Philosophy of Mathematics and Natural Science; Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics

Philosophy of Mathematics and Natural Science; Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics , Hermann Weyl  Princeton U. Press, Princeton, NJ, 2009 [1949]. $35.00 (311 pp.). ISBN 978-0-691-14120-6

Any reader of Physics Today probably knows Hermann Weyl (1885–1955) as an iconic mathematician who made fundamental contributions to diverse branches of mathematics—particularly analysis, number theory, and group theory. Readers also may well know about his work in mathematical physics: Together with Eugene Wigner, Weyl pioneered the application of group theory to quantum theory. He also initiated the gauge principle, in his unification of electromagnetism and gravity proposed in 1918 immediately after Albert Einstein’s formulation of general relativity.

Weyl’s genius had another aspect: his engagement with and writings on foundational, even philosophical, aspects of mathematics, physics, and the scientific worldview. His book Philosophy of Mathematics and Natural Science is a masterpiece. The original was written in German for the Handbuch der Philosophie and published in 1927. An authorized 1949 English translation by Olaf Helmer from Princeton University Press contained numerous corrections, additions, and six appendices spanning 80 pages. In view of the 20 years separating the German original and the American release, Weyl considered rewriting the book in English himself, but ultimately rejected the idea. In the preface, he writes, “How could I hope to recapture the faith and spirit of that epoch of my life when I first composed it … after due literary preparations dashing off the manuscript in a few weeks?” The translation has long been out of print, so this recent republication, with a very fine introduction by Frank Wilczek, is to be highly valued.

Of course, every book, even one by a genius, is a creature of its time. So Weyl’s discussions of some topics—especially in cosmology, molecular biology, and evolutionary theory, which have exploded since 1949—now seem either dated or too brief. But even in such cases, Weyl’s grasp of the contemporary science is thorough and his awe at the intricacy and unity of nature and his open-mindedness about unsolved scientific problems make inspirational reading. Besides, when we consider foundational topics of mathematics or physics that were center stage in Weyl’s time, his discussions are always both wise and thought-provoking. One example in the foundations of mathematics is Weyl’s discussion of his doctoral adviser David Hilbert’s formalism; to Hilbert’s chagrin, Weyl was drawn to L. E. J. Brouwer’s intuitionism. An example from physics is Weyl’s discussion of quantum theory, which is full of expository detail from spectroscopy to noncommutativity and entanglement—Weyl admits that “the meaning of quantum physics … is not yet clarified as thoroughly as … relativity theory” (p. 264). There are also wise yet very individual discussions of much more specific topics—for example, the nature of the continuum and why space is three-dimensional. Weyl’s Philosophy of Mathematics and Natural Science should be on every mathematician’s or physicist’s bookshelf.

Weyl’s Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics is of more interest to historians or to philosophers like me, but I still heartily recommend it to physicists and mathematicians. A selection of philosophical writings from the period 1921–55, it is beautifully edited with an introduction and scholarly endnotes by Peter Pesic. Mind and Nature includes several treasures: two lecture series, from 1932 and 1934; two letters from 1922—one by Weyl and the other by Einstein—about relativity theory’s bearing on the idea of mechanism; and a touchingly personal essay from 1955 about “the part which philosophical reflection … has played in my life” (p. 204). From that essay, as from several others, it emerges—to the dismay of any philosopher trained in the analytic tradition! — that Weyl was profoundly influenced by Edmund Husserl’s phenomenology and its roots in German idealism, a school of philosophy that tries to deduce substantial conclusions about what we know and how we know it by analyzing experience.

There are other texts about Weyl’s scientific legacy: His Classical Groups (1939), Algebraic Theory of Numbers (1940), and Symmetry (1952; all from Princeton University Press) and his Theory of Groups and Quantum Mechanics (Methuen, 1931; Dover, 1950) remain in print and are inspiring reads. Top-quality scientific and historical commemorations of his work and legacy continue to be published. Two examples are Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work, edited by Erhard Scholz (Birkhauser, 2001), and Groups and Analysis: The Legacy of Hermann Weyl, edited by Katrin Tent (Cambridge University Press and London Mathematical Society, 2008).

Reading Philosophy and Mind and Nature was a poignant experience for me, since so many pages reminded me of how Nazism destroyed German science and philosophy. Taken together, these books give a remarkably detailed and fascinating picture of Weyl’s philosophical outlook in the last 30 years of his life. What a pleasure, what a privilege, to read and contemplate Hermann Weyl’s monumental achievements.