If you wanted to explain how philosophy has been important to mathematics, and why it can and should continue to be, it would be hard to do it better than Jeremy Avigad. In this beautiful plea for a mathematically relevant philosophy of mathematics disguised as a book review he writes:

Throughout the centuries, there has been considerable interaction between philosophy and mathematics, with no sharp line dividing the two. René Descartes encouraged a fundamental mathematization of the sciences and laid the philosophical groundwork to support it, thereby launching modern science and modern philosophy in one fell swoop. In his time, Leibniz was best known for metaphysical views that he derived from his unpublished work in logic. Seventeenth-century scientists were known as natural philosophers; Newton’s theory of gravitation, positing action at a distance, upended Boyle’s mechanical philosophy; and early modern philosophy, and philosophy ever since, has had to deal with the problem of how, and to what extent, mathematical models can explain physical phenomena. Statistics emerged as a response to skeptical concerns raised by the philosopher David Hume as to how we draw reliable conclusions from regularities that we observe. Laplace’s

Essai philosophique sur la probabilités, a philosophical exploration of the nature of probability, served as an introduction to his monumental mathematical work,Théorie analytique des probabilités.

In these examples, the influence runs in both directions, with mathematical and scientific advances informing philosophical work, and the converse. Riemann’s revolutionary

Habilitationlecture of 1854,Über die Hypothesen welche der Geometrie zu Grunde liegen(“On the hypotheses that lie at the foundations of geometry”), was influenced by his reading of the neo-Kantian philosopher Herbart. Gottlob Frege, the founder of analytic philosophy, was a professor of mathematics in Jena who wrote his doctoral dissertation on the representation of ideal elements in projective geometry. Late nineteenth-century mathematical developments, which came to a head in the early twentieth-century crisis of foundations, provoked strong reactions from all the leading figures in mathematics: Dedekind, Kronecker, Cantor, Hilbert, Poincaré, Hadamard, Borel, Lebesgue, Brouwer, Weyl, and von Neumann all weighed in on the sweeping changes that were taking place, drawing on fundamentally philosophical positions to support their views. Bertrand Russell and G. H. Hardy exchanged letters on logic, set theory, and the foundations of mathematics. F. P. Ramsey’s contributions to combinatorics, probability, and economics played a part in his philosophical theories of knowledge, rationality, and the foundations of mathematics. Alan Turing was an active participant in Wittgenstein’s 1939 lectures on the foundations of mathematics and brought his theory of computability to bear on problems in the philosophy of mind and the foundations of mathematics.

Go and read the whole thing, please. And feel free to suggest other examples!

The book reviewed is *Proof and Other Dilemmas: Mathematics and Philosophy*, Bonnie Gold and Roger A. Simons, eds., Mathematical Association of America, 2008

[Photo: Bertrand Russell and G. H. Hardy as portrayed by Jeremy Northam and Jeremy Irons in *The Man Who Knew Infinity*, via MovieStillsDB]

Mathematics is really just the branch of first-order logic that deals with tautological statements.