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Princeton University Press
Plato's Ghost: The Modernist Transformation of Mathematics

Plato's Ghost: The Modernist Transformation of Mathematics

by Jeremy GrayJeremy Gray


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Plato's Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions.

Plato's Ghost evokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Richard Dedekind, Henri Lebesgue, Henri Poincaré, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naïve set theory and the revived axiomatic method—debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism.

Plato's Ghost is essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics.

Product Details

ISBN-13: 9780691136103
Publisher: Princeton University Press
Publication date: 09/22/2008
Edition description: New Edition
Pages: 528
Product dimensions: 7.40(w) x 10.10(h) x 1.50(d)

About the Author

Jeremy Gray is professor of the history of mathematics and director of the Centre for the History of the Mathematical Sciences at the Open University. His books include Worlds Out of Nothing and János Bolyai, Non-Euclidean Geometry, and the Nature of Space.

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Copyright © 2008 Princeton University Press
All right reserved.

ISBN: 978-0-691-13610-3

Chapter One


1.1 Modernism in Branches of Mathematics

The origins of modern mathematics can be found in the mathematical practices of the nineteenth century. It has become a commonplace that the nineteenth century saw the rigorization of analysis under the slogan, coined by Felix Klein in a public lecture in 1895, of the "arithmetization of analysis." Klein was then making his bid to be the leading mathematician in Germany, with a vision of the subject as a whole, and, as he was eager to point out, the arithmetization he was criticizing underestimated the flourishing nineteenth-century line in applied mathematics, but it is true nonetheless that analysis was rigorized, and indeed based on arithmetic. However, this was only intermittently the aim of mathematicians, and their motives were various (correct reasoning, proving theorems, resolving contradictory answers, obtaining good applications, and pedagogy belong among them).

The late nineteenth century saw many overlapping kinds of mathematics being done, as well as a growing awareness of the possibility of error in mathematical reasoning. Attempts to give some "established truths" the security of a decent proof can seem obscure or pedantic-as they did to many physicists-but once the point is successfully put across that a proof is lacking, the search for one can simply become part of accepted mathematical practice. Developments in geometry displaced old "certainties" about the nature of mathematics and mathematical objects. The discovery of non-Euclidean geometry by János Bolyai and Nicolai Lobachevskii (and, if one is charitable, Carl Friedrich Gauss) resolved ancient difficulties about the parallel postulate in the most contentious way. Mathematicians finally accepted their work after it was given foundations in differential geometry by Bernhard Riemann, who was widely recognized as the most original mathematician of the mid-nineteenth century, and Eugenio Beltrami. It was then speedily incorporated into projective geometry by Klein. This, the most famous "revolution" in mathematics, put an end to any idea that mathematics was, as it were, distilled science. If there are two distinct geometries, then neither can be necessarily true.

The situation was, however, even more complicated and difficult for mathematicians. A chain of writers from Gauss and August Crelle at the start of the nineteenth century to Moritz Pasch and Bertrand Russell at the end found problems with the very apparatus of Euclid's geometry: terms once obvious seemed harder and harder to define, gaps in reasoning harder and harder to fill. There were discussions of what counts as a plane, for example. Hitherto unnoticed gaps in reasoning (or missing axioms) were picked out and filled, most famously by Pasch as part of wholesale rewrite of elementary (projective) geometry. It was noticed, for example, that the opening definitions in Euclid's Elements do not actually succeed in defining anything. Consider the first: "A point is that which has no part." This is very evocative if you know what a point is, but it is of no use if you do not. Deliberate errors (such as the famous purported proof that all triangles are isosceles) were put forward to highlight the degree to which Euclidean arguments presupposed diagrams and might for that reason be erroneous. These, and other criticisms I shall look at, all cumulatively undermined previous confidence that Euclid's Elements were indeed the epitome of reasoning.

Projective geometry, by any standards the most remarkable success story of nineteenth-century geometry, in both its synthetic and analytic modes deployed points, lines, and ultimately hyperplanes at infinity in a manner unintelligible to classical geometry. Moreover, as the philosopher and historian Ernest Nagel argued many years ago, the use of duality in projective geometry plays havoc with intuition and, he argued, opens the door to purely logical reasoning. This claim has recently been contested in a nineteenth-century English setting by Richards, but I think it is on the mark and the English are best seen here and in algebra as a case unto themselves (and a good test case for the modernist thesis).

Modernism in geometry arrives with Hilbert, and the difference between his Grundlagen der Geometrie (Foundations of Geometry) and the earlier work of Pasch. As recent scholars have emphasized, from Hans Freudenthal to Elena Marchisotto and Michael Hallett and Ullrich Majer, the Italian contribution is if anything more abstract in its axiomatic approach, and it is currently being integrated into a full historical account in the work of Avellone et al. and Umberto Bottazzini. Several threads need to be disentangled. The absorption of non-Euclidean geometry into differential geometry need not have provoked a crisis in the foundations of mathematics (that it did is another matter, connected to the global character of Euclidean definitions). The break with any kind of geometric intuition in defining geometric terms is characteristically modernist, however. A similar shift took place in the presentation of projective geometry. Then there are the implications for other branches of mathematics-and so for mathematics as a whole-of the introduction of the abstract axiomatic method. A significant group here is the American postulation theorists, who also influenced the logician Alfred Tarski in the 1920s. Last, but indeed by no means least, there are the philosophers' responses and debates about the import of non-Euclidean geometry for philosophy in general and the philosophies of mathematics and science in particular. Poincaré's famous conventionalism arose in this setting, but the passions the topic aroused (contagious even among modern painters, as Henderson has described in her The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983) are worthy of independent attention.

A long-running topic of relevance to the development of proof is the convergence of series, especially Fourier series. Another is potential theory: the persistent failure of mathematicians to prove the existence of harmonic functions satisfying arbitrary boundary conditions, although physicists found strong heuristic grounds for accepting them. These often, and for once well-told, tales bear reexamination for two reasons. They exemplify the origins of rigorous analysis in broad lines of mathematical inquiry, and they culminate in what might be called the crisis of continuity. This is the widespread feeling among mathematicians around 1900, documented in many sources, that the basic topic of analysis, continuity, was profoundly counterintuitive. This realization marks a break with all philosophies of mathematics that present mathematical objects as idealizations from natural ones; it is characteristic, I argue, of modernism.

1.1.1 Ontology and Epistemology

There are indeed two foundational aspects at work-one largely ontological, the other largely epistemological. The first is the one usually called the arithmetization of analysis. It traces a path from Cauchy's novel theory of functions in the 1820s that was based on certain limiting processes defining continuity, differentiability, and integrability through its unclear notion of the real numbers to an eventual resolution somewhere in the use of limiting processes to define the real numbers. Such a destination presupposed a satisfactory theory of the integers, whence the familiar slogan. At least for mathematicians, ontological clarity was shed on the nature of the real numbers by the ways in which they were defined from the rational numbers (which were themselves constructed from the integers), and then, in a second phase, light was shed on the integers themselves. In the first phase it was the difficult and sometimes counterintuitive behavior of limit processes that forced the nature of irrational numbers onto the attention of mathematicians. In the second phase, formulations of arithmetic led inexorably to set theory and then to deep problems with set theory, one of the many instances where it was sometimes felt that problems had only been traded in for deeper problems.

The epistemological aspect stays with the notion of continuity: its separation from differentiability, the emergence of a class of phenomena sometimes called pathological and which required special techniques to handle that must (it was suggested) demarcate mathematics from science or philosophy. These too provoked deep problems that set theory was created to handle.

The third major domain of mathematics sees a similar transformation; indeed, the term "modern algebra" has greater resonance than modern analysis or modern geometry. The latter two sound like marketing terms today, but the first carries real weight: the arrival of structural algebra with the work of Emmy Noether and her school in the 1920s and then Bourbaki in the next generation. The historian Leo Corry has recently analyzed the history in considerable detail, laying great emphasis on the structural aspects. As with applied mathematics, there was also a strong sense that these developments grew out of real questions in mathematics and were not pursued merely for their own sake. It is well known to historians of mathematics that many of the key elementary terms of modern structural algebra can be found in nineteenth-century algebraic number theory and algebraic geometry, along with proofs of many of the key results. In view of this tradition, it is necessary to characterize the modernist aspect of the matter with greater precision, and again this can be done in ontological and epistemological terms. As with the analysis of continuity, the abstraction and generality of the new ontology promoted the new epistemology.

Ontologically, the erosion of the concept of number is crucial. The most famous example is that of the quaternions, a noncommutative system of numberlike objects discovered by William Rowan Hamilton in 1843. I shall argue that the asymmetrical responses to non-Euclidean geometry and quaternions are interesting and need fresh thought: geometry was put with mechanics and empirical matters, while numbers remained with logic. But the high road to structural algebra leads from the algebraic integers used even by Leonhard Euler to Ernst Kummer's more mysterious ideal numbers to Richard Dedekind's ideals, and then joins a complicated and previously not-well-understood topic concerned with Kronecker's vision of algebra and its reception. This last stands at the junction of algebraic number theory and algebraic geometry. The outcome is a concept of algebraic integer vastly more general than that of the usual integers, and an array of theorems that show just how clearly the usual integers form a special case. This aspect is the epistemological one. The two together made it possible for mathematicians to take the decisive modernist step of setting up the new theory in opposition to all previous ones, as the new autonomous domain that includes, in reworked form, all "that matters" of earlier approaches.

Unlike the developments in classical mathematics, contemporary changes in logic grew out of a moribund field. It is generally agreed that investigations into logic grew quietly to a halt after Leibniz, his followers like Christian Wolff, and finally Johann Heinrich Lambert at the end of the eighteenth century. They were revived by a remarkable British school, prominent among whom were de Morgan and Boole. After them, but independently, Gottlob Frege recast the traditional theory of logic insofar as it was an analysis of mathematical language, and the period of modern logic began. The nature of Frege's contribution and its significance (then or since) remains controversial, and so too does the work of his contemporaries. Some (see van Heijenoort 1980) see a bifurcation into algebraic logic, associated with George Boole, Charles Sanders Peirce, and Ernst Schröder, and the alternative offered by Frege and Peano. An early exponent of this view was Russell, who, it is often said, was influential in marginalizing the algebraic viewpoint. Others, such as Anellis (1995), find the algebraic school more active in its day, and more substantial in their points, than Russell appreciated, and consequently differ in their historical assessment of the period.

The first significance of the explosion of interest in mathematical logic in the second half of the nineteenth century is that it took place at all, given the hitherto quiet state of affairs. Gradually during the first half of the nineteenth century it had come to be appreciated that classical logic was inappropriate to classical mathematics: Euclidean geometry was simply not written in syllogisms. What then emerged as an acknowledged unsolved problem in the subject was posed by quantifiers. This problem would not in itself have caught the attention of mathematicians, who were in the process of giving up Euclidean geometry anyway. What caught their attention was both the algebraic side of the new logic, growing as it did out of the formal approach to problems in analysis, and the possibility that logic was a possible resting place for mathematics, deprived as it was increasingly of bedrock in the natural sciences.

The second significance of the explosion is that it promoted successive waves of investigation into logic itself. Logic, it transpired, was not a simple matter of thinking clearly, which, to be sure, might not be a simple matter to describe fully and accurately. There were levels of logic, requiring assumptions about modes of reasoning, which seemed arbitrary and did not at all have the force of the elementary logical laws (which could quite naturally be thought of as laws of thought). This recognition was slowly and painfully won, and consequently many of the earliest analyses of the relation between logic and mathematics are bedeviled by obscurities. In particular, the existence of models, and the nature of various kinds of models for various kinds of axiom systems, was obscure for a long time, an obscurity that can be measured by the distance from Hilbert to Thoralf Skolem and Tarski.

The third significance is that the enterprise was to have no satisfactory resolution. Many, some might say all, of the major problems that were thrown up have been solved; their answers are theorems in modern mathematical logic. But the answers collectively show both that elementary logic is inadequate as a foundation for any kind of sophisticated mathematics, and that no adequate foundation has enough intuitive feel or force to command assent. In short, the modernist transformation of logic itself meant that even logic no longer has a straightforward connection with simple clear thinking.

Taken together, this plunge into the depths of the nature of truth and proof in mathematics epitomizes the introspective, even anxious character of modernist mathematics. The initial possibility that logic could give naive but acceptable answers to problems in the foundations of mathematics, upon which mathematicians could then erect sophisticated theories of an entirely mathematical kind, was found to be barren. Once this was discovered, there was no alternative but to seek sophisticated foundations, adequate to the sophisticated mathematics of the day. Issues in the foundations of mathematics became issues in the very philosophy of mathematics, and because of their intimate connection to other branches of mathematics caught the attention of mathematicians.


Excerpted from PLATO'S GHOST by JEREMY GRAY Copyright © 2008 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

Introduction 1

I.1 Opening Remarks 1

I.2 Some Mathematical Concepts 16

CHAPTER 1: Modernism and Mathematics 18

1.1 Modernism in Branches of Mathematics 18

1.2 Changes in Philosophy 24

1.3 The Modernization of Mathematics 32

CHAPTER 2: Before Modernism 39

2.1 Geometry 39

2.2 Analysis 58

2.3 Algebra 75

2.4 Philosophy 78

2.5 British Algebra and Logic 101

2.6 The Consensus in 1880 112

CHAPTER 3: Mathematical Modernism Arrives 113

3.1 Modern Geometry: Piecemeal Abstraction 113

3.2 Modern Analysis 129

3.3 Algebra 148

3.4 Modern Logic and Set Theory 157

3.5 The View from Paris and St. Louis 170

CHAPTER 4: Modernism Avowed 176

4.1 Geometry 176

4.2 Philosophy and Mathematics in Germany 196

4.3 Algebra 213

4.4 Modern Analysis 216

4.5 Modernist Objects 235

4.6 American Philosophers and Logicians 239

4.7 The Paradoxes of Set Theory 247

4.8 Anxiety 266

4.9 Coming to Terms with Kant 277

CHAPTER 5: Faces of Mathematics 305

5.1 Introduction 305

5.2 Mathematics and Physics 306

5.3 Measurement 328

5.4 Popularizing Mathematics around 1900 346

5. Writing the History of Mathematics 365

CHAPTER 6: Mathematics, Language, and Psychology 374

6.1 Languages Natural and Artificial 374

6.2 Mathematical Modernism and Psychology 388

CHAPTER 7: After the War 406

7.1 The Foundations of Mathematics 406

7.2 Mathematics and the Mechanization of Thought 430

7.3 The Rise of Mathematical Platonism 440

7.4 Did Modernism'"Win"? 452

7.5 The Work Is Done 458

Appendix: Four Theorems in Projective Geometry 463

Glossary 467

Bibliography 473

Index 503

What People are Saying About This

Moritz Epple

A major addition to scholarship in the history of mathematics and in the history of science in general. Gray throws light on a major cultural transformation of mathematics. The book is written for a large readership of historians of science, philosophers, and scientists. It will have repercussions in broader debates on scientific culture, and will remain a reference work for many years to come.
Moritz Epple, Johann Wolfgang Goethe University

From the Publisher

"In this impressive synthesis, Gray brings, in a largely nontechnical way, the technical development of mathematics from the 1880s to the 1930s into the broader historical analysis of the concept of modernity. His argument promises not only to challenge historians of mathematics but also, finally, to bring mathematics into wider discussions of cultural history."—Karen Hunger Parshall, author of James Joseph Sylvester: Jewish Mathematician in a Victorian World

Karen Hunger Parshall

In this impressive synthesis, Gray brings, in a largely nontechnical way, the technical development of mathematics from the 1880s to the 1930s into the broader historical analysis of the concept of modernity. His argument promises not only to challenge historians of mathematics but also, finally, to bring mathematics into wider discussions of cultural history.
Karen Hunger Parshall, author of "James Joseph Sylvester: Jewish Mathematician in a Victorian World"

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