an objective notion of simultaneity. In the presence of even a small rotation this is prevented, irrespective of whether or not the possibility of time travel exists. A rotating universe does not allow us to speak of an objective passage of time. That is the crux of the matter. Yet why was Gödel interested in these solutions anyway?

8.2

Time and Reality

Kurt Gödel was a mathematician and became world-famous through his incompleteness theorems, which shook the foundations of mathematics. Which way led him from this work to his results in cosmology and on time? In contrast to the laws of physics, the laws of logic are timeless. When time and logic get intermingled as in Aristotle’s example of the sea battle that will happen the next day (Chapter 9 of his De interpretatione), the issue is rather one of language than of an alleged time-dependent logical law (as with the statement of the excluded third in Aristotle’s example which holds only for past and present—but not for the future). How then, are these timeless logical laws and the time-dependent cosmology connected? A treasure trove in the search for the origin of Gödel’s interests is offered by his notebooks, the publication of which has recently begun.8 Stylistically, they are at times very reminiscent of the Philosophical Investigations of his compatriot Ludwig Wittgenstein. Gödel calls them Maximen, i.e. guiding principles. For example, they contain the remark that mathematics is the only science that has something perfect about it (the exact proof and the exact concept). Elsewhere, Gödel emphasizes that logic is not about laws of nature.9 No wonder that Gödel professed a Platonic world view in mathematics several times. Its structures are simply there and do not develop in time. For this reason, his relationship to the Viennese circle around Moritz Schlick, with whom he was

7 Wheeler

and Ford (1998, p. 309). Gödel (2019, 2020). 9 Gödel (2019, p. 50). 8 See

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in contact in the 1920s, was also divided; Gödel simply could not warm up to the positivism propagated there. A Platonic view of things is still taken today; a famous representative is Roger Penrose.10 Of course it was clear to Gödel that mathematics and reality and, thus, also their cognition are different things. Thus, for example, in Gödel (2019, p. 57) the passage can be found: Mathematical and logical cognition is of a principally different kind than reality cognition. Only this is a real cognition and for this philosophy is interested in the first place.

There can be no doubt that Gödel’s philosophical interest did not stop at mathematics but was also directed towards reality. From a Platonic point of view, however, this primarily raises the question of time and the associated change in the world. Does such a change exist at all? How can the Platonic world view of mathematics be reconciled with the dynamic world view of physics? On the surface, Gödel’s preoccupation with this question is seen as a reaction to Paul Arthur Schilpp’s invitation to write a contribution for the “Festschrift” planned for Einstein’s 70th birthday.11 As Howard Stein points out in his introductory remarks to this contribution,12 Gödel was primarily concerned with providing arguments for certain philosophical ideas that deny the objectivity of change. Of course, he was already flirting with these ideas before Schilpp’s invitation. It is clear from the title of the article that it is about idealist philosophy, by which primarily Kant’s philosophy is meant. Yet Gödel, according to Stein, does not hold the view of some representatives of this philosophy that everything real is of a mental nature; rather, he is only concerned that our idea of a changing world is an illusion. This, of course, recalls a famous sentiment of Einstein’s from a letter of condolence to the family of his long-time friend Michele Besso, written a month before his own death, according to which the distinction between past, present and future only has the significance of an illusion, albeit a persistent one. So does time exist only as an illusion? One could argue that Gödel merely reproduced there what follows naturally from the concept of time in Einstein’s theory of relativity. Yet this is not the case. The cosmological solutions of Einstein’s equations commonly used in Gödel’s time and also today are of a very special nature. They contain a distinguished class of world lines describing observers at rest relative to the expanding universe. Importantly, they do not experience rotation. For such observers, one can introduce an objective “cosmic” time that bears resemblance to Newtonian notions. All processes in nature could then be related to this distinguished time. Many researchers saw this as a rescue of the old Newtonian concept of time. Einstein’s theory may no longer contain

10 The

debate between the two physicists printed in Hawking and Penrose (1998) begins with the following sentence by Hawking: “This series of lectures has revealed quite clearly the difference between Roger’s views and mine. He is a Platonist and I am a positivist”. 11 This contribution appeared in 1949; the German version is Gödel (1955). 12 See Gödel (1990, pp. 199–201).

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objective time, but this does not apply to the universe in which we live, according to this view. Gödel sought and found a solution with rotation, a solution that shows that there are cosmological spacetimes that prohibit the introduction of objective time. That he actually did not regard his solution as pure mathematics but as a possible model for our own universe can be seen from the fact that he made diligent efforts to find effects of a rotation in existing observations. To quote Wheeler and Ford: “He was so passionately interested in the subject and so desperate for facts and figures, it turned out, that he had taken down the great Hubble photographic atlas of the galaxies, lined up a ruler on each galactic image to estimate the galaxy’s axis of rotation, and compiled statistics of the orientation. He found no preferred sense of rotation”.13 Today’s observations are much more precise than was even conceivable in Gödel’s time. Effects of a possible rotation should be found, for example, in the anisotropies of the cosmic background radiation, whose spectrum was carefully measured by the Planck space telescope between 2009 and 2013. Saadeh et al. (2016) find no evidence for rotation in the data.14 On the other hand, the dynamics of galaxies and galaxy clusters do give evidence for possible rotation; see Wang et al. (2021). Thus, it cannot be ruled out that there is or has been a rotation, for example, at the time of the very early universe, of which only scanty information reaches us. In any case, much more important than the actual limits found from observation is the possibility in principle of a rotating universe, which would refute an objective concept of time as with Newton. Gödel made this clear once and for all by explicitly constructing a spacetime that describes a rotating universe. The most important property of the Gödel cosmos for the discussion of the concept of time is, therefore, rotation. The fact that there are additionally closed time-like world lines in it, and thus the possibility of time travel, seems to strengthen Gödel’s argument of the nonexistence of an objective time. Finally, paradoxes could arise from the possible influence of one’s own past. Einstein only addresses this point in his response to Gödel’s “Festschrift” contribution, expressing unease about the possibility of time travel. However, in the decades since Gödel’s work, it has been learned that it is perfectly possible to come up with consistent solutions to Einstein’s equations that allow time travel and that make physical sense, i.e. that do not involve paradoxes.15 Moreover, in his 1952 paper, Gödel was able to posit rotating spacetimes without time travel. The nonexistence of an objective time follows solely from the possibility of a rotating universe. We have Gödel to thank for this insight. A consistent picture of our world must take this into account.

13 See

Wheeler and Ford (1998, p. 310). give an upper bound of 4.7 × 10−11 for the so-called vector mode that can be associated with rotation. 15 See for example Thorne (1994). However, these spacetimes require exotic entities such as wormholes, whose existence in nature is questionable. 14 They

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Time and Quantum Gravity

From Gödel’s famous theorems follows the incompleteness of formal systems. The importance of this insight can hardly be overrated. Jürgen Schmidhuber put it this way in an article for the Frankfurter Allgemeine Zeitung: “Gödel sent shock waves through the academic community at the time when he revealed the fundamental limits of computation, AI16 , logic and mathematics itself. This had an enormous impact on the science and philosophy of the 20th century”.17 Incompleteness is also a property of Einstein’s general theory of relativity. In this case, too, it follows from sophisticated mathematical theorems. As Roger Penrose, Stephen Hawking, Robert Geroch and others were able to prove in the 1960s, Einstein’s theory inevitably contains so-called singularities, limits of spacetime beyond which the theory loses its validity.18 The most important examples are the singularities at the Big Bang and inside black holes. As in Gödel’s case, the theorems were developed from within the theory itself, rather than from the perspective of a higher-level (as yet unknown) theory. Mathematical theorems cannot do without assumptions, of course. So one of the assumptions has at least indirectly to do with Gödel’s cosmological solution. Most singularity theorems postulate as one of their assumptions the absence of closed time-like world lines, i.e. the impossibility of time travel. Conversely, this means that singularities can be avoided in the presence of such lines. This is exactly the case in Gödel’s cosmos—its spacetime is complete and free of singularities. The incompleteness of formal systems can be remedied by moving on to a more comprehensive system, which is then admittedly itself incomplete. This expectation also applies to Einstein’s theory—a more comprehensive theory should avoid the previous singularities. In contrast to mathematics, however, the new physical theory should ideally manage without new singularities, i.e. it should no longer require a more comprehensive theory. Whether this is the case or whether there is possibly an infinite hierarchy of physical theories is beyond our knowledge. An answer to this question would be of importance that should not be underestimated. It is generally assumed that a quantum theory of gravitation will remedy the incompleteness of Einstein’s theory. Such a theory is only in its beginnings, but features of a “quantum gravity” can already be clearly recognised. One approach goes back essentially to the work of the already mentioned John Wheeler in Princeton, which he published in the 1960s. Together with the work of his colleague Bryce DeWitt from Austin, Texas, the researchers developed the foundations of what is now called quantum geometrodynamics. The starting point here is a property of Einstein’s theory that had only been recognised a few years earlier. One cannot capture the dynamics of relativity only through four-dimensional spacetimes, but through the dynamics of three-dimensional geometries (i.e. “spaces”), which can be incor-

16 Artificial

intelligence. (2021). 18 See Hawking and Ellis (1973) for a detailed treatment. 17 Schmidhuber

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porated into four-dimensional spacetime in very different, but not arbitrary, ways. For Wheeler, the restrictions on incorporating three-dimensional geometries into the four-dimensional were the expression of the fact that these three-dimensional geometries contain “information about time”, i.e. information about their position in spacetime. Two different three-dimensional geometries should make it possible to determine the elapsed proper time for all observers travelling between these geometries. This is the picture of classical (Einstein’s) theory. For the resulting quantum gravity it has drastic consequences.19 It is known from quantum mechanics that the orbits of, for example, an electron known from classical physics no longer exist there. In Einstein’s theory, however, the four-dimensional spacetimes correspond to these orbits. The same quantum formalism that causes the disappearance of orbits in mechanics leads to the disappearance of spacetimes there—only the three-dimensional spaces remain. With the fourth dimension, time also disappears. The fundamental equations of quantum gravity turn out to be timeless. Unfortunately, no reaction to these developments has come down to us from Gödel. Together with his colleagues Kip Thorne and Charles Misner, John Wheeler paid Gödel a visit in the early 1970s. The three were in the process of completing their extensive book Gravitation, which was to become one of, if not the classics of relativity. The three wanted to know from the famous mathematician whether he saw a relationship between his incompleteness theorem and Heisenberg’s indeterminacy relations.20 Gödel was incensed by this question. He was only interested in what the three wanted to write about his cosmological solution in their book.21 At first, Wheeler could not explain Gödel’s negative reaction to the mention of Heisenberg’s inequalities. At a cocktail party hosted by Oskar and Dorothy Morgenstern, Wheeler finally had the opportunity to talk to Gödel in person. Gödel revealed to Wheeler the reason for his unwillingness to discuss the uncertainty relations with him and his colleagues: There, at last, Gödel confessed to me why he had been unwilling to talk with Kip Thorne, Charlie Misner, and me about any possible connection between the undecidability he had discovered in the world of logic and the indeterminism that is a central feature of modern quantum mechanics. Because, he revealed, he did not believe in quantum mechanics. Gödel was a friend of Einstein and apparently the two walked and talked so much that Einstein had convinced him to abandon the teachings of Bohr and Heisenberg.22

Einstein did not doubt the validity of quantum theory, but was absolutely convinced of its incompleteness. In an elaborate thought experiment that he devised in 1935 together with Podolsky and Rosen, and which achieved great fame as the EPR-

19 See,

e.g. Kiefer (2008) or Barbour (2001) for a detailed account. for instance, the account in Yourgrau (2005, 164 f.). 21 In fact, Gödel’s solution remains unmentioned in the book. In the bibliography, one only finds Gödel’s famous 1931 paper on formally undecidable systems of Principia Mathematica and related systems. See Misner et al. (1973). 22 Wheeler and Ford (1998, p. 310). 20 See,

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Experiment, the authors drew the conclusion that the wave functions of quantum theory cannot provide a complete description of reality.23 This conclusion arose in particular from the contradiction of an assumed completeness with Einstein’s notion of a locality of descriptions, i.e. the idea that the real state of a system is independent of what happens to another spatially separated system at the same time. So on the walks with Einstein, Gödel had gradually adopted Einstein’s rejection of this theory. This probably prevented him from taking Wheeler’s approach to quantum gravity seriously and also to accept the nonexistence of time on the quantum level. However, what is the relevance of this insight? What is its relation to Gödel’s cosmology?

8.4

Where Is the Path Going?

The disappearance of time is not a property that is unique to quantum geometrodynamics. Every classical theory of gravitation, which, like Einstein’s theory, contains a dynamic picture of space and time, is stripped of its fourth dimension when quantised. In John Wheeler’s words: “These considerations show that the concepts of ‘space-time’ and ‘time’ are not primary but secondary ideas in the construction of physical theory. [...] There is no space-time, no time, no before and no after. The question ‘what happens next’ loses all meaning”.24 This nonexistence of time at the fundamental level does not, of course, contradict the usual concept of time in physics, but it now proves to be only a more or less good approximation. Physicists have developed a number of sophisticated procedures for this approximation, but the idea of an intrinsically existing time remains an illusion. Gödel reached this conclusion much earlier – not because of quantum gravity, but because of the possibility of rotating universes in Einstein’s theory. In doing so, he opened the door to a world view that is even more radical with regard to time than he himself could have imagined. Gödel is most famous as a mathematician, but it should be remembered that he had first studied physics. In his Maximen he remarks: “What originally interested me was the explanation of everyday life from higher concepts and general laws, hence physics”.25 Einstein, who revolutionised the physical concept of time like hardly anyone else, was distressed throughout his life that there seemed to be no place for the concept of the present (as opposed to the past and future) in physics. From the perspective of quantum gravity described above, this concern is relativised, since past and future

23 See,

e.g. Kiefer (2022). (1968, p. 26). 25 Gödel (2020, p. 81). From these original interests a straight path leads to Gödel’s derivation of his cosmological solutions. His worldview, which encompassed mathematics and physics, was consistent, even if it did not seem so at first glance. As Yourgrau has pointed out, Gödel was a realist (as a Platonist) with regard to mathematics and an idealist with regard to time, which is precisely not a contradiction, since an idealist conception of time corresponds very well to a realistic conception of the world. 24 Wheeler

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also have no fundamental meaning. In a sense, all these concepts are subjective and without meaning from a fundamental point of view. The nonexistence of time at the fundamental level follows naturally from established physical theories that are empirically established. Nevertheless, even specialists find it difficult to accept this consequence. After all, the experience of time is, it seems, an elementary everyday experience that structures our lives and human history and makes them possible in the first place. That the timelessness of the world enters the general consciousness is not to be expected in the foreseeable future (sic!). However, it will gradually gain a foothold in science and from there at some point radiate into other areas, with effects on human life that are likely to far exceed the effects of the transition from the Ptolemaic to the Copernican world view. We would then be back to a world view as already propagated by Parmenides of Elea over 2000 years ago—only on a much higher level.

References Barbour, J. (2001). The end of time: The next revolution in physics. Oxford University Press. Dawson, J. (1997). Logical Dilemmas: The life and work of Kurt Gödel. A K Peters. Gödel, K. (1949). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 21(3), 447–450. https://doi.org/10.1103/RevModPhys. 21.447. Gödel, K. (1952). Rotating universes in general relativity theory. In Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30 to September 6, 1950. American Mathematical Society (pp. 175–181). Gödel, K. (1955). Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der idealistischen Philosophie. In von P. A. Schilpp (Eds.), Albert Einstein als Naturforscher und Philosoph (pp. 406–412). Gödel, K. (1990). Collected Works, Vol. II: Publications 1938–1974, ed. S. Feferman, J. W. Dawson, S. C. Kleene, G. H. Moore, R. M. Solovay, J. v. Heijenoort. Oxford University Press. Gödel, K. (1995). Collected Works, Vol. III: Unpublished Essays and Lectures, ed. W. Goldfarb, C. Parsons, S. Feferman, J. W. Dawson, R. N. Solovay. Oxford University Press. Gödel, K. (2019). Philosophische Notizbücher, Vol. I, ed. von E.-M. Engelen. Gödel, K. (2020).Philosophische Notizbücher, Vol. II, ed. von E.-M. Engelen. Gold, T. (Ed.). (1967). The nature of time. Cornell University Press. Hawking, S., & Ellis, G. (1973). The large scale structure of space-time. In Cambridge Monographs on Mathematical Physics. Cambridge University Press. https://doi.org/10.1017/ CBO9780511524646. Hawking, S., & R. Penrose (1998). Raum und Zeit. Dt. von Claus Kiefer. Rowohlt. Kiefer, C. (2008). Der Quantenkosmos. Fischer. Kiefer, C. (Ed.). (2022). Albert Einstein, Boris Podolsky, Nathan Rosen: Can Quantum- Mechanical Description of Physical Reality Be Considered Complete? Birkhäuser. Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. Macmillan Education. Saadeh, D., Feeney, S. M., Pontzen, A., Peiris, H. V., & McEwen, J. D. (2016). How isotropic is the universe? Physical Review Letters, 117(13), 131302. https://doi.org/10.1103/PhysRevLett.117. 131302. Schmidhuber, J. (14. June 2021). Als Kurt Gödel die Grenzen des Berechenbaren entdeckte. Frankfurter Allgemeine Zeitung. Thorne, K. S. (1994). Gekrümmter Raum und verbogene Zeit: Einsteins Vermächtnis. Dt. von Doris Gerstner (4th ed.). Droemer Knaur.

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Wang, P., Libeskind, N. I., Tempel, E., Kang, X., & Guo, Q. (2021). Possible observational evidence for cosmic filament spin. Nature Astronomy, 5, 839–845. Wheeler, J. A. (1968). Einsteins vision. Springer. Wheeler, J. A., & Ford, K. (1998). Geons, Black Holes, and quantum foam: A life in physics. Norton. Yourgrau, P. (2005). A World without time: The forgotten legacy of Gödel and Einstein. Basic Books. Claus Kiefer is Professor of Theoretical Physics at the University Cologne (Germany). He has studied physics and astronomy in Heidelberg and Vienna and received his PhD for work on the notion of time in quantum gravity (supervised by D. Zeh). His major research interests cover general relativity and its possible extensions in both the classical and the quantum regimes. He has authored numerous books on quantum mechanics and quantum gravity—some of them also for a general audience. In 2013 he has received the Hanno und Ruth Roelin-Preis for his work on the popularisation of modern physics.

9

Was bedeutet es für unser Weltbild, wenn wir mit Gödel die Nichtexistenz der Zeit annehmen? Claus Kiefer

Gewißheit giebt allein die Mathematik. Aber leider streift sie nur den Oberrock der Dinge. Wer je ein gründliches Erstaunen über die Welt empfunden, will mehr. Er philosophirt … Wilhelm Busch in einem Brief an Maria Anderson, 2. Mai 1875

9.1

Zeit und Gödel-Kosmos

Vom 30. Mai bis 1. Juni 1963 fand an der Cornell-Universität eine kleine Tagung zur Natur der Zeit statt, ausgerichtet von den aus Wien stammenden Astrophysikern Hermann Bondi und Thomas Gold. Die Tagung vereinigte eine illustre Runde von Physikern, Mathematikern und Philosophen, darunter die späteren Nobelpreisträger der Physik Subrahmanyan Chandrasekhar, Roger Penrose und Richard Feynman.1 Kurt Gödel war nicht anwesend. Dabei hatte der berühmte Mathematiker etwa 15 Jahre zuvor Arbeiten veröffentlicht, die ganz wesentlich und weitreichend mit dem Begriff der Zeit zu tun hatten. Es war ihm gelungen, eine exakte Lösung von Einsteins Feldgleichungen der Gravitation zu finden, die ein rotierendes Universum beschreibt, das geschlossene zeitartige Weltlinien aufweist.2 Vor allem Letzteres hatte für Aufregung gesorgt. Die Existenz solcher Weltlinien bedeutet, dass es in einem solchen

1 Letzterer

tritt in dem Tagungsband nur als „Mr. X“ auf, siehe Gold (1967). Die Originalveröffentlichung ist Gödel (1949); eine ausführlichere und ergänzende Darstellung ist Gödel (1952). Beide Arbeiten sind, versehen mit einführenden Worten von Stephen Hawking, abgedruckt in Gödel (1990).

2

C. Kiefer (B) Institut für Theoretische Physik, Universität zu Köln, Köln, Deutschland E-Mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_9

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Universum Beobachter gibt, die über ihre Zukunft in die eigene Vergangenheit reisen könnten, mit all den damit verbundenen Konsequenzen. Warum war dann Gödel auf dieser Tagung abwesend? Ein möglicher Grund hierfür deutet sich in dem kurzen Vortrag von Chandrasekhar an, der mit den sich anschließenden Diskussionen in dem Tagungsband abgedruckt ist, siehe Gold (1967, S. 68). Der Vortragstitel lautet Geodesics in Gödel’s Universe und bezieht sich daher explizit auf Gödels kosmologische Lösung. Chandrasekhar referiert zunächst einen Vortrag Gödels, den dieser etwa 15 Jahre zuvor in Princeton gehalten hat.3 Dann bringt er eine Kritik ein, die er bereits zuvor in einer Fachzeitschrift publiziert hat. Er zeigt, dass es einem Beobachter, der sich im Gödelschen Universum auf Geodätischen bewegt, also auf kräftefreien Bahnen des „freien Falls“, tatsächlich nicht gelingt, in seine eigene Vergangenheit zu reisen. Das ist korrekt, nur hat Gödel dies nicht behauptet. Seine geschlossenen zeitartigen Weltlinien, denen entlang man in seine Vergangenheit reisen kann, sind gerade keine Geodätischen, sondern beschreiben beschleunigte Bewegungen. Mit solchen Bewegungen sind Zeitreisen möglich. Man hätte erwarten können, dass Gödel die Gelegenheit bekommen hätte, auf Chadrasekhars Kritik zu antworten, doch das ist nicht geschehen. Es steht zu vermuten, dass Chandrasekhars Kritik mitverantworlich für das Desinteresse war, das lange Zeit gegenüber Gödels Lösung herrschte.4 In den Diskussionsbeiträgen zu Chandrasekhars Vortrag geht es dann hauptsächlich um die prinzipielle Möglichkeit von geschlossenen zeitartigen Weltlinien in Einsteins Theorie. Wie der Physiker Wolfgang Rindler dort betont, gibt es durchaus andere Lösungen mit diesen Eigenschaften. Eine davon ist der sogenannte Anti-De-Sitter-Raum, der sich heute in der Stringtheorie großer Beliebtheit erfreut.5 In dem Tagungsband findet sich ebenfalls ein Beitrag von John Wheeler, auch er wie Gödel und Einstein in Princeton arbeitend, allerdings nicht am Institute for Advanced Study, sondern an der Universität. Wheeler geht nur kurz auf Gödel ein.6 Es geht ihm dabei nicht um Gödels kosmologische Lösung, sondern um dessen philosophische Ansichten, vor allem um die Beziehung von Subjekt und Objekt. Dass diese Ansichten sehr wohl mit Gödels kosmologischen Interessen zu tun haben, werden wir später sehen. Wie bewerten wir Gödels kosmologische Lösungen aus heutiger Sicht? Der in Gödel (1949) vorgestellte „Gödel-Kosmos“ hat – neben der schon diskutierten Präsenz geschlossener zeitartiger Weltlinien – vor allem die Eigenschaft, dass er ein rotierendes Universum beschreibt. Im Unterschied zu dem Universum, in dem wir leben, gibt es im Gödel-Kosmos keine Expansion. Die Expansion unseres Universums kannte man bereits 1949, was wohl der Grund dafür war, dass Gödel in der späteren Arbeit Gödel (1952) auch eine rotierende Lösung mit Expansion vorstellte, bei

3 Es

handelt sich um den Vortrag vom 7. Mai 1949, der später nach einem im Nachlass gefundenen handschriftlichen Manuskript in Gödel (1995, S. 261 ff.) veröffentlicht wurde. 4 Siehe Dawson (1997) für eine ausführliche Darstellung der Rezeptionsgeschichte von Gödels Arbeit. 5 Die geschlossenen zeitartigen Weltlinien können dort allerdings vermieden werden, indem man mathematisch betrachtet zu dem sogenannten Überlagerungsraum übergeht. 6 Siehe Gold (1967, S. 91).

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der die Möglichkeit, in die Vergangenheit zu reisen, nicht besteht. Der Gödel-Kosmos beschreibt ein materiedominiertes Universum mit positivem Druck und negativer Kosmologischer Konstante, also negativer Vakuumenergie; letztere Eigenschaft teilt er mit dem schon erwähnten Anti-De-Sitter-Raum, der freilich keine Materie enthält und dessen empirische Relevanz fragwürdig ist. In der einschlägigen Literatur zur Relativitätstheorie hat Gödels Lösung ihren Platz gefunden, siehe etwa das Standardwerk Hawking und Ellis (1973, S. 168–170). Bei Abwesenheit eines absoluten Bezugssystems, und das ist eine der zentralen Eigenschaften von Einsteins Theorie, ist zunächst nicht klar, was Rotation bedeutet. Rotation gegenüber was? Gemeint ist hier, dass lokale Inertialsysteme, die man etwa durch frei fallende Kreisel realisieren kann, gegenüber weit entfernten Objekten, etwa Quasaren, rotieren. Oder in den anschaulichen Worten von Wheeler und Ford: „In Gödels theory, individual galaxies rotate more in one direction than another – just as the hands of clocks on a wall rotate more in one direction than another.“ 7 Gödels Lösung ist vor allem wegen ihrer Folgen für den physikalischen Zeitbegriff von Bedeutung. Das betrifft natürlich zum einen die Möglichkeit der Existenz von Weltlinien, denen entlang man in die eigene Vergangenheit reisen kann. Zum anderen führt die Anwesenheit von Rotation dazu, dass man nicht einmal näherungsweise mehr die Illusion einer globalen Zeit aufrechterhalten kann. Gewiss, diese Relativierung des Zeitbegriffs ist eine der wesentlichen Konsequenzen von Einsteins Theorie, doch kann man sich vor dieser Konsequenz drücken, wenn man an den kosmologischen Lösungen mit Expansion ohne Rotation als den einzig relevanten Lösungen festhält. Es gibt dann tatsächlich eine ausgezeichnete Zeit t, die für alle mitbewegten Beobachter gilt, und bezüglich der man objektiv von Gleichzeitigkeit sprechen kann. Bei Anwesenheit selbst einer kleinen Rotation gilt das nicht mehr, und zwar unabhängig davon, ob es die Möglichkeit von Zeitreisen gibt oder nicht. Ein rotierendes Universum erlaubt es nicht mehr, von einem objektiven Zeitverlauf zu sprechen. Das ist der springende Punkt. Doch warum interessierte sich Gödel überhaupt für diese Lösungen?

9.2

Zeit und Wirklichkeit

Kurt Gödel war Mathematiker und wurde weltberühmt durch seine Unvollständigkeitssätze, welche die Grundlagen der Mathematik erschütterten. Welcher Weg führte ihn von diesen Arbeiten zu seinen Einsichten über Zeit und Kosmologie? Im Unterschied zu den Gesetzen der Physik sind die Gesetze der Logik zeitlos. Wenn Zeit und Logik dennoch vermischt werden, wie etwa in Aristoteles’ Beispiel der morgen stattfindenden Seeschlacht im neunten Kapitel seines De interpretatione, so handelt es sich eher um ein sprachliches Problem als um eine Zeitabhängigkeit logischer Gesetze (wie der behaupteten Zeitabhängigkeit des Prinzips vom ausgeschlossenen Dritten in Aristoteles’ Beispiel, das nur für Vergangenheit und Gegen-

7 Wheeler

und Ford (1998, S. 309).

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wart, aber nicht für die Zukunft gilt). Wie führt der Weg von der zeitunabhängigen Logik zu den zeitabhängigen Gesetzen der Kosmologie? Eine Fundgrube auf der Suche nach dem Ursprung von Gödels Interessen bieten seine Notizbücher, mit deren Publikation vor Kurzem begonnen wurde.8 Stilistisch erinnern sie zuweilen sehr an die Philosophischen Untersuchungen seines Landsmanns Ludwig Wittgenstein. Gödel nennt sie Maximen, also Leitsätze. So findet sich darin zum Beispiel die Bemerkung, dass die Mathematik die einzige Wissenschaft sei, welche etwas Vollkommenes an sich habe (den exakten Beweis und den exakten Begriff).9 An anderer Stelle betont Gödel, dass es sich bei der Logik nicht um Naturgesetze handele.10 Kein Wunder, dass sich Gödel mehrfach zu einem platonischen Weltbild in der Mathematik bekennt. Deren Strukturen sind einfach nur da und entwickeln sich nicht in der Zeit. Aus diesem Grund war auch sein Verhältnis zu dem Wiener Kreis um Moritz Schlick, mit dem er in den zwanziger Jahren in Kontakt stand, gespalten; Gödel konnte sich für den dort propagierten Positivismus einfach nicht erwärmen. Eine platonische Sichtweise der Dinge wird auch heute gerne eingenommen; ein berühmter Vertreter ist Roger Penrose.11 Natürlich war Gödel klar, dass Mathematik und Wirklichkeit und somit auch deren Erkenntnis unterschiedliche Dinge sind. So findet sich beispielsweise in Gödel (2019, S. 57) die Stelle: Mathematische und logische Erkenntnis ist von prinzipiell anderer Art als Wirklichkeitserkenntnis. Nur diese ist eine eigentliche Erkenntnis und für diese interessiert sich die Philosophie in erster Linie.

Es kann kein Zweifel daran bestehen, dass Gödels philosophisches Interesse nicht bei der Mathematik stehengeblieben ist, sondern sich auch auf die Wirklichkeit richtete. Von einem platonischen Gesichtspunkt aus stellt sich dabei aber vornehmlich die Frage nach der Zeit und der damit verbundenen Veränderung der Welt. Gibt es eine solche Veränderung überhaupt? Wie lässt sich das platonische Weltbild der Mathematik mit dem dynamischen Weltbild der Physik in Einklang bringen? Vordergründig gilt Gödels Beschäftigung mit dieser Frage als Reaktion auf die Einladung Paul Arthur Schilpps, einen Beitrag für die zu Einsteins siebzigsten Geburtstag geplante Festschrift zu verfassen.12 Wie Howard Stein in seinen einleitenden Bemerkungen zu diesem Beitrag betont,13 ging es Gödel vor allem darum, Argumente für gewisse philosophische Ideen zu liefern, welche die Objektivität einer

8 Siehe

Gödel (2019, 2020). (2020, S. 214). 10 Gödel (2019, S. 50). 11 Die in Hawking and Penrose (1998) abgedruckte Debatte zwischen den beiden Physikern beginnt mit dem folgenden Satz von Hawking: „Diese Vortragsreihe hat ganz deutlich den Unterschied zwischen Rogers und meinen Auffassungen offenbart. Er ist Platoniker und ich bin Positivist.“ 12 Dieser Beitrag ist 1949 erschienen; die deutsche Version ist Gödel (1955). 13 Siehe Gödel (1990, S. 199–201). 9 Gödel

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Veränderung leugnen. Mit diesen Ideen liebäugelte er natürlich schon vor Schilpps Einladung. Aus dem Titel des Artikels wird klar, dass es sich um idealistische Philosophie handelt, womit in erster Linie Kants Philosophie gemeint ist. Dabei vertritt Gödel, so Stein, nicht die Ansicht einiger Vertreter dieser Philosophie, dass alles Wirkliche von mentaler Natur sei; vielmehr geht es ihm nur darum, dass unsere Vorstellung einer sich verändernden Welt eine Illusion sei. Das erinnert natürlich an eine berühmte Sentenz von Einstein aus einem Kondolenzbrief an die Familie seines langjährigen Freundes Michele Besso, einen Monat vor seinem eigenen Tod geschrieben, wonach die Scheidung zwischen Vergangenheit, Gegenwart und Zukunft nur die Bedeutung einer, wenn auch hartnäckigen, Illusion habe. Existiert Zeit also nur als Illusion? Man könnte argumentieren, dass Gödel hier nur wiedergibt, was sich auf natürliche Weise aus dem Zeitbegriff in Einsteins Relativitätstheorie ergibt. Dem ist aber nicht so. Die zu Gödels Zeit und auch heute üblicherweise benutzten kosmologischen Lösungen von Einsteins Gleichungen sind von sehr spezieller Natur. Sie enthalten eine ausgezeichnete Klasse von Weltlinien, welche Beobachter beschreiben, die relativ zu dem expandierenden Universum ruhen. Wichtig ist, dass sie keine Rotation erfahren. Für solche Beobacher kann man eine objektive „kosmische“ Zeit einführen, die Ähnlichkeit mit den Newtonschen Vorstellungen aufweist. Alle Vorgänge in der Natur könnte man dann auf diese ausgezeichnete Zeit beziehen. Viele Forscher sahen darin eine Rettung des alten Newtonschen Zeitbegriffs. Einsteins Theorie möge zwar keine objektive Zeit mehr enthalten; auf das Universum, in dem wir leben, so die Forscher, treffe dies aber nicht zu. Gödel hat eine Lösung mit Rotation gesucht und gefunden, eine Lösung, die zeigt, dass es kosmologische Raumzeiten gibt, welche die Einführung einer objektiven Zeit verbieten. Dass er seine Lösung tatsächlich nicht als reine Mathematik, sondern als mögliches Modell für unser eigenes Universum betrachtete, lässt sich daran erkennen, dass er sich fleißig bemühte, Effekte einer Rotation in vorhandenen Beobachtungen ausfindig zu machen. Um Wheeler und Ford zu zitieren: „He was so passionately interested in the subject and so desperate for facts and figures, it turned out, that he had taken down the great Hubble photographic atlas of the galaxies, lined up a ruler on each galactic image to estimate the galaxy’s axis of rotation, and compiled statistics of the orientation. He found no preferred sense of rotation.“ 14 Heutige Beobachtungen sind wesentlich genauer, als dies zu Gödels Zeiten überhaupt vorstellbar war. Effekte einer möglichen Rotation sollten sich zum Beispiel in den Anisotropien der kosmischen Hintergrundstrahlung finden lassen, deren Spektrum vom Weltraumteleskop Planck in den Jahren 2009 bis 2013 sorgfältig ausgemessen wurde. Saadeh et al. (2016) finden in den Daten keinen Anhaltspunkt für eine Rotation.15 Andererseits ergeben sich aus der Dynamik von Galaxien und

14 Siehe

Wheeler und Ford (1998, S. 310). geben für die sogenannte Vektormode, die mit einer Rotation assoziiert werden kann, eine obere Schranke von 4, 7 × 10−11 an. 15 Sie

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Galaxienhaufen durchaus Hinweise auf eine mögliche Rotation, siehe Wang et al. (2021). Somit ist nicht ausgeschlossen, dass es eine Rotation gibt oder gegeben hat, zum Beispiel zu Zeiten des sehr frühen Universums, von dem uns nur spärliche Informationen erreichen. Viel wichtiger als die aus der Beobachtung gefundenen Schranken ist jedenfalls die prinzipielle Möglichkeit eines rotierenden Universums, wegen der es einen objektiven Zeitbegriff wie bei Newton nicht mehr geben kann. Dies hat Gödel durch die explizite Konstruktion einer Raumzeit, die ein rotierendes Universum beschreibt, ein für allemal klargestellt. Die für die Diskussion des Zeitbegriffs wichtigste Eigenschaft des Gödel-Kosmos ist also die Rotation. Dass es in ihm zusätzlich noch geschlossene zeitartige Weltlinien und somit die Möglichkeit von Zeitreisen gibt, scheint Gödels Argument der Nichtexistenz einer objektiven Zeit noch zu verstärken. Schließlich könnten sich durch die mögliche Beeinflussung der eigenen Vergangenheit Paradoxien ergeben. Einstein geht in seiner Antwort auf Gödels Festschriftbeitrag nur auf diesen Punkt ein und drückt dabei sein Unbehagen über die Möglichkeit von Zeitreisen aus. Allerdings hat man in den Jahrzehnten seit Gödels Arbeiten gelernt, dass man durchaus konsistente Lösungen von Einsteins Gleichungen aufstellen kann, die Reisen in der Zeit erlauben und die physikalisch sinnvoll sind, also ohne Paradoxien auskommen.16 Zudem konnte Gödel in seiner Arbeit von 1952 rotierende Raumzeiten ohne Zeitreisen aufstellen. Die Nichtexistenz einer objektiven Zeit folgt alleine aus der Möglichkeit eines rotierenden Universums. Diese Erkenntnis haben wir Gödel zu verdanken. Ein konsistentes Bild unserer Welt muss ihr Rechnung tragen.

9.3

Zeit und Quantengravitation

Aus Gödels berühmten Theoremen folgt die Unvollständigkeit formaler Systeme. Die Bedeutung dieser Erkenntnis kann kaum überschätzt werden. Jürgen Schmidhuber drückte es in einem Beitrag für die Frankfurter Allgemeine Zeitung so aus: „Gödel sandte seinerzeit Schockwellen durch die akademische Gemeinschaft, als er die fundamentalen Grenzen des Rechnens, der KI17 , der Logik und der Mathematik selbst aufzeigte. Dies hatte enorme Auswirkungen auf Wissenschaft und Philosophie des 20. Jahrhunderts.“ 18 Unvollständigkeit ist auch eine Eigenschaft von Einsteins allgemeiner Relativitätstheorie. Und auch in diesem Fall folgt sie aus ausgeklügelten mathematischen Theoremen. Wie Roger Penrose, Stephen Hawking, Robert Geroch und andere in den sechziger Jahren des letzten Jahrhunderts beweisen konnten, gibt es in Einsteins Theorie unweigerlich sogenannte Singularitäten, Grenzen der Raumzeit, hin-

16 Siehe

etwa Thorne (1994). Allerdings benötigen diese Raumzeiten exotische Gebilde wie etwa Wurmlöcher, deren Existenz in der Natur fragwürdig ist. 17 Künstliche Intelligenz. 18 Schmidhuber (2021).

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ter denen die Theorie ihre Gültigkeit verliert.19 Die wichtigsten Beispiele sind die Singularitäten beim Urknall und im Inneren Schwarzer Löcher. Wie in Gödels Fall wurden die Theoreme aus der Theorie selbst heraus entwickelt, nicht etwa aus der Perspektive einer übergeordneten (noch unbekannten) Theorie. Mathematische Theoreme kommen natürlich nicht ohne Annahmen aus. Und so hat eine der Annahmen zumindest indirekt mit Gödels kosmologischer Lösung zu tun. Die meisten Singularitätentheoreme postulieren als eine ihrer Annahmen die Abwesenheit geschlossener zeitartiger Weltlinien, also die Unmöglichkeit von Zeitreisen. Im Umkehrschluss bedeutet dies, dass Singularitäten bei Anwesenheit solcher Linien vermieden werden können. Genau dies ist in Gödels Kosmos der Fall – dessen Raumzeit ist vollständig und frei von Singularitäten. Die Unvollständigkeit formaler Systeme kann behoben werden, indem man zu einem umfassenderen System übergeht, das dann freilich selbst wieder unvollständig ist. Diese Erwartung trifft auch auf Einsteins Theorie zu – eine umfassendere Theorie sollte die bisherigen Singularitäten vermeiden. Im Unterschied zur Mathematik sollte die neue physikalische Theorie idealerweise aber ohne neue Singularitäten auskommen, also keiner umfassenderen Theorie mehr bedürfen. Ob dies der Fall ist, oder ob es womöglich eine unendliche Hierarchie von physikalischen Theorien gibt, entzieht sich unserer Kenntnis. Eine Antwort auf diese Frage wäre von nicht zu unterschätzender Bedeutung. Gemeinhin nimmt man an, dass eine Quantentheorie der Gravitation die Unvollständigkeit von Einsteins Theorie behebt. Eine solche Theorie liegt erst in Ansätzen vor, doch lassen sich Züge einer „Quantengravitation“ bereits deutlich erkennen. Ein Ansatz geht wesentlich auf Arbeiten des schon erwähnten John Wheeler in Princeton zurück, die dieser in den sechziger Jahren veröffentlicht hatte. Zusammen mit den Arbeiten seines Kollegen Bryce DeWitt aus Austin, Texas, entwickelten die Forscher die Grundlagen dessen, was man heute als Quantengeometrodynamik bezeichnet. Ausgangspunkt ist dabei eine Eigenschaft von Einsteins Theorie, die erst wenige Jahre zuvor erkannt worden war. Man kann die Dynamik der Relativitätstheorie nicht nur durch vierdimensionale Raumzeiten erfassen, sondern durch die Dynamik von dreidimensionalen Geometrien (also „Räumen“), die sich auf sehr unterschiedliche, aber nicht beliebige Weisen in die vierdimensionale Raumzeit einbauen lassen. Die Einschränkungen, die Dreiergeometrien in das Vierdimensionale einzubauen, waren für Wheeler der Ausdruck dafür, dass diese Dreiergeometrien „Information über die Zeit“ enthalten, also Information über deren Lage in der Raumzeit. Zwei unterschiedliche Dreiergeometrien sollten es erlauben, die verstrichene Eigenzeit für alle Beobachter zu ermitteln, die zwischen diesen Geometrien unterwegs sind. Das ist das Bild der klassischen (Einsteinschen) Theorie. Für die resultierende Quantengravitation hat es drastische Konsequenzen.20 Man weiß aus der Quantenmechanik, dass es dort die aus der klassischen Physik bekannten Bahnen etwa eines Elektrons nicht mehr gibt. Diesen Bahnen entsprechen in Einsteins Theorie aber

19 Siehe 20 Siehe

Hawking und Ellis 1973 für eine ausführliche Behandlung. z. B. Kiefer (2008) oder Barbour (2001) für eine ausführliche Darstellung.

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die vierdimensionalen Raumzeiten. Der gleiche Quantenformalismus, der in der Mechanik das Verschwinden der Bahnen bewirkt, führt dort auf das Verschwinden der Raumzeiten – nur die dreidimensionalen Räume bleiben zurück. Mit der vierten Dimension verschwindet auch die Zeit. Die grundlegenden Gleichungen der Quantengravitation entpuppen sich als zeitlos. Von Gödel ist leider keine Reaktion auf diese Entwicklungen überliefert. Zusammen mit seinen Kollegen Kip Thorne und Charles Misner stattete John Wheeler Gödel Anfang der siebziger Jahre einen Besuch ab. Die drei waren gerade dabei, ihr umfangreiches Buch Gravitation zu vollenden, das sich zu einem, wenn nicht dem Klassiker der Relativitätstheorie entwickeln sollte. Die drei wollten von dem berühmten Mathematiker wissen, ob er eine Beziehung zwischen seinem Unvollständigkeitssatz und Heisenbergs Unbestimmtheitsrelationen sehe.21 Gödel war ob dieser Frage erbost. Ihn interessierte nur, was die drei in ihrem Buch über seine kosmologische Lösung schreiben wollten.22 Wheeler konnte sich Gödels ablehnende Reaktion auf die Erwähnung von Heisenbergs Ungleichungen zunächst nicht erklären. Auf einer von Oskar und Dorothy Morgenstern ausgerichteten Cocktailparty fand sich schließlich die Gelegenheit zu einem persönlichen Gespräch mit Gödel. Dabei verriet dieser Wheeler den Grund für seine Unwilligkeit, mit ihm und seinen Kollegen über die Unbestimmtheitsrelationen zu diskutieren: There, at last, Gödel confessed to me why he had been unwilling to talk with Kip Thorne, Charlie Misner, and me about any possible connection between the undecidability he had discovered in the world of logic and the indeterminism that is a central feature of modern quantum mechanics. Because, he revealed, he did not believe in quantum mechanics. Gödel was a friend of Einstein and apparently the two walked and talked so much that Einstein had convinced him to abandon the teachings of Bohr and Heisenberg.23

Einstein zweifelte nicht an der Gültigkeit der Quantentheorie, war aber von deren Unvollständigkeit vollkommen überzeugt. In einem ausgeklügelten Gedankenexperiment, das er 1935 zusammen mit Podolsky und Rosen ersonnen hatte, und das als EPR-Experiment zu großer Berühmtheit gelangte, zogen die Autoren die Schlussfolgerung, dass die Wellenfunktionen der Quantentheorie keine vollständige Beschreibung der Realität liefern können.24 Dieser Schluss ergab sich insbesondere aus dem Widerspruch einer angenommenen Vollständigkeit mit Einsteins Vorstellung einer Lokalität der Naturbeschreibung, der Vorstellung, dass der reale Zustand eines Systems unabhängig davon ist, was mit einem anderen räumlich getrennten System zur gleichen Zeit geschieht.

21 Siehe

etwa die Schilderung in Yourgrau (2005, 164 f.). bleibt Gödels Lösung in dem Buch unerwähnt. Im Literaturverzeichnis findet man dort nur Gödels berühmte Arbeit von 1931 über formal unentscheidbare Systeme der Principia Mathematica und verwandte Systeme. Siehe Hawking und Ellis 1973. 23 Wheeler und Ford (1998, S. 310). 24 Siehe z. B. Kiefer (2015). 22 Tatsächlich

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Auf den Spaziergängen mit Einstein hatte Gödel also allmählich Einsteins Ablehnung dieser Theorie übernommen. Das hat ihn wohl daran gehindert, Wheelers Zugang zur Quantengravitation ernst zu nehmen und die Nichtexistenz der Zeit auf der Quantenebene als relevant zu akzeptieren. Aber welche Relevanz hat diese Erkenntnis? Und was ist deren Beziehung zu Gödels Kosmologie?

9.4

Wohin geht der Weg?

Das Verschwinden der Zeit ist keine Eigenschaft, die nur der Quantengeometrodynamik zu eigen ist. Jede klassische Theorie der Gravitation, die wie Einsteins Theorie ein dynamisches Bild von Raum und Zeit enthält, wird bei der Quantisierung der vierten Dimension entledigt. Mit John Wheelers Worten: „Diese Betrachtungen zeigen, daß die Begriffe Raum-Zeit‘ und Zeit‘ nicht primäre sondern sekundäre Ideen ’ ’ im Aufbau der physikalischen Theorie sind. [...] Es gibt keine Raum-Zeit, keine Zeit, kein Vorher und kein Nachher. Die Frage was geschieht als Nächstes‘ verliert jeden ’ Sinn.“ 25 Diese Nichtexistenz der Zeit auf grundlegender Ebene steht natürlich nicht in Widerspruch zum üblichen Zeitbegriff der Physik, doch erweist sich dieser jetzt nur noch als mehr oder weniger gute Näherung. Die Physiker haben eine Reihe ausgeklügelter Verfahren für diese Näherung entwickelt, doch bleibt die Idee einer an sich existierenden Zeit eine Illusion. Gödel ist bereits viel früher zu diesem Schluss gelangt – nicht wegen der Quantengravitation, sondern wegen der Möglichkeit rotierender Universen in Einsteins Theorie. Damit hat er die Tür zu einem Weltbild geöffnet, das bezüglich der Zeit noch viel radikaler ist, als er selbst es sich vorstellen konnte. Gödel ist vor allem als Mathematiker berühmt, doch sei daran erinnert, dass er zunächst Physik studiert hatte. In seinen Maximen merkt er dazu an: „Was mich ursprünglich interessiert hat, ist die Erklärung des Alltagslebens aus höheren Begriffen und allgemeinen Gesetzmäßigkeiten, daher Physik.“ 26 Von diesen ursprünglichen Interessen führt ein gerader Weg zu Gödels Ableitung seiner kosmologischen Lösungen. Sein Weltbild, das Mathematik und Physik umfasste, war konsistent, auch wenn dies auf den ersten Blick nicht so schien. Wie Yourgrau betont hat, war Gödel in Bezug auf die Mathematik Realist (als Platoniker) und hinsichtlich der Zeit Idealist, was eben kein Widerspruch darstellt, da eine idealistische Auffassung der Zeit sehr wohl einer realistischen Auffassung der Welt entspricht.27 Einstein, der wie kaum ein anderer den physikalischen Zeitbegriff revolutioniert hat, war zeitlebens bekümmert, dass sich für den Begriff der Gegenwart (im Unterschied zu Vergangenheit und Zukunft) in der Physik kein Platz zu finden schien. Von der geschilderten Perspektive der Quantengravitation aus relativiert sich diese Sorge, da auch Vergangenheit und Zukunft keine fundamentale Bedeutung haben.

25 Wheeler

(1968, S. 26). (2020, S. 81). 27 Yourgrau (2005). 26 Gödel

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In gewissem Sinne sind alle diese Begriffe subjektiv und von einer grundlegenden Warte aus ohne Bedeutung. Die Nichtexistenz der Zeit auf fundamentaler Ebene ergibt sich auf natürliche Weise aus etablierten physikalischen Theorien, die empirisch etabliert sind. Dennoch fällt es selbst Spezialisten schwer, diese Konsequenz zu akzeptieren. Schließlich ist das Erleben der Zeit eine, so scheint es, elementare Alltagserfahrung, die unser Leben und die menschliche Geschichte strukturiert und überhaupt erst ermöglicht. Dass die Zeitlosigkeit der Welt in das allgemeine Bewusstsein eintritt, ist auf absehbare Zeit (sic!) nicht zu erwarten. Sie wird aber in der Wissenschaft allmählich Fuß fassen und von da aus irgendwann in andere Bereiche ausstrahlen, mit Auswirkungen auf das menschliche Leben, welche die Auswirkungen des Übergangs vom ptolemäischen zum kopernikanischen Weltbild bei Weitem übersteigen dürften. Wir wären dann wieder bei einem Weltbild angelangt, wie es schon Parmenides von Elea vor über 2000 Jahren propagiert hat – nur auf einer viel höheren Ebene.

Literatur Barbour, J. (2001). The end of time: The next revolution in physics. Oxford University Press. Dawson, J. (1997). Logical dilemmas: The life and work of Kurt Gödel. A K Peters. Gödel, K. (1949). An Example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 21(3), 447–450. https://doi.org/10.1103/ RevModPhys.21.447. Gödel, K. (1952). Rotating universes in general relativity theory. In Proceedings of the international congress of mathematicians, Cambridge, Massachusetts, U.S.A., August 30 to September 6, 1950, American Mathematical Society (S. 175–181). Gödel, K. (1955). Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der idealistischen Philosophie. In von P. A. Schilpp (Hrsg.), Albert Einstein als Naturforscher und Philosoph (S. 406–412). Gödel, K. (1990). Collected works, Bd. II: Publications 1938–1974, ed. S. Feferman, J. W. Dawson, S. C. Kleene, G. H. Moore, R. M. Solovay, J. v. Heijenoort. Oxford University Press. Gödel, K. (1995). Collected works, Bd. III: Unpublished Essays and Lectures, Hrsg. W. Goldfarb, C. Parsons, S. Feferman, J. W. Dawson, R. N. Solovay. Oxford University Press. Gödel, K. (2019). Philosophische Notizbücher, Bd. I, Hrsg. von E.-M. Engelen. Gödel, K. (2020). Philosophische Notizbücher, Bd. II, Hrsg. von E.-M. Engelen. Gold, T. (Hrsg.). (1967). The nature of time. Cornell University Press. Hawking, S., & Ellis, G. (1973). The large scale structure of space-time. In Cambridge monographs on mathematical physics. Cambridge University Press. https://doi.org/10.1017/ CBO9780511524646. Hawking, S., & Penrose, R. (1998). Raum und Zeit. Dt. von Claus Kiefer. Rowohlt. Kiefer, C. (2008). Der Quantenkosmos. Fischer. Kiefer, C. (Hrsg.). (2015). Albert Einstein, Boris Podolsky, Nathan Rosen: Kann die quantenmechanische Beschreibung der physikalischen Realität als vollständig betrachtet werden? Springer Spektrum. Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. Macmillan Education. Saadeh, D., Feeney, S. M., Pontzen, A., Peiris, H. V., & McEwen, J. D. (2016). How isotropic is the universe? Physical Review Letters, 117(13), 131302. https://doi.org/10.1103/PhysRevLett.117. 131302. Schmidhuber, J. (14. Juni 2021). Als Kurt Gödel die Grenzen des Berechenbaren entdeckte. Frankfurter Allgemeine Zeitung.

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Thorne, K. S. (1994). Gekrümmter Raum und verbogene Zeit: Einsteins Vermächtnis. Dt. von Doris Gerstner (4th ed.). Droemer Knaur. Wang, P., Libeskind, N. I., Tempel, E., Kang, X., & Guo, Q. (2021). Possible observational evidence for cosmic filament spin. Nature Astronomy, 5, 839–845. Wheeler, J. A. (1968). Einsteins vision. Springer. Wheeler, J. A., & Ford, K. (1998). Geons, Black Holes, and quantum foam: a life in physics. Norton. Yourgrau, P. (2005). Gödel, Einstein und die Folgen. Dt. von Susanne Kuhlmann- Krieg und Kurt Beginnen (2nd ed.). Beck. Claus Kiefer ist Professor für theoretische Physik an der Universität zu Köln. Er studierte Physik und Astronomie in Heidelberg und Wien und promovierte mit einer Arbeit über den Begriff der Zeit in der Quantengravitation (betreut von D. Zeh). Sein Hauptforschungsinteresse gilt der allgemeinen Relativitätstheorie und ihren möglichen Erweiterungen sowohl im klassischen als auch im Quantenbereich. Er hat zahlreiche Bücher über Quantenmechanik und Quantengravitation verfasst, von denen einige auch für ein allgemeines Publikum bestimmt sind. Im Jahr 2013 erhielt er den Hanno und Ruth Roelin-Preis für seine Arbeit zur Popularisierung der modernen Physik.

Consequences from the Impossibility of Objectively Identifying Change. Philosophical Considerations Following Kurt Gödel

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In A remark about the relationship between relativity theory and idealistic philosophy (1949a) Kurt Gödel argues that an objective identification of change and time is impossible.1 Yet, how can this non-objectivity and its argument be philosophically conceived? Further: what consequences can be drawn from this? The paper will regard these questions as follows. First, Gödel’s argument for the impossibility of an objective identification of change will be outlined (Sect. 10.1). Building on considerations of Gödel, the ontological and epistemological meaning of the non-objectivity of change and time will be developed. It comprises: 1.1) illusion as deception, 1.2) illusion as a process of discovery, 2) appearance and 3) a combination of 1.2) and 2) with Michael Friedman’s theory of relativized a priori (Sect. 10.2). As Gödel’s argument is neutral towards these interpretations a concept of our own will follow. In accordance with J. Ellis McTaggart’s distinction between A and B series of time and with reference to the work of Peter Rohs a differentiation in the concept of time is carried out. In combination with the pragmatic ideas of Peter Janich this leads to an enhancement of 2), which integrates versions of 1) and 3) (Sect. 10.3). Finally, in Sect. 10.4 an outlook is given to broaden the perspective and to show that Gödel’s argument can be expanded towards the general question of intersubjectivity.

1 If not stated otherwise, “possible” and “impossible” are taken as conceivable and as argumentatively

justifiable. T. Alles (B) University of Bonn, Bonn, Germany e-mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_10

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Gödel’s Argument for the Non-Objectivity of Change and Time

Kurt Gödel is famous for his revolutionary research in mathematics and logic. Furthermore, he has worked on numerous philosophical and theological matters. However, he kept his notes mostly unpublished, so they are still being edited in parts. While Gödel published some of his works on the philosophy of mathematics, A remark about the relationship between relativity theory and idealistic philosophy (1949a), which he wrote in honour of his friend Albert Einstein, seems to be the only other philosophical publication during his lifetime.2 Therein, Gödel deals with the question of whether change is objective or only an “illusion” or “appearance”. Change is here conceived as change over time. From relativity theory one can deduce an argument that time courses cannot be identified objectively but only dependent on the observer’s position, for simultaneity is dependent on the observer’s position in relativity theory. However, according to Gödel, this argument can be refuted by establishing a world time and by relating other time observations to this world time.3 That is why he develops his own argument against the objectivity of time, which aims to show the impossibility of objectively establishing a world time. For that, he refers to the physicomathematical explanations of his paper that was published shortly thereafter An example of a new type of cosmological solution of Einstein’s field equation of gravitation. There, he presents his relativity-theoretic conception of rotating universes. As he expresses in more detail in his Lecture on rotating universes, which underlies his paper, time travel is possible in these rotating universes.4 Gödel uses this physical possibility of time travel in A remark about the relationship between relativity theory and idealistic philosophy (1949a) as an argument against the objectivity of time; for every definition of a world time it is possible to travel once or several times into the past in these rotating universes. Hence, there could be observers whose time observation cannot be related to the world time. In extreme cases, from the point of view of the world time, these observers might only exist at one moment, i.e. without duration, while there would be duration from the point of view of these specific observers. From that, Gödel concludes that accepting objectivity of time is unjustified.5 It is important to note that the argument is not based on the supposition that our universe really complies with Gödel’s rotating universes. Gödel himself has even given an example for a solution of Einstein’s field equations that does not allow for closed time-like curves and, therefore, offers no possibility of time travel in his paper Rotating universes in general relativity theory from 1952. However, in the posterior German publication of 1949a in 1955, Gödel adds a reference to his

2 Cf.

Stein: Introductory Note to 1949a, 199. Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 557–559/CW II 202–204. Published works are first cited with the original pagination and then with the page number of the respective Kurt Gödel. Collected Works issue (abbreviated as “CW“). 4 Cf. Gödel: Lecture on rotating universes, CW III 285–287. 5 Cf. Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 560 f./CW II 204–206. 3 Cf.

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paper from 1952 and nonetheless sticks to his argumentation from 1949a.6 For using rotating universes, the demonstrated possibility that our universe might be formed in such a way that time travel is possible is totally sufficient. Possibility is not meant in a technical or practical sense here. This becomes clear when one considers the absurdly high amounts of energy and velocity that would be needed for such time travel.7 Instead, it is a physical or metaphysical understanding of possibility. For the supposition is that the objectivity of time should depend on natural laws of the universe and not on the contingent arrangement and movement of matter. However, as natural laws do not exclude the possibility of time travel—as has been shown with the idea of rotating universes—these contingent conditions would be the only possibility left for the objectivity of time.8

10.2

Philosophical Interpretations of Gödel’s Argument

Let us suppose that Gödel’s argument is correct and that objective time is a necessary condition for objective change: what are the consequences for our understanding of the world? This question shall be studied in the following paragraph. Hence, the physicomathematical and the cosmological suppositions, as well as the conclusiveness of the metaphysical argument shall be endorsed for the time being. Starting with the explanations in the foregoing section it is clear that objective time means that there is one position of reference for assigning and comparing moments and intervals of time, which applies to all observing positions. If such an object of reference is not possible, there are two theoretical alternatives. Gödel calls them 1) “illusion” and 2) “an appearance due to our special mode of perception”.9 These shall be further explained and differentiated in the following paragraphs. Furthermore, there will be a third alternative. 1) If objective time and change is conceived as pure illusion, one holds a staticdeterministic worldview. What is perceived as change would then either be mere deception or the process of discovery of what has always been true but not necessarily known. 1.1) The option of mere deception would presumably be difficult to defend. First, it does not seem possible to neglect the perception of change phenomenologically.

6 Cf.

Gödel: Eine Bemerkung über die Beziehungen zwischen der Relativitatstheorie und der idealistischen Philosophie, footnote 10. 7 Cf. Gödel: A remark about the relationship between relativity theory and idealistic philosophy, footnote 11. 8 Cf. Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 562 f./CW II 206 9 Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 557 f./CW II 202.

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Second, Gödel’s argument does not exclude subjective time. He even states: “Each observer has his own set of ‘nows”’.10 1.2) Putatively, Gödel would have consented to the second theoretical option, the process of discovery. Thereby, thinking, perceiving and knowing would be a process in the direction of a truth, that holds “always”. The inverted commas shall indicate that “always” does not mean a temporal extension because the truth that is directed at would be valid independently of temporal specifications. It might be called “eternal” instead. Such a conception could go well along with Gödel’s Platonism, which assigns reality to mathematical concepts. Thereby, it is necessary to assume a process of thinking and discovery. For according to Gödel’s first incompleteness theorem there are mathematical propositions that cannot be proved within the present mathematical-axiomatic system.11 Consequently, they need to be discovered by the human mind.12 For example, this might take place by mathematical intuition.13 Therefore, it is to be expected that Gödel conceives mathematics as fixed and at the same time not immediately and completely given, so that there is a process of discovery.14 Even though Gödel’s Platonism is reconstructed only in outlines and with incomplete consideration of the sources, the focus in this paper lies on his most important works, which are most relevant and for the most parts have been published or presented. The preceding considerations, therefore, show that a process of discovery towards a timeless truth goes well along with important aspects of others of Gödel’s philosophical thoughts. Furthermore, this option would be compatible with possible consequences of Gödel’s theological thoughts. His ontological proof of God, which relies on formal logic and has neither been published nor further interpreted by him, points at the necessary existence of a being—i.e. God—that comprises all positive properties.15 Therefore, all positive properties can be conceived as sharing in this necessarily existing being. Following the conversation reports of Hao Wang, Gödel intended to develop a system with God as the central monad like Leibniz.16 The idea of God as a timeless17 or unchangeable reality, which all thinking

10 Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 558/CW II 203. 11 Cf. Gödel: Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I, 173–191/CW I 144–180. 12 Cf., e.g. Gödel: Some basic theorems on the foundations of mathematics and their implications, 9–13/CW III 308–310. 13 Cf. e.g. Gödel: What is Cantor’s continuum problem?, 271/CW II 267 f. 14 As far as I know, Gödel does not speak of a process. However, “development” (“Entwicklung”) can at least be found in a note in his bequest. Cf. Gödel: Meine ph Ansicht, statement numbers 2, 7 and 8. 15 Cf. Gödel: Ontological proof, CW III 403 f. 16 Cf. Wang: A Logical Journey, 8. 17 For example, Otto Muck proposes to understand the modal operator in Gödel’s proof as temporal. Thereby, God’s necessary existence means that God exists at all times. Cf. Muck: Eigenschaften Gottes im Licht des Gödelschen Arguments, 70–72.

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shares in as a process and to which its discoveries converge, is at least one possible conceptual consequence. Jordan Howard Sobel showed that under certain conditions18 actuality, possibility and necessity coincide in Gödel’s proof of ´ ˛torzecka suggest that this God’s existence.19 Sre´cko Kovaˇc und Kordula Swie so-called modal collapse might be intended by Gödel.20 Thereby, the proof would itself be a process of thinking towards a last and unchangeable truth. These considerations, and particularly the last one, are, of course, mere speculations. Nonetheless, they show at least the possibility of Gödel’s theological ideas harmonizing with the abandonment of objective time in combination with a subjective process of discovery. 2) The alternative option of regarding change as appearance is more difficult to regard with reference to Gödel. In his discarded manuscripts for the paper 1949a, he put as the title: Some observations about the relationship between theory of relativity and Kantian philosophy.21 Consequently, the context of discovery is—as Gödel also remarks in his Lecture on rotating universes and as he at least hints at in a footnote in 1949a22 —an analogy between the Kantian subjectivitiy theorey with its a priori conception of time and the dependence on positions for time measurement in the theory of relativity. As is obvious, with the later change of title Gödel decided to present his ideas less centred around certain persons. This is sensible since Immanuel Kant’s transcendental philosophy in his Critique of Pure Reason differs in some aspects; there, he reflects on the conditions of “Erkenntnis” (among else: knowledge, cognition, perception, understanding)23 and finds time to be the a priori given form of subjects’ thinking and perception. According to Kant, time is the form in and with which all humans are thinking. Only due to time is the world describable at all.24 Even tough time does not exist as a thing-in-itself, a certain objective state still rests, namely the shared form of human perception and thinking—or even the shared form of every reasonable perception and thinking. It should not be understood as relativism that Kant calls this “subjektiv” (“subjective”) in opposition to “objektiven Bestimmungen” (“objective specifications”).25 Rather, it means “generally applicable” for all reasonable subjects. This corresponds to Kant’s so-called “Copernican Revolution”,26 which leads to no longer asking for the

18 The discussion about these conditions is surveyed in Alles: Gödels ontologischer Gottesbeweis im Kontext der Systematischen Theologie, 26–28. 19 Cf. Sobel: Gödel’s Ontological Proof, 250–253. 20 Cf. Kovaˇ ´ ˛torzecka: Gödel’s “Slingshot” Argument and His Onto-Theological System. c/Swie 21 Via the Collected Works issue the versions *1946/49/9-C1 and *1946/9-B2 are accessible. 22 Cf. Gödel: Lecture on rotating universes, CW III 274; id.: A remark about the relationship between relativity theory and idealistic philosophy, footnote 3. 23 As its semantics is so rich, “Erkenntnis” is left untranslated in the following. 24 Cf. Kant: Kritik der reinen Vernunft, A 30–32/B 46 f./AA III 57 f. 25 Kant: Kritik der reinen Vernunft, A 32/B 49/AA III 59. 26 Kant’s comparison to Copernicus can be found in: Kant: Kritik der reinen Vernunft, B XVI/AA III 12.

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objects but for the conditions of “Erkenntnis” that are conceived as generally valid. 3) Next to the distinction between illusion and appearance there are other options of which one, which specifically combines 1.2) and 2), shall be expounded in the following. It is a rather new combination of Kantian philosophy and relativity theory. Michael Friedman, referring to Hans Reichenbach, describes a “relativized a priori”, which shall preserve the constitutive function for thinking as in 2) while not maintaining unchangeability of the grounds of our thinking and our perception.27 This conception does not specifically rely on the concept of time but on the respective principles of knowledge, particularly of physical knowledge. The succession of relativized principles a priori is thereby understood as a progress in knowledge and an approximation to a last objective and unchangeable knowledge.28 Therefore, there are parallels to the process of discovery in 1.2). Friedman does not reflect on the conditions of recognizing and describing such a process of discovery. He uses time both as a concrete physical concept and as a description of the process of discovery as a whole. This indicates the necessity of a differentiation. Modifying the transcendental philosophical conception of 2), even the process of discovery cannot be described independently of the respectively holding principles a priori. This shall be furthered in the following section. The foregoing remarks show that different interpretations of the non-objectivity of time are feasible. It is possible to develop further specifications. Completeness has not been demonstrated. Instead, as 3) suggests, the existence of further possible options can be assumed. At the same time, the options 1.1), 1.2) and 2) could be called the basic ones, on which other options rely in one way or another. The decision as to which of these options one would prefer, would need further justifications as Gödel’s argument for the non-objectivity of change and time does not predetermine a choice. Instead, the philosophical possibilities of interpretation are varied.

10.3

Consequences of a Differentiation of the Concept of Time

The argument for the non-objectivity of change and time does not decide about its philosophical consequences but rather leaves room for different options. Hence, in the following paragraph, the concept of time shall be differentiated by using the work of Peter Rohs. Building on that, I will develop my own position. In his first footnote of 1949a, Gödel refers to J. Ellis McTaggart’s paper The Unreality of Time. There, one can find a differentiation between so-called A and B series of time.29 While A series are determined by the distinction of past, present and

27 Cf.

e.g. Friedman: Einstein, Kant, and the Relativized A Priori. is influenced by Charles S. Peirce’s pragmatic theory of truth. Cf. Friedman: Dynamics of Reason, 64. 29 Cf. McTaggart: The Unreality of Time, 458. 28 He

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future, B series are characterized by the distinction of earlier and later. McTaggart’s argumentation against the reality of change and time shall not be observed in detail in the following. It relies on showing that A series are necessary for time and at the same time are inconsistent.30 A problem in this argumentation is that McTaggart does not sufficiently differentiate between A and B series but unobtrusively switches them. It should be further specified that A series are phenomenological and contain a first-person point of view that is linked to the “now”, which is always becoming.31 In contrast, B series are a strict total ordering. Their relation can be named as “simultaneous” as well as “before” or “after”, and each one of them is the inversion of the other. In Peter Rohs’ field theoretic transcendental philosophy, which he worked through in Feld—Zeit—Ich for the first time, A and B series are two irreducible and relating aspects of time. B series are understood as undirected, so that the irreversibility of time exists only in the combination of A and B series.32 This is constructively grounded in Immanuel Kant’s idea of the world consisting of two realms, the realm of mind and the realm of nature. The region of mind is expounded with a theory of consciousness. It is centred around the first-person point of view, and its centredness to the “now”. Consequently, the A series can be used for the description here. The realm of nature is expounded within field theory as space-time and, consequently, relates to the B series. Both realms are connected via time, which encompasses both A and B series.33 This is why many temporal descriptions of our world have to refer to both A and B series.34 Therefore, it is not necessary to decide which one of them is basic or real or singular and so forth. Both represent important aspects of time. Rohs refers to the problem of simultaneity and invariance posed by relativity theory. He uses it as an argument for the connection of space-time as a field. Combining space and time would lead to an invariance with respect to reference system changes.35 In Geist und Gegenwart, he adds the idea that the field is more fundamental than space and time.36 Consequently, regarding the field there is no problem; objective identifications are possible. Yet, there is also no change—even no subjectively or individually perceived or thought change. B series are undirected and space-time is given as a four-dimensional system. Thus, Spohn uses the expression “block”.37 Ending here would lead to option 1.1), i.e. change as mere deception. As has already been noted, this option is theoretically problematic; there is the necessity to include experience to receive substantial and verifiable statements. Against this background, 1.1) seems to be no more than an intellectual game without explanatory force. Using the distinction between A and B series of time this can be specified:

30 Cf.

McTaggart: The Unreality of Time, 459–461, 466–470. uses the expression “nunc-centric”. Rohs: Feld—Zeit—Ich, 37. 32 Cf. Rohs: Feld—Zeit—Ich, 35–41. 33 Cf. Rohs: Feld—Zeit—Ich, 9–13. 34 Cf. Rohs: Feld—Zeit—Ich, 40. 35 Cf. Rohs: Feld—Zeit—Ich, 23 f. 36 Cf. Rohs: Geist und Gegenwart, 52. 37 E.g. Rohs: Geist und Gegenwart, 46. 31 Rohs

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option 1.1) denies the existence of A series and, that is, the existence of first-person perspectives and experience. Subjective or individual perception and thought of change is, hence, to be assumed at minimum. This presupposes the possibility of observing change. A series consequently have to be taken into account. It is important to note that by doing so, a mere physical description is abandoned. This is what Gödel does in his argument, too; taking different incommensurable perspectives into account means using A series as first-person perspectives that are centred around the respective “now”. However, Rohs transcends the options 1.2) and 3) by supposing a dualism of the realms of mind and nature, which are connected by time. The change perceived in A series is, therefore, not a historic or noetic process of approximation to objects. Rather, this should be described in B series. Yet, there are also objects of the realm of the mind or combinations of both realms that can only partly be described or not at all with B series. While objects that can be entirely described in B series do not undergo any changes, indeed, this is not true for the whole world. This is an expansion of 2) insofar next to transcendental philosophical reflexions on subjective conditions of “Erkenntnis” objective facts, which can be understood with 1.1), are acknowledged as another part of the world, too. Therefore, it is not a problem that change cannot be part of all descriptions of the world: “It does not have to be assumed that there is a present spreading out over the whole universe”.38 A problem then is, if and how the “Erkenntnis” of these parts of the world is possible at all. Rohs describes them as a “transcendent reality”.39 Therefore, he does not think they can be reached—not even through the sciences.40 However, then the question has to be raised as to what type Rohs considers his theory to be. For how does this meta-theory know about the structures, namely the B series, of the “transcendent reality”? Going beyond Spohn, my suggestion is: this philosophy is a theory and as such can be examined, questioned and discussed. Peter Janich illustrates this with the cultural development of time measurement. For the uniformity of measurement is an assumption that cannot be found directly in nature.41 Instead, it is pragmatically reached by comparisons and abstractions of comparing.42 Rohs knew Janich’s pragmatic theory of time measurement in his Protophysik der Zeit. He found it to be a convincing approach.43 Yet, he did not take into account that Janich’s philosophy developed towards acknowledging the importance of the cultural

38 “Es muss nicht angenommen werden, dass es eine sich über das gesamte Universum erstreckende

Gegenwart gibt”. Rohs: Geist und Gegenwart, 46 (transl. by TA). 39 Rohs: Geist und Gegenwart, 60. 40 Cf. Rohs: Geist und Gegenwart, 59. 41 Cf. dazu Janich: Was messen Uhren?; ders.: Zeit und Natur. 42 The groundwork for this is Janich: Die Protophysik der Zeit, Chap. 3. Strictly speaking, there should first be time durations and points in time are a further abstraction. Cf. ibid., 213 f. However, this will be not further discussed as the respective steps of construction are not so important for the argumentation of this paper. 43 Cf. Rohs: Ist eine ausweisbare Zeitmessung möglich?, 146–151.

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context.44 Hence, for Janich, the historic reconstruction45 is not supplementary but an essential component. With these considerations, any theory that describes B series and the field ontologically has to be located in the life world and in the history of its formation. This does not prevent an understanding of reality but liberates such an understanding from a static metaphysics. With regard to the options already mentioned, this approach would be rather option 1.2) than 1.1). However, 1.2) has to be modified so that the process of discovery is, as in 3), evaluated from each respective point of view. Exceeding 3), it is not necessary to assume a linear progress towards one last truth. In sum, it is a concentration of a variant of 2) that allows us to integrate forms of 1) and 3).

10.4

Outlook: The Problem of Intersubjectivity

After the develops his own account of philosophical consequences from Gödel’s argument about the impossibility of objectively identifying change in the previous section, the final section will expand the topic. As shall be shown, the question of objectively identifying change is not only an interesting problem in theory. Rather, it offers links to practically relevant fields. The problem of the non-objectivity of change and time gains scope by considering the A and B series. Gödel leaves the purely physical argumentation behind by including positions of observation, that is the A series. Hence, the question of objectivity can be re-formulated: is intersubjectivity in assigning and comparing phenomenal experiences of time or phenomenal experiences in general possible? Intersubjectivity is understood as the existence of a reference position that is valid for all observing positions. Phenomenal experiences in general can be considered in this regard because they are structured by A series. Insofar as a phenomenal experience of time cannot be objectified as described above, this is true for phenomenal experience in general and, thereby, for all experience. This is in a way analogous to Kant’s conception of time as a priori. However, the inverse conclusion that time is not objective, that one might conclude from the non-objectivity of experience or phenomenal experience, does not apply. Yet, neither is it needed for these considerations. The question of objectivity of change is rather theoretically abstract. However, taking the previous considerations into account, it entails an epistemologically and practically relevant question that prevails in so-called post-modern developments. For it directs at the desire of gaining an ultimate, certain truth and validity; of reaching security and orientation that can never be shaken. Using Gödel’s argument, such a desire cannot be satisfied as there is no absolute position of reference that allows for this kind of intersubjectivity. The direction of the implication is important here; from the possibility of time travel follows the non-

44 This is also the case for later stages of his theory. Cf. Rohs: Feld—Zeit—Ich, 194. For the programme of Methodological Culturalism cf. Janich/Hartmann: Methodischer Kulturalismus. 45 Cf. Janich: Die Protophysik der Zeit, Chap. 4.

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objectivity of time. The latter in turn implies the impossibility of intersubjectivity in the sense described above. By no means is it possible to conclude from this that the impossibility of intersubjectivity has the non-objectivity of time or even the possibility of time travel as a necessary condition. In addition, it is possible to make a further distinction using Gödel’s argument. As described above, Gödel makes the assumption that metaphysical necessity shall be presupposed for showing objectivity; it would not be enough, if the universe is presumed as one in which time travel is not possible and, in turn, objective time is possible. Rather, if time is objective, the universe should necessarily be (in a metaphysical sense) in such a way that this is obtained.46 If this assumption is embraced, intersubjectivity in the sense described above is not possible. Leaving this assumption aside, there is a further interesting result. Without metaphysics, the concept of intersubjectivity has to be adjusted a little. It is possible to state then that intersubjectivity may be objective but not ultimately certain. For, however small the possibility of the non-objectivity of time is, it is nonetheless real. There is no ultimate proof that there is intersubjectivity.47 Nonetheless, this need not mean that intersubjectivity is not possible at all. In the above sense, intersubjectivity is understood as absolute. However, intersubjective exchange and understanding in certain established frames may work out. Peter Rohs takes such a frame to be the conditions of successful communication. The reason for this is that exchanging and comparing one’s own and others’ points of view is necessary for reaching intersubjectivity. This communicative framework implies the lack of high velocities in particular.48 Even if we cannot prove the possibility of intersubjectivity without a doubt, this evades Gödel’s argument, as time travel is dependent on high velocities.49 A further set of conditions is important for the frame of communication. Among them should be at least considerations about discourses, arguments and reasons. To explain intersubjectivity more precisely these need to be taken into account. However, they exceed the current purpose of this paper. Here, it is sufficient to have shown that Gödel’s argument not only addresses interesting physical matters but may also be used to deal with further important philosophical problems.

References Alles, T. (2020). Gödels ontologischer Gottesbeweis im Kontext der Systematischen Theologie. Neue Zeitschrift für Systematische Theologie und Religionsphilosophie, 62(1), 1–40.

46 Cf.

Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 562/CW II 206 f. 47 There is a certain parallel to the consequences of Gödel’s second incompleteness theorem. While the theorem refers to logical systems, intersubjectivity is directed to the structures of experience. 48 Cf. Rohs: Geist und Gegenwart, 58. 49 Gödel calculates that the velocity must at least amount to √1 of light speed. Cf. Gödel: A remark 2 about the relationship between relativity theory and idealistic philosophy, Footnote 11.

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Friedman, M. (2001). Dynamics of reason (Stanford Kant Lectures 1999). CSLI Publications, 2001. pp. xiv+ 141. ISBN 1-57586-292-1. Friedman, M. (2009). Einstein, Kant, and the relativized a priori. In M. Bitbol, P. Kerszberg, & J. Petitot (Eds.), Constituting objectivity: Transcendental perspectives on modern physics (pp. 253–267). The western ontario series in philosophy of science 74. Springer Netherlands. https:// doi.org/10.1007/978-1-4020-9510-8_15. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In S. Feferman, J. W. Dawson, S. C. Kleene, G. H. Moore, R. M. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: Collected works. Vol. I: Publications 1929–1936 (pp. 144–195). Oxford University Press. (Original source: Monatshefte für Mathematik und Physik 38 (1931), 173–198). Gödel, K. (1946/49/9-C1 und 1946/9-B2). Some observations about the relationship between theory of relativity and Kantian philosophy. In S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: Collected works: Bd. III: Unpublished essays and lectures (Vol. 3, pp. 230–259). Oxford University Press (1995). Gödel, K. (1949a). A remark on the relationship between relativity theory and idealistic philosophy. In S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: collected works: Vol. II: Publications 1938–1974 (Vol. 2, pp. 202–207). Oxford University Press (1990). (Original source: Schilpp, P. A. (ed.): Albert Einstein, Philosopher-Scientist (Library of Living Philosophers 7) Evanston 1949, 555–562). Gödel, K. (1949b). Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der idealistischen Philosophie. In P. A. Schilpp (Ed.) Albert Einstein als Naturforscher und Philosoph (Translation from 1949a by Hans Hartmann and additions by Kurt Gödel) (pp. 406– 412). Kohlhammer (1955). Gödel, K. (1949c). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. In S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: Collected works: Vol. II: publications 1938–1974 (pp. 190–198). Oxford University Press (1990) [Original source: Reviews of Modern Physics 21 (1949), 447–450]. Gödel, K. (1949d). Lecture on rotating universes. In S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: collected works: Vol. III: Unpublished essays and lectures (Vol. 3, pp. 269–287). Oxford University Press (1995). Gödel, K. (1951). Some basic theorems on the foundations of mathematics and their implications. In S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: collected works: volume III: unpublished essays and lectures (Vol. 3, pp. 304–323). Oxford University Press (1995). [25. Josiah Willard Gibbs Lecture]. Gödel, K. (1952). Rotating universes in general relativity theory. In S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: Collected works: Vol. II: Publications 1938–1974 (Vol. 2, pp. 208–216). Oxford University Press (1990). (Original source: Proceedings of the International Congress of Mathematicians. Cambridge, Massachusetts, U.S.A. August 30–September 6, 1950. Volume I, Providence 1952, 175–181). Gödel, K. (1964). What is Cantor’s continuum problem? In S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: Collected works: Vol. II: Publications 1938–1974 (Vol. 2, pp. 254–270). Oxford University Press (1990). (Original source: Benacerraf, P., & Putnam, H. (Eds.). (1964). Philosophy of mathematics: Selected readings (258–273); revised and extended version of: Gödel, K. (1947). What is Cantor’s continuum problem? The American Mathematical Monthly, 54, 515–525). Gödel, K. (1970). Ontological proof. In S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay (Eds.), Kurt Gödel: Collected works: Vol. III: Unpublished essays and lectures (Vol. 3, pp. 403–404). Oxford University Press (1995). Gödel, K. (2016). Meine ph Ansicht. (Note from the Nachlass, Item 060168; Transcription according to: Engelen, Eva-Maria: What is the Link between Aristotle’s Philosophy of Mind, the Iterative Conception of Set, Gödel’s Incompleteness Theorems and God? About the Pleasure and the Difficulties of Interpreting Kurt Gödel’s Philosophical Remarks. In G. Crocco

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& E.-M. Engelen (Eds.), Kurt Gödel. Philosopher-Scientist, Aix-en-Provence: Presses Universitaires de Provence, 2016, pp. 171–188, here: 172]. Janich, P. (1980). Die Protophysik der Zeit. Konstruktive Begründung und Geschichte der Zeitmessung. Suhrkamp Verlag GmbH. Janich, P. (1997a). Was messen Uhren? In P. Janich (Ed.), Das Maß der Dinge. Protophysik von Raum, Zeit und Materie. Suhrkamp Taschenbuch Wissenschaft 1334 (pp. 131–138). Suhrkamp. (first published in: alma mater philippina 1982, pp. 12–14). Janich, P. (1997b). Zeit und Natur. In P. Janich (Ed.), Das Maß der Dinge. Protophysik von Raum, Zeit und Materie. Suhrkamp Taschenbuch Wissenschaft 1334 (pp. 253–268). Suhrkamp. (first published in: Hauskeller, M., Rehmann-Sutter, C., & Schiemann, G. (Eds.), Naturerkenntnis und Natursein. Für Gernot Böhme, 1997, pp. 107–122). Janich, P., & D. Hartmann (1996). Methodischer Kulturalismus. In P. Janich & D. Hartmann (Eds.), Methodischer Kulturalismus. Zwischen Naturalismus und Postmoderne. Suhrkamp Taschenbuch Wissenschaft 1272 (pp. 9–69). Suhrkamp. Kant, I. (1998). Kritik der reinen Vernunft, ed. J. Timmermann. Felix Meiner Verlag. (first edition 1781 = A, second edition 1787 = B); Kant’s gesammelte Schriften Bd. 3, ed. Königlich Preussischen Akademie der Wissenschaften, 1902 [= AA]. ´ ˛torzecka, K. (2015). Gödel’s “Slingshot” argument and his onto-theological Kovaˇc, S., & Swie ´ ˛torzecka (Ed.), Gödel’s ontological argument: History, modifications, and system. In K. Swie controversies (pp. 123–162). Warsaw. McTaggart, J. E. (1908). The unreality of time. Mind, 17(68), 457–474. Muck, O. S. J. (1992). Eigenschaften Gottes im Licht des Gödelschen Arguments. Theologie und Philosophie, 67, 60–85. Rohs, P. (1986). Ist Eine Ausweisbare Zeitmessung Moglich? Zur »Protophysik der Zeit«. Philosophische Rundschau, 33(1–2), 133–151. Rohs, P. (1996). Feld – Zeit – Ich. Entwurf einer feldtheoretischen Transzendentalphilosophie. Klostermann, Vittorio. Rohs, P. (2016). Geist und Gegenwart. Entwurf einer analytischen Transzendentalphilosophie. mentis. Sobel, J. H. (1987). Gödel’s ontological proof. In J. J. Thomson (Ed.), On being and saying. Essays for Richard Cartwright (pp. 241–261). Cambridge University Press. Stein, H. (1990). Introductory note to 1949. In S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Eds.), Kurt Gödel: Collected works: Vol. II: publications 1938–1974 (pp. 199–201). Oxford University Press. Wang, H. (1996). A logical journey. From Gödel to philosophy. The MIT Press. Thorben Alles is research assistant at the Department of Systematic Theology of the Protestant Faculty of the University of Bonn (Germany). He has studied protestant theology, mathematics, educational sciences and philosophy in Bonn and Strasbourg (France). His research interests are theories of reasons, ethics of argumentation, theological and religious epistemology and theological hermeneutics. These shall be combined in his ongoing dissertation project “A Logic of Reasoning”.

Konsequenzen aus der Unmöglichkeit einer objektiven Bestimmung von Veränderung. Philosophische Überlegungen im Anschluss an Kurt Gödel

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Kurt Gödel argumentiert in A remark about the relationship between relativity theory and idealistic philosophy (1949a) für die Unmöglichkeit1 einer objektiven Bestimmung von Veränderung und Zeit. Doch wie können diese Nicht-Objektivität und ihr Argument philosophisch verstanden werden und welche Erkenntnisse lassen sich daraus ziehen? Diesen Fragen geht der Aufsatz im Folgenden nach. Dazu wird zunächst Gödels Argument für die Unmöglichkeit einer objektiven Bestimmung von Veränderung im Zusammenhang weiterer seiner Texte vorgestellt (Abschn. 11.1). Ausgehend von weiteren Überlegungen Gödels, aber konstruktiv über diese hinausgehend, werden philosophische Anschlussmöglichkeiten vorgestellt, wie die NichtObjektivität von Veränderung und Zeit ontologisch oder erkenntnistheoretisch erklärt werden kann. Diese umfassen: 1.1) Illusion als Täuschung, 1.2) Illusion als Erkenntnisprozess, 2) Erscheinung oder 3) eine Kombination aus 1.2) und 2) in der Theorie des relativierten Apriori Michael Friedmans (Abschn. 11.2). Da Gödels Argument selbst gegenüber diesen Interpretationsmöglichkeiten neutral ist, soll anschließend eine eigene Positionierung erfolgen. Über J. Ellis McTaggarts Unterscheidung zwischen A- und B-Reihen und vor allem unter Bezugnahme auf die darauf aufbauenden Arbeiten Peter Rohs’ wird dabei eine Ausdifferenzierung des Zeitbegriffs vorgenommen. Dies resultiert unter Hinzuziehung pragmatischer Überlegungen Peter Janichs in einer Erweiterung von 2), die auch 1) und 3) in abgewandelter Form zu integrieren erlaubt (Abschn. 11.3). In einem Ausblick soll schließlich noch die Problem-

1 Wo

nicht anders kenntlich gemacht, werden hier und im Folgenden „Möglichkeit“ und „Unmöglichkeit“ ausschließlich als sinnvolle Denkbarkeit und argumentative Vertretbarkeit verstanden.

T. Alles (B) Universität Bonn, Bonn, Deutschland E-Mail: [email protected]

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_11

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perspektive ausgeweitet werden. Wie in Abschn. 11.4 ausgeführt wird, lassen sich die Überlegungen zu Gödels Argument erweitern auf die grundsätzliche Frage nach Intersubjektivität.

11.1

Gödels Argument für die Nicht-Objektivität von Veränderung und Zeit

Kurt Gödel ist für seine bedeutende und bahnbrechende Forschung in Mathematik und Logik bekannt. Darüber hinaus hat er sich mit vielfältigen philosophischen und theologischen Themen auseinandergesetzt. Die zugehörigen Arbeiten und Notizen ließ er jedoch größtenteils unveröffentlicht, weshalb sie bis heute Gegenstand intensiver Editionsarbeiten sind. Während Gödel einige seiner Arbeiten zur Philosophie der Mathematik veröffentlichte, scheint A remark about the relationship between relativity theory and idealistic philosophy (1949a), welches er zu Ehren seines Freundes Albert Einstein verfasste, die einzige Publikation zu seinen Lebzeiten zu sein, die sich mit einer anderen philosophischen Thematik befasst.2 Gödel thematisiert darin die Frage, ob Veränderung objektiv ist oder stattdessen entweder eine „Illusion“ oder eine „Erscheinung“. Veränderung ist hierbei als Veränderung in der Zeit zu verstehen. Das sich aus der Relativitätstheorie ergebende Argument, dass Gleichzeitigkeit nur beobachtungsabhängig festgestellt werden könne und daher Zeitverläufe nicht objektiv, sondern beobachtungsabhängig zu bestimmen seien, kann gemäß Gödel durch die Bestimmung einer Weltzeit und deren Relationierung zu anderen Zeitbeobachtungen widerlegt werden.3 Daher entwickelt er ein eigenes Argument gegen die Objektivität der Zeit, welches auf die Unmöglichkeit der objektiven Bestimmung der Weltzeit zielt. Dafür nimmt er Bezug auf die physikalisch-mathematischen Ausführungen seines wenig später erschienenen Aufsatzes An example of a new type of cosmological solution of Einstein’s field equation of gravitation. Dort stellt Gödel seine relativitätstheoretische Konzeption rotierender Universen vor, innerhalb derer auch Zeitreisen möglich sind, wie er in der diesem Aufsatz zugrundeliegenden Vorlesung Lecture on rotating universes genauer darlegt.4 Diese auf Lösungen mathematischer Gleichungen beruhende physikalische Möglichkeit von Zeitreisen nutzt Gödel in A remark about the relationship between relativity theory and idealistic philosophy (1949a) als Argument gegen die Objektivität der Zeit: Für jede Definition einer Weltzeit ist es demnach möglich, einmal oder mehrmals in die Vergangenheit zu reisen. Dadurch könnte es aber Beobachtende geben, deren Zeitbeobachtungen nicht in Relation zur Weltzeit gesetzt werden können. Im Extremfall könnten sie gemessen an der Weltzeit sogar nur zu

2 Vgl.

Stein: Introductory Note to 1949a, 199. dazu Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 557–559/CW II 202–204. Bei veröffentlichten Werken zitiere ich zuerst nach der Originalpaginierung und anschließend nach den Seitenzahlen der jeweiligen Kurt Gödel. Collected Works Ausgabe (abgekürzt als „CW“). 4 Vgl. Gödel: Lecture on rotating universes, CW III 285–287. 3 Vgl.

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105

einem einzigen Zeitpunkt existieren, also völlig ohne Dauer, während es sich für diese bestimmte Beobachtungsperspektive um eine Zeitdauer handeln würde. Daraus schließt Gödel, dass die Annahme einer objektiven Zeit unbegründet ist.5 Das Argument basiert dabei nicht auf der zwingenden Annahme, dass unser Universum der von Gödel beschriebenen Art rotierender Universen tatsächlich entspricht. Ein Beispiel für eine Lösung der Feldgleichungen, die keine geschlossenen zeitartigen Kurven und daher keine im Argument vorausgesetzten Zeitreisen zulässt, hat Gödel selbst in seinem Aufsatz Rotating universes in general relativity theory von 1952 präsentiert. Dennoch ergänzt Gödel 1955 in der späteren deutschen Veröffentlichung des Sammelbandes von 1949a einen Hinweis auf diesen Aufsatz und hält zugleich an der Argumentation fest.6 Denn es genügt hierbei die aufgezeigte Möglichkeit, dass unser Universum derart beschaffen sein könnte, dass Zeitreisen möglich sind. Die durch den Potentialis („könnte“) ausgedrückte Möglichkeit ist hier nicht im technischen oder praktischen Sinne gemeint. Gödel verdeutlicht dies an der Berechnung der absurd hohen Größen der theoretisch für eine solche Zeitreise benötigten Energie und Geschwindigkeit.7 Stattdessen geht es um ein physikalisch-metaphysisches Verständnis von Möglichkeit. Schließlich, so die Annahme, sollte die Objektivität von Zeit notwendig in den Naturprozessen und dadurch in den Gesetzen des Universums verankert sein und nicht kontingent von der Anordnung und Bewegung der Materie abhängen. Durch die in Naturgesetzen nicht ausgeschlossene Möglichkeit von Zeitreisen bliebe dies aber als einzige Option für objektive Zeit übrig.8

11.2

Philosophische Anschlussmöglichkeiten an Gödels Argument

Angenommen, Gödels Argument ist richtig, und angenommen, das Vorhandensein objektiver Zeit ist notwendige Bedingung für das Vorhandensein von objektiver Veränderung – was folgt daraus für unser Verständnis der Welt? Dieser Frage soll im Folgenden nachgegangen werden. Die physikalisch-mathematischen sowie kosmologischen Annahmen seien also ebenso hypothetisch vorausgesetzt wie die Schlüssigkeit des metaphysischen Argumentes. Aus den vorhergehenden Erläuterungen zu Gödels Aufsatz ist deutlich, dass unter objektiver Zeit zu verstehen ist, dass es einen für sämtliche Beobachtungspositionen gültigen Referenzpunkt für die Zuordnung und den Abgleich von Zeitpunkten und -intervallen gibt. Ist eine solche Bezugnahme nicht möglich, ergeben sich idealtypisch zwei theoretische Alternativen, die

5 Vgl.

dazu Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 560 f./CW II 204–206. 6 Vgl. Gödel: Eine Bemerkung über die Beziehungen zwischen der Relativitatstheorie und der idealistischen Philosophie, Fußnote 10. 7 Vgl. Gödel: A remark about the relationship between relativity theory and idealistic philosophy, Fußnote 11. 8 Vgl. dazu Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 562 f./CW II 206.

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Gödel mit 1) „illusion“ und 2) „an appearance due to our special mode of perception“ betitelt.9 Diese sollen im Folgenden genauer ausgeführt und ausdifferenziert und anschließend um eine weitere Alternative ergänzt werden. 1) Wenn objektive Zeit und damit objektive Veränderung reine Illusion ist, ergibt sich ein statisch-deterministisches Weltbild. Das, was als Wandel aufgefasst wird, wäre somit entweder bloße Täuschung oder der Weg zur Erkenntnis dessen, was „immer schon“ gilt. 1.1) Die Option der bloßen Täuschung wäre vermutlich schwierig zu vertreten. Denn erstens lässt sich phänomenologisch kaum negieren, dass Wandel wahrgenommen wird. Zweitens schließt Gödels Argument keineswegs subjektive Zeit aus, denn er stellt – philosophische Vorgänger seines Argumentes rekonstruierend – fest: „Each observer has his own set of nows‘“ 10 . ’ 1.2) Gödel hätte also vermutlich der zweiten theoretischen Option, dem Weg der Erkenntnis zugestimmt. Dabei würden das Denken und Erkennen als Prozess in Richtung auf eine „immer schon“ geltende Wahrheit verstanden. Die Anführungszeichen sollen dabei andeuten, dass es sich eigentlich nicht um eine zeitliche Erstreckung handelt, denn diese zu erkennende Wahrheit hätte ihre Gültigkeit unabhängig von zeitlichen Bestimmungen, sie könnte gewissermaßen „ewig“ genannt werden. Ein derartiges Konzept könnte mit dem von Gödel vertretenen Platonismus, der beispielsweise den mathematischen Konzepten eine Realität beimisst, übereinstimmen. Dafür ist es durchaus erforderlich, eine denkerische Entwicklung, also einen Erkenntnisprozess anzunehmen. Denn gemäß Gödels erstem Unvollständigkeitssatz gibt es mathematische Aussagen, die innerhalb des vorhandenen mathematisch-axiomatischen Systems nicht beweisbar sind.11 Sie wären also durch den menschlichen Geist eigens zu bestimmen.12 Dies könnte beispielsweise durch mathematische Intuition erfolgen.13 Es ist somit zu vermuten, dass Gödel die Mathematik als feststehend und zugleich dem Denken nicht unmittelbar vollständig gegeben auffasst, sodass ein Erkenntnisprozess vorauszusetzen wäre.14 Auch wenn Gödels Platonismus hier nur andeutungsweise und im Quellenbezug unvollständig rekonstruiert werden

9 Gödel:

A remark about the relationship between relativity theory and idealistic philosophy, 557 f./CW II 202. 10 Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 558/CW II 203. 11 Vgl. Gödel: Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I, 173–191/CW I 144–180. 12 Vgl. z. B. Gödel: Some basic theorems on the foundations of mathematics and their implications, 9–13/CW III 308–310. 13 Vgl. z. B. Gödel: What is Cantor’s continuum problem?, 271/CW II 267 f. 14 Auch wenn Gödel meines Wissens nicht von einem Prozess spricht, findet sich der Begriff der Entwicklung zumindest auf einem Zettel im Nachlass. Vgl. Gödel: Meine ph Ansicht , Punkte 2, 7 und 8.

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kann, liegt der Fokus auf wesentlichen, in der Regel veröffentlichten oder vorgetragenen Werken. Die vorangegangenen Überlegungen zeigen daher, dass die theoretische Option eines Erkenntniswegs hin zu zeitloser Wahrheit in wichtigen Aspekten mit weiteren philosophischen Überlegungen Gödels vereinbar ist. Darüber hinaus wäre diese Option gut mit möglichen Folgen aus Gödels theologischen Überlegungen vereinbar. Sein formallogisch formulierter und von ihm weder publizierter noch weiter ausgeführter ontologischer Gottesbeweis zielt auf die notwendige Existenz eines Wesens – bzw. Gottes –, welches alle positiven Eigenschaften umfasst.15 Sämtliche positiven Eigenschaften können daher als an diesem notwendig existierenden Wesen Anteil habend vorgestellt werden. Folgt man den Gesprächsnotizen Hao Wangs, dann wollte Gödel in Anlehnung an Leibniz ein System mit Gott als Zentralmonade entwickeln.16 Die Vorstellung von Gott als der zeitlosen17 oder unveränderlichen Realität, an der alles Denken als Prozess Anteil hat und auf die es als sein Ziel hinstrebt, liegt zumindest als eine Konsequenz nahe. Jordan Howard Sobel konnte zeigen, dass unter gewissen Voraussetzungen18 in Gödels Gottesbeweis Aktualität, Möglichkeit und Notwendigkeit zusammenfallen.19 Sre´cko Kovaˇc und Kordula ´ etorzecka argumentieren dafür, dass dieser sogenannte modale Kollaps von Swi¸ Gödel beabsichtigt worden sein könnte.20 Der Beweis wäre damit eine Form des sich in Veränderung befindenden Denkprozesses hin auf eine letzte Gültigkeit. Auch wenn diese Überlegungen – insbesondere die letzte – eher in den Bereich der Spekulation gehören, sollte deutlich geworden sein, dass Gödels theologische Ansichten mit einer Aufgabe objektiver Zeit bei gleichzeitiger subjektiver Veränderung des Erkenntnisprozesses durchaus harmonieren können. 2) Die Alternative Möglichkeit, Veränderung als Erscheinung zu betrachten, ist mit Gödel schwieriger zu erarbeiten. In seinen verworfenen Manuskripten zum Aufsatz lautete der Titel noch: Some observations about the relationship between theory of relativity and Kantian philosophy.21 Der Entdeckungszusammenhang ist also – wie Gödel auch in seiner Lecture on rotating universes anmerkt und in einer Fußnote von 1949a zumindest noch andeutet22 – eine Analogie zwischen der Kantischen subjekttheoretisch-apriorischen Konzeption der Zeit und der relativitätstheoretischen Standortgebundenheit von Zeitmessungen. Wie die Änderung des Titels andeutet, entschied sich Gödel jedoch

15 Vgl.

Gödel: Ontological proof, CW III 403 f. Wang: A Logical Journey, 8. 17 So schlägt Otto Muck vor, die modalen Operatoren in Gödels Beweis zeitlich zu interpretieren, sodass die notwendige Existenz Gottes bedeutet, dass Gott zu allen Zeitpunkten existiert. Vgl. Muck: Eigenschaften Gottes im Licht des Gödelschen Arguments, 70–72. 18 Zur Zusammenfassung der Diskussion über diese Voraussetzungen vgl. Alles: Gödels ontologischer Gottesbeweis im Kontext der Systematischen Theologie, 26–28. 19 Vgl. Sobel: Gödel’s Ontological Proof, 250–253. 20 Vgl. Kovaˇ ´ etorzecka: Gödel’s „Slingshot“ Argument and His Onto-Theological System. c/Swi¸ 21 Über die Collected Works Edition zugänglich sind die Versionen *1946/49/9-C1 und *1946/9-B2. 22 Vgl. Gödel: Lecture on rotating universes, CW III 274; Ders.: A remark about the relationship between relativity theory and idealistic philosophy, Fußnote 3. 16 Vgl.

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später für eine weniger personenzentrierte Darstellung. Dies ist auch insofern sinnvoll, als Immanuel Kant mit seinem transzendentalphilosophischen Ansatz in der Kritik der reinen Vernunft einen anderen Zugang wählt. Er reflektiert dort über die Bedingungen von Erkenntnis und erhält unter anderem Zeit als a priori gegebene Form des Denkens – genauer: der Anschauung – von Subjekten. Mit Kant ist Zeit die Form, in der alle Menschen denken und dank welcher Welt als solche erst beschreibbar wird.23 Auch wenn Zeit bei Kant nicht als Ding an sich existiert, würde sie somit einen gewissen objektiven Status behalten, nämlich als geteilte Form menschlichen Wahrnehmens und Denkens – oder gar als geteilte Form jeglichen vernünftigen Wahrnehmens und Denkens. Dass dies bei Kant „subjektiv“ genannt wird und „objektiven Bestimmungen“ gegenübergestellt wird,24 darf nicht als Relativismus verstanden werden, sondern ist als allgemein für vernünftige Subjekte gültig zu lesen. Dieses Vorgehen entspricht Kants „Umänderung der Denkart“,25 sodass nicht mehr nach den Gegenständen an sich gefragt wird, sondern die als allgemeingültig verstandenen und zu erweisenden Erkenntnisbedingungen zugrundegelegt werden. 3) Neben diesen an Gödels Unterscheidung zwischen Illusion und Erscheinung orientierten Einordungen soll nun noch eine dritte Möglichkeit thematisiert werden, die in spezifischer Weise 1.2) und 2) kombiniert. Es handelt sich um eine neuere Kombination aus Kantischer Philosophie und Relativitätstheorie. Michael Friedmans an Hans Reichenbach orientiertes relativiertes Apriori soll die konstitutive Funktion für das Denken wie in 2) bewahren, ohne dass eine Unveränderbarkeit der Erkenntnisgrundlagen postuliert werden müsste.26 Dies ist nicht spezifisch auf Zeit, sondern auf die jeweils aktuell geltenden Prinzipien der Erkenntnis, bei Friedman insbesondere der physikalischen Erkenntnis, zu beziehen. Die Sukzession relativierter Prinzipien a priori wird dabei verstanden als Fortschritt im Erkenntnisprozess im Sinne einer Annäherung an den objektiven, unveränderbaren und letztgültigen Erkenntnisgegenstand.27 Hierbei ergeben sich Anknüpfungen an den unter 1.2) beschriebenen, auf eine letztgültige Wahrheit bezogenen Weg der Erkenntnis. Die Bedingungen, einen solchen Erkenntnisprozess überhaupt erkennen und beschreiben zu können, werden bei Friedman nicht eigens bedacht. Dass Zeit begrifflich sowohl für konkrete physikalische Bestimmungen als auch zur Darstellung des Erkenntnisprozesses insgesamt in Anspruch genommen wird, deutet jedoch darauf hin, dass eine hier anschließende Differenzierung notwendig wäre. Es könnte daher durch Abwandlung der in 2) beschriebenen transzendentalphilosophischen Konzep-

23 Vgl. dazu Kant: Kritik der reinen Vernunft, A 30–32/B 46 f./AA III 57 f. Aufgrund der besseren Zugänglichkeit zitiere ich neben der ersten und zweiten Auflage auch die um Vorwort und Einleitung der ersten Auflage ergänzte und sonst mit der zweiten Auflage übereinstimmende Akademieausgabe. 24 Kant: Kritik der reinen Vernunft, A 32/B 49/AA III 59. 25 Kant: Kritik der reinen Vernunft, B XVI/AA III 11. 26 Vgl. z. B. Friedman: Einstein, Kant, and the Relativized A Priori. 27 Dies geschieht in Anlehnung an Charles S. Peirce’s pragmatische Wahrheitstheorie. Vgl. Friedman: Dynamics of Reason, 64.

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tion festgestellt werden, dass sich der Erkenntnisprozess selbst nicht unabhängig von den jeweils geltenden relativen Prinzipien a priori zeitlich beschreiben lässt. Anhand der vorausgehenden Bemerkungen konnte gezeigt werden, dass sich ausgehend von einer Ablehnung objektiver Zeit oder Veränderung verschiedene Optionen der Interpretation und entsprechend verschiedene Möglichkeiten für unser Verständnis von Welt ergeben. Dabei ist es möglich, diese noch weiter zu differenzieren. Auch wurde keine Vollständigkeit nachgewiesen. Vielmehr deutet 3) an, dass es weitere Möglichkeiten geben könnte. Zugleich dürften die mit Gödel entwickelten Optionen 1.1), 1.2) und 2) die Grundlagen benennen, auf denen andere Optionen aufbauen. Welche dieser Optionen theoretisch zu präferieren ist, wäre eigens zu begründen. Die Ablehnung einer objektiven Zeit liefert hierfür keine Vorentscheidung, sondern verbleibt in ihren philosophisch-theoretischen Ausgestaltungsmöglichkeiten plural.

11.3

Konsequenzen aus einer Ausdifferenzierung des Zeitbegriffs

Nachdem festgestellt werden konnte, dass ein Beweis der Unmöglichkeit objektiver Zeit und Veränderung selbst noch keine Entscheidung über philosophische Konsequenzen liefern kann, es vielmehr verschiedene Möglichkeiten gibt, soll im Folgenden insbesondere unter Hinzuziehung der Arbeiten Peter Rohs’ eine Differenzierung des Zeitbegriffes vorgenommen werden, um darauf aufbauend eine eigene Position gegenüber den oben genannten Anschlussmöglichkeiten einzunehmen. In seiner ersten Fußnote in 1949a verweist Gödel auf J. Ellis McTaggarts Artikel The Unreality of Time. Dort wird eine Differenzierung zwischen sogenannten Aund B-Reihen der Zeit vorgenommen.28 Während A-Reihen durch die Unterscheidung von Vergangenheit, Gegenwart und Zukunft charakterisiert seien, lassen sich B-Reihen durch die Unterscheidung von früher und später auszeichnen. McTaggarts Argumentation dafür, dass Zeit und Veränderung nicht real sind, soll im Folgenden nicht weiterverfolgt werden. Sie basiert im Wesentlichen darauf, dass A-Reihen zuerst als grundlegend für Zeit und anschließend als selbstwidersprüchlich dargestellt werden.29 Ein dabei auftretendes Problem ist die Unschärfe von und das Changieren zwischen den A- und B-Reihen in McTaggarts Argumentation. Zu präzisieren wäre, dass A-Reihen primär phänomenologisch zu verstehen sind und entsprechend eine Perspektive der ersten Person auf das ans „Jetzt“ gebundene Werden beinhalten.30 Demgegenüber sind B-Reihen als strenge Totalordnung zu verstehen, deren zugrundeliegende Relation als „gleichzeitig“ bezeichnet werden kann sowie als „vor“ oder als „nach“, wobei das eine jeweils die Umkehrung des anderen darstellt.

28 Vgl.

McTaggart: The Unreality of Time, 458. McTaggart: The Unreality of Time, 459–461, 466–470. 30 Rohs verwendet auch die Bezeichnung „nunczentrisch“. Rohs: Feld – Zeit – Ich, 37. 29 Vgl.

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Bei Peter Rohs’ erstmals in Feld – Zeit – Ich ausführlich entwickelter Konzeption einer feldtheoretischen Transzendentalphilosophie handelt es sich bei A- und B-Reihen um zwei irreduzible und aufeinander zu beziehende Aspekte von Zeit. B-Reihen sind hier als ungerichtet verstanden, sodass die Anisotropie, d. h. Irreversibilität, der Zeit erst durch die Kombination von A- und B-Reihen entstehe.31 Dies ist begründet in der sich konstruktiv auf Immanuel Kant stützenden Annahme, dass sich die Welt in das Gebiet des Geistes und das Gebiet der Natur ausdifferenzieren lasse. Das Gebiet des Geistes wird selbstbewusstseinstheoretisch bestimmt und ist auf die Perspektive der ersten Person in ihrer diachronen Identität und Jetztzentriertheit gerichtet. Hier können dementsprechend die A-Reihen zur Beschreibung verwendet werden. Das Gebiet der Natur wird feldtheoretisch als Raumzeit bestimmt und beinhaltet folglich die B-Reihen. Verbunden werden beide Gebiete über die A- und B-Reihen vereinende Zeit.32 Dies zeige sich auch daran, dass viele zeitliche Beschreibungen der Welt auf A- und auf B-Reihen gemeinsam Bezug nehmen müssen.33 Demnach gilt es nicht, zu entscheiden, welche von A- und B-Reihe die grundlegende, reale, einzige etc. ist. Beide stellen zentrale Aspekte der Zeit dar und sind nicht gegeneinander auszuspielen. Interessant ist, dass Rohs auf das im Kontext von Gödels Aufsatz bereits thematisierte, mit der Relativitätstheorie gegebene Problem der Gleichzeitigkeit und Invarianz als Argument für den Zusammenhang von Raumzeit als Feld rekurriert: Es zeige gerade die Notwendigkeit, Raum und Zeit zu kombinieren, da diese dann invariant gegenüber Bezugssystemwechseln seien.34 In Geist und Gegenwart korrigiert er sich dahingehend, dass das Feld sogar als das Grundlegendere gegenüber Raum und Zeit anzusehen sei.35 Für das Feld selbst ergibt sich somit gar nicht erst das zuvor thematisierte Problem; objektive Bestimmungen sind durchaus möglich. Zugleich gibt es aber auch keine – nicht einmal subjektiv oder individuell wahrgenommene oder gedachte – Veränderung: B-Reihen sind ungerichtet, und die Raumzeit ist als vierdimensionales System schlichtweg gegeben. Rohs spricht daher auch von einem „Block“.36 Würde dabei stehen geblieben, ergäbe sich die zuvor vorgestellte Möglichkeit 1.1) von Veränderung als bloßer Täuschung. Wie dort angemerkt, ist diese Option jedoch theoretisch schwierig zu halten: In Anbetracht der Notwendigkeit, Erfahrung einzubeziehen, um gehaltvolle und überprüfbare Aussagen zu erhalten, erweist sie sich als reines Gedankenspiel ohne erklärende Kraft. Anhand der inzwischen eingeführten Unterscheidung von A- und B-Reihen lässt sich dies präzisieren: Möglichkeit 1.1) negiert das Vorkommen von Perspektiven der ersten Person und damit auch von Erfahrung. Zumindest subjektiv oder individuell wahrnehmbare und denkbare Veränderung ist also sinnvollerweise anzunehmen. Dies setzt jedoch voraus, dass ein Werden

31 Vgl.

dazu Rohs: Feld – Zeit – Ich, 35–41. dazu Rohs: Feld – Zeit – Ich, 9–13. 33 Vgl. Rohs: Feld – Zeit – Ich, 40. 34 Vgl. Rohs: Feld – Zeit – Ich, 23 f. 35 Vgl. Rohs: Geist und Gegenwart, 52. 36 Z. B. Rohs: Geist und Gegenwart, 46. 32 Vgl.

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beobachtet werden kann. Es müssen also die A-Reihen mit einbezogen werden. Zu beachten ist, dass dann auch die rein physikalische Beschreibung verlassen wird. Und genau dies ist auch, was Gödel in seinem Argument durchführt: Die Problematisierung verschiedener inkommensurabler Perspektiven verweist darauf, dass die A-Reihen als Perspektive der ersten Person mit Bezug auf die Zentrierung im jeweiligen „Jetzt“ vorausgesetzt werden. Die Möglichkeiten 1.2) und 3) werden durch Rohs jedoch insofern überschritten, als er einen Dualismus der Gebiete des Geistes und der Natur annimmt, welche durch die Zeit miteinander verbunden werden. Die in A-Reihen wahrgenommene Veränderung wäre somit kein geschichtlicher oder noetischer Prozess der Annäherung an die letztendlich in B-Reihen zu verstehenden Objekte. Vielmehr würde es auch Objekte des Geistes oder Mischformen geben, die teilweise oder ganz nur durch A-Reihen beschreibbar sind. Während rein in BReihen beschreibbare Objekte tatsächlich keiner Veränderung unterliegen, würde dies für die Welt insgesamt also nicht gelten. Es handelt sich hierbei insofern um eine Erweiterung von 2), als neben der transzendentalphilosophischen Reflexion auf subjektive Erkenntnisbedingungen auch objektive Tatsachen als gleichberechtigt anerkannt, aber einem anderen Bereich der Welt zugeordnet werden, der im Sinne von 1.1) verstanden werden kann. Dass Veränderung somit nicht in allen Beschreibungen von Welt vorkommen kann, ist damit kein Problem: „Es muss nicht angenommen werden, dass es eine sich über das gesamte Universum erstreckende Gegenwart gibt.“ 37 Wie oder ob dann überhaupt Erkenntnis dieser Teile von Welt möglich sein kann, ist ein sich anschließendes Problem. Rohs spricht von einer „transzendenten Wirklichkeit“ 38 und nimmt entsprechend nicht an, dass sie vollständig erkannt werden kann, auch nicht durch die Naturwissenschaften.39 Es stellt sich damit jedoch die Frage, welchen Status Rohs seiner eigenen Philosophie zuschreibt. Denn woher hat diese als Metatheorie Kenntnis von den Strukturen der „transzendenten Wirklichkeit“? Über Rohs hinausgehend könnte ein Vorschlag lauten, dass es sich eben auch hierbei um eine Theorie handelt, die überprüft, hinterfragt und diskutiert werden kann. Peter Janich verdeutlicht dies an der kulturellen Entwicklung der für die Naturwissenschaften unabdingbaren Zeitmessung. Denn die Gleichförmigkeit der Messung sei als Annahme nicht in der Natur selbst zu finden.40 Stattdessen erfolge sie pragmatisch über Abgleichungen und die Abstraktion des Abgleichens.41 Rohs kannte Janichs pragmatische Theorie der Zeitmessung in der Protophysik der Zeit und hat sie für einen möglichen unter eventuell mehreren überzeugenden Ansät-

37 Rohs:

Geist und Gegenwart, 46. Geist und Gegenwart, 60. 39 Vgl. Rohs: Geist und Gegenwart, 59. 40 Vgl. dazu Janich: Was messen Uhren?; ders.: Zeit und Natur. 41 Die Grundlagen hierfür führt er aus in: Janich: Die Protophysik der Zeit, Kap. 3. Hierbei wäre streng genommen zuerst von Zeitdauern auszugehen, sodass der Abgleich von Zeitpunkten bereits eine Abstraktionsleistung darstellen würde. Vgl. ebd., 213 f. Da die einzelnen Konstruktionsschritte hier jedoch nicht weiter ausgeführt werden, sondern nur die Grundidee an dieser Stelle von Interesse ist, bleibt dies unberücksichtigt. 38 Rohs:

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zen gehalten.42 Allerdings berücksichtigt er auch später43 nicht, dass Janich mit der fortschreitenden thematischen Auseinandersetzung immer stärker die Bedeutung der kulturellen Einbettung auch für das Erarbeiten solcher Theorien selbst herausgearbeitet hat.44 Aus diesem Grund ist auch die historische Rekonstruktion45 bei Janich keineswegs nur Zusatz, sondern wesentlicher Bestandteil. Mit diesen Überlegungen wäre auch die Theorie, welche die B-Reihen und das Feld ontologisch beschreibt, in einen Kontext der jeweiligen lebensweltlichen und historisch bedingten Entstehung und Einbettung einzuordnen. Dies verhindert keineswegs ein Verständnis von Realität, aber dieses würde aus seiner statischen Metaphysik herausgelöst. Mit Blick auf die im vorhergehenden Abschnitt aufgeführten Optionen wäre daher weniger für 1.1) als für 1.2) zu argumentieren. 1.2) ist hierbei allerdings daraufhin zu modifizieren, dass der Erkenntnisprozess selbst wie in 3) aus dem jeweiligen Erkenntnisstand heraus beurteilt wird. Über 3) hinausgehend ist es jedoch nicht zwingend erforderlich, von einem linearen Progress des Erkennens auf eine letztgültige Wahrheit hin auszugehen. Insgesamt ergibt sich daraus eine Fokussierung auf eine Variante von 2), die 1) und 3) in modifizierter Form zu integrieren erlaubt.

11.4

Ausblick: Das Problem der Intersubjektivität

Nachdem zu den philosophisch-erkenntnistheoretischen Konsequenzen aus Gödels Argument zur Unmöglichkeit einer objektiven Bestimmung von Veränderung im vorhergehenden Abschnitt Position bezogen wurde, soll nun zum Abschluss der Problembereich noch etwas ausgeweitet werden. Wie sich zeigen wird, handelt es sich bei der Frage nach einer objektiven Bestimmung von Veränderung nicht nur um ein theoretisch interessantes Problem. Vielmehr lassen sich darüber hinaus auch Verbindungen zu praktisch relevanten Diskursen ziehen. Das von Gödel aufgeworfene Problem der Nicht-Objektivität von Veränderung und Zeit gewinnt durch den Einbezug der Differenz zwischen A- und B-Reihen der Zeit an Reichweite. Da auch in Gödels Argumentation die rein physikalische Ebene zugunsten des Einbezugs von A-Reihen bezüglich der Beobachtungspositionen verlassen wird, lässt sich die Frage nach der Objektivität von Veränderung unter Bezugnahme auf die Darstellung Rohs’ im vorhergehenden Abschnitt umformulieren in: Ist Intersubjektivität im Sinne von für sämtliche Beobachtungspositionen gültigen Referenzpositionen zum Zuordnen und Abgleichen von Zeiterleben oder gar von Erleben allgemein möglich? Der Ausgriff auf Erleben allgemein wird dadurch ermöglicht, dass es im Anschluss an Rohs strukturell als durch A-Reihen bestimmt zu verstehen ist: Erleben ist immer an die Struktur des „Jetzt“ und des Werdens gebunden, und Erlebtes ist als vergangenes „Jetzt“ zu verstehen. Sofern Zeiterleben

42 Vgl.

Rohs: Ist eine ausweisbare Zeitmessung möglich?, 146–151. Rohs: Feld – Zeit – Ich, 194. 44 Zum Programm des von ihm gegründeten Methodischen Kulturalismus vgl. Janich/Hartmann: Methodischer Kulturalismus. 45 Vgl. Janich: Die Protophysik der Zeit, Kap. 4. 43 Vgl.

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nicht im beschriebenen Sinne objektivierbar ist, gilt dies daher auch für Erleben allgemein und dadurch auch für Erfahrung überhaupt. Dies ist in gewisser Weise analog zur Kantischen Konzeption von Zeit als a priori. Die Umkehrung, dass aus der Nicht-Objektivität von Erleben oder Erfahrung die Nicht-Objektivität der Zeit folge, gilt indessen nicht, wird aber auch für die hier gemachte Argumentation nicht benötigt. Die zuvor eher theoretisch-abstrakt anmutende Frage nach der Objektivität von Veränderung erweist sich dadurch zugleich als erkenntnistheoretisch und praktisch relevante Frage, die insbesondere unter sogenannten postmodernen Fragestellungen eine Aktualität entfaltet. Schließlich verweist sie auf das Bedürfnis und das Ausgangsproblem, Gültigkeit und Wahrheit als letztlich gesichert ansehen zu wollen und dadurch im wesentlichen Sinne nicht mehr zu erschütternden Halt und Orientierung zu finden. Mit Gödels Argument ist einer solchen, als mit einer absoluten Referenzposition von Erleben im Einklang stehend verstandenen Intersubjektivität eine Absage zu erteilen. Zu beachten ist die Schlussfolgerungsrichtung der hier geführten Argumentation: Im Kontext von Gödels Argument folgt aus der Möglichkeit von Zeitreisen die Nicht-Objektivität der Zeit. Diese wiederum impliziert, wie im vorangegangenen Abschnitt dargelegt, die Unmöglichkeit von Intersubjektivität im zuvor genannten Sinne. Keineswegs lässt sich jedoch daraus schließen, dass das Annehmen der Unmöglichkeit von Intersubjektivität im genannten Sinne die Nicht-Objektivität von Zeit oder gar die Möglichkeit von Zeitreisen zur Voraussetzung hat. Des Weiteren kann ausgehend von Gödels Argument eine Differenzierung vorgenommen werden. Wie oben beschrieben, macht Gödel die Annahme, dass für den Erweis von Objektivität sinnvollerweise eine metaphysische Notwendigkeit vorauszusetzen sei: Es genüge nicht, dass das Universum vermutlich faktisch so beschaffen ist, dass Zeitreisen nicht möglich sind und Zeit deshalb objektiv bestimmbar sein kann. Vielmehr sollte, wenn Zeit objektiv ist, das Universum notwendigerweise (im metaphysischen Sinn) so beschaffen sein, dass dies gilt.46 Wird diese Annahme gemacht, bleibt es dabei, dass Intersubjektivität im oben beschriebenen Sinne nicht möglich ist. Aber auch ohne diese Annahme ergibt sich ein interessantes Ergebnis, bei dem der Objektivitätsbegriff unter Verzicht auf Metaphysik entsprechend abgewandelt wurde. Es wäre dann festzustellen, dass Intersubjektivität zwar objektiv sein kann, aber nicht zwingend sein muss. Denn die wie auch immer gering eingeschätzte Möglichkeit, dass Zeit nicht objektiv ist, besteht weiterhin. Durch Einbezug dieser Metaebene geht auch die durch Intersubjektivität als grundsätzlicher Möglichkeit des Abgleichens von Erleben erhoffte absolute epistemische Sicherheit verloren: Dass Intersubjektivität im genannten Sinne gegeben ist, lässt sich nicht zwingend beweisen.47

46 Vgl.

Gödel: A remark about the relationship between relativity theory and idealistic philosophy, 562/CW II 206 f. 47 Hier lässt sich eine gewisse Parallele zu den Konsequenzen von Gödels zweitem Unvollständigkeitssatz ziehen. Während dieser auf logische Systeme bezogen ist, geht es hier aber um die Struktur von Erfahrung.

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Es gilt jedoch zu berücksichtigen, dass Intersubjektivität damit nicht grundsätzlich unmöglich ist. Im oben genannten Sinne ist Intersubjektivität als absolut gesetzt. Es ist jedoch gleichwohl möglich, dass innerhalb bestimmter festgelegter Rahmen intersubjektiver Austausch und intersubjektives Verstehen gelingen. Peter Rohs nimmt an, dass ein solcher Rahmen in den Bedingungen gelingender Kommunikation besteht. Der Grund hierfür dürfte darin liegen, dass ein Austausch über und ein Abgleichen der eigenen Standpunkte unabdingbar sein dürfte, um Intersubjektivität herzustellen. Dieser kommunikative Rahmen impliziere insbesondere das Fehlen extrem hoher Geschwindigkeiten.48 Ohne dass dadurch die Möglichkeit von Intersubjektivität unbezweifelbar bewiesen wäre, ist zumindest festzustellen, dass Gödels Argument für diese Eingrenzung keine Gültigkeit hat, da die Zeitreisen auf hohe Geschwindigkeiten angewiesen sind.49 Mit dem Benennen von Kommunikation wird das Bestimmen weiterer Bedingungen relevant, zu denen insbesondere Überlegungen zu Diskursen, Argumenten und Begründungen zählen. Diese wären daher zur genaueren Erläuterung des Herstellens von Intersubjektivität zu vertiefen. Sie gehen jedoch über die aktuelle Themenstellung hinaus. Hier genügt die Vorführung, dass Gödels Argument nicht nur ein interessantes physikalisches Problem thematisiert, sondern darüber hinaus auch an wichtige weitere philosophische Problemfelder anschlussfähig ist.

Literatur Alles, T. (2020). Gödel ontologischer Gottesbeweis im Kontext der Systematischen Theologie. Neue Zeitschrift für Systematische Theologie und Religionsphilosophie, 62 (1), 1–40. Friedman, M. (2001). Dynamics of reason (Stanford Kant Lectures 1999). Stanford. Friedman, M. (2009). Einstein, Kant, and the relativized a priori. In von M. Bitbol, P. Kerszberg,& Petitot, J. (Hrsg.), Constituting objectivity: Transcendental perspectives on modern physics. The Western Ontario Series in Philosophy of Science 74 (S. 253–267). Springer Netherlands. https:// doi.org/10.1007/978-1-4020-9510-8_15. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In von S. Feferman, J. W. Dawson, S. C. Kleene, G. H. Moore, R. M. Solovay, & J. van Heijenoort (Hrsg.), Kurt Gödel: Collected works. Vol. I: Publications 1929–1936 (S. 144–195). Oxford University Press (1986). (Originalquelle: Monatshefte für Mathematik und Physik 38 (1931), 173–198). Gödel, K. (1946/49/9-C1 und 1946/9-B2). Some observations about the relationship between theory of relativity and Kantian philosophy. In von S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay (Hrsg.), Kurt Gödel: collected works: Vol. III: Unpublished essays and lectures (Bd. 3, S. 230–259). Oxford University Press (1995). Gödel, K. (1949a). A remark on the relationship between relativity theory and idealistic philosophy. In von S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay & J. van Heijenoort (Hrsg.), Kurt Gödel: Collected works: Vol. II: Publications 1938–1974 (Bd. 2, S. 202–207). Oxford Uni-

48 Vgl.

dazu Rohs: Geist und Gegenwart, 58. berechnet, dass die Geschwindigkeit mindestens √1 der Lichtgeschwindigkeit betragen 2 müsste. Vgl. Gödel: A remark about the relationship between relativity theory and idealistic philosophy, Fußnote 11.

49 Gödel

11

Konsequenzen aus der Unmöglichkeit einer objektiven . . .

115

versity Press (1990). (Originalquelle: Schilpp, Paul A. (Hrsg.), Albert Einstein, PhilosopherScientist (Library of Living Philosophers 7) Evanston 1949, S. 555–562). Gödel, K. (1949b). Eine Bemerkung über die Beziehungen zwischen der Relativitätstheorie und der idealistischen Philosophie“. In von P. A. Schilpp (Hrsg.), Albert Einstein als Naturforscher und Philosoph. (Übersetzung von 1949a durch Hans Hartmann und Ergänzung durch Kurt Gödel) (S. 406–412). Kohlhammer (1955). Gödel, K. (1949c). An example of a new type of cosmological solutions of einstein’s field equations of gravitation. In von S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Hrsg.), Kurt Gödel: Collected works: Vol. II: Publications 1938–1974 (Bd. 2, S. 190– 198). Oxford University Press (1990). (Originalquelle: Reviews of Modern Physics 21 (1949), S. 447–450). Gödel, K. (1949d). Lecture on rotating universes. In von S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay, (Hrsg.), Kurt Gödel: Collected works: Vol. III: Unpublished essays and lectures (Bd. 3, S. 269–287). Oxford University Press (1995). Gödel, K. (1951). Some basic theorems on the foundations of mathematics and their implications. In Feferman, von S., Dawson Jr., J. W., Goldfarb, W., Parsons, C., & Solovay, R. (Hrsg.), Kurt Gödel: Collected works: Vol. III: Unpublished essays and lectures (Bd. 3, S. 304–323). Oxford University Press (1995). (25. Josiah Willard Gibbs Lecture). Gödel, K. (1952). Rotating universes in general relativity theory. In S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Hrsg.), Kurt Gödel: Collected works: Vol. II: Publications 1938–1974 (Bd. 2, S. 208–216). Oxford University Press (1990). (Originalquelle: Proceedings of the International Congress of Mathematicians. Cambridge, Massachusetts, U.S.A. August 30–September 6, 1950. Volume I, Providence 1952, 175–181). Gödel, K. (1964). What is Cantor’s continuum problem? In von S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Hrsg.), Kurt Gödel: Collected works: Vol. II: Publications 1938–1974 (Bd. 2, S. 254–270). Oxford University Press (1990). (Originalquelle: Benacerraf, P., & Putnam, H. (Hrsg.), Philosophy of mathematics: Selected readings, New Jersey, 1964, S. 258–273; überarbeitete und erweiterte Version von: Gödel, Kurt: What is Cantor’s continuum problem? The American Mathematical Monthly, 54(1947), 515–525). Gödel, K. (1970). Ontological proof. In von S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons, & R. Solovay (Hrsg.), Kurt Gödel: Collected works: Vol. III: Unpublished essays and lectures (Bd. 3, S. 403–404). Oxford University Press (1995). Gödel, K. (2016). Meine ph Ansicht. (Zettel aus dem Nachlass, Item 060168; Transkription nach: Engelen, E.-M.: What is the Link between Aristotle’s philosophy of mind, the iterative conception of set, Gödel’s incompleteness theorems and god? About the pleasure and the difficulties of interpreting Kurt Gödel’s philosophical remarks. In G. Crocco& E.-M. Engelen (Hrsg.), Kurt Gödel. Philosopher-Scientist, Aix-en-Provence: Presses Universitaires de Provence, 2016 (S. 171–188, hier: 172). Janich, P. (1980). Die Protophysik der Zeit. Konstruktive Begründung und Geschichte der Zeitmessung. Suhrkamp Verlag GmbH. Janich, P. (1997a). Was messen Uhren? In von P. Janich (Hrsg.), Das Maß der Dinge. Protophysik von Raum, Zeit und Materie. Suhrkamp Taschenbuch Wissenschaft 1334 (S. 131–138). Suhrkamp. (zuerst erschienen in: alma mater philippina 1982, S. 12–14). Janich, P. (1997b). Zeit und Natur. In von P. Janich (Hrsg.), Das Maß der Dinge. Protophysik von Raum, Zeit und Materie. Suhrkamp Taschenbuch Wissenschaft 1334 (S. 253–268). Suhrkamp (zuerst erschienen in: Hauskeller, M., Rehmann-Sutter, C., Schiemann, G. (Hrsg.), Naturerkenntnis und Natursein. Für Gernot Böhme, 1997, S. 107–122). Janich, P., & Hartmann, D. (1996). Methodischer Kulturalismus. In von P. Janich & D. Hartmann (Hrsg.), Methodischer Kulturalismus. Zwischen Naturalismus und Postmoderne. Suhrkamp Taschenbuch Wissenschaft 1272 (S. 9–69). Suhrkamp. Kant, I. (1998). Kritik der reinen Vernunft, Hrsg. v. Jens Timmermann. Felix Meiner Verlag. (erste Aufl. 1781 = A, zweite Aufl. 1787 = B); Kant’s gesammelte Schriften Bd. 3, hrsg. von der Königlich Preussischen Akademie der Wissenschaften, Berlin 1902 [= AA].

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´ etorzecka, K. (2015). Gödel’s slingshot argument and his onto-theological sysKovaˇc, S., & Swi¸ ´ etorzecka (Hrsg.), Gödel’s ntological argument: History, modifications, and tem. In von K. Swi¸ controversies (S. 123–162). Semper. McTaggart, J. E. (1908). The unreality of time. Mind, 17(68), 457–474. Muck, O. S. J. (1992). Eigenschaften Gottes im Licht des Gödelchen Arguments. Theologie und Philosophie, 67, 60–85. Rohs, P. (1986). Ist Eine Ausweisbare Zeitmessung Moglich? Zur »Protophysik der Zeit«. Philosophische Rundschau, 33(1–2), 133–151. Rohs, P. (1996). Feld – Zeit – Ich. Entwurf einer feldtheoretischen Transzendentalphilosophie. Klostermann, Vittorio. Rohs, P. (2016). Geist und Gegenwart. Entwurf einer analytischen Transzendentalphilosophie. mentis. Sobel, J. H. (1987). Gödel’s ontological proof. In von J. J. Thomson (Hrsg.), On being and saying. Essays for Richard Cartwright (S. 241–261). Cambridge University Press. Stein, H. (1990). Introductory note to 1949a. In von S. Feferman, J. W. Dawson, S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort (Hrsg.), Kurt Gödel: Collected works: Vol. II: Publications 1938– 1974 (S. 199–201). Oxford University Press. Wang, H. (1996). A logical journey. From Gödel to philosophy. The MIT Press. Thorben Alles ist wissenschaftlicher Mitarbeiter am Lehrstuhl für Systematische Theologie der Evangelischen Fakultät der Universität Bonn (Deutschland). Er hat evangelische Theologie, Mathematik, Erziehungswissenschaften und Philosophie in Bonn und Straßburg (Frankreich) studiert. Seine Forschungsinteressen sind Theorien der Begründung, Ethik der Argumentation, theologische und religiöse Erkenntnistheorie und theologische Hermeneutik. Diese sollen in seinem laufenden Dissertationsprojekt „A Logic of Reasoning“ zusammengeführt werden.

How much Time Does a Logical Inference Take?

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12.1

Introduction

The Kurt Gödel Award 2021, presented by the Kurt Gödel Circle of Friends Berlin, focuses on the question of what it “mean[s] for our world view if, according to Gödel, we also assume the non-existence of time”. In a physical world, the concept of time— or as one should maybe say more carefully: the psychological illusion of time—has always been closely connected to the concept of causation. While Gödel himself regarded causation and time as two fundamental concepts in philosophy and metaphysics (see Fig. 12.1),1 the close interwovenness between the two has been stressed since ancient times, when Aristotle, both in his Physics and Metaphysics2 addressed causality in his account on the “Four Causes”.3 Moreover, whereas all four causes can be regarded as an explanation and an answer to the central question of why things are as they are, it is Aristotle’s third cause, the efficient cause, that, in a straightforward manner, underlines the temporal element of the concept of causality. In this paper, we now transfer the focus, which is usually seen in the connection between physical causation and time, into an epistemological setting, concentrating on the connection between logical inferences and the phenomenon of time. These

1 Kurt

Gödel Papers, Box 11b, Folder 15, item accession 060168, on deposit with the Manuscripts Division, Department of Rare Books and Special Collections, Princeton University Library. Used with permission of the Institute for Advanced Study. Unpublished Copyright Institute for Advanced Study. All rights reserved. Transcriptions and translations by the author. 2 Physics II 3 and Metaphysics V 2 3 For detailed information about Aristotle’s Four Causes, see Falcon (2022). T. Lethen (B) Department of Philosophy, University of Helsinki, Helsinki, Finland

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_12

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Fig. 12.1 An undated list, written by Gödel, of fundamental philosophical concepts including cause and time. Note that cause is immediately preceded by the concept reason. Additionally, matter and form remind us of Aristotle’s first and second causes

logical inferences, which will be represented by formal rules, play a central role in the evolution of our knowledge, and as the term “evolution” suggests, they should be closely connected to the concept of time, or at least—as was mentioned before—to a psychological illusion of something we would call “time”. Concerning the evolution of mathematical knowledge, Gödel writes in his notebook MaxPhil IX (p. 45):4 Bem. (Phil): Wenn man die Objekte der Mathematik als durch den Geist konstruiert ansieht, bringt man notwendig ein zeitliches Element herein. Sie existieren erst nach der Konstruktion [. . .]. [Remark (philosophy): If one regards the objects of mathematics as constructed by the mind, one necessarily brings in a temporal element. They exist only after the construction [. . .].]

If one agrees that a mental construction of mathematical knowledge is always based on logical inferences, then there appears to exist a striking analogy between Aristotle’s effective physical cause on the one hand side and an epistemological cause or “reason” on the other. The aim of this paper is now an analysis of the link between logical reasoning and the concept of time, thus supporting a remark in Gödel’s notebook MaxPhil IV,5 which clearly identifies “Zeit” (time) as one of the fundamental psychological (as well as physical) concepts. Gödel writes (p. 251): Fra. (Phil.): 1. Gibt es auch eine “reine” Psychologie, welche a priori ist und in welche die inneren Wahrnehmungen eingeordnet werden, ebenso wie eine reine Physik (Raum-ZeitLehre)? [Question (philosophy): 1. Is there also a “pure” psychology, which is a priori and into which the inner perceptions are embedded, as well as a pure physics (space-time theory)?]

4 Kurt

Gödel Papers, Box 6b, Folder 69, item accession 030095. The book was written between November 1942 and March 1943. 5 Kurt Gödel Papers, Box 6b, Folder 67, item accession 030090. This book was written between May 1941 and April 1942.

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As a footnote he adds: Begriffe: Zeit, actus, Erinnerung, Wahrnehmung [Concepts: time, actus, memory, perception]

As far as the title of this paper is concerned, we will certainly not be able to give a definite answer to the question how much time a logical inference takes. However, we will use the opportunity to shed light upon this question from at least three distinct angles, regarding this investigation as a collection of evidence, sometimes in favour of, sometimes against the just quoted “necessary temporal element”. We postpone an answer to the question of what the non-existence of time would mean to our worldview, to our conclusion at the end of this paper. In the course of this paper, we proceed as follows. In Sect. 12.2, we lay the basis for our discussion by fixing a set of rules of inference. In a system of natural deduction, these rules define an intuitionistic propositional logic, and we argue why this kind of logic is appropriate for our investigation. Section 12.3 analyses what are known as Kripke models, which, from the very beginning, have always been closely associated to the phenomenon of time. Section 12.4 then focuses on the close connection between rules of inference on the one hand and algorithms and their inherent dynamics on the other, established by the Curry–Howard correspondence. Finally, we sketch an analogy, mentioned by Gödel in unpublished notes, between biological evolution and the epistemic evolution of knowledge.

12.2

Rules of Inference

In this section, we introduce the rules of inference that form the basis for our discussion in the sections to come. These rules, presented in Fig. 12.2, constitute a proof system called intuitionistic natural deduction (NJ), in which trees represent formal proofs of a judgement   ϕ, which is always the root of its proof tree. The leaves of the tree are axioms (Ax), while the inner nodes follow the construction scheme defined by the given rules. Note that the actual logical inference is always read from the top to the bottom of a rule, thus interpreting the rules in a forward direction. Each judgement   ϕ (read “ proves ϕ”) consists of a set  of propositional formulas, which can be interpreted as a logical environment, and a single formula ϕ, which holds in this very environment. A formula ϕ is provable in our logical system if  ϕ is the root of a proof tree.6 For more details on intuitionistic natural deduction and examples of proof trees, see, e.g. Sorensen and Urzyczyn (2006). Before we start our discussion about the central question of how much time is needed for a single logical inference, it should be worth explaining why we concentrate on an intuitionistic setting, thus neglecting the rule of double negation elimination and closely related rules and axioms like the law of the excluded middle.

6 In this case, the environment (i.e. the left-hand side of the judgement) is taken to be the empty set.

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Fig. 12.2 The rules of inference for the intuitionistic natural deduction calculus NJ

First, the considered phenomena naturally call for an intuitionistic setting, either because the structures in question naturally constitute models for an intuitionistic logic (Sect. 12.3), or because we interpret logical formulas as simple types of certain algorithms represented by terms of combinatory logic. These types always correspond to theorems of an intuitionistic logic (Sects. 12.4 and 12.5). Secondly, it is Gödel himself who leads us into an intuitionistic direction as soon as propositions are regarded as a piece of knowledge. In his notebook MaxPhil V (p. 292) he writes about L.E.J. Brouwer’s intuitionistic interpretation of mathematics:7 Bem. (Gr.): In der Brouwerschen Interpret. der Mathematik werden gar nicht die mathema” (oder in einer schwächeren tischen Sätze P interpretiert, sondern die Sätze “ Form: P ist beweisbar). [Remark (foundations): In Brouwer’s interpretation of mathematics, not the mathematical ” (or in a weaker form: P is propositions P are interpreted, but the propositions “ provable).]

Therefore, following Brouwer, the application of a rule of inference represents an evolution of knowledge.8 Taking rule (→ E), i.e. modus ponens, as an example, the

7 Kurt Gödel Papers, Box 6b, Folder 67, item accession 030091. The book was started in May 1942. 8 Gödel’s

claim that intuitionistic logic is closely connected to the notion of knowledge is also strongly supported by the Gödel–McKinsey–Tarski translation (Gödel, 1933), which translates

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knowledge of both ϕ and the implication ϕ → ψ in the same logical environment inevitably leads to the knowledge of ψ within this environment. We will see in the following section that, apparently, this is by no means a matter of course. Gödel’s psychological view upon intuitionism is also expressed in a remark to be found in his notebook MaxPhil III (p. 149), where he states:9 Bem.: Nächstes Ziel für Lekt. & Arb. Unmath. sollte sein: die Grundbegriffe der Psychol. in Ordnung bringen (derart, dass man alle beschreibt und zumindest die “möglichen” Gesetze sieht, analog zu den kinematischen und Kraftbegriffen in der Physik). Rechtfertigung dafür: 1.) Anwendungen für Grundlagen (Int. ist eine schematisierte Psych.) 2.) Günstige Wirkung auf die Klarheit meines Denkens, die Arbeitseinteilung, Sprachbeherrschung, Arbeits-Max. ganz im Allgemeinen 3.) Das ist wahrscheinlich eine Voraussetzung und ein Weg zur Metaphysik und zu einer “Weltanschauung” zu kommen. Und zwar solltest du es systematisch tun. [Remark: The next goal for reading and working non-math. should be: to put the basic concepts of psychology in order (such that one describes all of them and sees at least the “possible” laws, analogous to the kinematic and force notions in physics.) Justification: 1.) Applications for foundations (intuitionism is a schematized psychology). 2.) Positive effect on the clarity of my thinking, organization of work, mastery of language, working maxims in general. 3.) This is probably a prerequisite and a way to come to metaphysics and to a “worldview”. Further, you should do it systematically.]

12.3

Time and the Kripke Model

As a basis for the discussion to follow, we first briefly define the notion of a Kripke model, which—in the case of intuitionistic logic—was first introduced in Kripke (1965). A Kripke model is a triple (W , ≤, ), where W is a non-empty set of possible worlds, ≤ is a partial order on W , and  (read “forces”) is a relation between W and the set of propositional variables, which, as a starting point, assigns atomic propositions to possible worlds, and for worlds w, w and propositions p satisfies the monotonicity condition If w ≤ w and w  p then w  p.

(12.1)

From the very beginning, Kripke regarded the relation ≤ as an “earlier-than” relation. In Kripke (1965, p. 98), he writes under the heading “Intuitive interpretation”:

intuitionistic propositions into a (classical) modal logic S4, in which the usual provability operator may now be interpreted as a knowledge operator. Details about the modal language of knowledge can be found, for example, in van Benthem (2010, Chap. 12). 9 Kurt Gödel Papers, Box 6b, Folder 66, item accession 030089.

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Fig. 12.3 An example Kripke model. (The model is the one given in Kripke (1965), the timeline having been added)

We intend the nodes H to represent points in time (or “evidential situations”), at which we may have various pieces of information.10

Moreover, Mints (2000, p. 47) writes in his introduction to Kripke models: The semantics for intuitionistic logic described in the following [i.e. Kripke models] reflects a more dynamic approach: Our current knowledge about the truth of statements can improve. Some statements whose truth status was previously indeterminate can be established as true. The value true corresponds to firmly established truth that is preserved with the advancement of knowledge, and the value false corresponds to “not yet true”.

Note how the idea of a “firmly established truth” corresponds to monotonicity as given in (1). An example of a Kripke model is depicted in Fig. 12.3. Finally, the forcing relation  is extended to include absurdity ⊥ and compound logical statements that have been constructed using the usual connectives ∧, ∨ and →, denoting conjunction, disjunction and implication, respectively. w w w w

 ⊥ never holds,  (ϕ ∧ ψ) iff w  ϕ and w  ψ,  (ϕ ∨ ψ) iff w  ϕ or w  ψ,  (ϕ → ψ) iff from w  ϕ follows w  ψ for all w with w ≤ w .

(12.2) (12.3) (12.4) (12.5)

If negation ¬ is understood as an abbreviation, i.e. ¬ϕ ≡ (ϕ →⊥), then we can also add w  ¬ϕ iff w  ϕ does not hold for any w with w ≤ w .

(12.6)

For the model in Fig. 12.3, we have, for example, H2  ( p ∧ q) and G  (r → p). The reader should also note that neither G  q nor G  ¬q are valid in this model.

10 Here, H

denotes some possible world. The “pieces of information” are represented by the propositional variables.

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We now turn to the question of how much time a rule of inference takes in a Kripke model. Considering conjunction and disjunction first, it turns out that no time is taken at all. As the stipulations (3) and (4) are restricted to a single world or—as we should say—to a single moment in time, the knowledge of the pieces of information p and q, for example, immediately leads to the knowledge of p ∧ q. Thus, the rules (∧ I) and (∨ I) seem to lose the necessity to be read in a forward direction. The knowledge of p, q, and p ∧q exists at the same time, and it thus certainly becomes more difficult to regard p and q as the reasons or the premisses for the information p ∧ q. The situation becomes more complex if implication is considered in (5) and in the special case of (6). An interpretation of both rule (→ I) and rule (→ E) (modus ponens) becomes much more complicated, of the former because the logical environment would have to include present and future situations, of the latter because the information ψ is no longer caused or preceded by ϕ and ϕ → ψ; rather, both ϕ and ψ now appear to be preconditions for the implication ϕ → ψ, thus turning the rule “upside down”. Time seems to collapse.11 Gödel clearly appears to have anticipated these problems. In unpublished (undated) notes12 concerning his 1951 Gibbs lecture (Gödel, 1951), he writes: wissen = mit Recht in jedem beliebigen Grad davon überzeugt sein (insbesondere also mit Recht beschließen, es unter keinen Umständen zu revidieren) Theorem: Ich weiß p. ⊃ Ich bin in unmittelbarem Kontakt mit den Gegenständen der Aussage p. [to know = to be rightly convinced of it to any degree (in particular, to decide rightly not to revise it under any circumstances). Theorem: I know p. ⊃ I am in direct contact with the objects of the statement p.]

Although his “definition” of the term “knowledge” (Wissen) is in perfect accordance with the monotonicity of the Kripke model, the “direct contact” (unmittelbarer Kontakt) appears to be highly problematic, as—in the case of implicational statements and negations—the future is concerned as well as the present moment. Apparently, Gödel was well aware of this fact. Addressing negation, he adds a footnote to his “definition” of knowledge, which reads: Weil sie Erkenntnis möglich machen, also sicherlich ihre Negation niemals anzunehmen wäre, denn das hieße erkennen wollen und gleichzeitig seine Möglichkeit zu negieren. (Für Wissen in diesem Sinne gilt das Th. nicht.) Aber woher weiß ich diese Implic.? (Zu kompliziert für eine direkte Schau. Außerdem gar nicht evident, denn ein Umlernen vielleicht möglich, und der Satz betrifft eine sehr allgemeine Aussage über unseren Erkenntnis-Appar., der gar nicht mit einem Blick zu übersehen [ist].) [Because they make knowledge possible, so certainly their negation would never be accepted, because that means to want to know and at the same time to negate its possibility. (For

11 These problems seem to be supported by the fact that, in Łukasiewicz’s many-valued logic, the epistemological compatibility of conjunction and implication, expressed by the inequality p∧( p → q) ≤ q, is not valid. On the other hand, we have p ∧ q ≤ p → q. For details, see Lethen (2021a). 12 Kurt Gödel Papers, Box 12, Folder 43, item accession 060573.

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knowledge in this sense the theorem does not hold.) But how do I know this implication? (Too complicated for a direct observation. Moreover, not evident at all, because a relearning may be possible, and the theorem concerns a very general statement about our cognitive apparatus, which cannot be overlooked by a glance).]

As an aside, it should be interesting to note that, if these remarks were indeed written in 1951 when the Gibbs lecture was given, Gödel would have anticipated the central elements of Kripke’s models—possible worlds and the fundamental monotonicity property—by more than a decade, even if he does not explicitly mention the close connection to intuitionistic logic in his notes.13 Finally, we take a brief look at the possible “re-learning” (Umlernen) mentioned in Gödel’s notes. How is it possible to gain new information in the course of time? Kripke himself writes (Kripke, 1965, S.p. 98):14 Now given a point in time G, there are various possibilities open for gaining further information about the propositions. [. . .] At point G (representing our present information) we have proved P. For all we know, we may remain “stuck” at G for an arbitrarily long time, without gaining any new information. But it is possible that we will gain enough information to “jump” to point H1 (in which case we have a proof of R in addition to P), or to the point H2 (where we get a proof of Q in addition to P), or even to the points H3 or H4 .

It is interesting to note that here a “jump” in time seems to rely on “proofs” of new pieces of information, and clearly these proofs cannot be based on proper rules of inference. Rather, “gaining further information about the propositions” seems to refer to the psychological feeling of evidence; as soon as a certain level of evidence has been reached, as soon as a certain threshold has been crossed, the piece of information is taken for granted within the present model. Alternatively, the new information might be a contingent fact, a revealed dogma, or a “new” plausible axiom that, up to that point of time, had not been included in the theory, but is now added for “aesthetical reasons” or “reasons of completeness”. Gödel himself, addressing the notion of evidence, writes in his notebook MaxPhil III (p. 54): Bem.: Alles, was irgendwie eingesehen werden kann, ist entweder15 1. vollkommen klar (das, was man wissen kann), 2. einigermaßen klar (Ersetzungssaxiom), 3. plausibel, d.h. annehmbar aus aesthetischen, Vollständigkeitsgründen, etc. [Remark: Everything that can be understood somehow is either 1. perfectly clear (that what can be known), 2. reasonably clear (axiom of replacement), 3. plausible, i.e. acceptable due to aesthetical reasons or reasons of completeness.]

13 The concept of branching time had been known to Gödel at least since 1935; see Lethen (2012b). 14 The 15 The

reader may want to compare this quote to Fig. 12.3. first two items are marked as “analytisch” at the right margin.

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Time and Algorithmic Dynamics

Algorithms have always played a central role in mathematics, and they have always been regarded as dynamic processes in which clearly defined distinct steps have to be carried out one after the other. This dynamic view was also preserved when the first formal concepts appeared, notably in Alan M. Turing’s introduction of the Turing machine in 1936, where Turing speaks of a “process” carried out in distinct “moments” (Turing, 1937, p. 231). Also, these dynamics reappear in other formal algorithmic concepts like Alonzo Church’s lambda calculus, and in combinatory logic, which was introduced independently in Schönfinkel (1924) and Curry (1930), and which we will take as a basis for our considerations in this section. The language of untyped combinatory logic (CL)16 can be defined by the following simple production rule, which introduces the two combinators S and K, as well as the notion of application, in which we call the left and right-hand terms the subject and the object, respectively, of the application. τ :: = S | K | (τ τ ).

(12.7)

Example CL terms are S, (SK), and ((KK)(KS)), of which the last one may be simplified to KK(KS), following an association to the left. Computation in CL is now reflected by two axiom schemes of weak reduction,17 which can be represented as Sx yz  x z(yz) Kx y  x,

(12.8) (12.9)

where the meta-variables x, y, z stand for arbitrary CL terms. We give an example reduction, in which the very last term cannot be reduced any further and, thus, constitutes a normal form. K(SKSS)K  K(KS(SS))K  KSK  S. The reader should note that we could also have chosen a different path of reductions, which would nevertheless have led to the same result, a property of CL known as the Church–Rosser property. K(SKSS)K  SKSS  KS(SS)  S. In typed combinatory logic, principal types can be assigned to terms. Two axioms assign the principal types (a → (b → c)) → ((a → b) → (a → c)) and

16 We restrict ourselves to a very brief introduction here. For details, see J. Hindley and Seldin (2008) and Bimbo (2011). 17 As we do not consider the notion of strong reduction in this paper, we will simply speak of reduction.

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Fig. 12.4 Assigning principal types to terms in CL. In rule (pt), sub is a “minimal” substitution that syntactically unifies ϕ and ϑ

Fig. 12.5 A tree assigning the principal type a → a to the CL term SKK using two applications of rule (pt). The substitution in the first step is {a  p, b  q, c  p}, and in the second step { p  a, q  (b → a)}

a → (b → a) (or any other alphabetic variants) to the terms S and K, respectively. If ϕ → ψ and ϑ are the principal types of terms σ and τ , one determines a “minimal” variable substitution sub that syntactically unifies ϕ and ϑ. This unification then enables an application of the rule modus ponens, which finally leads to the conclusion sub(ψ). These rules are summarised in Fig. 12.4. An example, which finally assigns the principal type a → a to the term SKK, is shown in Fig. 12.5.18 The reader who up until now has been unfamiliar with typing systems will have noticed that types are nothing but implicational propositions, and indeed, this close correspondence between propositions and types is often referred to as the propositions-as-types- or Curry–Howard correspondence.19 Note that, within this framework, the term may also be regarded as a proof for its principal type. Thus, the example in Fig. 12.5 also demonstrates that SKK is a proof for the proposition a → a. Without going into further details, we mention that the rules in Fig. 12.4 are, indeed, sufficient to prove the whole implicational fragment of intuitionistic propositional logic (without the absurdity ⊥). The details can be found in J.R. Hindley (1997). Before we move on, we mention as an aside that Gödel, in his notebook MaxPhil X,20 comments on the proposition a → (b → a), which is the principal type of the combinator K and is also known as the Positive Paradox, as follows. Note the close correspondence to the monotonicity property of the Kripke model and to Gödel’s definition of “wissen” (to know). Bem. (Gr.): Der psych. Sinn von a ⊃ (b ⊃ a) ist:21 Das Festhalten an einem einmal gefällten Urteil, wenn etwas Neues gefunden wird. Es besteht irgendwie die psych. Tendenz, wenn

reader should note that the term SKKx reduces to x for any CL term x. It, thus, represents an identity operator with principal type a → a. 19 In the context of propositional logic, rule (pt) is often called rule D or condensed detachment. 20 Kurt Gödel Papers, Box 6b, Folder 70, item accession 030096. The book was written between March 1943 and January 1944. 21 Gödel adds the comment “ = ‘wenn’ ” above the second ⊃. (“wenn” means both “when” and “if.”) 18 The

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man durch eine neue Erkenntnis zur Umstoßung einer alten veranlasst wird, das irgendwie nicht als tatsächliches Zurücknehmen gelten zu lassen. (“So wurde das nicht gemeint.”) Remark (foundations): The psychological sense of a ⊃ (b ⊃ a) is: Holding on to a judgment once made, when something new is found. There is somehow the psychological tendency, when one is prompted by a new finding to overturn an old one, to somehow not let that count as an actual withdrawal. (“That’s not how it was meant”.)

We now return to our central question, how much time a rule of inference takes, this time regarded in the light of algorithmic dynamics. Starting with rule (pt), which is obviously closely related to rule (→ E), modus ponens, one can see that it actually takes place before the algorithm starts running. If we regard an application of terms as nothing but a syntactical juxtaposition of the subject and the object of the application, we are rather confronted with a phenomenon that can be described as the stretching of a rubber band: time seems to stand still. Nevertheless, the stretching does trigger the actual run of the corresponding algorithm. In order to find out what kind of inference takes place while the algorithm is actually running, we first take a look at the rule of substitution, which has not explicitly been considered so far. This rule can be regarded as a way to specialise already proved knowledge. Moreover, while the calculi given in Figs. 12.2 and 12.4 do not include the rule of substitution, it is an admissible rule in intuitionistic propositional logic.22 The rule of substitution can simply be presented as follows, sub being any substitution that replaces variables by arbitrary propositions: ϕ for any variable substitution sub. sub(ϕ)

(12.10)

As an example, the substitution {a  (a → b)} would transform the proposition a → a into the specialised proposition (a → b) → (a → b). As a single step of an algorithm is now represented by a single reduction in combinatory logic, we consider the principal types of a CL term and of the corresponding reduced term. Taking I as an abbreviation for the term SKK, the example term SKSI reduces to KI(SI) in a first step, which, in a second step, can be reduced to the term I. However, while the original term SKSI has the principal type (a → b) → (a → b), the two following terms both have the principal type a → a, of which (a → b) → (a → b) is a substitution instance. The situation is depicted in Fig. 12.6. As it turns out, the development of the types shown in the example is not a coincidence, and it can be shown in general that for CL terms σ and τ with principal types ϕ and ψ, respectively, a reduction σ  τ always implies that ϕ is a substitution instance of ψ. Thus, knowledge seems to decrease over time while the

22 Admissibility means that any proof

without it.

of a proposition that uses this rule can be replaced by a proof

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Fig. 12.6 The example algorithm SKSI reduces to KI(SI) and finally to I. The knowledge, represented by the corresponding types, seems to decrease over time

algorithm is running; the rule of substitution (10) is turned upside down, the effect appears before the cause, and again time seems to collapse. As a closing remark for this section, we mention that the reason for the fact that a reduction in combinatory logic changes the corresponding principal types seems to be well hidden in rule (pt), which allows for a variable substitution in both the subject and the object of an application, heavily relying on a procedure known as (bidirectional) unification, which was first considered in Robinson (1965). Unpublished work (Lethen, 2022) has recently shown that a restriction to a unidirectional unification, generally known as pattern matching, does indeed preserve the principal type when a CL term is weakly reduced.23 This seems to indicate that the subject of an application should not be given the power to alter the object in any way. We will briefly return to this phenomenon in the following section, where the object will be interpreted as a fixed point of knowledge.

12.5

Time and the Evolution of Knowledge

This section, in which we take a look at the intimate connection between biological evolution and the evolution of knowledge, was inspired by a quote that can be found in Gödel’s “Aflenz” book on quantum mechanics.24 Here Gödel writes in item 263: Es ist tatsächlich bestechend, die zweckmäßigen Umstellungen des Individuums (Gedächtnis) aus demselben Prinzip zu erklären wie die zweckmäßigen Umwandlungen der Arten (Anpassung). [It is indeed convincing to explain the purposeful transformations of the individual (memory) in terms of the same principle as the purposeful transformations of the species (adaptation).]

In another place, he phrases the same analogy as follows:25 Es ist tatsächlich unbefriedigend, die zweckmäßigen Reaktionen der einzelnen Individuen (Lernen) durch ein vollkommen anderes Prinzip zu erklären als die zweckmäßigen Reaktionen der Arten (Anpassung).

23 For

the difference between uni- and bi-directional unification, see Knight (1989). Gödel Papers, Box 6a, Folder 59, item accession 030082. 25 Notebook Quantenmechanik II, Kurt Gödel Papers, Box 6b, Folder 78, item accession 030107. 24 Kurt

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[It is indeed unsatisfactory to explain the purposeful reactions of individuals (learning) by a completely different principle than the purposeful reactions of the species (adaptation).]

This very idea can already be found in Henri Bergson’s writings, who puts the emphasis on the close relation between nature’s creation of new species on the one hand and intellectual, i.e. human, creation and invention on the other. In Bergson (1904), he writes: Si la vie est une création, nous devons nous la représenter par analogie avec les créations qu’il nous est donné d’observer, c’est-à-dire avec celles que nous accomplissons nous-mêmes. [If life is a creation, we must imagine it by analogy with the creations that we are given to observe, that is to say with those that we ourselves accomplish.]

Bergson even goes so far as to claim that the analogy is not a mere coincidence (as cited in Hadamard, 1945, p. xii): The inventive effort which is found in all domains of life by the creation of new species has found in mankind alone the means of continuing itself by individuals on whom has been bestowed, along with intelligence, the faculty of initiative, independence and liberty.

In what follows, we will utilise Gregory Chaitin’s metabiological model of evolution as presented in Chaitin (2012). This model has often been criticised as being far too simple and, thus, as not being able to display any interesting behaviour.26 Notwithstanding this critique, Chaitin’s model will first enable us to bring out the just mentioned “inventive effort” in a precise and rigourous way, and we will suggest combinatory logic—interpreted as a prototype programming language—as a means to actually implement Chaitin’s algorithm. Secondly, the model will enable us to transfer the mechanism of the creation of new species into an epistemological setting, thus—as Gödel puts it—explaining “the purposeful transformations of the individual in terms of the same principle as the purposeful transformations of the species”. As it turns out, our formal approach leads to the insight that human creation does indeed simulate nature’s—or, if the reader prefers—divine creation in a very precise sense. If this simulation is taken for granted, the mental construction of ideas should take a certain amount of time, just as the phenomenon of time has always been inevitably connected to the process of Darwinian evolution. In fact, Gödel himself

26 For

a critique, see, for example, Siedliñski (2017) and Ewert, Dembski, and Marks (2013). This is certainly not the place to give a comprehensive answer to this critique. Nevertheless, we rather prefer to have the question, whether the model is “inspiring either for computer scientists [. . .] or for biologists” (Siedliñski, 2017, p. 143), be decided by the test of time.

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was concerned with the very question of how much time biological evolution would take. In a notebook entitled “Physik (1935)”, he writes:27 Biologie: Mögliche Anwendungen der Mathematik: a.) Wie viele Generationen seit Beginn des Lebens bis zum Menschen? Wie schnell muss die Entwicklung vor sich gegangen sein? (Zufällige Schwankungen wie groß?) b.) Anzahl der möglichen Molekülgruppierungen in einer Keimzelle daraus berechnen. Wie große Schwankungen der Eigenschaften des Tiers ruft die kleinstmögliche Schwankung der Eigenschaften der Zelle hervor? [Biology: Possible applications of mathematics: a.) How many generations from the beginning of life to human beings? How fast must the development have progressed? (How wide are the random variations?) b.) From this, calculate the number of possible molecule groupings in a germ cell. How much variation in the properties of the animal does the smallest possible variation in the properties of the cell cause?]

Chaitin’s Model of Evolution Evolution in Chaitin’s model is an evolution of a single piece of software.28 In order to run this software, we first fix a universal computer C, which always takes a program and some data for this program as its input. Evolution then starts with a randomly chosen program A, which, along with an empty set of data ε, is run on the computer C. The output is then interpreted as a natural number a, called the fitness of the program A. Next, we randomly choose a mutation M, which is nothing but another program for the computer C. M is then applied to the program A, yielding a new program A . Again, the fitness a  of the program A is computed, which is then compared to the current fitness a. If a  is greater than a, the program A is taken as our newly produced ‘organism”, A being discarded. Otherwise A itself is discarded: an implementation of the survival of the fittest. Afterwards, a new random mutation M is chosen, and the whole process starts again. The procedure is coded as Algorithm 1 below.

27 Kurt

Gödel Papers, Box 6b, Folder 77, item accession 030105. Chaitin follows the general idea that DNA may be regarded as software.

28 Here,

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Algorithm 1 Chaitin’s metabiological algorithm 1: A ← random program 2: a ← C(A, ε) 3: loop 4: M ← random program 5: A ← C(M, A) 6: a  ← C(A , ε) 7: if a  > a then 8: A ← A 9: a ← a 10: end if 11: end loop

Needless to say, we are concealing some of the details in our all-too-brief description of the process. What kind of “software” are we talking about, what is the programming language? How do we interpret an output of the computer C, sometimes as an integer, sometimes as a new program? How do we choose a program “at random”? However, before we address at least some of these questions, we briefly mention a truly fundamental drawback of Chaitin’s algorithm, which surfaces in lines 2, 5 and 6, or—in other words—whenever the computer C is evaluating its input. Here, one does not know whether or not the computation will halt; the computer underlies the famous halting problem. Thus, Chaitin’s model of evolution has to rely on an oracle which predicts whether a program gets stuck in an endless loop. If so, a new random mutation (or random organism) will have to be chosen. Again, the question of whether we encounter a divine element here is left to the reader. In any case, the dynamics of the underlying algorithm perfectly matches the dynamics of Darwinian evolution. Without going into the details here, Chaitin’s main result now states that the overall fitness would grow much more slowly if an algorithm simply produced random organisms (i.e. programs) until it finds a fitter one. Thus, the secret of a fast growing fitness seems to lie in the fact that the fittest organism is not discarded but “used” and “incorporated” in the next round of the algorithm. Turning to the question of the programming language, it is worth mentioning that the literature is largely silent about this decision. One of the exceptions, if not the only one, is the paper Ewert, Dembski, and Marks (2013), which uses the language P  introduced and analysed in Böhm (1964) and Böhm and Jacopini (1966). The programs of P  are strings built of the four symbols R, λ, (, ) only, following the production rule π :: = R | λ | π π | (π ) An example

P 

program might thus look like this: R Rλ(λR(Rλ))R.

(12.11)

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Fig. 12.7 The first 30 steps of an example run of Chaitin’s algorithm in CL. Whenever a fitter organism (i.e. a longer CL term) is found, the number of the current generation and the fitness of the organism are printed

Similar to Turing machines, P  programs act on symbols on an infinite tape, which serves as the computer’s memory.29 If the symbols R, λ, (, ) are now used as the possible tape symbols as well, one can achieve that P  programs can actually be applied to other P  programs. However, Ewert, Dembski, and Marks (2013) have not been able to reproduce Chaitin’s theoretical results about a fast-growing fitness, thus concluding that “metabiology does not demonstrate successful Darwinian evolution” and that “although elegant in conception, metabiology departs from reality because it pays no attention to resource limitations”. The authors, though, do not take into consideration a major drawback of the use of the language P  in a metabiological setting: if the currently fittest program is stored on the tape and the next random mutation starts to act on this program, the mutation really destroys—in a very crude manner—what has been reached so far, instead of incorporating and re-using the information gained so far. Thus, the overall fitness cannot grow any faster than in a naive algorithm that always starts from scratch. Having identified this fundamental problem with the P  approach, we now propose combinatory logic (CL) as a suitable programming language in Chaitin’s algorithm. While the application of one program (a mutation) to another program (the organism) is straightforward, we now define the fitness of a program simply as the number of applications appearing in its normal form.30 Again, we will have to rely on an oracle in order to find out whether a normal form exists. An example run of Chaitin’s algorithm in CL is shown in Fig. 12.7. As the fitness and, thus, the number of applications in the corresponding CL terms grows very fast indeed, only the number of iterations and the fitness itself are reproduced. As a comparison, Fig. 12.8 shows the first nine steps of an algorithm that produces random CL terms and simply keeps them until a fitter one is found. What is missing is an application of the newly produced term to a current individuum.

instruction R moves a read/write head one position to the right. The instruction λ moves it one position to the left, after altering the symbol on the tape in a cyclic way. Parentheses indicate a loop that is executed as long as the current symbol on the tape is not the first one in the list of possible tape symbols. 30 A more elegant solution would consider the information content of an organism A, defined as the length of the shortest CL term, which reduces to A. As this measure is not computable, we stick to the more naive (but practical) solution. 29 The

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Fig. 12.8 The first nine steps of an example run of an algorithm that simply produces random CL terms until a fitter one is found. Again, only the number of the current generation and the fitness of the organism are printed

The main advantages of using CL as the programming language in Chaitin’s algorithm can be summarised as follows: • The currently fittest organism is re-used and incorporated by the newly generated random mutation. While it might be cancelled by the mutation in certain rare cases, it may as well be duplicated or itself applied to parts of the mutation. The implementation, thus, supports the theoretical results. • The model can easily be extended to a population of several organisms, which may even interact through mutual “sexual application”.31 • Combinatory logic is a very well understood theory for which many different mathematical models exist; see Bimbo (2011) and the literature mentioned therein. • By using typed combinatory logic, one could even avoid the use of an oracle, as typable CL terms always have a normal form.

The Evolution of Knowledge After this brief digression into the field of biological evolution, we will now turn to the evolution of knowledge. The close analogy will suggest that if time is needed in order to create new species, a certain amount of time is also needed for the creation of knowledge, which—in the case of purely mental creativity—can be reduced to the phenomenon of having an idea. Moreover, whereas it is certainly difficult to give a precise definition of the term “idea”, at least three components should be present in order to describe this very concept: 1. Randomness 2. Truth 3. Applicability. Although we regard these components as necessary, we allow for a vague perception of these terms at this stage, and they will merely serve as a guide when constructing the formal system in question.

31 The fact that Chaitin reduces his model to one single organism has frequently been criticised; see, for example, Siedliñski (2017). However, Bergson, (1911, Chap. 1) already argues: “Strictly speaking, there is nothing to prevent our imagining that the evolution of life might have taken place in one single individual by means of a series of transformations spread over thousands of ages. Or, instead of a single individual, any number might be supposed, succeeding each other in unilinear series”.

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As to (1.), it should be noted that we do not construct or compute our everyday ideas algorithmically, nor do we choose them out of a given pool in a predefined manner, at least not as a conscious act. A real idea seems to crawl up from our subconscious and—at least viewed from our conscious perspective—seems to be carrying a clear element of chance. Jacques Hadamard, in his book about the psychology of invention (Hadamard, 1945, pp. 29–30) puts it this way: It cannot be avoided that this first operation take place, to a certain extent at random, so that the role of chance is hardly doubtful in this first step of the mental process. However, we see that that intervention of chance occurs inside the unconscious: for most of these combinations—more exactly, all those which are useless—remain unknown to us.

As to (2.), every idea has to be true in some sense. Any idea not fulfilling this condition would, when applied to our knowledge (which is, of course, considered to be true, following the principle of veridicality32 ), produce nothing but nonsense. Last but not least, an idea has to be applicable, as it is always an idea relative to some situation or to some knowledge, in order not to be “useless”, as Hadamard puts it in the quotation just given. If one is stuck in the middle of a mathematical problem, an idea has to be directed towards the solution of that very problem. It has to be applicable to a prepared set of propositions or—as we will call it later on—of points of knowledge. Hadamard (1945, p. 32) even connects applicability with beauty: So there remains only Poincaré’s final conclusion, viz., that to the unconscious belongs not only the complicated task of constructing the bulk of various combinations of ideas, but also the most delicate and essential one of selecting those which satisfy our sense of beauty and, consequently, are likely to be useful.

In what follows, we will represent both ideas and points of knowledge as implicational formulas of propositional intuitionistic logic. In order to meet the conditions of randomness and truth of an idea, we will repeatedly produce random proofs, which are represented by terms of (typed) combinatory logic. The actual principal type will then serve as a possible candidate for an idea. In the next step, we test whether this candidate (which can be considered as a pre-idea) can actually be applied to the current point of knowledge. Here, application is considered an application of modus ponens, following a minimal substitution either in both pre-idea and point of knowledge, or, if one prefers, in the pre-idea only.33 Let us consider an example. Suppose the current point of knowledge were the simple proposition a → a. In a first case, the random term might be the expression SII, where I again abbreviates the term SKK. As it turns out, this term is not typable. Thus, it is not a proof of a proposition and would simply be discarded, not even having reached the status of a pre-idea. In the next case, the random proof might

K a ϕ → ϕ states that what an agent a knows is always true, truth is included in the concept of knowledge. 33 In this connection, the reader may want to skip back to the closing remark of Sect. 12.4. 32 In epistemic logic, the veridicality property

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Fig. 12.9 The first 24 steps of an example run of the algorithm that produces random ideas and applies them to the current point of knowledge. Printed are the number of the current generation and the complexity of the knowledge, measured by the number of implications. In this case, the initial point of knowledge is the proposition a → (b → a)

Fig. 12.10 The first 9 steps of an example run of the algorithm that produces random proofs and keeps the according knowledge until more complex knowledge has been found. Note the enormous difference in speed compared to the example given in Fig. 12.9, where the current knowledge is incorporated in the process

be the term SI, which proves the proposition ((b → c) → b) → ((b → c) → c). However, this pre-idea cannot be applied to the point of knowledge a → a and is, therefore, discarded as well. Finally, we consider the simple random proof S with principal type (a → (b → c)) → ((a → b) → (a → c)). This pre-idea is indeed a proper idea as it can be applied to the current point of knowledge a → a, yielding the new knowledge ((b → c) → b) → ((b → c) → c), which is then subjected to a new idea in the following iteration of the algorithm. Figure 12.9 shows an example run of the algorithm just described. Again, only the number of the iteration, as well as the size (i.e. the number of implications) of the current point of knowledge, is printed. As a small modification, we only update the current point of knowledge if the new knowledge is more complex by at least ten implications. Finally, in order to demonstrate the enormous speed at which an evolution of knowledge now takes places, we compare the algorithm described with a version in which the current point of knowledge is always completely ignored and not incorporated in the production of new random knowledge. The modified algorithm now simply produces random proofs and checks whether the proved proposition is greater in size than the current one. What is missing is an application of the pre-ideas to the current knowledge. Figure 12.10 shows the resulting behaviour. Basically, we have been able to show that creativity can flourish even in complete isolation, i.e. without any contact with the outside world, just by tossing a coin. As Bergson (1911) puts it in the introduction to his Creative Evolution: [T]he intellect has only to follow its natural movement, after the lightest possible contact with experience, in order to go from discovery to discovery, sure that experience is following behind it and will justify it invariably.

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We have also been able to demonstrate that the current knowledge grows far faster if the new ideas are applied to this knowledge. In the present context, it is most important, though, to note the close analogy between natural evolution on the one hand, represented here by Chaitin’s metabiological algorithm, and the evolution of knowledge on the other, simulated by the application of random ideas to a point of knowledge, which have both been represented by logical propositions. This analogy can be regarded on two different levels. First, from an intuitive point of view, an obvious parallel between the two given algorithms meets the eye: random mutations correspond to random ideas, organisms to points of knowledge. Application and iteration each play a central role. Secondly, on a more technical level, combinatory logic, untyped and typed, serves as a convincing tool in the actual implementation of both algorithms, representing both software and proofs, linked through the Curry– Howard correspondence. As we have already mentioned, time has always been inseparably connected to Darwinian evolution, and we may, therefore, conclude that it should also be inseparably connected to human creativity and an evolution of knowledge. Bergson (1911, p. 11) states: The universe endures. The more we study the nature of time, the more we shall comprehend that duration means invention, the creation of forms, the continual elaboration of the absolutely new.

12.6

Conclusion

In his notebook MaxPhil V, Gödel writes (p. 350): Bem. (Phil.): Es gibt zwei Methoden der Philosophie, die intuitive und die kombinat[orische]. Die erste ist sehend, die zweite blind, die erste anstrengend,34 die zweite leicht, die erste verständnisvoll, die zweite mechanisch. Die erste hat zu tun mit dem Sinn der Sprache, die zweite mit der Sprache selbst. Die erste führt zu einem lebendigen Wissen, die zweite zu einem abstakten Wissen. (Das Richtige [ist] eine Kombin[ation] beider. Ich habe bisher die zweite vernachlässigt.) Nur für die zweite (axiomat[ische]) braucht man Papier und Bleistift. Die beiden Methoden entsprechen genau den beiden Anschauungen über die Erkenntnis, dass sie ein Wahrnehmen bzw. ein Konstruieren ist. [Remark (philosophy): There are two methods of philosophy, the intuitive and the combinatorial. The first is seeing, the second blind, the first exhausting,35 the second easy, the first understanding, the second mechanical. The first has to do with the meaning of the language, the second with the language itself. The first leads to a living knowledge, the second to an abstract knowledge. (The right approach is a combination of the two. I have neglected the second so far). Only for the second (axiomatic) one needs paper and pencil. The two methods correspond exactly to the two views of knowledge, that it is a perceiving and a constructing respectively.]

34 Gödel’s 35 Gödel’s

footnote: “erfordert Konzentration” footnote: “requires concentration”

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Without doubt, we have followed the second approach in this essay, having used “paper and pencil”, assisted by a computer executing the “mechanical” simulations proposed. Equipped with the considerations and the formal results of the different perspectives on the connection between logical inference and (an illusion of) time, we can now finally return to the question in which way the assumption of the nonexistence of time may influence our worldview. For this purpose, we have to include the human mind into our worldview and analyse how the brain and our thoughts reflect the physical world surrounding us. If a physical time does not exist, still, nature has obviously equipped man with an inner sense of time, a sense that helps to structure the outer physical world and its events, as well as our collections of knowledge. The—without doubt—diffuse picture that emerges from the different considerations presented in this paper may now be taken as a gentle hint at an inconsistency within this very perception of (a non-existing) time, an inconsistency that may be interpreted as an immediate consequence of the non-existence itself. It is worthwhile mentioning that the overall situation may be easily compared to a worldview in which the infinite set, especially the universal set, does not have any “real” existence. Still, man is apparently equipped with the ability to “perceive” infinite sets, possibly even the universal set. Again, this ability helps us to structure both the world surrounding us as well as our mental processes. Yet, the (formal) inconsistencies within naive set theory, for instance surfacing with Russell’s paradox, clearly hint at the assumed non-existence. Acknowledgements The research for this article was conducted within the Gödeliana project led by Jan von Plato, Helsinki, Finland. The project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 787758) and from the Academy of Finland (Decision No. 318066).

References Bergson, H. (1904). “Notice sur la vie et les oeuvres de Félix Ravaisson-Mollien”. In: H. Bergson (Ed.), La Pensée et le Mouvant. Essais et conférences, (pp. 253-291). Alcan. Bergson, H. (1911). Creative Evolution. Henry Holt and Company. Bimbo, K. (2011). Combinatory Logic - Pure, Applied and Typed. Chapman and Hall/CRC. https:// doi.org/10.1201/b11046. Böhm, C. (1964). “On a family of Turing machines and the related programming language.” ICC Bulletin, 3(3), 185–194. Böhm, C. & Jacopini, G. (1966). “Flow Diagrams, Turing Machines and Languages with Only Two Formation Rules”. Communication ACM, 9(5), 366–371. https://doi.org/10.1145/355592. 365646. Chaitin, G. (2012). Proving Darwin: Making Biology Mathematical. Pantheon Books. Curry, H. B. (1930). “Grundlagen der Kombinatorischen Logik”. American Journal of Mathematics, 52(3), 509–536. Ewert, W., Dembski, W. A., & Marks, R. J. (2013). “Active Information in Metabiology”. BIOComplexity, 2013(4), 1–10. https://doi.org/10.5048/BIO-C.2013.4. Falcon, A. (2022). “Aristotle on Causality”. In: E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. Spring 2022. Metaphysics Research Lab, Stanford University.

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Gödel, K. (1951). “Some basic theorems of the foundations of mathematics and their implications.” In: S. Feferman, W. Goldfarb & C. Parsons et al. (Eds.), Kurt Gödel, Collected Works. Vol. III. Unpublished Essays and Lectures (pp. 304–323). Oxford University Press. Gödel, K. (1933). Eine Interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematischen Kolloquiums, 4, 39–40. Hadamard, J. (1945). The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field. Princeton University Press. Hindley, J. R. (1997). Basic Simple Type Theory. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press. https://doi.org/10.1017/CBO9780511608865. Hindley, J. & Seldin , J. (2008). Lambda-Calculus and Combinators: An Introduction. Cambridge University Press. ISBN: 9780521898850. Knight, K. (1989). “Unification: A Multidisciplinary Survey”. ACM Computing Surveys, 21(1), 93–124. ISSN: 0360-0300. https://doi.org/10.1145/62029.62030. Kripke, S. A. (1965). “Semantical Analysis of Intuitionistic Logic I”. In: J. Crossley & M. Dummett (Eds.), Formal Systems and Recursive Functions. Vol. 40. Studies in Logic and the Foundations of Mathematics, (pp. 92–130). Elsevier. https://doi.org/10.1016/S0049-237X(08)71685-9. Lethen, T. (2021a). “Gödel on many-valued logic”. The Review of Symbolic Logic, 1–17. https:// doi.org/10.1017/S1755020321000034. Lethen, T. (2021b). “Monads, Types, and Branching Time-Kurt Gödel’s Approach Towards a Theory of the Soul”. In: O. Passon & C. Benzmüller (Eds.), Wider den Reduktionismus: Ausgewählte Beiträge zum Kurt Gödel Preis 2019, (pp. 13–24). Springer Berlin Heidelberg. https://doi.org/ 10.1007/978-3-662-63187-4_3. Lethen, T. (2022). “A (machine-oriented) Logic based on Pattern Matching”. In: submitted. Mints, G. (2000). A Short Introduction to Intuitionistic Logic. Kluwer Academic. Robinson, J. A. (1965). “A Machine-Oriented Logic Based on the Resolution Principle”. Journal of the ACM, 12(1), 23–41. https://doi.org/10.1145/321250.321253. Schönfinkel, M. (1924). Über die Bausteine der mathematischen Logik. Mathematische Annalen, 92, 305–316. Siedliñski, R. (2017). “Turing Machines and Evolution. A Critique of Gregory Chaitin’s Metabiology”. Studies in Logic, Grammar and Rhetoric, 48(1), 133–150. https://doi.org/10.1515/slgr2016-0059. Sorensen, M. and P. Urzyczyn (2006). Lectures on the Curry-Howard Isomorphism. Vol. 149. Studies in Logic and the Foundations of Mathematics. Elsevier. Turing, A. M. (1937). “On Computable Numbers, with an Application to the Entscheidungsproblem”. In: Proceedings of the London Mathematical Society s2-42.1, pp. 230–265. https://doi. org/10.1112/plms/s2-42.1.230 A correction, ibid. 43 (1937), 544–546. van Benthem, J. (2010). Modal Logic for Open Minds. Center for the Study of Language and Inf. Tim Lethen equipped with a degree in mathematics and computer science, has studied the Gabelsberger shorthand system in order to be able to read the personal and scientific notes in Kurt Gödel’s Nachlass, kept at the Institute for Advanced Study in Princeton. Since 2018, Tim has been working for the Godeliana project based in Helsinki, Finland, led by Jan von Plato. As part of this project, he has transcribed many of Gödel’s notebooks, including several books on theology and on the foundations of quantum mechanics. Tim is mainly interested in Gödel’s formal approach to metaphysics and theology, on which he has written several papers. In 2019, the Kurt Gödel Society (Vienna) awarded to him the Kurt Gödel Gold Medal for the best oral presentation at the Kurt Gödel’s Legacy conference.

Was bedeutet es für unser Weltbild, wenn wir mit Kurt Gödel die Nichtexistenz der Zeit annehmen?

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Guido Stemme

»Sie sind die Leser, ich bin der Schreiber, die Welt ist der Autor, und wir sind die Welt« Für Aaron Jan Ło´s, der zu kurz bei uns war, immer bei uns ist.

Vorüberlegendes Geleit 1968 spricht der US-amerikanische Literaturwissenschaftler Leslie Fiedler an der Universität zu Freiburg über die Notwendigkeit, die Scheidung von literarischer Hochkultur und Massenkultur zu beenden. Im selben Jahr erscheint der gedruckte Vortrag unter dem Titel »Das Zeitalter der neuen Literatur« in der Wochenzeitung »Christ und Welt«. Ein Jahr später findet er sich – in etwas kompakterer Version – unter dem Titel »Cross the border – close the Gap« im Männermagazin »Playboy«. Grenzen mögen überschritten, Lücken geschlossen werden. Diesen Gedanken legt auch die gestellte Frage nahe. Eine Beantwortung, Antworten suchend, müssen wir – um mit Heinrich von Kleist zu sprechen – die Reise um die Welt machen, das Paradies ist uns verriegelt. Ich nehme Rücksicht auf Sie, mache – wenn nötig – kleine Pausen auf unserem Spaziergang. Ich biete kein schwieriges Gelände, Sie müssen nicht klettern und können nicht abstürzen. Bitte gestatten Sie mir aber, dass ich nicht nach Ihnen suchen kann, wenn Sie sich verlaufen, verlaufen haben. Es obliegt Ihnen, wieder den Anschluss zur Gruppe zu finden. Und bitte zeigen Sie sich auch nicht verwundert – oder reagieren gar ablehnend – wenn die Form des Textes nicht Ihren Gewohnheiten entsprechen sollte. Es ist doch nur ein Versuch. Zu

G. Stemme (B) Mainz, Deutschland E-Mail: [email protected] URL: https://guidostemme.de/

© Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature 2023 O. Passon et al. (Hrsg.), On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit, https://doi.org/10.1007/978-3-662-67045-3_13

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dieser Form gehört es auch, dass ich mich fragen muss, was es denn für Kurt Gödel bedeutet, wenn wir seine Annahme nicht teilen, nicht teilen können oder wollen. Ernst Cassirer folgend, geht es hier doch um Geist und Leben, um beider Präsenz in symbolischen Formen, zusammengefasst in Mythos, Religion, Sprache und Kunst. Ich darf, ich muss mich wiederholen, wie der Teufel, »wenn es der Wahrheitsfindung dient«. Mich – mich wiederhole ich hier und da und kennzeichne mich nicht. Mein Text soll ja frisch bleiben. Wenn ich andere wiederhole, kennzeichne ich sie – mein Text soll ja redlich sein. Den großen Friedrich Schleiermacher aus seinen vertrauten Briefen über Friedrich Schlegels »Lucinde« zitierend, möchte ich meine bescheidene Summe zu diesem Aspekt bilden dürfen. »Wo so viel Schönheit und Harmonie ist, da muss auch zwischen dem Stoff und der Form, zwischen dem Dargestellten und der Darstellung ein so inniger Zusammenhang obwalten, daß die Einheit des Werks der einzig sichere Schlüssel zum Verständnis auch des Einzelnen bleibt und der einzige Standpunkt zur vollständigen Beantwortung mancher Fragen, was mit diesem und jenem gemeint oder warum gerade dieses und jenes dargestellt sei.« und »Die Liebe soll auferstehen, ihre zerstückten Glieder soll ein neues Leben vereinigen und beseelen, daß sie froh und frei herrsche im Gemüt der Menschen und in ihren Werken und die leeren Schatten vermeinter Tugenden verdränge.« Einstieg/e Wir wollen uns vergewissern, ob wir die Frage verstehen, verstanden haben, wie wir die Frage verstehen, verstehen wollen. Wenn wir die Frage und ihre Felder weiten, die Begriffe atmen lassen, ihre Magie nicht verwerfen, werden wir – vielleicht – transsektoral, finden wir den Gang in die Freiheit, den Angang der Freiheit zur Antwort. Wir reiben uns an den aufscheinenden Begriffen und gewinnen so Energie (Reibungsgewinn): Bedeutung, Wir, Weltbild (und Weltanschauung), Nichtexistenz, Zeit, annehmen Das Deuten erschließt uns Bedeutung, wenn wir um Bedeutungen der Bedeutung wissen. Es wird eine offene, eine offenbarte Bedeutung geben, das ist soweit klar. Es wird aber wohl auch eine verborgene Dimension der Bedeutung geben, eine, der wir nachspüren müssen, die sich nicht zeigt, vielleicht nur kurz zeigt, nur vermuten, nichts fassen lässt und doch zur Sprache finden sollte, finden will, finden muss – eine Reise um die Welt eben, der ganze Riemen. Von der einfachen Bedeutung, der wörtlichen, ziehen wir weiter zur indirekten, versuchen Hinweise zu sehen, auf sie zu achten. Schon finden wir ins interpretative Bedeutungsfeld, Quantität und Qualität liegen hinter, über, vor und unter uns. Diesem Weg folgend weist schließlich das Geheimnis über das Interpretative hinaus. Wir werden es kaum beziffern können, doch vielleicht können wir es umkreisen und einkreisen. So haben wir nun die Bedeutungsräume und Dimensionen benamt, die uns, unseren Weg formend, begleiten mögen. Wir sind jetzt hier. Und wir sind auch mit den Sinnen hier. Wir schmecken, fühlen, riechen, hören und sehen. Dieses »Wir« ist die Personaldeixis unserer Frage. Und ich erlaube mir(,) hier einfach(,) für uns zu sprechen. Ich bin jetzt das Sprachrohr eines (unseres) Wir – wir kommen zur Welt, indem ich uns zur Sprache bringe. Die

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Grammatik des Deutschen gibt uns kein Tertium, zwischen »Ich« und »Wir« findet sich nichts. So muss ich uns allen die Mutter sein, und ich nehme diese Aufgabe ernst – sehr ernst. Unser Weltbild spiegelt uns und anderen unsere Weltanschauung, wie die Spur eines, unseres Weges. Wie wir unser Weltbild haben und lesen, so hat das Weltbild auch uns, ist – mit uns – Epóche und Epoché. Die beiden Begriffe »Nichtexistenz« und »Zeit« sind für unseren Einstieg natürlich ganz erheblich. So erheblich, dass wir an dieser Stelle keine vorläufige Begriffsarbeit leisten wollen. Jetzt und hier den falschen Schluss zu ziehen, gefährdet das ganze Unterfangen, macht es – am Ende – gar hinfällig. Gedulden Sie sich, während Ihre Haltung zu den beiden Punkten durch Sie wirkt. Im Hauptteil wird Ihnen und ihnen ein Großteil unserer Aufmerksamkeit gelten – geschenkt. Annehmen heißt, eine Annahme zu machen, etwas in uns aufzunehmen, Teil von uns werden zu lassen, zu inkorporieren. So gesehen liegt mit dem Annehmen der Prozess einer bejahenden Akzeptanz vor, die Teil unserer selbst wird – ein grundsätzlicher Prozess, der nicht unterschätzt werden darf. Jetzt wollen wir die Frage erneut stellen, die Reise beginnen und uns an Antworten versuchen. Jetzt wird ein/e Geschichte in ihren und Ihren Lauf entlassen. Ja, auch in Ihren Lauf – mich interessiert doch Ihr Ausrollen von Welten, ohne indiskret werden zu müssen. [Hier wäre jetzt übrigens ein guter Punkt für Ihren Ausstieg, sofern Sie mich nicht weiter begleiten mögen.] Was bedeutet es für unser Weltbild, wenn wir mit Kurt Gödel die Nichtexistenz der Zeit annehmen?

0◦ Power deduced from powerless dust, Nurture from the infertile grave; Much the years may hold in trust, Space a thrall and Time a slave. Hark the boasting of the wise: “First are we of those that know!” But the little boy playing by the roadside cries, Trundling his hoop by the roadside cries, “I said it long ago.” John Buchan »Gleichviel ist mir’s aber, wo ich beginne. Denn ich werde dorthin wieder zurückkommen.« Parmenides, 3 (Übertragung von Hermann Diels)

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Die Zeit, das sind die anderen.

1◦

Zwei Proto-Axiome leiten die Geburt eines Dazwischen – ein und aus vorderes Proto-Axiom (Trennungsaxiom): Es ist zulässig, ein Ganzes zu unterteilen. A ∈ Welt ⇒ A ist Teil der Welt (ist wahr). hinteres Proto-Axiom (Fundamentalpunktaxiom oder Axiom des Maßpunkts): Jegliches System lässt das Setzen mindestens eines (Orientierungs-)Punkts zu. Von hier aus kann es nun in Raum und Zeit losgehen (solange wir uns klar darüber sind, dass unser Wesen auch im Bereich außerhalb fußt). Die Axiomatik der Axiomatik ist da – von nun an können sich sämtliche Axiome frei bilden und entfalten. Von der Götterwelt bis in den Teilchenzoo – der Kampf der Melkmaschinen um den Zugang zum Euter hat begonnen. Das Bestimmende und das Bestimmte scheinen auf. Damit das Unbestimmte via Bestimmung zu Bestimmtem werden kann, müssen diese Axiome gültig sein, sind sie nicht gültig, bleibt das Unbestimmte. Es ist wahr, dass Zeit nicht existiert, und es ist wahr, dass Zeit existiert. Zwei Aussagen – die eine der anderen gegenüber, die eine gegen die andere, die eine um die andere drehend – die eine in der anderen – ein namenloses Gemeinsam. Der Ordnung halber möchte ich erwähnen, dass Octavio Paz mit seiner »Custodia« der Pate für diese Passage ist – einfach schön. Chaos scheint überwunden, wird endgültig ungültig. »Von außen und innen heraus« fällt in ein »Innen« zusammen, ein Blitzableiter ersetzt die Erdung – Zeit bringt sich selbst zur Strecke – laisser faire, laisser aller. . . Von einer solchen Freiheit geht es hinab in die Logik. Alles zerfällt in reine Information. Hinauf, Hinauf! Parmenides und Heraklit – Einheit im Mantel der Dissense »Sie verstehen nicht, wie es [das Eine] auseinanderstrebend ineinander geht: gegenstrebige Vereinigung wie beim Bogen und der Leier.« Heraklit, Fragment 51 (Übertragung von Hermann Diels) »Die Zeit ist ein Knabe, der spielt, hin und her die Brettsteine setzt: Knabenregiment!« Heraklit, Fragment 52 (Übertragung von Hermann Diels) 2◦ Der homo illudens entwickelt sich weiter, illudiert – fortwährend gibt er sich dem inneren Spiel hin, schafft Illusionen, formt sie – Information, Weltanschauung. (Nicht die Illusion, das Illudieren ist uns das zentrale Phänomen.)

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Wollen wir richten, indem wir ordnen? Müssen wir ordnen, damit wir werden? Der Student in der Mansarde der Monade als hallende conditiones sine quibus ita vero. Das Licht ist (noch) an. Von Kollaps zu Kollaps setzen sie emsig ihre Bojen, ihr Geschichte – einzeln und im Kollektiv, übergeschichtlich wäre vermessen, ungeschickt ungeschichtlich. Und wie sie es lieben, aus falschen Voraussetzungen scheinbar richtige Schlüsse zu ziehen. Es ist so schön menschlich und klingt so kultiviert – Subreption. Mit Leibniz ist uns die Grenze des Leibes die Grenze der Welt und umgekehrt. Dreimeterbrett, ein Sprung ins Wasser – mit dem Wissen um die Höhe scheint man nicht mehr eintauchen zu können. Und zählt man beim Auftauchen Kacheln, ist man doch bereits aufgetaucht. Alle schöne Wassermetaphysik geht da baden. Es ist die Ewigkeit, die die Zeit ruft, die Welt in Angeln hebt – und sie bejahen, indem sie verneinen. Illusion ist, wo auch Fleisch ist. 3◦ Has everything already happened? Everything happens-now! now here (and/but here now nowhere) 5◦ Ein kleiner Zyklus – ein erstes passendes Zwischenstück Die Quelle aller Bedeutung sind Symbole auslösende Strukturen. wieder von der Kultur kommend LOGIK (der Mathematik koordiniert) Logik ist uns ein Grundmuster. Logisches Definieren schafft die Möglichkeit zum Finden und zur Untersuchung/Unterscheidung weiterer Muster. KRAFT, MATERIE, RAUM UND ZEIT (der Physik koordiniert) Wenn die Logik körperhaft wird – den Grundkräften der Schöpfung/Natur begegnet – entsteht Realität – granular und aus Schwerkraft. VITALITÄT (der Biologie koordiniert) Wenn sich (diese) Realität (iterativ) selbst begegnet, kann Vitalität entstehen. FREIHEIT (der Kultur koordiniert) Wenn sich Vitalität selbst begegnet, kann Freiheit (im Sinne der Schaffung symbolischer Räume) entstehen. von hier wieder in die Logik findend »Die große Freiheit des Künstlers ist, dass er keine hat; versteh’s wer kann.« Ernst Barlach

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8◦ »Es gibt eine andere Welt, aber sie ist (in) diese(r).« Paul Éluard zitiert in seinem Text »Physique de la Poésie« (Leib der Poesie) den in seinem Werk »L’âme romantique et le Rêve« (Die Seele der Romantik und der Traum) Ignaz-Vitalis Troxler zitierenden Albert Béguin. Éluard spielt zweimal über Bande, und doch ist er es, der mit dem Zitat verknüpft und verwoben wird. Entspricht das der besten aller Welten? Im Original, in Ausführlichkeit: »Il y a assurément un autre monde, mais il est dans celui-ci et, pour atteindre á sa pleine perfection, il faut qu’il soit bien reconnu et qu’on en lasse profession. L’homme doit chercher son état á venir dans le présent, et le ciel, non point au-dessus de la terre, mais en soi.« 13◦ Augustinus sagt »Was ist also die Zeit? Wenn mich niemand darnach fragt, weiß ich es, wenn ich es aber einem, der mich fragt, erklären sollte, weiß ich es nicht . . .« (Übertragung von Georg Rapp) Ich sage: »Wenn ich Zeit habe, habe ich keine Zeit, und wenn ich keine Zeit habe, habe ich Zeit« und denke dabei an das Doppelspaltexperiment. Welle oder Teilchen oder Welle oder Nicht-Welle oder Teilchen oder Nicht-Teilchen. Wer der Ungeheuerlichkeit dieses Experiments nicht folgen mag, verpasst Wesentliches. Welle oder Teilchen oder Seele oder Leib? Die Zeit existiert und die Zeit existiert nicht. Wir sind darauf zurückgeworfen, unser Schicksal, unsere Chance – ein Zauber. 21◦ Jetzt schon eine kleine Pause, ein kleiner, hoffentlich erheiternder Exkurs. Diesmal führt er uns in die physikalische Betrachtung eines Phänomens des Alltags. Der Augenblick: Wieviel fällt uns doch zu diesem Begriff ein, welche Erinnerungen und Erfahrungen können wir mit ihm verknüpfen. Die Physik ist, wie folgt, nüchtern interessant: Gesucht ist die Anzahl der Augenblicke vom Urknall bis zum Verdampfen des letzten großen Schwarzen Lochs. Die kleinste Zeiteinheit, die physikalisch (im Sinne der Disziplin »Physik«) relevant (weil noch sinnvoll) ist, ist die PlanckZeit: ca. 5 · 10−44 Sekunden. Sofern Protonen stabil sind, braucht es – mit dem Urknall beginnend – wohl rund 10122 Jahre, bis das letzte große Schwarze Loch verdampft ist (so zumindest Ervin László in »Wissenschaft und Wirklichkeit«). Ein Jahr entspricht ca. 3 · 107 Sekunden. Wir rechnen entsprechend und erhalten 10122 · 3 · 107 dividiert durch 5 · 10−44 ≈ 10173 . Es gibt also (nicht mehr als) rund 10173 (eine Eins, gefolgt von 173 Nullen) Augenblicke. [Ab da bleiben nur Elektronen-PositronenPaare, Neutrinos und Gammastrahlung.] 9 9 Wir zeichnen drei Kringel — 99 — und sind schon jenseits dieser Zahl. (99 hat mehr als 300.000.000 Stellen.)

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34◦ Wo bleibt Platon, wo ist Aristoteles? Wir sind inzwischen schon bei 34◦ angelangt. Jetzt sollte es doch daran sein, auf Platon zu sprechen zu kommen. »Ion«, von dort aus zum »Phaidon«, zum »Parmenides«, zum »Timaios«. Wir sind doch den üblichen Weg gewohnt und folgen ihm so gerne, weil wir wissen, was uns erwartet. Ach, lasst uns doch heute einfach nur in den Regen ein paar ihrer Gedanken stellen! Wege der Seele – Anamnesis, Partizipatio | Inhalt und Form, Form und Inhalt (a) | die Wesen der Schau der Zeit | Der Architekt schätzt die Luftschlösser nicht, Wasser und Feuer liegen dazwischen. | Wenn der Garten des Akademos von zu vielen Böcken bevölkert ist, dann . . . | aus Form reine Form machen | von Form zu reiner Form werdend | Götter sterben, damit wir leben. | Neue Ufer, Verdinglichung, verklärendes Klären frisst sein Geschichte. | Uns aus der zeitlosen Ewigkeit erblickend, wird Zeit als Zahl. | Der Gedanke der Bewegung stiftet diesen Blick, setzt aber voraus, dass unsere (beiden) Axiome gültig sind. | Urbild und Abbild wechseln die Stellung. | Geschichtetes wird Geschichte. | Das Unendliche wird unter Tränen aus der Ewigkeit geschlagen. | Hylemorphismus (b) | das Höhlengleichnis | die reinigende, die klärende Kraft des Unsauberen . . . 55◦ Zwischenspiel mit zwei Fragmenten, eines davon aus Fragmenten Dante macht den Anfang: »Dem Geometer gleich, der drauf geheftet Ganz ist, den Kreis zu messen, und, ob sinnend, Doch das Prinzip, des er bedarf, nicht findet, Also war ich bei diesem neuen Anblick. Sehn wollt’ ich, wie das Bild sich mit dem Kreise Vereint, und wie’s drin seine Stätte findet« (Dante – Paradiso – aus dem Ende des letzten Gesangs | Philalethes-Übertragung) –Pause– »die Vierecke und ihre Bewohner das Meer wird dir ganz gewiß das alles vermischen wird es dir durchnässen, es ertränken, illuminieren die Sprachen dienen als Farben was die Literatur anbetrifft setze zusammen, fahre fort: das Meer wird dir ganz gewiß das alles vermischen das alles bewegen, waschen, auflösen« (vermischte Fragmente | frei zusammengetragen aus Michel Butors – Illustrationen – übertragen von Helmut Scheffel)

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89◦ Wo bleibt eigentlich Leibniz? Die Frage beantworte ich nicht, hoffe, dass sie nicht ernst gemeint ist. Wo soll er bleiben, wenn er ist, wenn er überall ist? Gottfried Wilhelm Leibniz zieht sich durch den ganzen Text, nicht nur durch den Text – und Sie merken das nicht? Sie merken das doch? Jetzt merken Sie es? Wenn Sie es nicht merken, merke ich mir das. Also merken Sie sich bitte, dass Sie ab nun achtsam folgen und dass Sie aufmerken, wenn Sie Leibniz – in welcher Form und Art auch immer – bemerken. 90◦ Hurra, Hurra, Pythagoras! – oder ein der Jugend geschuldeter Einschub Die Geburt der Tragödie aus dem Sinnlich-Werden der Harmonie. 144◦ Wittgenstein Ja klar, Ludwig Wittgenstein darf hier natürlich auch nicht fehlen. Ihm obliegt es – und das nun schon seit längerer Zeit –, durch seine Erwähnung eine heitere und doch ernste, eine warme und doch auch nüchterne Atmosphäre in so viele Texte zu tragen. Das wollen wir teilen. Lassen wir ihn »Das Blaue Buch« aufschlagen und lassen Sie uns in »Eine Philosophische Betrachtung« und die »Zettel« schauen. Lassen Sie mich kurz, wild und unsauber zitieren. Bitte glauben Sie mir, dass auch so ein Schuh daraus wird (ein linker, ein rechter – das entscheiden Sie, aber seien Sie vorsichtig – in Wirklichkeit kann nichts gespiegelt werden). Die Seiten 49, 50 und 51 (gebundene Ausgabe), da sind wir doch schon mittendrin. »Was ist Zeit? [. . .] Und tatsächlich ist es die Grammatik des Wortes ›Zeit‹, die uns verwirrt [. . .] ein Ausdruck geistiger Unbehaglichkeit (,) und nicht notwendig eine Frage nach Ursache und Grund. [. . .] Das Kopfzerbrechen über die Grammatik des Wortes ›Zeit‹ entsteht aus dem, was man scheinbare Widersprüche in dieser Grammatik nennen könnte. [. . .] ›Zeit ist die Bewegung der Himmelskörper‹ [. . .] Jedoch sind wir nun, da wir feststellen, daß die erste Definition falsch ist, versucht zu denken, daß wir sie durch eine andere, die korrekte, ersetzen müssen. [. . .] Philosophie, so wie wir das Wort gebrauchen, ist ein Kampf gegen die Faszination, die die Ausdrucksformen auf uns ausüben.« »Im Augenblick, als ich es sagte, war ich davon überzeugt.« (Seite 223) »Die Rechnung als Ornament zu betrachten, das ist auch Formalismus, aber einer guten Art.« (Seite 427) 427. »Der Sessel ist der gleiche, ob ich ihn betrachte oder nicht. . . .« (Seite 373) (Ludwig Wittgenstein, The Blue Book/The Brown Book, Zettel, Schriften, Band 5 | Suhrkamp) –

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Ob der besessene Sessel der gleiche ist, ob ich ihn betrachte oder nicht, wäre zu erörtern. Sicher ist aber, dass seine Präsenz den Raum ähnlich aufwerten würde, wie die Zitate von Ludwig Wittgenstein diesen Text. – »Daraus scheint irgendwie hervorzugehen – was mir sehr einleuchtet – daß die Unendlichkeit der Länge keine Größe der Länge ist.« (Ludwig Wittgenstein, Philosophische Bemerkungen, Schriften, Band 2 | Suhrkamp) Bitte lassen Sie sich von mir mit der Dekoration aus einem Traum zurückholen, lassen Sie mich das Licht anmachen, und es wird dann gleich weitergehen. Achtung, ich mache jetzt das Licht an, indem ich Sie einer mehr oder minder sinngemäßen Zusammenfassung eines Gedankens von Karol Irzykowski aussetze. Keine Angst, es tut nicht weh, es erfrischt nur. Rehabilitation von Worten – begonnen mit ihrer Demaskierung An die Ästhetiker von heute: Das Wort ist ein Strahl, der aus dem Fiktiven die fernsten Welten der Unsichtbarkeit erreicht. Mit dem Spiegel der Sichtbarkeit, der realen, der imaginären, der bekannten und der zukünftigen, ist das Wort die metaphysische Erweiterung seiner eigenen Wurzeln. Verschmolzen sind ihm Objekt und Subjekt ein und dasselbe. Das sind zwei Gesichter des Lichts. 180◦ Jetzt betritt das Unbestimmende unsere Bühne Willkommen an der Schwelle in den dritten Quadranten! Willkommen an diesem Kipppunkt! Superposition – der Kreis scheint damit zu beginnen, sich zu schließen. In seinem kurzen Vortrag »The modern development of the foundations of mathematics in the light of philosophy« spricht Kurt Gödel Punkte an, die uns für das Verstehen wesentlich sind. Einleitend scheidet er zwei Gruppen von Weltanschuungen, die er – mit dem Skeptizismus, dem Materialismus und dem Positivismus – links und – mit dem Spiritualismus, dem Idealismus und der Theologie – rechts positioniert. Die linke Seite koordiniert er im Wesentlichen dem Pessimismus, die rechte dem Optimismus. Im Anschluss legt er uns dar, wie diese Positionen in ihren faktischen Mischformen in die mathematische und die philosophische Perspektive unseres Weltbilds wirken, wobei er »seit der Renaissance« ganz klar eine Tendenz zur linken Seite hin sieht. So weit, so gut. Der Abschluss seines Vortrags lässt uns hellhörig zurück. Gödel fragt mit dem letzten Satz, wieviel wir wohl von einem richtig verstandenen Kant erwarten können. Implizit lassen der Vortrag und sein Ausklang den Schluss zu, dass sich Gödel selbst sowohl linear als auch zyklisch zu den einleitend beschriebenen Positionen verhält. So, wie Kants »Kritik der Urteilskraft« als Kritik der »Kritik der praktischen Vernunft« (ihrerseits Kritik der »Kritik der reinen Vernunft«) – richtig verstanden, mit Gilles Deleuze gesprochen – die Gründung der Romantik ist, so ist Kurt Gödels Beweis der Nichtexistenz der Zeit die streng

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formalisierte Gründung der Zeit im Modus der Superposition; unsere Schau, das Geschaute und ihr Wesen sind nicht weiter das Resultat von Kollapsen, sie dürfen – endlich – unendlich, ewig als eines schillern. So wird übrigens auch Paul Feyerabend eingelöst. Der Methodenzwang hat ausgedient, die Träger des Fundaments sind verschwunden, Wahrheiten werden plural, ohne sich voneinander zu lösen. 190◦ Zeit für einen Anwalt Ich brauche Unterstützung, ich brauche eine Vertretung. Schnell, wer kann helfen? Wittgenstein vielleicht? Nein, der hat schon geholfen, der hat uns den zweiten Quadranten gemütlich gemacht. Chwistek, Leon Chwistek ist gerade frei und bereit, uns zu unterstützen. Er reißt sich in unseren Zusammenhang, und bevor es mir gelingt, Sie mit ihm bekannt zu machen, fällt er uns, den Abschluss seines Vortrags »Überwindung des Begriffsrealismus«, den er am 2. März 1936 im »Wiener Kreis« gehalten hat, zitierend – im Duktus eines Plädoyers – ins Wort. »Die Theorie der vier W irklichkeiten kann dazu b e n ü tz t werden, die begriffsrealistischen V orurteile nicht n u r aus der E rkenntnislehre, sondern auch aus den N aturw issenschaften und aus den sog. G eistesw issenschaften wegzuschaffen. [. . .] A nderseits erm öglicht uns die Theorie der vier W irklichkeiten, eine gewisse O rientierung in die G eschichte der Philosophie einzuführen, in dem sie uns den Irrationalism us als eine A rt V erm ischung verschiedener W irklichkeiten, anderseits aber die W idersprüche verschiedener rationaler System e als etw as N atürliches und Unverm eidliches erscheinen lässt. Analoges k ann in der K unstw issenschaft und insbesondere in [. . .] In der E th ik und in den Sozialw issenschaften w ird jede Tendenz, einen prinzipiellen U nterschied zwischen den Menschen einzuführen, in radikaler W eise überw unden, indem alles, was den gem einen M enschenverstand überschreitet, als relativ b etrachtet werden muss. Die individuelle Schöpfung muss au f den Bereich des inneren Erlebens beschränkt werden. Sie ist a u f die E ntdeckung neuer Tatsachen der W irklichkeit der V orstellungen zurückzuführen. [. . .] Alles, w as in einem gew issen T ypus der W irklichkeit als w illkürlich vorkom m t, k an n in einem höheren T ypus a u f gewisse Schem en der W irk u n g zu rü ck g efü h rt w erden. Doch w ird diese Schem atisierug in einem noch höheren Typus der W irklichkeit w ieder überw unden, in dem w ir zu dem un m ittelbaren E rleben der N orm en übergehen.

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Allatum est die 19. Octobris 1936« Allatum est die 11. Septembris 2021. Moment, ich muss raus aus seinem Fahrwasser, um über ihn sprechen zu können, um Sie über ihn informieren zu können – aber will ich das? Ich reiße es einmal an, und wir schauen gemeinsam, wie sich die Vorstellung entwickelt. Leon Chwistek formuliert Realität in vier Grundtypen – allgemeines Verständnis von Realität, physikalische Realität, phänomenologische Realität, visionäre (intuitive) Realität –, und mit dieser Typenlehre wendet er sich gegen sämtliche Verengungen/Vereinnahmungen von Wirklichkeit. Jeglicher Anspruch auf Deutungshoheit ist somit niemals überzeitlich und ständig verpflichtet, sich auch bei allgemeiner Betrachtung zu bewähren – aber ich wiederhole mich, wiederhole ich mich schon? Weshalb sein Zitat so merkwürdig gesetzt ist, was die verunglückt wirkende Spationierung bedeuten mag, erschließt sich mir nicht. Abschließen möchte ich Chwisteks Aufenthalt in diesem Text mit einer Anekdote. – Dieser Satz hat trei Feler. – Es wird kolportiert, dass Kurt Gödel und Leon Chwistek den Satz gemeinsam gelesen und im Anschluss – den ganzen Tag über – das Gesicht zu einem Grinsen verzogen haben, wie Tertianer, die bemüht und doch hilflos lautes Lachen unterdrücken wollen. Wenn Sie das Foto von Niels Bohr und Wolfgang Pauli kennen, das die beiden heiter über einem kleinen Spielzeugkreisel kniend zeigt, wissen Sie, welche Art Situation ich meine. Schade, dass die Quelle unklar ist. Die Passage durch Q3 ist gesichert, der Anriss abgerissen . . . 270◦ Ohne Borges kriegen wir den Sack nicht zu. Der Literat Jorge Luis Borges schildert unser Unterfangen anders aus. Seine Gedanken tränken uns an dieser Station, am Wegesrand. Wir begegnen anderen Symbolen, anderen Operatoren, zwei seiner Kurzgeschichten, »Das Aleph« und »Der Zahir«. Das Aleph ist ein kleines Objekt (von Form und Größe her einer kleinen Münze ähnlich), das alles, Alles in einem Punkt vereint und vereinigt. Der Zahir ist eine Münze, eine 20-Centavo-Münze aus dem Jahre 1929, die alles, Alles auf einen Punkt, auf sich, verengt. Kann man aus einem Zahir ein Aleph schöpfen? In grundsätzlicher Hinsicht ist der Zahir als das Gegenteil des Alephs zu verstehen – und doch (oder gerade deshalb?) lassen sich beide Ideen mathematisch – (a) mittels Riemannscher Zahlenkugel und Möbius-Transformation oder (b) mittels generischem Punkt, algebraischer Geometrie – ineinander überführen! Einerseits bestimmt, andererseits unbestimmt . . . Wie das Narrativ und das Deskriptiv, die Diesheit und die Washeit – wir haben es mit zwei Qualitäten der Unbegrenztheit zu tun. Wenn wir nun in der Literatur sind, können wir doch noch einen zweiten Poeten zur Sprache finden lassen, wieder Karol Irzykowski. Er wird uns – unkommentiert – über Zitate aus seinem Werk »Pa?uba | Sny Marii Dunin« (Pa?uba | Die Träume der Maria Dunin) anregen. »Der Begriff ›Hermina‹ lachte unverschämt und sagte: ›Lieber Doktor, viele unserer angesehensten Ingenieure sind bereits zu diesen und ähnlichen Gedanken gekommen. Es gibt eine Menge Literatur zu diesem Thema und noch mehr Durcheinan-

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der. In letzter Zeit wurden beispielsweise zwei Lager geteilt: einige behaupten, das Negative sei die Glocke der edlen Ideale, andere behaupten, es sei der Kelch des Gebrauchs. Aber mehr darf ich dazu nicht sagen‹ [. . .] Vor ihm lag eine große Karte mit vielen Kreisen, die durch entsprechende Linien verbunden waren. Jeder Kreis repräsentierte eine menschliche Seele. Und Herr Acheronta schrie herein: Diritto pelek, fa geniosa ilia, Usol mi cortu, pajto beni filia.∗ ∗ Es bedeutet: ›Losgelöst von der Weisheit, ungeboren fast ... (hier kommt eine verächtliche Phrase, die nicht ins Polnische übersetzt werden kann) ... eine Tochter des stillen Bösen.‹« Zum Abschluss legen wir dem Sack noch ein Zitat von Lewis Carroll bei. Das kann man ja immer gebrauchen. »He had bought a large map representing the sea, Without the least vestige of land: And the crew were much pleased when they found it to be A map they could all understand.« (Lewis Carroll, The Hunting of the Snark) 359◦ Es ist wahr, dass Zeit existiert, und es ist wahr, dass Zeit nicht existiert. Wieder zwei Aussagen – wieder die eine der anderen gegenüber, wieder die eine gegen die andere, wieder die eine um die andere drehend – wieder die eine in der anderen – wieder ein namenloses Gemeinsam. Und jetzt? Das Unbestimmende ist die (ganze) Welt und hebt uns das Bestimmte wieder und wieder ins Unbestimmte. Bevor der Zyklus in die Wiederholung geht, muss ich noch drei Anagramme präsentieren. Ich habe versprochen, ihnen Unterschlupf zu gewähren, und hier in der 359 ist noch Platz. Ich konnte mich nicht weigern, konnte aber aushandeln, dass ich den Sachverhalt so wie geschehen darlege und auch auf jegliche weitere Kommentierung verzichten werde. Kurt Friedrich Goedel | Chirurg kodierte Feld | Gedicht erfror Euklid | Euklid fordre Gericht! 360◦ An der Hand der Nymphe in Ruhe die Stufen hinauf, steigen wir wieder und wieder hinab. “Magic,” gasped the dull of mind, When the harnessed earth and skies Drew the nomads of their kind To uncharted emperies – Whispers round the globe were sped,

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Construed was the planets’ song. But the little boy playing in the orchard said, Conning his tale in the orchard said, “I knew it all along.” Ausblick/e Kurt Gödel blieb und bleibt sich – wie immer – treu. Albert Einstein zu dessen siebzigsten Geburtstag adressierend, schenkt er auch uns – ganz unauffällig, regelrecht am Rande – eine kleine Ungeheuerlichkeit. In seinem Werk »Prozeß und Realität« schreibt Alfred North Whitehead »Die sicherste allgemeine Charakterisierung der philosophischen Tradition Europas lautet, daß sie aus einer Reihe von Fußnoten zu Platon besteht« (Übertragung von Hans Günter Holl). Gödel hat Fußnoten gesetzt, die als Kopfnoten gesehen werden müssen, als Ergänzungen, die ganz elementar in die Wurzeln wirken – so bereichert er Platons »Phaidon«, »Parmenides« und »Timaios« lebendig, bricht eine Lanze für die Vernunft aus den Bedeutungsräumen nahe der Mathematik. Was nicht unterscheidbar gezählt werden kann, entzieht sich dem Verstand – das Maß ist verloren. Die Vernunft muss nicht zählen; so kann sie das Unendliche als Einheit erfassen, Unterscheidbarkeit hat keine Notwendigkeit. Ein Kreis mit unendlichem Umfang unterscheidet sich nicht von einer Geraden. Der Verstand wehrt sich – nur die Vernunft findet über das Paradox hinaus, lässt es als scheinbar zurück, Cusanus’ coincidentia oppositorum. Bestand formt sich neu. Geltungsgefüge erfahren eine fundamentale Modifikation, eine Erweiterung aus der Vernunft, durch die Vernunft. Logizismus und Psychologismus einen sich widerspruchsfrei unter ein Ganzes, heben die Welt in vereinende Angeln. Wahre Bedeutung findet sich im Zyklus des Linearen, in der Linearität der Zyklen. Ohne Unterlass zwischen Höhle und Sonne kreisend weichen die Deutungshoheiten der Vernunft, und die Vernunft ruft die Gelassenheit, die zu einem Miteinander in freier Annäherung finden/führen möge. Ein Stärken von Tugenden∗ aus er- und gelebter Solidarität, aus einem Motiv des Zusammenrückens, der Vergeschwisterung in und durch Freiheit, führt uns aus dem Würgegriff der Aufklärung und der überheblichen Mythologien. Ein System implodiert – die Schönheit der Schöpfung, die Mächtigkeit der Lebenswelt, der Lebensgeschichte, und die Beseelung des Kosmos scheinen auf. Ockham packt das Rasiermesser weg und arbeitet von nun an mit dem Löffel – aus Freude am Leben. »Die Formen des Wissens und die Bildung«, eine Begegnung in neuer Harmonie, reich an Obertönen, vernünftig – endlich. Alle Menschen sind gleich, weil alle Menschen verschieden sind. Max Scheler freut sich. ∗ Achtsamkeit, Aufrichtigkeit, Behutsamkeit, Bescheidenheit, Dankbarkeit, Demut, Freiheitssinn, Gerechtigkeit, Herzensgüte, Humor, Mitgefühl, Selbständigkeit, Wohlwollen, Ehrlichkeit, Redlichkeit, Toleranz – um ein paar zu erwähnen Epilog Bisher fanden weder Kurt Gödels Mutter noch seine Frau in diesem Text Erwähnung. Aber ich möchte mit meinem Text doch auch auf Marianne Gödel (geborene Hand-

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schuh) und Adele Gödel (geborene Porkert) eingehen. Die beiden Damen waren für Kurt Gödel sehr bedeutsam, sehr wichtig, bedeutsam wichtig, lebenswichtig. Auch sie dürften maßgeblich auf seine Positionen eingewirkt haben, seinem Genius – vielleicht gar kongenial? – Orientierung vermittelt haben. »Doch gnügten nicht dazu die eignen Schwingen, Bis daß mein Geist von einem Blitz durchzuckt ward, In welchem sein Verlangen sich ihm nahte. Der hehren Phantasie gebrach’s an Kraft hier, Doch schon schwang um mein Wünschen und mein Wollen, Wie sich gleichförmig dreht ein Rad, die Liebe, Die da die Sonne rollt und andern Sterne.« (Dante – Paradiso – Ende des letzten Gesangs | Philalethes-Übertragung) Sinn erschließt sich, Bedeutung entschließt sich. Wir haben uns verirrt und haben so hergefunden, sind gemeinsam angekommen. Das Leben ist schön, und »Die Welt ist vernünftig«. Guido Stemme is a generalist who fundamentally seeks to approach and present aspects of life and literature from a holistic hermeneutic perspective. The focus of his work lies in the overlapping fields of philosophy/pedagogy, art and digital technology. He regards teaching as a basically trans-sectoral process, with the idea of education as a constant exercise in developing and questioning one’s position at its core. The ”human human”, the human before any kind of classification, is the central starting point to which it is always necessary to return. His art projects enrich his work as a consultant for digital transformation in the area of culture—and vice versa.