Music, literature and the visual arts all underwent modernist transformations around the turn of the 20th century. Could it be that mathematics did the same thing? Just to give this question serious consideration, it is necessary to abandon the naive point of view of mathematics as a collection of necessary, eternal truths about the physical world, or even a universe of Platonic ideals.

In his book Plato's Ghost, Jeremy Gray provides a wide-ranging and thoughtful examination of the evolution of mathematical thought from about 1890 to about 1930. He argues that what happened to mathematics during this period was indeed a characteristically modernist transformation. He writes 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". Although this working definition is distilled from modernist views of art during the period around the First World War, it applies equally well to set theory or modern algebra as it does to Cubism.

Consider the case of geometry. The name itself means the measurement of the earth. The ancient Greeks believed they were describing physical reality when they proved geometrical theorems. Euclid synthesised the elementary geometry known in his time, about 300BC, into The Elements, considered to be the model for precision in mathematical proof for more than two millennia. When Augustin-Louis Cauchy tried to settle the foundational problems in calculus in his Cours d'analyse of 1821, he wrote of his methods: "I have sought to give them all the rigour which one demands from geometry."

Euclid's geometry accepted the Parallel Postulate. However, at about the same time that Cauchy was holding up Euclid as the standard for rigour, Janos Bolyai, Nikolai Lobachevsky and Carl Friedrich Gauss were developing versions of geometry in which the Parallel Postulate was false. Such non-Euclidean geometry was initially greeted with suspicion by mathematicians. However, largely through the work of Berhard Riemann and Eugenio Beltrami, the validity of non-Euclidean geometry became generally accepted with time. This paved the way for a critical change of focus in pure mathematics, away from necessary truths and towards derivability within a formal system. "If there are two distinct geometries," writes Gray, "then neither can be necessarily true."

One might wish to know which of the many possible geometries actually describes our physical universe, but such a question is not within the purview of modern mathematics. Gray points out that modernism in mathematics was strongest where the separation between mathematics and physics was most pronounced. In Germany, where in the late 19th century there was already a strong commitment to mathematical research for its own sake, modernism took root quickly. Mathematicians in the US were relatively few in number at that time, but a significant proportion of them had studied in Germany, so America was another early adopter of modernist ways. In the UK, where the emphasis had been on applied mathematics since the time of Newton, modernism came somewhat later, only after the formation of a community of pure mathematicians.

Gray points to Riemann in Germany and the American philosopher Charles Sanders Peirce as early prophets of mathematical modernism. He holds up David Hilbert's 1899 Foundations of Geometry as a manifesto of modernism. This masterpiece of the axiomatic method repaired various shortcomings in Euclid's logic and made clear how the various theorems of elementary geometry depended on the first principles he adopted. Even more importantly, Hilbert used the axiomatic method to create new mathematics, whereas axioms had generally been used before this time only to codify things that were already known.

Not limiting this work to a study of geometry alone, Gray considers the evolution of five fields of mathematics: geometry, analysis, algebra, set theory and logic. He examines many of the new branches of mathematics that arose during the period 1890-1930, including measure theory, topology, transfinite arithmetic and intuitionism. In addition, he pays considerable attention to the many philosophical issues raised by modern mathematics. Chronologically, Gray considers four periods: the period before modernism, culminating in what he calls "the Consensus in 1880", the arrival of modernism at the end of the 19th century, the embrace of modernism at the beginning of the 20th, and the period after the First World War.

Plato's Ghost places mathematics in a broad cultural context, but it is still essentially an internalist inquiry. It is probably best suited to readers with a good background in the mathematical sciences, but it is not overly technical and is sufficiently self-contained to be accessible to anyone interested in the history or philosophy of science. It is an absorbing and thought-provoking study in the history and philosophy of mathematics, and one that seems destined to frame future debate and research in these fields.

## Plato's Ghost: the Modernist Transformation of Mathematics

By Jeremy Gray Princeton University Press

526pp, £32.50 ISBN 9780691136103

Published 9 October 2008

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