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Axiomatics

Mathematical Thought and High Modernism

Axiomatics

Mathematical Thought and High Modernism

The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century.

Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. 

For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. 

Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart’s history ably demonstrates, mathematics is the social activity in which styles of truth—here, abstraction—become synonymous with ways of knowing. 

300 pages | 12 halftones | 6 x 9 | © 2023

History: History of Ideas

History of Science

Mathematics and Statistics

Reviews

“Steingart takes a wide-angle view on mid-twentieth-century mathematics, connecting the axiomatic movement with high abstraction in modern art, structuralism in the social sciences, the New Criticism in literary criticism, and the deep unease felt by many scientists and mathematicians in the wake of World War II as their research became ever more entangled with military applications. Unfailingly lucid and alert to sympathetic resonances between apparently disparate realms, Steingart positions modern mathematics squarely in the center of high modernism.”

Lorraine Daston, author of Rules: A Short History of What We Live By

“The push for axiomatic reasoning, so central to twentieth-century mathematics, extended by 1950 to elite social science. But the power of this abstract logic, never absolute, was in retreat by the 1990s. Although the most familiar of these challenges took form as a new cult of data, Steingart’s most engaging arguments explore a new fascination with mathematical historicism.”

Theodore M. Porter, author of Trust in Numbers: The Pursuit of Objectivity in Science and Public Life

Table of Contents

Note to Readers
Introduction
1. Pure Abstraction: Mathematics as Modernism
2. Applied Abstraction: Axiomatics and the Meaning of Mathematization
3. Human Abstraction: “The Mathematics of Man” and Midcentury Social Sciences
4. Creative Abstraction: Abstract Art, Pure Mathematics, and Cold War Ideology
5. Unreasonable Abstraction: The Meaning of Applicability, or the Miseducation of the Applied Mathematician
6. Historical Abstraction: Kuhn, Skinner, and the Problem of the Weekday Platonist
Epilogue
Acknowledgments
Archival Collections
Notes
Index

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