BOOK REVIEW

# Modernism in Mathematics

**PLATO’S GHOST: The Modernist Transformation of Mathematics.** Jeremy Gray. x + 515 pp. Princeton University Press, 2008. $45.

Modern mathematics—in the sense the term is used by working mathematicians these days—took shape in the period from 1890 to 1930, mainly in Germany and France. Strikingly new concepts were introduced, new methods were employed, and whole new areas of specialization emerged, while other themes were relegated to the dusty shelves of history. At the same time, the nature of mathematical truth and even the consistency of mathematics were put into question, as mathematicians, logicians and philosophers grappled with the subject’s very foundations. Is it an accident that the same period witnessed radical societal, cultural, technological and scientific changes, almost across the board, or is there some intrinsic connection between these things?

In *Plato’s Ghost,* Jeremy Gray contends that the best way to explain this remarkable transformation of mathematics is in terms of certain overarching “modernist” features or themes. Because the term *modernism* is usually reserved for the changes that took place in literature and the arts during the period in question—such icons as Pablo Picasso, Wassily Kandinsky, James Joyce, Virginia Woolf, T. S. Eliot, Ezra Pound, Igor Stravinsky and Arnold Schoenberg come to mind—and because there is no apparent connection between the content of the new mathematics and that of the representative artistic creations, we must ask what ties them together in the author’s view. His explanation is that

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.

To what extent does this accord with how cultural historians characterize modernism, and—independently of that—to what extent are these the best terms in which to describe the development of modern mathematics? Both are vexed questions to which I shall return, if only briefly, at the conclusion of this review.

Gray has worked since 1975 as a member of the Centre for the History of the Mathematical Sciences at the Open University in the United Kingdom and has made extensive contributions to the history of modern mathematics, especially concerning geometry and parts of analysis. In *Plato’s Ghost *he has moved beyond that to present us with an ambitious and in many respects remarkable synthesis of the modern transformation of mathematics via structural and set-theoretic notions, together not only with its logic and philosophy but also with related developments in artificial languages and psychology. In this he has drawn on an extensive literature that has been growing rapidly since the 1960s on more or less specific aspects of these same subjects; as reflected in the bibliography and its relations to the text, there appears to be hardly anything that Gray has overlooked in the process. In the introduction he lists the things that he believes are novel about his book, and one is that he has resurrected a number of forgotten individuals. This may appeal especially to historians of the era, but the reader who wants to concentrate on the thought and contributions of the better-known figures will find that the text is broken up so well into many independent sections that one can usefully pick and choose one’s way through the wealth of material.

In order to reach as wide an audience as possible, Gray provides very few detailed descriptions of the relevant mathematics; the more technical ones are set aside in brief italicized paragraphs. To give the general reader a sense of the ideas involved, these are supplemented by a few diagrams from geometry and analysis plus a glossary of mathematical terms. There are many photographs (with no separate listing) of both prominent and obscure figures in this period; reflecting the general nature of the profession at that time, not one of them is a woman, although Emmy Noether, among possible candidates, certainly deserved to be among those represented for her leading contributions to modern algebra.

Following an introduction and an initial overview chapter, the book is largely chronological in order: Chapter 2, “Before Modernism,” covers the period 1800-1880; chapter 3, “Mathematical Modernism Arrives,” takes us up to 1900; chapter 4, “Modernism Avowed,” covers the period up to 1914; some (but not enough of) the story of the further development of modern mathematics to 1930 is presented in the concluding chapter 7, “After the War.” Before that there are two chapters dealing with side issues: chapter 5, “Faces of Mathematics,” and chapter 6, “Mathematics, Language and Psychology.”

One main thread traced through all this is that of geometry: The initial high points in the 19th century were the discovery of a consistent non-Euclidean plane geometry by János Bolyai and Nicolai Lobachevskii, and the renewal of work on projective geometry by Jean Victor Poncelet featuring the duality between the notions of point and line. Later, on the road to abstraction, Felix Klein combined these together with Euclidean geometry under the single roof of his famous “Erlangen Program,” in which each geometry is classified according to the transformations leaving its basic notions invariant. Finally, in Gray’s view, the paradigmatic modernist contribution came with David Hilbert’s monograph on the foundations of geometry in 1899. This presented a new and more rigorous axiomatization of Euclidean geometry, for which the independence of various of the axioms—such as the parallel postulate—from the others was established by the construction in each case of a model satisfying all the axioms but the one in question. Famously, Hilbert spoke of the variety of possible interpretations of the basic notions by saying that one could replace points, lines and planes with tables, chairs and beer mugs, respectively, with corresponding relations between them. Hilbert’s view was that mathematical concepts are implicitly defined in structural terms by axiomatic systems, of which all one needs to know is that they are consistent and complete.

Incidentally, Gray places the unique ideas of Bernhard Riemann on the treatment of curvature for non-Euclidean manifolds of arbitrary finite dimension—which he presented in his astounding *Habilitationsvortrag* (higher doctorate lecture) in 1854—with the developments in geometry before modernism. But the nature of Riemann’s work is of quite a different character, employing as it does concepts from analysis in an essential way. Far ahead of its time, it would eventually provide the underlying mathematics for the general theory of relativity and lead to the modern subject of differential geometry.

The existence of a number of non-Euclidean geometries and the increased distancing of geometry from reality raised problems for the philosophers, especially the post-Kantians. The question was how to accommodate Immanuel Kant’s view that space and time are fixed in the intrinsic human structuring of experience and that this is what makes Euclidean geometry (and arithmetic) true. Also, according to Kant, mathematics, which proceeds by constructions in intuition, constitutes synthetic a priori knowledge. An attack on that general idea came from a different direction, in spirit going back to Gottfried Wilhelm Leibniz, via Gottlob Frege’s attempt to show that arithmetic, at least, is analytic in the philosophical sense (that is, its statements are true solely in virtue of the meaning of the concepts involved) through its reduction to logic. Although Bertrand Russell discovered a fatal contradiction in Frege’s system, his elaborate effort with Alfred North Whitehead to repair this approach to the foundations of mathematics in their *Principia Mathematica* proved to be enormously influential for the subsequent development of mathematical logic in the 20th century, even though it too was beset with problems.

In support of his modernist thesis in algebra, Gray takes special note of the novel attacks by Ernst Kummer on Fermat’s Last Theorem using factorization in certain classes of “ideal” complex numbers; this was to have a distinctive set-theoretical turn in the hands of Richard Dedekind later in the century. As another example, the abstract concept of a group originated in the work of Évariste Galois on the relations between permutations of the roots of a polynomial equation and its solvability or unsolvability by radicals. But then finite groups in general were studied for their own sake, with the investigation of all possible such systems meeting certain special conditions, such as being commutative, “simple,” solvable and so forth.

In analysis, Gray’s poster example is the replacement by Henri Lebesgue of the concept of integration—intuitively conceived of as determining the area under a curve or the volume under a surface—with the concept of the measure of a set of points, controlled by four abstract axiomatic conditions. Another example is the introduction by Maurice Frechét of concepts of distance between functions in the study of their approximation to minimal solutions of variational problems; he then moved on to deal with these in terms of a general concept of metric spaces. Hardly mentioned is the combination of this idea with the algebraic concept of vector space in the modern subject of functional analysis, which would lead after the war, at the hands of John von Neumann, to an abstract mathematical framework for the interpretation of quantum mechanics.

Increasingly, set-theoretical methods and concepts and nonconstructive arguments took a ubiquitous role; these originated with Georg Cantor’s adventuresome steps into the transfinite in the late 19th century. In the first decade of the 20th century, Ernst Zermelo isolated the controversial axiom of choice needed to establish basic properties of Cantor’s transfinite numbers and showed that it had been used implicitly by mathematicians for many arguments elsewhere, especially in analysis.

Philosophically, the justification for the axiom system of set theory that Zermelo introduced requires a Platonistic account of the nature of mathematics, according to which the objects of mathematics exist in an abstract realm outside of space and time independent of human ideas and constructions; concomitantly, mathematical truths hold whether or not they can be established by human beings. This accords with mathematical practice to the extent that mathematicians believe their work is a matter of discovery and verification, not of invention. As a philosophy of mathematics, Platonism is deeply problematic, but no effort thus far to replace it with one brand or another of logicism, formalism, constructivism, nominalism or fictionalism (to name only the main alternatives that have been proposed) has succeeded in accounting for the accepted body of mathematics while being philosophically convincing.

Gray is not the first to try systematically to hang the development of modern mathematics on the peg of modernism. But an earlier book by Herbert Mehrtens, *Moderne Sprache, Mathematik* (1990), with its focus on the role of language, is more limited in scope. In a trenchant essay, “How Useful Is the Term ‘Modernism’ for Understanding the History of Early Twentieth-Century Mathematics?,” which is included in a collection edited by Moritz Epple and Falk Mueller, *Modernism in the Sciences, ca. 1900–1940* (forthcoming from Akademie Verlag), the historian Leo Corry critiques both the work of Mehrtens and a 2006 article by Gray that served as an *avant-propos *for the present book. First of all, Corry notes, much has been written about the cultural phenomenon of modernism in literature and the arts, without any consensus being achieved about what its defining features are. Extending the appellation *modernism* to mathematics, he says, is like “shooting an arrow and then tracing a bull’s eye around it.” He also notes that “some of the most essential features usually associated with Modernism in art [such as abstraction and formalism] are ubiquitous in mathematics throughout history.” It is thus a genuine question how illuminating any such effort is. Space prevents me from going further into Corry’s stimulating essay, with which I am largely in agreement, so I can only suggest it as something to be read alongside Gray’s anatomization. In any case, I can certainly recommend *Plato’s Ghost* highly as a rich resource and point of departure for readers who want to learn more about this exciting period in the development of modern mathematics.

*Solomon Feferman is professor of mathematics and philosophy, emeritus, and Patrick Suppes Family Professor of Humanities and Sciences, emeritus, at Stanford University. He is the author of *In the Light of Logic* (Oxford University Press, 1998) and is coauthor with Anita Burdman Feferman of *Alfred Tarski: Life and Logic* (Cambridge University Press, 2004).*

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