Axiomatics: Mathematical Thought and High Modernism
by Alma Steingart
University of Chicago Press, 2023 Cloth: 978-0-226-82418-5 | eISBN: 978-0-226-82419-2 | Paper: 978-0-226-82420-8 Library of Congress Classification QA8.4.S74 2023 Dewey Decimal Classification 510.1
ABOUT THIS BOOK | AUTHOR BIOGRAPHY | REVIEWS | TOC | REQUEST ACCESSIBLE FILE
ABOUT THIS BOOK 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.
AUTHOR BIOGRAPHY Alma Steingart is assistant professor of history at Columbia University. Her work has appeared in the Los Angeles Review of Books, Grey Room, Representations, and Social Studies of Science.
REVIEWS
“This sophisticated and wide-ranging book examines mid-century American mathematics as a species of high modernism, both in its pure form and in applied mathematics. It looks at how it was supported, why it was advocated, how and why it was compared to contemporary abstract art, how the evolving ideas of abstraction played out in the Cold War, and how this even affected the writing of the history of mathematics. It is a major addition to and critique of the literature that presents modern mathematics as a species of modernism, and it should be read by every historian of modern science and indeed by anyone interested in how abstract ideas have shaped the modern world.”
— Jeremy Gray, author of Plato’s Ghost: The Modernist Transformation of Mathematics
“American mathematics was in the midst of a puzzling contradiction at midcentury: applied mathematics appeared triumphant even as many mathematicians promoted abstraction and rejected the idea that utility was important. Steingart’s brilliant book has finally resolved this puzzle. Far from standing in opposition, mathematics’ utility and idealism, its calculations and foundations, were historically intertwined with the concept of axiomatics. By masterfully weaving together the work of artists and mathematicians, mundane academic conference proceedings and philosophical treatises, Steingart has written an essential guide to the transformation of postwar mathematics.”
— Christopher J. Phillips, author of The New Math: A Political History
“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
“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
“Mathematics has undergone tremendous changes, especially during the twentieth century, when it pushed ever deeper into the realm of abstraction. This upheaval even involved a redefinition of the definition itself, as Steingart explains in Axiomatics. A historian of science, Steingart sees this revolution as central to the modernist movements that dominated the mid-twentieth century in the arts and social sciences, particularly in the United States.”
— Nature
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
REQUEST ACCESSIBLE FILE
If you are a student who cannot use this book in printed form, BiblioVault may be able to supply you
with an electronic file for alternative access.
Please have the accessibility coordinator at your school fill out this form.
Axiomatics: Mathematical Thought and High Modernism
by Alma Steingart
University of Chicago Press, 2023 Cloth: 978-0-226-82418-5 eISBN: 978-0-226-82419-2 Paper: 978-0-226-82420-8
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.
AUTHOR BIOGRAPHY Alma Steingart is assistant professor of history at Columbia University. Her work has appeared in the Los Angeles Review of Books, Grey Room, Representations, and Social Studies of Science.
REVIEWS
“This sophisticated and wide-ranging book examines mid-century American mathematics as a species of high modernism, both in its pure form and in applied mathematics. It looks at how it was supported, why it was advocated, how and why it was compared to contemporary abstract art, how the evolving ideas of abstraction played out in the Cold War, and how this even affected the writing of the history of mathematics. It is a major addition to and critique of the literature that presents modern mathematics as a species of modernism, and it should be read by every historian of modern science and indeed by anyone interested in how abstract ideas have shaped the modern world.”
— Jeremy Gray, author of Plato’s Ghost: The Modernist Transformation of Mathematics
“American mathematics was in the midst of a puzzling contradiction at midcentury: applied mathematics appeared triumphant even as many mathematicians promoted abstraction and rejected the idea that utility was important. Steingart’s brilliant book has finally resolved this puzzle. Far from standing in opposition, mathematics’ utility and idealism, its calculations and foundations, were historically intertwined with the concept of axiomatics. By masterfully weaving together the work of artists and mathematicians, mundane academic conference proceedings and philosophical treatises, Steingart has written an essential guide to the transformation of postwar mathematics.”
— Christopher J. Phillips, author of The New Math: A Political History
“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
“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
“Mathematics has undergone tremendous changes, especially during the twentieth century, when it pushed ever deeper into the realm of abstraction. This upheaval even involved a redefinition of the definition itself, as Steingart explains in Axiomatics. A historian of science, Steingart sees this revolution as central to the modernist movements that dominated the mid-twentieth century in the arts and social sciences, particularly in the United States.”
— Nature
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
REQUEST ACCESSIBLE FILE
If you are a student who cannot use this book in printed form, BiblioVault may be able to supply you
with an electronic file for alternative access.
Please have the accessibility coordinator at your school fill out this form.
It can take 2-3 weeks for requests to be filled.
ABOUT THIS BOOK | AUTHOR BIOGRAPHY | REVIEWS | TOC | REQUEST ACCESSIBLE FILE