Quick math-personality quiz: What is seven-and-one-fourth minus three-fourths, expressed as a mixed number (a whole number plus a proper fraction)?
What matters isn’t what answer you get but how you arrive at it; your thought-process will reveal what kind of thinker you are. So please stop reading now and continue once you’ve found the answer.
Got the answer? Here are two common ways of getting it:
You could convert 7 1/4 into 29/4, subtract 3/4 from that to get 26/4, and reduce that fraction to get 13/2, or 6 1/2.
Or, you could reason that, because increasing each of two numbers by 1/4 doesn’t change the difference between them (or to put it in daily-life terms, the height-difference between two barefoot people doesn’t change if they both put on 1/4-inch shoes), 7 1/4 minus 3/4 equals (7 1/4 + 1/4) minus (3/4 + 1/4), which equals 7 1/2 minus 1, or 6 1/2. Alternatively, you could reason that 7 1/4 minus 3/4 equals (7 1/4 − 1/4) minus (3/4 − 1/4), which is 7 minus 1/2, or 6 1/2; same idea, same answer. Pictorially:
The red lines are all the same length, and the length of each red line equals the difference between the two numbers on the ruler associated with its endpoints.
Did you solve the problem the second way, nudging the two numbers upward or downward? Congratulations: you’re thinking like a German. But if you solved the problem the first way, converting the mixed fractions into improper fractions, then I have bad news: you’re thinking like a Jew.1
That doesn’t mean you’re actually Jewish; it’s possible that some of your math teachers were. You might not have known that they were Jewish at the time; they might have had wholesome Aryan looks and deceptively Christian names. And you may have been too young to realize that they were infecting you with Jewish mathematics.
Of course, I’m messing with you here. My opening paragraphs parrot the ideas of a small but pivotal group of early twentieth-century German mathematicians, one of whom, Oswald Teichmüller, wrote:
“Many academic courses, especially the differential and integral calculus, have at the same time educative value, inducting the pupil not only to a conceptual world but also to a different frame of mind. But since the latter depends very substantially on the racial composition of the individual, it follows that an Aryan student should not be allowed to be trained by a Jewish teacher.”
Teichmüller was writing in the early years of Hitler’s ascendancy, explaining why it was necesssary (hey, nothing personal!) for the University of Göttingen to fire all its Jewish mathematicians, including his own former teacher Emmy Noether, upon whose advances some of his own work was founded.
I wrote about Noether earlier this year – see my essay “Plus and Times Set Free” – and in the course of writing that essay I learned a lot about Jewish mathematicians in Germany in the half-century leading up to the 1930s. The articles listed in the References of that essay and this one were extremely helpful. In the end I omitted most of the material I’d gathered, since my essay was already long-ish. But here I will share some of the most eye-opening things I learned. (If you spot any errors, please assume that they’re mine alone.) As you read, keep in mind that what I say about the experience of Jews in early 20th century Germany has applications to other people at other times in other places.
THE POWER OF TWO CLASHING STEREOTYPES
I won’t dwell on the long history of antisemitism in Europe preceding the mid-1800s. The stigma against Jews has proved to be remarkably versatile, able to mutate as circumstances required. Statistician Francis Galton and his fellow English eugenicists thought of Jews as being unintelligent; nowadays the counter-stereotype of Jews as being shrewd is dominant. This is an essay about stereotypes, but it should be kept in mind that often stigma precedes stereotype, with stereotypes being manufactured to retrospectively justify a stigma.
Mathematical historian David Rowe writes: “A standard stereotype presumed that Jews had a distinctly different way of thinking about mathematics stemming from a Talmudic tradition that favored abstract theorizing, while neglecting fields with close ties to the physical sciences.” At the same time, there was the stereotype I trotted out at the start of this essay: Jewish minds trudge along on the ground with plodding logic and computation while Germanic minds are more conceptual. The latter stereotype was codified by the German psychologist and ardent Nazi Erich Rudolph Jaensch, who divided mathematicians into J-type and S-type. (Perhaps confusingly, Jews were S-type, not J-type.) According to Jaensch, the “integrative” (J-type) Germanic mind is conceptual and geometric whereas the “dissolutive” (S-type) Jewish mind is merely logical.2 If you’re confused about the difference between “conceptual thinking” and “abstract theorizing”, that’s kind of the point. The handy thing about having two opposed stereotypes is that you can use whichever one the occasion demands. (“Your ideas are abstract; mine are conceptual.”)
The fact is that, just as an expert trial lawyer can cross-examine a witness for the defense so as to support the case for the prosecution (or vice versa as needed), the human mind is extremely good at suborning truth itself, twisting evidence so that it seems to support what we already believe. Suppose for instance that a detailed study of classroom pedagogy in early 20th century Germany had revealed that it was in fact mostly Jewish students, not their Christian classmates, who preferred the second, quicker method of computing 7 1/4 minus 3/4. A German race-theorist might initially be confounded by such a finding but then conclude that replacing the question “What is 7 1/4 minus 3/4?” by the question “What is 7 minus 1/2?” (answering a question with a question!) is a trick, and a shrewd one at that: Jewish mathematics. Benjamin Franklin wrote: “So convenient a thing it is to be a reasonable creature, since it enables one to find or make a reason for every thing one has a mind to do.” This applies especially to the activity of sorting people into groups; it’s easy for us to see patterns that aren’t there, especially the patterns we’re expecting to find.
Mathematicians spend a lot of their time hunting for patterns, but much of the day-to-day work of mathematical progress is deduction, and in the Teichmüller quote I gave earlier (“it follows that an Aryan student should not be allowed to be trained by a Jewish teacher”) one can hear a distorted echo of deductive reasoning. The certitude that adheres to mathematical facts can lead a mathematician to drop the habit of doubt that in the real, non-mathematical world can be helpful in pulling us back from the edge of doing terrible things. In 1923, mathematician Theodor Vahlen wrote “Mathematics becomes a mirror of the races and proves the presence of racial qualities in the intellectual realm so to speak with mathematical, therefore incontrovertible, certainty.” Ten years later he joined the SS.
THE WRONG SIDE OF THE EQUATION
Permit me to ruin for you a famous and lovely description of mathematics proposed by the German mathematician Karl Weierstrass, namely, “A mathematician who is not something of a poet will never be a complete mathematician.” What a wonderful way to convey the artistic side of mathematics, so invisible to 99.99% of the world but so important to those of us who court the mathematical muse! Here is a fuller quote, taken from a letter Weierstrass wrote to the Russian mathematician Sonya Kovalevsky which includes an assessment of the Jewish mathematician Leopold Kronecker: “He shares the shortcoming that one finds in many intelligent people, especially those of Semitic stock; he does not possess sufficient fantasy (intuition, I would prefer to say). And it is true, a mathematician who is not something of a poet will never be a complete mathematician. Comparisons are instructive: an all-embracing vision focusing on the loftiest of ideals distinguishes Abel from Jacobi, Riemann from his contemporaries Eisenstein and Rosenhain, and Helmholtz from Kirchhoff (although the latter is without a drop of Semitic blood) in an altogether splendid manner.” (Abel was Christian, Jacobi was Jewish, Riemann was Christian, etc.) A mathematical historian who knows the famous Weierstrass quote without knowing about the antisemitic bits will never be a complete mathematical historian.
But hang on: why is Weierstrass comparing Helmholtz with Kirchhoff to shore up his claim of a correlation between race and mathematical style when he himself admits that neither man is Jewish? Such inconsistencies are a hallmark of the typology game. Weierstrass himself, as a critic of careless intuition and a proponent of rigor, could have been assailed by Jaensch as an S-type, dissolution-loving mathematician. Indeed, the French mathematician Henri Poincaré complained, apropos of the counterintuitive, un-graphable function that Weierstrass had unleashed upon the world (and which now bears his name), “In the old days when people invented a new function they had something useful in mind; now, they invent them deliberately just to invalidate our ancestors’ reasoning.” Some might say that the Weierstrass function was a “hostile” piece of mathematics, to borrow a word that Jaensch’s disciple Ludwig Bieberbach used in his description of the S-type. But Bieberbach found a way to classify Weierstrass as J-type anyway. In the apt words of Paulo Mancosu, “Bieberbach was forced to do a lot of gerrymandering to make sure that important German mathematicians did not end up on the wrong side of the equation.”
German views about a distinct Jewish style of mathematics ranged from Felix Klein’s “It would seem as if a strong naive space-intuition were an attribute pre-eminently of the Teutonic race, while the critical, purely logical sense is more fully developed in the Latin and Hebrew races” to Alexander von Brill’s “The effect of the Jews on Germanic peoples is like alcohol on the individual; in small doses they are stimulating and invigorating, but in large quantities devastating like poison” and onward to Ludwig Bieberbach’s “There is a German and a Jewish mathematics, two worlds, separated by an unbridgeable chasm.” Note that Klein lumped together Jews and “Latin” peoples; other writings of the period identified more specific national styles. Some of the writings of the period eschew a simplistic “Our math good; your math bad” vibe and instead advocate a kind of academic apartheid. Let a hundred styles of mathematics bloom around the world, but within the borders of the Fatherland let the flowering of German mathematics be uncorrupted by alien influences! (Curiously, the precise location in which Jewish mathematics could have its own flowering was never specified.)
I’m focussing my essay on (some) Germans’ assessments of Jewish mathematics, but to understand why those Germans felt so comfortable classifying mathematicians into national and racial types, it’s helpful to know that this kind of classification was a well-established European intellectual pastime. Here’s an example from the French historian of science Pierre Durhem who, writing in between the two World Wars, contrasted French and German styles of science: “To start from clear principles … then to make progress step by step, patiently, painstakingly, at a pace that the rules of deductive logic discipline with extreme severity: this is what German genius excels at … . The Germans lack the spirit of subtlety.” This game of comparing Our Culture to Their Culture was all the rage in the fanciest European salons back before National Socialism gave nationalism a bad name.
If it were possible to put an entire culture on the psychoanalytic couch, I might say to 1930s-era German academia, “Ah, so it hurts when other people call you overly logical? Perhaps you wish to project this description onto other people?”
ONE GERMAN-JEWISH FAMILY
The Noether family only slightly fit into the image of Jewish science promulgated by the Nazis. The father, Max, did work in both pure mathematics and physics. Max’s first child, Emmy, became the famous pure mathematician I wrote about in “Plus and Times Set Free”. Max’s second child, Alfred, obtained a PhD in chemistry (though he died in 1918 before his career got underway). Max’s third child, Fritz, pursued applied mathematics.
For a time Emmy’s main professional obstacle was her sex, but some German mathematicians regarded her love of abstraction as “Hebraic”. Meanwhile, German antisemitism was at a high simmer, ready to become a low boil if the temperature rose just a little bit higher. This was especially true in Göttingen, a university town whose undergraduates were predominantly right-wing and whose humanities professors sympathized with the undergraduates.
In April of 1933, the Reich Ministry of the Interior issued a Law for the Restoration of the Professional Civil Service, excluding racial and political “enemies” of the Nazi regime from all forms of government service. The edict did not automatically expel Jewish professors from their jobs, but it gave universities the right to expel them. Noether’s students were worried that she would be fired, and they wrote a letter on her behalf, arguing that “It is no coincidence that all her students are Aryans; this is due to her essential conception of mathematics, which corresponds entirely to an Aryan way of thinking.” Likewise, her collaborator and friend Helmut Hasse wrote that “In no sense can one call her mathematics ‘alien’. On the contrary, it has a quality much like the typical German mindset, which in its nature favors the intellectual, the theoretical, and the ideal rather than such qualities as purpose, material success, or the real. That this is so can be seen from the fact that the vast majority of German mathematicians who have found their way to her school over the past two decades are of Aryan descent.” Suddenly Noether’s mathematics was not Hebraic after all.
It didn’t matter; Noether lost her position that spring, as did most of the other Jewish mathematicians at Göttingen. Her esteemed colleague Edmund Landau held on for a bit longer, but in the fall Teichmüller led a student boycott of Landau’s lectures, and by February of 1934 Landau too was gone. Later that year the Nazi Minister of Culture asked David Hilbert, Göttingen’s most eminent mathematician and Noether’s champion, whether it was true that the Mathematics Institute had suffered because of the expulsion of its Jewish faculty. Hilbert replied: “Suffered? It hasn’t suffered, Herr Minister. It just doesn’t exist anymore.”
Hilbert’s dark quip was not literally true; people still showed up for work at the Institute. Its new director was Hasse, an old-school German nationalist who disliked the antisemitism of Teichmüller and his crowd but who felt that Hitler was the only person who could restore Germany to its former glory. Hasse kept the lights on at the institute in a physical sense, but the torch of mathematical learning that had once been Göttingen had been snuffed out, never to be rekindled.
During that year, a young American mathematician named Saunders Mac Lane, doing doctoral work at Göttingen, got to view the death of the Mathematics Institute from a front-row seat, and his reminiscence Mathematics at Göttingen under the Nazis is well worth reading in its entirety. Two passages stood out for me. One was his concluding assessment: “Now in retrospect, the whole development is a decisive demonstration of the damage done to academic and mathematical life by any subordination to populism, political pressure and proposed political principles.” The other was a passage from a letter he wrote to his mother in 1933. “I have recently become impressed with the great variety of opinions within the Nazi movement. All Nazis do not think alike, even though it may externally seem as if they did!” If I had been in young Saunders’ shoes, I would have taken comfort in that diversity of opinion, expecting that a party made of such disparate bedfellows would dissipate its energy in internal squabbling and not accomplish much.
But in this I would have been tragically mistaken. Despite the many contradictions of Nazi ideology, the party’s hold on power only solidified with the passage of time. Resentment of Germany’s humiliation twenty years earlier and a thirst for vengeance proved sufficiently cohesive to bind the movement together, especially as the regime learned which groups served as the best targets of the people’s rage. Bieberbach thought that the chasm between Jewish mathematics and German mathematics could not be bridged, but meanwhile, the chasm between old-fashioned nationalism, rabid antisemitism, pagan mysticism, and cynical opportunism proved to be eminently bridgeable.
Emmy Noether was one of the lucky ones; she moved to the U.S. where she had a happy year before complications following what should have been routine surgery unexpectedly cut her life short. Her brother Fritz was not so lucky. Although his service in the First World War qualified him for an exemption from the 1933 law, some students at the University of Breslau where Fritz taught told the University’s Rektor that having Noether on the faculty “in large measure contradicts the Aryan principle.” Later that year the students turned up the heat, accusing Fritz of holding left-wing political views. Sensing that he had no future in Germany, Fritz moved to Russia. He was imprisoned in 1938 and executed in 1941 on a trumped-up charge of anti-Soviet agitation (reversed as baseless in 1988).
Ironically, after the war, the style of abstract algebra Noether had pioneered wasn’t called Jewish; it was called German. Still later, after Bourbaki got ahold of it, the approach was deemed French. Building on the work of Noether and Bourbaki, Saunders Mac Lane built a new super-abstract approach to mathematics called category theory. As far as I know, nobody has called it Scottish mathematics. (Yet.) Then again, Mac Lane’s collaborator in this enterprise was Samuel Eilenberg, so, someone’s bound to call category theory Jewish mathematics, eventually.
HOW SHALL WE DISCUSS DIFFERENCES?
Taste plays as big a role in mathematics as it does in art, determining a mathematician’s style and choice of projects, and young mathematicians look to their elders as they develop their own personal mathematical taste. This tendency, writ large, has given rise to schools of mathematical thought, and Jewish mathematicians have been just as subject to the dictates of fashion as anyone else. The quest for abstraction has been one such fashion. Some early twentieth century mathematicians derided the quest as a fad; if so, it has proved to be a durable one. Perhaps there has been a tendency for Jewish mathematicians to be in the vanguard of this fashion, as they have been in the vanguard in various political and artistic movements, perhaps for reasons having to do with Jewish culture. One might try to follow this train of thought, but if it travels from the terrain of cultural studies into the land of biological determinism, I would advise jumping off the train as quickly as possible. That particular track leads to a wide depot with a squat tower in the middle.
Let’s circle back to my opening example about the “German” and “Jewish” ways of computing 7 1/4 minus 3/4. The example can be traced back to a student named Steckel (apparently more interested in math education than math research) who had done some teaching in Eastern Europe and who described his experiences in Felix Klein’s seminar “Psychological Foundations of Mathematics” sometime around 1910. Mathematical historian David Rowe says that Steckel’s presentation “was, no doubt, meant to illustrate the usual stereotype that Jews excelled in logical thinking, whereas Germans thought intuitively,” and that seems likely. But what if (hypothetically) Steckel had seen his observation not as a cause for triumphant celebration of German superiority but as an argument for racially differentiated instruction? What if Steckel had dreamed of an idyllic future in which each student would be schooled in accordance with his or her own ancestors’ unique and valued way-of-knowing? What if Steckel had no intention of any teachers losing their jobs or anyone losing their lives?
My answer is, it would have made no difference at all. Essentialist ideas like Steckel’s have a life of their own, and can travel into contexts far from the one in which they originated, taking on meanings not intended by their originators. (Indeed, some of the Nazi Party’s official race-ideology was borrowed from American race-theorists of earlier decades.) Steckel’s stereotype was just one rivulet in a great stream. No single rivulet makes an appreciable contribution to the stream, but there would be no stream if there were no rivulets.
Social scientists debate when and how stereotypes lead to persecution, and I’m no social scientist, but it seems to me that when we say that certain other people Aren’t Like Us, it may sometimes help us see them for who they are, but it can also make us see them as Less-Than people, or even see them as less than people. And dehumanization is often the final step that authorizes people to move beyond hateful words to hateful actions.
So, my proposed (not very helpful!) answer to the question “How shall we discuss differences?” is: very, very carefully. No matter how noble your intentions may be, when you broadcast some generalization about a group of people, it ceases to be yours and becomes the world’s, and you can’t know what your words might mean someday.
Thanks to Sandi Gubin.
ENDNOTES
#1. There are other approaches: you could for instance have rewritten 7 1/4 as 6 5/4 and then subtracted 3/4 to get 6 2/4, or 6 1/2. Maybe that makes you Slavic?
#2. In a future essay I’ll argue that the most integrative German mathematician of the late 19th century was actually the incontrovertibly Jewish Hermann Minkowski, who not only gave us a geometry of numbers and an arithmetic of shapes but also laid the foundation for the unification of space and time! But you’ll have to wait to read that story.
REFERENCES
Antonio Durán, Pi and the Nazis, Blog del Instituto de Matemáticas de la Universidad de Sevilla.
Abraham Fraenkel, Hitler’s Math. Tablet (February 8, 2017). From Fraenkel, Recollections of a Jewish Mathematician in Germany, Springer 2016.
Saunders Mac Lane, Mathematics at Göttingen under the Nazis. Notices of the AMS 42 No. 10 (1995), 1134–1138.
Paulo Mancosu, Mathematical Style. From the Stanford Encyclopedia of Philosophy, 2009 (revised 2021).
David E. Rowe, “Jewish Mathematics” at Göttingen in the Era of Felix Klein. Isis 77 No. 3 (1986), 422–449.
Sanford L. Segal, Mathematics and German Politics: The National Socialist Experience. Historia Mathematica 13 (1986), 118–135.