Mathematicians under the Nazis

Mathematicians under the Nazis

by Sanford L. Segal


Want it by Monday, November 26 Order now and choose Expedited Shipping during checkout.

Product Details

ISBN-13: 9780691164632
Publisher: Princeton University Press
Publication date: 11/23/2014
Pages: 558
Product dimensions: 6.10(w) x 9.30(h) x 1.60(d)

About the Author

Sanford L. Segal is Professor of Mathematics at the University of Rochester and the author of Nine Introductions in Complex Analysis.

Read an Excerpt

Mathematicians under the Nazis

By Sanford L. Segal


Copyright © 2003 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-6538-3


Why Mathematics?

Mathematics under the Nazi regime in Germany? This seems at first glance a matter of no real interest. What could the abstract language of science have to say to the ideology that oppressed Germany and pillaged Europe for twelve long years? At most, perhaps, unseemly (or seemly) anecdotes about who behaved badly (or well) might be offered. While such biographical material, when properly evaluated to sift out gossip and rumor, is of interest—history is made by human beings, and their actions affect others and signify attitudes—there is much more to mathematics and how it was affected under Nazi rule. Indeed, there are several areas of interaction between promulgated Nazi attitudes and the life and work of mathematicians. Thus this book is an attempt at a particular investigation of the relationships between so-called pure (natural) science and the extra-scientific culture. That there should be strong cultural connections between the technological applications of pure science (including herein the social applications of biological theory) and various aspects of the Industrial Revolution is obvious. Social Darwinism, and similar influences of science on social thought and action, have been frequently studied. It is not at all clear at the outset, however, that theoretical science and the contemporary cultural ambience have much to do with one another. Belief in this nonconnection is strengthened by the image of science proceeding in vacuo, so to speak, according to its own stringent rules of logic: the scientific method. In the past thirty years, however, this naive assumption of the autonomy of scientific development has begun to be critically examined.

A general investigation of this topic is impossible, even if the conclusion were indeed the total divorce of theoretical science from other aspects of culture. Hence the proposal to study one particular microcosm: the relationship between mathematics and the intensity of the Nazi Weltanschauung (or "worldview") in Germany. Although 1939 is a convenient dividing line in the history of Hitler's Reich, nonetheless the prewar Nazi period must also be viewed as a culmination; the Germany of those years was prepared during the Weimar Republic, and both the cultural and scientific problems that will concern us have their origins at the turn of the century. World War I symbolized the conclusion of an era whose end had already come. Similarly, World War II was a continuation of what had gone before, and a terminal date of 1939 is even more artificial and will not be adhered to.

The concentration on mathematics may perhaps need some justification. At first glance, a straw man has been set up—after all, what could be more culture-free than mathematics, with its strict logic, its axiomatic procedures, and its guarantee that a true theorem is forever true. Disputes might arise about the validity of a theorem in certain situations: whether all the hypotheses had been explicitly stated; whether in fact the logical chain purporting to lead to a certain conclusion did in fact do so; and similar technical matters; but the notion of mathematical truth is often taken as synonymous with eternal truth. Nor is this only a contemporary notion, as the well-known apocryphal incident involving Euler and Diderot at the court of Catherine the Great, or the Platonic attitude toward mathematics, indicate. Furthermore, there is the "unreasonable effectiveness" of mathematics in its application to the physical and social scientific world. Even so-called applied mathematics, concerning which Carl Runge remarked that it was merely pure mathematics applied to astronomy, physics, chemistry, biology, and the like, proceeds by abstracting what is hypothesized as essential in a problem, solving a corresponding mathematical problem, and reinterpreting the mathematical results in an "applied" fashion.

Mathematics also has a notion of strict causality: if A, then B. It is true that the standards of rigor, the logical criteria used to determine whether or not a proof is valid, that is, to determine whether or not B truly follows from A, have changed over time; nevertheless, the notion that it is conceivable that B can be shown always to follow from A is central to mathematics. As the prominent American mathematician E. H. Moore remarked, "Sufficient unto the day is the rigor thereof." Both the necessary process of abstraction and the idea of mathematical causality separate mathematics from more mundane areas. Somewhat paradoxically, perhaps, they are also partly responsible for the great power of mathematics in application. Mathematical abstraction and mathematical causality seem to elevate mathematics above the sphere of the larger culture.

Twin popular illusions incorrectly elaborate upon this view and make mathematics seem even more remote from the general culture. The first of these is that the doing of mathematics is only a matter of calculation, or, more sophisticatedly, of logical step-by-step progress from one eternal truth to another via intermediate truths. This view is enhanced by the way mathematicians publicly present the results of their investigations: exactly as such logical progressions. In fact, however, the discovery of mathematics, as opposed to the presentation of it, is more like the reconnoitering of some unknown land. Various probes in various directions each contribute to the forming of a network of logical connections, often even unconsciously. The realization of this network, the a posteriori checking for logical flaws, and the orderly presentation of the results, do not reflect the process of mathematical creativity, whatever that may be, and however ill it is understood.

The second illusion is that all that counts for a mathematician is to distinguish the correct from the incorrect. Correctness is indeed the sine qua non of mathematics, but aesthetic considerations are of great importance.

Among the various aesthetic factors influencing mathematical activity are economy of presentation, and the logic (inevitability) of often unexpected conclusions. While correctness is indeed the mathematical essential, some correct proofs are preferable to others. Proofs should be as clear and transparent as possible (to those cognizant of the prerequisite knowledge). A good notation, a good arrangement of the steps in a proof, are essential, not only to aid the desired clarity, but also because, by indicating fundamentals in the problem area, they actually incline toward new results. Clarity, arrangement, and logical progression of thought leading to an unexpected conclusion are well illustrated in an incident concerning no less a personage than the philosopher Thomas Hobbes:

He was 40 yeares old before he looked on Geometry; which happened accidentally. Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the 47 El. libri I. He read the Proposition. By G——, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that truth. This made him in love with Geometry.

A simple example of an "unbeautiful truth" is a list of positive integers. Mathematics is not frozen in time like a Grecian urn; solutions of old problems lead to new considerations. Though truth may not necessarily be beauty, beauty is truth, and for the mathematician impels to its own communications. As Helmut Hasse (who will be met again) remarks:

Sometimes it happens in physics again and again, that after the discovery of a new phenomenon, a theory fitted out with all the criteria of beauty must be replaced by a quite ugly one. Luckily, in most cases, the course of further development indeed reveals that this ugly theory was only provisional....

In mathematics this idea leads in many instances to the truth. One has an unsolved problem, and, at first, has no insight at all how the solution should go, even less, how one might find it. Then the thought comes to describe for oneself what the sought-for truth must look like were it beautiful. And see, first examples show that it really seems to look that way, and then one is successful in confirming the correctness of what was envisaged by a general proof.... In general we find a [mathematical] formulation all the more beautiful, the clearer, more lucid, and more precise it is.

As Hasse puts it elsewhere, truth is necessary, but not sufficient for real (echt) mathematics—what is also needed is beautiful form and organic harmony.

One result of this aesthetic is that the mathematician thinks of himself as an artist, as G. H. Hardy did:

The case for my life, then, or that of anyone else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.

Or, as Hasse says even more forcefully, "The true mathematician who has found something beautiful, senses in it the irresistible pressure to communicate his discovery to others."

Mathematics is the "basic science" sine qua non. At the same time, it is quite different from basic experimental science by being divorced from laboratory procedures. Even so-called applied mathematics only takes place on paper with pencil. The hallmark of mathematics is logical rigor. However important or suggestive or helpful heuristic or analogical arguments may be, it is only the mathematical proof according to accepted standards of logical rigor that establishes a mathematical result. Those logical standards may be and are disputed (and were in Nazi Germany), but given an accepted set of such standards, mathematical proofs according to them establish mathematical results that are true without qualification. On the one hand, a mathematical result is "sure"; on the other, however, all but the final results with proofs are, at best, incomplete mathematics: the mathematician's "experiments" are usually eminently unpublishable as such. This removal of mathematics from the concrete world contributes to the mathematical aesthetic. While there are notions of a "beautiful" experiment in the experimental sciences, in mathematics the aesthetic is purer for its removal from the natural irregularities of concrete life. "As for music, it is audible mathematics," writes the biologist Bentley Glass, and perhaps the traditional musical aesthetic is the one most closely resembling the mathematical; here, too, given the underlying assumptions, there is a purity of form that is part of the notion of beautiful. Deviations like Mozart's Musikalischer Spass or some of the less slapstick efforts of P.D.Q. Bach (Peter Schickele) are jokes because of their introduction of irregularities into a presumed form. Similarly, Littlewood presents as humorous an unnecessarily cumbrous presentation of a proof that can be expressed quite clearly and elegantly. The papers of Hasse and Archibald cited earlier also stress the analogy between the musical and the mathematical aesthetic.

In some sense, then, mathematics is an ideal subject matter; it is, however, made real by the actions of mathematicians. In Russell's well-known words:

Mathematics possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Nevertheless, mathematicians make tremendous emotional investments in the doing of mathematics. Mathematicians, despite their pure aesthetic, the divorce of their actual work from concrete reality, and the surety of their results, are not like petty gods in ivory towers playing at abstruse and difficult, but meaningless, games. The final piece of mathematics is abstract, aesthetically beautiful, and certain; but it is not (nor could it be) an instantaneous or automatic creation. The doing of mathematics is as emotionally involved, often clumsy, and uncertain as any other work that has not been reduced to a purely automatic procedure.

Thus, the nature of mathematical abstraction and mathematical causality, coupled with the popular ignorance of the nature of mathematical research and the removal of mathematics from the everyday world, seem to make mathematics one of the least likely subjects for the sort of investigation proposed. Yet some Nazi mathematicians and psychologists stood this reasoning on its head. At the same time, they emphasized with a peculiarly Nazi bias the often neglected roles of aesthetics and inspiration in creating mathematics. They argued that exactly the apparent culture-free nature of mathematical abstraction and mathematical causality makes mathematics the ideal testing-ground for theories about racially determined differences in intellectual attitudes. As E. R. Jaensch wrote in 1939:

Mathematics can simply have no other origin than rational thinking and mental activity (Verstandestätigkeit). "Irrational" mathematics would be a wooden iron, a self-contradiction. If, therefore, one discovers something worth exposure about the ways of thought (Verstandeskräfte) that still command the field on this area—and that happens in many respects with complete justice—so one can hereby only obtain help by bringing other forms of rational thought in more strongly—in no case however, through the conjuring up of irrationality. This way is simply excluded in mathematical thought. Even if in other scientific and educational disciplines it is possible artificially still to maintain the appearance that Reason (Verstand), as treated through that radical cure, still lives—in Mathematics it is impossible.

Hereby, the question of mathematical thought attains the character of an especially instructive example—an "illuminating case" in Baco's [sic] sense—for the forms of logical thought and rationalism above all, but also in other areas of knowledge and in everyday life.

What is important to note here is the insistence that the supposed autonomy of mathematics from irrational influence makes it exactly appropriate for investigating various intellectual types. Just because of the rationality of its results, mathematics was deemed an excellent medium for perceiving the various important differences between different peoples' ways of thought. It did not prove difficult to discover, for example, a Nordic type, a Romance or Latin type, a Jewish type, and, in fact, several subvarieties of these. Jaensch's theory of types could be elaborated independent of or in conjunction with Rassenseelenkunde, or the theory of the "racial soul." This was done most prominently by the distinguished mathematician Ludwig Bieberbach, who will be discussed particularly in chapters 6 and 7. By delineating a "Nordic" mathematics distinct from French or Jewish mathematics, great emphasis could be placed (necessarily) on the mode of intellectual discovery as opposed to its fruits, and, therefore, on feeling and attitude toward the world.

However important this inversion of the usual attitude toward mathematics may be for investigation, there are at least two other reasons arguing for a study of mathematics in the Nazi period. The first is that among the substantial number of mathematicians who were sympathetic in varying degrees to the Nazi cause were several who attempted to associate the political argument with various philosophical differences within mathematics. This did not alter the truth of any mathematical fact, but it did declare that certain mathematical disciplines were "more equal" than other varieties. Nor was this simply a question of "pure" versus "applied," of theory versus immediately usable results. Both these beliefs and the ones about the salience of psycho-racial differences within mathematics also argued for the distinction in differences of pedagogical style. Put succinctly, a Nazi argument promoted by Bieberbach was that because Jews thought differently, and were "suited" to do mathematics in a different fashion, they could not be proper instructors of non-Jews. Indeed, their presence in the classroom caused a perversion of instruction. Thus an elaborate intellectual rationale for the dismissal of Jews was established, discussed, and defended.


Excerpted from Mathematicians under the Nazis by Sanford L. Segal. Copyright © 2003 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

  • Frontmatter, pg. i
  • Contents, pg. ix
  • PREFACE, pg. xi
  • ABBREVIATIONS, pg. xxi
  • CHAPTER ONE. Why Mathematics?, pg. 1
  • CHAPTER TWO. The Crisis in Mathematics, pg. 14
  • CHAPTER THREE. The German Academic Crisis, pg. 42
  • CHAPTER FOUR. Three Mathematical Case Studies, pg. 85
  • CHAPTER FIVE. Academic Mathematical Life, pg. 168
  • CHAPTER SIX. Mathematical Institutions, pg. 229
  • CHAPTER SEVEN. Ludwig Bieberbach and “Deutsche Mathematik”, pg. 334
  • CHAPTER EIGHT. Germans and Jews, pg. 419
  • APPENDIX, pg. 493
  • BIBLIOGRAPHY, pg. 509
  • INDEX, pg. 523

What People are Saying About This

Walter Noll

Segal must be commended for the enormous amount of research he has done in arriving at an accurate picture of the complexity of the situation faced by mathematicians during the Nazi regime. Avoiding stereotypes and oversimplification, he presents fascinating information and valuable insights to those interested in mathematicians and to people interested in the history of Nazi Germany.
Walter Noll, Carnegie Mellon University


"Segal must be commended for the enormous amount of research he has done in arriving at an accurate picture of the complexity of the situation faced by mathematicians during the Nazi regime. Avoiding stereotypes and oversimplification, he presents fascinating information and valuable insights to those interested in mathematicians and to people interested in the history of Nazi Germany."—Walter Noll, Carnegie Mellon University

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews