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Levels of Infinity: Selected Writings on Mathematics and Philosophy

Hermann Weyl
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Jeremy Gray
, on

Another bumper crop of essays by Hermann Weyl, some freshly translated, that between them discuss the life and work of two of the great mathematicians of the 20th century (David Hilbert and Emmy Noether), what it is to think mathematically (whether in algebra, topology, or logic), axiomatic and constructive work in mathematics, and the impact of relativity theory on mathematics. The first of these essays, on the nature of the mathematical infinite, gives the book its title and dates from 1930; the last, on why the world is four-dimensional, from 1955, the year in which Weyl died.

The essay ‘Levels of infinity’ is translated into English for the first time, and does not appear in Weyl’s Gesammelte Abhandlungen. As those familiar with Weyl’s career might guess, it is the most overtly philosophical. It compares action and reflection as modes of knowledge, and the potential (intellectually accessible) infinite with an actual (mentally inaccessible) infinite that we are driven to represent “as a closed Being by means of a symbolic construction”. This is not immediately clear; it is not how we write today, and it shows how much Weyl was immersed in the philosophical traditions of European thought, but one of the many merits of this book is that the selection of essays brings us some of Weyl’s later formulations. Thus the essay ‘Axiomatic versus constructive procedures in mathematics’ written 23 years later (and also translated and published here for the first time) also contrasts the constructive, open-ended approach to mathematics with reflective, axiomatic formulations. The constructive approach makes numbers out of our ability to make marks on paper, geometry from our ability to make marks on objects and so, in a sense, on space. It is axiomatic thinking that systematises and inter-relates our various constructive activities. To a considerable extent, these contrasts underlie Weyl’s comparison of topology and algebra, the one flexible and intuitive, the other more rigid and precise.

Weyl famously found it difficult to play the role of Hilbert’s successor. There were many differences between them, and the more algebraic and axiomatic character of Hilbert’s thought was only one of them. Leadership, the competitive Göttingen atmosphere, and of course the increasingly poisonous Nazi presence among the students by 1930, when Weyl finally did accept a position at Göttingen, were others. But it is particularly valuable in this book to have reprinted Weyl’s obituaries of Hilbert and Emmy Noether, and one can see in what ways he lived his life in the context of theirs, and created his mathematics while they created theirs — algebraically and axiomatically in their cases. But construction and axiomatisation are not a pair of mutually exclusive alternatives, and Weyl understood group theory profoundly and used it in his account of quantum mechanics and his major work on the structure of Lie groups. Rather, it is a fruitful tension that Weyl presented, and one that can be profitably transcribed to other dichotomies today, such as pure and applied mathematics.

The essays presented here carry some useful, discrete footnotes by Peter Pesic and a fine essay that rightly introduces Weyl not only as a remarkably versatile and creative mathematician but also as a fine writer. It is evident from the range of topics on offer how versatile Weyl was, and we do not even see here his work on either mathematical analysis or groups and quantum mechanics. The quality of his thought makes these essays as vital today as ever, and the book an excellent introduction to this most humane of mathematicians.

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 most recent book is Henri Poincaré: A Scientific Biography.


Levels of Infinity (1931)
Topology and Abstract Algebra as Two Roads of Mathematical Comprehension (1932)
Emmy Noether (1935)
The Mathematical Way of Thinking (1940)
David Hilbert (1862-1943) (1944)
David Hilbert and His Mathematical Work (1944)
Mathematics and Logic (1946)
Relativity Theory as a Stimulus in Mathematical Research (1949)
A Half-Century of Mathematics (1953)
Axiomatic versus Constructive Procedures in Mathematics (1953)
Why is the World Four-Dimensional? (1955)