Musings on Mathematics in Mid-20th-century America

Axiomatics: Mathematical Thought and High Modernism. By Alma Steingart. The University of Chicago Press, Chicago, IL, January 2023. 300 pages, $35.00.

As author Alma Steingart writes in the book’s introduction, Axiomatics: Mathematical Thought and High Modernism is not a history of mathematics. The book contains practically no theorems, proofs, calculations, or conjectures; and despite the title, there are very few axioms. It does include capsule biographies of many mathematicians—which describe how they studied, got hired, taught, wrote textbooks and reports, gave keynote talks on methodology, and raised funds—but there are no accounts of them actually doing mathematics (or any details about their personal lives). Rather, Axiomatics is a history of “mathematical thought”—that is, thoughts about mathematics—primarily by American mathematicians but also by social scientists, physicists, historians, and others.

The text’s six chapters and epilogue comprise seven loosely connected essays, each of which addresses a different aspect of mathematical thought. Though the titular subject of axiomatics is a recurrent theme, it is only central in two or three of the essays; meanwhile, the subject of high modernism merely makes an occasional appearance.

Chapter 1 offers a history of frameworks for algebraic topology between 1930 and 1960 and documents its progression through different levels and forms of abstraction, illustrated largely by the career of Norman Steenrod. Steenrod first studied point-set topology with Raymond Louis Wilder, a student of Robert Lee Moore; his notes and letters from that period are full of carefully drawn geometric diagrams. He then went to Princeton University to work with Solomon Lefschetz, the leading figure in the development of algebraic topology. Lefschetz penned Algebraic Topology, which does not contain a single diagram, in 1942. George David Birkhoff described the text as “a culmination of the abstract phase” of algebraic topology. In 1945, Samuel Eilenberg and Saunders Mac Lane introduced category theory as a unifying framework for a wide range of algebraic and geometric theories. Seven years later, Eilenberg and Steenrod published Foundations of Algebraic Topology, which begins with a purely algebraic axiomatization and only later comments on the geometric interpretation. At each stage of topology’s development, some members of the older generations found themselves somewhat resentful of the new approaches. Steingart identifies “high modernism” in mathematics as this unending process of constant revising, restructuring, and recasting.

Chapters 2 and 3 discuss how researchers came to view mathematical abstraction generally and axiomatization specifically as powerful analytic tools across a wide range of fields — including the increasingly abstract mathematics in 20th-century physics, Kenneth Arrow’s axiomatization of social choice theory, and Oskar Morgenstern and John von Neumann’s development of game theory. Some of these topics are more axiomatic in style than in actual content. For instance, Steingart quotes the following description from a chapter entitled “Choosing an Appropriate Axiomatics” in Yona Friedman’s 1975 book, Toward a Scientific Architecture:

The work of architects and planners produces enclosures — separations in space.

A separation in space (an enclosure) cannot be the work of architects and planners if it does not have at least one access.

In a system of spatial separations, there must be at least one enclosure that differs from the others in some respect, whether as a result of physical qualities or of others.

These are certainly not axioms in the sense that Euclid or David Hilbert used the term.

Chapter 4 describes a vogue among pure mathematicians in the 1940s and 1950s for describing pure mathematics as akin to abstract art; at the time, abstract expressionism in painting—as practiced by artists like Jackson Pollock and heralded by critics like Clement Greenberg—was the last word in the avant-garde. Steingart notes that math’s association with art served a number of purposes. It was an easy (if not particularly accurate) way of explaining pure math research to lay people, it justified pure math research outside of applications, it allowed pure math to separate itself from its association with the military-industrial complex, and it promoted a Cold War vision of American intellectuals as individualists who carved their own paths. Many mathematicians discussed the comparison, although some—like von Neumann—were opposed to it. The discourse reached an apex in a 1951 speech entitled “Mathematics and the Arts” by topologist Marston Morse, which included this impassioned cri de cœur:

Often, as I listen to students as they discuss art and science, I am startled to see that the ‘science’ they speak of and the world of science in which I live are different things. The science that they speak of is the science of cold newsprint, the crater-marked logical core, the page that dares not be wrong, the monstrosity of machines, grotesque deifications of men who have dropped God, the small pieces of temples whose plans have been lost and are not desired, bids for power by the bribe of power secretly held and not understood. It is science without its penumbra or its radiance, science after birth, without intimation of immortality.

The creative scientist lives in ‘the wildness of logic’ where reason is the handmaiden and not the master. I shun all monuments that are coldly legible. I prefer the world where the images turn their faces in every direction, like the masks of Picasso. It is the hour before the break of day when science turns in the womb, and waiting, I am sorry that there is between us no sign and no language except by mirrors of necessity. I am grateful for the poets who suspect the twilight zone.

Chapter 5 of Axiomatics addresses the struggle between pure mathematicians like Marshall Stone and applied mathematicians like Richard Courant in the 1950s and 1960s. At that time, applied math received more government funding while pure math attracted more doctoral students and led to faculty positions. Eugene Wigner’s famous 1959 speech, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” was a pivotal event; both sides of the debate tried to use the lecture to support their points of view.

In chapter 6, Steingart reflects on her own discipline and recounts mathematicians and intellectual historians’ growing interest in the history of mathematics during the 1960s and 1970s. She describes the profound, disturbing effects of Thomas Kuhn’s The Structure of Scientific Revolutions and Quentin Skinner’s essay on “Meaning and Understanding in the History of Ideas.” Kuhn argued that the history of science was marked by large conceptual revolutions, while Skinner argued that the notion of an eternal conceptual framework was a myth. How could one reconcile either of these perspectives with the stability of mathematics over the millennia? Whereas the earlier chapters of Axiomatics are almost purely descriptive, Steingart explicitly shares her own opinions in this section; the phrase “I think” appears more than once. She contends that the issue—which is still not resolved—is not primarily a historical problem, but rather a problem with the conceptualization of mathematics itself. Though the Platonic view of mathematics is intellectually unsustainable and historically naive, there is not yet a coherent alternative.

The book concludes with a short epilogue that describes mathematicians’ retreat from abstraction and re-engagement with the concrete through activities like Thomas Banchoff and Charles Strauss’ work with computer graphics.

To a marked degree, Axiomatics is a history of institutions: universities, committees, professional organizations, and government agencies. Chapter 1 opens with the 1950 decision by Harvard University’s Department of Mathematics to give a faculty appointment to Claude Chevalley, an algebraist with an abstract approach, rather than Arne Beurling, an analyst who used classical techniques for classical problems. Steingart provides long, detailed accounts of reports, white papers, keynote addresses, and efforts to lobby government funding agencies.

The scope of Axiomatics is limited in a number of dimensions. The focus is strictly American; Steingart acknowledges some French members of Bourbaki but ignores almost all British, Russian, or other European mathematicians later than Hilbert. In fact, Harvard, Princeton, and the Institute for Advanced Study are much more present than the rest of U.S. academia combined. G.H. Hardy’s A Mathematician’s Apology, the best-known defense of pure mathematics, is omitted. In terms of mathematical fields, Steingart barely discusses mathematical logic — the area that deals most directly with axiomatics. She mentions analysis, number theory, and probability theory even less, and most of the pure mathematicians that appear are algebraists or topologists. The fraught issue of grade-school math education is ignored; the omission of the “New Math” of the 1960s, a famous overreach by the mathematical establishment, is particularly striking.

Yet within its scope, Axiomatics is a groundbreaking accomplishment and a major contribution to the history of mathematics in the 20th century. The extent and care of its research and the depth of its analysis are truly impressive.