March 28th, 2018, Keio University

Lecture 1 (15:00-16:00): ``Poincaré on proof and understanding''

Poincaré 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 Poincaré’s fundamental position is that the use of mathematics in science is close to Wittgenstein’s idea of a language game.

Lecture 2 (16:00-17:00): ``The philosophy of Hermann Weyl''

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.

April 2nd, 2018, Kyoto University

``The philosophy of Hermann Weyl''

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.

April 7th, Nagoya University: Symposium on Modernism and Modernisation of Mathematics

Lecture 1: Jeremy Gray, ``Poincaré and Weyl: two dissenters from mathematical modernism''

Around 1900 a characteristic form of modern mathematics took over the subject, which we associate with Georg Cantor, Richard Dedekind, and David Hilbert among others. It sees mathematics as an autonomous system of ideas, emphasises the formal or axiomatic aspects, and brings about a complicated relationship with the sciences. Henri Poincaré (1854?1912) and Hermann Weyl (1885?1955) preferred a much more intimate connection between mathematics and physics and argued in different ways for a symbiotic approach, which they also linked to a broader philosophical vision.

Lecture 2: Susumu Hayashi, ``How was Mathematics modernized?''

I have been developing a historical view on the modernization, in the sense of Max Weber sociology, of mathematics for the last 19 years. I will outline it in this presentation. The starting point of my research was an enigmatic (to me) claim by Kurt Godel in his unpublished philosophical essay. His claim may be interpreted as “World-views have been disenchanted (modernized) through the history since the Renaissance. Particularly in physics, this development reached a peak in the 20th century. However, mathematics alone went in the opposite direction as set theory was introduced into it.” My historical view was slightly changed from Godel’s to make it fit into Weber’s modernization theory. I will discuss how important Hilbert’s program and Godel’s incompleteness theorems were for the modernization of mathematics, and how they made the process of modernization of mathematics somewhat different from the process of modernization of physics.

Jeremy Gray short bio

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 Poincaré: 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.

Contact Information

Minao Kukita (久木田水生), minao.kukita@i.nagoya-u.ac.jp