Axiomatics and Abstraction in the Mid-20th Century

Axiomatics: Mathematical Thought and High Modernism

The University of Chicago Press, 2023, 291 pp.

By Alma Steingart

What is the source of meaning in mathematics? This sounds very philosophical, but it is a hard-to-escape question for reflective mathematicians—or even for those who are writing a report for a nonmathematical audience. Is mathematics just a matter of abstract structure, of pure axiomatic formalism? Should meaning and content be evacuated, retaining only the formal? We all know that there will be two kinds of answers here: there are some who will be thrilled with extreme generality and abstraction, and those who prefer content and details, the rich particularities of concrete problems.

Steingart gives many examples of both attitudes in her book, but the main topic revolves around the “high modernist” trends that dominated (in the USA) after WWII. Her book is an interesting exploration of the many sides and meanings of two keywords—axiomatics and abstraction—in mid-20th century mathematics and beyond. The ambition is to present mathematics as a central element of high modernism and a decisive influence on the configuration of American intellectual thought. Thus the work ranges widely, including chapters on the impact of axiomatic thinking on the social sciences (“the mathematics of man”), on the relations between math and abstract art (also related to Cold War ideology), on the state of applied mathematics and mathematization after WWII, and on the subsequent rise of an interest in the history of math.

In the modernist spirit of the 1950s, “axiomatics” signified the great unifications of general theory, the search for universality, and category theory was perhaps the prime example of this abstractionist philosophy. The names of Steenrod, Eilenberg, and Mac Lane feature prominently in Chapter 1, the only chapter with some complex mathematical content and where topology is the main area of “pure math” discussed. But in this chapter, one can also find the reaction of Mordell to the abstractionist tendencies, in a scathing review of a textbook by Lang, which according to Mordell defaced the theory of Diophantine geometry; and the famous critical remarks of Siegel, who expressed very openly his preference for the classical style of the 19th century and was disgusted by “the abstract dreams of these people,” the young generation.

Chapter 2 discusses the new meaning of mathematization and how it affected the idea of “applying” mathematics. Here we can follow experts like Stone and Weaver debating what kind of mathematician could be more helpful in the war effort (72ff). Math was now explicitly conceived as theory construction (based on hypothetical assumptions) and thus mathematization was no longer mere quantification and induction of regularities from the observable. It was not a matter of intuition or visualization, but a more abstract (top down) proposal of principles and models. The dominance of theoretical physics in the 20th century (think of general relativity theory) was related to this shift, promoted also by the focus of applied mathematicians on the nonobvious formulation of problems by creating mathematical models. In this chapter, some key names are Birkhoff, Stone, Morse, and von Neumann (vs Hilbert, see 75).

Chapter 3 deals with a shift in the social sciences from their former focus on measurement to theory construction, which was heavily influenced by math and led to an emphasis on social systems and their structures. Axiomatic thinking was deemed key for model building, and efforts were made to provide fruitful social models. People such as Luce and Raiffa, H. Simon, Arrow, and Rapoport are some of those to whom it was obvious that math goes beyond quantification. Truly, the main feature of mathematics is creative theory construction, which leads to clarity about complex phenomena. H. Simon is quoted in the book as saying, “Mathematics has become the dominant language not because it is quantitative—a common delusion—but primarily because it permits clear and rigorous reasoning about phenomena too complex to be handled in words” (100).

Chapter 4 deals with the perceived links between pure math and abstract art, a topic emphasized by many mathematicians. Here, Steingart finds connections to Cold War ideology: by claiming to be “pure” and associating themselves with art, mathematicians were avoiding the “bad company” of the physical sciences, held responsible for ominous developments. There is also the issue of Cold War rationality, algorithmic and formal (see 57), with the “game theory phenomenon” as a prototype. Chapter 5 discusses the way in which there emerged a contradiction, insofar as math was well-funded in the USA mostly because of its utility and applications, but the funding went to “pure math” departments. At the time, statistics, computing, operations research, and modelling were still perceived to be directly linked to math, and they enjoyed rapid development. But applied research remained far from well served.

There were heated debates between practitioners of “applied” math (like Courant, J. Weyl, etc.) and “pure mathematicians” (Albert, Morse, Stone, etc.) concerning the institutional and funding situation of the respective fields. Central to the debate was, again, the role of abstraction and axiomatics in modern math—a defining characteristic for some, but with a “danger of deterioration” for others who criticized those who went “down the road of completely detached, self-motivated abstraction.” In their turn, pure mathematicians appealed to the mysterious nature of math’s amazing applicability (often with a reference to Wigner), insisting on the idea that mathematical training needed to be abstract, and applications would eventually emerge, no matter what. Thus they were able to counter the main tendencies of science policy in their time, dismantling the notion of utility insofar as mathematical research was concerned and keeping their idiosyncratic direction. (Steingart sometimes talks about this in a contradictory (hence productive, she would say) way; see 23.)

Chapter 6 shifts gears by elaborating on how an interest in the history of math emerged during the same period. Chapter 6 is an interesting read, although perhaps a bit disconnected from the other topics and a bit complex in its way of superposing different layers of argument. The author tries to present this historical turn as just another aspect of the high modernism in the postwar, but it could be seen as a reaction against. Disagreeing with the extreme formalism and abstraction that had become common, some mathematicians, interested in meaning and content and continuity (of development), turned to history (names include Weil, van der Waerden, Kline, etc.). Soon after that, professional historians of math were coming to the scene, with goals and criteria of their own, leading to disputes (most famously, Unguru arguing against the idea of “geometric algebra” in ancient Greece). Here, as in other places, Steingart draws on the record of debates held at the time (1974, the American Academy of Arts and Sciences), and this enlivens her narrative. For instance, she includes debates such as: Are there revolutions in the development of mathematics? Can one defend the atemporal nature of mathematical results? And, is there an inevitable tension between historicity and Platonism?

Summarizing, to this reader, the chapters on social sciences, abstract art, the Cold War debates about mathematization and pure vs. applied math, and on the history of math (Chapters 3, 4, 5, 6) are perhaps the most interesting. The introduction to the book is a kind of musical overture, where all the themes that will later appear are sounded in a sort of preview. The style in which it’s written will certainly appeal to some readers, but I confess that was not my case. Too many things were being said about axiomatics, abstraction, modernism, science, and mathematics, in a presentation that seemed confusing or even contradictory to me. This initial impression was later corrected by reading the book chapters—but it took some time to overcome.

Discussing modernism is a difficult topic, and it’s even more difficult to illuminate its changing relations with math. Thus it should be no surprise that I was left with the impression that the book’s rich materials were asking for a more detailed and nuanced study. Let me try to explain by way of examples. Some marks of high modernism (in math) detected by the author are: the search for general, unified theory; abstraction as an ongoing project, constant rewriting of ideas; an ideology of maximum generality as progress; the popularity of group theory in the 1930s; and in some cases, unification through interdisciplinary collaboration, not through a global approach.

However, most of these features can be found in the (lower) modernist mathematics developed before the 1930s, particularly in Europe. To give a couple of examples: constant rewriting in search of general theories is what one finds, de facto, from the 1870s (some key names are Dedekind, Jordan, Dini, Pasch, Weber, Hilbert, Noether); and the impact of group theory can be felt very clearly in the 1890s at the community level (among algebraists), and even in the 1870s in singular cases.

I say this just to suggest that there’s still work to do⁠¹ and definitely not to diminish or undervalue Steingart’s work. She may be very perceptive in choosing the above marks and noticing that they were indeed particularly characteristic of high modernism. Perhaps what is needed is a more detailed explanation of how they were shaded in earlier times, say, around 1900, and what connotations they gained after the 1930s. Perhaps, too, we should study how they interacted with very different social and institutional conditions—in Europe around 1900, in the USA after 1940—and how they affected different subcommunities of mathematicians at different times (for instance, she’s thinking about the impact of group theory among topologists, not algebraists, in the 1930s).

¹

A recent work that includes some chapters discussing modernism is K. Chemla et al., The Richness of the History of Mathematics, Springer Verlag, 2023 (Archimedes series, 66).

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In any event, the text itself makes clear a well-known point: the role of abstract unification (and of axiomatics) varies greatly from one area of mathematics to another. This is exhibited immediately in the examples of Siegel and Beurling in Chapter 1.

The book is well written, and employs many articles, speeches, memoranda, and national reports in which mathematicians were addressing the public at large, especially by those in charge of science policy decisions. This adds greatly to the value of the work. The discussion is kept accessible for a relative large readership, avoiding any kind of technical details. Readers will come away with an enriched picture of the world of mathematics.