# Plato's Ghost

The Modernist Transformation of Mathematics

Publication Year: 2008

*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.

Published by: Princeton University Press

#### Cover

#### Title Page, Copyright Page, Poem

#### Contents

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pp. vii-viii

#### Introduction

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pp. 1-17

In this book I argue that the period from 1890 to 1930 saw mathematics go through a modernist transformation. Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated—indeed, anxious—rather than a naïve relationship with the day-to-day world, ...

#### 1 Modernism and Mathematics

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pp. 18-38

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.’’^{1} ...

#### 2 Before Modernism

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pp. 39-112

Among the many lasting consequences of the French Revolution was its effect on higher education. Although the École Polytechnique was and is very different from a modern university, its creation marks the decline of the learned academy as a central focus for research, and the start of the system of high-level teaching coupled with the production of new knowledge. ...

#### 3 Mathematical Modernism Arrives

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pp. 113-175

Beltrami’s disk model of the geometry of Lobachevskii gave it mathematical rigor and respectability. Where there had been neglect, distrust, hostility, now there was a quiet chorus of agreement among mathematicians: there was, after all, a geometry that differed from Euclid’s only over the definition of parallels, and it was the intrinsic geometry of a disk with constant negative curvature. ...

#### 4 Modernism Avowed

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pp. 176-304

We have already seen that Italian mathematicians were energetic students of projective
geometry, which some, such as Corrado Segre and Federigo Enriques, extended
to *n* dimensions and treated abstractly, the better to allow it to be interpreted
in a variety of ways. ...

#### 5 Faces of Mathematics

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pp. 305-373

As mathematics advanced in the late nineteenth and early twentieth centuries, mathematicians fashioned for themselves a new image of the subject: autonomous, abstract, largely axiomatic, and unconstrained by applications even to physics. At the same time, they often valued and cherished the link to physics, whether from a genuine appreciation of, and interest in, science, ...

#### 6 Mathematics, Language, and Psychology

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pp. 374-405

It was the painter and aspiring but unsuccessful poet Edgar Degas who complained to his friend Stéphane Mallarmé that he had many ideas for poems but could not write the poems, to which Mallarmé replied, ‘‘Poems, my dear friend, are made of words, not ideas.’’ ...

#### 7 After the War

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pp. 406-462

The origins of modernism in mathematics, and its eventual acceptance by large sections of the mathematical profession, have now been traced in numerous disciplines of mathematics as well as in the philosophy of mathematics and the relations of those subjects with logic. ...

#### Appendix: Four Theorems in Projective Geometry

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pp. 463-466

#### Glossary

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pp. 467-472

#### Bibliography

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pp. 473-502

#### Index

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pp. 503-515

E-ISBN-13: 9781400829040

E-ISBN-10: 1400829046

Print-ISBN-13: 9780691136103

Print-ISBN-10: 0691136106

Page Count: 526

Publication Year: 2008

Edition: Course Book