Notes on Purity Math 111, History of Math, Duchin 1. Pure versus applied Pure math, as distinct from applications, came only relatively recently to Europe. The Oxford English Dictionary dates the very first use of the word in this way to 1648, with the statement, Mathematicks..is usually divided into pure and mixed; and though the pure doe handle only abstract quantity..that which is mixed doth consider the quantity of some particular determinate subject. It was arguably only in the nineteenth century, though, that pure mathematics became its own subject of study. Sociologist Sal Restivo locates the emergence of pure math in Germany, roughly coincident with the professionalization and institutionalization of math. Sal Restivo, from Mathematics in Society and History: References to pure or mixed mathematics were not uncommon in the seventeenth century; and by the eighteenth century France had established university chairs in pure and applied mathematics. Germany, however, was the center of activities that led to the crystallization of mathematics as a profession and of pure mathematics as a distinct form of mathematical work. [...] [By the first half of the nineteenth century,] Crelle [the founder of the first modern journal of pure mathematics] was arguing in opposition to the practical orientation at the Ecole Polytechnique in Paris that the essence of mathematical work was to foster spiritual enlightenment and mental power. In the 1850s and 1860s, the split between pure and applied continued to widen. This was reflected in the development of mathematics at the universities in Göttingen and Berlin. At Berlin, Kummer and Weierstrass promoted a formalist approach to rigorous mathematics. At Göttingen, by contrast, pure mathematics remained tied to the intuitive perspective and applications. [...] The University of Göttingen was founded by King George II in 1737. Gauss was called there in 1807. Gauss felt that the more deeply he understood mathematics, the more clearly he recognized the significance of its applications in everyday life and in the natural sciences. [...] Moreover, mathematics brought him closer to the spiritual world. [...] For Gauss, mathematics helped to reveal the immortal nucleus of the soul. It was a source of recreation and consolation, and in his later years a source of confidence. This illustrates some of the functions of pure ideas for individuals. And it suggests one reason why, perhaps, pure mathematicians seek timeless truths; timebound truths would undermine the function of pure mathematics in some cases as a source of assurance about an eternal after-life. By the 1870s and 1880s, the Göttingen-Berlin split was so great that a young mathematician could for the first time seriously consider a career in pure mathematics.
2 As pure mathematics emerged as its own profession, attitudes about the relationship to applications varied widely, from finding motivation there, to finding it irrelevant, to finding it impure or unworthy. Here are a few examples, cited from Segal s book Mathematicians under the Nazis. The celebrated mathematician Ernst Kummer (1810-93), from the University of Berlin, called applied mathematics dirty mathematics. Edmund Landau of Göttingen associated applications with grease (Schmierol). Landau is discussed in more detail below. Felix Klein taught at Erlangen, Munich, Leipzig, and Göttingen. Circa 1921, he wrote the following in a letter on the occation of founding a new journal of applied mathematics. The goal of theoretical natural science should be not only a passive understanding, but also an active domination of Nature. This is notable not just for its appeal to domination, but also for its explicit call for applied mathematics as a force pulling pure mathematics. The Oxford/Cambridge number theorist G.H. Hardy wrote at length about his rather dismissive attitude towards applications of mathematics. G.H. Hardy, from A Mathematician s Apology: [N]either physicists nor philosophers have ever given any convincing account of what physical reality is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls real. Thus we cannot be said to know what the subject-matter of physics is; but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics. A mathematician, on the other hand, is working with his own mathematical reality. Of this reality, as I explained [earlier], I take a realistic and not an idealistic view. At any rate (and this was my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more than what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but 2 or 317 has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way. [...]
3 I hope that I need not say that I am trying to decry mathematical physics, a splendid subject with tremendous problems where the finest imaginations have run riot. But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. Imaginary universes are so much more beautiful than this stupidly constructed real one; and most of the finest products of an applied mathematician s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts. 2. Ideological and racial overtones By coincidence or not, German academia was both the center of the emergence of the pure mathematician (along with the host of often-negative attitudes towards the impure world around math) and, not very much later, the center of increasingly bizarre and elaborate ideologies of purity connecting the (racialized) nature of minds to the nature of the mathematics they produced. Here is Felix Klein again, from a speech at Northwestern University (Chicago) in 1893: Finally, it must be said that the degree of exactness of the intuition of space may be different in different individuals, perhaps even in different races. It would seem as if a strong racial space intuition were an attribute primarily of the Teutonic race, while the critical, purely logical sense is more fully developed in the Latin and Hebrew races. A full investigation of this subject somewhat along the lines suggested by Francis Galton in his research in heredity might be interesting. Klein, of course, wasn t the first to make nationalistic claims about minds and math, but this speech was fairly influential. Later, the psychologist E.R. Jaensch and the prominent mathematican Ludwig Bieberbach would develop Klein s thinking into a theory of mental types, using mathematics as an especially instructive example. The J-type was German; the S-type was French and Jewish. Bieberbach felt that pure German mathematics was intuitive and geometric and organic, as opposed to the formal, analytic, logical style of Jewish mathematics. In order to praise a mathematician like Lagrange despite his Frenchness, Bieberbach resorts to laughable claims, like his appearance revealed discernible Nordic features. By the time Hitler was coming to power, the universities were home to very extreme theories about minds and their products. As with the emergence of pure math, the distinction may have begun in a value-neutral way, but it crept towards vicious disparagement. A leaflet distributed at German universities in mid 1930s read as follows: A Jew cannot write German. Were he to write German, he is lying. We demand that Jews write only in Hebrew; German editions are to be considered a translation from the Hebrew, and to be denominated as such.
4 Ludwig Bieberbach argued that pure mathematics must be rooted in the national people. To that end, he founded a journal called Deutsche Mathematik (German mathematics) which contained math of what Bieberbach considered a pure German character. There were also articles directly about racial theories, and some of these were devoted to increasingly contorted claims that various intellectual heros were in fact of Aryan heritage. For instance, one author named Thuring writes of the North-German feeling for Nature at the basis of Kepler s and Newton s work, in contrast to Einstein (a German Jew who had fled for the United States in the early 1930s). Bizarrely, Newton is explicitly called a German researcher. 2.1. The Landau boycott. Edmund Landau was a distinguished analyst and a senior professor at Göttingen at the time who, despite being one of the only practicing Jews on the mathematics faculty (there were many ethnic Jews, but mostly non-religious), had been extremely dismissive of how hostile the new nationalism would be towards Jewish Germans. Landau, remember, was the one whose textbook had defined π analytically (setting π/2 as the smallest positive root of the function cos x, itself defined via power series), causing Bieberbach to label that viewpoint typically Jewish, inorganic, foreign to matter, and inimical to life. So an interesting shift has happened in what is considered pure; the German style comes from Nature and human experience, while the impure Jewish style is excessively abstract. The boycott of Landau s classes invited the following comment from Bieberbach, in print:...the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-german style of this man in his research and teaching is unbearable to German feelings. A people who have perceived,... how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture. Oswald Teichmüller was a brilliant young student at Göttingen during Hitler s rise to power and a true believer in the Nazi cause; later he would die (at age 30) fighting on the Eastern Front during his third time volunteering for combat duty. His comments in explaining his (organizing) role in the boycott echo Bieberbach, but with some bizarre nuance, giving a fascinating glimpse into the interlocking ideologies of purity. (January 1933) Hitler elected chancellor of Germany. (April 7, 1933) Civil Service Law is passed, subsequently used to dismiss Jews from their academic positions at German universities. Exceptions were made for those with the most seniority and/or who served for Germany in World War I. (November 2, 1933) Teichmüller leads boycott of Edmund Landau s calculus lecture. They subsequently discuss this in person and Landau requests that Teichmuller put his views in writing.
5 Here is an excerpt from Oswald Teichmüller s letter: You [Landau] expressed the assumption yesterday [in our conversation] that it had been an anti-semitic demonstration. I stood and stand by the view (Standpunkt) that a special action inimical to Jews should be directed against almost anyone else before you. It was, for me, not about makign difficulties for you as a Jew, but solely about protecting German students in their second semester from being instructed by a teacher of a completely foreign race precisely in differential and integral calculus, while sparing as much as possible all others therefrom. I dare as little as any other person to doubt your capability for pure international-mathematical-scientific teaching of suitable students of whatever heritage. However, I also know that many academic lectures, especially also differential and integral calculus, at the same time have educational value and lead the student not only into a new conceptual world, but also to a different mental viewpoint (geistige Einstellung). Again, since the mental viewpoint of an individual depends on his mentality (Geist); which thus should become transformed; this mentality, again, according to fundamental rules, not only contemporary ones, but already long recognized, depends completely substantially on the racial composition of an individual; allowing Aryan students to be educated by a Jewish teacher, for example, ought not in general be recommended. I can speak here from my own experience. For the student [taught by a teacher of a foreign race] remains really only two paths perhaps (entweder) he draws out of the teacher s lecture only the international-mathematical skeleton and clothes it with his own flesh. That is mathematically-philosophically productive work, to which only the fewest have grown... The third path, to take over the material in its foreign form, leads to a spiritual (geistigen) degeneration that you could not well expect of a student today and also do not wish. The possibility, however, that you transmit to your hearers the mathematical kernel without your own national coloration is so small as it is certain that a skeleton without flesh does not run but falls into a heap and disintegrates. From this, my view, also follows that there were little to argue against it if you wish to hold more advanced lectures, building on the already present mental viewpoint worked out for application or knowledge of important mathematical facts, now as before in the best relationship with the students in our university. This is a view that only a few of my comrades have joined. 3. The Point Purity in math has meant many different things at different moments, setting apart for instance types of curves and types of numbers in the Hellenistic era, and types of proofs in modernity. This handout juxtaposes the ideologies of mathematical purity in German academia before and during the Nazi period to highlight some of the arbitrariness in the distinctions and to show that they can connect to dangerous disparagement of difference.