Jeremy Gray is an Emeritus Professor of The Open University and an Honorary Professor in the Mathematics Department at the University of Warwick. His research interests are in the history of mathematics, specifically the history of algebra, analysis, and geometry, and mathematical modernism in the 19th and early 20th Centuries. The work on mathematical modernism links the history of mathematics with the history of science and issues in mathematical logic and the philosophy of mathematics.
He was awarded the Otto Neugebauer Prize of the European Mathematical Society in 2016 for his work in the history of mathematics, and the Albert Leon Whiteman Memorial Prize of the American Mathematical Society in 2009 for his contributions to the study of the history of modern mathematics internationally. In 2012 he was elected an Inaugural Fellow of the American Mathematical Society. In 2010 he was one of the nine founder members of the Association for the Philosophy of Mathematical Practice (APMP).
He is the author of eleven books, of which among the most recent are Plato’s Ghost: The Modernist Transformation of Mathematics (Princeton U.P. 2008), Henri Poincare: a scientific biography (Princeton 2012), and The Real and the Complex (Springer 2015). Two more books are to be published in 2018: Under the Banner of Number: A History of Abstract Algebra, by Springer, and Simply Riemann in the Simply Charly series of e-books.
Poincare has an exaggerated reputation not being rigorous in his work. In this talk I shall show that he cared about rigour in mathematics, but had justified criticisms of it. However, the more important task was to understand mathematics and physics, and this meant to be enabled to discover new ideas. Certainty in abstract mathematics was provided by the principle of recurrence, which imposed limits on any theory of sets. Thereafter, a pragmatic sense of certainty was provided in applied mathematics and physics by the use of conventions. Conventions, he believed, govern our choice of a geometry for space and the choice of the laws of mechanics and other branches of physics. Objectivity, he said, depended on discourse, and I shall argue that Poincare’s fundamental position is that the use of mathematics in science is close to Wittgenstein’s idea of a language game.
By 1910, the year he turned 25, Weyl was developing a finitist philosophy of mathematics, based on a logical theory of relations. He also believed that the human mind can understand ideas only sequentially. He developed this approach on his book The Continuum (1918), and for a time came close to agreeing with Brouwer’s intuitionism, but he abandoned them in the mid-1920s when he became involved in exploring the theory of Lie groups. He then had to turn back towards Hilbert’s ideas about mathematics and physics, and developed his own theory of what he called the symbolic universe in which mathematics and physics supported each other in complementary ways. Weyl sought a unified philosophy that would govern not only his scientific practice but be rooted in a theory of knowledge and an understanding of how it is acquired.