The republication of Weyl’s Philosophy of Mathematics and the Natural Sciences is a welcome event. Frank Wilczek has written a useful short introduction that quickly reminds the reader of some of the scientific and mathematical advances since Weyl wrote the book in 1949, and invites us to speculate on what Weyl would have made of them. It would be wonderful indeed if someone of Weyl’s calibre did exactly that.
Sixty years ago Weyl ranged over mathematical logic, and axiomatic, topics to do with the concepts of number, the continuum, and the infinite, and mostly metrical geometry. These were all subjects on which he had worked personally and on which he knew most, if not all, of the leading figures. Then he turned to science, mostly Einsteinian relativity and to a certain extent quantum mechanics, where again he could draw on decades of experience. All this is held together by his highly developed instinct for philosophy, here less idealistic than he had been when standing on German soil. There are still compelling reasons to believe that every educated scientist and mathematician should read this book, properly and slowly. One might hope the same for every numerate student at university, but the level of mathematical knowledge is quite demanding. They would get a fine, reflective account of what claim mathematics and physics have on our attention.
Wilczek’s introduction naturally reports on contemporary physics and notes the arrival of mathematical biology and the foundational problems being raised in computer science. One might add that Weyl himself could have laid greater emphasis on probabilistic questions and approaches. The category-theoretic framework of mathematics, with its implications for physics, could be added to the list of topics for a 21st century follower of Weyl (and it might be best not to mention the topic of consciousness). Perhaps these topics taken together would significantly change what one would want to say away from the picture presented by Weyl, but two important facts would remain. First, that Weyl took up this vital educational task, and second that he did it so lucidly.
Jeremy Gray is a Professor of the History of Mathematics at the Open University, and an Honorary Professor at the University of Warwick, where he lectures on the history of mathematics. In 2009 he was awarded the Albert Leon Whiteman Memorial Prize by the American Mathematical Society for his work in the history of mathematics. His latest book is Plato’s Ghost: The Modernist Transformation of Mathematics.