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Plato's Ghost

Plato's Ghost: The Modernist Transformation of Mathematics

Copyright Date: 2008
Pages: 526
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  • Book Info
    Plato's Ghost
    Book Description:

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

    eISBN: 978-1-4008-2904-0
    Subjects: Mathematics, History

Table of Contents

  1. (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, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve.

    This brisk definition is certainly...

  2. (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 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...

  3. (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. In its early days this, the first of the Grandes Écoles, was animated by Gaspard Monge’s vision of the central place of mathematics in the advanced education of the citizen.

    When the revolution began, Monge was drawn into radical politics.¹ After...

  4. (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. Beltrami (1868b) showed that the same approach could be made to work in any number of dimensions. Non-Euclidean geometry had arrived among the mathematicians.

    There was still to be a quarter-century of debate about what it was that had been...

  5. (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 tondimensions and treated abstractly, the better to allow it to be interpreted in a variety of ways. Another of Segre’s students, who followed him in this work but was more sympathetic to the axiomatic approach, was Gino Fano. When Fano wrote aboutn-dimensional geometry, he tried systematically to show that each new postulate is independent of the previous ones, and in this way he made some interesting discoveries almost without noticing.

    For example, Fano’s fourth...

  6. (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, or from a shrewd sense that the utilitarian arguments for science and technology could only benefit mathematicians if they were somehow conspicuously tied to those subjects. On the other side of the divide that was growing up, albeit in rather different ways in different countries,...

  7. (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.” The same is true, and to the same extent, of a mathematical paper, as every contemporary mathematician knew. But in what language should mathematicians write, and is, or could there be, a language best suited to their purposes?

    The international mathematical arena in 1900 was an arena of competing nationalisms, chiefly those of Europe, the...

  8. (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. At least another book could be written describing developments after the First World War, but this is not the place. Accordingly, this chapter takes those topics in the foundations of mathematics that simply cannot be left in midair and traces their implications for the subject as a whole in the interwar period.¹

    It would also take a book...

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