Axiomatics: Mathematical Thought and High Modernism

300 Pages·2023·3.023 MB·English

by  Alma Steingart

Axiomatics

Axiomatics
Mathematical Thought and
High Modernism

alma steingart

the university of chicago press chicago and london
The University of Chicago Press, Chicago 60637
The University of Chicago Press, Ltd., London
© 2023 by The University of Chicago
All rights reserved. No part of this book may be used or reproduced in any manner
whatsoever without written permission, except in the case of brief quotations in critical
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1427 E. 60th St., Chicago, IL 60637.
Published 2023
Printed in the United States of America
32 31 30 29 28 27 26 25 24 23 1 2 3 4 5
isbn- 13: 978- 0- 226- 82418- 5 (cloth)
isbn- 13: 978- 0- 226- 82420- 8 (paper)
isbn- 13: 978- 0- 226- 82419- 2 (e- book)
doi: https://doi.org/10.7208/chicago/9780226824192.001.0001
Library of Congress Cataloging-in-Publication Data
Names: Steingart, Alma, author.
Title: Axiomatics : mathematical thought and high modernism / Alma Steingart.
Description: Chicago : The University of Chicago Press, 2023. |

Includes bibliographical references and index.
Identifiers: lccn 2022026640 | isbn 9780226824185 (cloth) | isbn 9780226824208 (paperback) |
isbn 9780226824192 (ebook)
Subjects: lcsh: Mathematics—Philosophy—History—20th century. | Mathematics—

United States—History—20th century. | Knowledge, Theory of—United States—
History—20th century. | Axiomatic set theory. | BISAC: MATHEMATICS / History &
Philosophy | PHILOSOPHY / Epistemology

Classification: lcc qa8.4 .S74 2023 | ddc 510.1—dc23/eng20220829
LC record available at https://lccn.loc.gov/2022026640

♾ This paper meets the requirements of ansi/niso z39.48- 1992 (Permanence of Paper).
Contents
Note to Readers vii
Introduction 1
chapter 1 Pure Abstraction: Mathematics as Modernism 25
chapter 2 Applied Abstraction: Axiomatics and the Meaning
of Mathematization 55
chapter 3 Human Abstraction: “The Mathematics of Man” and
Midcentury Social Sciences 84
chapter 4 Creative Abstraction: Abstract Art, Pure Mathematics,
and Cold War Ideology 116
chapter 5 Unreasonable Abstraction: The Meaning of Applicability,
or the Miseducation of the Applied Mathematician 142
chapter 6 Historical Abstraction: Kuhn, Skinner, and the Problem of
the Weekday Platonist 171
Epilogue 199
Acknowledgments 211

Archival Collections 215

Notes 217

Index 271

Note to Readers
This book is a history of mathematical thought rather than of math-
ematics per se. The distinction is an important one, not only because
my aim is to trace the history of axiomatic thinking outside of the sole
confines of mathematics, but also because I have tried to limit to a mini-
mum the mathematical content in the book. The reader does not need to
have any prior knowledge of, or even affinity for, mathematics in order to
follow the arguments and analysis I make in these pages. The only chap-
ter that asks you to follow a mathematical procedure is chapter 1. Here
I felt that outlining some of the basic ideas behind the development of
algebraic topology was necessary to appreciate the sea change that oc-
curred when mathematical texts reversed the relation between structure
and content in the 1950s and 1960s. The reader who is strongly averse to
mathematics can skip the chapter.

Introduction
To see things not abstract as if they were. And vice versa.
There is no abstraction like the present, and the present is entirely concrete. (False.)
Abstraction makes nothing happen.
That’s no abstraction, that’s my husband.
That’s no abstraction, that’s the symbolic order writ large.
— Charles Bernstein, “Disfiguring Abstraction”
On the evening of Thursday, December 19, 1946, the great German math-
ematician Hermann Weyl, an émigré to the United States, pleaded
with eighty of the leading lights of modern mathematics while they were
trying to eat their dinner. The room before him was packed with math-
ematicians who had descended upon Princeton University three days ear-
lier for the first international mathematical gathering since the Second
World War’s end. The tone was meant to be triumphant, a return to intel-
lectual comity after the ravages of war. Now that the fighting was finally
over, mathematicians were eager to celebrate their past achievements
and announce the start of a new age of research. Those in attendance fell
roughly into three groups. The first included old guard American math-
ematicians like Oswald Veblen and Marston Morse, who had worked tire-
lessly in the decades before the war to promote research mathematics

in the United States and place it on the international stage. The second
consisted of European émigrés who poured into the US throughout the

1930s seeking shelter from persecution and the devastation of war (Weyl,