From Brouwer to Hilbert 0195096312

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From Brouwer To Hilbert The Debate on the Foundations of Mathematics in the 1920s

PAOLO MANCOSU

New York

Oxford

OXFORD UNIVERSITY PRESS 1998

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Copyright © 1998 by Oxford University Press , Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 1001

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Library of Congress Cataloging-in-Publicatio n Data

From Brouwer to Hilbert : the debate on the foundations of mathematics in the 1920s / Paclo Mancosu. D.

CrL.

Includes bibliographical references and index. ISBN 0-19-509631-2 {cloth). — ISBN

0-19-509632-0 (paper)

i, Mathematjcs—Europe—Phjlcnsophy —Hi I. Mancosu, Paoio. QAB.6.F757

511".2—dc20

stary—ED'th century,

1998

96-32913

CIp

ISBN 0-19-509631-2 ¢l 0-19-509632-0 pb

Printing (last digit):

| 357 98 6 4 2 Printed in the United States of Americ a on acid-free paper.

CONTENTS

Preface

vii

Part L. L. E. J. Brouwer Brouwer’s Intuitionist Programme

1

Walter P. van Stigt

1. Brouwer, Intuitionist Set Theory

23

2. Brouwer, Does Every Real Number Have a Decimal Expansion?

28

3. Brouwer, Proof that Every Full Function Is Uniformly

Continuous

36

4. Brouwer, Intuitionist Reflections on Formalism

40

L . Brouwer, Mathematics, Science, and Language

45

6. Brouwer, The Structure of the Continuum

54

Part I1. H. Weyl Hermann Weyl: Predicativity and an Intuitionistic Excursion

65

Paolo Mancosu

7. Weyl, On the New Foundational Crisis of Mathematics

8. Brouwer, Comments on Weyl 1921

86

119

9. Weyl, The Current Epistemological Situation in Mathematics

123

10. Holder, The Alleged Circulus Vitiosus and the So-Called

Foundational Crisis in Analysis

143

Part IIL P. Bernays and D. Hilbert

Hilbert and Bernays on Metamathematics

149

Paolo Mancosu

11. Bernays, Hilbert’s Significance for the Philosophy of

Mathematics

189

12. Hilbert, The New Grounding of Mathematics: First Report

198

13. Bernays, On Hilbert’s Thoughts Conceming the Grounding of

Arithmetic

215

14. Bernays, Reply to the Note by Mr. Aloys Miiller, “On Numbers as Signs”

223

15. Hilbert, Problems of the Grounding of Mathematics

227

Contents

Vi

16. Bernays, The Philosophy of Mathematics and Hilbert's

Proof Theory

7.

234

Hilbert, The Grounding of Elementary Number Theory

266

Part IV. Intuitionistic Logic Intuitionistic Logic

275

Paolo Mancosu and Walter P. van Stigt 18. Brouwer, Intuitionist Splitting of the Fundamental Notions of

Mathematics (Dutch Version)

286

19. Brouwer, Intuitionist Splitting of the Fundamental Notions of

Mathematics (German Version)

290

20. Brouwer, Addendum to “Intuitionist Splitting of the Fundamental

Notions of Mathematics”

293

21. Borel, Concerning the Recent Discussion between Mr. R. Wavre and Mr. P. Levy

296

22. Glivenko, On Some Points of the Logic of Mr. Brouwer 23. Heyting, On Intuitionistic Logic

306

24, Heyting, The Formal Rules of Intuitionistic Logic

311

25. Kolmogorov, On the Interpretation of Intuitionistic Logic Index

335

301

328

PREFACE

The 1dea for the present book arose out of a pedagogical need. After the end of my graduate studies I found myself teaching a course in philosophy of mathematics at

Oxford University. When covering the central foundational debate between intuitionists and formalists, I realized that only a very few contributions of that debate

had been translated into English. I thus convinced myself that a volume of translations into English of these sources would be of help to students and scholars alike. With this aim in mind 1 contacted Professor Walter P. van Stigt, Dr. Benito Miiller,

Professor Amy Rocha, and later Professor William Ewald. It is thanks to their enthusiastic response that the idea of the project has become a reality.

We present here translations into English of 25 major articles in the debate on

the foundations of mathematics that took place in the 1920s. The articles are translated from Dutch, French, and German and have not been previously translated. In

particular the anthology consists of four sections devoted, respectively, to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each of the four sections is prefaced by an introduction that gives the nec-

essary histortcal and technical context of the essays and is by no means confined to

the essays In the present volume. For a detailed list of the articles translated, see the table of contents. The choice for the essays to be translated was dictated in the first place by whether they had been previously translated into English or not. At first I thought

that this might create an imbalance in the sections, but 1 soon convinced myself that

the abundance of material to be translated made the project into a unity rather than a scattered collection of articles.

The articles have been selected with a strict chronological criterion in mind.

They were all written before the groundbreaking discoveries by Godel (the only ex-

ception being Kolmogorov 1932). Whereas this is not of great consequence for the selections and introductions in Parts I, II, and IV, the situation is rather different for the discussion of Hilbert’s programme as presented it my introduction to Part IH.

It should be pointed out that my introduction to Hilbert’s programme only attempts

a characterization of the programme up to Hilbert 1931, and does not take into con-

sideration the consequences of Godel’s theorems for it. Furthermore, I did not even attempt to provide an exposition of the programme as it emerges in the Grundla-

gen der Mathematik (1934, 1939). And this is why [ did not get involved in some

of the contemporary discussions, for example, whether finitism can be identified with Primitive Recursive Arithmetic (see Tait 1981) or whether Godel’s theorems show the nonviability of Hilbert’s programme. Although this might be seen as a limitation, there is an advantage in restricting the focus of attention. It is often the Vil

viil

Preface

case that conflicting accounts of Hilbert’s programme rely on a very narrow selec-

tion of passages for their understanding of the programme. The consequence is often that conflicting accounts stand side by side with much confusion on the part of the reader. I hope that many of these confusions, but certainly not all of them, can be dispelled by an acquaintance with the primary literature, which shows, as I argue 1n my introduction to Part III, that Hilbert and Bernays developed their con-

ception of the programme in response to different pressures and sometimes changed their minds on some of the key issues.

Several choices had to be made at the editorial level, and I am aware that each

decision brings advantages and disadvantages. For instance, we present no elaborate commentary for each article. By contrast, our introductions are rather detailed and present the contributions in the general context of the foundational research at

the time. Part of the goal of the introductions should be that of pointing the reader to the extensive literature in the area. However, much had to be excluded. I thought it would be a good idea to focus

on Intuitionism and Hilbert’s programme rather than to attempt to include the numerous contributions by the logicists and other workers in the foundations of math-

ematics such as Fraenkel, Skolem, and others. In the end I think the volume has gained from such exclusions in that it has preserved a certain unity that might have been iost had its boundaries been extended.

Many people have helped bring this project to its final form. Professor Walter P. van Stigt (Wolfson College, Oxford) has provided all the Brouwer translations, the introduction to Part 1, and 81 of the introduction to Part 4. His expertise on Brouwer has been invaluable. Dr. Benito Miiller (Wolfson College, Oxford) has translated Weyl’s articles from the German. Dr. Amy Rocha (San Jose State University) has translated Borel, Glivenko, and Heyting from the French. Professor

William Ewald (University of Pennsylvania) has kindly agreed to our use of his translations of Hilbert 1922 and 1931, which have just appeared in his volume From

Kant to Hilbert, Oxford University Press, 1996. I want to express my gratitude to all of them for their excellent work and active participation in this project. I'would also like to thank several people who have helped with the stylistic re-

visions of the texts: Michael Halberstam, Ingo Kraus, and Jale Okay. I am grateful to Jale for her extensive help in the translation of Bernays 1930a. My research as-

sistant Richard Zach has been of the greatest help in a variety of tasks ranging from bibliographical research to detailed improvements on the translations. Finally, I am grateful for their support and suggestions to Aldo Antonells,

George Boolos, Charles Chihara, Solomon Feferman, Michael Hallett, Daniel Isaac-

son, Jerry Katz, Eberhard Knobloch, Arnie Koslow, Volker Peckhans, Michael

Resnik, Wilfried Sieg, Hourya Sinaceur, and Christian Thiel. It is a pleasure to acknowledge the financial support of the Humboldt-Stiftung. It would have taken me much longer to complete this project were it not for the

work I was able to carry out during my tenure in Berlin in 1993-1994 as HumboldtStipendiat. Moreover, I would like to mention the financial support of the Griswald Facuity Research Fund (Yale University) and the Committee on Research (Unive r-

sity of California, Berkeley), which helped cover the costs for stylistic revisi ons of

some of the translations and other clerical tasks. Finally, a very gener ous grant from the Hellman Family Faculty Fund has provided financial support both for further

Preface

ix

research in the Gottingen and Zurich archives as well as for the final revisions and

proof-checking of the book.

The project could only be brought to completion during a leave, generously granted by the Department of Philosophy of the University of California at Berke-

ley, which released me from my teaching duties during the spring semester 1996. During my leave the Department of Philosophy at CUNY Graduate Center provided

an exciting and friendly atmosphere.

Let me mention the conventions followed before putting an end to the preface. Each text is translated following very closely the notation of the original texts. Only one significant exception should be noted. Instead of using Fraktur characters, we

use bold letters. A * sign after the author of each article refers to a footnote in which is speci-

fied the original source, the translator, and the language from which the article is translated.

Page references in the introductions are to the English translation, if available; to the original if no English translation is available. Thus Weyl (1921, p. 88) refers to p. 88 of the English translation of Weyl 1921. Reference to the English transla-

tion of a paper, if available, are given in the bibliography together with the biblio-

graphical specifications of the original paper. However, in the introduction to Brouwer the references to page numbers of aritcles not included in this book are always to the original. The reader will be able to locate easily the passages in Brouwer’s Collected Works since the original pagination is always given there, as well as in all pieces by Brouwer (such as Brouwer 1933) translated in van Stigt 1990.

We use square brackets to make an occasional remark in the translation or, most

of the time, to give the original text when it was felt that having the original might

help the reader decide on the meaning of a problematic passage. In the case of Brouwer some articles have a Dutch and a German version, and Professor van Stigt has taken both into account (the translation being usually from the Dutch—Dutch being Brouwer’s mother tongue). In such cases if an original text appears in square brackets without specification of the language, this is to be understood as the language from which the paper is translated. If attention needs to be drawn to the ter-

minology of both the Dutch and the German version, this is specified inside the brackets, for example, [{Dutch: Van gelijke uitgestrektheid, German: Ausdehnung].

Notes by the author of the article follow the usual numeration and will be found

as endnotes; translator’s notes follow the author’s notes and are denoted by lowercase Latin letters a, b, ¢, etc. (the exception being item 11, where the notes in lowercase Latin letters are by Bernays).

As an editor I have not tried to force a uniform terminology in the translations

or the introductions. In some cases, as Professor van Stigt’s preference for the adjective “intuitionist” versus mine for “intuitionistic,” this matters little. However, there were cases where the choice resulted in a more noticeable lexical difference.

For instance “Widerspruchslosigkeit” is translated as “consistency” in most of the Bernays—Hilbert selections, but “noncontradictority” is often used in the Brouwer translations. My feeling is that these divergences do not pose a major problem, given

that each translator was responsible for a rather uniform set of texts, With the exception of Chapters 12 and 17, which are here reprinted from Ewald 1996, the rest

X

Preface

of the book follows Oxford’s house standards concerning matters of punctuation and usage.

Each introduction is followed by its own bibliography. I felt this would be more usefu] than having a long bibliography at the end of the book. The references after each section are divided into primary and secondary literature. We use a standard system of reference with the name of the author followed by the year (e.g., Weyl 1921). When more articles were published by the same author in the same year, we

use lower-case Latin letters to distinguish between the publications (e.g., Bernays

19222 and 1922b). Brouwer’s articles are referred to using the convention and biblography given by Heyting in Brouwer's Collected Works, Vol. 1 (e.g., B1919DI1)

and in van Stigt 1990. We use the following abbreviations: KNAW = Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam; and JDMV = Jahresbericht der Deutschen Mathematiker-Vereinigung. New York

P.M.

May 1996 (Added in January 1997) While I was working on the copy-edited version of the manuscript, Ewald 1996 appeared. I have thus changed some of the references and quotes to take this into account. In addition I would like to express my thanks to Trudy Brown and Robert Miller of Oxford University Press for their excellent work and support throughout the production of the book.

From Brouwer To Hilbert

PART |: L. E. ]. BROUWER Brouwer’s Intuitionist Programme WALTER P. VAN STIGT

Luitzen Egbertus Jan Brouwer (1881-1966) is a central figure in the history of con-

temporary mathematics and philosophy.! His main contributions are in the field of topology and the foundations of mathematics. It is Brouwer’s contribution to the foundations of mathematics, the intuitionist programme, that has made him known

to the wider scientific and philosophical community. His influence is very much alive today, as is witnessed by the ongoing mathematical research in intuitionist and constructive mathematics (see Bridges, Richman 1987, and Troelstra, van Dalen,

1988) and by the variety of philosophical contributions that have their roots in Brouwer’s intuitionism (see, for instance, Detlefsen 1990, Dummett 1973, 1977,

McCarthy 1983). However, although many of these contributions take their start from Brouwer’s Intuitionist approach, it is also true that they have departed to a great extent from Brouwer’s original formulation of the programme. In van Stigt

1990 I have endeavored to present Brouwer and his intuitionist programme in their historical setting. The following introduction is conceived in the same spirit. I shall begin with a section on the intuitionisi—formalist controversy. Section 1.2 is about

Brouwer’s intuitionist philosophy of mathematics. Sections 1.3 and 1.4 present

Brouwer’s views on the nature of mathematics and on the relationship between mathematics, language, and logic. Section 1.5 gives an account of Brouwer’s new set theory and his conception of the continuum. Finally, Section 1.6 gives a short in-

troduction to the selected contributions.

1.1

The Intuitionist—Formalist Controversy

In 1920 Hermann Weyl diagnosed “a new crisis in the foundations of mathematics” (Weyl 1921), sparked off by the publication of Brouwer’s “Foundations of Set

Theory Independent of the Principle of Excluded Middle” (B1918B and B1919A). In a series of lectures at the Mathematical Colloquium of Ziirich, he dramatically

renounced his own Das Kontinuum and hailed Brouwer’s set theory and interpretation of the continuum as “the revolutionTM: “ ... und Brouwer—das ist die Revolution!” (Weyl 1921, p. 99), the one mathematician who at last had solved the problem of the continuum, which since ancient times had defeated even the greatest 1

2

Walter P. van Stigt

minds. At the same time, in “Intuitionist Set Theory” (B1919D) Brouwer set out

the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: “the use of the

Princtple of the Excluded Middle is not permissible as part of a mathematical proof ... [1t] has only scholastic and heuristic value, so that theorems which in their proof cannot avold the use of this principle lack all mathematical content.” (p. 23)

Both Brouwer’s challenge and Weyl’s support raised the alarm among the Can-

tortan and Formalist establishment of Géttingen. Hilbert, who had recognized Brouwer’s major contribution to topology and had welcomed him as a member of

his tnner circle, grew increasingly impatient with his old friend and alarmed by the implied threats to Cantorian set theory and his own programme. He launched a counterattack in 1922: What Weyl and Brouwer do amounts in principle to following the erstwhile path

of Kronecker: they seek to ground mathematics by throwing overboard all phenomena that make them uneasy . . . if we follow such reformers, we run the dan-

ger of losing a large number of our most valuable treasures. (Hilbert 1922, p. 200)

The ensuing Intuitionist—Formalist “debate” dominated the foundational scene throughout the 1920s. Brouwer and Hilbert remained the main protagonists, each drawing support for his cause beyond national frontiers and an even greater audi-

ence of interested observers and commentators. The debate centered on two different, though related, issues:

1. The nature of mathematics: either human thought-construction or theory of

formal structures;

.

2. The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic.

Brouwer’s main concern was the nature of mathematics as pure, “languageless” thought-construction. He had set himself the task of bringing the mathematical world around to his view, convincing them of the need for reform, and had started the programme of reconstructing mathematics on an Intuitionist basis. Most of his publications in the period 1918-1928 were part of this programme;: only a few dealt di-

rectly with the “negative” aspects of his Intuitionist campaign: the misuse of logic, in particular the Principle of the Excluded Middle, and the flaws in the Formal ist

programme. Understandably these papers aroused greater interest and further controversy. His excursion into the field of logic (“Intuitionist Splitt ing of the Fundamental Notions of Mathematics,” B1923C), in which he drew the immediate con-

clusions from his strict interpretation of negation and his rejection of the Princi ple of the Excluded Middle, created considerable excitement among logici ans and started a debate about an alternative, “Brouwer Logic.” This debate was joined by

Kolmogorov, Borel, Wavre, Glivenko, Heyting, and others (see Part IV). Brouwer himself did not take a further active part, remaining true to his convi ction that logic and formalization were “an unproductive, sterile exercise” with no direct relevance to mathematics and its foundations.

The main Intuitionist-Formalist “debate” was a contest betw een the leaders of two opposing philosophies of mathematics, each with its own progr amume and com-

peting for the suppert of the mathematical world. Apart from the occasional direct

Brouwer’s Intuitionist Programme

3

exchange, each camp concentrated on its own programme. Hilbert’s Programme, retaining the “whole treasure of classical mathematics” and basing its validity on a proof of the consistency of its formalization, attracted widening support and an able

team of collaborators. Brouwer’s constructive interpretation of mathematics, much

in line with the natural outiook of the working mathematician, was enthusiastically received and raised early hopes. However, his austere programme of reconstruction

within the Intuitionist constraints failed to gather momentum. His increasing isolation was partly due to his inability to work with others, but more important, Brouwer’s and Weyl’s hopes that the “natural” Intuitionist approach would lead to

a simplification of reformed mathematics did not materialize. Indeed, it proved “unbearably awkwardTM in comparison with traditional mathematics relying on the meth-

ods of classical logic. Even Weyl had to accept this with regrets: Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the

contact with Inteition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an alimost un-

bearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice, which he believed to be built of concrete blocks, dissolve into mist before his eyes. (Weyl 1949, p. 54}

The Brouwer—Hilbert debate grew increasingly bitter and turned into a personal feud. The last episode was the “Annalenstreit,” or, to use Einstein’s words, “the frog-and-mouse battle.” It followed the unjustified and illegal dismissal of Brouwer from the editorial board of the Mathematische Annalen by Hilbert in 1928 and led

to the disbanding of the old Annalen company and the emergence of a new Annalen under Hilbert’s sole command but without the support of its former chief editors, Einstein and Carathéodory.

For Brouwer it was the last straw. His failure to “simplify’” Intuitionist meth-

ods and make Intuitionism the universally accepted mathematical practice had

eroded his self-confidence. The conspiracy of his fellow Arnalen editors and “lack of recognitionTM left him bitter and disillusioned. He abandoned his Intuitionist Programme and withdrew into silence just about the time when the Formalist Programme was shown to be fundamentally flawed. Some “books” were left uncompleted and unpublished. The 1928 “Vienna Lectures,” The Structure of the

Continuum (B1930A) and “Mathematics, Science and Language” (B1929), and his paper “Intuitionist Reflections on Formalism” (B1928A2) mark the end of Brouwer’s creative life and his Intuitionist campaign. They reflect the stage his programme had reached and the mood of its founder at the time. The Structure of the Continuum summarizes his Intuitionist vision and analysis of the continuum. In “Mathematics, Science and Language” he returns to the pessimism of his philosophy of science

and language, which had inspired his Intuitionist rebellion. “Intuitionist Reflections on Formalism” is Brouwer’s final assessment of the state of play in the contest between Intuitionism and Formalism and an emotional outburst at the lack of recognition. It lists outstanding differences as well as “the Intuitionist Insights” adopted by Formalists “without proper mention of authorship,” such as the notion of metamathematics.

4

Walter P, van Stigt

1.2

Intuitionism and Brouwer’s Intuitionist Philosophy of Mathematics

Brouwer's Intuitionist reform of mathematics and his revolutionary views on the use of logic can only be fully understood in the context of his particular philosophy of mathematics. Indeed, his Intuitionism is first and foremost a philosophy of mathematics from which these new ideas emerge guite naturally. Most of Brouwer’s philosophical views on life in general and on the nature of mathematics were formed during the years of undergraduate and doctoral studies, and they remained virtually the same throughout his life. They are expressed most

clearly in his early publications: his doctoral thesis On the Foundations of Mathematics (B1907) and Life, Art and Mysticism (B1903), in some of his post-1928 papers such as “Mathematics, Science and Language” (B1929), “Will, Knowledge and

Speech” (B1933), and “Consciousness, Philosophy and MathematicsTM (B1948C), and in unpublished papers. This section 15 a brief introduction to Philosophical Intuitionism and the main aspects of Brouwer’s philosophy as are relevant to his Intuitionist practice.

A more

detailed analysis 15 given in van Stigt 1990.

Intaitionism is a philosophical trend that places the emphasis on the individual con-

sciousness as the source and seat of all knowledge.® Besides the faculty and activity of reasoning, it recognizes in the individual mind a definite faculty and act of

direct apprehension, intuition, as the necessary foundation of all knowledge, both in the grasping of first principles on which a system of deductive reasoning is built

and as the critical link in every act of knowing between the knower and the object known. Intuttionism stands in contrast to a more general rationalistic and deter-

ministic trend that dentes the possibility of knowing things and facts in themselves and restricts human knowledge to what can be deduced mechanically by analytical reasoning, ultimately from self-evident facts and principles that result from common sense or are based on the authority of collective wisdom.

Elements of Intuitionism can already be found in classical philosophies, for example, in the Aristotelian voto, a special faculty of direct apprehension, an active

faculty that is indispensable in the creation of primary concepts and first principles

as well as at every step of the thought process. Elements of Intuitionism are also

found in the systems of some of the modern German and English philosophers such as Kant, Hamilton, Whewell, and even Russell (for the Kantian roots of Brouwer's

philosophy of mathematics, see Posy 1974). But it is in the revolutionary and libertarian chmate of Holland and France that Intuitionism took root and developed

into a full and coherent philosophy. Descartes, the father of modern philosophy, can rightly be claimed to be the father of modern Intuitionism. A Frenchman by birth, Descartes settled in Holland, “a country”—as he wrote o Balzac—"where complete liberty can be enjoyed.” His rebellion was the fundamental break with the traditional reliance on authority, reli-

gious and otherwise, as the ultimate source of truth and placing the origin and seat of knowledge firmly in the individual mind of man. He starts from “self-awareness” and distinguishes between various faculties in the process of acquiring knowledge,

Brouwer’s Intuitionist Programme

5

but insists that every form of knowing ultimately requires an act of immediate mental apprehension, “intuition.” He insists on the need for rational argument and sets out rigorous rules of correct reasoning, but points out that logical deductive rea-

soning does not produce any new truths, that true knowledge comes from intuition. Descartes’ intuitionist lead was followed in the nineteenth century by a number of French philosophers such as Maine de Biran, Ravaisson, Lachelier, and

Boutroux. It was developed into a full and comprehensive philosophy by Henri Berg-

son, who raised Intuition to the faculty of grasping the spiritual and changing reality, distinct from Reason, the analytical mind, which probes the material and static

reality. Bergson’s living reality, however, did not include the mathematical universe; his concepts of number and the mathematical continuum are spatial, products of the analytical intellect.

As to the precise nature of Intuition as the foundation of mathematics, Descartes remains somewhat vague: The fundamental mathematical truths are “indubitable”

because they are “clearly and distinctly perceived” by the mind’s eye. Yet in his argument for the existence of God, for which he claims “the same level of certainty

as the truths of mathematics,” he concludes that these truths, such as the essence and nature of the triangle, are “immutable and eternal and not invented by me nor dependent on my mind” (Descartes, 5th Meditation). Equally vague. as to the nature

of Intuition are the French “New Intuitionists” Poincaré, Borel, and Lebesgue.® It was not until the beginning of the twentieth century that an attempt was made at a precise interpretation of mathematical Intuition, when Brouwer took Descartes’ in-

tuitionist thesis to its radical subjective and constructive conclusion. Brouwer’s outiook on life and general philosophy can best be described as a blend

of romantic pessimism and radical individualism. In Life, Art and Mysticism (B1903) he rails against industrial pollution and man’s domination of nature through his intellect and against established social structures, and promotes a return to “Nature” and to mystic and solitary contemplation.

,

In his Foundations of Mathematics (B1907), especially its original version, it

is the application of mathematics in experimental science and logic that 1s exposed as the source of all evil and analyzed as “the causal” or “cunmng act,” superimposing a mathematical regularity on the physical world. Both works express his con-

viction of the opposition between mind and matter, the individual consciousness and the exterior world. Reflectifig on the nature of man, Brouwer identifies personal identity, the “SelfTM

or “the Subject,” with the pure-spiritual “Soul” in his later work referred to as “Consciousness in its deepest home” (“Consciousness, Philosophy and Mathematics,”

B1948C). The life of the Soul is the complex of thought processes in response to its awareness of the world outside. They are analyzed as distinct mental states, “phases of consciousness” in a process of evolution, each resulting from a definite “happening” and each producing its characteristic form of knowledge and human activity. It is a “deteriorative” process moving consciousness further and further away “in its exodus from its deepest home” on a sliding scale from “beautiful,” that is, good, to evil.

The original preperceptional stage of “stillness” is followed by “the naive phaseTM of receiving images through physical sensations and reacting spontaneously to them.

6

Walter P. van Stigt

The momentous event of the Subject linking isolated sensations, becoming aware

of time, referred to by Brouwer as “the Primordial Happening” or “the Pnmordial

Intuition of Time,” brings about a transformation of the Naive Consciousness to the rational “Mind” and at the same time generates the fundamental concepts and tools of mathematics. The Primordial Intuition of Time is the fundamental single act of isolating and linking distinct moments in time, creating mathematical “Two-ity” and the ordinal numbers as well as the continuum. It is first mentioned in Brouwer’s analysis of science, Chapter 2 of his Foundations, where it is used to show the priority of mathematics with respect to science and expose the ideal nature of science, no more than man’s mathematical interpretation of the world. The mathematical

power to generate sequences enables man to create in his individual thought-world an interpretation of “Nature,” the outside world, which 1s manmade and mathematical. “Things,” including other human beings, are no more than repeated sequences or sequences of sequences, manmade, as is indeed the so-called scientific or *“causal” coherence of the world. And because of the individual nature of human thought, this universe of “things” is wholly private. Brouwer refers to it as “the Exterior World of the Subject.” The scientific observation of regularity in Nature, linking things and events in time as sequences, 18 a creative, mathematical process of the

individual Mind and is referred to as “mathematical viewing” or “causal attention.” Causality is an artificial, mind-made structure, not inherent in Nature. Indeed, Brouwer rejects any universal objectivity of things as well as their “causal coherence,” basing his argument on the essential individuality of thought and mind. In

“Consciousness, Philosophy and MathematicsTM (B1948C) he emphatically denies the existence of a collective or “plural” mind. The Brouwerian evolutionary “exodus from its deepest home of consciousness”

enters a moral phase when man takes advantage of and acts upon his causal knowledge by setting in motion a causal sequence of events, selecting a first element of the sequence in order to achieve a later element, the desired “end.” Such mathematical or causal acting is “calculated” and “cunning,” condemned as “sinful” and “not-beautiful,” that is, morally evil.

Even more remote from the “deepest home of consciousness” is the next and final phase of *“social acting,” described as “the enforcement of will” in social interaction and organization, in particular by the creation of language. Brouwer’s philosophy of language starts from the conviction that direct communication between human beings is impossible. His chapter on “Language” in Life, Art and Mysticism

(B1905) starts as follows: “From Life in the Mind follows the impossibility of communicating directly with others . . . never has anyone been able to communicate directly with others soul-to-soul.” (p. 37). The privacy of mind and thought and the hypothetical existence of minds in other human beings, who are no more than the

Subject’s mind-creations, “things in the exterior world of the Subject” rule out “any exchange of thought” (B1948C, p. 1240). In line with his “genetic” principle of ontological analysis, Brouwer searches for the nature of language in the process that

brought it into being. He traces the origin of language to a particular form of cunning or mathematical acting, the “imposition of will through sounds,” forcing another human being to act m pursuance of the end desired by the speaker: “At the most primitive stages of civilization . . . the transmission of will to induce labour or

Brouwer’s Intuitionist Programme

7

servitude is brought about by simple gestures of all kinds especially and predominantly the emotive natural sounds of the human voice” (B1933, p. 51). As social

iteraction develops and grows more complex, language becomes more sophisti-

cated, but its essence, as of all instruments, is determined by its purpose: the trans-

mission of will. Used as a means of communicating thought to others, language is bound to remain defective, given the essential privacy of thought and the nature of the “sign,” the arbitrary association of a thought with a sound or visual object.

Within the private world of the Subject, language may have a function as “an aid to memory,” helping the Subject to recall his past thought. In “Will, Knowledge

and Speech” (B1933), when Brouwer had to accept human frailty, the limitations of the flesh-and-blood mathematician, even the stability of such private language, was called into question: “The human power of memory . .. is by its very nature

Iimited and fallible” (p. 58), even when it calls in the help of linguistic signs. It is

at this point that Brouwer introduces his notion of the “Idealized Mathematician.”

1.3

The Nature of Pure Mathematics

The mathematical nature of causality and the Primordial Intuition.of Time as the fundamental creative act of mathematics are the central theses of Brouwer’s analysis of science and language. In Chapter 2 of The Foundations of Mathematics they are treated as closely related, and the “mathematical” appears somewhat tainted by its association with “causal” or “cunning acting.” There are, however, signs that as

early as 1907 Brouwer had established an independent and redeeming role for pure mathematics. His Foundations ends with a summary that starts: “Mathematics is a

free creation, independent of experience; it develops from one single a priori Primordial Intuition . . .”(B1907, p. 179). In the original plan of the thesis, moreover, there is an additional chapter entitled “The Philosophical Significance of Mathematics,” in his Preparatory Notes referred to as “Mathematics and the Liberation

of Mind.” The “Liberation of Mind” is a favorite theme of Life, Art and Mysticism.

In the mathematical context it refers to the elimination of all exterior, phenomenal elements and causal influences from the creative mathematical act. It allows the Primordial Intuition as an abstraction of pure time awareness, eliminating also the con-

tent of sensations, to be a pure and a priori basis of mathematics and its defining act. The Primordial Intuition is not only necessary and sufficient for the creation of

two-ity; it dlSo holds the continuum as “its inseparable complement” and contains the fundamental elements and tools from which and with which the whole of math-

ematics is to be constructed. Indeed, mathematics is identified with the whole of the

constructive thought-process on and with the elements of the Primordial Intuition alone. Brouwer's preferred term is “building” (Duich: bouwen) rather.than “construction,” a building upwards from the ground, a time-bound process, beginning at some moment in the past, existing in the present, and having an open future ahead. Indeed, mathematics is the life of what Brouwer calls “the Subject,” “the Creating

Subject,” or the “Idealized Mathematician.” Its characteristics are determined by the time-bound and individual nature of mind as the sole creator and seat of mathematical thought and by the limits of Intuition.

8

Walter P. van Stigt

Mathematical Existence and Truth Mathematical existence then in its strictest sense is “having been constructedTM and

rematning alive in the mind or memory. The whole of the Subject’s constructive

thought-activity, past and present, constitutes mathematical reality and mathematical truth: “Truth is only in reality, i.e. in the present and past experiences of consciousness” (B1948C, p. 1243). Mathematical entities are identified with the whole

of their constructive pedigree, whether they be single concepts, such as, for example, an ordinal number, or more complex, such as a mathematical theorem, which combines various constructions. Past, completed constructions consist of sequences of constructive steps and as such are fimte. Mathematical existence can be claimed for “the infinite”TM within an

interpretation that is based on completed constructions and the freedom of the Subject to proceed. In the case of a denumerably infinite sequence such as “the fundamental sequence” of ordinal numbers, the completed construction is the algorithm or “lawTM by which each element of the sequence is uniquely determined. The “free” power of the live Subject 1o proceed ensures that the elements be generated “in-

definitely.” The essential active role of the Subject in constructing his procedure for determining elements and in the continued generation of these elements allows the possibility of extending the traditional notion of infinite sequence. Brouwer took

this step in 1917, when he introduced the “free-choice sequenceTM and his new set concept as the procedure for generating “points on the continuum” (see further Sec-

tion 1.5). The established concept of the continuum as a set, the totality of existing

points, was rejected outright in The Foundations (B1907) and “On Possible Powers’ (B1908A). The Brouwer notion of the continuum-as-a-whole, “the Intuitive

Continuum,” is & primitive concept generated in the Primordial Intuition of time. It is abstracted from the time interval, the mathematical “between” that is never ex-

hausted by division into subintervals.

1.4

Mathematics, Language, and Logic

Within the Brouwerian conception of mathematics as pure, individual thoughtconstruction on the basis of the Primordial Intuition alone there is clearly no place for language in any form. The emphatic “languageless” in nearly all his definitions

of mathematics reflects the need for express denial of the role language plays in almost all alternative philosophies of mathematics, even that of his favorite “preintu-

itionist,” Poincaré. Freeing mathematics from its traditional reliance on language and logic was the objective of Brouwer’s first Intuitionist campaign, the “First Act

of Intuitionism,” in his historical surveys described as “the uncompromising separation of mathematics and mathematical language and thereby of the phenomena

described by theoretical logic.”

In the genesis of mathematics, wholly confined to the private thought-world of the Subject, no use is made of any aspect of language, either as the carrier of common “objective” concepts or as primitive symbols with no meaning. For the sake

of “aiding the memory” the flesh-and-blood mathematician may resort to recording

Brouwer’s Intuitionist Programme

9

his constructions in symbols, linking a thought-construction to a name, “an aural or visual thing”; such recording, however, is a posteriori and not part of the mathematical process itself. Moreover, it is essentially private language since both the

thought-construction and the assignation to a particular symbol are exclusively acts of the individual mathematician. As to language as a means of communicating mathematics to other individu-

als, there is no basis for agreement betwgen the constructive thought-processes of different individuals represented in a “common language.” To the Subject other individuals are “things,” creations of his Exterior World, and the existence of other

minds similar to his own “mere hypothesis.” And even if the existence of other

minds were to be conceded, there is no guarantee that common words would represent the same thought-construction in the private worlds of different individuals.

Brouwer’s “mathematical language” is the report, the record of a completed

mathematical construction; its truth and noncontradictority are due solely to the con-

structions it represents.

Logic The Foundations (B1907) defines “theoretical logic” as an application of mathematiics, the result of the “mathematical viewing” of a given mathematical record, seeing a certain regularity in the symbolic representation: “People who want to view

everything mathematically have done this also with the language of mathematics .. . the resulting science 1s theoretical logic . . . an empirical science and an application of maihematics . . . to be classed under ethnography rather than psychology”TM (p. 129).

The classical laws or principles of logic are part of this observed regularity;

they are derived from the post factum record of mathematical constructions. To interpret an instance of “lawlike behavior” in a genuine mathematical account as an application of logic or logical principles is “like considering the human body to be

an application of the science of anatomy” (p. 130). The cunning application of such principles in the verbal or symbolic domain produces nothing but “verbal edifices” outside the mathematical reality: Linguistic edifices, sequences of sentences which follow one another according to

the laws of logic. . .. Even if it appears that these edifices can never show up the linguistic figure of a contradiction, they are only mathematics as linguistic constructions and have nothing to do with mathematics, which is outside this edifice. (B1907, p. 132)

Brouwer reiterates Descartes’ observation that logic cannot generate new mathematical truths. In his “Unreliability of the Principles of Logic” (B1908C) he goes one step further and questions the validity of the principles of logic when applied to mathematics. To prove his general point he singles out the Principle of the Excluded Middle (PEM)—identified with the principle of the solvability of every math-

ematical problem-—as flawed and an obvious misstatement of fact. The argument

here is based on the lack of guarantee of a solution for an infinite system and on his strict interpretation of affirmation and negation. Trme mathematical statements,

affirmative and negative, express the completion of a constructive proof; in partic-

Walter P. van Stigt

10

ular the negative statement expresses what Brouwer calls “absurdity,” the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence. In B1908C Brouwer simply states that for an infinite system there is no guarantee that such a construction can be completed and “that for infinite systems the Principle of the Excluded Middle is not a reliable principle” (p. 157). His major campaign against the use of the PEM starts after the publication of “The Foundations of Set Theory Independent of the Logical Principle of the Excluded Middle” (B1918B and B1919A), when in a number of papers he challenges the proofs of certain classical theorems, analyzes the logic of negation, and tries to prove the invalidity of the PEM by means of his counterexamples of essentially unsolvable mathematical problems (see further Section 4.1).

Formalism and the Formalization of Mathematics and Logic Brouwer’s interpretation of mathematics and his views on language and logic are in direct opposition with the Formalist definition of mathematics as the theory of abstract structures or formal systems. In his Foundations (B1907) he analyzes the processes of logic and formalization as stages of linguistic engineering, further and further away from the pure mathematical thought-construction. In particular, the systern as created by Hilbert’s formal axiomatic method 1s descrnibed as a mathematical application in the domain of language, “mathematics of the second and third order.” He frequently expresses his doubts about the success of the Formalist

Programme of creating a language free from contradiction. His fundamental disagreement, however, is with the Formalist identification of mathematics with formal consistent system: It does not follow from the consistency of the axioms that the supposed corresponding mathematical system exists. Neither does it follow from the existence of such a system of mathematical reasoning that the linguistic system is alive, i.e,, that it accompanies a chain of thought, and even less that this chain of thought is a mathematical construction. (B1907, p. 138)

Following his analysis in 1923 of the logic of negation, there appears to be some shift in Brouwer’s views on logic and formalization. When one of his research students, Arend Heyting, expressed an interest in Intuitionism, Brouwer suggested to him the “Intuitionist Axiomatics of Projective Geometry” as the subject of his

doctoral dissertation (1925). He also approved of Heyting’s formalization of Intuitiomst logic and supported its publication (“The Formal Rules of Intuitionist Logic,”

Heyting 1930b). The character and function of such formalization in Brouwer’s view

remained strictly limited: a post factum analysis without any contributory role in the construction of mathematics proper. In the opening lines Heyting expressly con-

forms to Brouwer’s view: Intuitionistic mathematics is a mental activity [Denktdtigkeit] and for it every language, including the formalistic one, is only a tool for communication. It is in prin-

ciple impossible to set up a system of formulas which would be equivalent to inturtionistic mathematics, for the possibilities of thought cannot be reduced to a finite number of rules set up in advance. (Heyting 1930b, p. 311)

Brouwer’s Intuitionist Programme

11

As to the value of formalization, Brouwer remained sceptical: an interesting but “sterile,” that 1s, unproductive, exercise with no benefit to mathematics itself, as Heyting testified: “Answering your question about Brouwer’s attitude towards for-

malization, I would like to add that he always maintained that formalizing mathematics is unproductive, since mathematics is constructing-in-the-mind, of which language, and therefore also a formal system, can give only inadequate representation.”

(Letter of Heyting to van Stigt; October 29, 1969).

1.5

Brouwer’s New Theory of Sets and the Continuum

Brouwer’s new set theory as published in 1917 was the result of years of critical

study of classical set theory and a continued search for an alternative, constructive tool in the analysis of the continuum. His rejection of the standard set-theoretical

treatment of the continuum is argued at length in his Foundations (B1907), “On Possible Powers” (B1908A), and “Intuitionism and Formalism” (B1912A). Cantor’s

definition of a set as “comprehension (German: Zusammenfassung) of definite and separate objects of our intuition or thought into a whole” is readily accepted as a legitimate mathematical construction; indeed, the primordial construction of two-ity and the further construction of what Brouwer calls “separable mathematics” are par-

ticular cases of such construction. However, for the characterization of the continuum and real number, the set—element relation i1s deemed 1nappropriate and wholly inadequate. According to Brouwer, in the genesis of mathematical sets the existence

or the completed construction of the elements is primary. Any definition of continuum as “the set or totality of all its points or real numbers” would require the preexistent construction of each point. The only acceptable constructive interpretation

of real number so far, however, is its identification with the constructed algorithm

or law generating convergent sequences of rationals, and Brouwer argues that the totality of such constructed algorithms is denumerable.

The extension of the set concept by means of language, property, or comprehension is rejected as nonmathematical in particular in “Intuitionism and Formal-

ism” (B1912A). Even the preintuitionists are taken to task for “not seeking an In-

tuitive origin for the continuum, one which lies outside the domain of language and logic,” and for not realizing that such a system of real numbers “generated by an ever-unfinished and ever-denumerable system of laws” remains ever denumerable

“and incapable of fulfilling the mathematical functions of the continuum for the simple reason that it cannot have a measure positively differing from zero” (BMS49, p. 3). Brouwer’s topological investigation into the nature of dimension had confirmed his earlier conviction of the dimensional gulf between sets of points and the

continuum. Recent publications by Bergson on the nature of movement and time reinforced that conviction and highlighted the “becoming” and indeterminate aspects of the live continuum.

His own editorial involvement in the re-edition of Schoenflies’ standard work on set theory during the years 1912-1913 made him even more aware of the inadequacy of current set theory as a tool in the analysis of the continvum. In his “Review of Schoenflies and Hahn” (B1914) he elaborates on its deficiencies, in par-

12

Walter P. van Stigt

ticular its failure as a coherent theory founded on one philosophical “Grundan-

schauung.” The significance of the “Review,” however, lies in its launching of the notion of a “*well-constructed set,” which contains all the new elements of Brouwer's alternative set theory of 1917,

The new set theory, “The Second Act of Intuitionism,” introduces two distinct

set concepts: the “Set” or “SpreadTM and the “Species.” They are based on what Brouwer calls his “New InsightsTM:

I. The continued conviction that the continuum is given n Intuition, that 1s, 1s a primitive concept, and that the appropriate method of investigation is “analysis,”

that is, decomposition into homogeneous parts; the continuum is the “between’ never to be exhausted by division into subintervals. “Elements of the continuum” reflect the nature of the continuum and share its properties. In the “Review” they are introduced as “sequences of nested intervals whose measures converge to zero.” Unlike the classical interpretation of real point, Brouwer's real point remains identified with the sequence of intervals itself: We call such an indefinitely proceedable sequence of nested A intervals a point P

or a real number P. We must stress that for us the point £ is the sequence (1) A, Ay, Ay, ... itself,

not something like “the limiting point” to which, according to the classical view, the A intervals converge. or something to be defined as the unique accumulation point of midpoints of these intervals. Every one of these A intervals (1) is therefore part of the point P. (BMS37, p. 3)

2. This idenuification of the sequence with the process of generation in progress

marks a fundamental change from Brouwer’s earlier concept of infinite sequence, hitherto justified by its identification with the completed algorithm or law of gen-

eration. As an ongoing process of construction in time, the infinite sequence has its own mathematical legitimacy, derntved ultimately from the constructive activity of

the Subject Mathematician. In the case of the “lawlike” sequence, the Subject de-

termines each element uniquely by means of an algorithm constructed by him, a

self-imposed restriction of his freedom. Acceptance of the constructive act of the Subject as the ultimate source of mathematical legitimacy, however, allows the broadening of the concept sequence to include those whose elements are determined by immediate free choice of the Subject: “sequences of mathematical systems pro-

ceeding indefinitely in complete freedom or in freedom subject to (possibly changing) restricttons” (BMS32, p. 7). The admssion of such “as-yet-uncompleted” constructions as legitimate mathematical objects extends the power of the range of sequences beyond the denumerable, in particular that of the sequences defined as “elements of the continuum.” Brouwer was well aware that as incomplete sequences the “elements of the contin-

uum” would be an inadequate basis for analysis and function theory, especially in the light of his strict interpretation of mathematical existence as “having been con-

siructed.” He found a solution by carefully analyzing the infinite sequence as an ongoing process in time, Viewed as such the sequence is split by the present moment of choice into a completed initial segment and a yet-to-be-determined tail. The initial finite segment is a completed construction and as such can claim full mathe-

Brouwer’s Intuitionist Programme

13

matical existence; the yet-to-be-constructed tail is governed by guarantees of proceedability and convergence. Brouwer hints at a solution in an added note to his unpublished Lecture Notes 1916. Referring to the free-choice sequence f as a possible argument of a real function, he finds: “with such a sequence one can work very well

as long as at every stage [of the function assignment] one can work with a suitable

initial segment of £ (BMS15, p. 1). The assertion that the function assignment must be wholly dependent on an initial segment of the Indefinitely Proceeding Sequence became the Fundamental Hypothesis of Brouwer’s analysis and function theory and, in particular, of his wellknown Uniform Continuity Theorem. “The Foundations of Set Theory” (B1918B) describes 1t 1n terms of an integral-valued function: “A law which assigns a natural number # to each element g of C must have determined 2 completely after a cer-

tain initial segment « of the sequence g has become known . ..” (p. 13).

Identification of real number with the whole of the Indefinitely Proceeding Se-

quence of nested intervals ensures the nondenumerability and measurability of the continuum; the Fundamental Hypothesis makes it a “workable” basis of analysis. Moreover, the choice sequence highlights the continuing active role of the Creating

Subject and is perhaps his most characteristic product, reflecting as it does “the Life

of the Subject” both in its past, completed initial segment and in the-free and open future of its yet-to-be-determined tail. 3. “Intuitionist Set Theory” (B1919D2) confirms Brouwer’s earlier total rejection of Cantor’s Principle of Comprehension: “The Axiom of Comprehension, on the basis of which all things with a certain property are joined into a set (also in the

restricted form given to it later by Zermelo), is not acceptable and cannot be used as a foundation of set theory” (p. 23). His introduction of an alternative, constructive interpretation of “property” follows his growing awareness that the use of some form of comprehension cannot be avoided. In his Lecture Notes 1916 he admits that

in his own work he had made use of it: “a point set is in fact a set defined by com-

prehension . . . further in this text we shall meet a vanety of point sets” (p. 23); but he is reluctant to give them the full status of constructive set; they are “pseudosets, better referred to as ‘sorts’ ” (p. 1).

The new Sort or Species was introduced in 1917 as part of the Second Act of Intuitionism: the insight that a constructive interpretation of property can be given

by restricting its domain to existing, that i1s, completed, mathematical constructs:

“the admission as a modality of the self-unfolding of the Primordial Intuition of mathematics .". . at each stage of this construction of mathematics of properties which can be presupposed for mathematical entities [lit. thought-constructions,

Dutch: denkbaarheden] already conceived, as new mathematical entities [thoughtconstructions] under the name Species” (BMS32, p. 7). As any legitimate mathematical entity, “property” is a construction; in particular it is a construction considered as a substructure, part of another construction: “the fitting-in of one construction into another” (B1908C, p. 156). Candidates for elementhood are only “previously acquired” mathematical entities or systems. The

Species or property does not by itself partition the domain of previously acquired mathematical entities into those that possess and those that do not possess the property. Elementhood is established by the successful fitting in of the property construction, and its negation by the absurdity of its fitting in.

14

Walter P. van Stigt

On the other hand, a property is considered to be a definite mathematical entity, irrespective of whether all its elements have been or can be established. As le-

gitimate mathematical objects they can be used in the construction of new species, in particular the construction of species of species, enabling the use of general statements in analysis and function theory. (This “species principle” was one of the points of disagreement between Brouwer and Weyl; see the introduction to Weyl in Part II.)

In order to avoid self-reference, Brouwer further introduced the notion of “higher-order species,” based on a hierarchy of domains at various levels of construction. Restricting the definition of “mathematical entities” to the fundamental

generating procedures and their elements, he defines the Species of the first order as “a property which only a mathematical entity can possess” (B1918B, p. 3). Species of higher order are then defined inductively: “By a Species of the second order we understand a property which only a mathematical entity or a species of the first order can possess” (ibid.). The fundamental role of species is acknowledged by

Brouwer when he stresses its use “at every stage of the construction of mathematics.” It is particularly evident in his definition and use of equivalence and negation. “Equivalence” plays a role in the generation of the fundamental “mathematical entities.” Even the abstraction of “two-ity” 15 “the common substratum of all two-

ities” constructed by the Subject at different points in time. The property of coincidence of points with a given point P generates or rather is a fundamental “mathematical entity” and is a species, the “point core P.” At first Brouwer refers to it as “a species of order zero” (BMS32, p. 8); in a letter to Heyting he later suggests

“Perhaps a better way of treatment 1s to introduce along with the things themselves the “species of identical things’ and then mainly use the latter, in the same way that

topological set theory considers point cores rather than the points themselves” (Brouwer to Heyting; July 17, 1928).

The Brouwer Negation In the generation of the fundamental “mathematical entities,” such as the natural numbers and the Brouwer set or spread and its elements, there is no place nor immediate need for negation. The question of negation only arises at the level of species

construction, at the point where the Subject is attempting to establish elementhood of a species S over a given domain of existing mathematical entities. Such attempt

may lead to “successful fitting in”; that is, a particular mathematical entity is established as an element of S. The alternatives to “successful fitting in” are: (1) the

constructed impossibility or “absurdity” of fitting in; and (2) the simple absence of the construction of elementhood or of its absurdity. Only negation in the first sense,

of constructed impossibility, meets Brouwer’s strict requirements and can claim to be an act of mathematical construction. The implications of such strict interpretation of negation are far reaching. It im-

mediately calls into question the use of double negation and the logical principle of

the excluded “third” or middle. In Brouwer’s own reconstruction of set theory and

mathematics the use of the Principle of the Excluded Middle is carefully and expressly avoided. In the foundational “debate,” however, it becomes the most contentious issue, the clearest manifestation of the fundamental differences between the

Brouwer’s Intuitionist Programme

15

two opposing philosophies of mathematics (see further the introductions to Parts III and IV),

As to the Brouwer negation, the question still remains as to what constitutes “absurdity” or “constructed impossibility.” His definitions remain somewhat vague, and 1n their use of terms such as “impossibility,” “incompatibility,” “difference,”

and “contradiction” they seem to be circular. “Contradiction” or “the impossibility of fitting in” is first described in B1907: “I just observe that the construction does

not go further [Dutch: gaat niet, that is, it does not work], that in the main edifice there is no room to be found for the posited structure.” (p. 127). The impossibility

is due to some “incompatibility,” a term Brouwer uses in his later work. But incompatibility—latent and inherent in the structures concerned—is not sufficient by itself; he insists that negation is “a construction of incompatibility” (B1954A, p. 3)

or “the construction of the hifting upon the impossibility of the fitting in” (B1908C, p. 3). Proof of the “absurdity of” or the incompatibility of two complex systems is identified, in particular, in the post-Brouwer Intuitionist tradition, with the reduction to a simple contradiction such as 1 = 0 or the logical p & — p. But these con-

tradictions, as tndeed all descriptions of “absurdity,” make use of some notion of

negation or difference. Their absurdity can ultimately only be justified by some intuitive, primitive relation of distinctness, an element of the Primordial Intuition, the fundamental recognition of the Subject of distinct moments in time. Brouwer also uses other, weaker forms of negation, in particular, where he moves outside the domain of mathematics proper into the realm of “mathematical

language” and mathematical *“assertions,” where, for example, he speaks of “unproven hypotheses,” “the case that « has neither been proved to be true nor to be absurd.” Negation in this case expresses the simple absence of proof, which in the world of mathematics as construction in time may well be reversed: Unsolved prob-

lems may one day become proven truth or absurdity. Moreover, “a mathematical

entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property it did not possess before” (BMS39, p. 1), leading to further distinctions, in particular, between “cannot now” and “cannot now and ever,” the

latter term frequently used by Brouwer in his later work as an alternative descrip-

tion of “absurdity.”

The Brouwer.Set or Spread

The concept of Brouwer Set or Spread* is the result of Brouwer’s search for a constructive procedure for generating elements of the continuum. “The Foundations of Set Theory” (B1918B) and the here-published “Intuitionist Set Theory” (B1919D) introduce the notion first in its generalized form. Later papers, in particular, Real Functions (BMS37), start with the definition of a special case, the “Point-Set,” which “genetically precedes and is more easily understood than the definition of the general case,” which Brouwer admits in (B1925A) “regrettably, has to be rather longwinded.”

Brouwer likened the Spread to a tree, a living, growing organism that produces

its elements in the form of “nodes” selected by the creator of the tree, the Subject, branching in various directions and “proceeding indefinitely.” In the tree each spread

16

Walter P, van Stigt

element is represented by an upward path, a sequence of nodes selected by the cre-

ator of the tree, the Subject. Each node is the last element of an initial segment of such a path and leaves the Subject sufficient options to select the next node of his path. Options are labeled or “assoctated withTM elements of a fundamental sequence

such as the natural numbers, so that a path of nodes 1s represented by a choice sequence of these labels. The Spread then is the whole of the internal organization: the rules that govern the domain of choice: the Spread law proper, as wel! as its labeling: the complementary law.

In the case of Point Spreads the domain of choice 1s that of A intervals, and the Spread law provides guarantees of indefinite proceedability and convergent nesting: By a point set we understand a law by means of which with the numbers #(, n5, na, . .. freely and in sequence selected from the sequence of natural numbers 1, 2,

3, ... either l. A A square 1s associated in such a way that of two subsequent A squares the sec-

ond lies strictly within the first; or 2. Nothing is associated with the chosen first number »y, or, if A squares have been assoclated with ny, na, . .., ny—| (A = 2), nothing is associated with the choice

of the Ath number ny, and at the same time all intervals generated so far are destroyed and the process is terminated. In the last case, however, there must be

at least one n;, with which, when chosen, the law does associate a A square. (BMS37, p. 5}

The defimtion of Point Spread allows restrictions on the options available to the

Subject. If at every stage the number of choices available is finite, the spread is

called a finite set or spread.” Freedom of choice can also be “narrowed to the point of complete determination,” resulting in a fundamental sequence, a sequence each term of which s uniquely predetermined. The Spread generates its elements, unlike

the Species, which is a property “to be posited for entities previously acquired.”

“Having the same genitor,” however, is a property and insofar the Spread can be considered to be a species: “A point set is always also a species, but a point species

15 not necessarily a point spread.” The Brouwer-Set or Spread as defined in *“The Foundations of Set Theory” and “Intuitionist Set Theory” is a generalization of point set: A Set is a law on the basis of which, whenever an arbitrary digit-complex is chosen from the sequence 1, 2, 3, 4, 5 .. ., each of these choices produces either a

definite syrubol or nothing, or causes the arresting of the process together with the definite destruction of its result; moreover, after every nonarrested sequence of n — I choices (for every n > 1) at least one digit-complex can be indicated that, chosen as the nth digit complex, does not lead to the arresting of the process. Every

sequence of symbols so generated by an indefinite sequence of choices (which therefore in general is unfinished in character) is called an element of the set, The common mode of generating elements of the set M is also referred to in short as the set M. (B1919D2, p. 24)

As in the case of the Point Spread, the spread law governs the choices from a given

“fundamental sequence.” Brouwer uses the term “digit-complexes” (Dutch: cijfercomplexen), that is, numerals as distinct from numbers. In BMS15 he clearly refers

Brouwer’s Intuitionist Programme

17

to numerals in their decimal composition: “We start from the sequence r of the digitgroups 1,2, ..., 10,11, ..., which can be continued without restriction according to a well-known law that enables us to derive from each digit-group the next one.”

The Complementary Law of the general spread associates with each chosen numeral “a definite symbol,” or it produces “nothing.” In the latter case, if at the nth selec-

tion as a result of a particular choice “nothing” is produced, the whole of this nterm sequence is eliminated. However, the spread law requires that there be at least one alternative choice at the ath selection that does not produce “nothing” and allows the sequence to proceed indefinitely.

1.6

The Selected Publications

In this section we present an anthology of Brouwer’s Intuitionist Programme dur-

ing the period 1920-1928, papers that set out his Intuitionist principles as well as

his attempts at reconstructing set theory and mathematical analysis. The following is a synopsis of the papers presented. “Intuitionist Set Theory” (B1919D2)

This paper marks the beginning of Brouwer’s

Intuitionist campaign of reform of general mathematical practice. It is the first time he uses the term “Intuitionist” to describe his own conception of mathematics and its reconstruction. The *“Set” and “Species” are announced as the new, improved alternative to the “classical theory of setsTM as the basis of analysis; only definitions are given, and the reader 1s referred to “The Foundations of Set Theory” for further

detail, More attention, however, is given to the reasons for the proposed change and its impact on traditional mathematics. Brouwer describes it as “a consequence of his earlier theses rejecting the Axiom of Choice and the Principle of the Excluded Middle,” identified with Hilbert’s axiom of the solvability of all mathematical prob-

lems. He further lists the radical changes in classical set theory and mathematics, resulting from the new constructive interpretation of set: the abandonment of some

notions and theorems and the redefinition and considerable change of proofs of others.

On only a few occasions during the following 10 years did Brouwer speak out

directly on the philosophical controversy of the Principle of the Excluded Middle. One such occasion was the Antwerp Congress of Natural and Medical Science 1n August 1923, when he delivered his paper on “The Splitting of the Fundamental Notions of MathematicsTM (1923C1), starting the debate on the “Brouwer Logic” (see

Part TV). Most of his publications are part of his Intuitionist programme of reconstructing mathematics: redefining fundamental notions of algebra and analysis and providing new, constructive proofs of classical results. Brouwer was well aware of what he himself describes as the “destructive and debilitating” effects of hus Intuitionist restrictions on classical mathematics. He was confident, however, that his

new set theory and what Weyl calls “his natural treatment of the continuum” would make up for these losses and lead to new developments and a new simplified analy818.

Promising new developments were made in particular in the following two se-

lected papers:

Walter P. van Stigt

18

“Does Every Real Number Have a Decimal Expansion?” (B1921A) In this paper Brouwer explores the effect of his new characterization of “point of the continuum” on the notion of “real number.” Real number now is a special case of “linear pointTM; in the definition of linear point, “indefinitely proceeding sequence” is replaced by “fundamental sequence,” that is, a sequence whose terms are determined by a law. He finds that this interpretation “comprises considerably more” than the classical

definition of real number as Dedekind cut, in particular, in their respective representations by decimal expansion.

“Proof That Every Full Function is Uniformly ContinuousTM (1924D1) In his analysis of the Brouwer continuum Weyl observed: “The concept of continuous function in a bounded interval cannot be defined without simuitaneously including uniform continuity and boundedness in the definition. ... What nowadays 1s called a discontinuous function actually consists of several functions in separate continua’ (Weyl 1921, p. 114). Brouwer marked his full approval in the margin of Weyl's manuscript: “Very true! Underline, because this is the main and most important

point!” (see further Part 2 of this book). Continuity and uniform continuity of all full functions, Brouwer claimed, “is the immediate consequence of my Inturtionist view and has been frequently mentioned in my lectures ever since 1918” (B1927B, p. 62; see also Parsons 1967). The theorem that “every full function is uniformly

continuous” is first stated, and a skeich of proof given in the first pages of “The Foundations of the Theory of Functions Independent of the Principle of the Excluded Middle” (B1923A). During the following years he made a number of attempts at a full proof, first in the here-published B1924D1, and further in B1927B,

BMS32, and BMS37. In all these proofs uniform continuity follows directly from

a proof of what we referred to above as “an insight” and “Brouwer’s Fundamental

Hypothesis,” the assertion that the function assignment is wholly dependent on an initial segment of the Indefinitely Proceeding Sequence. A proof is given for a func-

tion defined on a “finite set” and is now called “the Fundamental Theorem of Finite Sets” (in his postwar papers also the “Bunch Theorem” or “Fan Theorem”TM). In BMS66 Brouwer describes it as “a most wonderful theorem whose importance would justify calling it ‘the Fundamental Theorem of Intuitionism’ .. .” (p. 6). The rela-

tive importance of the Uniform Continuity Theorem is clear from BMS32, where it is relegated to the third place in “the applications of the Fundamental Theorem,”

The proof of the Fundamental Theorem in tumn is based on what Brouwer later calls the “Bar Theorem,” which asserts that if a function f assigns each element of

a spread M to a naturai number, then M is split into subspreads M, so that all elements of one M, have the same initial segment and are all assigned by f to the same

natural number 8,. In the proof of the Bar Theorem Brouwer introduces a form of regressive induction, which he justifies on the basis of “profound intuitionist re-

flection.” That this proof of the Fundamental Theorem did not wholly satisfy Brouwer is clear from a remark he made in a lecture in 1952: *. . . a wonderful theorem . . . but one whose absolutely rigorous proof till now has not been sufficiently stmplified” (BMS66, p. 0).

The remaining three selected papers are major statements made at the end of

Brouwer's creative period, each concentrating on a particular aspect of his Intu-

Brouwer’s Intuitionist Programme

19

itionist Programme: his philosophical views on the nature of mathematics and its relation to logic and science, his reconstruction of mathematical analysis, and his campaign to have [ntuitionism universally accepted as the right interpretation and practice of mathematics. “Intuitionist Reflections on FormalismTM (B1928A2)

This paper is Brouwer’s sum-

mary of the state of play in the Intuitionist—Formalist debate by the end of 1927, “the battle about the Foundations of Mathematics,” which was turning into a bitter personal feud between the two main protagonists, It lists the Intuitionist achieve-

ments, Brouwer’s “insights,” some wholly or partly adopted, “copied,” by the op-

posttion, others still to be ceded. There are, however, signs of battle fatigue, resentment at the lack of universal recognition, disappointment at the falling support

for his cause, and, possibly, the progress of his Intuitionist Programme. Particularly resented 1s the annexation of the notion of “meta-mathematics” “without proper

mention of authorship.” Brouwer’s terms for an end to the foundations “battle,” however, are uncompromising; they include the total surrender of the fundamental hypothesis of the For-

malist Programme, the justification of mathematics by a proof of its consistency. As a peace offering he concedes a modified Principle of the Excluded Middle and

its usefulness for those “endeavoring to work out a proof of the consistency of Formalist meta-mathematics.” “Mathematics, Science and Language” (B1929)

The first of two lectures given by

Brouwer in Vienna on March 10 and 14, 1928, It focuses on the philosophical theses that underlie his Intuitionism and clearly reflects his mood and concems at the time. There is a return to the pesstmism and isolationism of his student years and to his moralistic stance on science and language. New 1s the strong Schopenhauer-

ian emphasis on the role of the will in the interpretation of both language and sci-

ence. “Imposition of will” is the origin and nature of language. Science 1s the causal interpretation of the world; both the causal, mathematical attention and man’s acting upon this knowledge, the mathematical or “cunning act,” are acts of the will.

The major part of the paper is devoted to the analysis of language. Its origin and

therefore its nature are determined by its function of “will transmission,” the imposition of one’s will on others. Even as an instrument of will transmission language lacks exactness and certainty; it is wholly inadequate as a carrier of thought and in particular as a representation of the pure mathematical thought-construction.

Both language and logic, the analysis of mathematical language, are fundamentally flawed. (“Will, Knowledge and Speech” (B1933) restates substantial parts of “Mathematics, Science and Language,” but also includes new elements such as the notion of the “Idealized Mathematician.”)

“The Structure of the Continuum” (B1930A) In his second Vienna Lecture Brouwer starts with a brief critical survey of the traditional interpretations of the continuum. He then introduces his notions “species,” “set,” “finite set,” and continuum and proceeds with an investigation of its properties, in how far the topological properties as defined for the classical continuum apply to his Intuitionist con-

20

Walter P. van Stigt

tinuum. All the properties investigated are found to be inapplicable. “Discreteness”TM in any form is rejected out of hand. Most properties, however, can be “re-established” after some “logical” transformation of definition, some though in substantrtally weaker form.

Notes 1. Brouwer was professor of mathematics at the University of Amsterdam from 1912

until his retirement in 1951, For further detatls of his life and career, see van Stigt 1990. A Brouwer biography by D. van Dalen is in preparation. Franchella 1994 contains an extended bibliography on Brouwwer and intuitionism. 2. On the topic of this section, see also Largeault 1993b. 3. On the French Inwitionists, see references given in the introduction to Part 2 and Largeault 1993a and 1993b,

4. In his German and Dutch publications Brouwer continues to use the general word

“Set” (German; Menge, Duich: verzameling), the term “Spread’ ts first used in the Dutch version of (BMS32). In this introduction we refer to it as “Brouwer-Set” or “Spread” except in quotations, where we give the literal translation “Set.” 5. Note that “finite” here does not refer to the number of elements of the spread. To avoild confusion modern Intuitionisis have adopted the term finitary spread, intro-

duced by Kleene. Fintte spreads were used by Brouwer. in particular, in his proof of uniform continuity.

Bibliography Works by L. E. ). Brouwer

Publications (All listed books and publications, except (B1905), (B1923C1), and {B1933), are republished in A. Heyting, Brouwer Collected Works I, North-Holland, Amsterdam, 1975) B1905, Leven, Kunst en Mystiek, Waltman, Delft 1905. An introduction and English translation by W. P. van Stigt in Notre Dame Journal of Formal Logic 37, 1996, p. xx.

B1907, Over de Grondslagen der Wiskunde, Maas & van Suchtelen, Amsterdam, 1907, Engfish translation in CWIL.

B1908A. Die moglichen Michtigketten, Ari IV Congr. Intern. Mat. Roma 111, pp. 569-71. B1908C, De onbetrouwbaarheid der logische principes, Tijdschrift voor Wijsbegeerte 2, pp.

152-58. English translation in CWI. BI912A, Intuitionisme en Formalisme, Clausen, Amsterdam. English translation: Intuition-

ism and Formalism, Bull. Am. Math. Soc. 20, pp. 81-96.

B1914, [Review of] A. Schoenflies und H. Hahn, Die Entwickelung der Mengeniehre und ihrer Anwendungen, Leipzig und Berlin 1913, JDMYV 23, pp. 78-83.

B1918B, Begriindung der Mengentehre unabhéngig vom logischen Satz vom ausgeschlosse-

nen Dritten. Erster Teil: Allgemeine Mengenlehre, KNAW Verhandelingen, le Sectie, deel XII, no. 5, pp. 1-43. B1919A, Begriindung der Mengenlehre unabhdngig vom logischen Satz vom ausgeschlosse-

nen Dritten. Zweiter Teil: Theorie der Punktmengen, KNAW Verhandelingen, le Sectie, deel XII, no. 7, pp. 1-33.

Brouwer’s Intuitionist Programme

21

B1919D1, Intuitionistische Mengenlehre, JDMV 28, pp. 203-8.

B1919D2, Intuitionistische Verzamelingsleer, KNAW Verslagen 29, 1921, pp. 797-802. Eng-

lish translation in this volume, Chapter 1. B1921A. Besitzt jede reelle Zahl eine Dezimalbruchentwickelung? KNAW Verslagen 29, pp.

803-812; also in KNAW Proceedings 23, p. 955, and Mathematische Annalen 83, pp. 201-210. English translation in this volume, Chapter 2.

B1923A, Begriindung der Funktionenlehre unabhiingig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Stetigkeit, Messbarkeit, Derivierbarkeit, KXNAW Verhandelingen, le sectie, dee] XIII, no. 2, pp. 1-24.

B1923C1, Intuitionistische splitsing van mathematische grondbegrippen, KNAW Verslagen

32, pp. 877-80. English translation in this volume, Chapter 18. B1923C2, Intuitionistische Zerlegung mathematischer Grundbegriffe, JOMV 33, 1925, pp. 251-56. English translation in this volume, Chapter 19, B1924D1, Bewijs dat iedere volle functie gelijkmatig continu is, KNAW Verslagen 33, pp. 189-93. English translation in this volume, Chapter 3. B1924D2, Beweis dass jede volle Funktion gleichméssig stetig ist, KNAW Proceedings 27,

pp. 189-93, B1924F1, Bewijs van de onafhankelijkheid van de onttrekkingsrelatie van de versmeltingsrelatie, KNAW Versiangen 33, pp. 479-80. B1924F2, Zur intuitionistischen Zerlegung mathematischer Grundbegriffe, JDMV 36, 1927, pp. 127-29. English translationin this volume, Chapter 20. B1925A, Zur Begriindung der intuitionistischen Mathematik 1, Mathematische Annalen 93, pp. 244-257.

B1927B, Uber Definitionsbereiche von Funktionen, Mathematische Annalen 97, pp. 60-75. English translation of §1-3 in van Heijenoort 1967, B1928A2, Intuitionistische Betrachtungen iiber den Formalismus, KNAW Proceedings 31, pp. 371-74; also in Sitzungsberichte der Preussischen Akademie der Wissenschaften

zu Berlin, 1928, pp. 48-52. English translation in this volume, Chapter 4. B 1929, Mathematik, Wissenschaft und Sprache, Monaishefte fiir Mathematik und Physik 36, pp. 153-64. English translation in this volume, Chapter 5. B1930A, Die Struktur des Kontinuums, Wien 1930 (Sonderabdruck). English translation in this volume, Chapter 6.

B1933, Willen, Weten, Spreken, Euclides 9, pp. 177-93. English translation in van Stigt 1990 (Appendix 5).

B1948C, Consciousness, Philosophy and Mathematics, Proceedings of the 10th International

Congress of Philosophy, Amsterdam 1948, 111, pp. 123549, B1954A, Points and Spaces, Canadian Journal for Mathematics 6, pp. 1-17

Unpublished . BMS3B, The Rejected Parts of Brouwer’s Dissertation; published in Dutch original and English translation by W. P. van Stigt in Historia Mathematica 6, 1979, pp. 385-404; also in van Stigt 1990; pp. 404415,

BMS 15, Notes for a course on set theory given in Amsterdam between 1914 and 1916, BMS32, Berliner Gastvorlesungen. A course on Intuitionism given by Brouwer in Berlin in 1927 and intended for publication. Published by D. van Dalen in L. E. ). Brouwer intuitionismus, Wissenschaftsverlag, Mannheim, 1991. English translation of the last part of Chapter 1 p. 7, in van Sugt 1990, pp. 481-85. BMS37, Reelle Funktionen. A 126-page manuscript of a partly completed book. English translation of the table of contents and extracts in van Stigt 1990, pp. 469-480.

BMS49, Disengagement of mathematics from logic. A 5-page manuscript of a lecture probably given in 1947. Published as an appendix in van Stigt 1990.

Walter P. van Stigt

22

BMSS51, The Cambridge Lectures. A course on Intuitionism given by Brouwer at Cambridge University regularly between 1946 and 1951. Now published in van Dalen 1981. BMS39, The Influence of Intuitionist Mathematics on Logic (alt. Changes in the Relation

between Classical Logic and Mathetmatics); a lecture given in London on November 2, 1951, published as an appendix in van Dalen 1981. A slightly different version in a different manuscript published as an appendix in van Stigt 1990, pp. 453-38. BMS66, Second Lecture. A 9—page manuscript dated May 1952 [University College London], published as an appendix in van Stigt 1990, pp. 459-068.

Secondary Literature Bridges, D., Richman, F., 1987, Varieties of Constructive Mathematics, London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge.

Detlefsen, M., 1990, Brouwerian Intuitionism, Mind 99, pp. 501-34. Dummett, M., 1973, The Philosophical Basis of Intuiticnistic Logic, in Truth and Other Enigmas, Harvard University Press, Cambridge (Mass.). Dummett, M., 1977, Elements of Intuitionism, Oxford University Press, Oxford. Franchella, M., 1994, L. E. J. Brouwer Pensatore Eterodosso. L intuizionismo tra matematiica e filosofia, Guerini, Milano. Heviing, A., 1930b, Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, phys. math. K., pp. 42-56. English translation 1n this volume, Chapter 24,

Heyting, A., 1975, Brouwer Collected Works I, North-Holland, Amsterdam. Hilbert, D., 1922, Neubegriindung der Mathematik, Abhandlungen aus dem mathematischen

Seminar der Hamburgischen Universitit 1, pp. 157-77. English translation in this volume, Chapter 12.

Largeault, J., [993a, L’ Intuitionisme des mathématiciens avant Brouwer, Archives de Philoso-

phie 56, pp. 53-68.

|

Largeault, J., 1993b, Intuition et Intuitionisme, Vrin, Paris. McCarthy, C. D., 1983, Introduction, Jourrnal of Philosophical Logic (special issue on intuittonism) 12, pp. 105-49. Parsons, C., 1967, Introductory note to B1927B 1n van Heijenoort 1967, pp. 446-57.

Posy, C. 1., 1974, Brouwer’s Constructivism, Synthese 27, pp. 125-59. Troelstra, A. 8., 1977, Choice Sequences, Oxford University Press, Oxford. Troelstra, A. S., and van Dalen, D., 1988, Constructivism in Mathematics. An Introduction, Vol. I, North-Holland, Amsterdam.

van Dalen, D., 1981, Brouwer’s Cambridge Lectures, Cambridge University Press, Cambridge.

van Heijenoort, 1., 1967, From Frege to Gddel, Harvard University Press, Cambridge (Mass.).

van Stigt, W. P., 1990, Brouwer’s Intuitionism, North-Holland, Amsterdam.

Weyl, H., 1921, Uber die neue Grundlagenkrise in der Mathematik, Mathematische Zeitschrift 10, 1921, pp. 39-79. English translation in this volume, Chapter 7. Weyl, H., 1949, Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton,

Intuitionist Set Theory' LUITZEN EGBERTUS JAN BROUWER*

The following is intended as a report and introduction to the two parts of my paper

“Foundations of Set Theory Independent of the Logical Principle of the Excluded Middle,” which I presented to the Academy in November 1917 and October 1918.

Since 1907 I have in various publications® defended the following theses: 1. The Axiom of Comprehension, on the basis of which all things with a cer-

tain property are joined into a set (also in the restricted form given to it later

by Zermelo?), is not acceptable and cannot be used as a foundation of set theory. A reliable foundation of mathematics is only to be found in a constructive definition of a set.

2. The axiom of the solvability of all problems as formulated by Hilbert in

1900* is equivalent to the logical Principle of the Excluded Middle; therefore, since there are no sufficient grounds for this axiom and since logic is based on mathematics—and not vice versa—the use of the Principle of the

Excluded Middle is not permissible as part of a mathematical proof. The Principle of the Excluded Middie has only scholastic and heuristic value, so that theorems that in their proof cannot avoid the use of this principle lack

all mathematical content.

In the publications quoted in note 2 I have so far mentioned only some of the consequences of the Intuitionist conception of mathematics as condensed in these two theses. In my philosophy-free mathematical papers published during this pe-

riod I still made regular use of the old methods, but | did take as much care as pos-

sible to derive only such results as could be expected also to find a place in the new system after the systematic construction of an intuitionist set theory, perhaps with

some modification, but in essence the same [met behoud van hun waarde lit. “retaining their valueTM].

Only with the publication quoted at the beginning of this paper did I start this

systematic construction of an Intuitionist set theory. In the following I shall give a *Intuitionistische verzamelingsleer,” KNAW Versiagen 29, 1921, pp. 797-B02 (paper presented to the

Royal Academy of Sciences on December 18, 1920). Translaied from the Dutch by Walter P. van Stigt.

The German version, “Intuitionistische Mengenlehre,” was published in JDMV 28, 1921, pp. 203-8 and

in the KNAW Proceedings 23, 1922, pp. 94954, Published by permission of Koninklijke Nederlandse Akademie van Wetenschappen. 23

24

Luitzen Egbertus Jan Brouwer

brief survey and focus attention on some of the most radical changes it has brought about in the classical theory of sets, changes that not only affect tts form, but also its content in an essential way.

The new system is based on the following definition of a set: A Set is a law on the basis of which, whenever an arbitrary digit-complex [cijfercomplex] is chosen from the sequence 1, 2, 3, 4, 5, . . ., each of these choices produces either a definite symbol or nothing, or causes the arresting of the process to-

gether with the definite destruction of its result; moreover, after every nonarrested sequence of 1 — | choices (for every n > 1), at least one digit-complex can be indicated that, if chosen as the nth digit-complex, does not lead to the arresting of

the process. Every sequence of symbols so generated by an indefinite sequence of choices (which therefore in general is unfinished [enaf] in character) is called an element of the set. The common mode of generating elements of the set M is also referred 1o in short as the set M.

On this set concept is also based the definition of the concept of Mathematical Species, which holds the set concept as a special case. In the theory of cardinal numbers it is first of all the splitting of the concept of equipotency that demands spectal attention.

Two sets or species that in a classical set theory are equipotent can in Intuitionist set theory be: equipotent, half-equipotent, equivalent, having the same range [van gelijke omvang)], having the same extension [Dutch: Van gelijke uitgestrek-

theid, German: Ausdehnung), or having the same weight [gewicht].> Accordingly, among the sets or species that are denumerable in the classical theory, one can in Intuitionist Set theory distinguish between sets, respectively, species, that are denumerably infinite, denumerable [Dutch: aftelbaar, German: abzdhlbar), numerable

[Dutch: telbaar, German: zdhlbar], out-numerable [Dutch: uittelbaar, German: auszdhlbar), through-numerable |Dutch: doortelbaar, German: durchzéhibar], and

Jull-numerable [Dutch: voltelbaar, German: aufzdhlbar]. The classical cardinal numbers @ and ¢ are retained, but abandoned is the example of a cardinal number > ¢

generated in classical set theory by the set of all functions of one variable. In the theory of ordered sets that now follows, the ordering of a species requires the existence of the ordering relation only for two arbitrary elements that

have been shown to be different. Some parts of this theory, such as the characterization of the ordinal numbers ¥ and ¢, become much more complicated than they

are in the classical theory. The characterization of 9 takes the following form: The ordered species P has the ordinal number ¥ if P contains a subset M that is

denumerably infinite and everywhere dense in the stricter sense and such that be-

tween any two elements® of P there are elements of M, such that the species of elements of M that lie before an arbitrary element of P is a separable subspecies of M that is either empty or has at least one clear [aanwijsbaar, i.e., indicable] element, and such that for every fundamental sequence of relations “after” and “not

after” w.r.t. elements of M,” which corresponds with the ordering relation, an element of P can be indicated to which these relations apply.

In the theory of well-ordered sets we have first of all to abandon the two main theorems that are the most important means of proof in the classical theory, that is,

the theorem that two arbitrary well-ordered sets are comparable and the theorem

Intuitiontst Set Theory

25

that every subset ofa well-ordered set has a first element. There are therefore practically no points on which the new constructive theory and its predecessor agree either in form or in content. Instead of the preceding main theorem we put the fol-

lowing theorem: A law that in a well-ordered species determines an element and that to every ele-

ment already determined adjoins either the arresting of the process or a preceding element definitely determines an element to which it adjoins the arresting of the process.

The theory of planar point sets starts from a set Q of squares that in relation to a right-angled coordinate system have the coordinates a/2” and #/2" as one of

their corner points and whose sides (parallel to the axes) have lengths 1/2* and

172", Further, a point of the plane is defined as an indefinitely proceeding sequence of squares of Q of which each lies in the interior of its predecessor.

Taking this as the basis of point set theory we again have to abandon a large number of theorems of the classical theory. For exampie, of Cantor’s main theorem only the following negative part remains valid: There cannot exist any closed ordered point set whose power is greater than the denumerably infinite, one in which every point has a “next point” and that also has

a denumerable order, respectively, has a finite distance from the species of points ordered later.

Moreover, this partial property must be proved by a method that is quite dif-

ferent from the usual one, based on the Principle of the Excluded Middle. One could, for example, argue as follows: A point set 71 whose power 1s greater than the denumerably mfinite can only be ordered in such a way that finite sets {;, is, . . . of finite choice sequences that are not

extensions of one another are successively ordered, while for each k the certainty must exist that S{i. 4|, ix+7, . . .} of every final segment of each infinite choice se-

quence contains an initial segment. By retaining in 8{i;4{, {43, ...} only those choice sequences that do not contain any preceding choice sequence as an initial segment, we determine a denumerable set ji of choice sequences. By means of the

proceeding construciion of the j,, one can in the construction of an arbitrary j, for

every one of its choice sequences construct for at most one single point generated as its extension (i.e., for the one determined by the condition that in every follow-

ing j,, for 44 > v, always the last ordered continuation of the choice sequence in

hand is to be chosen) the certainty that it has a “next point” in the resulting ordering of . Therefore, the cardinal number of the species of points for which this cer-

tainty can be acquired cannot be greater than the denumerably infinite.®

The positive part of Cantor’s main theorem 1s in Intuitionist Set Theory replaced by an extensive characterization of point sets and point species that have the

property expressed by this positive part.”

The interior boundary sets of the classical theory, that is, the intersections of

fundamental sequences of “areas” [arealen],!® are introduced in Intuitionist Set The-

ory as interior boundary species, since they do not necessarily have the character of set. In this way the classical theorem that the intersection of two interior boundary species is again an interior boundary species remains valid. However, the cor-

responding thesis for the union of sets must be abandoned. Of the main property of

26

Luitzen Egbertus jan Brouwer

interior boundary sets of the classical theory, that is, that for an arbitrary point set O there exists an interior species that apart from ( contains only boundary points of the final coherence of Q, only the following part remains valid: For every point set o that can be completely broken off, there exists an interior boundary species that is locally congruent with the union of 7 and a subspecies of the closure of the final coherence of # and that contains a point set locally con-

gruent with 7 as a subspecies.

The classical definition of measurability is subject to only a minor modifica-

tion in Intuitionist set theory. However, there is no certainty as to the measurability of “areas,” closed point species, and interior boundary species. The main property of the classical notion of measurability, that is, that the union of a denumerable

set of measurable sets without common points is measurable and that its measure is equal to the sum of the measures of its components, is formulated in Intuitionist set theory as follows: If F 1s a fundamental sequence of measurable point species such that the contents of the unions of their initial segments form a limited sequence {, then the union of F 13 also measurable, and its content is equal to .

The notion “point of the plane” is obviously subject to considerable restriction if 1n its definition “indefinitely proceeding sequence” is replaced by “fundamental sequence.” It is, however, remarkable that its analogue in the linear continuum, even

of this more restricted notion of point, comprises considerably more than the classical notion of linear point, which is based on the cut. This is further explained in my paper on the decimal expansion of real numbers, which has been presented to the Academy at the same time as this paper.

Notes 1. A paper of the same title and substantially the same content appeared in the Jahresbericht der Deutschen Mathematiker-Vereinigung 28, 1922,

2. Cf. The Foundations of Mathematics, Doctoral Dissertation, Amsterdam 1907, in particular also the appended list of theses: “The Unreliabulity of the Principles of Logic,” Tijdschrift voor Wijsbegeerte

2, 1908, reprinted in Mathematics, Truth and Reality, Groningen 1919, “On the Foundation of Mathematics,” Nieuw Archief voor Wiskunde (2) 8, 1908.

“Review of Mannoury ‘Methodologisches und Philosophisches zur Elementarmathematik’,” Nieww Archief voor Wiskunde, (2) 9, 1910.

“Intuitionism and Formalism,” Inaugural Address, Amsterdam 1912, reprinted in Mathematics, Truth and Reality, Groningen 1919, and published in English in Am. Bull, 20 (1913).

Review of Schoenilies/Hahn “Die Entwickelung der Mengenlehre und ihrer Anwendungen,” Jahresbericht der Deutschen Mathematiker-Vereinigung 23 (1914), “Addenda and Corrigenda—On the Foundations of Mathematics,” Verslag der Koninklijke Akademie van Wetenschappen, Amsterdam 25 (1917), reprinted in

Nieuw Archief voor Wiskunde (2) 12, (1918).

Intuitionist Set Theory

27

3. Cf. Mathematische Annalen 65, p. 263. [The German version adds “beschrinkteren,” Le., “the more restricted form given to it by Zermelo” Transl.]

4, Cf., e.g., Archiv der Mathematik und Physik 1 (3), p. 52. According to the view expressed by Hilbert in this paper, this axiom reflects a conviction shared by all math-

ematicians. However, in his more recent lecture Axiomatic Thinking, published in M.A. T8, Hilbert qualifies the question of the solvability of all mathematical prob-

lems by calling it a question still to be solved (p. 412). The following remarks, in which he speaks of the finiteness of-a complete system of algebraic invariants, mean—in my terminology—that from the impossibility of the infiniteness of a set we can by no means deduce that it is necessarily finite.

In my opinion both the axiom of solvability and the Principle of the Excluded Middle are incorrect; they are dogmas that have their origin in the practice of first abstracting the system of classtcal logic from the mathematics of subsets of a definite finite set, and then attributing to this system an a priori existence independent

of mathematics, and finally applying it wrongly—on the basis of its reputed a priori nature—to the mathematics of the infinite sets. . In my earlier writings I could allow the notion of equipotency only in the case of

very special sets. Ontby this “splitting” of the concept has enabled me to extend it to all species, and so in some sense to restore the existence claim of a comprehen-

sive conception of species. i.e. between any two elements that have been shown to be different. ~J . 1.e., Tor every indefinitely proceeding sequence of relations “after” and “not after” w.r.t. elements of M enumerated in a fundamental sequence. . This proof 1s already given in the last two papers quoted in note 2, although the terminology used is not the same as that introduced in my paper. Moreover, there are some typographical errors in the relevant passage of my “Review of Schoenflies”

(p. 81, lines 3 and 19, should read

*nondenumerable subsets of the second kind”

instead of “subsets of the second kind”; line 10 should read “overlapping regions

€y €ap - - . Instead of “regions ey, ez, . . .," and line 11 should read “i,, iq,, .. .” instead of “iy, i, .. .").

In my papers quoted in note 2 (with the exception of the last paper), in which the full implications of Intuitionism were as yet not so clear to me, there are still two unnecessary restricting suppositions attached to the constructive definition of set: The point sets discussed in these papers are in my present terminology first of all

defined as locally individualized, and second they allow a complete internal breaking off [afbreking). The result is that, e.g., in my “Review of Schoenflies” the main

theorem is presented as obvious instead of incomrect, and that the distinction made there between well-constructed point sets and point sets in general is now—after

the removal of the above restrictions—practically covered by the present distine-

tion between point set and point species (the splitting of a well-constructed point set, which I introduced at the same time [in the “Review of Schoenflies,” Transl.] into a set of the first and a set of the second kind of which the first is a special case of the second, is nonessential and should not be retained).

As to the example of a not-well-constructed point set given in the quoted passage, I should note that the underlying function f{x) does not have the whole continuum as its existence domain (cf. my paper on the decimal expansion of real numbers, published at the same time as this paper); further, that line 12 should read “which can be represented by a finite dual fraction” instead of “rational,” and that a much simpler example of a not-well-constructed point set is given by the species of finite definable points of the plane.

10.

The notion “area” [areaal] (Bereich) contains the notion of “region” as a special cdse.

2 Does Every Real Number Have a

Decimal Expansion?’ LUITZEN EGBERTUS JAN BROUWER*

§1

The Existence Domain of Infinite Decimal Expansions on the Continuum

We define in the set of finite dnal fractions =0 and =1:

A A, interval as the closed interval with the two binary fractions /2 and (a + 2)/2" as end elements; A point of the continuum as an indefinitely proceeding sequence of A intervals, each of which contained in the interior of its predecessor?;

X as a variable point on the continuum; Fn(x) as a decimal fraction expanded to n places with the property that every

point on the continuum lying to the left of it lies to the left of an interval of x while F,(x) + 107" lies to the right of an interval of x: F(x) as the unique decimal expansion of x. Then F,.(x) has the property (in common with all noncontinuous functions) that its domain G, cannot coincide with the continuum.? Therefore, the domain G =

D(Gy, Gy, . . .} of F(x) cannot coincide with the continuum although it fits so closely into the continuum (as does indeed the domain of the regular expansion of x) that

they are locally concordant* and have the same content.>S The definition of a point of the continuum, however, becomes severely restricted if “the indefinitely proceeding sequence” is replaced by “fundamental sequence.”” The purpose of the following sections is to establish for these points of the con-

tinuum in the stricter sense in how far an infinite decimal expansion can be said to exist,

*Besitzt jede reelle Zahl eine Dezimalbruchentwicklung?,” KNAW Verslagen 29, 1921, pp- 803—12 (Presented to the Royal Academy on December 18, 1920). Also published in Proceed ings of the KNAW 23, 1921, pp. 935-64 and in the Mathematische Annalen 83, 1921, pp. 201-10. Translat ed from the German by Walter P. van Stigt. Published by permission of Koninklijke Nederla ndse Akademie van Weten-

schappen.

28

Does Every Real Number Have a Decimal Expansion?

§2

29

The Completion Elements [Ergdnzungselemente] of Denumerably Infinite, Everywhere-Dense Ordered Sets

Let H be a denumerably infinite ordered set that is everywhere-dense in the stricter

sense.® Let g, g2, g3, . . . be the elements of H numbered according to a particular law vy that characterizes [ as a denumerably infinite set, and let S(g[, g2, .. ., &) = sp- We define an i,, respectively, a j,, as a closed interval of H (possibly consisting

of a single element)” whose end elements belong to s, but whose interior contains at most one, respectively, not a single, element of s,.

We detine a filling element [Ausfiillungselement] r of H: first as a species— having at least one element—of indefinitely proceeding sequences Fa, Fot1s Fat2s - - ( 1s a positive whole number determined for r) where every £, is an {, and every

Fa+v+1 18 contained in £, ,, while £, for every » belongs to a species S, (determined for r) of which every two elements have an element of s, in common; and second, as

a species—having at least one element—of indefinitely proceeding sequences {1, {,

{3, . . . of separable subsets® of H that have at least one determinable element while in every sequence each {1 is contained in £, and a fundamental sequence of whole positive numbers ny, #y, Ha, ..., (B4 = n,) is determined as well as a filling ele-

ment rg of H of the first kind with the property that for every element £, 3, &, . .. of r there exists an element £, Fo 11, Fata, - .. Of 1y such that £, belongs to £, +,. We define a completion element of order zero or in short a completion element

r of H: first as a fundamental sequence Fa, Fo+1, Fatz, .- - (@ 18 a positive whole

number determined for r}, where every £, is an i, and each fF,+,+; 15 contained in Fo+s Second, as a species—having at least one element—of indefinitely proceeding sequences £y, {7, {3, . .. of separable subsets of H that have at least one determinable element while in every sequence each {4+ 1s contained in £, and a fun-

damental sequence of whole positive numbers ny, np, n3, ..., (Hye) =n,) 18

determined as well as a completion element fo, Fa+1, Fa+2, - - - Of H of the first kind

such that each £, of r belongs to Foy )" If yr and ,r are filling elements of H and every 1, has an element in common

with , F,, we say that 7 and »r coincide in H. A filling element of H that coincides in H with a completion element of H will also be referred to as a completion element of H.

If the element g of H belongs to every £, of the filling element r of H, we say that r and g coincide in H.

If |r and ,r are filling elements of H and if one can indicate a | £, and a »£,

that have no element in common, we say that r and or are locally distinct [drtlich verschieden].

If an F, of the filling element r of H can be indicated to which the ¢lement g of H does not belong, we say that r and g are locally distinct n H. The completion element or the filling element r of H 1s called a first-order completion element of H if for every g of H either the relation g = r (i.¢., every clement of H to the right of g lies to the right of a definite £, of r) or the relation g =

r (i.e., every element of H to the left of g lies to the left of a definite £, of ) can be established, or—what amounts to the same—if r coincides with a completion element ' of H of which every £, 1s a j,.

30

Luitzen Egbertus Jan Brouwer

The completion elements of the first order correspond to the Dedekind cuts of H.

The first-order completion element r of H is called a second-order completion element of H if for every element g of H the relation g = r can either be establhished

or be reduced to absurdity or—what amounts to the same—if ' can be chosen such that there cannot exist a u with the property that the right end elements of £, and F., are identical for every » > u.

The second-order completion element r of H is called a third-order completion

element of H if for every element g of H either the relation g > r (1.e., one can In-

dicate a £, of r lying to the left of g) or the relation g = r can be established, or— what amounts to the same—if ' can be chosen such that for every £, an £, can be determined whose right-hand end element lies to the left of the right-hand end element of £,,.

A third-order completion element of H is called a fourth-order completion element of H if for every element g of H either the relation g > r can be established or the relation g = r (i.e., g and r coincide in H} or finally the relation g < r (L.e., one can indicate a £, of r that lies to the nght of g) or—what comes to the same—

if # can be so chosen that for every /; a ¥ > w can be indicated such that the two end elements of £, are distinct from the two end elements of 7. These definitions of the filling elements as well as those of the completion el-

ements of order zero and of order one, two, three, and four of H are for given ordering relations in H clearly independent of the numeration law [Abzdhlungs-

gesetz] . Let M be a finite set or a fundamental sequence of fourth-order completion el-

ements of H,, of which every two are locally distinct in H, and of which each one 1s locally distinct in H,, from every element in H,. Then the unton of M and H, forms a denumerably infinite, in the stricter sense everywhere-dense, ordered set H, .. Every completion element of H,, is at the same time a completion element of H,.\, and every completion element of order i of H,,; coincides in H,,, with a completion element of order A of H,. The above relation exists between the ordered set of finite dual fractions Hy

and the ordered set of finite decimal fractions H| as well as between H; and the or-

dered set of rational numbers H-.

§3

Completion Elements, Decimal Expansions, and Continued Fractions

A first-order completion element of H allows a positioning [Ortsbestimmung} of or-

der one that, if H 1s interpreted as the set of finite decimal fractions, appears as the nonunique [mehrdeutig] infinite decimal expansion. Conversely, every filling ele-

ment of H that in H allows a first-order positioning is a first-order completion element of H. The tirst-order positioning in H can turn out to be different for coinciding com-

pletion elements of H.

A second-order completion element of H allows'in H a second-order positioning that, if A is mterpreted as the set of finite decimal fractions, will be the unigue

Does Every Real Number Have a Decimal Expansion?

31

infinite decimal expansion (in which the existence of a last numeral other than 9 is excluded). Conversely, every filling element of H that allows in H a second-order positioning 1s a second-order completion element of H.

Two completion elements of H for which the second-order positioning turns out to be distinct cannot coincide in H. A third-order completion element allows in H a third-order positioning that, if

H is interpreted as the set of rational numbers, will be the infinite reduced-regular continued fraction. Conversely, every filling element of H that allows in H a third-

order positioning will be a third-order completion element of H. Two completion elements of H for which the third-order positioning turns out to be distinct are locally distinct in H. A fourth-order completion element of H allows in H a fourth-order positioning that, if A is mterpreted as the set of rationals, will be the unique regular continued

Sfraction (which may possibly turn out to be finite). Conversely, every filling element of H that in H allows a fourth-order positioning is a fourth-order completion

element of H. Two completion elements of H for which the fourth-order positioning turns out

to be distinct are locally distinct in H.

§4

The Existence of the Decimal Expansion of Real Algebraic Numbers

Let r; and r, be arbitrary real algebraic numbers; that is, each is a filling element of the ordered set of rational numbers H satisfying an algebraic equation with whole

rational coefficients. One can then determine an algebraic equation

FO=agx*+ax* '+ - +a,—ix+a,=0 with whole rational coefficients and nonvanishing discriminant D that satisfies r| as well as r.

Let wy, wa, ..., wy, be the roots of F(x} = 0 (to be approximated to whatever

degree of accuracy one chooses). Then w, and w; cannot coincide in Hj if r # s. Let p be a rational number that is greater than the modulus of every root of F(x) = 0 and let b = 2p. Then

|wr-w5‘ a flz.n—zbngr-n—z e

|2

32

Luitzen Egbertus Jan Brouwer

so that by means of sufficiently accurate approximations of r; and r, we can either establish with certainty that ry and r; coincide with the same root w, or determine a rational interval that separates r, and r;. If we then consider this result for the spe-

cial case that r, is a rational number, it will be quite obvious that r| in H> either

coincides with an element of H, or that it is locally distinct from every element of H>, so that ry is shown to be a fourth-order completion element of H,, and therefore can be expanded as a unique infinite decimal fraction and also as a unique infinite regular continued fraction.

If we now presuppose further that neither r| nor r» coincides with an element of Hj, then they either coincide in H, or they are locally distinct 1n . It follows then that the species of real algebraic numbers forms a denumerably infinite, in the stricter sense everywhere-dense, set H3 that stands to H; in the rela-

tion of an H,4 to a corresponding H, as discussed at the end of §2.

§5

The Existence of the Decimal Expansion of

Let a and & be positive whole numbers and a < b. We define K to be the absolutely

[unbedingt] convergent'? infinite continued fraction E

B

'

az

o

Qv+ b

:

and K, as the absolutely convergent infinite continued fraction a?

fl2

[(2m+ Db’ Qv+ Db ]

oo

m+

Then the following relations appiy:

a

a

tg—~=K b— K, 0 y=—— g 3 &2

m =

Om+ Db — Kppey©

’

=]

Let xp, x1, x2, . . . be real variables related by:

a

“y

.

x

Xy = "

T

=

@

Zm + Db — Xy ,

m =] | B o

and let x; be a rational number between 0 and 1 and therefore 2a from the

end values of n,. We can then determine a non-negative whole number s < r such

that xp, x1, ..., x;—; are contained one after another in %, 7, ..., M- while x, lies at a distance > a from K.

Let 8’ be a positive rational number such that for every x; belonging to 7 the inequality dxo/dx; > ' holds. Then xj lies at a distance >af’ from X. Let x be a rational number such that the corresponding number xJ, (on the basis of (1)) 1s =0 or =1. We can then determine a non-negative whole number ¢
« from K.

Let 8" be a positive rational number such that for every xy belonging to g the iequality dxp/dx, > " holds. Then xj lies at a distance >af8” from K. One can therefore for any arbitrary positive rational number i; < 1 and an arbitrary positive

rational number i determine a positive rational number i, i. In particular, one can for an arbitrary positive rational number {; i»/5 (since in the range of y between the values 0 and 2 the inequality ‘a‘ arctg y;’dy\ = % holds). The number 1 is therefore shown to be a fourth-order completion element'? of H>, and can therefore be developed as a unique infinite decimal fraction and aiso

as a unique regular continued fraction. The two preceding paragraphs provide examples of characterization of com-

pletion and filling elements r of H as fourth-order completion elements of H, respectively, by means of positive rationality proofs in H (which determine an ele-

ment of H with which r coincides) or by means of positive irrationality proofs in H (which show r to be locally distinct from every element of H). We should add

that from a negative rationality, respectively, irrationality, proof in H (one in which

the assumption that r is locally distinct from every element of H, respectively, coincides with one element of H, leads to absurdity), it in no way follows that ris a

first-order completion element of H. It is for this reason that in this section we have given an appropriate reworking of Lambert’s negative iirationality proof of 7 and turned it into the above positive proof. Other classical proofs of the same theorem can also be changed and reconstructed in a similar way.

§6

Real Numbers that do not Have a Decimal Expansion

Let ¢, be the nth figure in the infinite decimal expansion of 7. We shall say that » is in case one if Cn, Cpi1s . . ., €n+4 are all the same, in case two it ¢, Cpa1, - . -, Cn+o are all different, and in case three if neither case one nor case two applies.

34

Luitzen Egbertus Jan Brouwer

We define a completion element r of the ordered set H) of infinite decimal frac-

tions by means of the infinite sequence ? a,107""1, where a,= 0 if n is in case

one, a, = 10 1f n 1s 1n case two, and Dtl‘f;er:wise a, =9,

This completion e¢lement then would only represent a first-order element of H

(i.e., would allow an infinite decimal expansion) if one would have a method by means of which for every arbitrary number # in case three one could either prove

the absurdity of the existence of an m in case two {m > n) with the property that any whole number between n and m would be in case three, or prove the absurdity

of the existence of a number m in case one {(m > n) with the property that any whole number between n and m would be in case three.

We further define a first-order completion element » of H, by means of the in-

finite sequence Z a,107" 1 where every a, is either equal to 0 or to 9, while =1

¢y = 9 and a, 41 18 distinct from a,, if and only if » is in case two. This first-order completion element then would only represent a second-order

completion element of H, (i.e., would allow a unique infinite decimal expansion as defined in §3) if one had a method by means of which—for every positive whole number n with the property that either none or only an even number of positive

whole numbers < n would be in case two—one could prove either the absurdity of the existence of an m in case two (m > n) or the absurdity of the absence of such an m In case (wo. A third-order completion element of H; then would only represent the same

completion element if one had a method by means of which one could—for every positive whole number » with the property that either none or only an even number of positive whole numbers = 1 would be in case two—either prove the absurdity of the existence of an m (m > n) in case two or could indicate a particular m

(m > n) that is in case two. Finally, we define a third-order completion element r of H, by means of the

infinite sequence Z a,107"" ! where a, = 9 if n is in case two; otherwise a, = 0. n=1

Thas third-order completion element would only represent a fourth-order completion element of H, if one had a method by which one could for every positive

whole number » either prove the absurdity of the existence of an m in case two such that m > n, or could indicate an m that is in case two such that m > n.

Moreover, all the examples of this section coincide in H, with fourth-order completion elements of the ordered set of dual fractions Hj,. As to the examples of real numbers without decimal expansion, there is with the further development of mathematics always the possibility that one day these no longer apply, but they then can always be replaced by others that will remain valid.

Notes I. The contents of this paper were the subject of a lecture given to the Naturforscherversammlung in Bad Neuheim on September 22, 1920.

Does Every Real Number Have a Decimal Expansion?

35

Cf. pp. 3 and 4 of my paper “Foundations of Set Theory Independent of the Logical Principle of the Excluded Middle,” Part 2, published in Verslagen der Koninklifke Akademie van Wetenschappen te Amsterdam, Vol. XII, No. 7. As was shown

in footnote 1 on p. 4 of that paper and as will also be made clear in this publication, the two notions of “real number” used on p. 9 of Part 1 (of the same paper) are considerably narrower than the notion “point of the continuumTM defined there. I am using the term “real number” in.the title and in the last section of this paper

because of its expressiveness but in a completely different sense, as will be clear from the context.

AR

Op. cit. Part 2, p. 5. Op. cit. Part 2, p. 6. Op. cit. Part 2, pp. 29, 30. Of course, the existence domain of a function of x expressed by means of the infinite decimal expansion of x cannot be wider than . E.g., the function fix) as defined by me in the Jahresbericht der D.M.V. 23, p. 80 has precisely G as its existence domain. But whereas the function F(x} of this text 15 uniformly continuous n the everywhere-dense point set G on the continuum and on the basis of this prop-

erty can be extended into a function @(x) = x existing on the full continuum, any such extension on the full continuum is out of the question in the case of fix). . Cf. “Foundations of Set Theory” etc,, Part 1, p. 14.

~oA—]0

. Op. cit, Part 1, p. 16. . Op. cit. Part 1, p. 13.

Ny[

. Op. cit. Part 1, p. 4.

We leave aside the question of whether the notion of completion element can be derived from that of filling element.

. Cf. Pringsheim, Miinchener Berichte 28 (1898), pp. 299 ff.

13. Op. cit., p. 318.

14, That the number e has the same property follows immediately from the regular continued fraction

e — 1 _

2

i

1

1”2 + 4p

o0

3 Proof that Every Full Function Is Uniformly Continuous LUITZEN EGBERTUS JAN BROUWER*

§1 Let M be an arbitrary set and let g be the countable [relbaar] (also denumerably in-

finite, but that is not relevant here) set on which M is based of finite (arrested and nonarrested) choice sequences Fy,, ... ,,, where s and the n, represent natural num-

bers. Further, let a natural number 8 be assigned to each element of M. Then there is contained in u a separable, countable subset u; of nonarrested finite choice sequences such that to an arbitrary element of p,—for all elements of M 1ssuing from it—a natural number £ is assigned, while moreover one has at one’s

disposal a proof A that shows for an arbitrary nonarrested element of w that every nonarrested infinite choice sequence issuing from it has an initial segment belong-

ing to u;. (A nonarrested element of . can be said to belong to w, if and only if at the generation of this element only—and not of any of its initial segments—the de-

cision as to B in accordance with the algorithm of the assignment law is not postponed until further choices are made. One should, however, not in any way rule out

the possibility that one can subsequently indicate an element of w that does not belong to w; nor has an initial segment that belongs to u; and vet has the property that the same natural number is assigned to all elements of M issuing from it.) We call an element of w secured if it is either arrested or if it has a (proper or not proper) initial segment that belongs to u,. Then g is split into a countable set T of secured finite choice sequences and a countable set o of nonsecured finite choice sequences; the proot 4, moreover, shows for an arbitrary element of o that it is se-

curable, that is, that any nonarrested infinite choice sequence in M issuing from it possesses an imtial segment that belongs to . Let A, ..., be the specialized proof

*“Bewijs dat iedere volle functie gelijkmatig continu is,” KNAW Verslagen 33, 1924, pp. 189-93 (Presented to the Royal Academy at the meeting of March 29, 1924). German version: “Beweis dass jede volle Funktion gleichmissig stetig ist,” KNAW Proceedings 27, pp. 286-90. Translated from the Duich

by Walter P. van Stigt. Published by permission of Koninklijke Nederlandse Akademie van Wetenschappen.

36

Proof that Every Full Function Is Unifermly Continuous

37

of A that derives the securability of the element Fg, ... n, Of 0. This proof then is exclusively based on existing relations b between elements of w. These relations b, however, can all be split into elementary relations e, each existing between a sin-

gle pair of elements F,,,_ ... mg A0 Fipppy . m, m,., ©f which the one is the immediate extension of the other. Since any proof in which the relations used are reducible to elementary relations can always be “canonicized” (although at the cost of short-

ness) in such a way that in the canonical form only elementary relations are used,

it 1s clear that in the canonical form kg, ... ,, of the proof Rsn, - n_the securability of Fy, ..., can be derived ultimately and exclusively from the elementary relations

e that connect F, ..., with F, ...,

and with the various Fy,, ..., .. In order to

be able to complete the proof kg, ... ,_ one must therefore first have proved the securability either of Fy,, We refer to

...,

orofall Fg ...,

an elementary inference that derives the

securability of an

g from that of F,,,, .. mg_, 88 @ { inference and fo an elementary inference that concludes the securability of F,,, ... m, from the securability of all Fy,y,, ... g as an F inference. Let f, ... ; be the well-ordered species of elements of o whose

securability is proved in order by kg, ... . It is then clear that the securability of the first element of f, ..., cannot possibly be proved by means of a { inference and that therefore it is proved by means of an £ inference. We further find by transfinite induction along f.,, ... ,_that at each stage of the proof ke, ... ,_the extensions

of all elements of ¢ that have already been found to be securable also have already been found to be securable, so that the securability of every element of f;, ..., In the proofs kg, ..., is derived by means of an £ inference. If then (in accordance with union relations between the corresponding sub-

sets of M) every time when in kg, ... , we prove the securability of an element

T

g of o we consider this element as the ordered sum of the F,,, ... m

01 -

dered by the index »—(and consider every arrested Fryn, ... m,» t0 be an empty elementary species), then (as is clear from transfinite induction along fs. ... »)

Fgy, ... n, emerges from this transfinite sequence of sum-formations as a well-ordered

species @gy, -.. p,. In this well-ordered species @y, ... n, the elementary species correspond to secured elements of o that do not have a secured proper initial segment,

and the remaining constructive underspecies correspond to elements of o. Con-

versely and in the same way an element of & or a secured element of 7 that does not have a secured proper initial segment corresponds to a constructive underspecies or elementary species of gy, ... , 1f and only if it has an index sequence Su, - m Py - Pu- In this case indeed the constructive underspecies or elementary species has such a sequence, namely the sequence p) ... p,. The result of this section can be summarized as follows: Theorem 1. If to each element of a set M a natural number S is assigned, then M

is split by this assignment into a well-ordered species § of subsets M, each determined by a finite initial segment of choices. To all elements of the same M, the same natural number B, is assigned. The species S can be constructed by means of generating operations of the second kind @ of which each corresponds to the continuation of a certain finite initial segment of choices that is nonarrested for M by means of a free new choice, while to a new choice that is arrested for M there corresponds for the corresponding operation w an empty elementary species.

38

Luitzen Egbertus Jan Brouwer

§2 In the case that M is a finite set, the well-ordered species @gy, ...

18 similar [Dutch:

gelijkvormig, German: dhnlich) to a well-ordered species tr,, ... ,, which can be constructed without using empty elementary species and vet in parallel with the above construction of @, ..., in such a way that to every operation w used in the construction of @, ..., there corresponds a finite number of generating operations of the first kind y for the construction of s, ..., _so that 5, ..., can be constructed

exclusively with generating operations of the first kind. From this it follows that Win, - n, a8 well as @, ... ,, are finite, and in particular that for every natural num-

ber s the well-ordered species o, is finite. We can therefore determine a natural number z such that an arbitrary element of w that is secured but does not have a se-

cured proper initial segment has at most z indices. Hence the natural number 3, assigned to an arbitrary element e of M is completely determined by the first z choices generating e. We have therefore proved: Theorem 2, If to cach element e of a finite set M a natural number S, is assigned,

then a natural number z can be determined such that 3, is completely determined

by the first z choices generating e.'

§3 We now determine on the X axis for every natural number » the «, intervals k., &,

, k59 (ie., the Agpss intervals), ordered from left to right and covered wholly or in part by the unit interval of the X axis. Then the finite set 7 of contracting infinite

sequences

of

intervals

[Dutch:

contraheerende,

German:

Inter-

vallschachtelungen)] ki*v, ké‘"?), ki, ... (where every subsequent interval is contained 1n the strict sense in the preceding one) forms a point set that coincides with the closed unit interval of the X axis, that is, with the species of poinis belonging to the closed unit interval of the X axis.

We define a full function y = f(x) as a function whose species of definition (i.e., the species of points for which it is defined) is identical with the closed unit interval of the X axis. Thenin the case of such a full function f to every contracting in-

finite sequence of intervals ki, &2, . . . is assigned a contracting infinite sequence of A intervals on the ¥ axis )l’., A”, A", .. .. Indeed, on the basis of Theorem 2 there exists for every natural number v a natural number m, (which we can assume does

not decrease for an increasing v) such that A*?is determined by the choice of k{*,

k(’”“fl .+,

k=), For every v, therefore, only a finite number 7, of A intervals of

the Y axis can appear as A, These have maximal widths b,, that for an indefinitely increasing v converge to zero.

Let ¢ ) represent the interval of the X axis concentric with k% and havmg = of the length of k{®). We choose two points P; and P, of the closed unit interval of

the X axis whose distance apart is n'if 7'+

7"isimpossible, and 7' < 7" if both m*

° 7" and o + «". This

ordering relation indeed does satisfy the five axioms of virtual ordering.

3. Density in itself (extended to virtual ordering) in the sense given above does

not exist for the Intuitionist continuum, and this because the above characterization of elements as main elements does not hold. Let us consider, for example, a char-

acterization of % as main element of the basis of a convergent sequence a; < a see 2l We now construct a sequence d|, d, . . . in the following way: We choose successtvely d; = a5, dr = ap, .. ., putting 4, = a,, as long as we have not yet found that a particular fleeing property has a critical number nor that the existence of such a critical number 1s absurd; if, however, between the choice of d, and that of d,, we do find that such a critical number exists or that its existence is absurd, we put

d, = d, = a,for u > v. The element d of the continuum belonging to this sequence

dy, ds, ... 18 < % and vet no a, can be indicated such that g, > d. To establish density in 1tself for the Intuitionist continuum is therefore, to say the least, a hopeless

task (see note 6); the same ts to be said for the reduced continuum.

To re-establish density in itself as a property of the Intuitionist continuum or reduced continuum, we submit the definition of density in itself to some logical

transformation; that is, we put it in another form (which in the classical view is equivalent to the one given above, but not equivalent in the Intuitionist view). We

first have to clarify the notion of interval of a virtually ordered species. For any two elements ¢ and b of a virtually ordered species S we define the closed interval ab as the species of all elements ¢ of S for which neither the relation both ¢ > a and

¢ > b nor the relation both ¢ 277 for a

suitable natural number n. Let p; be the first element of F that follows after p; and that lies between the zero point and p;; let p; be the first element of F that follows

after p; and that lies between p; and py; let p4 be the. first element of F that follows

after ps and lies between p; and ps, etc. We now construct a convergent sequence

The Structure of the Continuum

61

my, my, . .. of elements of F as follows: We put m,, = p, as long as we have not

learned of the existence of a critical number of an arbitrary fleeing property nor of the absurdity of such critical number; if, however, between the decisions as to m; and my+ a critical number is found or its absurdity is proved, we put m, = p; for v > k. The element p of K that belongs to this convergent sequence is distinct from p1, and yet no element of X can be indicated that lies between p and p;. To estab-

lish separability in itself as defined above for the full Intuitionist continuum is there-

fore, to say the least, a hopeless task (see note 6); the same is to be said for the reduced continuum., Separability tn itself can also be re-established as a property of the Intuitionist continuum by means of a logical transformation of the definition. To show this we

define two elements a and b of a virtually ordered species to be sharply different

and the interval ab to be spread out [ausgedehnt] if the complementary species k(ab)

of the open interval ab is split” (in the sense to be explained under (5)) into a subspecies ky(ab) whose elements are both ¢ of the reduced continuum should be-

long to B; on the other hand, b" > ¢ for every element b’ of 8. The subspecies £ therefore is identical to the subspecies of elements > e, and the subspecies a is 1dentical to the subspecies of elements = e, so that every element of the reduced con-

Luitzen Egbertus Jan Brouwer

62

tinuum must either be e, which is not the case. That the assertion of sharp connectedness for the reduced continuum 1s false follows from the same exampie as that for the full continuum.

To re-establish connectedness as a property of the Intuitionist continuum by means of a logical transformation of definition, we introduce the notion of exhaustive division. The virtually ordered species § is said to be exhaustively divided into the ordinally separated subspecies « and 8 of which it is composed, if for any two sharply different elements a and b (@ < b) either all elements =a belong to « or all

elements =b belong to 8. We call the virtually ordered species S freely connected, respectively, boundedly connected [gebunden zusammenhdngend], 1f for every ex-

haustive division of S, respectively, for every exhaustive division of § determined

by a law into two ordinally separate subspecies « and B, there exists an element ¢ of S such that every element >e belongs to 3. The full Intuitionist continuum then is freely connected, and the reduced continuum is boundedly connected.

6. Everywhere densiry as defined above is out of the question as far as the Intuitionist continuum and the reduced continuum are concerned for the simple rea-

son that this property requires first of all that for every two different elements a and b either the relation a b exists; that is, it presupposes the order of the species in question. If, however, we interpret the word “between” for the virtually ordered species in the sense defined under (3), then 1t is possible to determine between any two elements of the full continuum X a further element of X; in this way everywhere density is re-established as a property of the full continuum.

The same is done for the reduced continuum in a similar way, 7. Compactness as defined above (extended to virtual ordering) does not exist as a property of the Intuitionist continuum or of the reduced continuum. This is clear from the following example:

Let Asbe the critical number of the parity-free fleeing property f; let I, = (——, + ') for v 1 and if none of the finitely many numerical products a - & of the following table

(T) n—D-1L,r—1-2,...,0-1)-(n—1)

(which we can run through in the same way in which we read the lines of a book) coincides with a. This law specifies the property of being a prime number as a char-

On the New Foundational Crisis of Mathematics

103

acter. The following judgment instruction is justified: The product of two numbers =1 18 never a prime number. We define a law that from 1 generates nothing, yet

that from each number n > 1

generates two numbers

#(n),

x(n),

such that

(n) - k(n) = nis generally true and o (n) is always a prime number: If # is a prime number, then let 7w(n) = n and &(n) = 1; but if # is not a prime number, then let

w(n), k(n) be the first pair of numbers in the table (T) whose product is = n. With regard to this law we can formulate the following statement (abstract of a judgment

instruction): Every number > 1 is divisible by a prime number. According to Euclid, the sequence of prime numbers p, itself is defined through the following law

(using the symbol 7 in the sense we have just introduced): 1 generates the prime number p; = 2; p, + 1 is the first prime number in the sequence of the numbers

from 1to 7(p;po---p, + 1) that is different from py, ps, . . . and p,,. Finally the following example. Leaving open the question whether Fermat’s “last” theorem is valid

or not, it was nonetheless self-evident according to pre-Brouwerian logic that, if » 1s any number, either there are three numbers x, y, z such that x* + y”* = z”, or x"* + y" # z" is true for any natural numbers x, y, z. Let us now formulate more precisely this self-evidence of the old logic according to our new view. The claim then is that: There 1s a law (in our strict sense) that from every number n generates either noth-

ing or three numbers x,,, ¥,,, Z, such that, in the first case, we always have x" + y* # z" for any three numbers x, y, z, whereas in the second it is true that x; + y7 = 2. Not only is this claim now far from self-evident, it does not even make sense to ask

whether it 1s, or it 1s not 80, in the hope of facing a state of affairs providing a def-

inite answer 1o this question. We are dealing with a judgment abstract that is valid provided that the law has been constructed and that the properties demanded of 1t (i.e., properties that are universal statements) exist justifiably. If this law u is given, then we can construct from it another law assigning to the number »n, on the one

hand, 1 if u does does not generate anything from #, and, on the other hand, 2 if u assigns three numbers x,, y,, Z,, to n. This law then 1s the “character” that distin-

guishes the Fermatian numbers n (for which Fermat’s theorem holds) from the nonFermatian ones.

if the values of two functiones discretae coincide for every argument, we say that the functions themselves coincide; if, however, there is a number » from which the first law generates a different number than the second one, then we say that two sequences differ from one another. The former is a universal statement; the latter

an existential one; neither of them is a judgment in the proper sense. Thus we may

not ask with respect to two given sequences whether they coincide or differ, meaning that one or the other must be the case.

As long as we make only universal statements about numbers and not about sequences developing through free choice—and thus take into account only laws that

coordinate numbers and not laws that, from a developing choice sequence generate a number or again a developing sequence, are dependent on the outcome of the choices—we are moving in the field of pure arithmetic and algebra. Those higher levels are characteristic for analysis. What has been said so far ought to be sufficient to clarify the spirit in which arithmetic and algebra must be carried out according to the new view. The radical consequences of the new theory that essentially alter the present face of mathematics, however, become apparent only in the field of analysis.

Hermann Weyl

104

b. Functio Mixta

A function that assigns a sequence to every number m

(i.e., assigning a law, determined by s, that from every number # generates a num-

ber @(m, n)) is nothing but a double sequence, and thus falls under the concept of the functio discreta, But how can, on the contrary, a sequence, that is, in this case a sequence of numbers v = {n|, nz, ...} developing through free choices, generate an individual number? The simplest case is obviously the one where the generated number merely depends on a limited number k of places of the developing sequence. In this case one is certain that the number is determined as soon as the development of the sequence has reached the kth number. The place number £ is in this case in-

dependent of the outcome of the individual acts of choice. Example: f(¥) = n + ny + 13 + ny,

k=4

A more complicated case is the following one:

f(V) =m tnr+...F ”n|+n2+n3

The situation here is as follows: As soon as three places of the sequence are given, it is known up to which place (namely, n; + ny + n3) the sequence must be con-

tinued before it determines the generated number; this place depends on the outcome of the first three choices. This complication can be iterated; for example, the

first ten places determine the number s of those places that themselves must be

known in order to determine the place up to which the development must have proceeded before it generates the assigned number; etc. It is also not required that the complication involves a twofold, or threefold, or fourfold iteration, for the number of iterations can again be dependent on the outcome of the first choices. If, for example, one sets f(k, F) =M + N> + e 4 iy

and then, through iteration, forms

fitks v) = flk; v) fatks ) = fi( flks v); v) falks v) = L(flk v); v)

one derives from this, say, the following functio mixta (for short, fm.):

fnl*n21n3 (7; v) The following is the general principle underlying these construction possibilities. 1. If k is a natural number and ¢(n,, 12, . . ., 1x) any function of k£ arguments, then f(2) = @{ny, ng, .. ., ny) defines a “primitive” f.m. if nq, no,. . ., By are the first k places of a sequence v developing through free choices.

On the New Foundational Crisis of Mathematics

105

2. Primitive f.m. are the starting point for the formation of higher f.m. that is carried out according to the following principle: If f(k; v) is an already con-

structed f.m. still depending on an arbitrary natural number &, and if g(v) is likewise an already constructed f.m., then one obtains a new f.m. f" according to the substitution formula

') = Hgw; v) This rule, it must be emphasized, is not a construction principle comparable to the one discussed in Part I, because no assumptions are made here as to how the dependency of the f.m. on the parameter k comes about. Iteration is one, but not the only, possibility; the construction remains completely free in this respect. Inciden-

tally, we shall not investigate the question of whether an f.m. can only be formed in this way, or whether certain generating laws that are built up differently may not

equally be tied to the essential insight that, according to this law, there will always be a moment in a developing sequence where the sequence, regardless of how it develops, creates a number. This feature is all that is essential for the concept of

the f.m. The special case of a character is given, if, according to the wording of the law, only the values from 1 to & are available for the generated number. If, on the

other hand, we also wish to include the possibility of a function not being defined

for all sequences (“scattered” f.m.), then we have to admit that there might be “unfruitful” sequences that do not generate a number. Yet, in this case, the law must imply for every sequence », however it may develop, that up to a certain place (dependent on v) we either have the generated number or the certainty that we are deal-

ing with a barren sequence, remaining unfruitful for all eternity.

Several arguments (1.e,, several choice sequences developing side by side) can

always be regarded as a single one; the transference of concepts is thus given. Instead of “character” we will then speak of “relation.”

c. Functio Continua

'We now move on to the functiones continuae (f.c.). We have

already looked at a particular case of f.c. when the thought of a developing sequence appeared before our eyes for the first time. The growth of the sequence occurring as functional value kept pace with the growth of the argument. Removing this par-

ticular assumption, we get the following definition:

An f.c. is a law according to which every step adding another member to a sequence of natural numbers developing through free acts of choice determines a par-

ticular number or nothing. What happens at the kth step is not only dependent on the outcome of the choice at the kth step, but, in general, on the whole past of the argument sequence at that moment of development. However, this formulation does not yet ensure that the sequence yielding the function value actually grows in infinitum when the development of the argument sequence proceeds in infinitum. So the following requirement V must still be added: If k is any natural number, and » a developing sequence the development of which we follow from the kth place onwards, then there must occur a moment when a new number (and not nothing) 1s generated. We must furthermore generalize the concept of the f.c. also to include those

Hermann Weyt

106

cases where the function is not defined for all possible sequences. This is achieved by admitting that at every step the developing choice sequence generates a number,

or generates nothing, or brings about the termination of the process, that 1s, ifs own death (and the destruction of what it has previously generated). The abovementioned requirement V can easily be transferred: A choice sequence v that has progressed without termination of the process up to the kth place must after &, however it be continued, again produce a number or be killed by the law of the f.c. not later than a certain place dependent on & and v. However, we must add yet another requirement. Let, for example, g(») be an f.m. and P(») a “scattered” f.c. such as the ones we are envisaging here. Let a developing sequence v progress up to a cer-

tain point without the process being terminated by @, in such a way that the segment already created by the associated sequence v = ®(v) determines the number g(v') {= g(®(v))}; say, it is =2. If for a developing sequence v the termination of the process according to the law ® occurs at some point, then the function g(®) = g’ will not be defined for such a »; it will hence mean nothing for such a v. Can

we claim under these circumstances that there is a sequence v for which g(®(v)) =

2? Clearly only if we are certain that the sequence v that has progressed up to a certain point without being terminated by the law @ can also be continued indefinitely, that is, that this will never be its fate. Together with @ a second law X must therefore be given, according to which v generates, as long as it is not yet terminated by ®, at each of the further steps in its development a number of the following kind: If one chooses as the (k + 1)st number of v the number that is generated by X in the kth choice step of v, then, given that it has not happened up to then, & will also not hamper the development process of » at the (k + 1)st step. With this the concepts of function are sufficiently fixed. But let me emphasize

again that individual, determined functions of this sort occur from case to case in

the theorems of mathematics, yet one never makes general theorems about them. The general formulation of these concepts is therefore only required if one gives an account of the meaning and the method of mathematics; for mathematics itself, and for the content of its theorems, it does not come nto consideration at all.

§3

Mathematical Propositions, Properties, and Sets

For these theorems, as far as they are self-sufficient and not purely individual judgments, are generai statements about numbers or choice sequences, but not about “functions.” We therefore distinguish the following kinds of statements:

I.

Judgments in the proper sense.

I1.

Universal statements. Their type: For every natural number » and every freely developing choice sequence v, the relation C(xz; ») holds; relation in the strict sense of an n-dependent character of the choice sequence ».

III. Abstracts from judgments or universal statements. The “there 1s” can, in this case, occur in conjunction with numbers, sequences, functiones mixtae, and functiones continuae. With regard to a sequence, the “there 18" can even refer to it in two ways, 1n that the sequence either occurs as a sequence in the proper

sense, or as a law of correspondence [Zuordnuhgsgesetz]. The first case is, for

On the New Foundational Crisis of Mathematics

107

example, given when ((») is a character of the freely developing sequence v

and one subsequently formulates the proposition: There is a sequence v (obviously a lawfully determined one) that has the character C(#). The second one

is given if the following proposition is put forward on the basis of a relation R(m, n) between arbitrarily natural numbers m, n: There is a sequence ¢ that from every number m generates a number () such that every number m has

the property R(m, ¢(m)). However, given that the condition that the mth number of a choice sequence v stands in relation R to the number m is an m-dependent character of v (indeed, it is even a primitive f.m.), the second case is

thus a spectal case of the first one. The type of the statements of (IlI) can thus

be characterized with sufficient generality by the schema: There is a number Ry, & sequence v, and furthermore a law f that from every sequence v gener-

ates a number f(#), and a law P that from every freely developing sequence » generates a developmg sequence v' = ®(v), such that for every number # and every sequence v developing through free acts of choice the relation C(igy vy,

n, V), v, ®(v}) holds; C(ng vy n 1’5 v ¥') here is a given relation between the arbitrary numbers ng, n, n’ and the choice sequences g, ¥, V', If, in addition, indeterminate numbers or choice sequences enter into statements

of these three kinds, then we get statement schemata of properties and relations be-

tween numbers and sequences. In the domain of properties we must therefore distinguish the same three kinds. The properties falling under (I) are nothing but the “characters” in our sense; that is, those properties that in themselves hold or do not hold of anumber or a sequence. We could set them as “extensionally definite” properties against

the “extensionally vague” ones falling under (II) and (III). Let us furthermore refer to

extensionally definite properties, that is, characters, as definite sets (sets of type I). Of such a set one may say that it is definite whether an element belongs to it or not. I M, N are two number characters (definite sets of numbers), then M is a subset of NV if every number of character M also has the character N. Or, more precisely, we specify a law

(M; N), which from an arbitrary number # generates the number 2 (the “No”} if M as-

signs the number 1 and N the number 2 to it—but that, in every other case, generates the number 1. Themeaning of the “subset” statement is that every number possesses the character (M, N). It is thus a statement of form (II). if M is a subset of N and N 15

a subset of M, then we say that M is a subset of N and N i3 a subset of M, then we say

that M and N are identical. The same is to be said of definite sets of sequences, or of the “multidiménsional” sets of numbers and sequences corresponding to relations. One can speak of the cardinal number of a definite set, but one must then be clear about the fact that for these cardinal numbers not even the following proposition is true: A cardinal number is either = 0 or =1, that is, a number character M either generates from every number the number 2 (the set M is empty, every number has the property

@, or there is a number from which the law M generates the number 1 (there is an el-

ement in M, there is a number that possesses the property M). So, according to our present view, the doubt about there being no gaps between the Cantorian Ns does not set

in only at 8}, and also not only at X, but rather at the very first beginning of the number sequence. The claim that 1 is the smallest cardinal number that succeeds 0 must be rejected as unfounded. This seems to me to demonstrate the mathematical worthlessness of this concept of cardinality. Finite cardinal numbers naturally retain their

108

Hermann Weyl

good old rights where they are not referring to “definite sets” but to the totality of given individual elements (the number concept of daily life).

We now turn to the “extensionally vague” properties of kind (II}. In doing so,

the following must be noted. Let C(n; v) be a relation between the number 7 and the choice sequence v (e.g., the nth number of » is odd). The statement “C(n; v) holds for every n (all the numbers of » are odd),” which formulates an extension-

ally vague property E of n, is then obviously no longer meaningful for a choice seguence v, but it remains meaningful only for a sequence determined by a law in infinitum. Now, if C’(v) is a character, how should the following statement be interpreted: “Every sequence that possesses the property E also has the character C'”? As we know, the expression “every sequence” can only mean every sequence

developing through free choice; yet the property E cannot be meaningfully attrib-

uted to a choice sequence, but only to a finished sequence determined in infinitum. Only the following interpretation is possible: If C(n; v) is satisfied for every n up to a certain v-dependent limit, then so i1s C'(»). Or, more precisely: Let C*(n, v) signify the relation obtaining between n and v if C(n; v) holds for all numbers m from 1 to n; then there is a functio mixta f{») such that every choice sequence of the character C*( f(v); ) is also of the character C'(v). We are thus dealing with a statement of type (III). If, however, the expression “every” in the specification of a property E of kind (II) does not, as in this case, refer to “natural number” but to “sequence,” then an analogous interpretation is, given the nature of the subject matter, not possible. However, the concepts of property and set only have a mathe-

matical meaning as far as the identity principle reaches, that is, as far as one can

associate a sense with the staiement “every element of property E has the property E' (E and E' being any two properties).” Statements of form (II) consequently only

give rise to sets if the expression “every” occurs merely in conjunction with that of

“natural number.” We shall thus specify: If e denotes an arbitrary number or an arbitrary sequence, and C{e; n) a character that, apart from e, is also dependent on the number x, then C gives rise to a “vague set” [C]. The statement that an element e belongs to [C] means that every number » stands in the relation C(e; n) to e (set of type II). According to the above instruction, we are thus able to fix the sense of the term “subset” for sets of type (I) and (I1). In the more complicated case of two sets

of sequences falling under type (II}, it would have the following sense: Let “» belongs to [C]” mean: every number n stands in refation C(v; n) to p;

Let “v belongs to [C']” mean: every number n stands in relation C'(»; n} to v;

“[C’} is a subset of [C]” means: there is a functio mixta f(n; ) such that the following judgment instruction is justified: each number n and each sequence v developing through free acts of choice that stands in relation C(»; m) to all numbers m from 1 to f(n; v) together satisfy the relation C(3; n). The statement 1s thus one of form (1II). The following syllogism, the transitive law, holds for the concept of

a subset; If M is a subset of M', and M’ a subset of M”, then M 1s a subset of M". We now turn to the properties of the third kind containing a “there is” in their specification. Let us, for example, take a property of the following structure. Let

C{e, e’} be a relation in which we have that either ¢ is a number and ¢’ is a num-

ber, or ¢ is a sequence and ¢’ is a number, or ¢ is a sequence and ¢’ is a sequence; and let “e possesses the property (C)’ mean that: there is a function f(¢) such that

On the New Foundational Crisis of Mathematics

109

C(e, f(e)) holds. Depending on the three cases to be considered, this function will naturally be a functio discreta, mixta, or continua. What, if (C') is a property of the same structure, is the meaning of the statement: “Every element e of property (C)

also possesses property (C')” or “(C) is a subset of (C'y’? Obviously this: There is a law that from every function f generates a function ' in such a way that if (e, f(e)) holds, then so does C'(e, f'{e)); the generating law itself can in this case still depend on e. However, one can only talk of a law of this kind if fis a functio discreta; the place of the arbitrary function fis in this case taken up by the sequence

developing through free choices. We will thus apply the term set only io those prop-

erties of type (IIT) where the “there is” occurs in conjunction with “sequence” but not with functio mixta or functio continua. The following is hence the typical form

of specifications of sets of the third type: Let E(e, nv) be a relationship of type (1) (relation) or a relationship of type (II) to which corresponds a set, that is, a rela-

tionship 1n whose definition the “every” occurs only in conjunction with “number,” and not with “sequence”; ¢ can denote a number or a sequence; n denotes an arbi-

trary natural number and » a sequence. Let “e possesses a propeity (E), or belongs

to the set (£)” mean that there is a number n and a sequence » such that the rela-

tionship E(e; nv) obtains. The general formulation of the concept of subset for sets of type (I), (II), and (III) is left to the reader. The syllogism turns out to be justified in all cases. These are the boundaries laid down by the identity principle within which ex-

tensionally vague properties of numbers and sequences are also to be referred to as sets. Sets of functions and sets of sets, however, shall be wholly banished from our minds. There is therefore no place 1n our analysis for a general set theory, as littie

as there is room for general statements about functions.” The new conception, as one can see, brings with it far-reaching restrictions with

respect to the generality that enthusiastically leads into vagueness and that we have become used to through traditional analysis. We must learn again to be modest.

With the intention to storm the heavens we merely piled up mists upon mists, un-

able to support anyone who was seriously trying to stand on them. What remained tenable might, at first glance, appear so insignificant that the possibility of analysis could generally be put into question. This pessimism, however, is ill-founded, as

will be shown in the next section. Yet we must firmly and with all energy hold on

to the fact that mathematics is through and through, even as concerns the logical forms in which it moves, dependent on the nature of natural numbers.

As far as I understand, I no longer completely concur with Brouwer 1n the rad-

ical conclusions drawn here. After all, he immediately begins? with a general theory of functions (the name “set” is used by him to refer to what I call here functio continua), he looks at properties of functions, properties of properties, etc., and apphies the identity principle to them. (I am unable to find a sense for many of his statements.) From Brouwer I borrowed (1) the basis that 1s essential In every respect, namely, the idea of the developing sequence and the doubt in the principium tertii exclusi, and (2) the concept of the functio continua. [ am responsible for the concept of the functio mixta and the conception I summarize in the following three theses: (1) The concept of a sequence alternates, according to the logical connection in which it occurs, between “law” and “choice,” that 1s, between “Being” and “Becoming”’; (2) universal and existential theorems are not judgments in the proper

sense; they do not make a claim about a state of affairs, but they are judgment in-

Hermann Weyl

110

structions and judgment abstracts, respectively; (3) arithmetic and analysis merely contain general statements about numbers and freely developing sequences; there 1s no general theory of functions or sets of independent content!

Having finished this accounting for the logical structuring of the science of the

infinite, we draw in the next section the consequences for the continuum problemn. §4

The Continuum

Traditionaliy, one has used several specifications of the concept of real numbers, the equivalence of which one thought could be proven. From our present standpoint,

however, these definitions no longer appear as equivalent, and one can easily convince oneself that it is not Dedekind’s cut but rather the definition introduced at the beginning of Part II that now remains the only possible one (and of which one can

surely say that it also, in itself, expresses more purely the nature of real numbers).

We distinguished dual intervals from one another by giving two integer characters [m; #]. They can easily be replaced by a single character that is a natural number by ordering the pairs of integers with a simple law in a denumerated sequence. Moreoever, if { is any dual interval, it is possible to order naturally those dual intervals lying in the inside of i in a denumerated sequence. If i is of the Ath level, we put in the first place the unique dual interval of level (2 + 1) that lies completely in the inside of i. Then we put the 5 such intervals of level (b + 2), according to

how they follow on the number line from left to right, then the 13 intervals of level (h + 3), etc. We thus know what it means if we talk of “the nth” dual interval inside of i. A real number is determined by a law that from every natural number n

generates a dual interval i, such that /") always lies within /9. If, in this context, we wish to make ourselves independent of the nesting condition, we replace

the reference to the (n + 1)st interval by a reference to its number in the order of the dual intervals inside i*”. The real number is then determined by giving the first interval i and by this sequence of ordering numbers, that is, by a law that from every

natural number n generates a natural number ns. The interval sequence begins with i' = i, followed by the (1+)th of the dual intervals lying within ', referred to as i,

and this by the (2+)th one lying within i”, referred to as {”, etc. The “indeterminate real number,” or “real variable,” is represented by a developing sequence of dual intervals in which the intervals are successively freely chosen apart from the restriction that the interval to be picked in the following step must always be within the last chosen one. If one wishes to replace this choice bound by a stipulation by a completely free one, one must choose not the intervals, but the oidering numbers as in the above case. Interval sequence is from now on always to be understood as a sequence of nested dual intervals.

Two real numbers a, 3 coincide if i (i.e., the nth interval of the sequence a) and the nth interval of the sequence S8 always overlap one another wholly or par-

tially; they are disjoins [liegen getrennt] if there is a natural number n such that i§” and .r,'g‘) are completely disjoint. These two possibilities do not form a complete al-

ternative; after all, none of the two constitutes a definite relation between the arbi-

trary real numbers « and 8. This is completely appropnate for the character of the

intuitive continuum where, in the process of moving closer together, the separatedness of two places turns, so to speak, gradually, or1n vague gradations, into indis-

On the New Foundational Crisis of Mathematics

111

‘tinguishability. What does hold, however, is the following proposition: If & coin-

cides with 8 and S with v, then « also coincides with y. Admittedly, of the three

intervals {7, ig”, igf), where the first two and the last two overlap, the first and the last one need not overlap. Yet, if this occurs at a certain place » of our interval se-

quences, it will have to be that in the further continuation of their development ei-

ther the interval sequences « and 3, or the sequences 8 and v become separated

from one another. Explicitly formulated, our theorem claims: There is a function Jin; affy) of three developing choice sequences a, 3, v that is dependent on the natural number # and generates a certain natural number m from the arguments when-

ever iJY and {$”, on the one hand, and % and i, on the other, overlap one an-

other, yet i and " are disjoint; and this holdsin such

way that, however the

interval sequences fl.’ £. vy may develop beyond the nth place, we have at the mth

place that either i{ is disjoint from i, or :'J‘ém} is disjoint from {¥TM. The construction of this function f naturally is very simple. The theorem in question, as one immediately realizes, is not based on the fact that «, 8, y are “approximation numbers” [Ndherungszahlen], but that the approximation can be pushed beyond every lrmit. Insofar as this is not true in some intuitively given continuum (take, say, the well-known example of localizing pressure experiences induced by touching the

hand’s surface with the points of a pair of compasses), the “transitivity” of coinci-

dence reveals the mathematical idealization carried out on reality. The continuum appears here as something that develops inwardly in infinitum [ein nach innen hinein ins Unendliche Werdendes). In the intuitively given reality, the developing process has only progressed to a certain point (because what is given is, it does not develop

[es wird nicht]), and, on top of this, it gradually leads into complete unseparatedness. In mathematics, by contrast, we regard this developing process as progressing in infinitum. It is, however, in any case nonsense to regard the continuum as a fin-

ished being [ein Fertig-Seiendes]. One can thus in all seriousness (indeed, one must)

claim that what is present [das Gegenwdrtige] 1s not something that is in itself finished and determined, but something that still develops inwardly [nach innen hinein wird] by enfolding itself in the future. Every piece of the world reality, including the one I am currently living through, is, so to speak, only precisely determined in

itself at the “end of all times.” This point seems to me very important for the eval-

uation of the metaphysical meaning of the causality of nature, but this is not the place to elaborate on this.

As we have seen before, if we pick out a specific point, say, x = 0, on the num-

ber line C (i.e.; on the variable range of a real variable x), then one cannot, under any circumstance, claim that every point either coincides with it or is disjoint from it. The point x = 0 thus does not at all split the continuum C into two parts U : x
0, in the sense that C would consist of the union of C~, C" and the one point O (i.e., that every point would either coincide with 0 or belong to €TM or belong to C*). If this appears offensive to present-day mathematicians with their

atomistic thought habits, it was in earlier times a self-evident view held by everyone; Within a continuum, one can very well generate subcontinua by introducing boundaries; yet it is irrational to claim that the total continuum is made up of the boundaries and the subcontinua. The point is, @ genuine continuum 1s something connected in itself, and it cannot be divided into separate fragments; this conflicts with iis nature. C* is a continuum in the same sense as C: a medium of free Be-

Mermann Weyl

112

coming. This means that its mathematical treatment must also start out with intervals and not with points. It is based on the system 2,* of those dual intervals whose first characteristic m is positive. A law, which from every natural number generates an interval of this system such that the intervals of the sequence are nested, produces a certain number in the continuum C*. Choice acts that are tied to the sys-

tem %% and the nesting condition, but that otherwise are free, generate a developing sequence representing the “variable ranging over the domain C . Here one

becomes aware that “point sets” that are to be considered as variable ranges for arguments of functions can always only be disguises of “interval sets,” more precisely of definite sets of intervals. Moreover, these point sets are also the only ones for which a general theory is possible within analysis, because they fall under the heading of functiones discretae. Apart from the system 2" we also encountered the sys-

tem S~ of the dual intervals with a first characteristic m < 0, and a third system 29 of the dual intervals characterized by m = 0. X7, 27, and 3" determine the con-

tinua of the points that “lie to the right of 0,” of the points that “lie to the left of

0,” and of the points that “coincide with 0,” respectively.

Let us now look at a simple operation applicable to a real variable x, for example, x2. From finitely many dual fractions @, a’, . . . one can easily construct the single dual interval of highest level that contains all those dual fractions; we are denoting this interval by (a, &', .. ). If

a:

m— 1

and

o

a

, _m+t 1 =

3

are the endpoints of an interval ¢, then we have that the squares of ail dual fractions falling within ¢ fall themselves within the interval

i2={a% aa,a? If a real number « is given by a sequence of nested binary intervals, then one gets

the number o by forming the “square interval” i? of every interval { of the sequence. The formation of a? out of « is thus not based on an assignment of inferval sequences, but simply on an assignment of infervals: We are dealing here with a law

that from every interval { generates the interval /%, we call this law “the function x2.” If a sequence of nested dual intervals i is, step by step, created through free choice, then there is, according to this law, a corresponding developing sequence

also of nested intervals %, In a similar way we specify the function x - y (the operation of multiplication) within the range of two variables x, y. At the basis of this variable range is the system of “dual squares”TM with endpoints

I:

m*1

gL

_nZxl

Y=

h

to be distinguished from one another by three integer characteristics m, n; A. Putting m— 1

,

m+1

a =

_n—1

b=

2'}3'-'

, _nt+1

b’ =

2k

On the New Foundational Crisis of Mathematics

113

one thus generates from this square J the interval

w(J) = (ab, a'b, ab’, a'b") This law 7 is the function x - y. If J runs through a developing sequence of nested squares, then #r(J) runs through a developing sequence of nested intervals.

Let us now finally interpret the identity

x+y)(x—y)=x—

(*)

valid In the range of two variables x, y. Let us refer to a pair of dual fractions as

“the point of intersection of a with b.” If a, &', . . . are several (for example, three) given dual fractions, and if, apart from this, a second sequence of finitely many (e.g., four) dual fractions b, &', . .. is given, then we can form the smallest dual square contamning all the (3 - 4) points of intersection of @, ', ... with b, b’, . . .:

J=(ad ...|bb .. ) The function

y =x—y

x =x+ty,

* S

I

W

T

»

b

{

.

b 1t

J)

a

a'

a”

Figure 2

is the law that [Figure 2], from every dual square J with corners that are the points of intersection of a, a’ with b, b’, generates the square

I=@(@+ba +tba+tb,ad+b|a—ba —ba—b,a—b) From this the interval i is to be formed according to the law x” - ¥' (i.e. the one we have just referred to as 7). With this we have constructed the left-hand side of the equation (*). Analogously for the right-hand side: first one forms the square

J2 = (a2 aa', a’® | b2 bb', b'%)

Hermann Weyl

114

out of J (this is the function x” = x%, ¥" = y°) and from this the interval : according to the law x” — y”. The claim of the equation (*) is that, whatever the dual square J may be, i and 1 will always overlap.

These examples suggest the general concept of a continuous function of a real variable. Such a function is certainly not determined by some arbitrary law assigning a developing interval sequence, but by a law according to which each dual interval (once it has become sufficiently small) generates an interval. This also corresponds completely to the sense in which this concept is used in the applications of mathematics: As soon as the argument is given with a certain degree of accuracy—and in applications it is never given in any other way—the value of the function is also known with a corresponding degree of accuracy. The latter sinks with the former below every boundary (if the function is investigated in a bounded interval). Continuous functions are thus merely “functiones discretae” in disguise, which is why a general theory of them can be given in analysis. A continuous function, we have said, is determined by a law ¢ that from every dual interval { either generates nothing, or a dual interval (7). This furthermore calls for a law generating from every dual interval i a natural number #; of the following kind: If { is any dual interval, and # any number = n;, then the nth one of those dual intervals in the inside of i certainly generates, according to the law ¢, an interval (and not nothing), an interval that, which is more, will be in the inside of @{(i), if ¢(f) exists. This nesting condition has the consequence that two overlapping intervals i, i’ will always correspond to two overlapping intervals ¢(f), ¢(i'); for-according to this condition,

one can always construct a dual interval 1 contained in the inside of both intervals i, i’ with an image (1) that exists and lies in the inside of both ¢(i) and ¢(i’).

If « is a single real number (i.e., a nested sequence of intervals i, i', i, . .") deter-

mined in infinitum by a law, we form the sequence ¢(i), ¢(i'), ¢(i"), . . . . Naturally, the nonexisting image intervals can be left out, and furthermore we shall cross out any interval that is not included in the immediately preceding one. Due to the number n;, which belongs to every interval i, this still leaves an infinite sequence. The image sequence thus prepared is hence again a real number 8 = ¢(«): the value of

the continuous function for the argument value . If the two numbers « and a’ coincide, then so do the corresponding functional values 8 and B8’'. Two continuous functions coincide, 1if they are determined by laws that assign to every dual inter-

val two overlapping intervals.’ One can see that the concept of a continuous function in a bounded interval cannot be defined without simultaneously including uniform continuity and boundedness in the definition. Above all, however, there cannot be any functions in a continuum other than continuwous functions. The fact that in the old analysis it was possible to form discontinucus functions shows most clearly how far it is removed from

grasping the nature of the continuum. What nowadays is called a discontinuous function actually consists of several functions in separated continua (and this again is basically nothing but a return to older intuitions). Let us, for example, look at the

above-mentioned continua C, C* (x > 0) and C~ (x < 0). The function f;(x) = x in C* is the law that assigns to every dual interval with positive endpoints that same interval. The function f>(x) = —x in CTM is the law that assigns to every dual inter-

val { with negative endpoints a, a’ the interval : = (—a’, —a). For these two func-

On the New Foundational Crisis of Mathematics

115

tions there is a single function |x| in all of C that in C* coincides with f,, and in C~ with f3; it assigns to a dual interval i the interval i if both endpoints of i are posi-

tive, —i, if both endpoints are negative, and the interval (—a’, —a, a, a’) if i contains the point 0. If, however, we take the two functions +1 in C*, —1 in C, then there 1s no function defined in the whole of C that would coincide with the one in

C* and with the other in C~. In traditional analysis, the continuum appeared as the set of its points; it was considered merely as a special case of the basic logical relationship of element and

set. Who would not have already noticed that, up to now, there was no place in

mathematics for the equally fundamental relationship of part and whole? The fact, however, that it has parts, is a fundamental property of the continuum; and so (in

harmony with intnition, so drastically offended against by todays “atomism’) this relationship 1s taken as the basis for the mathematical treatment of the continuum by Brouwer’s theory. This is the real reason why the method used in delimiting subcontinua and in forming continuous functions starts out from intervals and not points

as the primary elements of construction. Admittedly a set also has parts. Yet what distinguishes the parts of sets in the realm of the “divisible” is the existence of “elements” in the set-theoretical sense, that is, the existence of parts that themselves

do not contain any further parts. And indeed, every part contains at least one “element.” In contrast, it is part of the nature of the continuum that every part of it can

be further divided without limitation. The concept of a point must be seen as an idea of a limit [Grenzidee], “point” is the 1dea of the limir of a division extending in in-

finitum. To represent the continuous connection of the points, traditional analysis, given its shattering of the continuum into a set of isolated points, had to have recourse to the concept of a neighborhood. Yet, because the concept of a continuous manifold remained mathematically sterile in the resulting generality, it became nec-

essary to introduce the possibility of “triangulation” as a restrictive condition.® In contrast to this construction it became already clearly apparent in the short introductory explanations to Brouwer’s famous proofs of the fundamental theorems

of the analysis situs’ that the simply connected pieces that make up the manifold are the originally given building blocks. This is the only route open for the new analysis.

Let us briefly sketch how, according to this, the concept of a two-dimensional closed manifold is formed. We must first of all give the schema of iis topological

structure, which 1 shall refer to as a “two-dimensional frameworkTM [zweidimensionales Geriist]. It consists of finitely many “comers” ep (elements of level 0), “edges” e; (elements of first level), “surface pieces” e; {elements of second level), which may be denoted by any arbitrary symbols. Each sutface piece is “limited” by certain edges, and each edge by certain comers; these details make up the essential content of the schema. It has to satisfy certain requirements, which can easily be stated. From the surface pieces of the framework one arrives at the points of the manifold by a process of division, which is to be repeated infinitely many times. Let us carry out this process by dividing every edge into two edges by means of one of their points, and then divide each surface piece into triangles by means of

lines from a center that is arbitrarily chosen within it to the corners of its boundary [Figure 3]. This process can in abstracto be described in the following manner: To

116

Hermann Weyl

epleoy (Epe); (e

Figure 3

every element e; of the original framework G there corresponds an element (e;)o of level O of the framework G’ arising from the division; two elements e;, ¢ (i > k) of the original framework, of which one delimits the other, generate an element of first level (¢; ep)} of the new framework G’ delimited by (e)p and (eg)q; three elements e, ¢; ¢ of G’ that delimit one another generate a second-level element (e; e1 eo)s of G’ limited by (€3 €)1, (e2 ep)l, (€1 €0)1. The surface pieces of G, and the ones generated through successive divisions of G', G7, . . ., play the same role here as the one played by those intervals of the linear continuum ansing from its division by dual fractions of the form m/2, m/2%, mi2%, ... (where m runs through all

whole numbers). Of these intervals we joined together any two adjoining ones to form a “dual interval” in order to get at every level of division a covering of the linear continuum by means of overlapping pieces. Analogously we now join together to form a “star” those surface pieces of any one of the frameworks G, G, G", ... generated by division, and which are delimited by a common corner. By a point of the manifold we are now to understand an infinite sequence of such stars such that each star is fully contained in the inside of the immediately preceding one. The sense of this nesting condition between two stars can be easily formulated. An open manifold is different from a closed one only in that the underiying framework does not consist of finitely many, but rather of an infinite sequence of, elements. The linear continuum that has already been discussed falls under this con-

cept if we accept as dual intervals only the intervals ((m — 1)/2%, (m + 1)/2")) with a positive characteristic A. This modification can be applied without any problem to all our developments so far. The concept of a continuous function has already been formulated in such a way that it can be transferred to any manmifold: A continuous mapping of one manifold onto another one i1s determined by a law that assigns to each star of the first one either nothing or a star of the second one; added to this is

the same nesting condition as earlier on. Now here, it is genuinely essential that the alternative of “nothing” is kept open, since it need after all not be that the image region of a star of the first manifold fits into a single star of the second one. As soon as we are dealing with a vanable ranging over a continuum of any

kind, we must, so to speak, hover above the continuum, and 1if we operate according to the new theory, we do not have the possibility, as before, to settle down on

On the New Foundational Crisis of Mathematics

117

an individual, even though arbitrary, point. Those who are used to the latter method may initially find this demand to be uncomfortable, but everyone will again feel

how truly the new analysis conforms to the intuitive character of the continuam. Brouwer’s view ties together the highest intuitive clarity with freedom. It must have

the effect of a deliverance from a nightmare for whoever has maintained any sense for intuitively given facts in the abstract formalism of mathematics. Finally, let me

point out how perfectly both parts of the new theory, the intuitive conformity to the continuum and its logical position concerning the universal and existential state-

ments, interlock by necessitating one another.

Notes 1. See, in particular, the dissertation Over de grondslagen der wiskunde (Amsterdam,

1907), the essay in the Tijdschrifi voor wijsbegeerte 2, pp. 152-58 (1908), the talk “Intuitionism and Formalism,” published in the Bull. Amer. Math. Soc. 20, pp. 81-96 (1913), the treatises “Begriindung der Mengenlehre unabhéngig vom logischen Satz vom ausgeschlossenen Dritten,” Verh. K, Akad. Wetensch. Amsterdam, 1918, 1919,

and the article “Intuitionistische Mengenlehre,” Jber. dtsch. math.-Vereinigung 1919, pp. 203-8. 2. Cf. also my essay “Der Circulus vitiousus in der heutigen Begriindung der Analysis,” Jber. disch. math.-Vereinigung 1919, pp. 85-92. 3. I do not wish to say with this that general statements about sets and functions (mixtac et continuae} are generally impossible. Thus it is certainly true that for every sequence » and every fm. that f(v) + 1 = 1 + f(»). Yet the generality of this statement is derived; it 1s gained through formal specialization from the generality of arithmetic

and analysis (the generality in the above example is based on the validity of the equation n + 1 = 1 + n for all numbers x). The gencrality of arithmetic and analysis, in

contrast, is a truly original one; based in each case on a distinct intuitive foundation (cf. pp. 159-60), and thus also filled with independent intuitive content. One may collect this kind of propositions about functions and sets (isolated points of support in an otherwise boundless ocean) to form a special discipline under the name of “set

theory,” but it in no way constitutes the foundation of mathematics. Analogously one can naturally also establish isolated particular classes of assignments between func-

tiones mixtae {or continuae). If, say, ¢ is a given f.d., then every f(») gives rise to

another.f'(»} according to the following rule: f'(») for v = {ny, ny, n3, ...} 1s equal to f(¢'), where ¥ = {©(n)), ¢(n2), ¢(ns), ...}. But this assignment is only a “disguise” of the sequence @, where it is said that “there is a mapping of this kind,” 1t is meant that: “there i1s a sequence ¢.” We will very soon get to know examples for this.

4. In the first of the above-quoted treatises on the “Begrindung der Mengenlehre unabhingig vom logischen Satz vom ausgeschlossenen Dritten,” Verh. K. Akad. Wetensch. Amsterdam, 1918, 1919,

5. We have talked here about the individual determined continuous function. The general theorems about them, however, are dealing with the continuum within which they are embedded, that is, with the arbitrary continuous function (to be regarded as

a developing one). To discuss this concept more thoroughly would, however, lead

too far at this point.

Hermann Weyl

118

6. Cf., for example H. Weyl, Die Idee der Riemannischen Fliche (Leipzig 1913), §4. 7. Cf.. above all, L. E. J. Brouwer, Math. Ann. 71, p. 97 (1912). Translator’s Notes

a. Even though [ shall translate Weyl's “WerdenTM as “becoming,” I decided to use the translation “(freely) developing (choice) sequence” as opposed 10 the more cumbersome “(freely) becoming (choice) sequence,” for Weyl’s “(frei) werdende { Wah!-) F{‘]fge-!'l

8 Comments on Weyl 1921 LUITZEN EGBERTUS JAN BROUWER*

1

Marginal Notes on Weyl 1921

(p. 87, lines 28 -29)

“neither by chains of reasoning”

(p. 98, lines 18-30) “Hilbert’s Theorem of Invariants. Without a method of calculation there is only the certainty that this scale does not

contain [...] This kind of existence theorem does not appear to be discussed here at all.” (p. 103, lines 7-9)

“this rule certainly falls within my notion of Set.”

(p. 105, knes 21-24) “this—from

my

point

of view—is

unnecessary”

(p. 103, lines 5-2 from below) “From my point of view this restriction 1s unnecessary; it excludes discontinuous

functions wholly unnecessarily.” (p. 106, lines 14-16)

“for me again an unnecessary restriction” *In Brouwer’s scientific estate there are three documents in which he comments on Weyl’s “Uber die Neue Grundlagenkrise der MathematikTM (Weyl 1921):

1. A copy of Weyl's typescript of Weyl 1921 with some added comments by Brouwer in the margin, some in Dutch some in German. 2. A (very) rough draft of a letter of Brouwer to Weyl (in German), most likely written after receipt of the MS and before the publication in 1921. This letter is now published almost in its entirety in van Dalen 1995,

3. A short, handwritten note by Brouwer (in German) in which he comments on pp. 109-110 of the publication Weyl 1921.

References to pages in the MS have been changed in this translation to the page numbers of Chapter 7. Translated from the Dutch and the German by Walter P. van Stigt. 119

Luitzen Egbertus Jan Brouwer

120 (p. 106, line 27) “former?”’

{(p. 106, lines 29-30)

“except to confute theorems like the Dirichlet Principle a la Hilbert.”

(p. 107, lines 26-28)

“hut not if one does not know whether or not the element in question belongs to the

numbers or sequences.”

~

{(p. 108, line 2 from below—p. 109, line 3}

“to me the ‘becoming sequence’ is neither the one nor the other; one should consider the sequence from the point of view of the powerless observer who does not know in how far the generating Subject has been free.” (p. 109, line 2 from below—p. 110, line 2) “I wholly disagree with (2} and (3).” (p. 114, lines 12-16 from below)

“Better to say: the function is not everywhere defined. But the domain of definition of a function needs to be ‘spliitable’ as is shown by the case of pointwise discontinuous functions.”

(p. 1135, lines 3-6)

“Very true. Underline! because this is the vital point”

2

Draift of a Letter by Brouwer to Weyl (Spring 1920)

Your unreserved support has been a source of infinite joy. Reading your manuscript 1s a constant pleasure. Your exposition to me seems to be clear and convincing, The fact that we dis-

agree on some minor points can only have a stimulating effect on the reader. Of course, you are fully entitled to formulating these differences of opinion. [For example] in your restrictions of the object of mathematics you are in fact much more radical than I am. These are not matters of discussion; they can only be a matter of

decision through individual concentration. But following your explanation of the notion “continuous function,” I would like to draw your attention to my notion of “a fully defined function of the continuum,” By this I understand a law that to every point of a point species in local agreement with the continuum assigns a further

point of the continuum. Such a function may well be discontinuous without in any way being the product of combining continuous functions on separate continua. In fact one can work with these functions in various ways (e.g., one can integrate them in certain cases without knowing whether they are continuous or discontinuous.) Apart from our points of disagreement, I would like to make the following com-

ment: Nonexistence theorems. In your listing of mathematical judgements you do not mention nonexistence theorems (to which, e.g., belong the “power theorems”

Comments on Weyl

121

on pp. 13 and 43 of Part One of my paper (1918A) and also Hilbert’s theorem on

the finiteness of the full invariant system in its first derivation. On p. 87, line 31 and similarly on p. 13 of Das Kontinuum, the meaning of the word Sachkenntnisse [factual knowledge] is not clear to me.

The final paragraph of part 2 (p. 106) seems to me to endanger the whole purpose of your paper. The reader, having just been roused from his sleep, then says

to himself: “Ah, therefore Weyl admits that real mathematical theorems are not af-

fected by his arguments. I shall therefore not bother any further!” And he turns round and goes back to sleep. But by saying this you do less than justice to our

cause, because together with the existence theorem of the condensation point of an infinite point set many of the classical theories of minimal functions (such as the

existence theorem of the geodesic line [without the] conditions of differentiability) lose the basis on which they were built. Your assertion on p. 108§, lines 19-22—which, as you know, conflicts with my view—requires, I believe, some further explanation. I think that the reader who has

followed you so far correctly will also have difficulty with this passage. As to your functio discreta and functio mixta, it seems to me that these are con-

tained in my notion of Set just as much as your functio continua. My set law can

very well provide in advance the certainty for every choice sequence that, after a sign has been generated, further always nothing will be generated. Einstein, who will give his inaugural address [Antrittsrede] in Leiden on June

4, tells me that you will definitely accept ¢ither Berlin or Géttingen. I am terribly interested in your decision whether you choose Goéttingen, Berlin, or Ziirich. I hope you will see things clearly and take the right decision. It will not be easy. I assume I can keep the copy of the MS you sent me. You need not send any-

thing back to me. Some of my older papers are being reprinted; please let me know

of which of the following papers you hold a copy:

1. Imtuitionisme en Formalisme (Dutch); 2. Intuitionism and Formalism (English)

3. De onbetrouwbaarheid der logische principes (Dutch); 4. Het wezen der meetkande (Dutch).

I can now supply any copy you might not have.

Again my sincere thanks for the joy and satisfaction your paper has given me. Best wishes also to your wife and auf Wiedersehen. Yours, Egbertus Brouwer

3

Note on Weyl 1921

Weyl [1921] p. 109 speaks for itself

1. Functio Mixta. But this is a special case of the functio continua (my “Set”); in my papers this specialization is explicitly mentioned in the proof that the set C is greater than the set A.

Luitzen Egbertus Jan Brouwer

122

2 Distinction between “law” and “choice.” This is also found throughout my work (passim), where 1 make a sharp distinction between “‘fundamental seguences” and “indefinite sequences.”

3. Referring to general and existence theorems as, respectively, “judgment instructions” and “judgment abstracts.” This 1s only a matier of name and certainly does not reflect any lacking insight on my part.

4, Limiting arithmetic and analysis to general statements about numbers and free-becoming sequences. This restriction of mathematics to mathematical entities and species of the lowest order is totally unjustified. This clearly refers to p. 109 [line 25], where he dismisses my theory of species as meaningless, and it shows that in the end Weyl only half understands what Intuitionism 18 about.

9 The Current Epistemological Situation in Mathematics HERMANN WEY[*

1

From Anaxagoras to Dedekind

Mathematics is the science of the infinite. To have made fertile the tension between

the finite and the infinite for knowing reality is the great achievement of the Greeks. The Onent has the feeling, the calm and unquestioning acceptance of infinity; yet for it, the infinite remains a merely abstract awareness that indifferently leaves the concrete diversity of what exists [das Dasein]—unstructured and unpenetrated—by

the wayside. Coming {from the Orient, the religious feeling of the infinite, of the

amewpov, takes hold of the Greek soul in the Dionysic-Orphic period that precedes the Persian wars. Here again the Persian wars mean a severance of Occident from

Orient. Now this tension and its resolution become for the Greeks the driving motive for knowledge. Yet every synthesis, having barely been achieved, brings about the old conflict in a new, more profound sense. And in this sense it determines the history of theoretical knowledge up to the present day.

Anaxagoras is the first to give a version of the concept of infinity through which it was able to influence science. A fragment of his that has come down to us states: “Neither is there a smallest part of what is small, but there is always a smaller (for it is impossible that what 1s should cease to be (through division, no matter how far

it is carried out)).”* The fragment is about space or body. He claims that the con-

tinnum cannot be composed of discrete elements that “are separated from one an-

other as if cut off with an axe.”® Space is not only infinite in the sense that one does not reach an end anywhere in it, but also in that at each place it is, so to speak, in-

wardly infinite. It is only through a process of division proceeding ad infinitum that

a point can be determined, step by step, more and more precisely. This is in contrast to the finished, calm existence of space as given to intuition. Space 1s the principle of separation for any guale filling it, actually creating the possibility of a di-

versity of what is qualitative. Yet (it is the principle of) separation as well as contact,

that is, of continuous connection, such that no piece can be separated from another

one “as if cut off with an axe.” The mathematical significance of Anaxagoras’ in*“Die heutige Erkenntnislage in der Mathematik,” Sympesion 1, 1925-1927, pp. 1-32. Translated from the German by Benito Miiller. 123

124

Hermann Weyl

finitesimal principle becomes evident in his solution of the “squaring of the circle,” that is, by his proof that the area of a circle 18 proportional to the square or the radius.

Rising up against Anaxagoras is Democritus’ strictly atomistic theory. One of his arguments against the unlimited divisibility of bodies runs along the following line: “One says that the division is possible; well, let it be done. The division Is possible in infinitum; assume the possible has occurred. What remains? Not bodies; for they would be further divisible, and the dissection would not have been carried out to the end. It could only be points, and bodies would have to consist of points, which

obviously is absurd.” The difficulty the continuum poses to thought appears in somewhat different form in the well-known paradoxes of Xeno about the race between Achilles and the turtle. Aristotle comments (Physics, Chap. VIII): “In the act of dividing the continuous distance into two halves one point 1s treated as two, since we make it a starting point and a finishing point. [ . .] But if divisions are made in this way, neither the distance nor the motion will be continuous . . . and though what s continuous contains an infinite number of halves, they are not actual but potential halves.” Tt is well known how these antinomies, almost untouched by the development of mathematics and losing, rather than gaining, in the precision of formulation, continue to have an effect in the more recent philosophy and how they play

a decisive role in the foundation of epistemological idealism. Leibniz—not to mention lesser figures such as Bayle and Collier—thus testifies that it was the desire to find a way out of the “labyrinth of the continuum” that first led him to the view of space and time as the order of phenomena. Even in the Kantian system these antinomies still occur in a prominent place as the two first antinomies of pure reason. We shall return to this later.

.

In the pure geometry of the Greeks as handed down to us by Euclid in his “Elements” [—a geometry] operating with ideal entities [—] a line segment a cannot only undoubtably be bisected continually, but with it there always exasts, and can be gained constructively from it, the line segment that is to a like 5:3 or like any two natural numbers m:n. The discovery of the irrational follows; spatial magnitudes, such as the side and the diagonal of a square, which do not stand in a rational proportion to one another and which do not have a common measure at all, are exhibited. With this, the atomistic theory has obviously become impossible for space. In the Platonic dialogues one feels the deep impression this discovery made on the emerging scientific consciousness of that time. Eudoxus recogmzed the general foundations of the phenomenon independently of the particular geometrical con-

structions that initially yielded some single irrationalities like V2. (1) In the place

of commensurability, which has become untenable, he puts forth the following axiom: If a and b are any two line segments, then it is always possible to add a so often to itself, say n times, until the sum #-a has become greater than b. This means that all line segments are, amongst one another, of comparable order of magnitude [Grifenordnung], that there is neither something actual infinitely small, nor something actual infinitely large in the continuum. For [ would call infinitely small with respect to b a line segment a that would remain smaller than b no matter how often [ add it to itself. (2) If it is not possible, in general, to express a ratio between

line segments by means of a fraction such as %, in what manner can it be done? Eu-

doxus replies: Two ratios between line-segment ratios, say a:b and a":b’, are equal

The Current Epistemological Situation in Mathematics

125

to one another, if for arbitrary natural numbers m and # that satisfy the relation given

in the first line [of the following schema], the relation set below it in the second line also always obtains: na > mb

()

na = mb

(II)

na' = mb’

na < mhb

(11D

na' = mb'

na’ The remedy is the axiomatic method, which, for decades, has been worked out in minutest detail and put to the test by him in the most diverse areas of mathematics and physics. However, one should not take his promises too literally. For, despite the vehement polemics against the intuitiomstic Brouwer—Weylean standpoint, he too is completely convinced that the power of contentual thought does not reach further than is claimed by Brouwer, that it is incapable of supporting the “transfinite” modes of inferences of mathematics, and that there is no justification for all the

transfinite statements of mathematics qua contentual, understandable [einsichtig] rruths. What Hilbert wants to secure 18 not the truth, but the consistency of the old analysis. This would, at least, explain that historic phenomenon of the unanimity

amongst all the workers in the vineyard of analysis. To furnish the consistency proof, he has first of all to formalize mathemarics. In the same way 1n which the contentual meaning of concepts such as *“‘point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all

interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely

logical one. The statements become meaningless figures built up from signs. Mathematics 1s no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess. Corresponding to the

chess pieces we have a limited stock of signs in mathematics, and an arbitrary con-

The Current Epistemological Situation in Mathematics

137

figuration of the pieces on the board corresponds to the composition of a formula out of the signs. One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess. And in the same way in which here a configuration occurring in a game is transformed

into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or

“deduced,” from formulae. By a game-conforming [spielgerecht] configuration in

chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game. The analogue in mathematics is the

provable (or, better, the proven) formula, which follows from the axioms on grounds

of the inference rules. Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration in which there are 10 queens of the same color. Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess

players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game. Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in “metamathematics,” this game now becomes the object of knowledge: What is meant to be recognized is that a contradiction can never occur as an end formula of a proof. Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration. One can see this in the following way: The rules are teaching us that a move can never increase the sum of the

number of pawns and queens of one color. In the beginning this sum = 9, and thus—

here we carry out an intuitively finite [anschaulich-finit] inference through complete induction—it cannot be more than this value in any configuration of a game. It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds guite analogously to the one

just carried out for chess, although it is, obviously, much more complicated. It follows from our account that mathematics and logic must be formalized to-

gether. Mathematical logic, much scorned by philosophers, plays an indispensable role in this context. The sign o occurs symbolizing the one-place number operation

that from a number g of the sequence 1, 2, 3, ... generates the successive one oq,

and furthermore the one-place propositional operation [Aussagenoperation] that

transforms the proposition a into the proposition non a, symbolized by a.!® The twoplace number relation & = b with the familiar equality sign occurs, and also the two

place propositional relation @ — b (denying that a is true and at the same time b false; read: from a follows b). The sign Z means the property of being a number (i.e., of belonging to the sequence 0, 1, 2, .. .); Za is to be read as: a is a number. However, like o, we understand properties and relations as operations; the operation —, for example, generates from two statements g, b the new statement a — b.

Consequently, these signs could also be written in front of the elements to which the operations in question are meant to be applied. We have no scruples about acting as if these operations could be applied indifferently to all possible kinds of things. For if the use of their signs is consistent in this scope, then it will also be so in the more narrow one where we can attribute a contentual interpretation to that use. Yet the formalism is simplified tremendously by letting go of such restrictions.

Apart from the operation signs we still need two other Kinds of signs, namely, con-

Hermann Weyl

138

stants (such as Q) and variables (a, b, x, . . .). Like the pieces of chess, they differ

in virtue of the rules that are valid for them. What constitutes a formula 1s recursively specified: (a) every constant or variable by itself is a formula; (b) a new formula is formed out of one or two (or several) already created ones by writing down a one-place, or two-place (or many-place), operation sign, respectively, and by adding on the formulae in question. One can always decide whether a given combination of signs is a formula or not, provided the signs are written sutficiently clearly and their sequential order is clearly indicated. It is easy to give some examples of axioms; however, following the usual habut,

let us put the signs =, — again berween the elements, and the negation sign “—"

over the sign of the negated proposition. This means that the use of brackets can no longer be avoided.

b— (a— b) A valid sentence b remains valid if a superfluous premise ¢ is added. Formula of the syllogism. (b—>c)— ((la—b)y—>(a—c)) a— (a— b) Basis of indirect proofs. i —

Z0 Za — Z(oa)

(a = b) —> (ga = gb) It is more convenient to regard these not as axioms, but as schemas for the forma-

tion of axioms. An axiom is obtained if one substitutes the letters in one of these schemas by any formulae—naturally, the same formula 1s to be substituted for the same letter where it occurs in the schema. The rule of inference 1s: From two formulae a and a — # where the first formula stands to the left of the sign — in the

second one, the formula b is created. A contradiction is given if, of two proof games played through in concreto, one ends with a formula a and the other with the opposite a. As long as one remains within the domain of the axioms just described (even though only incompletely enumerated), the consistency proof can easily be

given. For it is possible, by means of a recursive procedure, to attribute to every formula in accordance with its genesis one of the “valuesTM true or false in such a way that all the axioms get the value true, such that a formula ¢ — & only gets the

value false if a is counted as true and b as false, and, finally, such that a becomes false or true depending on whether a is true or false. This leads to the recognition that: The syllogism, or the deductive method, remains completely powerless as long as the transfinite remains excluded, the truth or falsity of the premise ¢ — & is only ever decided after the sentence b has been evaluated. To introduce the transfinite modes of inference, we need a new type of sign.

If, for example, we use a property, or a propositional schema a(x) that contains a

variable or empty place x (such as this: the person x is corruptible) to create the proposition: All x satisfy the proposition a(x) (all people are corruptible), then this

is achieved through a certain logical operation that eliminates the variable x in the statement formula (nothing can be substituted for x anymore). Let us call such an

operation an integration with respect to x. In formalized mathematics a sign with the index x would correspond to it. The difficulty connected with the unrestricted

The Current Epistemological Situation in Mathematics

139

usage of the terms “there is” and “all” in contentual analysis is now formally overcome in the following way. If, in a first instance, one relies on [sich stiitzen ouf]

the old alternative, disputed by Brouwer, that either all people are corruptible or

that there is an incorruptible one, then let “AristidesTM be understood as referring to an arbitrarily picked out person in the first case, and one of the incorruptible ones,

in the second. According to Brouwer, however, we must generate constructively this Aristides from the property of incorruptibility. Let us thus fictitiously consider a “divine antomaton”: If a propositional formula a(x) with the free variable x is thrown

into 1t, then it indicates to us an individual 7,4, which, with respect to the property a, can serve as a proxy for all, on account of the fact that the following proposition

holds: If this individual has the property a, then they all have it. 7, is the sign for an integration with respect to x. If we had such an automaton at our disposal, then we would be relieved of all our troubles; but believing in its existence is naturally purest nonsense. Yet mathematics acts as if it did exist. This can be expressed in an axiom schema, and if this schema does not lead to an inconsistency, then its formation 1s legitimate in formalized mathematics. That schema is

“(a) =2 (s) T Ll

b

B

that is, take two formulae a, b, write to the left of the sign — that formula that arises from a if you replace all the occurrences of the variable x by the formula 74; on the right side, however, put the formula that arises from a by substituting b for x in the same way; let the formula so obtained be an axiom. The schema, of course, does not achieve the same as the antomaton, because it does not tell for a given formula

a what 7.a 15, 1t only says that a formula such as m,a = 0 can, under certain cir-

cumstances, occur as the end formula of a proof beginning with the axtoms. Hilbert

succeeded in proving the consistency even after the transfinite scheme (*) was incorporated amongst the axioms. However, a recursive classification of all formulae into “true” and “false” ones will no longer do. This means that the syllogism only

becomes fertile in connection with the transfinite; yet this connection carries us way

beyond the region accessible to intuitive insight Brouwer tried to delineate more precisely. However, the one transfinite axiom (*) is not sufficient; the free formation of sets and functions still requires a second one. The Russellian levels remain an obstacle to setting up set theory in the full Cantorian generality, yet the destiny of analysis is not that closely tied to that of set theory. Analysis seems {o require only the first Ievel, where only arguments and elements of the number sequence 1, 2, 3, ... are employed. The definite formulation and the proof of the consistency

are here still to be given. At any rate, it is clear that all the valuable thoughts of the whole development of the problem of analysis also carry weight, in a modified form, in formalized mathematics.

Similar formalistic tendencies, such as the one realized by Hilbert, presumably already underlie the “general characteristic” of Leibniz. And many of the Leibnizian statements sound as if his view of the infinitely small is that it—like the integration r.—cannot be given a reasonable contentual interpretation, yet that nonetheless all

things stand as if it [the infinitely small] did actually exist. All that matters for the mathematicians is that it fits into the calculus of signs without contradiction.

L

140

Hermann Weyl

Yet perhaps the reader has long been full of trepidation, like a living soul a spell has transported into the realm of shadows. After all, where are we? Is it not just the bloodless ghost of the old analysis, robbed of all content, that walks anew? Without doubt, if mathematics is to remain a sSerious cultural concern, then some sense must be attached to Hilbert’s game of formulae, and [ see only one possibiiity of attributing it (including its transfinite components) an independent intellectual meaning. In theoretical physics we have before us the great example of a [kind of] knowledge of completely different character than the common or phenomenal knowledge that expresses purely what is given 1n intuition. While in this case every judgment has its own sense that is completely realizable [vollziehbar] within intuiticu, this is by no means the case for the statements of theoretical physics. In that case it is rather the system as a whole that is in question if confronted with experi-

ence. Theories permit consciousness “to jump over its own shadow,” to leave behind the matter of the given, to represent the transcendent, yet, as is self-evident, only in symbols. Theoretical creation is something different from intuitive insight; its aim is no less problematic than that of artistic creation. Over idealism, which is called to destroy the epistemologically absolute naive realism, rises a third realm,

which we see Fichte, for example, enter in the final epoch of his philosophizing. Yet he still succumbs to the mystical error that, ultimately, we can nonetheless apprehend this transcendent within the luminous circle of insight. But here, all that remains for us is the symbolic construction. It never leads, I believe, to a final result—Ilike phenomenal knowledge, which, although subject to human error, is nonetheless by its nature immutable. The symbolic construction rather continues to be supported by the mind’s vital process [LebensprozefS des Geistes] taking place in us, and it must always begin anew. It is not a reproduction of the given, but 1t is

neither the sort of arbitrary game in the void proposed by some of the more extreme branches of modern art. Of the principles of reason [Vernunfiprinzipien] by which this construction is governed, we can so far only grasp that of consistency with some clarity; yet it is hardly the only relevant one. It is part of the task of the mathematician to ensure that at least this conditio sine qua non is satisfied throughout. If phenomenal insight is referred to as knowledge, then the theoretical one 1s based on belief—the belief in the reality of the own I and that of others, or belief in the reality of the external word, or belief in the reality of God. If the organ of the former is “seeing” in the widest sense, so the organ of theory is “creativity.” If Hilbert is

not just playing a game of formulae, then he aspires to a theoretical mathematics in contrast to Brouwer’s intuitive one. But where is that transcendent world carried by belief, at which its symbols are directed? I do not find it, unless I completely fuse mathematics with physics and assume that the mathematical concepts of number, function, etc. (or Hilbert’s symbols), generally partake in the theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc. The history of physics shows that intuition and theory must constantly go hand in hand. On the one hand, it cannot be denied, for example, that Mach’s phenomenalism is defeated by the theory of atoms, but Einstein’s theory of relativity teaches, on the other hand, how important is the going back from theoretical construction (geometry) to intuitive meaning, and the elimination of all too arbitrary elements (absolute space). Even if the development in the direction outlined by Hilbett continues, one day 7, may possibly be rejected as a means of theoretically construct-

The Current Epistemological Situation in Mathematics

141

ing the continuum in the same way in which Newton’s absolute space was rejected. Thus it is certainly greatly beneficial that Brouwer has strengthened again the sense

1n mathematics for the intuitively given. His analysis expresses in a pure manner

the content of the mathematical basic intuition and is therefore shone through by clarity without mystery. Yet beside Brouwer’s way, one will also have to pursue that of Hilbert; for it is undeniable that there is a theoretical need, simply incom-

prehensible from the merely phenomenal point of view, with a creative urge directed upon the symbolic representation of the transcendent, which demands to be satis-

fied. Being myself in the middle of the war of the factions, I have tried, sine ira et studio, to describe the current situation. One can see how deeply mathematics is

tied in its foundations to the general problems of knowledge. The old opposites of realism and idealism, of the Beirng of Parmenides and the Becoming of Heraclitus,

are here again dealt with in a most pointed and intensified form.

Notes

1. “The incomprehensibility of mathematics” is a favorite expression at the beginning of the eighteenth century. 2. Dedekind, Was sind und was sollen die Zahlen?, Braunschweig 1888. 3. A book has recently appeared that is well suited to introduce philosophers to the mathematical way of thinking: O. Hélder, Die mathematische Methode, Berhin 1924, As concerns set theory, see, in particular, A. Fraenkel, Einleitung in die Mengenlehre, second edition, Berlin 1923; concerning the reduction of mathematics to logic:

B. Russell, Einfiihrung in die mathematische Philosophie, German edition, Munich 1923,

4. Thus, tf we only use the two principles as examples, one. generates through Number of Applications

The Numbers

0

1

1

2

2

3, 4

3

5,6, 8

4

7,9, 10,12, 16

In this example, 4 would be = 11.

5. For in this case one could actually go from 1 to @ in only three steps: twice a multiplication by 2 and then an application of R.

6. More detailed information about Russell’s theory, which has found a most meticulous implementation in his and Whitehead’s Principia Mathematica (3 volumes, Cambridge 1910/13) can be found in his previously quoted publication, 7. The “static” theory discussed in Section II, which can logically so eastly be shaken, has initially endured probably only because the intuitively given resting existence of the continuum qua whole was covering the fact that the possibilities of picking out single points from the continuum do not form an extensionally definite totality. 8. Brouwer, “Intuitionism and Formalism,” Bulletin of the American Mathematical Society 20 (1913); “Begriindung der Mengenlehre unabhiingig vom logischen Satz vom ausgeschlossenen Dritten,” Verhandel. d. K. Akad, van Wetensch., Amsterdam

1,

Hermann Weyl

142

1918/19. Weyl, “Uber die neue Grundlagenkrise der Mathematik,” Muathem. Zeitschrift 10 (1921). Compare also O. Becker, “Beitrdge zur phinomenologischen Begriindung der Geometrie und ihrer physikalischen Anwendungen,” Husserls Jahrbuch fiir Philosophie 6, in particular, pp. 398463, and the “Philosophische Untersuchungen liber Limiten und Idealgebilde,” pp. 459-77.

A similar standpoint was already supported by L. Kronecker; yet, unlike Brouwer,

he did not provide a reconstruction over and above the criticism. 10. il

What I am describing here is not a faithful account of Brouwer’s standpoint, but one to which I was naturally led when [ was captivated by Brouwer's ideas. The property “even,” incidentally, must be defined by recursion as on p. I1 [129] and not as on p. 15 [131].

12. “The number £ falls into the interval o, & means a < § < b. 13. See: Weyl, Was ist Materie? Berlin, 1924, 14.

15,

Some ideas on this can be found in Boscovich, Theoria philosophiae naturalis (Venice 1763}; the problem how real matter can in its behavior be dependent on something merely “possible” is also alive in Euler’s remarks on absolute space. Hilbert, “Neubegriindung der Mathematik,” Abhandlungen aus dem mathematischen Seminar der Universitdt Hamburg 1 (1922); “Die logischen Grundlagen der Mathematik,” Mathematische Annalen 88 (1922).

16.

My description here is based on a system that is greatly simplified in companson to that of Hilbert and is due to v. Neumann, a young mathematician living in Zurich; nothing has been published on this so far,

Translator’s Notes Fr. 3, Simplicius Phys. 164, 17. The translation from the Greek is taken from G. S. Kirk and J. E. Raven, The Presocratic Philosophers, Cambridge, 1957, p. 370. The square brackets contain my translation of a passage Weyl seems to have added to the fragment for explanatory purposes.

. “the things in one world order are not separated one from the other nor cut off with an axe (...)" Fr. 8, Simplicius Phys. 175, 12, and 176, 29; Kirk and Raven, p. 381. . Physics, Bk. VIII: (Chap. 8, 263a 23). Translated from the Greek by R. P. Hardie and R. K. Gaye.

. “so divisibility may be looked at in two ways. For it is a question either of objective or of subjective resolubility. Objectively, that is inasmuch as matter effectively depends on divine representations, it is resoluble to the extent that the unbounded in-

tellect sees its resolubtlity. Subjective divisibility of matter does not extend bevond our perceptions.” e.

F. Nietzsche, Thus Spake Zarathustra, Third Part, “Of Old and New Tables,”

10 The Alleged Circulus Vitiosus and the So-Called Foundational Crisis in Analysis OTTO HOLDER*

Weyl has stated that the definition of the upper bound?® contains a circulus vitiosus.

Since this assertion has made quite a stir, and even among the mathematicians a

certain uncertainty in the comprehension of the foundations of their science has taken place, it would be worthwhile to investigate the matter further and to refute,

if possible, the assertion.

1

Conceptual Types

In particular Weyl thinks that in the construction of the upper bound the Russellian

type structure [Stufenbildung] of the concepts has been faultily ignored.! Weyl has devised an interesting example for the elucidation of the “types” that I would like to take over here with a small modification and to develop further. He considers

natural (integer, positive) numbers and introduces to begin with two operations A and B. A is the addition of 1 to the number given in each instance, and B is the duplication of the number given in each instance. 1 now want to ask for the least number greater than k that cannot be reached through at most four applications of the

operations A and B starting with %.? The production of that least number, starting from k, should be seen as an application of the operation Ny to k. If now, for example, the operations

A, B, A Ny

are to be carried out in this order, starting from 1, then the operations A, B, A, first lead from 1 to 2 and 4, and then to 5. But starting from 5 one could reach, with at

most four operations A or B, only the numbers 6-16, 20-26, 28, 4042, 44, 48, 80, so that now if one starts from 5, the operation N4 leads to the number 17.

*“Der angebtiche circulus vitiosus und die sogennante Grundlagenkrise in der Analysis,” Sitzungsberichte der Leipziger Akademie T8, 1926, pp. 243-50. Translated from the German by Paolo Mancosu. Published by permission of Akademie Verlag, Berlin. 143

144

Otto Holder Clearly, one can now designate A and B as operations of type | and define, in

the way shown in the example of N, infinitely many operations of type 2

N15N25 N}!----

(1)

If now one adds one of the operations in (1) to the operations A and B, say, the operation N, just discussed itself, then one can construct anew similar concepts.

Thus one can, for example, ask for the least number greater than & that cannot be reached, starting from k, through a sequence of at most 5 operations of the series A, B, Ny

whereby one would arrive at an operation N?}. One would have thereby achieved an operation of a still higher type. At the same time it is clear that in this way one

could construct innumerably many definitions, one after another, that do not even

separate in a simple sequence of types; for, one could add, with the objective of

[forming] a new definition, not merely any one, but at once several of the operations (1), and thus define a new operation N, and then begin over again from the start.

Weyl now adduces as an example of a construction principle based on a cir-

culus vitiosus the following: “Starting from n one constructs the smallest number that cannot be obtained by applying, starting from 1, the given principles,? includ-

ing this construction principle itself. in succession at most # times.” Everyone sees that this would be a wholly inadmissible circle; but the question is whether the de-

finitton leading to the upper bound can be put in parallel with the one just mentioned.

2

The Principle of the Excluded Middle in Infinitely Many Instances

In order to prepare for the concept of the upper bound, I would like to point to a state of affairs about which Weyl also seems to be in agreement with me and with the majority of mathematicians. This state of affairs concerns the application of the

principle of the excluded middle to infinitely many instances (infinitely many ob-

Jects, infinitely many relations, infinitely many computational processes). First of all one should recall that infinitely many instances can only be defined by a law.

Suppose we are dealing with certain instances, infinite in number, and with a prop-

erty that either belongs or does not belong to each instance in a completely determinate way, provided it is given; something that can be established, for example, by performing a computation. If now a determinate infinite numbe r of these in-

stances are again selected (i.e., they are given by a new law), then we proceed ac-

cording to the logical principle that we consider as compl etely determined, by the principle of the excluded middle, the answer to the quest ion of whether among these Instances there is one with the mentioned property or not, and the answer to the

question of whether all the instances now defined have the property or not. We do this even if we are not actually capable, on the ground of general principles, to de-

The Alleged Circulus Vitiosus and the So-Called Foundational Crisis in Analysis

145

cide* the answer to the relevant questions for the infinite totakity of cases now set

off. All sorts of classifications [Einteilungen] are based on such an application of the principle of the excluded middle, and on these in turn new concepts, which we

form. It is in this sense that the Weylean prescription that the words “all” and “there 18" may only refer to the totalities that were already formed before in an unobjectionable way is to be understood. If then new totalities are formed in the way described, then of course the principle of the excluded middle can hereafter be applied to these, a further type of concept can be formed and so on.

In the described sense the construction of concepts according to types is certainly necessary. I do not believe, however, that the recognition thereof should be

claimed as especially new; indeed, such a procedure has always been followed by anyone who may be called a mathematician.

3

Construction of the Upper Bound

It shall now be investigated whether the construction of the upper bound is actually

in opposition to the procedure described previously. Let the infinite totality A of absolute real numbers be given—say, in sequential form, although nothing hinges upon that:

A, As, As,

L

(2)

First of all one should now consider that an absolute “real number” is given by a division of the absolute rational numbers in such a way that all the absolute ratio-

nal numbers are completely partitioned into two classes whereby every number of

the first class i1s less than every number of the second class, and the first class does

not contain® a greatest rational number. In this division one then has a so-called Dedekind cut in the domain of the absolute rational numbers and one can speak of

the “lower” and the “upper” numbers® of the cut. Of course such a cut can only be given by a law. Weyl expresses himself on this issue as follows: The “lower” numbers of the cut are given by a “property.” All the more, infinitely many absolute real mumbers can only be given by a law, as it were, by a law of laws.

If now the numbers In (2) are all below a fixed bound [Schranke], say, below 100, then one ean construct a number -y, which wiil turn out to be an upper bound,

as follows. One forms a new totality of absolute rational numbers to which a num-

ber x belongs if and only if x is a lower number in any of the cuts (2). Granted that one can define this totality of numbers x, then one immediately recognizes that each

number less than a number x is also a number x, and that not all numbers belong to the numbers x, since 100 does not belong to it. Likewise, there cannot be a greatest number x, and it is now clear that these numbers x represent the lower numbers of a new cut y constructed by us. Weyl now objects to the fact that the number x belongs to the new aggregate

of numbers if and only if among the cuts (2) there is one 1n which x 1s a lower number or, as Weyl says, if there is a property of type A that holds of x. However, it is now a matter not just of a property of any type but of a property from a domain well defined by the law of the sequence (2); namely, it is a question of the prop-

Otto Halder

146

erty of being a lower number in one of those given cuts (2). It s only claimed that in addition to this lawlike totality of numbers (2) there exists an upper bound 7. Clearly to this end we have abided by the aforementioned logical principle. If x is given and in addition one of the cuts (2), say A, then by the given law of this cut

it is determined whether one of the infinitely many lower numbers of this cut coincides with x or not. But since the mentioned relation between the given x and any given cut from (2) is completely determined, and there is a law for the infinitely many cuts (2), then it is also determined, according to the law of the excluded middle, whether among the cuts (2) there is one to which the given rational number x stands in the mentioned relation, namely, that it is a lower number of this cut. I cannot find here at all a circulus vitiosus that in the construction of the number v (.., in the construction of the totality of the numbers x) already uses this very totality, for example, in the way of the crass circular definition put forward by Weyl and cited above.

If it is now assumed that a real number ' less than v is given, then this means nothing else than that cut y' possesses an upper number that is a lower number in

v (i.e., that belongs to the above-mentioned numbers x). This number xp would thus have to be a lower number in a cut A, of the sequence (2) (i.e., ¥ would be exceeded by A ), while v itself, as one likewise easily sees, is not exceeded by any of the numbers (2). The two properties through which one usually defines the “upper bound” thus hold, with respect to the numbers (2), of the constructed number y: 1.

|

v is not exceeded by any of the numbers (2).

2. Any number less than vy 1s exceeded by the numbers (2).

Supposing that a real number " is given of which the same two properties could be proved, then clearly the assumption v” # y would lead to a contradiction sintce of two noncoinciding cuts one must necessarily be the smaller. One thus usually says that the upper bound is a number uniquely defined by these two properties of it.

If we now say that in the domain of real numbers there exists one number that possesses, with respect to the numbers (2), the above-mentioned two properties, then certamnly the Weylean reproach seems to be correct in that the word “existsTM is ap-

plied to a totality that is not given constructively.’ But, basically, here is a case in which even Weyl considers the use of “it existsTM as legitimate, since the existence

is here claimed only after the noncircular construction of the number v.% The way in which the number y has been constructed can be approximately

characterized by saying: “The number vy is defined by the totality of rational num-

bers x that are less than y.” Should these words be considered as the actual definition, as of course they are not at all meant, then this definition would in fact present a circulus vitiosus in the worst form.

Thus only the one disadvantage remains, if one wants to call it so, that the upper bound b

a*b

Z(a) are prime formulae.

If one places a prime formula or a variable formula (I 3)? on both sides of an implication sign, then an implication formula anses. If one places a prime formula

or a variable formula or an implication formula on both sides of an implication sign, then the resulting line is also called a formula. And in general A—B

is to be a formula, if A and B are variables or already constructed formulae. Certain formulae, which serve as the building-blocks of the edifice of mathematics, are called axioms.

In the treatment of the axioms, and in operating with them, the following gen-

eral rules are to be observed:

Individual signs are irreplaceable; basic variables can be replaced at will by

functionals.

Parentheses are used in the ordinary way to separate out parts of signs; they serve to mark empty positions and they lend certainty and precision to the insertion of lines.

206

David Hilbert

The all-sign (I 6) is a logical sign: a parenthesis with a variable inside; the following subtormula, which in general contains this variable, is marked off by a special parenthesis and is thereby made recognizable as the scope of the all-sign. The

following special rules hold for the ali-sign: A variable in a formula is called “free” if it does not occur in an all-sign of this

tormula; an all-sign containing a free variable may be prefixed to any formuta, so that the entire formula is the scope of the all-sign. Conversely, an all-sign whose

scope 18 the rest of the formula can always be omitted.

A variable occurring in an all-sign may be replaced there and simultaneously

in the corresponding scope by any other variable that does not occur in that scope. Two all-signs may be interchanged if they occur in immediate succession and if their scopes extend equally far.

If a part of a formula takes the form

(b) (A — B(b)) where A does not contain the variable b, then () may be placed after the sign —

so that we obtain the formula:

A — (b6)B(b) Using our new formal standpoint, we shall now show how we acquire the theorems of elementary calculation. For this we need a table of axioms, Wthh begins

as follows:

hSali

a =

g,

T4+@a+Dh=0U+a)+ 1,

,a=b—sa+1=b+1,

el

a+l=b+1—a=h, a=c¢—=>b=c—a=b).

Moreover, we make use of the following inference schema:

S S—T T

Then the formal proofs for the number equations can be given in the manner shown by the following special example:

From axiom 1 by substitution we get 1=1; moreover, using the abbreviatory sign 2 for 1 + 1 and the abbreviatory sign 3 for

2+1

2=2

(1)

The New Grounding of Mathematics

207

3 =3

(2)

and

From axiom 2 we also get by substitution I+ (1 +DH={0+1)+1 or

1+2=2+1

or

1+2=3

(3)

From axiom 5, we obtain by substitution 3=3-5(1+2=3->53=1+2), and because of (2) we get, by an application of the inference schema, the formuia 1+2=3-53=1+2

and finally by (3) we get, by an application of the inference schema, the formula 3=1+2 This is therefore a formula that can be proved from the axioms we have already

introduced. Since we do not yet get all the formulas we need from the axioms we have hitherto introduced, the path is open to us to introduce additional axioms. But first we need a stipulation of what a proof is, and a precise description of the use of the ax10ms.

A proof is a figure, which we must be able to view as such; it consists of inferences according to the schema

] 5§5—>T

T

where at each stage each of the premises—that is, each of the formulae S and § — T —is either an axiom, or results directly from an axiom by substitution, or agrees with the end-formula T of an inference that occurs earlier in the proof, or results from such an end-formula by substitution. A formula is said to be provable if it is an axiom or results from an axiom by L,

208

David Hilbert

substitution or 1s the end-formula of a proof or results from such an end-formula by substitution. Thus the concept “provable” is to be understood relative to the underlying axiom system. This relativism is natural and necessary; it causes no harm,

since the axiom system is constantly being extended, and the formal structure {Aufbau], In keeping with our constructive tendency, is always becoming more complete. To reach our goal, we must make the proofs as such the object of our investi-

gation; we are thus compelled to a sort of preof theory which studies operations with the proofs themselves. For concrete—intuitive number theory, which we treated first, the numbers were the objectual and the displayable, and the proofs of theorems about the numbers fell into the domain of the thinkable. In our present investigation, proof itself is something concrete and displayable; the contentual reflections follow the proofs themselves. Just as the physicist investigates his apparatus

and the astronomer investigates his location; just as the philosopher practices the

critique of reason; so, in my opinion, the mathematician has to secure his theorems by a critique of his proofs, and for this he needs proof theory.

Recall now in particular our intention to prove the consistency of the axioms. From the present standpoint this problem seems to be meaningless, since at present

the only “provable” formulae that arise are formulae that are as it were equivalent to purely positive assertions, and which can accordingly produce no contradiction: we could allow 1 = 1 + 1 to count as a formula along with 1 = 1, provided it were

a provable formula yielded by our rules of inference. But if our formalism is to offer a full replacement for the earlier, real theory consisting of inferences and assertions then a contentual contradiction must have its formal equivalent. In order that

this should be so, we must take inequality to be a positive expression like equality,

and we must introduce it as a new sign # with new axioms; this sign is then operated in a manner that accords with our earlier rules. And then we declare an axiom system to be consistent if the formulae

a=>b

and

a¥x*bh

are never simultaneously provable formulas, where a and b designate functionals.

In accordance with this general plan, we introduce the new axiom

6. a+ 1+ 1 for the sake of simplicity, we now delete axiom 2. Then the first test of a genuine proof of consistency in our new proof theory lies in the proof of the following theorem:

The axiom system that consists of the Jollowing five axioms:

il

a=a,

a=b—sa+1=5b+1, at+tl=b+1—qg=5h,

ca=c—>{b=c-—>a=b),

6.a+1#1. IS consistent.

The New Grounding of Mathematics

209

The proof of this theorem takes several steps; first we prove:

Lemma. A provable formula can contain at most two occurrences of the sign — .

For suppose we had a proof for a formula with more than two — signs. Then we proceed through this proof until we find the first formula that has this property,

1.e. such that no previous formula in the proof of this formula contains — more than twice. This formula cannot result directly from an axiom by substitution, for the letters a, b, ¢ appearing in the axioms can be replaced only by functionals, and these

do not introduce any new — sign. But neither can that formula appear as the endformula T of an inference; for then the second premise 8 — T of this inference would be an earlier formula with more than two — signs; and therefore T would

not be the first formula with this property.

Next we prove:

Lemma. A formula a = b is provable only if a and b are the same sign. To prove this, once again we distinguish the two cases. First, suppose the for-

mula is the direct result of substituting into an axiom. Then only axiom 1 comes into constderation, and in this case our theorem obviously holds. Second, we as-

sume we have a proof with the end-formula a = b, such that a and b are not the same sign and such that no such formula occurs earlier in the proof. Then in our

inference schema T must agree with a = b and S must be a provabile formula; so the second premiss would have had the form S—a=b

(4)

This formula in turn must either be the result of substitution into an axiom, or be the end-formula of a proof. In the first case, only axioms 3 and 4 come into considera-

tion; if axiom 3, then a would have to beof the forma" + 1 and b of the form b’ + 1, and § would have to be the formula a’ = b’'. But if a" and b’ were the same sign,

then a and b would have to be as well, contrary to our hypothesis. But if, on the

other hand, a’ and b’ were not the same sign, then § (that is, a’ = b’) would be a formula of the sort under discussion and would appear in the proof before T; and this too cannot be. But if axiom 4 is involved, then S8 must be the formulaa + | = b + 1, in which we do not have the same sign on both sides of the equality sign; but this too is impossible, as S occurs earlier in the proof. 5o the only remaining possi-

bility is that.(4) is the end-formula of a proof whose last inference would have to have the form A

A—-8—>a=h

S—a=Db

Accordingly we investigate the origins of the second premiss A—-{(S—a=bhb)

If this premiss resulted directly from substitution into an axiom, then only axiom 5 would come into consideration, in which case S would have to be of the form

210

David Hilbert

b = ¢ and A would have to be of the form a = ¢. Now, if ¢ were the same as b, then A would just be a = b, and this formula would therefore already have appeared at an earlier place in the proof, But if ¢ were not the same as b, then the formula b = ¢ 1s a formula occurring earlier in the proof having the property originally required of T. Accordingly, the only remaining possibility is that (5) is the endformula of an inference; but then the second premiss of this inference must be a

formula with at least three — signs, and by our previous lemma, this could not be a provable formula.

‘

With this, our second lemma has been proved. We earlier declared an axiom-system to be consistent if

a=>b

and

a+b

are never simultaneously provable formulae. Now since according to the lemma just

proved a = b is a provable formula only if a and b are the same sign, our proof of the consistency of our axioms reduces to the problem of showing that our axiom system can never give rise to a provable formula with the form

a+a

(6)

We prove this as follows,

To obtain a formula of the form of (6) and containing the sign # directly from an axiom by substitution, it would be necessary to use axtom 6; but a formula arising from axiom 6 by substitution always has the form

a’' +1+#1 and here a” + 1 is certainly not the same sign as 1. If, on the other hand, (6) should arise as the end-formula of an inference, then the second premiss of this inference

must have the form

S—>a+a

(7)

and since such a formula cannot possibly arise from substitution into an axiom, this formula (7) must itself be the result of an inference. The second premi ss of this inference would then have to be

T—>(8—a+a) and this formula too would for the same reason have to come from an inference whose second premiss would necessarily have the form

A—-(T— (S > —a*a) But by our first lemma, such a formula is not provable beca use it certainly contains more than three — signs. This excludes the possibilit y that (6) is a provable forraula, and this completes our proof of the consiste ncy of our axiom-system.

The New Grounding of Mathematics

211

A new goal would be to carry out the corresponding investigation after having reintroduced the previously excluded Axiom 2. And in fact it is possible in this way

to demonstrate the consistency of the axiom-system

hAied

a=a,

l+a+1)=( +a)+ 1, a=b—at+tl1=5p+1, atl=>b+1—>a=b, a=c—>b=c—a=2b),

a-+1+F1.

We have hitherto introduced no logical sign apart from the — sign and the all-

s1gn; in particular, we have avoided the formalization of the logical operation “not.” This way of treating negation is characteristic for our proof theory: A formal equivalent for the missing negation lies solely in the sign # ; by introducing this sign, inequality is expressed just as positively and treated in the same way as its coun-

terpart equality. Contentually we use negation only in the proof of consistency and

only in so far as it corresponds to our basic point of view. In light of this circum-

stance, it seems to me that our proof theory also yields us an epistemologically important insight into the meaning and the essence of negation. The logical concept “all” i1s exhibited in our theory by the variables that ap-

pear there and by the rules we have laid down for operating with them and with the all-sign.

The only logical concept still to be formalized is the concept “there exists”— a concept which, as is well known, can be expressed in formal logic by negation and the concept “all.” But since negation cannot have any direct representation in

our proof theory, the formalization of “there exists” 1s achieved here by introduc-

ing individual function-signs through a kind of implicit definition; so that “that which exists” is as it were actually produced by a function. The simplest example of this is the following:

To express: “If a is not 1, then ‘there exists” a number which precedes a,” we

introduce the function-sign 3(*) with one empty position as an individuval sign, and we lay down as an axiom the formula 7. a#1—>a=6d6a +1

I shall here merely mention that it can then be proved by contentual consider-

ations that the axiom-system consisting of axioms 1-7 is consistent. Although these explanations contain only the very beginning of my proot theory, we can nevertheless perceive in them the general tendency and direction in which the

new grounding of mathematics ought to proceed. Two points emerge in particular. First: everything that hitherto made up mathematics proper 18 now to be strictly

formalized, so that mathematics proper, or mathematics in the strict sense, becomes a stock of provable formulae. The formulae of this stock are distinguished from the usual formulae of mathematics only by the fact that, besides the mathematical signs, they also contain the — sign, the all-sign, and the sign for statements. This cir-

cumstance corresponds to a conviction I have long maintained,” namely, that a si-

multaneous construction of arithmetic and formal logic is necessary because of the close connection and inseparability of arithmetical and logical truths, L

212

David Hilbert

Secondly: 1n addition to this proper mathematics, there appears a mathematics that 1s to some extent new, a metamathematics which serves to safeguard it by pro-

tecting 1t from the terror of unnecessary prohibitions as well as from the difficulty

of paradoxes. In this metamathematics—in contrast to the purely formal modes of inference in mathematics proper—we apply contentual inference, in particular, to the proof of the consistency of the axioms. The development of mathematical science accordingly takes place in two ways that constantly alternate: (i) the derivation of new “provable” formulae from the ax-

ioms by means of formal inference; and, (ii) the adjunction of new axioms together with a proof of their consistency by means of contentual inference. In keeping with the principles and tendencies we have just described, let us now carry out the new grounding of mathematics.

Our previous stock of axioms is merely the axioms 1-7 that we have already

mentioned. These axioms are of a purely arithmetical character; the provable formulae that follow from them do not in the least supply a foundation for the theory

of real numbers, and they make up only a small part even of arithmetic. A glance

at these axioms [-7 shows that the only variables that appear (small Latin letters without empty positions) are basic variables. But even for the grounding of arithmetic axioms of such a sort are utterly inadequate. Rather, we need a sequen ce of

axioms that contain variable formulae (large Latin letters): in particular, we lay down the following arithmetical axioms, each with a variable formula: Axtom of mathematical equaliry

8. a=b— (A(a) — A(b)) Axiom of complete induction

9. (a)Ala) — Ala + 1)) = {A(]) — (Z(b) — A(b))} We furthermore need a stock of such axioms corresponding to the usual

patterns of logical inference; they are the following four axioms with variable for-

mulae:

Axioms of logical inference 10.

A— (B— A),

1. {A-A—-B)}—>(A4A->B,

2. {A>B->0}—> {B>(A—O), B B->0->{A-B->B-0).

Moreover, we also need two axioms for mathematical inequ ality; these axioms

serve as an equivalent for certain modes of inference that are indispensable in con-

tentual reflections, namely, the following axio ms:

Axioms of mathematical inequality 4. a #a— 4,

I5. (a=b—>A)— {(a # b—A)—> A)}.

As | have already mentioned, the axioms 1-7 are only some of the arithmetical axioms that are necessary. To complete them, we need above all to introduce the logical function-sign Z (“to be a positive integer”). On the other hand, it is nec-

The New Grounding of Mathematics

213

essary to restrict axiom 6. If at the same time, for the sake of orthographic. unifor-

mity, we use the sign *—1 instead of the function-sign 8(*); and if we generalize and complete axioms 2 and 7; and if we discard axioms 3, 4, and 5 (because they are now provable formulae), we finally obtain the following axioms in place of 1-7:

Arithmetical axioms 16. Z(1), 17. Z(a) — Z{a + 1),

18. Z{a) > (a # | — Z(a — 1)),

19. Z(a) —» (a+ 1 # 1),

20 (a+1)—1=g, 2.

(a— 1)+ 1=aq,

22.

a+ b+ 1)=( b))+ a+ 1,

23

a—-b+ 1)=(@a@a—-»b

— 1.

If we take this axiom-system 1, 8-23 as a basis,‘ then we can obtain the entire

stock of formulae and theorems of arithmetic purely by application of our rules, i.e. in a formal manner,

The first important goal is to prove the consistency of this axiom system 1, 8-23. This proof can in fact be carried out, and the mode of inference of complete

induction (axiom 9), which is characteristic of arithmetic, is thereby secured.? But the most essential step still remains to be taken, namely, the proof of the applicability of the logical principle tertium non datur in the sense of the admissi-

bility of the inference, for infinitely many numbers, functions, or functions of functions, that a statement either holds for all these numbers, functions, and functions

of functions, or that there necessarily exists one among them for which the statement does not hold. Only with the proof of the applicability of this principle will

the grounding of the theory of real numbers have been accomplished, and the bridges be erected to analysis and set theory. This proof can be carried out on the basis of the fundamental 1deas I have de-

scribed. 1 introduce certain functions of functions 7 and « by laying down axiom

systems, and I prove the consistency of these axiom-systems.* The simplest example of a function of functions serving this end is the function of functions «( f), where the argument f is a variable number-theoretic function of the basic variable a, so that

Za)— {fla) # 1 — 1= Z(f(a)} holds, where «( f) = 1 — 1 if f has the value 1 for all a; otherwise, x(f) 1s the least argument for which f1s not 1. The axiom system for this «(f} is:

24. (k( f)=1—-1)— (Z{a)— f(a) = 1),

f), 25. (k(f)F 1 — 1) — Z«(

26. (k(f)#F1 - D> (f(x(f))F 1), 27. Zla) > {Z(k(f) —a)—= flx(f)y —a)= 1}, In a similar manner, a certain pair of functions of functions r,a that belong together can be introduced, by means of which the grounding of the theory of real

214

David Hilbert

numbers—and in particular the proof of the existence of upper bounds for any arbitrary set of real numbers—becomes possible. To conclude this first report, I should like to remark that P. Bernays has been

of the greatest assistance to me in working out the ideas presented here.

Notes l. In this sense, I call signs of the same shape “the same sign” for short, 2. The variable in question can still have one or more functionals as arguments. Thus,

for example, ((1, a) is a variable formula,

3. See my lecture, “Uber den Zahlbegriff,” Jber. disch. Math.-Ver., vol. 8, 1900, pp. 18084, reprinted as Appendix VI of my “Grundlagen der Geometrie.”

Notes by Bernays a. The expression “sign without meaning” caused offense to the philosophers. (See, for

example, the note of Aloys Miiller, “Uber Zahlen als Zeichen,” and the reply by P. Bernays, both in Math. Ann. 90 (1923).) In Hilbert's later writings on the foundations of mathematics, the term “number-sign” was replaced by “numeral” [“ZifferTM]. b. Hilbert here uses the word “formula” in the narrow sense, i.e. for the formulas of formalized mathematics. But one could of course equally well speak here of formu-

lae with meaning, just as one speaks of signs with meaning.

¢. A schema for the introduction of functions by recursion equations must also be added. d. As maiters have turned out, the mentioned proof is valid only if one excludes the all-

sign and replaces Axiom 9 by the induction schema.

¢. Hilbert here refers to his attempt to treat the transfinite functions in his consist ency proof. But it is still uncertain whether one can attain the desired goal by these means.

13 On Hilbert’s Thoughts Concerning the Grounding of Arithmetic PAUL BERNAYS*

Hilbert’s new methodological approach for the grounding of arithmetic, which I would like to address, presents a modified and more definite version of the plan that

Hilbert already had in mind for a long time and to which he first gave expression in his Heidelberg lecture. A sharply outlined and comprehensible programme, the

beginnings of which have already been carried out, has now succeeded the previ-

ous quite obscure suggestions, The problem whose solution we are seeking here is that of the proof of the consistency of arithmetic. First we have to bring to mind how one arrives at the formulation of this problem.

The construction of arithmetic (in the wider sense, i.e., encompassing analysis

and set theory), as it has been proceeding since the introduction of the rigorous methods, 1s axiomatical. This means that, analogously to the axiomatic grounding of geometry, one begins by assuming a system of objects with determinate relational properties {Verkniipfungseigenschaften). In Dedekind’s grounding of analysis what

is taken as a basis is the system of the elements of the continuum, and in Zermelo’s construction of set theory it is the domain of operations B. And also in that grounding of analysis that proceeds from the consideration of numerical sequences, the number series is conceived of as a closed, surveyable system, perhaps akin to an

infinite keyboard.

In the assumption of such a system with determinate relational properties there lies something transcendent, as it were, for mathematics, and there the question arises as to which fundamental position one should take in this regard. An appeal to an intuitive grasp of the number series as well as to the manifold

of magnitudes is certainly to be considered. But this could not be a question of an

intuition in the primitive sense, for certainly no infinite mamfolds are given to us in the primitive intuitive mode of representation. And even though it might be quite

rash to contest any farther-reaching kind of intuitive evidence from the outset, we will nevertheless make allowance for that tendency of exact science that aims to **[Jber Hilberts Gedanken zur Grundlegung der Arithmetik,” JDMV 31, 1922, pp. 10-19. (Lecture delivered at the Mathematikertagung in Jena, September, 1921). Translated from the German by Paoclo

Mancosu, Published by permission of B. G. Teubner GmbH. 215

216

Paul Bernays

eliminate the finer organs of cognition [Organe der Erkenntnis] as far as possible, and to rely only on the most primitive means of cognition, According to this viewpoint we will examine whether it is possible to ground those transcendent assumptions in such a way that only primitive intuitive cogni-

tions come into play. On account of this restriction of the means of cognition we cannot, on the other hand, demand of this grounding that it allow us to recognize as truths (in the philosophical sense) the assumptions that are to be grounded. Rather, we will be content if we succeed in proving the arithmetic built on those assump-

tions to be a possible (i.e., consistent) system of thotight. We have hereby already arrived at Hilbert’s formulation of the problem. But

before we look at the way in which the problem must be tackled, we must first ask ourselves whether there is not a different and perhaps more natural sort of attitude towards the transcendent assumptions.

In fact two different kinds of attempts suggest themselves and have also been undertaken. The first attempt also aims at a demonstration of consistency, not by

the means of primitive intuition, but rather with the help of logic. One will recall that the consistency of Euclidian geometry was already proved by Hilbert by the method of reduction to arithmetic. Thus it now also seems ap-

propriate to prove the consistency of arithmetic by reduction to logic. Especially Frege and Russell vigorously attacked the problem of the logical grounding of arithmetic. As regards the actual aim, the result was negative.:

First of all the famous paradoxes of set theory showed that no greater certainty of operating whatsoever was achieved by going back to logic. The contradictio ns of naive set theory could be seen [liefen sich wenden] logically as well as'set theoretically. And even the control of inferences through the logical calculus, which

had been constructed precisely for securing the mathematical inferences, did not help to avoid the contradictions. When Russell then introduced the very cautious procedure of the calcul us of

types, it turned out that analysis and set theory in their usual form could not be ob-

tained in this way. And thus Russell and Whitehead, in Principia Mathe matica, saw themselves forced to introduce an assumption about the system of predicates “of the first type,” the so-called “axiom of reducibility.” But hereby one again returned to the axiomatic standpoint and gave up the goal of the logical grounding, Incidentally, the difficulty already appears within the theory of whole numbers. Indeed, by defining the Numbers [Anzahlen] logically accor ding to Frege’s fundamental idea, one here succeeds in proving the computatio nal laws of addition and multiplication as well as the determinate numerical equat ions as theorems of logic. However, through this procedure one does not obtain the usual theory of numbers, for one cannot prove that for every number there exist s a larger one, unless one expressly introduces some sort of axiom of infinity,

Even though the development of mathematical logic did not in principle lead beyond the axiomatic standpoint, an mnpressive syst ematic construction of arithmetic as a whole, equal in rank to the system of Zermelo, has nonetheless emerged in this way.

On Hilbert’s Thoughts Concerning the Grounding of Arithmetic

217

Moreover, symbolic logic has taken us further in methodological knowledge: While previously one only justified the assumptions of the mathematical theories,

now the inferences are specified as well. And it turns out that one can replace mathematical inference—insofar as it is only a matter of the results proceeding from it—

by a purely formal manipulation according to determinate rules in which actual thinking is completely eliminated.

However, as was already said, mathematical logic does not achieve the goal of a logical grounding of arithmetic. And it is not to be assumed that the reason for this failure lies in the particular form of Frege’s approach. It seems rather to be the case that the problem of reducing mathematics to logic is in general wrongly posed, namely, because mathematics and logic do not really stand to each other in the relationship of particular to general.

Mathematics and logic are based on two different directions of abstraction. While logic deals with the contentually most general [das inhaltlich Allgemeinste], (pure) mathematics 1s the general theory of the formal relations and properties, and so on the one hand each mathematical reflection is subject to the laws of logic, and

on the other hand every logical construct of thought falls into the domain of mathematical reflection on account of the outer structure that is necessarily inherent in it.

[n view of this situation, one is impelled to attempt an investigation that is, in

a certain way, opposed to the logical grounding of arithmetic. Because we are unsuccessful in proving the mathematically transcendent basic assumptions as logically necessary, we then ask ourselves whether these assumptions cannot in fact be dispensed with.

Indeed, a possibility for the elimination of the axiomatic basic assumptions

seems to consist in removing, without exception, the existential form of the axioms and putting construction postulates in place of the existential assumptions. Such a replacement procedure is not new to the mathematician; especially in

elementary geometry the constructive formulation of the axioms is often applied.

For example, instead of laying down the axiom that any two points determine a straight line, one postulates as a possible construction the connection of two points by a straight line.

One can also proceed in the same way with the arithmetical axioms. For ex-

ample, instead of saying “each numbers has a successor,” one introduces progression [Fortschreiten] by one, or the affixing of +1, as a basic operation. One thus arrives at the attempt of a purely constructive development [rein kon-

struktiver Aufbau] of arithmetic. And indeed the goal for mathematical thought is a very tempting one: Pure mathematics ought to construct its own edifice and not be dependent on the assumption of a certain system of things. This constructive tendency, which was first brought to bear very fDrcefull}" by Kronecker, and later by Poincaré in a less radical form, is currently pursued by Brouwer and Weyl in their new grounding of arnithmetic. Weyl first checks the higher modes of inference in regard to the possibility of

a constructive reinterpretation; that is, he investigates the procedures of analysis, as

well as those of Zermelo’s set theory, as to whether or not they can be interpreted

as constructive. He finds that this is not possible, for in the attempt.to carry out a

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Paul Bernays

thoroughgoing replacement of the existential axioms by constructive methods, one

falls into logical circles at every turn. Thus Weyl draws the conclusion that the modes of inference of analysis and set theory have to be restricted to the extent that no logical circles come about in their constructive interpretation. In particular, he feels compelled to give up the the-

orem of the existence of the upper bound.

Brouwer goes even further in this direction by also applying the constructive principle to large numbers. If one wants, as Brouwer does, to avoid the assumption of a closed given totality of all numbers and takes as a foundation only the unlimited performable act of progressing by one, then statements of the form “There are

numbers of such and such a type . . .” do not necessarily have a meaning. And thus one 1$ also not in general justified in putting forward, for every number theoretical statement, the alternative that either it holds for all numbers or that there is a num-

ber (respectively, a pair of numbers, a triple of numbers, . ..) by which it is re-

futed. This sort of application of the “tertium non datur” is then at least questionable.

Thus we find ourselves in a great predicament: The most successful, most elegant, and most established modes of inference ought to be abandoned just because,

from a specific standpoint, one has no grounds for them. The considerations through which Weyl tries to show that the concept of the

mathematical continuum, which lies at the basis of usual analysis, does not correspond to the pictorial [bildhaft] representation of continuity, also does not help us

get over the unsatisfactoriness of such a procedure. For an exact analogy to the content of perception is not at all necessary for the applicability and the fruitfu lness of analysis; rather, it is perfectly sufficient that the method of idealization ahd con-

ceptual interpolation used therein be consistently practicable. Conce rning the ques-

tion of pure mathematics, what matters is only whether the usual, axioma tically characterized mathematical continuum is in itself a possible, that is, a consistent, structure [Gebilde].

This question could only be rejected if there was at our disposal a simpler and clearer conceptual structure that would supersede the current mathe matical continuum. But if one examines more closely the new approaches by Weyl and Brouwer, one notices that a gain in simplicity cannot be hoped for here; rather, the compli-

cations required in the concepts and forms of inference are only increased instead of decreased. There 1s thus no justification in rejecting the question of consistency of the usual axiom system of arithmetic. And what we are to draw from Weyl’s and Brouwer’s investigations is the result that a consistency proof is not possible by way of re-

placing existential axioms by construction postulates. Hereby we come back to Hilbert’s idea of a theory of consistency based on a primitive—intuitive foundation. And I would now like to describe the plan, according to which Hilbert conceives of the construction of such a theory, and the guiding principles he follows to this end. Hilbert adopts what is positively fruitful in the two foundational attempts discussed above. From the logical theory he takes the method of the rigorous formal-

1zation of inference. The necessity of this formaliz ation follows directly from the formulation of the problem. For the mathematical proofs are to be made the object

On Hilbert’s Thoughts Concerning the Grounding of Arithmetic

219

of a concrete-intuitive form of consideration. To this end it is, however, necessary that they are projected, as it were, into the domain of the formal. Accordingly, in Hilbert’s theory we have to distinguish sharply between the formal image [Abbild] of the arithmetical statements and proofs as object of the theory, on the one hand,

and the contentual thought about this formalism, as conrent of the theory, on the other hand. The formalization is done in such a way that formulas take the place of

contentual mathematical statements, and a sequence of formulas, following each other according to certain rules, takes the place of an inference. And indeed no meaning is attached to the formulas; the formula does not count as the expression of a thought, but it corresponds to a contentual judgment only insofar as it plays, within the formalism, a role analogous to that which the judgment plays within the contentual consideration.

More basic than this connection to symbolic logic is the contiguity of Hilbert’s

approach to the constructive theories of Weyl and Brouwer. For Hilbert in no way wants to abandon the constructive tendency that aims at the self-reliance of math-

ematics; rather, he is especially eager to bring it to bear in the strongest way. In light of what we stated with respect to the constructive method, this appears at first to be incompatible with the goal of a consistency proof for arithmetic. In fact, however, the obstacle to the unification of both goals lies only in a preconceived opin-

ion from which the advocates of the constructive tendency have until now always

proceeded, namely, that within the domain of arithmetic every construction must indeed be a number construction (set construction, respectively). Hilbert considers this view to be a prejudice. A constructive reinterpretation of the existential axioms is possible not only in such a way that one transforms them into generating princi-

ples for the construction of numbers, but the inference rule made possible by such

an axiom can be replaced as a whole by a formal procedure in a such a way that determinate signs stand for general concepts such as number, function, etc. Whenever concepts are missing, a sign is introduced at the right moment. This

is the methodological principle of Hilbert’s theory. An example should explain what is meant. The existence axiom ‘“‘each number has a successor” holds in number the-

ory. In keeping with the restriction to the concretely intuitive, the question 15 now

to avoid the general concept of number as well as the existential form of the statement.

The usual constructive reinterpretation in this case consists (as already re-

marked) in replacing the existential axiom by the procedure of progression by one. This is a procedure of rumber construction. Hilbert, on the other hand, replaces the concept of number by the symbol Z and puts forward the formula: Z(a)— Z(a + 1)

Here a is a variable for which any mathematical expression can be substituted, and the sign — represents the hypothetical propositional connective “if—then,” that is, the rule “if two formulas A and A — B are written down, then B can also be written down,” holds.

On the basis of this stipulation, the mentioned formula accomplishes, within the framework of the formalism, exactly what the existence axiom accomplishes in the contentual inference.

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Paul Bernays

Here we see how Hilbert utilizes the method of formalization of inferences in

keeping with the constructive tendency; in no way does it constitute for him merely a too] for the consistency proof. This method at the same time also provides the approachto a rigorous constructive development [streng konstruktiver Aufbau) of arith-

metic. And indeed the methodological idea of construction is here so broadly conceived, that all higher mathematical modes of inference can also be included in the constructive development.

After having characterized the aim of Hilbert’s theory, I would now like to describe the main features of the structure of the theory. The following three questions are to be answered: 1. The constructive development must represent the formal image [Abbild] of the system of arithmetic and at the same time must constitute the object for

the intuitive theory of consistency. How does such a development take shape? 2. How is the consistency statement to be formulated? 3. What are the means of contentual consideration through which the consistency proof is to be carried out?

First, the constructive development takes place in the following way. To begin with the different kinds of signs are introduced, and thereby the substitution rules

are layed down. Furthermore, certain formulas will be put forward as basic formulas. And the question is now that of forming “proofs.”.

What counts as a proof is a concretely written-down sequence of formul as in

which for every formula occurring in the sequence the following holds: Either the

formula is identical with a basic formula, or it is identical with a prece ding formula, or results from such a formula by a valid substitution; or, altern atively, it constitutes the end formula in an “inference,” that is, in a sequence of formulas of the

type

A A—B _.}

B

Hence a “proofTM

is nothing else than a figure with determinate concrete properties and the formal image [Abbild] of arithmetic consists of such figures. This answer to the first question makes the urge ncy of the second quite evident. For what should the statement of consistency express within the pure formal-

1sm? Is it not the case that mere formulas cannot contradict themselves? The simple reply goes as follows: The contradictio n is simply formalized as

well. Faithful to his principle Hilbert introduc es the letter 3 for the contradiction; and the role of this letter within the formalis m is determined by putting forward ba-

sic formulas so that from any two formulas to whi ch contrary statements correspond, {} can be deduced. More precisely, by adding two such formulas to the basic for-

mulas, a proof can be constructed with £} as the end formula.

Specifically the following basic formula

a=b—(a+b-0 o

On Hilbert’s Thoughts Concerning the Grounding of Arithmetic

221

where =+ 1s the usual sign of inequality, serves us here. (The relation of inequality is taken by Hilbert as a genuine arithmetical relation, just as equality is, and by no means as the logical negation of equality. Hilbert does not introduce a sign for negation at all.)

The statement of consistency is now simply formulated as follows: (3 cannot be obtained as the end formula of a proof. It 1s then necessary to provide a proof for this claim.

Now the only question still remaining concerns the means by which this proof should be carried out. In principle this question is already decided. For our whole

problem originates from the demand of taking only the concretely intuitive as a basis for mathematical considerations. Thus the matter is simply to realize which tools are at our disposal in the context of the concrete—intuitive mode of reflection.

This much is certain: We are justified in using the elementary ideas of sequence

and ordering, as well as the usual counting, to the full extent, (For example, we can determine whether there are three occurrences of the sign — in a formula or fewer.) However, we cannot get by in this way alone; rather, it is absolutely necessary to apply certain forms of complete induction. Yet, by doing so we still do not go

beyond the domain of the concretely intuitive.

In this regard, two types of complete induction are to be distinguished: the narrower form of induction, which relates only to something completely and concretely given, and the wider form of induction, which uses either the general concept of whole number or the operating with varniables in an essenttal manner.

Whereas the wider form of complete induction i1s a higher form of inference

whose justification constitutes one of the tasks of Hilbert’s theory, the narrower form of mference belongs to the primitive intuitive mode of cognition and can therefore be applied as a tool of contentual inference.

As typical examples of the narrower form of complete induction, as they are used in the argumentations of Hilbert’s theory, let us adduce the foilowing two inferences:

I. If the sign + occurs at all in a concretely given proof, then in reading the

proof one finds a place where it occurs for the first time. 2. If one has a general procedure for eliminating from a proof with a certain concretely describable property E the first occurrence of the sign Z, without

the proof losing the property E in the process, then one can, by repeated ap-

plication of the procedure, completely remove the sign Z from such a proof, without its losing the property E.

(Notice that here it is exclusively a question of formal proofs, i.c., proofs in the sense of the definition given above.)

The method the theory of consistency must follow is hereby set forth in its essentials. The development of this theory is currently still in its beginnings; most of it still has to be carried out. Certainly the basic possibility and the feasibility of the modes of reflections demanded can already be recognized from what has been said so far; and one also sees that the considerations to be empioyed here are mathematical in a very genuine sense.

The great advantage of Hilbert’s procedure rests precisely on the fact that the problems and difficulties that present themselves in the grounding of mathematics

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Paul Bernays

are transferred from the epistemologico-philosophical domain into the domain of what is properly mathematical. Mathematics here creates a court of arbitration for itself, before which all fundamental questions can be settled in a specifically mathematical way, without having to rack one’s brain about subtle logical dilemmas [Gewissensfragen] such as whether judgments of a certain form have a meaning or not.

Therefore, it is also to be expected that the enterprise of Hilbert’s new theory will soon find resonance as well as participation in the circles of mathematicians.

" Received October 13, 1921

14 Reply to the Note by Mr. Aloys Miiller, “On Numbers as Signs” PAUL BERNAYS*

Mr. Aloys Miiller’s criticism of the conception of number theory as a theory that deals with meaningless signs (“number signs”) consists basically in three objections. A discussion of these objections is useful for clarifying the standpoint of intuitive number theory, and thereby also that of Hilbert’s proof theory.

1. The first objection is terminological and is directed against the use of the

word “sign” for something meaningless [bedeutungslos]. If the objects of number theory, such as 1, 1 + 1, have no meaning, then they are not signs, so the objection

goes, but rather figures or, “as we would rather want to say,” shapes [Gestalten). The first part of this objection must be conceded: Indeed, it corresponds better to linguistic usage to say that the objects of intuitive number theory, the “number

signs,” are figures. On the other hand, we must thoroughly aveid using the word “shape” in the same sense as “figure.” Figures are not shapes; rather, they have a

shape. (Moreover, they also have individuality.) We must be able to speak about the fact that a figure @ has the same shape as another figure b. 2. The second argument is the following. Since the figures under consideration

have no meaning, nothing can hinge on the particular form [Form] of the individ-

ual constituents. For example, instead of the figures I+ 1,

1+1+1

one might just as well choose different ones, for example, OeO

ol o

0

Moreover, the cobjection proceeds, we are not bound to the *serial form of the arrangement” [Reihenform der Zusammensetzung], and just as little to the number +FErwiderung auf die Note von Herrn Aloys Miiller: Uber Zahlen als Zeichen,” Mathematische Annalen

90, 1923, pp. 159-63. Translated from the German by Paclo Mancosu. Published by permission of Springer-Verlag GmbH & Co. KG, Heidelberg. 223

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Paul Bernays

of elementary shapes. Any specification relating to this would already introduce a contentual element [inhaltliches Moment]. (“One thinks of members of the sequence, of the position of members in the sequence, and thereby a meaningful content [Be-

deutungsgehalt] again unnoticeably attaches itself.”) Thus completely arbitrary kinds of arrangements of discrete constituents are admissible. However, the examination

of such shapes does not lead to number theory.

This much is correct about the foregoing, namely, that the special shapes [Formen] “1” and “+ are inessential. If we disregarded the connection to habit, it would even be advisable, in order to emphasize the principle, to take as numerical s1gns

figures of the type

------

(which are thus constituted merely of points). And, of course, stars, vertical strokes, circles, and other shapes could just as well be chosen instead of points. One could also take a time sequence, say, of similar noises, instead of a spatial sequence.

But it is essential that specimens of equal shape be Joined in the same sort of

arrangement [Zusammensetzung]. In this respect we are indeed bound to the serial

form of the arrangement. However, thereby we are in no way surreptitiou sly ob-

tatning a meaning for the numerical signs. For the intuitive—contentu al elements that occur in the description of a figure need not be ascribed as meaning to the figure

itself.

In order to indicate what kind of figures the “number signs” shoul d be, we need

the idea of a determinate, concretely exhibitable, form of succession. The inessential elements of the shape and of the arrangement that occur here will then be, as it

were, eliminated by the sort of consideration that is applied to the number signs, for

they play no role in the occurring relations. Thus intuitive number theory can indeed be viewed as a funda mental chapter of the theory of shapes. Its delimitation and problematic is also by no means arbitrary from the point of view of the shape, but rests on a natural abstra ction, on a choice of

certain, simplest elements. The necessary abstraction is here not at all obtained sur-

reptitiously but rather is displayed by means of manifest objec ts (whereby, of course, the possibility of communication through language is a prere quisite). The principle of abstraction is however something contentual—an intellectu al discovery—but this content is thereby not yet a meaning of the objects about which we think. The claim “that mere shapes do not suffice as a basis for number theory” may

be conceded. But the meaningless figures are not supp osed to constitute the basis but only the objects of number theory. 3. To elucidate his standpoint, Mr. Aloys Miiller adva nces the following third

objection: “3 > 2” means, according to Hilbert’s explanatio n, that the number sign 3, thatis, 1 + 1 + 1, extends beyond the number sign 2, that 1s, 1 + 1, or that the latter figure is a segment of the former. However, according to this spatial interpretation, the claim 3 > 2 is not correct under all circ umstances. For example, the objection goes, if one writes the two figures in the form

1+

1

1+1+1

Reply to the Note by Mr. Aloys Miiller, “On Numbers as Signs”

225

one beneath the other, then the second does not extend beyond the first, and also the first is not a segment of the second. “Whoever contests that is again secretly at-

tachingasenseto1 + land 1 + 1 + 1, e.g,, that the first shape contains two units, and thus one unit less than the second.” To this it should first be remarked that, according to Hilbert’s explanation, the claim 3 > 2 indeed has a spatial sense, but not thereby a metric sense. In charac-

terizing the attitude required in the case of intuitive number theory, Hilbert does

stress (in a passage quoted by Mr. Aloys Miiller himself) that “insignificant differences 1n the construction [Ausfiihrung) of the figures should be disregarded.” The

separation of the constituents of the figure 1 + 1 from one another by larger or smaller distances is such an insignificant difference. That this difference is to be

viewed as “insignificant” already follows from the fact that the same shape must always be denoted by “2.” This shape is completely described by the fact that 1

stands first, after that “+,” and after that again “1.” The figure 1 + 1 + 1 is to be described correspondingly. And that the figure 1 + 1 coincides with a constituent of the figure 1 + 1 + 1, in such a way that the latter resuits by affixing something, namely, “+1,” to the former, is now a fact that can be grasped intuitively. But by this observation it is surely not the case that a sense is “secretly” conferred upon the figures. Indeed, it 1s only a question of a purely external relation-

ship between the figures. Moreover, the circumstance addressed by Mr. Aloys Miiller in the quoted sentence, viz. that the shape 1 + 1 contains two units (by unit the component 1 is probably meant), and thus one less than | + 1 + 1, does not con-

stitute the sense of these signs, just as the observation that the word “chair” contains five letters contributes nothing to the sense of the word “chair.” However, when he speaks of a meaning of the number signs, Mr. Aloys Miiller is apparently not at all thinking of the kind of meaning as it occurs in the words of

a language. Rather he is thinking of the meaning that befits the number signs within the formalism of number theory. But senseless figures are equally capable of such meaning, because of the external properties that are found in them and of the external relationships that can be observed between them.

It should also be noted that the contentual character of the Number [Anzahli]

concept is indeed compatible with the purely figural character of the number signs. The figures are used as tools for counting, and by counting one arrives at the determination of number. Incidentally, this way of introducing the concept of Number [Anzahlbestimmung], together with the required intuitive considerattons, can al-

ready be found in the earlier literature (e.g., in Helmholtz, “Zihlen und Messen”). One here has to recognize that the Numbers are only defined in connection with

the entire Number statement. For example, it will not be explained what “the num-

ber five” is, but only what it means for the Number five to apply to a given totality of things.

|

Of course, it cannot be claimed that the method of intuitive number theory and the Number definition assoctated with it represents the only grounding of number

theory compatible with Hilbert’s basic methodological direction. (Indeed, in the construction of his more comprehensive theory, Hilbert himself replaces this ground-

ing with a different and more formalized one.)

It is by no means compatible, however, with Hilbert’s basic thoughts to introduce the numbers as ideal objects “with quite different determinations from those

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Paul Bernays

of sensory objects,” “which exist entirely independently of us.” By this we would go beyond the domain of the immediately certain. In particular, this would be evi-

dent in the fact that we would consequently have to assume the numbers as all existing simultaneously. But this would mean assuming at the outset exactly that which Hilbert considers to be problematic.

However, the objects of intuitive number theory, the number signs, are, according to Hilbert, also not “created by thought.” But this does not mean that they

exist independently of their infuitive construction, to use the Kantian term that is quite appropriate here. But the construction always only yields either a single determinate figure or a procedure for obtaining a further figure from a given one (e.g.,

by affixing “+1”). But it does not lead to the idea of a simultanecus existence of “all” the number signs. That the idea of the number series as a closed totality [In-

begriff] can be applied in mathematical inferences without danger of a contradiction is precisely what is shown by Hilbert’s proof theory.

Hilbert’s theory does not exclude the possibility of a philosophical attitude that conceives of the numbers as existing, nonsensory objects (and thus the same kind

of ideal existence would then have to be attributed to transfinite numbers as well,

and in particular to the numbers of the so-called second number class). Neverthe-

less the aim of Hilbert’s theory is to make such an attitude dispensable for the foundation of the exact sciences. Received April 7, 1923

15 Problems of the Grounding of Mathematics DAVID HILBERT*

For the mathematical sciences the most recent decades were a period of greatest flourishing. I may remind you how in anithmetic, and especially in the theory of algebraic

number fields, the most difficult problems have been solved, and the completion of

this magnificent edifice of thought has been achieved. At the same time, the longsought transcendental functions, which belong to the number field, have been dis-

covered, and through them various number theoretical truths, which had so far been hidden, have come to light. On the other hand, the concept-formations {Begriffshil-

dungen] of the theory of ideals were carried, with greatest success, far beyond the boundaries of number theory, and into algebra and function theory, and thereby a great complex of mathematical knowledge has been turned into a uniform structure. In the past period no small progress was made in the theory of functions of a

complex variable as well. In particular, we owe to the formulation of the principle

of conformal mapping the magnificent methods for obtaining automorphic functions and the solution to the uniformization problem. The extremely difficult proofs of the existence theorems have reached the highest degree of simplicity and trans-

parency through the application of the methods of the calculus of variations.

And what abundance of aspects geometry yields! Topology alone has been enriched so much by new problems and methods of treatment that one must see in it

the emergence of a new independent branch of knowledge. And the old disciplines closely associated with geometry, group theory and invariant theotry, have reached

an unforeseen expansion and deepening. Finally, physics has erected before our eyes edifices of mathematical thought

whose halls are of impressive grandeur. Moreover, let us alsc think of the applica-

tions. It is not the worst fruit that mathematical research harvests in the field of applications, whether they sprung from neighboring fields of knowledge or from prac-

tical needs. And the territory into which mathematics penetrates expands steadily.

+“Probleme der Grundlegung der Mathematik,” Mathematische Annalen 102, 1929, pp, 1-9, Translated from the German by Paolo Mancosu. Published by permission of Springer-Verlag GmbH & Co. KG, Heidelberg.

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David Hilbert

On account of such a happy state of affairs, the special duty to secure mathematics in its foundations arises for the mathematician. What is then the generally held opinion about mathematics and mathe matical thought? It runs as follows: The mathematical truths are absolutely certain, for they

are proved on the basis of definitions through infallible inferences. Therefore they must also be correct everywhere in reality. As a result of this popular opinion math-

ematics is supposed to serve as the model for all science in general. We now want

to see whether this obtains in mathematics. How was the state of the foundational enquiry three decades ago? The great classics and creators of foundational research

were Cantor, Frege, and Dedekind. They had found in Zermelo a congenial inter-

preter. Zermelo had laid down the assumptions necessary for the axiomatic con-

struction of set theory and had thereby specified the means that had been used only unclearly [unbestimmi], and in some cases unconsciously, by Cantor and Dedekind. Moreover, these Zermelian axioms are such that a serious doubt about their cor-

rectness could not arise. Zermelo’s procedure was by all means legitimate and cor-

responds to the axiomatic method. However, Zermelo’ s paths were abandoned under the pressure of influential circles. Kronecker’s old objec tions against Cantor and Dedekind, which we believed to have been long overcome , and to which Kronecker

himself had not adhered in his own work, were revived [vorgesucht]. An unfortu-

nate view of Poincaré conceming the inference from » to n + 1, which had already been refuted by Dedekind through a precise proof two decades earlier, barred the way to progress. A new prohibition, the prohi bition of impredicative propositions,

was issued and maintained by Poincaré, althou gh Zerm elo immediately gave a striking example against this prohibition, and although this prohibition contravened Dedekind’s results as well. Unfortunately, Russell’ s logic, otherwise so excellent, also gave support to the erroneous doctrine in its application to mathematics. So it happened that our beloved science was, in regard to the question of its most inner arithmetical nature and basis, haunted for two deca des as by a nightmare. I welcome as an awakening, as a luminous auro ra, the recent return of a series of younger mathematicians back to Zermelo’s ideas. These mathematicians have completed Zermelo’s axioms and in addition have treated successfully a series of

very important and deep questions.

Of course, a definitive solution of the foundati onal problems is never possible

through this axiomatic procedure. For the axio ms taken as a basis by Zermelo contain

genuine contentual assumptions, and to prov e the latter is, I believe, precisely the main

task of foundational research. Indeed, by that time , proving the consistency of the axioms of arithmetic had already become a burn ing question, However, if we use contentual axioms as starting points and foundati ons for the proofs, then mathematics thereby loses the character of absolute certaint y. With the acceptance of assumptions

we enter the sphere of what is problematic. Indeed, the disagreements among people are mostly due to the fact that they pro ceed from different assumptions. In a series of

lectures over the course of the past year, I have therefore taken a new path to the treat-

ment of the foundational problems. Wit h this new grounding of mathematics— which one can appropriately call a proof theory —I believe to dispose of the foundational ques-

tion in mathematics as such once and for all by turning every mathematical statement Into a concretely exhibitable and rigorous ly derivable formula and thereby transfer ring the whole complex of questions into the dom ain of pure mathematics.

Problems of the Grounding of Mathematics

229

Of course, for the complete realization of this task the devoted collaboration of the younger generation of mathematicians is needed. In this sense, I would now like to make more detailed comments. The most important problems all center on the so-called € axiom, which 1 have put forward:

Ala@) = A(eA) When applying the axiom, one has to observe above all the type of variables to

which the € is to be related. With the natural numbers the same axiom serves the

formulation of the common inferences with “any”: We understand by €A any natural number for which the statement A certainly holds, if there is any nataral number at all for which A holds.

I would like now to mention some problems. The consistency proof of the e-axiom for the natural numbers has been ac-

complished by the works of Ackermann and von Neumann. Thereby the following three problems are settled: 1. The tertium non datur for numbers, that is, if a statement does not hold for

all integer numbers, then there is a number for which it does not hold. For example, according to Kronecker it was inadmissible to define an integer rational function of a variable x with integer rational coefficients as irreducible, if there is no decomposition of the same as the product of two such functions. 1 prove by means of proof theory that, on the contrary, this

is a completely rigorous definition in a purely mathematical sense. Therefore, Kronecker’s assertion was not merely logically unfounded but incor-

rect in a purely arithmetical sense, incorrect in the same sense as any false arithmetical sentence or a false number-theoretic formula.

2. The solution of an assertion about the existence of a natural number, for this number: “the least natural number that.”

3. The inference from n to n + 1, if one moreover adds as an axiom the formiula

(eA = b') > —Ab

As one may notice, an essential tool for my proof theory is the conceptual notation. We owe to Peano, the classic of this conceptual notation, the most scrupu-

lous care and the most far-reaching development of it. The form in which I use the conceptual notation is essentially the one introduced by Russell. The following problems are as yet not settled.

Problem |

The consistency proof of the e-axiom for the function variable f. We have the outline of a proof. Ackermann has already cartied it out to the extent that the only remaining task consists in the proof of an elementary finiteness theorem that is purely arithmetical.

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David Hilbert

With the solution of this Problem I a very big complex of basic questions ts already removed [behoben]. For this consistency proof yields:

1. The tertium non datur for functions of whole numbers and thereby also for the real numbers.

2. The definitional processes, which had been opposed by Poincaré as impredicative, and which Russell and Whitehead could justify only with the

help of the very problematic axiom of reducibility, and in reference to which Weyl once spoke of a circulus vitiosus in analysis.

3. The axiom of choice in the weaker form. Concerning (3), the following should be remarked. In more recent considerations of the axiom of choice, distinctions have been frequently made between weaker

and stronger forms of the axtom of choice. These distinctions concern, for the most part, the cardinality of sets of sets and of their sets of representatives, and in particular the difference between countable and uncountable. Through proof theory we are induced to regard as essential, above all, a fur-

ther distinction, namely, whether it is required that the choice of the representative of a set must be uniquely determined by the elements of the set independently of the way In which the set is defined, or whether this is not required.

For example, let a one-parameter collection of sets M(z) be given. For every value of the real parameter ¢, M(¢) destgnates a determinate set of real numbers that contains at least one element. The principle of chotce in the weaker form then requires that there is a single-

valued function of such a kind that for every value of ¢, the value fr) belongs to M(t). Moreover, in the stronger form the principle of choice requires the existence

of a function f{1), such that

) = At) if the sets M(¢;) and M(z;) have the same elements.

With the help of the e-axiom for the function variable f, we obtain the princi-

ple of choice for sets of real numbers in the weaker form. Through the solution of our Problem I the following theories, in particular, are mastered. 1. The theory of real numbers (Dedekind cut, least upper bound of a bounded

set of real numbers). 2. Peano’s grounding of number theory. In this theory one does not need any principle of choice, but one does need the impredicative definition, such as the definition of a =< b: viz., every set that contains ¢ and with # contains at the same time n + 1, also contains b. Previously what had always been seen to be problematic with the set-theoretical grounding of number theory

was only the assumption of an infinite domain of individuals. On the basis of the things said so far, this assumption can already be recognized as con-

sistent. The greater difficulty lies with the consistency proof of the impredicative definitions.

Problems of the Grounding of Mathematics

231

With the solution of Problem I Dedekind’s ingenious procedure in his “Was sind und was sollen die Zahlen?” is also justified. 3. Cantor’s theory of the well orderings of the number series. Hereby the theory of numbers of the second number class, which can be constructed ax1omatically, in complete analogy to number theory, is reduced to the theory

of functions of number variables.

Problem II For the further development of analysis, in particular, for the theory of point seis

(set-theoretical topology) as well as for the theory of the second and higher number classes, one needs the consistency of the e-axiom relative to higher types of variables and first of all relative to the type of the functions of real variables. Furthermore, for the proof of the well-ordering theorem and also for some proofs

in measure theory (proofs of nonmeasurability), one uses the axiom of choice in the stronger form. The latter is expressed in proof theory by the formula

—> €A = €B (NA(f) s B(f)) (axiom of cheice identity); here, eA and eB are functions of number variables and

the equality means identity [Ubereinstimmung] for all the values of the arguments. For the inclusion of this formula the consistency proof is again necessary.

Problem IlI It is generally maintained that the axiom systems for number theory as well as for

analysis are complete. However, the usual argument with which one shows that any two realizations of the axiom system for number theory, respectively, of analysis, must be isomorphic, do not satisfy the demands of finitistic rigor. What is at issue here is to transform the usual proof of isomorphism finitisti-

cally, in particular, for number theory whose domain can be delimited precisely, so that the following is thereby demonstrated: If for a proposition 8 the consistency with the axioms of number theory can be

proven, then the consistency with those axioms cannot also be proven for § (the opposite of S).

And moreover in close connection with this: If a statement is consistent, then it is also provable.

B

The case of the consistency of S as well as of § would be conceivable in higher domains. In that case the assumption of one of the two statements S and S as axiomatic is to be justified by methodological advantages (principle of the permanence of laws, further possibilities of construction, etc.). Objections to my proof theory have been raised. These objections rest essentially on a misunderstanding of my proof theory. I therefore take the liberty to make a few more explanatory comments here.

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David Hilbert

If a statement or a proof is given, then it must be surveyable in all its parts. The exhibition, recognition, distinction, and succession of their individual parts are immediately intuitable for us. Without this approach thought or even a scientific activity 1s altogether impossible. When investigating consistency the question is whether it is possible to present a proof that leads to a contradiction. If such a proof

cannot be presented 1o me, so much the better, since I am then spared a careful analysis. If, however, the proof is presented to me, then I may select certain indi-

vidual parts and consider them in themselves. In particular, [ may then also decompose again the special numenical signs that occur in these parts and are thus pre-

sent as produced and constructed. Hereby the inference from »# to # + 1 is in no way employed, rather, as we already know since Dedekind, and as my proof the-

ory confirms anew, from here it is still a long way and an essentially different task to analyze the validity of this inference from n to n + 1.

The consistency proof for the inclusion of a statement must be carried out every

time according to the principles just discussed. If one succeeds in carrying out this proof, then for that statement this means that if a numerical statement that is finitistically interpretable is derived from it, then it is indeed correct [richtig] every

time. At the same time, whenever a proof leads to a false result, the consistency proof shows us how to find the place where the error lies. It 1s obvious that purely logical problems also fall within the compass of the proof theory I just sketched. Let the following problem serve as an example.

Problem IV The assertion of the completeness of the axiom system for number theory can also

be stated in this way: If a formula belonging to number theory, but not provable in it, 18 added to the axioms of number theory, then a contradiction can be derived from the extended axiom system. Since in proof theory we are always dealing with formalized proofs, the statements we just made about the completeness of number theory contain at the same

time the assertion that the formalized rules of logical inference are sufficient, at least in the domain of number theory. The question conceming the completeness of the system of the logical rules, stated

in the general form, constitutes a problem of theoretical logic. Starting from number theory, we arrive at this more general formulation of the problem if we remove from

the domain of the considered formulas, and especially also from the axioms, all those in which the sign “+ 1" occurs, while admitting, however, the occurrence of predicate

variables. This essentially means that we disregard the ordering [Ordnungscharakter] of the system of numbers, and we only treat it as any system of things to which predicates with one or more terms can refer. Among these predicates only the equalit y

(1dentity) is fixed as a determinate relation through the usual axioms of equality a=a

a = b— (Ala) — A(b))

while the remaining predicates can be chosen freely.

Problems of the Grounding of Mathematics

233

In this domain of formulas one sets off those that cannot be refuted by any determinate specification of the variable [wdhlbar] predicates. These formulas repre-

sent the logical statements that are universally valid. Now the question arises as to whether all these formulas are provable from the

rules of logical inference with the addition of the mentioned equality axioms, in other words, whether the system of the usual logical rules is complete.

By trial and error we have so far arrived at the conviction that these rules are sufficient. An actual proof of this is only available in the domain of pure proposi-

tional logic. In the domain of monadic predicate logic a proof for the completeness

of the rules can likewise be obtained from the method of solution of the decision problem (Schrider’s elimination problem). This was shown in connection with

Schréder’s first attempts, first by Lowenheim and later in definitive form by Behmann.

My lecture today shows how many problems still await solution. However, in a general and basic sense, not even the faintest trace of obscurity is possible any

longer: Every basic question can be answered in a mathematically precise and univocal manner on the basis of the proof theory I have sketched. Now the theorems

at issu€ can in part be proved, in an absolutely certain and purely mathematical fash-

ion, with the help of the present results, and they have therefore been removed from the dispute. Whoever wants to confute me must show me, as has always been customary in mathematics and will continue to be so, exactly where my supposed error lies. I reject a limine an objection that does not do so. I believe that my proof theory additionally renders us a more general service.

For what would be the state of the truth of ocur knowledge in general, and of the existence and of the progress of science, if not even in mathematics there was certain

truth? Indeed, nowadays scepticism and pustllanimity are not infrequently expressed against science even in specialized publications and in public lectures. This 1s a cer-

tain form of occultism I consider deleterious. Proof theory makes such an attitude

impossible and brings us the exaltation of the conviction that at least the mathematical understanding encounters no limits and that it 1s even capable of discover-

ing the laws of its own thought. Cantor said: The essence of mathematics consists

in its freedom, and I would like to add for the skeptics and the pusillanimous: In mathematics there is no ignorabimus. On the contrary, we can always answer meaningful questions. And it is confirmed, as Aristotle perhaps already foresaw, that our understanding does not practice any secret arts, but rather always proceeds according to well-determined and presentable [aufstellbar] rules. And this is at the same time the guarantee for the absolute objectivity of its judging [seines Urteilens]. Received March .25, 1929

Notes

1. Lecture delivered at the International Congress of Mathematicians in Bologna on September 3, 1928.

16 The Philosophy of Mathematics and

Hilbert’s Proof Theory PAUL BERNAYS*

1

The Nature of Mathematical Knowledge

Anyone not familiar with mathematical activity may think, when reading and hear-

ing today about the foundational crisis in mathematics or of the debate between “formalism” and “intuitionism,” that this science is shaken to its very foundations. In

reality mathematics has been moving for a long time on a smooth wake, so that one senses more a lack of bigger sensations, although there is no lack of significant sys-

tematic progress and brilliant achievements. In fact, the current discussion about the foundations of mathematics does not have its origins in a predicament of mathematics itself. Mathematics is in a com-

pletely satisfactory state of methodological certainty. In particular, the concern caused by the paradoxes of set theory has long been overcome, ever since it was discovered that for the avoidance of the contradictions encountered, one only needs restrictions that do not encroach in the least on the claims of mathematical theories on set theory. The problematic, the difficulties, and the differences of opinion begin rather at

the point where one inquires not simply about the mathematical facts, but rather about the grounds of knowledge and the delimitation [Abgrenzung] of mathemat-

ics. These questions of a philosophical nature have received a certain urgency since the transformation the methodological approach to mathematics experienced at the

end of the nineteenth century. The characteristic moments of this transformation are: the advance of the concept of set, which aided the rigorous grounding of the infinitesimal calculus, and

further the rise of existential axiomatics, that is, the method of development of a mathematical discipline as the theory of a system of things with determinate operations whose properties constitute the content of the axioms. In addition to this we

have, as the result of the two aforementioned moments, the establishment of a closer connection between mathematics and logic. **Die Philosophie der Mathematik und die Hilbertsche Beweistheorie,” Bléitter fiir deutsche Philosophie 4, 1930-1931, pp. 326-67. Reprinted in P. Bernays, Abhandlungen zur Philosophie der Mathe-

matik, Wissenschaftliche Buchgesellschaft, Darmstadt 1976, pp. 17-61, Translated from the German by Paolo Mancosu. Published by permission of Wissenschaftliche Buchgesellschaft, Darmstadt,

234

The Philosophy of Mathematics and Hilbert’s Proof Theory

235

This development confronted the philosophy of mathematics with a completely new situation and entirely new insights and problems. The discussion about the foun-

dations of mathematics has never since come to rest. The debate concerning the difficulties caused by the role of the infinite in mathematics stands in the foreground in the present stage of this discussion. The problem of the infinite, however, is not the only nor the most general ques-

tion with which one must come to terms in-the philosophy of mathematics. The first task is to gain clarity about what constitutes the peculiarity of mathematical knowl-

edge. We would like to concern ourselves first with this question, and, in order to

do so, recall the development of [the different] views, even though only in broad

strokes and without an exact chronological order.

1

The Development of the Conceptions of Mathematics

The older conception of mathematical knowledge proceeded from the division of

mathematics into arithmetic and geometry; according to this conception, mathematics was characterized as a theory of two kinds of specific domains, that of num-

bers and that of geometric figures. However, this division was already unsustainable in the face of the advance of the arithmetical method in geometry. Moreover,

geometry did not content itself with the study of properties of figures, but rather it expanded to a general theory of manifolds. Klein’s Erlangen Program, which sys-

tematically combined the vanous branches of geometry from the points of view of a group-theoretical formulation, gave concise expression to the completely differ-

ent situation of geometry. Out of this situation arose the possibility of incorporating geometry into arithmetic, and since the rigorous grounding of the infinitesimal calculus by Dedekind, Weierstrass, and Cantor reduced the more general number concept, as required by

the mathematical theory of quantities [Grdfienlehre] (rational number, real number), to the usual (“natural”) numbers 1, 2, . . ., the conception arose that the natural num-

bers constitute the true object of mathematics and that mathematics consists precisely of the theory of numbers.

This conception has many supporters. This view 1s supported by the fact that all mathematical objects can be represented through numbers or combinations of numbers or through formations of higher sets [hdohere Mengenbildungen], which are

derived from the number sequence. In a fundamental aspect, however, the charac-

terization of mathematics as a theory of numbers is unsatisfactory, if only because it remains undecided what one regards as the essence of number. The inquiry into

the nature of mathematical knowledge is thereby shifted io the inquiry into the nature of numbers.

However, this question scems completely pointless to the declared proponents

of the view of mathematics as the science of numbers. They proceed from the view, which is the common one in mathematical thought, that numbers form a category of things, which by their nature are completely familiar to us, and to such a degree that an answer to the question concerning the nature of numbers could only consist in reducing something familiar to something less familiar. One perceives the reason for the special position of numbers from this point of view in the fact that numbers make up an essential component of the world order; and this order is compre-

236

Paul Bernays

hensible to us in a rigorous scieatific way exactly insofar as governed by the element [Moment] of number. Opposing this view, according to which number is something completely absolute and fundamental [Lezztes], soon emerged, in the aforementioned epoch of the development of set theory and axiomatics, a completely different conception, which

completely disputes the existence of a particular, peculiar kind of mathematical knowledge and holds that mathematics is to be obtained from pure logic. One was naturally led to this conception on the one hand through axiomatics and on the other

hand through set theory.

_

The new methodological turn in axiomatics consisted in giving prominence to the fact that for the development of an axiomatic theory the epistemic status [Erkenntnischarakter] of 1ts axioms is irrelevant. Rigorous axiomatics demands that in the

proofs no other facts [Erkenntnisse] be used from the field that is to be considered than those that are expressly formulated in the axioms. This is already the meaning

of axiomatics found in Euclid, even though at certain points the program is not completely carried out. In accordance with this demand, the logical dependence of the theorems on the

axioms is shown through the development of an axiomatic theory. For this logical dependence it does not matter whether or not the axioms placed at the beginning are true statements. The logical dependence represents a purely hypothetical con-

nection: If things are as the axioms claim, then the theorems hold, Such a detachment [Loslosung] of deduction from the assertion of the truth of the initial state-

ments is in no way idle hair splitting. An axiomatic devélopment of theories, which occurs without regard for the truth of the principles assumed at the starting point, can be of high value for our scientific knowledge, since in this way assumptions,

on the one hand, whose correctness is doubtful, can be made amenable to a test by means of the facts through the systematic pursuit of their logical consequences and,

on the other hand, the possibilities of theory-formation |Theorienbildung] can be explored a priori through mathematics according to the points of view of system-

atic simplicity, in advance and all at once [auf Vorrar durch die Mathemarik]. With the development of such theories mathematics takes over the role of that discipline,

which was earlier called mathematical natural philosophy. By completely ignoring the truth of the axioms within the axiom system, the

contentual conception of the basic concepts becomes irrelevant, and in this way one

arrives at abstracting, in general, from all intuitive content of the theory. This abstraction 1s also supported by a second element, which appears in the recent axiomatics, a prime example being Hilbert’s “Foundations of Geometry,” and which 1s essential for the development [Gestaltung] of more recent mathematics, namely ,

the existential formulation of a theory [die existentiale Fassung der Theori e]. Whereas Euclid always thinks of the figures to be studied as constructed, con-

temporary axiomatics proceeds from the idea of a system of object s fixed from the outset. In geometry, for example, one considers the points, straight lines, and planes

i their totality as such a system of things, Within this system one thinks of the relationships of incidence (a point lies on a straight line, or on a plane) , of betweenness (a point lies between two others), and of congruence as being determined from

the outset. These relationships can, without regard for their intuit ive meaning, be

The Philosophy of Mathematics and Hilbert’s Proof Theory

237

characterized purely abstractly as certain basic predicates (we want to use the term

“predicate” also in the case of a relationship between several objects, so that we also speak of predicates with several subjects).! In this way, in Hilbert’s system the place of the Euclidian construction postu-

late that demands the possibility of the connection of two points by means of a straight line 15 taken by the following existence axiom: For any two points there is

always a straight line that belongs to each ‘of the two points. “Belonging” [Zusammengehiren] 18 here the abstract expression for [the relationship of] incidence.

In the sense of this formulation of axiomatics, the axioms as well as the theo-

rems of an axiomatic theory present themselves as statements about one or several

predicates, which refer to the things of an underlying system, and the knowledege,

provided to us by the proof of a theorem L, which is carried out by means of the axioms Ay . ... A—for the sake of simplicity we assume that in this case only one predicate 1s at 1ssue—consists in the realization that if the statements 4, . . . A; hold

of a predicate, then the statement L also holds of it.

What we have before us is, however, a very general proposition about predicates, that is, a proposition of pure logic. In this manner, the results of an axiomatic theory, in the sense of the purely hypothetical and existential formulation of ax-

iomatics, present themselves as theorems of logic.

|

These theorems, though, are only meaningful if the conditions formulated in

the axioms can at all be satisfied by means of a system of objects with certain predicates that are related to them. If such a satisfaction is unthinkable, that is, logically impossible, then the axiom system does not lead to a theory at all, and the only logically important statement about the system then is the statement [Feststellung] of

the contradiction that results from the axioms. For this reason there exists for every axiomatic theory the requirement of a proof of satisfiability, that is, of the consistency of its axioms,

This proof is accomplished in general, unless one can make do with direct fi-

nite model constructions, by means of the method of reduction to arithmetic, that is, by exhibiting objects and relationships within the realm of arithmetic that satisfy

the axioms that are to be investigated. In this way one again faces the question about the epistemic status of arithmetic.

Even before this question became acute in connection with aXiomatics, in the connection described, set theory and logistics had already taken a position on this issue in a new manner. Cantor showed that the number concept in the sense of car-

dinal number (Number [Anzah{]) as well as 1n the sense of ordinal number (order

number {Ordnungszahl]) can be extended to infinite sets. The theory of natural num-

bers and the theory of real numbers [Mafizahlen] (analysis) were included in general set theory as subdomains. If, in this way, the natural number forfeits something of the essence of its distinct role, nonetheless the number sequence constitutes, also for Cantor’s standpoint, something immediately given, from the examination of which set theory originated.

This was not the end of the matter; rather, the logicians soon moved on to the far-reaching claim that sets are nothing other than extensions of concepts [Begriffsumfiinge] and that set theory is equivalent to extensional logic [Umfangslogik]; in particular the theory of numbers is also to be derived from pure logic.

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Paul Bernays

With this thests that mathematics is to be obtained from pure logic, an old cher1shed thought of rational philosophy, which had been suppressed by the Kantian

theory of pure intuition, was taken up again.

Now the development of mathematics and theoretical physics already demon-

strated that the Kantian theory of experience was, in any case, in need of a fundamental revision, and the moment seemed to have arrived for the radical opponents

of the philosophy of Kant to refute this philosophy in its initial thesis, namely, in the claim of the synthetic character of mathematics. This [attempt] was, however, not completely successful. A first symptom that

showed that the subject is more difficult and entangled than the leaders of the of logistic movement thought became evident in the discovery of the famous set-

theoretic paradoxes. This discovery historically constituted the signal for the onset of the critique. If we want to explain the subject philosophically today, then it is more satisfactory to carry out the consideration directly, without introducing the dialectical argument of the paradoxes.

2

The Mathematical Element in Logic—Frege’s Number Definitions

In order to deal with the essential points of view, we need only consider the new

discipline of theoretical logic, the creation [Gedarnkenwerk)] of the important logicians Frege, Schroder, Peano, and Russell and observe what it teaches us about the relationship of mathematics to logic [des Mathematischen zum Logischen). A peculiar double sidedness of this relationship, which is revealed in the dif-

ferent version of the task of theoretical logic, becomes immediately apparent: While Frege strives to subordinate the mathematical concepts to the concept-formations [Begriffsbildungen] of logic, Schroder attempts conversely to emphasize the math-

ematical character of logical relationships and develops his theory as an “algebra of

logic.”

However, this is only a difference in emphasis. In the various systems of lo-

gistics the specifically logical point of view nowhere rules alone: rather it is per-

meated from the beginning with a mathematical mode of consideration. The mathematical formalism and the mathematical concept formation prove to be, in a way completely analogous to the area of theoretical physics, the proper aid in the rep-

resentation of the connections and in the achievement of a systematic overvi ew. It 1s not the usual formalism of algebra and analysis that is applied here, though

,

but rather a newly created calculus, which theoretical logic develops with the aid

of the language of formulas, by means of which it represents the logica l connections. No one familiar with this calculus will doubt that both this caleul us and its theory have a pronouncedly mathematical character.

This state of affairs shows that the concept of the mathematical needs to be delimited, independently of the factual stock of mathematical discip lines, by means of a fundamental characterization of the mathematical way of knowl edge [Erkenni-

nisart]. It we pursue what we mean by the mathematical character of a considera-

tion, it becomes apparent that the typical characteristic is located in a certain mode of abstraction that comes into play. This abstraction, which may be called formal

or mathematical abstraction, consists in emphasizing and exclusively taking into account the structural elements of an object—*object” here is meant in its widest

The Philosophy of Mathematics and Hilbert's Proof Theory

239

sense—that is, the manner of its composition from its constituent parts, One may, accordingly, define mathematical knowledge as that which rests on the structural consideration of objects.

A study of theoretical logic further teaches us that in the relationship between mathematics and logic the mathematical way of consideration, in contrast to the contentual logical way, under certain circumstances constitutes the standpoint of higher abstraction. The aforementioned analogy between theoretical logic and the-

oretical physics extends in such a way, that just as the mathematical lawlikeness of theoretical physics is contentually specialized by means of its physical interpretation, so the mathematical relationships of theoretical logic also experience a spe-

cialization through their contentual logical interpretation. The lawlikeness of the logical relationships appears here as a special model for a mathematical formalism.

This peculiar relationship between logic and mathematics, that is, that not only can one subject mathematical judgments and inferences to logical abstraction, but

also the logical relationships to a mathematical abstraction, has its reason in the special position the area of “the formal” [des Formalen] occupies vis-a-vis logic. Namely, whereas in logic one can usually abstract from the specific determinations

of any domain of logic, this is not possible in the area of the formal, because for-

mal elements enter essentially into logic.

'

This is especially true for logical inference. Theoretical logic teaches that one can “formalize” a logical proof. The method of formalization consists first of all in the representation of the initial propositions of the proof with the aid of the logical

language of formulas by means of specific formulas, and further in the replacement

of the principles of logical inference by rules that specify certain procedures, according to which one proceeds from given formulas to other formulas. The result of the proof is represented by an end formula, which expresses the proposition to

be proven on the basis of the interpretation of the logical language of formulas.

In this way it becomes evident that all logical inference, observed according to

its course [Verlauf], can be reduced to a limited number of logical elementary processes that can be exactly and completely enumerated. In this way it becomes possible to pursue systematically the questions of provability. There results here a field of theoretical research, within which the theory, developed in traditional logic, of the various possible forms of categorical syllogisms constitutes only a very specific, special problem.

The typically mathematical character of the theory of provability manifests itself especially clearly through the role of logical symbolism [Symbolik]. Symbolism in this case is the tool for the accomplishment of the formal abstraction. The transition from the contentual logical to the formal approach takes place in such a way that we disregard the original meaning of the logical symbols and make the symbols themselves representatives of formal objects and connections. If, for example, the hypothetical relationship “if A, then B”

is symbolically represented by A— B

240

Paui Bernays

then the transition to the formal point of view consists in abstracting from the meaning of the symbol — and in taking the connection by means of the “signTM — itself

as that which is to be contemplated. In this way a figural specialization takes the place of the oniginal contentual specialization of the connection; this is. however, harmless to the extent that it is readily grasped as an inessential element. Mathe-

niatical thought accomplishes the formal abstraction just by means of the symbolic

figure,

The method of formal consideration is not artificially introduced, but rather it appears almost by necessity if one wants to pursue more closely the process of log-

ical inference with respect to its outcome.

If we consider, then, why it is that the examination of logical inference is in such need of the mathematical method. we find the following [fact]. In the process of demonstration, there are two significant moments that work together: the clarifi-

cation of the concepts, that is, the moment of reflection. and the mathematical moment of combination. As long as inference rests only on the clarification of the meanings, it is in the strictest sense analytical; the progress to something new takes place only through mathematical combination.

This combinatorial element seems so obvious that it is not at all regarded as a special factor. Especially in philosophy it was always customary to consider as epistemologically problematic and in need of explication only that aspect of a theorem

lan einer deduktiv gewonnenen Erkennmis] that is the given for the proof: the initial propositions and the rules of inference. This point of view is, however, inadequate for the the philosophical understanding of mathematics, because the typical

aclnevement of a mathematical proof only begins after the starting propositions and the rules of inference are already fixed, and the astounding thing of mathem atical results does not disappear if we modify the provable propositions contentually by

introducing the highest assumptions of the theory as premuses and besides by also

specitying explicitly the rules of inference, in the sense of the formal point of view, For the clarification of the facts of the matter, Weyl's comparision of a proof carried out purely formally to a chess game can be helpful to us: the initial position

In the game corresponds to the initial sentences in the proof, and the rules of the game correspond to the rules of inference. Let us now assum e that an astute chess

champion has discovered for a certain initial position A4 the possibility to checkmate

his opponent in ten moves. From the point of view of the customary mode of consideration, we have to say that this possibility is logically deter mined by the initial

position and the rules of the game. On the other hand, howev er, one cannot maintain that the claim that in ten moves the opponent can be check mated is logically contained [sinnesmdflig enthalten] in giving [Angabe von] the initial position A and the rules of the game. The appearance of a contradiction betwe en these claims dis-

appears if we make it clear to ourselves that the “logi cal” outcome of the rules of

the game rests on combination, and it comes to light there fore not by means of a mere analysis of meaning but only through a real demon stration [Vorfiihrung].

Every mathematical proof [Beweis] is, in this sense, a demonstration [Vor-

fihrung]. Let us show, by means of a simple special case, how the combinatory element appears in a proof.

We have the inference rule: “If A holds and if from A follows B, then B holds.”

The Philosophy of Mathematics and Hilbert's Proof Theory

241

In a formalized proof [ins Formale iibersetzen] to this inference principle corresponds the rule that from two formulas A,

A — B the formula B can be derived.

Now let this rule be applied in a formal derivation and indeed let us assume that A and A — B do not belong to the initial formulas. Then we have an inference sequence § that leads to A, and a sequence T that leads to A — B; and so the formu-

las A, A — B yield, in accordance with the aforementioned rule, the formula B. If we want to analyze what takes place here, we must take care not to anticipate the decisive point through the manner of denotation. Namely, the end formula

of the inference sequence T is first of all only determined as such, and it is a new

step for knowledge to establish the coincidence of this formula with the one originating from the other given formula 4 and from the formula B that is to be derived,

by means of connection [Zusammenfiigung] through “ — .

The establishment of an identity is in no way always an identical (tautological)

establishment. The coincidence, which is to be found in the present case, cannot be directly read off from the content of the formal inference rules and from the structure of the initial formulas, but rather it can only be read off from that structure ob-

tained through the application of the inference rules, that is, through the carrying

out of the inferences. There exists here in fact a combinatorial element.? If we clearly understand in this way the role of mathematics [des Mathematis-

chen] in logic, the possibility of the inclusion of arithmetic into the system of theoretical logic will not seem astonishing. However, this inclusion also loses its epistemological significance for the standpoint we have reached. For we know in advance

that the formal element is not eliminated by means of the inclusion of arithmetic into logical systematics [logische Systematik). With reference to the formal [sphere], mathematical consideration represents, as we found, the point of view of higher ab-

straction when compared to conceptually logical consideration. Therefore we cannot gain a higher generality for mathematical knowledge [mathematische Erkennt-

nisse] through its inclusion into logic, but rather on the contrary only a specialization through logical interpretation, which is a kind of logical clothing. A typical example of such a logical clothing is represented by the method ac-

cording to which the natural numbers are defined by Frege, and following him, with a certain modification, by Russell. Let us recall briefly the train of thought of Frege’s theory. Frege introduces the

numbers [Zahlen] as Numbers [Anzahlen] (cardinal numbers). His starting theses are as follows:

The Number applies to a predicate as determination [Die Anzahl kommt als Be-

stimmung einem Pridikat zu]. The Number concept originates from the concept of equinumerosity. Two predicates are said to be equinumerous if the things to which

the one predicate applies can be reversibly [umkehrbar] and uniquely associated to | those to which the other predicate applies. If the predicates are partitioned into classes with respect to equinumerosity in such a way that all predicates of a class are equinumerous to one another and predicates of different classes are not equinumercus, then each such class represents the Number, which applies to the predicates that belong to it. In the sense of this general Number definition, the individual finite numbers | such as {0, 1, 2, 3 are defined as follows:

0 is the class of predicates which do not apply to anything. 1 is the class of

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“one-valued” [einzahlig) predicates, and a predicate P is called “one-valued” if there 1s a thing x to which P applies, and no other thing (different from x), to which P

applies. Correspondingly, a predicate is two-valued [zweizahlig], if there is a thing

x and a thing y that differs from x, so that P applies to x and to y, and if there is nothing that differs from x and y to which P applies. 2 is the class of two-valued predicates. The numbers 3, 4, 5 and so on are to be defined analogously as classes, Frege defines the general concept of a finite number, after he previously introduced the concept of a number that immediately follows another number [the successor], in the following way: A number # is said to be finite if any predicate that applies to ( and that, when it applies to the number a, also applies to the number

that immediately follows it, also applies to n. The concept of a number that belongs to the number sequence from 0 to » is explamned in a similar way. The derivation of the principles of number theory from

the concept of finite number is based on these concept-formations. Now we want to examine in particular Frege’s definitions of the individual finite numbers. Take the definition of the number 2, which is defined as the class of two-valued predicates. Against this definition goes the objection that the member-

ship of a predicate in the class of two-valued predicates depends on extralogical conditions, and therefore the class does not constitute a logical object.

Thas objection is taken care of, however, if we adopt the point of view of Russell’s theory regarding the conception of the classes (respectively, sets, extensions of concepts). According to this [theory] the classes (extensions of concepts) do not

at all constitute real objects; rather, they function only as dependent expres sions within a paraphrased proposition. If, for example, K is the class of things with the

property E, that 1s, the extension [Umfang] of the concept E, then we are to con-

sider, according to Russell, the statement that a thing a belongs to class K only as

a paraphrase of the statement that a thing a has the property E. If we combine this view with Frege’s Number definitions, we arrive at the point of defining the number 2, not by the class of two-valued predicates, but rather by

that concept, whose extension constitutes this class. The number 2 is then identified with the property of two-valuedness of a predicate, that is, with the proper ty of a predicate to apply to a thing x and to a thing y, which is different from x, but not to a thing that is different from x and y.3

For the evaluation of this definition it is essential in which sense the definition [das Definieren] is meant and what it is supposed to achieve [mit welch em Anspruch

es geschieht]. What should be shown here is that this definition cannot be regarded as an account of the true meaning of the Number concept “two,” by which this concept, freed from all inessential elements, would be unveiled in its logical purity, but

rather that exactly the specifically logical element of the definition is an inessential addition, Namely, the two-valuedness of a predicate P means noth ing other than that there are two things to which the predicate P applies. Here three conceptual elements are separate from each other: the conc ept “two things,” the existential ele-

ment, and the fact that the predicate P applies [das Zutreffen des Priidikates P).

The conceptual content of “two things” is not logic ally dependent on one of the

two other conceptual contents. “Two things” already means something, even without the claim of existence of two things, and also without reference to a predi-

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cate, which applies to the two things; it means, simply, “a thing and one more

thing.” With respect to this simple definition, the Number concept turns out to be an elementary structural concept. The appearance that this concept is obtained from the elements of logic emerges in the logical definitions of Number we have considered by joining the concept with logical elements, namely, the existential form

and the subject—predicate relationship, which in themselves are inessential for the number concept. Here we actually intend a logical clothing of a formal concept.

The result of these considerations is that the claim of the logicists [Logisten] that mathematics is purely logical knowledge turns out to be blurred and mislead-

ing upon closer observation of theoretical logic. That claim is only correct when we take over the concept of the mathematical in the sense of the historical definition and we systematically extend, in contrast, the concept of the logical. However, through this definition what is episterologically essential is concealed, and what is

peculiar to mathematics 1s overlooked.

3

Formal Abstraction

We have established formal abstraction as the defining characteristic of the mathematical mode of knowledge, that 1s, the focusing on the structural side of objects,

and thereby we have delimited the field of mathematics in a fundamental way. If we likewise want to grasp epistemologically the concept of the logical, then we are

prompted to select from the entire field of the theory of concepts, judgments, and inferences, which is generally denoted as logic, a more narrow subfield, reflective or philosophical logic, which 1s the area of properly analytical knowledge, that is,

knowledge that originates from the pure consciousness of meaning. Systematic logic is connected to this philosophical logic in that it gathers its initial elements and prin-

ciples from the results of philosophical logic and develops from these a theory according to mathematical method.

In this way the part of genuine analytical knowledge is clearly separated {rom that of mathematical knowledge, and in this way it is emphasized what is justified, on the one hand, in Kant's Theory of Pure Intuition, and, on the other hand, in the

claim of the Iogicists. We can separate Kant’s basic idea that mathematical knowl-

edge, and in general the successful application of logical inference, rests on intuitive evidence, from the particular formulation that Kant gave to this idea in his theory of space and time. In this way we simultaneously gain the possibility of doing

justice to the very elementary character of mathematical evidence and to the level of abstraction {Abstraktionshohe] of the mathematical attitude. The claim concerning the logical character of mathematics aims at emphasizing these two aspects. The view we have reached also gives a simple piece of information about the role of number in mathematics: We have defined mathematics as the knowledge that rests on the formal (structural) consideration of objects. The numbers, however, as Numbers constitute the simplest formal determinations and as ordinal numbers the simplest formal objects.

The Number concepts offer a particular difficulty to philosophical discussion on account of their categorial special position, which also shows itself in language in the necessity for a separate category of terms, that is, the numerals [Zahlworte].

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We do not need to get more deeply involved with the discussion, but rather we only need to pay attention to the fact that the determinations of Number [Anzahibestimmungen] concern the composition from components of a total complex of that which

Is given or represented, that is, exactly what constitutes the structural side of an object. And indeed it is the most elementary structural characteristics that are given

by the Numbers. In this way the Numbers appear in all areas that are accessible to formal consideration; in particular, we come across the Number in the most diverse ways within theoretical logic, for example, as Number of subjects of a predicate (or as 1s said: as Number of arguments of a logical function), as the Number of the variable predicates that go into a logical proposition, as the Number of the applications of a logical operation within a concept-formation or within a proposition, as a Number of the propositions within an inferential figure, as the number of the type [Stufen-

zahl] of a logical expression, that is, as the maximum of nestings of the subject— predicate relationships that occur in it (in the sense of the rise from the objects of a theory to the predicates, from the predicates to the predicates of predicates, from

these again to their predicates, etc.).

The Numbers give us, however, only formal determinations but not yet formal objects. For example, in the representation [Vorstellung] of the number three the

combination of the three things into one object is still not present. The connection of several things into one object requires a form of ordering. The simplest ordinal form [Ordnungsform] is that of the simple succession, which leads to the concep t of ordinal number. The ordinal number is in and of itself also not determined as object; it is only a place marker {Stellenzeiger]. We can objectively [gegenstindl ich) standardize it by choosing as a place marker the simplest structure Jrom those that

originate in the form of the succession. Corresponding to the twofol d possibility of

beginning the number sequence with 1 or with 0, two kinds of standardizat ion may be considered. The first one is based on a sort of things and a form of addition of a thing; the objects are figures that begin and end with a thing of the appropriate sort, and in which every thing, which does not yet constitute the end of the figure, is followed by an added thing of that kind. In the other kind of standardization we have an initial thing and a process; the objects are then the initial thing 1itself and further the figures one obtains, beginning with the initial thing, through a single or repeated application of that process.

If we want to have, in the sense of the one or the other standardiz ation, the ordinal numbers as definite [eindeutig] objects free from all iness ential clements, then in each case we have to take the mere scheme of the relevant figure of repetition

[Wiederholungsfigur] as an object; this requires a very high abstraction. We are free

to represent these purely formal objects through concrete objects (“numerals”

[Zahlzeichen]); these contain then inessential, arbitrarily added properties that, how-

ever, are also easily grasped as such.* This proc edure takes place each time on the

basis of a certain convention, which must be adhered to in the context of one and the same consideration. Such a convention, in the sens e of the first standardization, 1s that according to which the first ordinal numbers are represented by the figures I, 11, 111, 1111. In accordance with one of the conv entions corresponding to the second standardization, the first ordinal numbers are represented by the figures 0,

0’, 07, 0", 0.

By thus finding easy access to the numbers from the structural side, our con-

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ception about the character of mathematical knowledge receives a new validation. For the dominating role of number in mathematics becomes comprehensible from

this conception, and our characterization of mathematics as the theory of structures seems to be the appropriate extension of the claim mentioned at the beginning, that

the numbers constitute the proper object of mathematics. The satisfactoriness of the point of view we have attained must not tempt us to

the opinion that we have already achieved all required basic insights required for the problem of the foundations of mathematics. We have indeed so far only treated

the preliminary question, about which we first wanted to gain clarity, namely, where

the specific character of mathematical knowledge is to be seen. Now, however, we must turn to that problem that causes the main difficulties in the foundations of mathematics, the problem of the infinite.

Il

1

The Problem of the Infinite and Mathematical Idea-formations [/deenbildungen]

The Postulates of the Theory of the Infinite; the Impossibility of Their Grounding by Means of Intuition—the Finitistic Point of View

The mathematical theory of the infinite is analysis (infinitesimal calculus) and its extension by means of general set theory. We can limit ourselves here to the con-

sideration of the infinitesimal calculus, since the step from it to general set theory requires the addition of assumptions but no fundamental modification of the philo-

sophical conception. The grounding of the infinitesimal calculus by Cantor, Dedekind, and Weier-

strass shows that the rigorous construction of this theory succeeds if, in addition to the elementary inference modes of mathematics, the following two are included:

1. The application of the method of existential inference [das existentiale Schliefien] to the whole numbers, that is, taking the system of whole numbers as basic in the manner of a domain of objects of an axiomatic theory—

as 1t is explicitly brought to expression in Peano’s axiom system of number theory,

2. The idea of the totality of all sets of whole numbers as a combinatorially surveyable [ibersehbar] manifold. A set of whole numbers 15 determined

by a distribution [Verteilung] of the values 0 and 1 to the positions in the number sequence. The number » either does or does not belong to the set depending on whether 1 or 0 is in the nth position of the distribution. Now, just as the totality of the possible distribution of the values 0, 1 1s completely surveyable for a finite number of positions, say five, this is analogously assumed for the whole number sequence.

In particular, the validity of Zermelo’s principle of choice [Auswahlprinzip] for collections of sets of numbers results from this analogy. We wish, however, to put aside the discussion of this principle for the time being; it will easily fit in later.

If we study these demands from the point of view of our general characterization of mathematical knowledge, then it seems, at first, that no kind of fundamental difficulty exists for their grounding through mathematical knowledge. For with

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the number sequence as with the set formations derived from it, it is a question of structures that differ from the structures dealt with in elementary mathematics only

in that they are structures of infinite manifolds. Existential inference applied to the numbers also appears to be justified through the object character of the numbers as formal objects, whose existence, however, cannot depend on the contingency of the

factual representations of the numbers.

However, in opposition to this argument it is to be noted that it is rash to infer from the character of the formal objects, that is, from the detachment [Losidsung] present in them from empirical contingencies, that the formal objects must be re-

lated to a realm of the formal existent [des existierenden Formalen]. We could mention the set-theoretical paradoxes as an argument against this view: however, it is simpler to refer directly to the fact that in primitive mathematical evidence there is no positing of such an area of existing formal objects, but rather that the tie to that

which was actually represented is essential as a point of departure for the formal abstraction. In this sense the Kantian proposition, that pure intuition is the form of

empirical intuition, is valid.

It also corresponds to this that in the disciplines that proceed from elementary mathematical evidence, existential statements only have an improper meanting. Par-

ticularly in elementary number theory we are concerned only with existential statements

that

relate

to

a quite

determined,

presentable

collection

[vorweisbare

Gesamtheit] of numbers or to a determined, intuitively performable [vorfiihrbar]

process, or with the two together, that is, to a collection of numbers that is to be arrived at by means of a performable process.

Examples of this kind of existential claims are: “There is a prime number between 5 and 107; namely, 7 is a prime number.

“For every number there is a larger one”; namely, when n is a numbe r, then one forms n + 1; this number is larger than .

“For every pnme number there is a larger one”; namely, if a prime number p is given, then one forms the product of this prime number with all prime numbers smaller than it and adds 1; if & is the resulting number, then amon g the numbers

from p + 1 to k there exists in any case a prime number. [u each of these cases the existential statement is made precise by means of further information; the existential claim keeps to the forma tion processes that can

be carried out [vollziehbar] in intuitive representation and does not refer to a manifold of all numbers. We wish to follow Hilbert and designate this elementary way of looking at things, linked to the conditions of basic representa bility, as the finitistic point of view [der finite Standpunkt] and in the same sense speak of finitary methods, finitary considerations, and finitary inferences [von finiten Methoden,

finiten Uberlegungen und finiten Schliissen).

It is now easy to determine that the existential inference goes beyond the fini-

tistic point of view. This already takes place in every existentia l proposition that is established without further specification of the existential claim, as, for example, in the statement that every unbounded arithmetic sequence a-n+b,

n=20,123,...

in which a, b are relatively prime, contains at least one prime number.

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A particularly common and important case of going beyond the finitistic point of view is the inference from the failure of the universal validity of a proposition (for all numbers) to the existence of a counterexample, or, in other words, the principle, according to which for every number predicate P(n) the alternative holds: Either the universal proposition is valid (i.e., P (r) holds of all numbers n), or there

1s a number #, such that P(n) does not hold. This principle results, from the point of view of existential inference, as a direct application of the principle of the excluded middle, that is, from the meaning of negation, That this logical consequence does not hold for the finitistic point of view lies in the fact that here the claim of the validity of P(r) for all numbers has the purely hypothetical meaning of validity for each given number, so that the negation of this claim does not yield a positive

existential statement. However, with this the discussion of the possibilities of an evident [einsichtig]

mathematical grounding of the assumptions of analysis is still not closed; granted,

that positing a total domain of formal objects as a basis does not correspond to the point of view of primitive mathematical evidence, however, the requirements of the

infinitesimal calculus could be motivated by the fact that the totalities of numbers

and of sets of numbers are structures of infinite sets. In particular, the application

of existential inference to the numbers should, accordingly, not be derived from the idea of the totality of numbers in the realm of formal objects, but rather from the consideration of the structure of the number sequence in which the single numbers occur as members of the sequence. We have in fact not yet dealt with the afore-

mentioned argument [to the effect] that mathematical knowledge could also concern structures of infinite manifolds. In this way we come to the question of the actual infinite. For the infinite that

is at issue in infinite manifolds is the proper actual infinite, in contrast to the “potential infinite,” which does not signify an infinite object but rather merely the un-

boundedness in the progress from the finite to an always new finite [zu immer neuem Endlichen], as 1s the case, for example, for the numbers even from the finitistic point of view, insofar as for every number one can always form a larger nember.

The question that we have to pose first of all as regards the actual infinite refers

to whether the actual infinite is given to us as an object of intuitive mathematical knowledge.

One could be of the opinion, in agreement with our previous statements, that

we are actually capable of an intuitive knowledge of the actual infinite. For even

though it is certain that we have a concrete representation only of finite objects, an accomplishment of formal abstraction could consist exactly in the fact that it frees itself from the limitation of the finite and that it completes, as it were, the passage to the limit in certain processes that can be arbitrarily continued. In particular, one will be tempted to refer to geometric intuition and adduce examples of intuitively given infinite manifolds from the domain of geometric objects. However, first of all, geometric examples do not prove anything. One is easily mistaken here in interpreting the intuitive-spatial [das Anschaulich-Réumliche] In

the sense of an existential conception. A segment, for example, 1s given intuitively not as an ordered manifold of points but rather as a homogeneous whole, though as an extended whole within which positions can be differentiated. The representation of a position on the segment is an intuitive representation, but the totality of ail po-

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sitions on the segment 18 only a conceptual totality [gedanklicher Inbegriff]. By

means of intuition we come here only to the potential infinite, in that to each posi-

tion on the segment corresponds a division into two parts [Teilstrecken], in which every part can again be divided into parts.

What further concerns infinitely extended entities, such as the infinite line, the infinite plane, the infinite space, is that these cannot be exhibited as objects of an

intuitive representation. In particular, space as a whole is not given to us in intuition. We imagine every spatial object as if it were contained in space. But this relationship of the single three-dimensional object [des einzelnen Raumlichen] to the whole of space is only objective in intuition as far as with every spatial object a

spatial environment is intuitively represented at the same time. Moreover, the inclusion in the whole of space [Gesamtraum] is—we have to claim this in Opposi-

tion to Kant—only intellectually graspable [gedanklich fassbar). The mam argument, which Kant adduces in favor of the intuitive character of our representation of space as a whole, proves in fact only that one cannot arrive at

the concept of a single inclusive space by means of a mere generalizing abstraction. But with the claim of the merely intellectual accessibility of our representation of the whole of space [Raumganzes], it should not be suggested that we are concerned

here with some kind of mere general concept.

What 1s meant 1s rather 2 more complicated state of affairs, namely, that in the representation of the whole of space two kinds of different thought-form ations

|Gedankenbildungen] are given, both of which go beyond the standpoint of intuition as well as beyond that of reflective [reflektierende] logic. The one relies on the concept of the connection of things to the world totality [Weltganzen] , it origi-

nates then from our belief in reality. The other is a mathematic al idea-formation

[{deenbildung] that certainly starts out with experience, but, however, does not remain in the reaim of that which can be intuitively represented; it is the representation of space as a point manifold subject to the laws of geometry.’

In these two ways of representing space as a whole, this totality is not recognized as given but rather only experimentally posited [versuchsweise angesetzt]. The representation of the physical whole of space is fundamentally problematic; never-

theless, exactly from the standpoint of present-day physics the possibility exists of giving this very vague idea a more narrow and precise formulatio n, through which it can become accessible and systematically important for resea rch. The geometric idea-formations of spatial manifolds are, to be sure, from the outset precise; how-

ever, they do require proot of their consistency.

We have then no reason to assume that we possess an intui tive representation of space as a whole. We cannot directly exhibit such a representation, and there is also no necessity to introduce this assumption as an explanatory ground. But if we deny the intuitiveness of the whole of space, then we also will not claim that infinitely extended spatial entities are intuitively represented.

It should also be noted that the original intuitive conception of elementary Euclidian geometry does not even require a represen tation of infinite entities. There we always have to deal only with finitely exte nded figures. Also, infinite manifolds never occur, since no general existenc e assumptions are used as a basis; rather every existence claim consists in the claim of the possibility of a geometrical construction.* For example, from this standpoint, that every segment has

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a midpoint says nothing other than that for every segment a midpoint can be con-

structed.8

With this the appearance of the exhibitability of the actual infinite in the realm of objects of geometric intuition turns out to be deceptive. However, we can make clear to ourselves in a more general way that a removal of the condition of finiteness through formal abstraction, as would be required for the intuition of the actual

infinite, is here out of the question. The condition of finiteness is not an accidental empirical restriction, but rather an essential characteristic of a formal object. ‘The empirical restriction still lies within the realm of the finite, where the formal

abstraction must aid us beyond the borders of the factual faculty of imagination {faktische Vorstellungskraft]. A clear example of this is the unlimited divisibility of a seg-

ment. Our actual imaginative faculty fails here, as soon as the division exceeds a certain degree of fineness. This threshold is physically accidental, and we can get beyond

it with the aid of optical apparatus. However, all optical apparatus fail at a certain de-

gree of minuteness, and in the end our spatial-metric representations become physically meaningless. With the representation of the unlimited divisibility we already abstract from the conditions of factual representation as well as from those of physical reality. Things are similar with the representation of unlimited addition in number theory. Here also there exist thresholds for the execution of the iterations both in the sense of actual representability as well as in the sense of physical realization. Let

us consider, for example, the number 1019190 We can arrive at this number in a finitistic way as follows: We start from the number 10 which we represent, according to one of our earlier standardizations, by the figure

1111111111

Let now z be any number that is represented by a corresponding figure. If in the previous figure we replace every 1 by the figure z , then we obtain again, as we can

intuitively make clear to ourselves, a number figure that for the purposes of com-

munication is denoted by “10 X z.”7 In this manner we obtain the process of a decuplication of a number. From this we arrive at the process of transition from a

number a to 102, as follows. We let the number 10 correspond to the first 1 in & and to every affixed 1 we apply the process of decuplication, and we keep going until we exhaust the figure a. The number obtained by means of the last process of decuplication is denoted by 107

This procedure offers basically no difficulty at all for the intuitive view. If we

want to visualize the process in detail, then our representation already fails with quite small numbers. We can aid ourselves somewhat with apparatus or with the usc [Heranziehen)] of objects of external nature in which very large Number determinations occur. But even with all this we soon come to a threshold: We can eas-

ily represent the number 20; 10%° exceeds by far our actual power of representation

but lies entirely in the realm of physical realization; finally, it 18 highly question-

able whether the number 1010 exists in any way as a ratio of quantities or as a

Number determination in physical reality. Intuitive abstraction is not concerned with such thresholds for the possibility

of the realization. For these thresholds are accidental from the point of view of the formal consideration. Formal abstraction, as it were, does not find an earlier

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Paul Bernays

position for a delimitation in principle than the difference between finite and in-

finite. This difference 1s in fact fundamental. If we consider more carefully how an

infinite manifold can be characterized as such, we find that this is not at all possible 1n the form of an intuitive exhibition but rather only by way of a claim (or an assumption or determination) of a lawlike relationship. Infinite manifolds are there-

fore accessible to us only through thought. This thought is, to be sure, also a manner of representation, but in this way what is represented is not the manifold as an object but rather conditions that are satisfied (or more precisely, are to be satisfied)

by a manifold.

The essential dependence of formal abstraction on the element of finiteness particularly asserts itself in that in considerations of totalities and of figures the property of finiteness does not constitute a particularly restrictive characteristic from the point of view of intuitive evidence. The restriction to the finite is accomplished automatically from this point of view, as it were, tacitly. We do not need here any

spectal definition of finiteness, for the finiteness of objects is a matter of course for

tormal abstraction. So, for example, the intuitive—structural introduction of the

numbers is appropriate only for the finite numbers. From the point of view of

intuitive—formal consideration, “iteration” is eo ipse finite iteration.

This representation of the finite, implicitly given in the formal view, contai ns the epistemic grounds for the principle of complete induction and for the admiss i-

bility of definition by recursion, both procedures being understood in their elemen tary form as “finite induction” and “finite recursion.”

Of course, this introduction to the representation of the finite does not belon g to that which necessarily enters from intuitive evidence into togical inference. It corre-

sponds rather to a point of view, according to which one already reflects on the general characteristic features of the intuitive objects. The application of the intuitive

representation of the finite can be avoided for number theory if one forgoes treating

this theory 1n an elementary way. However, the intuitive representati on of finiteness shows up of necessity as soon as one makes formalism itself the object of consideration, 1n particular in the systematic theory of logical inference. In this way it is ex-

pressed that finiteness is an essential element of the entities [Gebil de] of every formalism. The boundaries of the formalism are, however, none other than those of representability, particularly of intuitive compositions |[Zusammen setzungen).

Thus our answer to the question about the intuitive knowabilit y of the actoal infinite is negative. Another result is that the method of finiti stic consideration is the appropriate method from the point of view of intuitive mathe matical knowledge. However, in this way we do not achieve a verification of the mentioned assumptions for the infinitesimal calculus.

2

Intuitionism—Arithmetic as Theoretical Frame

How should we react in the face of this state of affairs? The views are divided as

to how to answer this question. A similar conflict of views takes place to the one we have encountered in the question of the characteri zation of mathematical knowledge. The proponents of the point of view of primitive intuition [primitive

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251

Anschaulichkeit] easily conclude from the circumstance that analysis and set theory go beyond the finitistic point of view on account of their postulates that these mathematical theories are to be abandoned in their present form and must be revised from their very foundations. The supporters of the point of view of theoretical logic, on the other hand, seck either to ground those postulates of the theory of the infinite by means of logic, or they dispute in general the problematical nature of the postulates in that they do not attribute any fundamental significance to the differ-

ence between the finite and the infinite.

The former view was, already at the time of the first appearance of the method of existential inference, represented by Kronecker, who was probably the first to examune closely and strongly emphasize the importance of the methodical point of view, which we term the finitistic. His attempts towards the fulfillment of this

methodological demand in the area of analysis remained, however, fragmentary;

moreover, there was a lack of an exact philosophical explanation of the point of view. In particular, the often-cited remark of Kronecker, that God created the whole

numbers and that everything else is the work of man, is not at all suited to the mo-

tivation of the demands represented by Kronecker®: If the whole numbers are created by God, then one should think that the existential inference is-admissible for application to the numbers, while Kronecker, however, excludes exactly the existential way of consideration in the case of the whole numbers.

Brouwer extended Kronecker’s point of view in two directions: on the one hand

with regard to philosophical motivation by putting forward his theory of “intu-

itionism,”” and on the other hand by showing how one can apply the finitistic point of view in the area of analysis and set theory and how one can ground these theories at least to a considerable extent in a finitistic way through a radical reformula-

tion of the concept formations and the modes of inference. The result of this investigation naturally has a negative side in that it becomes

apparent that in this process of finitistic treatment of analysis and set theory one has to accept not only considerable complications but also heavy losses in systematics

[Systematik]).

The complications appear already in the first concepts of the infinitesimal cal-

culus, as in those of boundedness, the convergence of a number sequence, the difference between rational and irrational. Let us take, for example, the concept of

boundedness of a sequence of whole numbers. According to the usual view the following alternative exists: Either the sequence exceeds every bound, it is then un-

bounded, or all numbers of the sequence lie below a bound, then the sequence is bounded. In order to retain a finitistic conceptual determination, we must sharpen

the definition of boundedness and unboundedness in the following way: A sequence is said to be bounded if we can exhibit a bound for the numbers of the sequence ei-

ther directly or by specifying a procedure; a sequence 1s said to be unbounded if there is a rule, according to which every bound of the sequence is necessarily exceeded, that is, if the assumption of a bound for the sequence leads to an absurdity. By means of this formulation of the concepts the finitistic character of the definitions is obtained, but we now do not have a complete disjunction between the case of boundedness and the case of unboundedness. Therefore, from a proof that shows the untenability of the assumption of the unboundedness of a sequence, we

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still cannot derive the boundedness of the sequence. By the same token. a proposition that 1s proved on the one hand under the assumption of the boundedness of a certain number sequence and, on the other hand. under the assumption of its un-

boundedness, cannot be considered as proven. In addition to these kinds of complications, which pervade the whole theory. there is aiso the more important disadvantage that the general theorems through which mathematics gains its systematic clarity become for the most part invalid.

Thus, for example, in Brouwer’s analysis, not even the following theorem is valid: Every continuous function has a maximum (in a finite, closed interval). It appears as an unjustified demand upon mathematics on the part of philosophy that mathematics should give up its simpler and more powerful methods in favor of an inconvenient method that is lacking in systematics [Svstemarik]. without being led to do so by an inuer necessity. The point of view of intuitionism becomes suspicious to us because of this unreasonable reguest. Let us consider what the main points of the philosophical view developed by Brouwer amount to. It contains first of all a characterization of mathematical evidence. Our preceding remarks about formal abstraction are in agreement with this characterization on the essential points, particularly in taking its starting point in Kant’s theory of pure intuition.

A difference, of course, consists in the fact that according to Brouwer's view. the temporal element belongs essentially to mathematical objectivity. However, we do not need to go into a discussion of this point here, since which way we decide

on this issue has no influence on the formulation of the methodologic al question concerning mathematics: What for Brouwer follows as a conse quence of the time dependence [Zeir-Gebundenheit] of mathematics is nothing other than what we have derived from the dependency of formal abstraction on the concre te—intuitive point of departure, namely, the methodical limitation of the finitistic proce dure, The crucial consequences of intuitionism follow from the furth er claim that any mathematical thinking that should be able to claim scientific validity must be carried out on the basis of mathematical evidence: that is, that the boundaries of mathematical evidence are simultaneously boundaries for mathematic al thinking, This demand of the restriction of mathematical thinking to that which is intuitively evident seems at first to be completely justified. Indeed, it corresponds to the view of mathematical certainty familiar to us. We must . however, consider that this

common view of mathematics originally belonged together with a philosophical

view, for which the intuitive evidence of the foundati ons of the infinitesimal calculus was not in question. We have departed from such a view because we found that the postulates of analysis cannot be verified by intui tion, that is. that the idea of an infinite totality, taken as a basis in analysis, is not graspable in intuition but rather only in the sense of an idea-formation. We cannot expect that this new view of the boun daries of intuitive evidence is readily consistent with the traditional view of the epis temic character of mathemat-

ics. Indeed, our remarks are grounds for suspicion that the widely held view of mathematics represents the facts of the matter too simp listically, and that we cannot account for that which takes place in mathematics from the point of view of evidence alone, but rather we must still grant thoughf its own role.

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We thus come to a differentiation between the elementary mathematical point

of view and a systematic point of view that goes beyond it. This differentiation is not drawn artificially or merely ad hoc, but rather it corresponds to the duality of

the points of departure that lead to arithmetic, namely, on the one hand the combinatorial activity with ratios in discrete quantities [mit Verhdltnissen im Diskreten]

and on the other hand the theoretical demand that is placed on mathematics from

geometry and physics. ! The system of arithmetic does not emerge only from a constructive and intuitively contemplating activity (konstruierende und anschaulich betrachtende Tdtigkeit], but rather mostly from the task to grasp exactly and master theoretically the geometric and physical ideas of set, area, tangent, velocity, etc. The

method of arithmetization is a means to this end. In order to serve this purpose, however, arithmetic must extend its methodical point of view from the original el-

ementary point of view of number theory to a systematic view in the sense of the aforementioned postulates. Arithmetic, which forms the large frame in which the geometric and physical disciplines are incorporated [eingeordnet], does not simply consist in the elementary—

intuitive treatment of the numbers, but rather it has itself the character of a theory in that it takes as a basis the idea of the totality of numbers as a system of things as well as of the idea of totality of the sets of numbers. This systematic arithmetic fulfills its task in the best way possible, and there is no reason to object to its pro-

cedure, as long as we are clear about the fact that we do not take the point of view of elementary intuitiveness but that of thought-formation, that is, that point of view Hilbert calls the axiomatic point of view. The reproach of arbitrariness is not justified against this axiomatic process, for

in the foundations of systematic arithmetic we are not dealing with an arbitrary system of axioms, set up according to need, but rather with a natural systematic extrapolation of elementary number theory. And the analysis and set theory that de-

velop from this foundation constitute a theory that is already purely intellectually

distinguished [rein gedanklich ausgezeichnet), that is, suited to be taken as the the-

ory kot é€oymy in which we incorporate the systems and the theoretical statements of geometry and physics.

We therefore cannot recognize the veto that intuitionism directs against the pro-

cedure of analysis. The statement, about which we are in agreement with mntuitionism, that the infinite is not given to us intuitively, compels us to modify our philosophical view of mathematics but not to reshape mathematics itself. However, the problem of the infinite returns. For by taking a thoughtformation as the point of departure for arithmetic we have introduced something

problematic. An intellectual approach, however plausible and natural from the systematic point of view, still does not contain in itself the guarantee of its consistent realizability [Durchfiihrbarkeit]. By grasping the idea of the infinite totality of numbers and the sets of numbers, it is still not out of the question that this idea could lead to a contradiction in its consequences. Thus it remains to investigate the question of freedom of contradiction [Widerspruchsfreiheit], of the “consistency” [Kon-

sistenz] of the system of arithmetic.!!

Intuitionism wants to spare us this task by limiting mathematics to the realm of finitistic consideration; this elimination of the problem, however, asks too high

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a price: The problem disappears, but the systematic simplicity and clarity of analy-

sIs are also lost.

3

The Problematic of Logistic Theory—Value of the Logistic Incorporation [Einordnung) of Arithmetic

The proponents of the logistic point of view believe to be able to come to terms with this problem in a completely different way. With the discussion of this point of view we take up our earlier reflections on logistics. There it was a matter of recognizing that intuitive evidence aiready enters into deductive logic and that the log-

ical Number definitions do not prove the Number concepts as such to be of a specifically logical nature (as pure concepts of reflection), but are rather only logical standardizations of elementary structural concepts.

These considerations concern the separation of the [realm of the] logical in the more narrow sense from the [realm of the] formal. But with the recognition of the

formal element in logic the methodological question of logistics is still by no means resolved. Logistics is not content with the theoretical development of the theory of

inferences, but rather, as already mentioned, it takes as its task moreover to incorporate all of arithmetic into the logical formalism. This incorporation takes place in the following way: First of all, the Numbers are introduced as properties of predicates in the manner described earlier, and in addition—in a way that cannot be explained i more detail here—one expresses the modes of formation of the sets of numbers by means of the logical formalism, in doing so replacing every set with a

predicate that defines it. The totality of number predicates thus takes the place of

the totality of all sets of numbers.

'

In this way one indeed succeeds in assigning to every arithmetic statement a statement from the realm of theoretical logic, in which apart from the variables only “logical constants” occur, that is, logical basic operations, such as conjunction, nega-

tion, the form of universality, etc.

It is now clear that the problem of the infinite still cannot be solved only by means of this translation of arithmetic into the logical formalism. If theoretical logic deductively produces the system of arithmetic, then either explicit or hidden as-

sumptions, through which the introduction of the actual infinite is brought about, must be contained in its procedures.

The justification of these assumptions and the positions on them consti tuted right from the beginning the weak point of logistics. In this way Frege and Dedekind, whose arguments and considerations are otherwise marked by extreme precision and

nigor, were uncritical in basing the point of view of general logic on a supposedly evident assumption, namely, the idea of a completed totality of all conceivable log1cal objects. This idea, if it were tenable, would of course be systematic ally more satisfactory than the specific postulates of arithmetic. As is well known , this idea had to be

dismissed on account of the contradictions to which it gave rise. Since then logistics does not attempt to prove the existence of an infinite totali ty; rather, it explicitly posits an axiom of infinity. This axiom of infinity still does not suffice as an assumption for obtaining arithmetic as conceived logically. With it we would only obtain that which results from

The Philosophy of Mathematics and Hilbert's Proof Theory

255

the application of our first postulate, that is, from the admissibility of existential inference with regard to the whole numbers. In order to be in keeping with our second postulate, something more is required, namely, the application of existential in-

ference with regard to the predicates. The justification of this procedure may appear at first to be logically obvious, and for the view Frege and Dedekind took as a point of departure, it does not actually come into question. With the abandonment of the idea of the totality of all logical objects, the idea of a totality of all predicates also

becomes problematic, and upon closer examination a particular fundamental difficulty becomes apparent. Namely, it corresponds to the logical point of view that we conceive the totality of predicates as something, which for the most part come into being only in the

context of the system of logic in such a way that the logical formation processes are applied to certain prelogical initial predicates that are derived, say, from intuition. By referring to the totality of predicates, more predicates are in turn obtained. An example of this is Frege’s definition, mentioned earlier, of finite number: “A number #r 1s said to be finite, if every predicate that applies to the number ¢ and

that, 1f it applies 1o the number a, also applies to its successor, also applies to n.” Here the predicate of fmiteness is defined with reference to the.totality of all

predicates.

Such definitions—termed “impredicative”!>—appear everywhere in the foundations of arithmetic, and indeed in decisive places. In itself there is nothing to object to in the fact that one determines a thing from a totality by a property that is related to this totality. So, for example, in the totality of numbers, a certain number is defined by the property that it 1s the largest of

all the prime numbers, whose product by 1000 is larger than the product of the previous prime number by 1001.1? But here it is presupposed that the totality concerned is determined independently of the definitions that refer to it; otherwise we enter into a vicious circle.

This precondition cannot easily be fulfilled, however, and precisely in the case

of the totality of the predicates and of the impredicative definitions that refer to them, because the field of predicates is determined—according to the view discussed

here—by means of the logical laws of formation [Bildungsgesetze], and the impredicative definitions also belong to this group.

In order to avoid the vicious circle, 1t would, however, suffice if it could be shown that eévery predicate introduced by an impredicative definition can also be defined in another way “predicatively.” Indeed, one would even manage with a weaker statement. Since within the logical foundation of arithmetic a predicate is only considered with regard to its extension, that is, with regard to the set of things

to which it applies, we would only need to know that every predicate introduced by an impredicative definition is co-extensional with a predicate that is predicatively defined.

Russell, who recognized with complete clarity the difficulty present in the impredicative definitions, posited this postulate as “axiom of reducibility” next to the axiom of infinity.

How, then, are we to understand this axiom of reducibility? It does not emerge from its formulation whether what is supposed to be expressed by it is a logical law or an extralogical assumption.

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In the first case, that is, if the reducibility axiom were the expression of a logical law, its validity would have to be independent of what kind of domain of prelogical initial predicates is used as a basis—provided at least that this domain satisfies the axiom of infinity. That would, however, mean that an axiomatic theory in which the forms of universal and existential judgment (existential inference) are ap-

plied only to the objects and not to the predicates is not capable of an extension of its predicate domain by means of the introduction of impredicative definitions provided only that the system of axioms is such that it requires an infinite system of

objects to be satisfied.

However, the validity of such a statement is out of the question. One can easily construct examples that disprove this claim.

Such an example is provided by Dedekind’s introduction of the number concept. Dedekind proceeds from a system in which an object 0 is distinguished and admits a reversible, one-to-one mapping onto a subset, to which that thing 0 does

not belong. If we represent this mapping by means of a predicate with two subjects and formulate the required properties of this predicate as axioms, then we obtain an elementary axiom system, which in its axioms contains no reference to the totality of predicates and that furthermore can only be satisfied by an infinite system of ob-

jects. Let us now consider Dedekind’s concept of number; its definition can be formulated quite analogously to Frege’s definition of finite number, by translating it

from the language of set theory into the language of theory of predicates: “a thing n of our system is a number, if every predicate that applies to 0 and that, if it applies to a thing a of our system, also applies to that thing that is assigned to the

thing a in the reversible one—one mapping, also applies to #.” This definition is impredicative; and one can see that it is not possible to obtain a co-extensive predicate to that just defined of “being a Number by means of a predicative definition

from the basic elements of the theory.”'4 We find consequently that we only need to consider the second Interpretation

of the reducibility axiom, according to which the axiom expresses a deman d on the iitial domain of prelogical predicates [eine Anforderung an den Ausga ngsbereich der vorlogischen Priidikate].

With the introduction of such an assumption, however, one renounces the idea that the domain of the predicates is produced by the logical processes. The goal of an essentially logical theory of predicates is thereby abandoned.

If one decides to do this, then it appears more natural and more appropriate to

return to that idea of lugical function that corresponds to Schroder’s point of view:

One considers a logical function as a distribution [Verteilung) of the values “true” and “false” to the things in the domain of individuals. Every predicate define s such a distribution, but the totality of the value distributions is conceived, in analogy to the finite, as a combinatorial manifold existing independently of the conceptual definitions. By means of this conception, the circularity of the impredicat ive definitions of theoretical logic is removed; we only need to replace every state ment about the totality of predicates with the corresponding statement about the totality of logical functions. Consequently, the axiom of reducibility becomes dispe nsable.

The logicist school took this step at the suggestion [Anregung] of Wittgenstein and Ramsey. In particular, they pointed out that in order to avoid the contradictions that are connected to the concept of the set of all mathematical objects, it is not nec-

The Philosophy of Mathematics and Hilbert's Proof Theory

257

essary o carry out a differentiation of the predicates according to the form of their

definition as Whitehead and Russell did in “Principia Mathematica”; rather, it is

sufficient to delimit clearly the domain of definition of the predicates, so that one differentiates between predicates of individuals, predicates of predicates, predicates

of predicates of predicates, and so on. In this way one returns from the type theory [Stufentheorie] of “Principia Math-

ematica” to the simpler conceptions of Cantor and Schroder. One should, however, not deceive oneself into thinking that one thereby has

not left the point of view of logical self-evidence. The assumptions, which in this way are put at the basis of theoretical logic, are in principle of the same type as the basic postulates of analysis and are completely analogous to them in content: The

axitom of infinity in the logical theory corresponds to the idea of the number se-

quence as infinite totality, and instead of the totality of all sets of numbers, the to-

tality of all logical functions (related to the “domain of individuals” or to a definite domain of predicates) is postulated here. Thus by mcorporating arithmetic in the system of theoretical logic, one does not cut down on assumptions. This incorporation does not at all have, as one might

have thought at first, the significance of a reduction of the posiulates of arithmetic to more basic assumptions; its value lies rather in the fact that the mathematical theory is placed on a wider basis by means of its union with the logical formalism. In the first place, the theory gains, in this way, a higher degree of methodological

distinction in that it is shown that its assumptions are obtained not only from the intuitive number theory by means of a natural extrapolation, but rather also equally result by extrapolating extensional logic in the sense of an extension to infinite totalities.

In addition, through the joining of arithmetic to theoretical logic we gain an insight into the connection of the process of set formation with the logical basic op-

~ erations, and the logical structure of the concept-formation and of the inferences stands out more clearly. In particular, the meaning of the principle of choice becomes in this way completely comprehensible only through the logical formalism. We can express the prin-

ciple in the following form: If B(x, y) is a predicate with two subjects (defined in a certain domain) and if for every thing x of the domain of definition there is at least

a thing y of this domain for which B(x, y) holds, then there is (at least) one function y = f{x) with the property that for every thing x of the domain of definition of B(x, y)

the value f(x) is again a thing of this domain, and indeed such that B(x, f(x)} holds. If one considers what this claim means for the special case of a subject domain with two individuals, whose things we can represent by the numbers 0, 1 and for

which only four different courses of value of functions y = f (x) come into consideration, then one finds that the claim turns out to be a simple application of one of the distributive laws that is valid for the relationship between conjunction and disjunction, namely, an application of the following elementary—logical proposition:

“If A holds and if B or C holds as well, then either A and B hold, orA4 and C hold.”!>

In the case of any given finite Number of things of the subject domain, the statement of the principle of choice also follows from this distributive law. The universal statement of the principle of choice is then nothing other than the extension of an elementary-logical law for conjunction and disjunction to infinite totalities, and the principle of choice constitutes thus a completion of the logical rules that

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concerns the universal and the existential judgment, that is, of the rules of existen-

tial inference, whose application to infinite totalities also has the meaning that certain elementary laws for conjunction and disjunction are transferred to the infinite, In comparison with these rules of existential inference, the principle of choice is entitled to a special position only to the degree that the concept offunction is re-

quired for its formulation; and this concept in turn receives an adequate implicit characterization only through the principle of choice.

This concept of function corresponds to the concept of logical function but with the difference that “true” and “false” are not taken as function values, but rather the

things of the subject domain. The totality of functions, with which we are concerned here, s then the totality of all possible self-coverings [Selbsthelegungen) of the sub-

ject domain,

In the sense of this concept of function, the existence of a function with property £ 1n no way guarantees the existence of a concept-formation through which a determinate function with property E is uniquely fixed. In considering this circumstance the usual objections against the principle of choice become invalid: these ob-

jections are for the most part based on the fact that one is misled by the name “principle of choice” to believe that the principle claims the possibility of a choice. Simultaneously we recognize that the assumption, which finds expression in the principle of choice, does not fundamentally go beyond the conception upon

which we otherwise already had to base the procedure [Verfahren] of theoretical logic in order to be able to interpret it in a noncircular way without the introduction of an axiom of reducibility. Naturally we can give this statement the opposite emphasis: The contentiousness of the principle of choice, which is in keeping with the aim of a consistent presentation of the point of view of theoretical logic, makes us realize the proble matic

nature of this point of view in an especially forceful fashion. This is the result to which the consideration of the logicist foundation of arithmetic has led us as well, that is, that this procedure of incorporating arithmetic into

theoretical logic does certainly create a wider foundation for arithmetical theory and contributes to the contentual grounding [Motivierung] of its assump tions; it does not, however, go beyond the methodical point of view of the ideal approa ch, which is beyond the point of view of axiomatics.

In this way, the problem of the infinite is formulated, but not solved . For it remains to be seen whether the analogies, postulated as assumptions for the construction of analysis and set theory, between the finite and the infini te form an admissible, that is, a consistent and feasible, theoretical approach |Gedankenansatz]. This question, which intuitionism wants to avoid by eliminating the problematic assumptions, and whose justification is for the most part challenged by the logicists, in that they do not at all recognize a basic difference betwe en the finite and the infinite, is tackled positively by Hilbert's Proof Theory.

4

Hilbert's Proof Theory

In order better to grasp the main ideas of proof theory, we need first of all to recall what kind of problem we have to solve. It is a matter of prov ing the consistency of the mathematical idea-formation on which the edifice of arithmetic rests.

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From the side of philosophy the question has been repeatedly raised whether a proof of consistency suffices as a justification for this idea-formation. This formulation of the question is misleading; it does not take into account the fact that the

sctentific grounding of the theoretical approach to arithmetic has been achieved for the most part through science, and that the proof of consistency is indeed the only desideratum that still needs to be fulfilled.

The edifice of arithmetic is constructed on the basis of thoughts that are of important significance for scientific systematics in general, namely, on the principle of the conservation (“Permanence”) of the laws [Prinzip der Erhaltung (“Permanenz” ) der Gesetzlichkeiten], which here appears as the postulate of the unlimited

applicability of the usual logical forms of judgment and inference, and on the de-

mand of a purely objective formulation of the theory through which the latter is detached from every reference to our knowledge. In the fundamental methodological meaning of these demands lies the inner grounding and specificity of the approach of the arithmetical theory.

To this inner grounding is added the magnificent way in which the conceptual system of arithmetic in the sense of its deductive fruitfulness, its systematic suc-

cess, and the unanimity of its consequences has proved its worth. The suitability of

this conceptual system for the mastering of ratios, both of Numbers and of magni-

tudes, 1s spectacular. The order [Systematik] of the magnificent edifice, which emerges through the union of function theory with number theory and algebra, is unequalled. And as a comprehensive conceptual apparatus for theory-formation in the natural sciences, arithmetic turns out to be not only suitable for the formulation

and development of the laws, but it is also invoked with great success, to a degree earlier undreamt of, in the search for the laws. As for what further concerns the unanimity of the consequences, this is tested,

in the best possible way, through the intensive theoretical development and the frequent numerical application of analysis.

What is still missing here is the achievement of a true insight into the consis-

tency of arithmetic, that is, into the constant agreement of its results, in place of the simple empirical confidence on the consistency obtained by repeated testing. To

bring this about is the task of the consistency proof.

The situation is not that the conceptual system of arithmetic should be first established by means of the proof of consistency, but rather the task of the proof con-

sists exclusivély in creating for us the complete, insightful certainty that this conceptual system, which is already motivated by inner reasons of systematics and

tested in its implementation as intellectual apparatus in the best possible way, cannot collapse [zu Falle kommen] on account of an inconsistency of its consequences.

If this is successful, then we know that the idea of the completed infinity can be carried out in a consistent way. And we can then rely on the results of the application of the basic postulates of arithmetic just as if we were 1 the position to verify these intuitively. For by recognizing the consistency of the application of these postulates, it is established at the same time that an intuitivé proposition that is interpretable in the finitistic sense, which follows from them, can never contra-

dict an intuitively recognizable fact. In the case of a finitistic proposition, however, the determination of its irrefutability is equivalent to the determination of its truth. What in particular emerges from this consideration about the requirement and

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the purpose of the consistency proof is that this proof is only a maiter of seeing [einsehen] the consistency of arithmetic theory in the literal sense of the word, that 18, the impossibility of its immanent refutation. The novelty of Hilbert’s approach s that he limited himself to this formulation

of the problem, while formerly one had always carried out the consistency proof for an axiomatic theory in the sense that by means of it one showed positively at the same time the satisfiability of the axioms by means of certain objects. For this

method of demonstration [Aufweisung] the case of artthmetic gives no handle; in particular, Frege’s idea, to take the objects to be exhibited {aufzuweisen] from the

field of logic, therefore falls short of its goal, because, as we have made clear, the application of ordinary logic to the infinite is just as problematic as arithmetic, which 18 to be proven consistent. The basic postulates of the theory of arithmetic concern exactly the extended application of the usual forms of judgment and in-

ference. With the realization of this fact, we are led directly to the first guiding principle of Hilbert's proof theory: This states that in the consistency proof of arithmetic we are to include the laws of logic, as they are applied in arithmetic, into the domain of that which is to be proven consistent, so that the proof of consistency applies jointly to logic and arithmetic.

The first important step in the implementation of this idea has been already taken by means of the incorporation of arithmetic in the system of theoretical logic.

On the basis of this incorporation, the task of the consistency proof of arithmetic amounts to recognizing theoretical logic as consistent, or, in other words, estab-

lishing the consistency of the axiom of infinity, of impredicative definitions, and of the principle of choice. '

It is here advisable to replace Russell’s axiom of infinity by Dedekind’s characterization of the infinite.

Russell’s axiom of infinity demands for every finite number # (in the sense of Frege’s definition of finite Number) the existence of an n-valued predicate, where by the infinity of the domain of individuals (of the starting domain of things) is also implicitly demanded. It is an unnecessary complication that ought to be critici zed

in principle, that three infinities occur here side by side at different levels: that of the infinitely many things of the domain of individuals; furthermore, that of the infimtely many predicates; and then finally, that of the infinitely many Numbe rs resulting from the above, which are, after all, defined as predicates of predica tes.

We can avoid this multiplicity by determining the infinity of the domain of in-

dividuals by means of a single predicate with two subjects rather than by an infinite series of predicates with one subject. Such a predicate provides a reversible

one-to-one mapping of the domain of individuals onto a proper (that is, at least ex-

cluding one thing) subset of the domain. The introduction of this Dedekindia n char-

acterization of the infinite takes shape in the most simple and most eleme ntary way, if we postulate the reversible, one-to-one mapping not by means of an existence axtom but rather introduce it immediately in an explicit way, by takin g an initial object and a basic process as basic elements of the theory,

In this way it is achieved that the numbers occur not only as predicates of pred-

1cates but already as things of the domain of individuals.

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This consideration already refers to the peculiar kind of implementation of the systematic construction, concerning which numerous ways are open. However, we must still orient ourselves in general as to how a consistency proof might be car-

ried out in the desired sense at all. This possibility is not immediately apparent. For how can one survey all possible inferences that result from the assumptions of arithmetic, or from those of theoretical logic?

Here the investigation of mathematical proofs by means of the logical calculus comes

to

bear

decisively.

The

logical

calculus

has

shown

that

the

concept-

formations and inferences that are applied in the theories of analysis and set theory are reducible to a limited Number of processes and rules, so that one succeeds in com-

pletely formalizing these theories in the context of a precisely defined symbolism. From the posstbility of this formalization, which was originally pursued only

for the purpose of a more exact logical analysis of the proofs, Hilbert has drawn the conclusion—this is the second leading thought of his proof theory—that the task of

the consistency proof of arithmetic is a finitistic problem. A contradiction in the contentual theory must show itself on the basis of the

formalization in such a way that according to the rules of the formalism two formulas are derivable, such that one of them originates from the other by means of that process that forms the formal representation [Abbild] of negation. The consistency statement 1s therefore equivalent to the statement that two formulas, which are 1n the aforementioned relationship, cannot both be derived according to the rules

of the formalism. However, this statement has fundamentally the same character as

any general statement of finitistic number theory, for example, the statement that it is impossible to give three whole numbers 4, b, ¢ (different from ) for which the

relationship a® + b* = ¢’ holds. So the consistency proof for arithmetic amounts indeed to a finitistic problem

of the theory of inference. Hilbert calls the finitistic investigation, which has as object the formalized theories of mathematics, metamathematics. The task, which

is the role of metamathematics vis-d-vis the system of mathematics, is analogous to that which Kant assigned to the critique of reason vis-a-vis the system of philosophy.

in the sense of this methodological program, proof theory has already made

considerable progress,'® though there are still considerable mathematical difficulties to be overcome. Through the proofs carried out by Ackermann and von Neu-

mann, the consistency for the first postulate of arithmetic, that is, the applicability of existential inference to whole numbers, is established. There is also a more developed attempt by Ackermann for the further problem of consistency of the general concept of sets of numbers (of numerical function, respectively), including the associated principle of choice.

With the solution of this problem, almost the entire domain of existing mathe-

matical theories would be proven to be consistent.!” In particular, this proof would

be sufficient to recognize the geometrical and physical theories as consistent. One can proceed further in the formulation of the problem and investigate the

consistency for more comprehensive systems, for example, for axiomatic set theory. Axiomatic set theory, as first put forward by Zermelo and supplemented and extended by Fraenkel and von Neumann, reaches beyond what s factually needed

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in mathematics in its formation processes, and with the establishment of its consistency, the system of theoretical logic would also be proven consistent. But even this does not achieve an absolute completion of concept-formation. For, formalized set theory gives once more occasion to a metamathematical con-

sideration, which has as its objects the formal formations of set theory and thereby

also goes beyond these formations. '8 This possibility of the extension of the concept-formation notwithstanding, a

formalized theory can nevertheless have the character of completeness, namely, if by means of the extension of the concept-formation, no new results in the domain

of the laws that can be formulated by the concepts of the theory come into being. This condition is then in any case fulfilled if the theory is in general deductively complete [abgeschlossen], that is, if it is impossible to add to it a new axiom, ex-

pressible in the concepts of the theory that is not already derivable in such a way that no contradiction emerges—or what amounts to the same thing: if every statement that

can be formulated in the context of the theory is either provable or refutable.’®

We believe that number theory, as it is defined by Peano’s axioms with the addition of definition by recursion, is deductively complete in this sense; the problem

of an actual proof for this is, however, still completely unsolved. The question becomes even more difficult if we proceed beyond the domain of number theory, to

analysis and the further concept-formations of set theory. In the area of this and related questions, a considerable field of problems still remains open. These problems, however, are not of the kind that they represent an

objection to the point of view we have adopted. We must only keep in mind the fact that the formalism of statements and proofs, with which we represent our ideaformation, does not coincide with the formalism of that structure we intend in the

concept-formation. The formalism is sufficient to formulate our ideas about infinite manifolds and to draw from these the logical consequences, but it is, in general, not capable of producing the manifold, as it were, combinatorially from within.

The view at which we have arrived concerning the theory of the infinite can be

seen as a kind of philosophy of the “as if.” However, it differs entirely from the so-

calied philosophy of Vaihinger in the fact that it emphasizes the consistency and the stability [Bestdndigkeit] of the idea-formations, whereas Vaihinger regard s the demand of consistency to be a prejudice and straightforwardly states that the con-

tradictions in the infinitesimal calculus are “not only not to be argue d away, but rather . . . exactly, the very means by which progress is achieved.”?¢ Vailhinger’s observation is exclusively focused on scientific heuris tics. He

knows only “fictions,” which appear as mere temporary tools of thought, throug h whose introduction thought violates itself and whose contradictory charac ter (if it is a matter of “true fictions”) is rendered harmless only by means of skillfu l com-

pensation of the contradictions.

The idea-formations in our sense are the enduring property of the mind [bleiben-

des Eigentum des Geistes). They are outstanding forms of systematic extrapolation and of the idealized approximation to the factual {das Tatséichliche}. They are in no way something arbitrary, nor are they forced upon thoug ht: on the contrary, they

form a world, in which our thought is at home and from whic h the human mind that becomes absorbed in them draws satisfaction and joy.

The Philosophy of Mathematics and Hilbert’s Proof Theory

263

Appendix On the basis of various insights that have emerged since the publication of the above essay, there are a few corrections to be made to what has been stated here.

First of all, as regards intuitionism, it was thought at first that the methodology of mntuitionistic proofs coincided with that of Hilbert’s “finitistic point of

view.” However, it has become clear that the methods of intuitionism go beyond the finitistic proof procedures intended by Hilbert. In particular, Brouwer makes

use of the universal concept of contentual proof, to which the concept of “absurdity” is also connected, and which, however, is not made use of in finitistic

inference. As for what then concerns Hilbert’s proof theory, the opinion that the con-

sistency proof for arithmetic boils down to a finitistic problem, is well founded only in the sense that the statement of consistency can be formulated in a finitistic sense. However, from this it does not follow at all that the problem is solvable with finitistic methods. On the basis of a theorem of Godel, the possibility

of a finitistic solution has been made highly implausible for number theory, if not

completely ruled out, and moreover it turned out that the mentioned consistency proofs that were at hand at the time did not suffice for the total formalism of num-

ber theory. The methodical point of view of proof theory was consequently extended, and different consistency proofs have been carried out, first of all for for-

malized number theory and then also for formal systems of analysis, whose proof methods are certainly not limited to finitistic, that is, to the elementary, combinatorial consideration, but which, however, also do not require the usual methods of existential inference or, on the other hand, the general concept of contentual

proof.

In connection with the mentioned theorem of Godel, the assumption that the axiomatically defined and formalized number theory is deductively complete turned out to be wrong. More generally, it has been demonstrated by Godel that formal-

ized theories, which satisfy certain, very general conditions of expressiveness as well as sharpness of formalization, as long as they are consistent, cannot be de-

ductively complete.

On the whole the situation is such that Hilbert’s proof theory, in connection

with the uncovering of the possibilities of formalizatton of mathematical theones, has created a rich area of research, and, however, the epistemological points of view,

from which its establishment started out, have become problematic. This gives cause to revise the epistemological observations of this essay. Of course, the positive observations, in particular the demonstration of the mathematical element in logic and the emphasis on elementary arithmetical evidence, are

hardly in need of revision. However, the sharp distinction between the mtuitive and the not intuitive, as it is employed in the treatment of the problem of the infinite, can apparently not be implemented so strictly, and the consideration of mathemat-

ical idea-formations is in this regard indeed in need of closer elaboration. The following essays contain different considerations for such elaborations. [This refers to the remaining essays in the collection Bernays 1976. Transl.]

Paul{ Bernays

264

Notes . This mode of denotation follows a suggestion of Hilbert. It offers certain advan-

tages over the usual distinction between “predicates” and “relations” for the con-

ception of wha is logical in principle [des prinzipiell Logischen] and is also in agreement with the customary meaning of the word “predicate.”

P. Hertz made the claim. in his work “On Thought” [Uber das Denken] (1923) that the logical imference contains “synthetic elements.” His grounding for this claim, which will be expounded in a forthcoming treatise on the nature of logic, includes

the point of view articulated here, but relies also on other considerations. . For the sake of simplicity the discussions about the concept of difference, or more

precisely the concept of identity, that contradicts it will be passed over. The philosopher is inclined to identify this relationship of representation as a mean-

ing connection. One should, however, be aware that in contrast to the customary relationship of word to denotation, the important difference consists here in the fact

that the representing object contains in its composition the essential qualities of the represented object, so that the relationships of the represented objects that are to be investigated are to be found in the representatives and can themselves be established by means of an examination of the representatives.

- In the view of nature of Newtonian physics these two representations of space are united with each other and do not clearly contrast with one another. Euclidian geometry constitutes here the law for the spatial association of things in the universe [Weitganzen]. Only through the present development in geometry and physics did

the necessity arise to differentiate between space as something physical and space as an ideal manifold, determined by geometric laws. ‘ In Euclid’s axioms this point of view is, however, not always consistently carried

out as the concept of the sufficiently large extension [Verlingerung] of a segment appears in them. This concept is in fact avoidable; one only has to give a differe nt

formulation of the parallel axiom.

We are here dealing with a sign “with meaning.” . The methodical point of view that is appropriate to this remark is the point of view that Weyl has taken in his text “The Continuum” ¢(1918).

It seems to me appropriate in the interest of the clarification of the discus sion to use the expression “intuitionism” to denote a philosophical view, in contras t to the term “finitistic,” which signifies a certain mode of inference and concep t formation.

10. It 1s remarkable that Jakob Friedrich Fries, who attributed to mathematical evi-

dence a realm that went far beyond the finite—in particular, according to his view the “continuous sequence of the larger and smaller” is given in pure intuition—

nevertheless drew a methodical differentiation between “arith metic as a theory,” which conceptualizes the intuitive representation of magnitude and develops it scientifically, and the “theory of combination or syntax [Syntaktik], ” which rests

solely on the postulate of arbitrary arrangement of given elements and their arbitrary repetition; it requires no axioms since its operations are “in themselves immediately comprehensible” (cf. J. F. Fries, Mathematical Philosophy of Nature,

1822),

11. It may be suggested here to use this expression, which is used by Cantor specifically with regard to set formations, in general, in refer ence to any theoretical approach,

12. The term stems from Poincaré, who in contrast to the other critics of set theory, almost all of whom were only focusing on the axiom of choice, brought the view-

The Philosophy of Mathematics and Hilbert’s Proof Theory

265

point of the impredicative definition into the discussion and attached importance to it. However, his critique was open to criticism insofar as he represents the use of

impredicative definitions as an innovation introduced by set theory. Zermelo could object to him that the impredicative definitions substantially appear in the usual inferential modes of analysis, fully recognized by Poincaré. Since then Russell and Weyl, in particular, have thoroughly explained and brought to full clarity the role

of impredicative definitions in analysis, 13. The example is chosen so that the reference to the totality of numbers cannot be easily eliminated, as is usvally the case in most simpler examples.

14. Waismann has given another example {in a note on “The Nature of the Reducibility Axiom,” 1928). This, however, is in need of modification. 15. The “or’” 1s meant here 1n both cases not in the sense of an exclusive “or,” but rather

in the sense of the Latin “vel.” However, the proposition is valid also for the exclusive “or.”

16. Hilbert delivered a first outline of a proof theory in 1904 in his Heidelberg lecture “On the Foundations of Logic and Arithmetic.” The first leading thought of the joint treatment of logic and arithmetic 1s already explicitly formulated in it; the methodical principle of the finitistic point of view 1s also intended but not explicitly artic-

nlated. Between this lecture and Hilbert’s later publications on proof theory falls the investigation of Julius Koenig, “New Foundations of Logic, Arithmetic and Set TheoryTM (published in 1914), which closely approximates Hilbert’s point of view and in which a consistency proof in the sense of proof theory is already carried out.

This proof concerns only a very narrow domain of the formal realm [des formalen Operierens], so that its significance is only methodological. 17. Cantor’s theory of numbers of the second number class is also included here. 18. The more precise discussions of this fact refer to Richard’s paradox, which has re-

cently received a sharper version by Skolem. Insofar as these considerations take

place in the context of a nonfinite mathematics, they do not possess a definitive character. A final clarification of the question discussed here would only be brought

about if it were possible to give a set of numbers in a finitistic way, of which it could be demonstrated that it does not occur in axiomatic set theory. 19, One should observe that this demand of deductive completeness does not go as far

as the demand of decidability of every question of the theory, which means that

there should be a procedure [Verfahren] for deciding for any given pair of two as-

sertions of the theory contradicting each other, which of the two is provable (“correct”). 20. Vaihinger, “Philosophy of As If.,” 2nd edition, Chap. XII.

Translator’'s Notes a. Konstruktion; the 1976 reprint reads Konjunktion but this is clearly a misprint.

17 The Grounding of Elementary Number Theory DAVID HILBERT*

It in the realm of mathematics we investigate the two sources of our knowledge, experience and pure thought, we come across a number of ideas that are perhaps also of philosophical interest. In particular, all these ideas point to similarities be-

tween these two sources of knowledge which, in themselves, are constituted so differently. For instance, we observe the unity of material in matter?; on the other hand,

the unity of foundations certainly crops up in our thought as a demand which we seek to fulfill, and often also achieve. The unity of the laws of nature, which we so

often encounter in such surprising ways, can serve as an example for both sources of knowledge. But even more striking than this point of view of unity i1s a phenomenon that we call pre-established harmony, and that clearly attests a connection

between nature and thought. The most wonderful and magnificent example of preestablished harmony is Einstein’s famous theory of relativity. Here the quite com-

plicated differential equations for the graviiational potential are uniquely derived from the general requirement of invariance alone. And this derivation would not have been possible without the profound and difficult mathematical investigations

of Riemann, which existed long before. Even in mathematical analysis it is an isolated occurrence that such a complicated special formal system with numerical coefficients arises from a general thought. My proof theory, which I shall discuss later in this article, is also an example of pre-established harmony. For it uses the so-

called logical calculus, which was devised earlier and for quite differ ent purposes, namely, solely for the shortening and communication of statements.

However, attentive reflection leads us to see that, besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea

of the Kantian epistemology retains its significance: to ascertain the intuit ive g priori mode of thought [Einstellung], and thereby to investigate the condit ion of the possibility of all knowledge. In my opinion, this is essentially what happens in my Investigations of the principles of mathematics. The a priori is here nothing more **Die Grundlegung der elementaren Zahlentheorie,” Mathematisch e Annalen 104, 1931, pp. 485-94. Translated from the German by William Ewald. Reprinted from William Bragg Ewald, From Kant to Hitbert: Readings in the Foundations of Mathematics (Oxfor d: Oxford University Press, 1996) by per-

mission of the publisher.

266

The Grounding of Elementary Number Theory

267

and nothing less than a fundamental mode of thought [Grundeinstellung], which I also call the finite mode of thought: something is already given to us in advance in

our faculty of representation: certain extra-logical concrete objects that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed

next to one another is immediately and in:cuitively given to us, along with the objects, as something that neither can be further reduced to anything else, nor needs such a reduction. This is the fundamental mode of thought that I hold to be necessary for mathematics and for all scientific thought, understanding, and communication, and without which mental activity is not possible at all. In this way, I believe myself to have recognized and characterized the third source of knowledge that accompanies experience and logic.

The a priori insights [Einsichten] are those intuitive insights as well as the logical insights that are achieved within the frame of the finite mode of thought. In par-

ticular, we see: There are propositions that Kant regarded as a priori, and that we ascribe to

experience. For example, all the basic facts of geometry, as well as the elementary properties of space and matter. But there are also propositions that are generally held to be a priori, but which cannot be achieved within the frame of the finite mode of thought—for example the principle of the tertium non datur, as well as the so-

called transfinite statements generally. The most obvious application and the first appearance of transfinite statements

occurs in number theory, and with this we come to the principal topic of the present lecture. It is already remarkable and philosophically significant that the first and simplest questions about the numbers 1, 2, 3, . .. present such profound diffi-

culties. These difficulties must be overcome; for how can knowledge be possible at all if not even number theory can be given a firm foundation, and if full unity and

absolute correctness cannot be demanded even here! It would be too great a digression and also superfluous to discuss the many

false paths that are today recognized as such: some tried to define the numbers purely logically; others simply took the usual number-theoretic modes of inference to be

self-evident. On both paths they encountered obstacles that proved to be insupera-

ble. One path was not yet trodden, which seemed the most obvious to a mathe-

matician. Before I describe this path, which in fact leads to the goal, I should like

to make some remarks about the most important dates in the pre-history of this problemn.

In the year 1888 as a young Privatdozent from Konigsberg I made a tour of the German universities. At my first stop, in Berlin, I heard people in all mathematical circles, both young and old, discuss Dedekind’s Was sind und was sollen die Zahlen?, which had then just appeared; their remarks were mostly critical. Thas essay is, along with the investigation of Frege, the most important and profound early

attempt to ground elementary number theory. At roughly the same time, and so more than a generation ago, Kronecker clearly expressed a conception which he illustrated with numerous examples; this conception today essentially coincides with our finite mode of thought.

In those days we young mathematicians, Privatdozenten and students, played

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David Hiibert

the game of transforming transfinite proofs of mathematical theorems into finite terms, 1n accordance with Kronecker’s paradigm. Kronecker only made the mistake

of declaring the transfinite mode of inference to be inadmissible. He issued prohibitions against the transfinite mode of inference; in particular, according to him, one was not allowed to infer that, if a statement A(r) does not hold for every integer n, then there must exist an integer »n for which that statement is false. At the time, the whole of mathematics unanimously rejected his prohibitions and went on to the busi-

ness of the day. How in fact do matters stand with the use of the transfinite modes of inference? The theory of number fields, for example, is a finely articulated edifice which

has been built up into the heavens, and which is bound up with the most fully developed theories of analysis. In beauty and perfection, it towers far above all other products of the human mind, and every step of the way it uses the tertinm non datur and transfinite modes of inference of the sort forbidden by Kronecker. All the heroes of the mind prior to Gauss—just like those from Gauss, Hermite, Jacobi, until Poincaré—have used the transfinite mode of inference in the boldest and most

manifold ways, and at no point did even the slightest discrepancy appear. Finally, if we just think of all the applications and make it clear to ourselves what a multitude of transfinite inferences of the most difficult and arduous sort are contained in

for example the theory of relativity and quantum theory, and how nature nevertheless precisely conforms to these results—the beam of the fixed star, the planet Mercury, and the most complicated spectra here on earth and at a distance of hundreds

of thousands of light years—how, in this situation, could we even for a moment doubt the legitimacy of applying the rertium non datur, just because of Kionecker’s pretty eyes and just because a few philosophers disguised as mathematicians have put forward reasons that are utterly arbitrary and not even precisely formulable?

Every piece of scientific knowledge whatsoever rests on a reasonable estimation of probability, invoking agreement and reciprocal relationships: think of theo-

ries in physics or astronomy (for example, the construction of the world of stars), or, in hology, of the laws of heredity or of the idea of evolution—all results that

we today view as firmly settled truths. It would be the death of all science and the end of all progress if we could not even allow such laws as those of elementary

arithmetic to count as truths. Nevertheless, even today Kronecker still has his followers who do not believe in the admissibility of the tertium non datur: this is probably the crassest lack of faith that can be met with in the history of mankind. However, a science like mathematics must not rely upon faith, however strong

that faith might be; it has rather the duty to provide complete clarity. Now, since the applicability of the tertium non datur to finitely many statements is self-evident, our entire attention immediately turns to the concept “infinite.” I have already be-

gun an extensive investigation into the infinite, but can here give oniy the upshot

of this investigation.

Physics teaches that a homogeneous continuum that would allow continued divisibility and would thus realize the infinitely small is nowhere encou ntered in reality. The infinite divisibility of a continuum is an operation that only exists in thought—is only an idea, which is refuted by our observations of nature and the ex-

periences of physics and chemistry. On the other hand, in astr onomy there are grave doubts about the existence of infinite space, and thus of the infini tely large. And all

The Grounding of Elementary Number Theory

269

of our action is finite; the infinite has no place in it. The infinite is realized nowhere: 1t does not exist in nature, nor is it admissible as a foundation of our rational thought. And yet we cannot dispense with the unconditional application of the tertium non

datur and of negation, since otherwise the gapless and unified construction of our science would be impossible. So operation with the infinite must be secured in the

finite; and precisely this occurs in my proof theory. [ pursue an important goal with this new grounding of mathematics. I should

iike to nd the world of the question of the foundations of mathematics once and for all by making every mathematical statement into a formula that can be concretely exhibited and rigourously derived, and thereby bring mathematical concept-formations and inferences into such a form that they are irrefutable and yet furnish a model [Bild] of the entire science.

The fundamental idea of my proof theory is as follows:

Everything that makes up mathematics in the traditional sense is rigorously formalized, so that mathematics proper {or mathematics in the narrow sense) becomes a stock of formulae. These are distinguished from the usual formulae of mathematics only in the following way: that besides the usual signs, the logical signs appear

as well—in particular, the signs for “implies” { — ) and for “not” (7). Certain formulae that serve as a foundation for the formal edifice of mathematics are called axioms. A proof is a figure, which must be intuitively presented to us as such; it congsists of inferences, where each of the premises is either an axiom, or agrees with

the end-formula of an inference that comes earlier in the proof, or results from such a formula by substitution. Instead of contentual inference, in proof theory we have an external action according to rules, namely, the use of the inference schemata and of substitution. A formula shall be called provable if it is either an axiom or the end-formula of a proof.

This proper, formalized mathematics is accompanied by a mathematics that is to a certain extent new-—a metamathematics that is necessary to secure formalized mathematics. In this metamathematics—in contrast to the purely formal modes of

inference of mathematics proper-—contentual inference is applied, but only to prove the consistency of the axioms. The axioms and provable theorems, i.e. the formulae that arise 1n this alternating game [Wechselspiel], are the images of the thoughts that make up the usual procedure of traditional mathematics.

The choice of axtoms for our proof theory is already indicated by this pro-

gramme. In selecting the axioms, we distinguish between qualitatively distinct groups, just as we do In geometry. 1. Axioms of implication: A—(B— A)

(Adjunction of a presupposition);

(A->B)— {(B—»()—(4A— O)} (Elimination of a statement); {A— (A—> B)}

- (A— B).

II. Axioms concerning “and” (&) as well as “or’” (v).

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David Hilbert

III. Axioms of negation: {A— (B&B)}

- A

(Principle of contradictiony);

A

A

{(Principle of double negation).

These axioms of groups {, II, and III are none other than the axioms of the propo-

sitional calculus. [V. Transfinite axioms:

(x)A(x) — A(b) (Inference from the general to the particular,

Aristotelian axiom); Converse via the schema:

A — B{a) A — (X)B(x) Aa) — (ExAX)

converse again via schema. Further formulae are derivable, for example:

A 2 (Ex)AX) (if a predicate does not hold for all arguments then there is a counterexample, and conversely);

(EAR) = (DA() (if there is no example for a statement, then the statement is false for all arguments, and conversely),

The axioms of this group IV are those of the predicate calculus. Next come the special mathematical axioms: V. Axioms of equality:

a=a

a = b— (Ala) — A(H))

and V1. Axioms of number: at+ 1+

as well as the axiom of complete induction and the schema of recursion. The proof of consistency has recently been carried so far by Ackermann and von Neumann that the axioms just listed for elementary number theory have been

shown to be consistent, and therefore the transfinite modes of inference—in partic-

The Grounding of Elementary Number Theory

271

ular the mode of inference of tertium non datur—can be seen to be admissible in the domain of elementary number theory. Our most important further task is to show the following (compare Mathematische Annalen 102, p. 6)*:

1. If a statement can be shown to be consistent, then it is also provable; moreover,

2. If a proposition S can be proved consistent with the axioms of number theory, then 8 cannot be proved consistent with those axioms as well. [ have succeeded in proving these theorems at least for certain simple cases. I obtained this result by adding to the already given rules of inferences (substitution

and inference schema) the following equally finite new rule of inference: If it has been proven, for any given numeral z, that the formula A(z)

is always a correct numerical formula, then the formula

(X)A(x) can be laid down as a starting formula [Ausgangsformel)]. Recall that the statement (x)A{x) extends far wider than the formula A(z), where

z 1s an arbitrary given numeral. For in the former case not merely a numeral, but any expression of our formalism having a numerical character can be substituted

for x in A(x); moreover, the negation can be formed in accordance with the logical calculus. First we observe that even with the addition of the new rule the axiom system remains consistent.

For suppose we are given a proof-figure which ends in a contradiction,

The previous proof of consistency consists in transforming all formulae of the given proof into numerical formulae according to a determinate procedure; then it is a matter of checking that all starting formulae are correct. Now, under

our procedure, the formulae that are written down in accordance with our new rule also are transformed into numerical formulae; in particular, from (x)A(x) we

obtain A(z), where z is a determinate numeral. But by the presupposition of the

new rule, this formula is correct as well. So as before our procedure converts all starting formulae of the proof-figure into correct formulae. The proof of consistency has thus been given.

Now let § be a formula of the form

(x)A(x)

that contains no variables other than x, and let it be consistent with the axioms. Then A(z) is certainly correct as soon as a numeral is substituted for z; for otherwise A(z) would be correct and therefore provable, and this would contradict (x)A(x), contrary to our hypothesis.

Therefore our formula 8 is proved by the new rule of inference. So Theorem 1 holds for every statement S of the form (x)A(x) that contains no variable other than

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David Hilbert

x. And for these statements of the form 8 the validity of Theorem 2 follows from

that of Theorem 1, which has just been proved. I we now consider a statement T of the form

T : (Ex)A(x) then clearly the negation of this statement

T : (x)z(x) is of the previously considered form 8. So by Theorem 2 it is not possible to give a proof of the consistency of both statements T and T. So let us suppose that the

proof of theconsistency of T has been given; it follows that the proof of the consistency of T cannot be given as well. Thus Theorem 2 has also been proved for every statement of the form T. To be sure, we cannot infer from this that T is provable. Objections of various sorts have been raised against my proof theory; they are

all unjustified. Let me make the following remarks on this matter: 1. The critics of my theory should indicate the precise spot in my proof where my alleged error is supposed to vecur. Otherwise, 1 shall refuse to examine

their arguments. . My theory has been subjected to the reproach that, although the theorems are indeed consistent, they are not for that reason proved. To be sure, they

are provable, as I have shown here in simple cases. More generally, it turns out (as I was convinced from the outset) that the attainment of consistency

s the essential thing in proof theory, and the question of provability (possibly with a suitable extension of the conditions that preserves the finite character) 1s settled at the same time. However, it cannot be demanded of a theory that all the relevant questions which it poses be fully solved at the outset;

it suffices if the path to that goal has been indicated. - Critics of my theory should use concepts such as “consistent” the way I use

them, and not the way other authors imagine them to be defined. On these points, my interpretation is authoritative because it is the only one that comes

into consideration for my theory.

. The objections to my theory often fasten on incidental and wholly indiffe rent matters; for example, when they are directed against the term “ideal”— which I use and which, despite all objections, I consider perfectly apt and

an aid to understanding. And in many other instances one-sided prejud ices and slogans are cheerfully introduced into the fray. I have already discus sed

the reproach of formalism in earlier essays. Formulae are a necessary aid to logical investigation. To be sure, their use demands precise mental labor, and makes empty twaddle impossible. . Until now there has existed no other theory (and indeed, in my opinion no other theory is conceivable) that has been equally successful. For my proof

theory does nothing other than to imitate thé intimate activity of our understanding, and to make a protocol of the rules whereby our thinking actually

The Grounding of Elementary Number Theory

273

proceeds. Thought takes place parallel to speaking and writing: by the for-

mation and placing together of sentences [Sdrzen]. And for justification I need neither God, like Kronecker, nor the assumption of a special capacity

of our understanding directed towards the principle of complete induction, like Poincaré, nor some ur-intuition like Brouwer, nor, like Whitehead and Russell, the axioms of infinity and reducibility, which are real, contentual presuppositions, not compensated for by proofs of consistency, and of which the latter is not even plausible.

In a recent philosophical lecture I find the sentence: “The nothing is the absolute negation of the allness of being.” This sentence is instructive for the following reason: in spite of its brevity, it illus-

trates all of the principal offences against the principles that are laid down in my

proof theory. Concepts like “the allness of being” contain a contradiction in themselves and already endanger the sense of every statement. But apart from this, nega-

tion is now applied to the problematic concept of the allness of being. It is precisely one of the most important tasks of proof theory to present clearly the sense and ad-

missibility of negation: negation is a formal process by means of which, from a

statement S, another arises, which is bound to § by the axioms of negation mentioned above (essentially, the principle of contradiction and tertium non datur). The process of negation is a necessary means of theoretical investigation; 1ts uncondi-

tional application first makes possible the completeness and closure of logic. But in

general the statement that arises through negation 1s an ideal statement, and to take this ideal statement as being in itself a real statement would be to misunderstand the nature and essence of thought.

I believe that in my proof theory 1 have fully attained what I desired and promised: The world has thereby been rid, once and for all, of the question of the foundations of mathematics as such.

The philosophers will be interested that a science like mathematics exists at all.

For us mathematicians, the task is to guard it like a relic, so that one day all human

knowledge whatsoever will partake of the same precision and clarity. That this must and will occur is my firm conviction.

Translator’s Notes a. die Einheit des Stoffes in der Matene. b. Hilbert 1928¢, [Chapter 15. Ed.]

PART IV: INTUITIONISTIC

LOGIC Intuitionistic Logic PAOLO MANCOSU AND WALTER P. VAN STIGT

In this fourth part of the book we introduce several texts related to the emergence

of intuitionistic logic. The introduction is divided into two sections. The first sec-

tron, written by W. P. van Stigt, describes Brouwer’s contributions to what soon became called the “Brouwer logic.” The second section, written by P. Mancosu, ana-

lyzes several further contributions to the formalization of intuitionistic logic due to Glivenko, Heyting, and Kolmogorov.

4.1

Brouwer

“Intuitionist Splitting of the Fundamental Notions of Mathematics” (B1923C1) is an important landmark in Brouwer’s Intuitionist campaign and in the history of Intuitionism. Brouwer had fought his “foundational battie” so far mainly on philo-

sophical grounds: a defense of the true nature of mathematics as constructive thoughtconstruction against the logicist—formalist confusion of mathematics

with

1ts

symbolic representation. His “first act of IntuitionismTM was “the separation of mathematics and language,” exposing the true nature of logic as no more than a science, a mathematical analysis of the symbolic record of a mathematical thoughi-

construction. His paper “The Unreliability of the Logical Principles” (B1908C) in

particular condemned the traditional practice of using logical principles—distilled from past mathematical records—as operators on words and sentences to generate new mathematical truths. The Principle of the Excluded Middle was simgled out as

obviously flawed when applied to the mathematics of the infinite, its use in his reconstruction of mathematics expressly avoided.

With the “Intuitionist Splitting of the Fundamental Notions of Mathematics,”

Brouwer’s approach to logic enters a new phase. While maintaining his fundamental stand on the separate identities of mathematics and logic and the nonproductive role of logic in mathematics, he now embarks on an investigation in the field of logic proper. During the past few years there had been a marked softening of his 275

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negative appraisal of language. He had become actively involved in the “Signific

movement” of linguistic reform. Despair of the possibility of human communication had (temporarily} made place for a more optimistic conviction that the instru-

ment of language is capable of improvement. The Manifesto of his newly created

“Signific Circle” expressed as its aim “‘the coining of new words expressing spiritual values for the languages of western nations.” The creation of new words was

particularly relevant to Brouwer’s programme of reconstructing mathematics; new words were needed to represent liis new notions and distinctions—and communicate his message to the mathematical world. “Intuitionist Splitting” is in fact such an exercise of creating new words, in this case words expressing the various relations between points and between points and

species of points. In line with his own rules of correct logical practice, Brouwer

starts from his concepts of mathematical truth and absurdity (i.e., proven impossibility), resulting immediately in the inapplicability of the Principle of the Excluded Middle and of what he calls “The Principle of Reciprocity of Complementary Species,” which asserts the equivalence of truth and double negation. He replaces the latter principle by a restricted form of complementarity: “Truth implies absurdity-of-absurdity, but absurdity-of-absurdity does not imply truth.” He then proceeds to analyze the effects of these distinct “correctness predicates,” considered as operators on “assertions of property,” that is, propositions. He states and proves one theorem: “Absurdity-of-absurdity-of-absurdity is equivalent with absurdity.”

(Brouwer seems reluctant to use symbols for logical operations as if to emphasize

their distinct nonmathematical nature.) The corollary of this theorem and his restricted form of complementarity is “that a finite sequence of absurdity predicates can be reduced to either absurdity or absurdity-of-absurdity . . . by striking dut pairs

of absurdity predicates, provided that the last absurdity predicate of the sequence is never included in the cancellation.”

The logical analysis of absurdities was intended as an introduction to the main part of the paper, sections 2, 3, and 4, where the absurdity and absurdity-of-absur-

dity operators are applied to terms expressing relations between points and point species, so generating a variety of new relations. Although it takes up less than one

page of the paper, this introductory paragraph is the most important part of “The Intuitionist Splitting.” The inapplicability of the traditional principle of complementarity and double negation was implied in Brouwer’s earlier rejection of the

Principle of the Excluded Middle. In this paper, however, Brouwer spelled it out in public and drew the attention of mathematical logicians to its implications for tra-

ditional logic.

There was a quick response from Rolin Wavre, who coined the term “Logiq ue Brouwerienne,” and from Kolmogorov, Others joined the debate or made their contributions to the development of Brouwer’s alternative, Intuitionist logic: Lévy,

Avstidisky, Barzin and Errera, Borel, Khinchin, Glivenko, and Heyting (see Thiel 1988).

Brouwer himself stayed aloof from the debate, remaining true to his convic tion that engagement in logic “is an interesting but irrelevant and sterile exercise.” He approved of the contributions of his loyal student Arend Heyting, “Die formalen

Regeln der intuitionistischen Logik” (1930b) and “Die formalen Regel n der intuitionistischen MathematikTM (1930c). When asked by the editor of the Bulletin de

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"Académie Royale de Belgique in 1930 to round off the debate, Brouwer referred him to Heyting’s papers as the authoritative and “masterly” account of Intuitionist Logic. By that time he had already retired from the foundational battle field and effectively passed the Intuitionist leadership to Heyting, who kept the cause of Intu-

itionism alive and changed its course. Let us now turn to the debate on intuitionistic logic.

4.2

The Emergence of the Intuitionistic Propositional Calculus

We have seen that in 1923 Brouwer explicitly formulated a number of principles whose validity could not be accepted by the intuitionist mathematician. In particu-

lar the excluded middle and the principle of double negation were singled out as especially problematic. By contrast, Brouwer remarked that the intuitionist accepts the

following principles: A — ——A and (A — B) — (-B — —A).! From these two principles and the rule of modus ponens, implicitly accepted as valid, Brouwer was able to show that ———A — —4 and that -4 — ———A, that is, —A is equivalent to —-——A. From this it follows that in general one can always strike out-an even num-

ber of negations from a formula provided the last negation is not cancelled. The first interesting reflections after Brouwer 1923 concerning the principles

of intuitionistic logic are found in an article by Wavre published in 1926a.2 Wavre’s article was devoted to a comparison of classical logic with the principles accepted by the intuitionist or, in his terminology, of the “logique formelle” and the “logique

empiriste.”> Wavre interprets the assertion of a statement A in an empirist context as asserting the provability of A, The negation of A is seen as stating that the as-

sertion of A leads to a contradiction.* Wavre proceeds to enumerate the similarities and dissimilarities between formal and empiricist logic. Similar principles:

. A—-B&(B->0—=(A4—0); 2. From A and (A — B) one can infer B;

4. (A— 1)— —A.

Among the different principles we find, as in Brouwer, the principle of the excluded

middle and the principle of double negation. Wavre remarks that only A — ——A has an equivalént in empirical logic. Having listed these principles, he then goes on to show Brouwer’s theorem on the equivalence of —A and ———A and to show that only some forms of reductio ad absurdum can be preserved in empiricist logic.

Although Wavre’s article does not push the analysis of intuitionistic logic much further than Brouwer’s remarks in 1923, it had, however, the merit of starting a discussion in the pages of the Revue de Métaphysique et de Morale on the nature of intuitionistic mathematics. In particular Wavre’s defense of the intuitiomstic claims brought about the impatient reply of Paul Levy, who defended a more platonistic conception of mathematics (see Wavre 1924, 1926a, 1926b, Levy 1926a, 1926b, and Borel 1927). The contribution by Borel is included, as a sample of this debate, in this book.

The next contribution to a clarification of intuitionistic logic came from the philosophical analysis proposed by O. Becker. Since in 1931 Heyting gave an in-

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terpretation of intuitionistic logic that appealed to Becker’s phenomenological interpretation of intuitionistic logic found in the 1927 book Mathematische Existenz,

it is worthwhile to describe Becker’'s approach. Becker’s interpretation rests on Husserl’s theory of judgments as exposed in the Logical Investigations (1900, 1913).

Becker begins by remarking that there are two possible ways of denying the propo-

sition “p holds.” We can have “—p holds” or “p does not hold.” More formally, he writes “+p,” “+(—p),” and “—(+p).” He also claimed that the three cases excluded any other possibility so that either +p or +(—p} or —(+p), quartum non datur. In order to give a concrete interpretation of the situation, Becker interprets the three

cases as fulfillment [Erfiillung], frustration [Enttéuschung), and nonfulfillment of an intention. Consider the proposition “the book is on the tableTM and call it p. To

the case +p corresponds a fulfillment, given for instance in perception, of (1) the intended table in question and (2) of the state of affairs that the intended book is on the table. +{—p) corresponds to a situation in which the table is seen but the book is not on it. Finally —(+p) corresponds to a situation in which, for instance, neither the table nor the book is seen. The second case, unlike the last, gives a posi-

tive experience of a “conflict” [Widerstreit]. In the mathematical realm Becker translates the above by means of various examples involving becoming free choice sequences and lawhke sequences.

The appendix (1927, pp. 335-40) contains several interesting reflections on the

1ssue of the compatibility of the phenomenological interpretation with the Brouwerian interpretation. Becker gives a clear interpretation of the meaning of true and false in intuitionistic logic: First of all it must be remarked that intuitionistic logic does not distinguish between “true” and “false” but between “trueTM and “absurd.” In this context “true” means:

actually provable (constructively), and absurd: “provably contradictory” (from the phenomenological point of view one can in principle say: true = there is the syn-

thesis of the fulfillment of the judgment intention, the agreement between what is intended and what is perceived; *absurdTM:= there is the “synthesis” of the frustration of the judgment intention, the “synthesis” of the conflict. . . . Clearl y there is no complete disjunction, in the sense of the tertium non datur, betwee n fulfillment and frustration, agreement and conflict. (Becker 1927, p. 775)

This was at the source of the quartum non datur stated above. Howev er, Becker remarks that Brouwer and Wavre seem flatly to deny this quart um non datur. The problem arises as follows. Becker asserts that phenomenclogically —(+p) is equiv~ alent to —(—p). But Wavre in his article (1926a, p. 72) rema rked that intuitionistically we do not have (p v —p v ——p). In order to avoid the confli ct with the phenomenological interpretation, Becker interprets the third disjunction in his quartum non datur principle as —(p v —p), that is, neither p nor — p is given. This, he claims, 18 1n accord with intuitionistic principles. However, he was wron g in his claim since @ v —p v = (p v —p)) is not intuitionistically valid. The princ iple is equivalent to the claim that if one rejects the principle of the excluded middl e, then one has to accept 1ts negation. It is unfortunate that after making such a good start in distinguishing the two meanings of negation, classical and intuit ionist, Becker fell prey

to the same confusion he was trying to clarify. The quartum non datur was to emerge again as a problem in the debate started

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by Barzin and Errera to which I will soon turn. There is no need to list here in detail Becker’s phenomenological interpretation of the principles stated by Wavre and

the interpretation of double negation. Suffice it to say that Becker managed to ground

all the principles in Wavre’s list according to his phenomenological interpretation. At the end of the appendix he also stated the problem of finding a calculus for in-

tuitionistic logic.

.

But could there be such a calculus? Not everybody agreed that it was possible consistently to develop a Brouwerian logic. In particular Barzin and Errera 1927 made the claim that Brouwerian logic was inconsistent, thereby starting a long de-

bate on intuitionistic logic. It will not be possible here to survey the numerous contributions to this debate, and we have to refer the reader to Thiel 1988 and Franchella

1995 for an overview. I will simply mention the objections raised by Barzin and Errera and expose Church’s and Glivenko’s answers to their claims. Barzin and Errera

interpreted Brouwer’s position as claiming that there are propositions that are neither true nor false. These propositions are “tierce.” Their aim was to show that as-

suming a “tierce” led to formal contradictions.® The strategy followed by Barzin and Errera was that of assuming the principles of intuitionistic logic and the principle of

quartum non datur for a “tierce.” If we denote “p is tierce” by “p’,” then the princi-

ple of quartum non datur states “p v —p v p’.” Under this assumption their aim was to show that one could prove the collapse of the truth values, that is, that in the calculus the same proposition could be true and “tierce,” or “tierce” and false. In particular, they provided proofs that showed that if a proposition is “tierce,” then it is

false (1927, p. 67) and that if a proposition is true, then it is “tierce” (1927, p. 68). It could be objected to thewr proof that i order to obtain their desired conclusions, p'— —p and p — p', they made use of a law that is classically but not intuitionis-

tically valid, that is, —(p & g) — (—p v —q).% Clearly, it would have been essential to their goals to make sure that only “intuitionistically” acceptable principles should appear in the procof. However, even granting the proof, it was not clear that the con-

clusion established their claim as to the impossibility of a logic with a third value.’ In “On the Law of the Excluded Middle” (1928) Church gave a clear analysis

of the situation by making the following three points. First of all, Church empha-

sized that if one drops the law of the excluded middie, then only a subset of the previously provable theorems can be proved, and thus no contradiction can emerge, unless the calculus we began with was already inconsistent. Thus the only possibility for the emergence of a contradiction is assuming a principle that contradicts

the law of the excluded middle. The second point argues that the argument by Barzin and Errera is not effective against those who simply refuse to accept the excluded middle:

The method of the argument [by Barzin and Errera] is the method of reductio ad absurdum. It is assumed that if the law of the excluded middle is not accepted, then it must be explicitly denied by asserting the existence of tiers [sic] propositions, and on this basis contradictory results are obtained. This argument is clearly not effective against one who merely omits the law of the excluded middle from his system of logic without assuming any contrary principle, because the msistence that one who refuses to accept a proposition must deny it can be justified only by an appeal to the law of the excluded middle, the very principle in doubt. The method

280

Paolo Mancosu and Walter P. van Stigt of reductio ad absurdum, in fact, necessarily employs the law of excluded middle and cannot be used against one who does not admit this law. (1928, p. 77)

Finally, even if one accepts the existence of “tierce” propositions, it does not follow that Barzin and Errera arc correct in their alleged proof of the inconsistency of

the logic so obtained. This only follows if one accepts the quartum non datur. However, this principle is justified by interpreting the “tierce” proposition as one that is neither true nor false. But this definition is, according to Church, contradictory: It is not possible, as an alternative to the law of the excluded middle, to assert that some proposition is neither true nor false, because by so doing not only the law of

the excluded middle would be denied but also the law of contradiction. In fact, to assert that a proposition p is not true and is also not false is to assert at once not-p and not-(not-p) and consequently to assert that not-p is both true and false. (1928, p. 75)

However, this does not follow, and Church’s faux pas shows the intricacies of the conceptual issues involved in this debate. Thus Church defended the possibility of a logic with “tierce” propositions, However, he recognized that if the concept of “tierce” is admitted, and together with it the quartum non datur, then the argument by Barzin and Errera is correct. But he saw no reason for accepting the quartum non datur.

Glivenko’s contribution to this debate (1928) consists in his claim that Brouwerian logic is not a three-valued logic [logique tripartite], that is, that the introduc-

tion of “tierce” propositions in Brouwerian logic is illegitimate. Starting from acceptable intuitionistic principles, he proves the following propositions:

3. ((-pvp)—-—g)-> g

In order to show that in Brouwerian logic one could not admit a “tierce,” Glivenko argued as follows. If p is false, then its “tierce” is also false, that is, —p — —-p’; and 1f p is true, its “tierce” is false, that is, p — —p’. By using the intuitionistically

valid principle (VIII} to the effect that p—rN—(g— PN —((p v q) — r)), Glivenko was able to deduce, appealing to VIII and 3, that —p’

I shall leave now the debate provoked by Barzin and Errera to pursue the more technical contributions to intuitionistic propositional logic due to Glivenko, Heyting, and Kolmogorov. Let us begin with Glivenko 1929. If we ignore Kolmogorov

1925, which remained unknown until later, Glivenko 1929 contains the first seeds of a development that will yield a long series of interpretations of classical logic Into intuitionistic logic known as double negation interpretations or negative trans-

lations (Kolmogorov 1925, Godel 1933a, Gentzen 1933).% Apart from some interesting additions to the axioms he had adopted in the previous article, the main two theorems of this paper can be given in Glivenko’s clear words: 1. If a certain expression in the logic of propositions is provable in classic al logic,

it is the falsity of the falsity of this expression that is provable in Brouwe rian logic.

2. If the falsity of a certain expression in the logic of propositions is provable in classical logic, that same falsity is provable in Brouwerian logic. (1929, p. 301)

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Ghivenko’s work cannot be considered a translation of classical logic into intu-

itionistic logic, but it certainly paved the way to that result. Although Glivenko was Russian, he does not seem to have been aware of Kolmogorov 1925 until after he

had completed his 1929 paper. He mentions Kolmogorov’s article in a letter to Heyting dated October 13, 1928 (the letter is published in Troelstra 1990).°

At the same time, Heyting had been working on a formalization of the principles of intuitionistic logic. The occasion for the paper was a prize question pub-

tished in 1927 by the Dutch Mathematical Association on the logic of Brouwerian mathematics. Heyting was awarded the prize in 1928, and his paper appeared in 1930. There were actually three papers, the first one of which, and the only one pre-

sented here, was concerned with the intuitionistic propositional calculus (for details on first-order intuitionistic logic, see Troelstra 1990). The other two papers dealt

with the formalization of intuitionistic mathematics. In a letter to Becker, dated September 23, 1933, Hevting described how he

found his axiomatization simply by going through the axioms and theorems of Principia Mathematica and including the “admissible” ones into a system of indepen-

dent axioms (for a list of the axioms, see the appendix to Heyting 1930b). The letter contains some other interesting reflections. For example, Heyting claims that “it

is in principle impossible to encompass with certainty in a formal system all the ‘admissible’ modes of inference” (original German in Troelstra 1990, p. 8). He also

distinguished between two different interpretations of the intuitionistic calculus, in terms of problems and of expectations. In the 1930b paper the “admissible” principles were only stated, and Heyting was not explicit on what the interpretation of the logical connectives in intuitionistic logic was. Heyting 1930a and 1931 provide an explicit interpretation for the con-

nectives — and “or.””1° Let us look at 1930a. Heyting says that a proposition p expresses a problem or an expectation: A proposition p like, for example, “Euler’s constant is rational,” expresses a prob-

lem, or better vet, a certain expectation (that of finding two integers ¢ and & such that C = a/b), which can be fulfilled [réalisée] or disappointed (décue]. (1930a, p. 307)

The interpretation in terms of expectations reminds one of Becker’'s phenomeno-

logical interpretation for intuitionistic logic. Although Becker and Heyting coire-

sponded about matters of intuitionistic logic in 1933, I do not know whether they

had any perstmal contact at the time of Heyting’s first attempts at providing an interpretation of intuitionistic logic. However, Heyting had certainly read Becker’s Mathematische Existenz, since in 1931 he quotes it approvingly. Heyting 1931 also reveals that the phenomenological interpretation was very much the one favored by Heyting at this stage: We conclude our treatment of the construction of mathematics in order to say something about the intuitionist propositional calculus. We here distinguish between propositions and assertions. An assertion s the affirmation of a proposition. A mathematical proposition expresses a certain expectation. For example, the proposition “Euler’s constant C is rational” expresses the expectation that we could find two integers @ and & such that C = a/b. Perhaps the word “intention,” coined by the

phenomenologists, expresses even better what 1s meant here. . .. The affirmation of a proposition means the fulfillment of an intention. (1931, pp. 58-59) L)

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Becker 1s explicitly quoted in connection with the interpretation for negation: A logical function is a process for forming another proposition from a given proposition. Negation is such a function. Becker, following Husserl, has described its meaning very clearly. For hum negation is something thoroughly positive, viz. the

intention of a contradiction contained in the original intention. The proposition “C 1s not rational,” therefore, signifies the expectation that one can derive a contra-

diction from the assumption that C is rational. (1931, p. 59)

As for disjunction, he observes that a proposition like “p v —p”* expresses the ex-

pectation of a mathematical construction that will either prove p or —p.

In addition to containing the above interpretation for the intuitionistic connectives — and “or,” Heyting 1930a and 1931 also contain several interesting features. For example, Heyting argues against Levy, who had identified “p is true” (in the intuitionistic sense) with “p is provable.” He objects that Levy’s distinction does not satis{y the intuitionistic demands: One does not escape this criticism by replacing, with Mr. Levy, “p is true” by “p is provable,” since this last sentence, being equivalent to “there exists a proof of p.” implies again the idea of transcendent existence. To satisfy the intuitionistic demands, the assertion must be the observation of an empirical fact, that is, of the re-

alization of the expectation expressed by the proposition p. Here, then, is the Brouwerian assertion of p: It is known how to prove p. (19304, p. 307)

Heyting also distinguishes carefully between the statement p and p is provab le (denoted by +p). However, at the end of Heyting 1931 he claims that developing the

intuitionistic calculus with the function + would lead to useless complications (see

pp. 60-61) and that his calculus should be interpreted as valid only for propos itions

of the form +p.1!

Let me conclude this introduction with a few words about Kolmogorov ’s interpretation of intuitionistic logic. We have seen that Heyting menti oned the possibility of interpreting a proposition p as a problem, although he seems to have favored the interpretation in terms of expectations. As Troelstra (1990 , p. 7) remarks, Heyting and Kolmogorov seem to have considered the interpretati ons in terms of problems and expectations as distinct. Indeed, in his introduction to intuitionism (1934), Heyting describes both interpretations, although he will later consider them as essentially equivalent.

Kolmogorov 1932 first introduces the notion of problem by mean s of examples and then proceeds to delineate a logic of problems: If @ and b are two problems, then a & b designates the probl em “to solve both problems ¢ and b, while a v b designates the problem “to solve at least one of the

problems ¢ and 5.” Furthermore, @ D b is the problem “to solve b provided that the solution for a is given” or, equivalently, “to reduce the solution of b to the solution of a” [. ..] —a designates the problem “to obtain a contradiction provided that the solution of ¢ is given.”(1932, p. 329)

Kolmogorov then set up a system of axioms for the calculus of problems and remarked that “the calculus of problems is formally ident ical with the Brouwerian in-

tuittonistic logic, which has been recently formalized by Mr. Heyting.” However, in a note added in proof, Kolmogorov pointed out that although his interpretation

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283

was similar to the one given by Heyting in 1931, the distinction between proposi-

tions and problems was missing in Heyting’s article.

With the articles by Glivenko, Heyting, and Kolmogorov, three major events

in the history of intuitionism had been accomplished. Heyting had provided a complete formalization of intuitionistic logic, Glivenko had pointed the way to the neg-

ative interpretations to follow, and the combined work of Heyting and Kolmogorov had given an explicit formulation to what is now known as the BHK interpretation

of intuttionistic logic. In the years to follow many more results would be obtained, and we have to refer the reader to Troelstra and van Dalen 1988 for more infor-

mation on the mathematical and metamathematical results obtained in the wake of the groundbreaking works of Brouwer, Glivenko, Heyting, and Kolmogorov.

Notes 1. In the following introduction we do not make a syntactic distinction between intuitionistic and classical connectives. I will use &, —, — , and v, Moreover, Brouwer does not use any symbolismin this connection. 2. I am ignoring here Kolmogorov 1925, which remained largely unknown until [ater, Kolmogorov’s 1925 will not be analyzedin this introduction. See Wang 1967, 3. The use of the word empirist in this context goes back to the characterization of Du Bois-Reymond. 4.

“Distinguons la logique formelle, logique de la négation ou encore du vrai et du faux, dans laquelle I’ alternative s’ impose entre le vral et le faux, et la logigue mathe-

matique empiriste du vrai et de Uabsurde, dans laquelle ’alternative entre le vrai et 1’absurde ne s’imposerait plus; vrai signifiant I’effectivement démontrable et ab-

surde, ce qui peut étre effectiverment réduit A une contradiction, la contradiction étant prise dans le sens formel” (Wavre 1926, p. 69).

5. “L’objection que nous faisons 3 M. Brouwer est d’une autre nature: elle veut établir qu’il est impossible de raisonner en admittant un tiers, sans tomber aussitdt dans une

contradiction que nous allons tdcher de mettre en lumiére” (Barzin, Errera 1927, p. 60). 6. This was remarked in Thiel 1988, = Indeed, the consistent developments of many-valued logics 1s already found in 1920

in Y.ukasiewicz, but none of the participants in the debate mention his work.

8. For a contemporary exposition see Troelstra and van Dalen 1988, Chap. 2, Sect. 3. For historical information on Kolmogorov 1925, Godel 1933a, and Gentzen 1933, see Wang 1967, Troelstra 1986, and Gentzen 1969, respectively. 9. The correspondence between Heyting and Glivenko is published in Troelstra 1990,

10. The explicit interpretation of the intuitionistic meaning of implication 1s given in correspondence with Freudenthal and in print in 1934,

11. Heyting’s position was influenced by an exchange with Freudenthal. See Troelstra 1990 and Troelstra 1983.

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Paclo Mancosu and Walter P. van Stigt

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ford.

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106-15. English translation in P. Benacerraf and H. Putnam, Eds., Philosophy of Mathematics, 2nd ed., Cambridge University Press, Cambridge (Mass. ) 1983, pp. 52-61.

Heyting, A.,

1934, Mathematische

Grundlagenforschung.

Intuitionismus.

Beweistheorie,

Springer, Berlin. Extended French translation, Les Fondements des Mathématiques. Intuitiomisme, Théorie de la Démonstration, Gauthier-Villars, Paris, 1955.

Khintchine, A., Objection & une note de MM. Barzin et Errera , Académie Royale de Belgigue, Bullerin 33, pp. 253-58. Kolmogorov, A. N., 1925, O principe tertium non datur, Matematiceskij Sbornik 32, pp.

646-67. English translation in J. van Heijenoort, ed., From Frege to Gdodel, Harvard University Press, Cambridge (Mass.), 1967, pp. 416-37.

Intuitionistic Logic Kolmogorov,

A.

N.,

285

1932, Zur Deutung der intuitionistischen Logik,

Mathematische

Zeitschrift 35, pp. 58-65. English translation: this volume, Chapter 25, Levy, P., 1926a, Sur le principe du tiers exclu et sur les théorémes non-susceptibles des dé-

monstration, Revue de Métaphysique et de Morale 33, pp. 253-38,

Levy, P., 1926b, Critique de la logique empirique, Réponse 4 M. Robin Wavre, Revue de Métaphysique et de Morale 33, pp. 545-51. Levy, P., 1927, Logique classique, logique broawerienne et logique mixte, Académie Rovale

de Belgique, Bulletin 13, pp. 256-66. Wavre, R., 1924, Y-a-t-il une crise des mathématiques? A propos de la notion d’existence et

d’une application suspecte du principe du tiers exclu, Revie de Métaphysique et de

Morale 31, pp. 435-70, Wavre, R., 19263, Logique formelle et logique empiriste, Revue de Métaphysique et de Morale

a3, pp. 65-75. Wavre, R., 1926b, Sur le principe du tiers exclu, Revie de Métaphysique et de Morale 33, pp- 425-30.

Secondary Literature Franchella, M., 1995, L. E. 1. Brouwer: Toward intuitionistic Logic, Historia Mathematica

22, pp. 304-22.

'

Thiel, C., 1988, Die Kontroverse um die intuitionistische Logik vor ithrer Axiomatisierung

durch Heyting im Jahre 1930, History and Philosophy of Logic 9, pp. 67-75. Troelstra, A. S., 1983, Logic in the writings of Brouwer and Heyting, in V. M Abrusci, E.

Casari and M. Mugnai, Eds., Atfi del convegno internazionale di storia della logica, San Gimignano, 4-8 Dicembre 1982, CLUEB, Bologna, pp. 193-210. Troelstra, A. S., 1986, “Introductory note to Godel 1933e,” in Godel 1986, pp. 282-87.

Troelstra, A. S., 1990, On the early history of intuitionistic logic, in Mathematical Logic, P. P. Petkov, Ed., Plenum Press, New York and London, pp. 3-17.

Troelstra, A. S., and van Dalen, D., 1988, Constructivism in Mathematics. An Introduction, Vol. I, North-Holland, Amsterdam:.

van Stigt, P. W_, 1990, Brouwer’s Intuitionism, North-Holland, Amsterdam. Wang, H., 1967, Introductory Note to Kolmogorov 1925, in J. van Heijenoort, ed., From Frege to Godel, Harvard University Press, Cambridge (Mass.), 1967, pp. 414-16.

13 Intuitionist Splitting of the Fundamental

Notions of Mathematics LUITZEN EGBERTUS JAN BROUWER*

1

Correctness Predicates

The Intuitionist conception of mathematics not only rejects the Principle of the Excluded Middle altogether but also the special case, contained in the Principle of Reciprocity of Complementary Species, that is, the principle that for any mathematical system infers the correctness of a property from the absurdity of its absurdi ty. This rejection is shown to be justified by the following example: Let &, be the

vth figure after the decimal point in the expansion of # and let m = k, if in the continued expansion of 7 at d,, it happens for the first time that the section s At 1, - - . » Ao forms the sequence 0123456789. Further let ¢, = (— i v =k

and ¢, = ( —%)" otherwise. Then the sequence ¢y, ¢, c3, . . . converges to a real num-

ber r.

If we define a real number g to be rational if one can calculate two whole rational numbers p and g whose quotient equals g, then r is not ration al. On the other hand the rationality of r cannot possibly be absurd, for in that case k) could not posstbly exist, which would imply that » = 0 and therefore would be rational, If we further define a real number g to be comparable with 0 if either g > 0 or

g < 0, then r is not comparable with 0; on the other hand the comparability of r

with O cannot possibly be absurd since in particular r > 0 woul d be absurd, in which

case r = () and r would be comparable with 0.

The classical tradition postulates for every property corre ctness and absurdity as the two only alternatives and therefore postulates the equivalence of correctness and absurdity-of-absurdity. In the Intuitionist view absur dity-of-absurdity follows from correctness but is not equivalent with correctness. Neither does it recognize the alternatives of absurdity and absurdity-of-absurdi ty to be, respectively, comrect-

ness and absurdity.

In the classical tradition a sequence of n absurdity predi cates: “the absurdity of the absurdity of . . . the absurdity” can, by repeated strik ing out of every pair of sub*“Intuttionistische splitsing van mathematische grond begrippen,” Verslagen der KNAW 32, 1923, PP.

877-80. (Presented to the Royal Academy on November 24, 1923.) Translated from the Dutch by Wal-

ter P. van Stigt. Published by permission of Konin klijke Nederlandse Akademie van Wetenschappen.

286

Intuitionist Spiitting of the Fundamental Notions of Mathematics

287

sequent absurdity predicates, be reduced either to absurdity or to correctness. One

might for a moment think that in accordance with the Intuitionist conception of mathematics such “striking out” would be completely ruled out and that therefore sequences

of different numbers of absurdity predicates must be nonequivalent. This, however, 1s not the case; on the contrary, such “striking out” of absurdity predicates is acceptable also to the Intuitionist provided that the last absurdity predicate of the sequence is never included in the cancellation, and this because of the following theorem: Theorem. Absurdity-of-absurdity-of-absurdity is equivalent with absurdity, Proof. a. If the property y follows from the property x, then from the absurdity of y follows the absurdity of x. Therefore, since absurdity-of-absurdity follows from correctness, absurdity must follow from absurdity-of-absurdity-of-absurdity.

b. Since from the comrectness of a property follows the absurdity-of-absurdity of that property, then as a special case from correctness of absurdity, that is, from absurdity, must follow absurdity-of-absurdity-of-absurdity.

In the Intuitionist view a finite sequence of absurdity predicates can on the ba-

sis of this theorem be reduced either to absurdity-of-absurdity or to absurdity.

2

The Relations of “Fusion” [versmelting] of Two Points’

The two fundamental relations here are coincidence and apariness. Two points Py

and P, comcide if every square of P contains a square of P, and every square of

P, contains a square of P;.! Two points P, and P; lie apart from each other (or are locally distinct), if one

can indicate a square of P and a square of P, that lie outside one another.! Application of the predicates of absurdity and absurdity-of-absurdity to the relation of coincidence generates only in the first case a new relation, to which we shall refer

as deviation [afwijking); in the latter case the result is again the relation of coincidence.

Application of the predicates absurdity and absurdity-of-absurdity to the apart-

ness relation results again in the relations, respectively, of coincidence and deviation.

We use the notation ¢ «Db.

Proof. [(z) (” ";”) 4.11,2.2, 2.22] F (@ A—a) D [4.1] + - 4.4. 441. F-an—-a-vbDb

Proof. [4.4, 2.21, 3.12].

442 Fravb-—=a-Db.

Proof. [341]1 F-avb -Ar—aDran—-a Vv bar-a*D: b a—-a- -2 [22]5b. [4.4]1] 443. ++—a 2D - —lb:}fl(fl v b).

Proof. [4.42,227F-=aDavbDb-D

{42 ~bD =(av b).

4.44. | - —{a v b) DC —a A —b.

F - =(av b)D-a. Proof. [3.1, 4.2] [3.1, 3.11,4.2] F - —=(a v b) 3 —b. [(1), (2), 2.24]

F + =(a v b) D —a A —b.

[443, 2.27]

Fr—aaAn—-bD —|(fl v b).

1,

(1) (2)

320

Arend Heyting

The following formula can be considered as a special case of 4.42.

445. F-av—a-2 - ——aDa. Proof. [442]F - —wava-A——a-Da:D:[227]F - 445, The tormula states: “If for a particular mathematical statement A the principle of the excluded middle holds, then for A the principle of the reciprocity of the complementary species holds as well.” This theorem cannot be reversed. By contrast the following propositions are equivalent: (a) For every mathematical statement and every mathematical system the principle of the excluded middle holds, and (b) for

every mathematical statement and every mathematical system the principle of the

reciprocity of the complementary species holds. (Brouwer, Jahresber. der D.M.V.

33, 1925, p. 252, footnote 4). The latter can be shown formally as follows. From F - g v —a + one immediately obtains by 4.45., F - —a D a. From +

——a 3 a- one first obtains by substitution

——(a Vv —a)

Jav —a,

and then by 48., F a v —a.

Two theorems mentioned in the introduction follow from 4.42446.

F - —av b-D-a2b.

Proof. [43,212]

v —mav - D-—m bavhAn -A a —ma-2 [4.42]

b [2.27] 447,

F - 446,

F'aVbD'fiflDb.

Proof. [4.42, 2.27]. We now give the formulas that lead to the proofs of those formu las that result from 4.44 by exchanging v and A (4.53, 4.54); of course, in 4.54 one has to add the restrictive conditions @ v —a, which cannot be replaced here by —a D a. It is noteworthy that 4.52 holds without such a condition.

45 F-brad—a-D- b

Proof. [222, 224]F:bAraDd—a:D:bralan—oa: [2.27]

bDadan—-a:D:

2.2, 2.22,4.11,2.291] b D —a. 431 F-a A ={a A b) D —b.

Proof. [226] FraD - -bDa Ab*D-[4.2]—1(a Ab)D—b:D: [2.27] + - 4.51,

The Formal Rules of Intuitionistic Logic

321

The theorem can be strengthened to 4511, F - ——a A "_I(fl FAN b) 2 =b.

Proof. (4.51,2271 F - —=(a A b) D *bD=a-D[4.2) v—a D —=b:D:

[2.27] F - 4.511.

452. F+—=(aAb)D-aD-b.

Proof. [4.51, 2.27] 4.521. F-a2D—-b -2 =(a A b). Proof. [2121F-aD=b-D-anbDdD=abab- -2 [4.11] —(a A b). 453, F»—av b 2D "—1({.1 A b) Proof. [2.2,42] F « —a D —(a A b). [2.22,42] F » —b D —(a A b).

1) 2)

(1), (2), 3.12] F 4.53.

4.54, |“—i(fl f\b) "AcTavar-—av b, Proof. [451]1F - a A —{a A b) D —b.

(1)

[22] F « —a A =la A b)D —a

(2)

[3.41, (1), (2), 3.3] F - 4.54.

Theorems on Double Negation 4.6. F - ——a-sAn——b 2D —(a A Db).

Proof. [452] F-—(aAb)D - aD=b D [42] —-bD—a:D: [2.27] F=(a A DA ——b D —a:D:[423] F 4.06.

4,61. F '—l—:(fl Ab):)—l—lflf\—r-"'lb. Proof. (2.2, 2.22, 4.22, 2.24].

462 F-——av—bD —|—|(a v b) Proof. [3.1, 3.11, 4.22, 3.12]

322

Arend Heyting

463 - —I“1({I W b) A=AV —d

D ——a v —b.

Proof. [444, 42] Fo- '—1—r(fl W b) - —l(—lfl Fal _Ib) [454]

F o —l(—lfl A —|b} A

(1)

md Vo —d

* D =g v b

(2)

[(1), (2), 2.12, 2.13] + - 4.63.

The condition —a v ~—a can of course be replaced by a v —a but not by ——a J 4.

47. F-ad—~bAac)-Danb—c

Proof. [4.52, 2.291, 2.27].

471. F-adbv—-c-D-ancDbh.

Proof. [2.12] F:aDbv—ac:D:iarcDbv—=c-anc:D: [3.41]

brc

v -=cac:D:[441)bAac:D:[22] 5.

On the “Law of the Excluded Middle” 48.

- —:—1(£I v _Ifl)-fi Proof. [3.1]

FaJav—ar3D-[42]F (g v —a) D a. 3L 311l F=aDav—a-2D-

(1)

[42]

(2)

F —l(fl v —nfl) J —d.

[(1}, (2), 4.11] + - 4.8, 4.81. F =—(—a 2 a). Proof. [4.45, 4.8, 4.22]. 482. Fav—-aalb - -D b

Proof. [4,22]

|-'flv'—lfl:]b':)'—T—l(aV_lfl)D_l_lb':)'

[4.8, 2.15] ——b.

4.83. F - av —ad—=b: D_Ibfi

Proof. [4.82, 4.32].

This formula states that for the proof of a negative theorem one can always use the law of the excluded middle. Furthermore, we prove some of the theorems mentioned

in the introduction.,

4.9, F‘fl:)b':)"fl(fl !\'—15)

The Formal Rules of Intuitionistic Logic

Proof. [43,2201] F-aDb

491. F-av b

—(—a

D

aD——b"

323

D -

b))

Proof. [4.47, 4.9].

492. F-a A b D al(—av b Proof. [4.53, 4.21].

Appendix

The Independence of the Axioms; The Unprovability of the Law of the Excluded Middle

In the preceding we took as axioms: 21.FFraDa A a.

2Al. FFra A bDb A a. 212. F+-aDb-D-ancDb rc. 213.

v F-aDdb A bDc-D-alec. 214 FFE-bDadb.

215 FFran-aDb 3.

Db,

FFa2av b.

3 1. Frav b2 bva 312. FF-aDdc-An-bIc-Dav bDe.

411.FFradDb-A--a>=b-2a.

We prove the independence of these axioms in the way indicated by Bernays

(p. 316 of the mentioned article). D, A, v are considered as types of composition in a group, and — as a function in the group. Hereby every formula receives a determinate value whenever elements of the group are substituted for the vanables. The formula “holds” if this value is always 0. The groups are constituted in such a way that in each one of them all the axioms but one hold. Moreover, we ensure that

valid formulas always follow from valid formulas by the application of the rules of operation of Section 1. To this end it is necessary and sufficient that sy

324

Arend Heyting

L.

0A0=0

l. 0 2 a has the value 0 only when a = 0. In the tables of composition the first element is above and the second on the left.

Group | Elements: 0, 1. 2. D10

1L

2

A0

1

2

v]0

1

olo oo

2

ol1o 1

olo

12

o

0

o

111 1le1 1 3

212

000

2

211

11

210

2

Sj0

1

2

|1 o 1

2

2.1 doesno for2 22 th A2=2 ol 21=2 d, .

Group Il Elements: all positive whole numbers and 0.

adb=1ifa b ¥ 0; a =0 Db = an 0 otherwd ise. an2=0anb=a ifb +b * 2.

avb=0,ifa=0o0rb b =0; =1 otherw ,av ise. —0=1—a=0,ifa + 0.

2.11 does not hold, since 3A222A3=0D5=1.

Group 111 Elements: all positive and negative whole numbers and 0. a2b=11 b+ a= 0;a2Db 0a =0 other nd wise.

arnb=a+b

avb=0,ifa=00rb=0av b= 1 otherwise, -0 = 1;-—1a=0,ifa¢0.

212 does not hold, for 1 .

22 -2+ 1 A —122A-1=02-021=0321=

(One should pay attention to the difference between — | and —1 ]

Group IV Elements: all positive whole numbers, 0, and the cardi nal number & of the countable infinite species. a2b=b—a ifb a2> b=0 2 othe a rwis ; e.

anb=a-+b.

avb=2aifa0

1

2

AlO0

1

2

vio

1

2

-0

1

2

0(0

0

0

0lo

1

0

0lo

0

o

1

0

1

1|11

0

1

1{1

1

1

1|10

1

0

211

0

1

210

1

©

210

0

0

2. 14 doesnothold, for 2 2-0222=2D21=1. Group VI

Elements: 0, I, 2, 3.

510

olo

1

2

oo

3

o

AJ0

1

2

11

0

1

1

212

o0

312

0

2

2

3

2

v|[0

olo

1

2

o o0

3

1|11

1

1

1]0

1

2

3

0

2121

2

2

210

2

2

2

0

31212

2

3|0

3

2

3

olo1

o

S|0

1

23

1 0 1 1

2.15doesnothold, forO A - 0D3:23=0A223=223=2.

Group VII Elements: 0, 1.

D)

! |— 1

1

0 0

>O

0

‘0

1

1

11

v

0‘ 1

3.1 doesnothdld, for0 D0v1=0>21=1

(The tables for 2 and A are those one obtains if O represents an arbitrary correct

proposition and 1 represents an arbitrary false one. We will call such tables “normal.!!')

Group VI Elements: 0, 1, 2. 2

0 | 2

=oO YBIeo i[8I-e0 >Ob— DO— R [—=oN Be cQo| B|ed RT(SAo -t

~]0 1 2

1 0 o

326

Arend Hevyting

311 doesnotholdforOv22D2v0=022=28

Group IX

Elements: 0, 1. For D and A the normal tables (see Group VII). avb=0always. =0=1--1=0.

312 doesnot hold: 1D 1+ A - 121 D-1vI1D31=0A0D-0D1= 0>1=1t,

Group X Elements: 0, 1.

For D, A, and v the normal tables. -0 = —1 = 0. 4.1 does nothol for =0 Dd, -0D1=0D1 =1,

Group XI Elements: 0, 1.

For D, A, and v the normal tables. -0 = —1 = 1. 411 doesnotho forl ld, D1 -A-12D51- 241 =0A0D1=0D21=1. Group XII

Elements: 0, 1, 2.

oUb= b|=t So-0| oO=k ODM= —[TSR To OM= )u(BTeoi oB]= TYBo)Ge A

0 1

2

Here all axioms hold; the statement ——a O @ does not because 222 = =12 2=0232=2. According to 4.45 we obtain the theorem: the formula that represents the principle of the excluded middle cann ot be derived from the axioms.

Group XII can be interpreted as follows: 0 denotes an arbitrary correct state-

ment, 1 an arbitrary false statement, 2 a sentence that cannot be false but whose correctness is not proven. One then obtains precisely the above tables.

Notes 1. This essay constitutes a reworking of the first part of an essay that was given an award by the “Wiskundig Genootschap” in Amst erdam at the beginning of 1928.

2. Cmp. Hilbert and Ackermann, Grundziige der theor etischen Logik, p. 33. 3. The more common ~ is used for several purposes; moreover the formulas that contain the negation depart most from classical logic. There fore, I choose a new stgn for the negation,

The Formal Rules of Intuitionistic Logic

327

. I owe the insight that this formula, originally set up by me as an axiom, follows from the remaimng axioms to the work of V. Glivenko. wh Cmp. Brouwer, Jahresber. der D.M.V. 33, 1925, p. 253. 4.8 contains the “law of the absurdity of the absurdity of the law of the excluded

middle,” which is due to Brouwer. The proof of 4.8 and Theorem 4.83 are taken from the work of V. Glivenko cited above.

-

. 'The tables for D and A of this group are constructed from those for v and - in Group V by Bernays by applying the formulas aJdb==—avbanb=-a(—av b

. The tables for v and — in Group VIII are identical with those in Group VI of Bernays; those for 2 and A are constructed as indicated in the previous footnote.

Translator’s Notes . Here indicated by lines in boldface type.

25 On the Interpretation of Inturtionistic Logic ANDRE! KOLMOGOROV*

The present essay can be considered from two quite different standpoints. I. If one does not accept the intuitionistic epistemological assumptions, then only the first section is relevant. The results of this section can be summarized ap-

proximately as follows:

In addition to theoretical logic, which systematizes the proof schem ata of theoretical truths, one can systematize the schemata of the solution of problems, for example, of geometrical construction problems. For example, corresponding to the principle of syllogism the following principle occurs here: If we can reduce the solution of b to the solution of a, and the solution of ¢ 1o the solution of b, then we can also reduce the solution of ¢ to the solution of a.

One can introduce a corresponding symbolism and give the forma l computational rules for the symbolical construction of the system of such schemata for the

solution of problems. Thus in addition to theoretical logic one obtains a new calculus of problems. One needs here no special epistemolo gical, for example, intu-

itionistic, assumptions.

Then the following remarkable fact holds: The calculus ofproblems is formally identical with the Brouwerian intuitionistic logic, whic h has recently been formal-

ized by Mr. Hevyting.!

2. In the second section intuitionistic logic is critically investigated while accepting the general intuitionistic assumptions. It will be shown that infuitionistic

logic should be replaced by the calculus of prob lems, for its objects are in reality not theoretical propositions but rather problems.

81 We do not define what a problem is; rather we expl ain this by some examples. Problems are: *Zur Deutung der intuitionistischen Logik,” Math ematische Zeitschrift 35, 1932, pp. 58-65. Trans lated from the German by Paclo Mancosu. Published by permission of Springer-Verlag GmbH & Co. KG, Heidelberg.

328

On the Interpretation of Intuitionistic Logic

329

1. To find four whole numbers x, v, z, n for which the relations

x"+ Yyt =

n>=>2

hold. 2. To prove the falsity of Fermat’s theorem.

3. To draw a circle passing through three given points (x, v, z).2 4. Provided that one root of the equation ax® + bx + ¢ = 0 is given, to find the other one.

3. Provided that the number 7 is expressed rationally, say, 1 = m/n, to find an analogous expression for the number e. That the second problem is different from the first is clear, and this does not yet

constitute a particular intuitionistic claim.* The fourth and the fifth problem are examples of conventional problems; yet the assumption in the fifth problem is impossible, and consequently the problem itself is without content [inhaltsios]. In what follows, the

proof that a problem is without content will always be considered as its solution. We believe that according to these examples and explanations the concepts

“problem”TM and “solution of a problem” can be employed without misunderstandings in all cases that occur in the concrete areas of mathematics.* From now on problems will be designated with lower-case italic letters a, b, ¢, .. . . If a and b are two problems, then a A b designates the problem “to solve both problems a and b,” while a v b designates the problem “to solve at least one of the problems & and ».” Furthermore, a O b is the problem “to solve b provided that the

solution for « is given” or, equivalently, “to reduce the solution of b to the solution of a.”

In the preceding we did not assume that every problem is solvable. If, for ex-

ample, Fermat’s theorem is true, then the solution of the first problem would be

contradictory. Correspondingly —a designates the problem “to obtain a contradic-

tion provided that the solution of a is given.”” Ifa, b, c,d,. .. arethe problems, then, according to the above definitions, every

formula p(a, b, ¢, . . .) constructed with the help of the signs A, v, 2, — also designates a problem. However, if a, b, c, . . . are only symbols for indeterminate prob-

lems, then one says that p(a, b, c, .. .) is a function of the problem variables a, b, ¢, ....If x is a variable (of any sort) and a(x) designates a problem whose meaning depends on the value of x, then (x)a(x) stands in general for the problem “to

give a general method for the solution of a(x) for every single value of x.” One should understand this as follows: To solve the problem (x)a(x) means to be able to solve for any given single value x; of x the problem a(xp) after a finite number

of steps known in advance (prior to the choice of xp).8

Furthermore, for the functions p(a, b, c, . ..) of indeterminate problems a, b, c, . .. one simply writes Fpla, b,c,...)

instead of

(@)b)c). . . pla, b, c, .. )

330

Andrei Kolmogorov

Thus F pla, b, ¢, . ..) designates the problem “to give a general method for the so-

lutien of pla. b, ¢, ...} for any single choice of problems a, b, ¢, . . ..

LR

The problems of the form F p(a, b. ¢, . . .}, where p is expressed by means of the signs v. A, D | and —, form the object of the elementary calculus of problems.’ The corresponding functions p(a, b, ¢, . . .) are the elementary functions ofprobfems.

The fact that [ have solved a problem is a purely subjective fact that in itself has as yet no general interest. However, the logical and mathematical problems possess the special property of the general validitv of their solutions: If [ have solved

a logical or a mathematical problem, then I can present this solution in a way that

1s intelligible to all and it is necessary that it be recognized as a correct solution although this necessity has to a certain extent an ideal character, for it presupposes a sufficient intelligence on the part of the listener.® The proper goal of the calculus of problems consists in giving a method for the solutton of problems of the form Fp(a, b, c, .. .), where p(a. &, ¢, ...) is an elementary function of problems, by means of the mechanical application of some sim-

ple computational rules. However, in order to reduce everything to these computational rules, we must assume that the solution of some elementary problems are already known. We postulate that we have already solved the following two groups of problems A and B. The further presentation addresses only a reader who has al-

ready solved all these problems.?

~

-

21.

F.aDanal®

21l. F.anbDbaa. 212. t.aDb.D.ancDbAc

213 F.aldb.A.bDc.D.alec.

214, F.bD.aDb. A.

Y213

31.

F.an.aDb.2b,

+t.adavbh.

31l. F.avbDbva

J12. F.adc.A.bDc.D.avbDe. 41 4ll.

F.flaD.an. F.a2b. A.al—=b.D—aq?

We thus assume that the reader can solve all the problems appearing here after the sign + for any choice of problems a, b, c. This also presents no difficu lties

at all. For example, in problem (2.12) one must, under the assumption that the solution of a has already been reduced to the solution of b, reduce the soluti on of b Ac to the solution of aac. Let the solution of @aac be given; this means that both the solution of @ and the solution of ¢ are given. From the solution of ¢ we can infer, according to the assumption, the solution of b. And since the solution of ¢ is already given, we obtain the solution of both problems b and ¢, and thus the solution b c. In this consideration is contained a general method for the soluti on of the problem aJdb.JD.ancODbh Ac

On the Interpretation of Intuitionistic Logic

331

which is valid for arbitrary a, b, c. We thus have the right to consider the problem

213.F.aDb.D.arcI bAac (with the universal sign ) as solved.

‘

Regarding Problem 4.1 in particular: As soon as —a is solved, then the solution of a is impossible and the problem a D b is without content. The second group of problems B for which we postulate the existence of solu-

tions contains only three problems.!! Namely, we assume that we are always able to (or that we possess a general method for) solving the following problems for ar-

bitrary elementary function problems p, ¢, 7, 5, . .. :

I. If F p A g is solved, to solve I p. II. If F p and F p D g are solved, to solve I g.

il If + p(a, b, c, ...) is solved, to solve | p(g, 7, s, .. .). We can now give the rules of our calculus of problems: 1. First of all we put the problems of the group A on the list of solved prob-

lems, 2. If F p A g is already on our list, then + p can also be added to it.

3. If both formulas | p and F p 2 g are there, then | ¢ may also be added. 4. If + pfa, b, c, ...) 1s already on the list and g, r, s, . . . are arbitrary functions of problems, then + p(q, r, s, . ..) can also be added to it. On the basis of the previously assumed postulates, one convinces oneself easily that these formal computations actually guarantee the solution of the corresponding problems.

We abstain here from further carrying out these computations, since all the

above formal rules of computation and a priori written formulas coincide with the

rules of computation and axioms of Heyting’s first essay'?; consequently we can interpret all the formulas of his essay as problems and regard as solved all these problems.

Among these problems we only make a note here of some especially interesting ones (which are to be considered as already solved). 4.3.

2D —a. F.a

432, F . —+—a D 1a.

The solution of 4.3 and 4.2 is clear without any computation. The solution of 4,32 is obtained from 4.3 and 4.2 by substituting in 4.2 the problem o by ——a. If one adds to the formulas B assumed a priori the formula

F.avoa.

(1)

(in propositional logic the principle of the excluded middle), then one obtains a complete axiom system of classical propositional logic. In our problem interpretation

332

Andrei Kolmogarov

the formula (1) reads as follows: to give a general method that allows, for every problem g, either to find a solution for 4, or to infer a contradiction from the exis-

tence of a solution for «!

In particular, if the problem a consists in the proof of a proposition, then one must possess a general method either to prove or to reduce to a contradiction any

proposition. If our reader does not consider himself to be omniscient, he will probably determine that the formula (1) cannot be found on the list of problems solved by him.

‘

It is, however, remarkable that one can solve the problem

48 F.——=.av oalt’ as Heyting’s calculus shows. Likewise, the formula F.—a 2 a.

(2)

(in the propositional calculus the principle of double negation) cannot appear in our calculus of problems, for the formula (1) follows from it by means of 4.8, Therefore we see that, unlike Heyting’s formulas of intuitionistic logic, already very simple formulas of classical propositional logic cannot appear in our calculus

of problems.

Let us also remark that if a formula - p is false in the classical propositional cal-

culus, the corresponding problem F p cannot be solved. Actually, from such a formula p one can infer by means of the previously assumed formulas and rules of

computation of the calculus of problems the obviously contradictory formula F. —a.14

§2 The basic principle of the intitionistic critique of logical and mathematic al theories is the following: Any proposition that is not without content shoul d refer to one or more completely determinate states of affairs accessible to our experience.'>

It @ is a general proposition of the form “any element of the set K posses ses

the property A,” and if in addition the set X is infinite, then the negation of a, “a is false” does not satisfy the above principle. In order to avoid this situation Brouwer gives a new definition of negation :“a is false” should mean “g leads to a contra-

diction.” Thus the negation of a is transformed into an existential propo sition: “There exists a chain of logical inferences that, under the assumption of the correctness of a, leads to a contradiction.” However, the existential propositio ns are also subjected

by Brouwer to a deep critique.

Namely, from the intuitionistic standpoint it makes no sense at all simply to say: “There is among the elements of an infinite set K at least one element with the

property A” without exhibiting this element,

However, Brouwer does not want to throw the existentia l propositions com-

pletely outside of mathematics. He only declares that one should not state an existential proposition without giving a corresponding const ruction. On the other hand,

On the Interpretation of Intuitionistic Logic

333

an existential proposition does not, according to Brouwer, merely consist in stating that we have already found a corresponding element in K in the latter case the ex-

istential proposition would be false before the invention [Erfindung] of the construction, and only afier that would it be true. Thus arises this quite special type of proposition that is supposed to have a content that does not change over time but even so may only be stated under special ¢onditions. Of course, one can ask whether this special type of proposition is perhaps only

a fiction. Actually, we are given a problem: “to find in the set XK an element with the property A”; this problem has really a determinate meaning that is independent

of our knowledge; if one has solved this problem, that is, if one has found a cotresponding element x, then one obtains an empirical proposition: “Now our problem

is solved.” Thus what Brouwer understands by an existential proposition is completely broken down into two elements: the objective element (the problem) and the subjective element (the solution). As a consequence one finds no object left that one would have to call an existential proposition in the proper sense.

Therefore, the main resulit of the intuitionistic critique of negative propositions should simply be formulated as follows: For any universal proposition it is in gen-

eral meaningless to consider its negation as a determinate proposition. However, with this disappears the object of intuitionistic logic, for now the principle of the excluded muddle holds for all the propositions for which the negation i1s meaning-

ful in general.!® Hence it follows that one should consider the solution of problems as the in-

dependent goal of mathematics (in addition to the proofs of theoretical propositions). As was shown in the first section the fonmulas of intuitionistic logic also receive a

new meaning in the field of problems and solutions.!” Gottingen, January 15, 1931 Received February 6, 1931

Notes 1. Heyting, Die formalen Regeln der intuitionistischen Logik, Sitz. 4. Preus. Akad. 1, 1930 p- 42; H, p. 57; 111, p. 158.

2. To be very precise, in the formulation of this problem one should indicate the allowed means of construction.

3. On the other hand, the propositions “Fermat’s theorem is false” and “four numbers

satisfying (1) exist” are equivalent from the standpoint of classical logic. 4. The main concepts of propositional logic, “proposition,” and “proof of a proposition” are in no better position.

5. Let us observe that —a should not be understood as the problem “to prove the unsolvability of a.” If one considers in general “the unsolvability of ¢” as a well-defined concept, then one only obtains the theorem that from —a follows the unsolv-

ability of a, but not the converse. For example, if it were proven that the well-ordering of the continuum surpasses our abilities, one could still not claim that a contradiction follows from the existence of such a well-ordering. 6. Here, too, as earlier, we hope that these definitions cannot lead to any misunderstandings in the concrete mathematical domains. . #

L

334

Andrei Koimogorov Ha. This explanation of the meaning of the sign t is very different from that of Heyting, although it leads to the same computational rules. . This definttion is analogous to the definition of the elementary propositional cal-

culus. However, in the propositional calculus the logical functions analogous to A, V, 2, — can be expressed by two among them. In the calculus of problems all four

functions are independent. . All this holds word for word also for the proof of theoretical propositions. It is, however, essential that every proven proposition is correct [richtig]; for problems one has no concept corresponding to this correctriess [Richtigkeit].

- In the case of the propositional calculus, one must first convince oneself of the correctness of the axioms, if one wants to determine the correctness of the consequences.

10. On the rumbering of the formulas and the use of separation marks (dots), see Heyting [,

11. They cannot, however, be expressed symbolically with the signs of the elementary calculus of problems.

12. Heyting 1.

13. In propositional logic (4.8) represents the Brouwerian proposition on the consistency of the principle of the excluded middle.

14. See V. Glivenko, Acad. 1. de Belgique, 5° série, 15, 1929, p. 183, 15. See H. Weyl, Uber die neue Grundlagenkrise der Mathematik, Math. Zeitschr. 10,

1921, p. 39. The whole further investigation of negative and existential propost tions essentially follows this work by Weyl.

16. However, a new question arises: Which logical laws are valid for propositions whose negation has no sense?

17. Remark inserted in proof. This interpretation of intuituionistic logic is closely con-

nected with the ideas Mr. Heyting has developed in the last volume of Erkennt nis

2, 1931, p. 106; yet in Heyting a clear distinction between propositions and problems is missing.

Translator’s Notes a. The original text contains a misprint.

INDEX

Abrusci, Michele, 150-51, 179n.16

Cantor, Georg, 25, 54, 66, 92-93, 127-28,

Ackermann, Withelm, 154, 17677, 229,

158, 202, 228, 231, 233, 235, 237,

261, 270, 326n.2

245, 257, 264n.11

Anaxagoras, 123-24

Carathéodory, Constantin, 3

Archimedes, 125

Carnap, Rudolf, 76

Aristotle, 124

Cassirer, Ernst, 165, 181n.31

Avstidisky, Sergei, 276

Cauchy, Augustin-Louis, 68, 125 Cayley, Arthur, 54

Barzin, Marcel, 276, 279-80,

Chevalley, Claude, 81n.1

306

Chihara, Charles S., 68

Bavle, Pierre, 124

Church, Alonzo, 279-80

Behmann, Heinrtch, 233

Cipolla, Michele, 181n.31

Becker, Oskar, 81, 165-67, 171, 173,

Clausius, Rudolf, 191

180n.26, 277-79, 281-82

Courant, Richard, 174

Beltrami, Eugenio, 54 Bergson, Henri, 5, 11

Dedekind, Richard, 55, 66-70, 86—87, 91,

Bernays, Paul, 75. 8081, 149-51, 154,

127, 130, 132, 151-52, 156-58,

159-61, 163-76, 177n.4, 214n.a,

165, 200-202, 215, 228, 231, 2335,

292n.4, 323

245, 254-56, 260

Betsch, Christian, 181n.31

Democritus, 124, 135

Blumenthal, Otto, 177n.1

Descartes, René, 4-3, 9

Bolyai, Janos, 54

Detlefsen, Michael, 1, 160-61, 16364,

Bolzano, Bernhard, 165, 196n.1

177n.1, 180n.21, 180n.23, 180n.26

Boole, George, 153

Diedonné, Jean Alexandre, 81n.1, 177n.1

Borel, Emile, 3, 35, 76

Dirichlet, Peter Gustav, 120

Boutroux, Pierre, 5

Du Bois-Reymond, Paul, 283n.3

Breger, Herbert, 82n.15

Dummett, Michael, 1

Bridges, Douglas S., 1 Brouwer, Luitzen Egbertus Jan, 1-20, 66,

Edwards, Harold M., 179n.11

7680, 86, 9499, 109, 115,

Einstein, Albert, 3, 121, 141, 196, 266

133-36, 139-41, 147, 147n.8, 154-56, 16063, 166, 168, 180n.22,

Errera, Alfred, 276, 279-80, 306

180n.28, 198-200, 217-19, 251-52,

Eudoxus, 124-27, 130

263, 273, 275-71, 3012, 306, 308,

Euler, Leonhard, 142n.14

320, 332--33

Ewald, William Bragg, 179n.11

Euclid, 103, 124-27, 150, 190, 236

335

336

Index

Fang, Joong, 177n.1

Jahnke, Hans Niels, 82n.16

Feferman, Solomen, 73, 74, 81, 81n.1,

83n.19, 177n.1, 182n.36 Fermat, Pierre de, 103 Fichte, Johann Gottlieb, 140 Fraenkel, Abraham Adolf, 180n.24,

181n.31, 261 Franchella, Miriam, 20n.1, 279

Frege, Gottlob, 67, 151, 156-58, 178n.8,

195, 200-202, 216-17, 228, 24143, 253436, 260 Freudenthal, Hans, 283n.10-11

Fries, Jakob Friedrich, 170, 173, 264n.10 Galilei, Galileo, 99 Gauss, Carl Friedrich, 52

Gauthier, Yvon, 179n.11

Geiger, Moritz, 165 (Gentzen, Gerhard, 181n.28, 280, 283n.8 Giaguinto, Marcus, 160 Gillies, Donald, 82n.12

Glivenko, Valerii Ivanovich, 276, 280-81, 283n.9, 306-7, 312, 326n.4

Gadel, Kurt, 66, 167, 177, 181n.28, 263, 280, 283n.8

Goldfarb, Warren D., 176 Grelling, Kurt, 173

Hahn, Hans, 11, 54

Hallett, Michael, 82n.9, 152-53, 160, 17/nd, 1780n.7, 179n.8

Hamilton, William, 4 Heidegger, Martin, 165

Heinzmann, Gerhard, 68, 83n.19, 179n.13 Helmholtz, Hermann Ludwig von, 191,

225 Heraclitus, 141

Herbrand, Jacques, 177, 180n.27 Heriz, Paul, 264n.2

Hessenberg, Gerhard, 173 Heyting, Arend, 1011, 14, 166, 179n.16,

27677, 281-83, 283n.9, 283n.11, 304n.3, 328, 331, 334n.17 Hilbert, David, 2-3, 10, 27n.4, 44n.2, 55,

65-66, 75-76, 80-81, 12021, 136-41, 149-77, 189-197, 215, 218-22, 223-26, 236-37, 246, 253,

25861, 263, 326n.2

Hoider, Otto, 75-76 Husserl, Edmund, 70, 83n.18, 91, 165, 171,

278, 282

Kant, Immanuel, 4, 54, 58, 124, 161, 16975, 189, 243, 248, 252, 261, 26667

Kaufmann, Felix, 76, 81

Khinchin, Aleksandr Yakovievich, 276 Kitcher, Phillip, 160-61, 177n.1

Klein, Fritz, 54, 192, 235 Kluge, Fritz, 181n.32 Kreisel, Georg, 177n.1

Kronecker, Leopold, 97, 142n.9, 151-53,

155-56, 168, 179.12, 200-201,

217, 228-29, 251, 26768, 273 Kdonig, Tulius, 81n.2, 265n.16

Kolmogorov, Andrei Nikolaevich, 280-83, 283n.8 Kummer, Emst Eduard, 300n.1

Largeault, Jean, 20n.3, 81n.1

Lachelier, Jules, § Lambert, Johann Heinrich, 33 Lauener, Henr, 177n.1

Lebesgue, Henri, 5

Leibniz, Gottfried Wilhelm, 124-26, 140, 195 . Lévy, Paul, 276-77, 282, 296-300, 3067

Levi-Civita, Tullio, 54 Lobachevsky, Nikolai Ivanovich, 54 Lorenzen, Paul, 81, 83n.19

Lowenheim, Leopold, 233

Lowy, Heinrich, 180n.19 Yukasiewicz, Jan, 283n.7

Mach, Emst, 141, 160-61

Mannoury, Gerrit, 180n.22 Marion, Mathieu, 179n.11 McCarthy, Charles D, 1 Mahnke, Dietrich, 18111.31

Maine de Biran, Pierre, 5 Majer, Ulrich, 161, 172, 181n.29

Meyerhof, Otio, 173 Mocoi}, Jan Johann Albinn, 179n.13 Moriconi, Enrico, 169, 177n.1

Miiller, Aloys, 159, 169-73, 175, 214n.a, 223-26 Miiller, Gert H., 177n.1

Natorp, Paul, 126

Nelson, Leonard, 170-75, 181n.30, 181n.34, 182n.35

Index Newman, Maxwell Herman Alexander,

8ln.1

337

Skolem, Thoralf, 65, 180n.24, 265n.18

Smorynski, Craig, 180n.21

Newton, Isaac, 125, 141

Specker, Emst, 177n.1

Nye, Mary Jo, 180n.20

Steiner, Mark, 180n.26 Strohal, Richard, 178n.8

Parmenides, 141

Pasch, Moritz, 192 Peano, Giuseppe, 55, 163, 195, 229-30,

245, 262, 313

Peckhaus, Volker, 150-54, 173-74, 177n.2,

179n.9 Ploucquet, Gottfried, 128

Poincaré, Henn, 5, 8, 41, 35, 68, 82n.5,

Tait, William W., 177n.1 Thiel, Chnistian, 68, 81n.2, 276, 279, 283n.6 Toepell, Michael-Markus, 177n.2 Tonietti, Tito, 81

Troelstra, Anne Sjerp, 1, 281-283, 283n.8-9, 183n.11

152, 15436, 162, 166, 201, 217, 228, 264n.12, 273

Posy, Carl J., 4 Prawitz, Dag, 160, 177n.1, 182n.36

Pythagoras, 100

Vaihinger, Hans, 262

van Dalen, Dirk, 1, 79, 81, 81n.1, 82n.13,

83n.18, 119n.2, 168, 283, 283n.8 van Heijenoort, Jean, 180n.24, 180n.26

van Stigt, Walter P., 4, 11, 20n.1, 66, Ramsey, Frank Plumpton, 66, 76, 256

Ravaisson, Felix, 5

82n.13 von Neumann, Johann, 142n.16, 154, 163,

Reid, Constance, 155, 177n.1

168, 176-77, 180n.27, 229, 261,

Resnik, Michael, 160

270

Riemann, Bermhard, 54, 266

Richard, Jules, 130-31

Waismann, Friedrich, 2, 65n.14

Richman, Fred, 1

Wang, Hao, 283n.2, 283n.8

Rickert, Heinnch, 171

Wavre, Rolin, 276-79, 296-97, 308-9

Rolf, Bertil, 177n.1

Webb, Judson Chambers, 177/n.4, 179n.11

Russell, Bertrand, 55, 68, 130-32, 153,

Weyl, Hermann, 1-3, 14, 18, 65-81,

156, 171, 195, 202, 216, 228-30,

119-22, 143-47, 154-56, 160-61,

241-43, 260, 273, 310n.2, 312

177n.1, 180n.24, 196, 198-200, 202, 217-19, 230, 240, 334n.15

Schlick, Emst, 171

Weierstrass, Karl, 155, 233, 245

Schmid, Anne Frangoise, 179n.13

Weil, André, 81n.1

Schoenflies, Arthur, 11

Wittgenstein, Ludwig, 66, 256

Scholz, Heinrich, 165

Whewell, William, 4

Schopenhauer, Arthur, 54, 58

Whitehead, Alfred North, 153, 156, 216,

Schroder, Ernst, 153, 233, 238, 256-57

230, 273, 316n.2, 312

Schiiler, Wolfgang, 177n.2

Sieg, Wilfried, 151, 175-76, 179n.11, 181n.28, 182n.36

Simpson, Stephen G., 182n.36

Sinaceur, Hourya, 169, 179n.14

Zach, Richard, vi Zeno, 124, 135

Zermelo, Emst, 23, 55, 65-70, 82n.5, 163, 193, 202, 215, 217, 228, 245, 261