Philosophy of Mathematics in the Twentieth Century

Charles Parsons

• Publisher: Harvard University Press
• Year: 2014
• ISBN:
  9780674419506
• Topic: Philosophy
• Subtopic: Modern Philosophy

In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century.Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl's idea of a "vicious circle" in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell's mathematical logic—Gödel's first mature philosophical statement and an avowal of his Platonistic view. Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine's early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference. Parsons attempts to tease out Putnam's views on existence and ontology, especially in relation to logic and mathematics. Wang's contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait's axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures.

PART I

SOME MATHEMATICIANS AS PHILOSOPHERS

1

THE KANTIAN LEGACY IN TWENTIETH-CENTURY FOUNDATIONS OF MATHEMATICS

Since my title refers to “twentieth-century” foundations of mathematics, you may think I haven’t quite got the news that we live in the twenty-first century. The fact is, however, that what I could find to talk about are some tendencies in the thought about the foundations of mathematics of the twentieth century, indeed more prominent in its first half than in its second. I will try to say at the end what relevance these tendencies might still have today.

There is a picture of the foundations of mathematics with which many English-speaking philosophers grew up in the period between roughly 1920 and the 1950s. To some extent this picture was formed in reaction to Kantian views. A prominent part of the picture was that Kant’s philosophy of mathematics was shown to be inadequate by developments in mathematics and physics, of which the most decisive are the development of non-Euclidean geometry and the theory of relativity, which applied non-Euclidean geometry to physics. Stated in this way, this is a claim with which one could hardly argue, but the picture goes on to hold that essential parts of Kant’s apparatus, especially the idea of pure or a priori intuition and of space and time as forms of intuition, no longer have any relevance to mathematics. This, according to the picture, has been shown by developments in logic and mathematics, first various aspects of the late nineteenth-century revolution in mathematics, but especially by the logicist construction of mathematics in Whitehead and Russell’s Principia Mathematica. (A present-day commentator on logicism might pay greater attention to Frege, but when the picture I am describing was ascendant, Frege tended to be viewed as a precursor of Russell and not much read.¹)

One could do much to fill in the details of this picture, but what is striking to anyone with more than a passing knowledge of the foundations of mathematics in the twentieth century is what it leaves out. Systematically, it leaves out the fact that aspects of Principia were contested from the beginning, especially the use of the axiom of reducibility and the axiom of infinity. Historically, it leaves out the two principal rivals of Russellian logicism as general points of view, the intuitionism of L. E. J. Brouwer (1881–1966), and what was called “formalism,” in effect the point of view underlying the foundational program pursued in the 1920s by David Hilbert (1862–1943). One does not have to look far into these views to see that they did not agree that the ghost of Kant had been quite laid to rest.

It is another question how Kantian their views actually were. I propose to examine this question with reference to Brouwer, Hilbert, and Hilbert’s junior collaborator Paul Bernays (1888–1977), who after the effective end of the Hilbert school in the 1930s had a long career during which he wrote quite a number of philosophical essays. Since Brouwer and Hilbert were the major Continental European figures in foundations for a fairly long period, it is natural to include them in a discussion of this kind. Bernays is chosen for another reason: He was philosophically the best trained of the three; it was probably partly for that reason that Hilbert chose him as his assistant in 1917. And starting as a disciple of the neo-Kantian Leonard Nelson, Bernays eventually came to a position rather removed from that of Kant, Nelson, or the more mainstream neo-Kantians. But there is still what might be described as a Kantian residue. I have written elsewhere on Bernays’s later views and will concentrate in this essay on other elements of his thought.²

1. Neo-Kantianism

Given the importance of neo-Kantianism in German philosophy from 1870 into the 1920s, one might expect the “Kantian legacy” in foundations during this period to have been mediated in an important way by neo-Kantianism. That is true only in a very limited way. The principal neo-Kantians were on the sidelines of the developments that interest me. Paul Natorp and Ernst Cassirer did write about the philosophy of mathematics.³ What is most interesting about Cassirer’s writing is the fact that in discussing contemporary developments, he began by concentrating on Russell’s Principles of Mathematics and contemporary writings of Louis Couturat. Cassirer argues that this early logicism is largely compatible with his own point of view and thus (in particular against Couturat) with some Kantian theses. Influenced by the general developments undermining Kant’s philosophy of geometry, he makes no attempt to defend the idea of space as an a priori form of intuition. Cassirer is also largely persuaded that the work of mathematicians leading up to and including Russell’s had succeeded in eliminating geometrical intuition even from the theory of real numbers.⁴ He makes the more general and striking remark:

It is important and characteristic that the immanent further construction of the Kantian theory has led by itself to the same result that is demanded more and more distinctly by the progress of science. Like “logistic”, modern critical logic has also progressed beyond Kant’s theory of “pure sensibility.”⁵

Similarly, Natorp remarks:

The post-Kantian philosophy that took its point of departure from him, as well as the present-day neo-Kantian tendency, has more and more taken exception to the dualism of intuition and pure thought and finally broken decisively with it.⁶

Unlike many mathematicians commenting on Kant, Cassirer was fully aware that according to Kant the synthetic activity of the understanding has a fundamental role in mathematical cognition, although he tends to interpret it at the expense of the forms of intuition.⁷ Concerning time and its role in arithmetic, he says that for Kant “it can only be a matter of the ‘transcendental’ conceptual determination of time, according to which it appears as the type of an ordered sequence.”⁸ In other words, it is the abstract structure embodied by time that matters for mathematics. Cassirer grants to “intellectual synthesis” much more power in mathematical cognition than Kant does, although he agrees with Kant that to be genuine cognition mathematics must be applicable to objects given in space and time, in particular in physics. Cassirer does not consider at this time that this view could be challenged by higher set theory.⁹

In Substanzbegriff und Funktionsbegriff Cassirer expresses views on the philosophy of arithmetic that are strongly influenced by Dedekind’s Was sind und was sollen die Zahlen? There Cassirer takes a position that could be called structuralist, as Jeremy Heis argues at some length.¹⁰ Some remarks are close to the slogans of recent structuralism, such as the following:

What is here expressed [by Dedekind] is that there is a system of ideal objects whose whole content is exhausted in their mutual relations. The “essence” of the numbers is completely expressed in their positions.¹¹

His main claim could be expressed by saying that the structure of the natural numbers is prior to other facts about them, and that in particular their application as cardinals can be derived. This is, I think, the main objection that he has to Frege’s and Russell’s treatment of number. Given certain assumptions, of course, the structure can be recovered from their definitions, but Cassirer thinks that that gets the conceptual order wrong.¹² Curiously, even in this slightly later work Cassirer does not take note either of the problem that the paradoxes pose for their treatment of number or of the theory that Russell develops in the face of them. Dedekind’s use of set theory apparently does not bother him; he seems prepared to accept what Dedekind uses as belonging to logic.

Cassirer does not seem concerned to work out the structuralist view as a general ontology of mathematical objects or to deal with the problems it faces; a criticism he makes of Frege and Russell, that in their definition of the cardinal number 1 the number one is presupposed, does not seem to me to be in accord with structuralism.¹³

Cassirer is an interesting commentator on developments in foundations just before and after the turn of the century. But I don’t know of evidence he or others of the main neo-Kantian schools had a substantial direct influence on Hilbert, Bernays, or Brouwer.¹⁴ However, neo-Kantianism did have a more direct influence on Hilbert and Bernays through Leonard Nelson (1882–1927). Nelson owed his position in Göttingen in large part to Hilbert’s patronage, and he was Bernays’s principal teacher in philosophy. He saw his mission in life as reviving and developing the Kantian philosophy of Jakob Friedrich Fries (1773–1843). In the philosophy of mathematics he wrote mainly about geometry, and here his position was a much more orthodox Kantianism, even at the end of his life.¹⁵ There is no attempt in his writing to minimize or interpret away pure intuition as a decisive factor in geometrical cognition. We shall see how this position left its traces in the thought of Bernays, and possibly also in that of Hilbert, although neither could ignore the obvious reasons why a Kantian philosophy of geometry would clash with contemporary physics.

2. Brouwer

Intuitionism begins with L. E. J. Brouwer’s dissertation of 1907, Over de grondslagen der wiskunde.¹⁶ This is a most unusual mathematical dissertation, because a lot of its content is philosophical. I will not comment on it directly. There is one element of the above picture that Brouwer fits perfectly, the rejection of the idea that a pure intuition of space has any relevance to the foundations of geometry or of mathematics generally. Although the point is remarked on in the dissertation,¹⁷ it is expressed most clearly in a lecture of 1909 on the nature of geometry.¹⁸ In the first part he traces a series of steps that undermined and eventually made untenable Kant’s philosophy of geometry, in the context of characterizations of geometric concepts by invariance under groups of transformations. The mathematical foundations of what one might loosely call an arithmetization of pure geometry existed already with Descartes’s coordinatization of geometry.

Consequently it is natural to consider the geometry determined by a transformation group in a Cartesian space as the only exact Euclidean geometry and to see its approximative realization in the world of experience as a physical phenomenon, which in principle has the same character as, for instance, Boyle’s law.¹⁹

The development of projective geometry was already a step toward alternatives to Euclidean geometry; in particular the group of projective transformations contains the classical non-Euclidean groups. On purely mathematical grounds Brouwer sees no reason for not giving “equal rights” to Euclidean and non-Euclidean geometry (p. 9, trans. p. 114), but, alluding to the special theory of relativity, he holds that the view that Euclidean geometry holds “physical apriority” is also untenable. He also dismisses the view that projective geometry is a priori, while experience determines the curvature and number of dimensions of s...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Contents
• Preface
• Introduction
• Part I: Some Mathematicians as Philosophers
• Part II: Contemporaries
• Bibliography
• Copyright Acknowledgments
• Index