Purity of mathematical language and (German) mathematics in the 20th century

1) Norbert Schappacher: Helmut Hasse’s Hygiene for Writing Mathematics

I plan to start the talk with two examples of Hasse’s interactions with colleagues.

First example: Hasse arranged to improve his English through daily conversations with Harold Davenport. This would result in Hasse's proof of the (analog of the) Riemann Hypothesis for function fields of elliptic curves over finite fields. The process involved a sequence of translations and reformulations.

Second example: In order to try and generalize this theorem from the elliptic case to curves of higher genus, Hasse liked Max Deuring’s idea to use algebraic correspondences. However, Hasse insisted on having Deuring’s first paper on the subject rewritten to comply with his methodological preferences, and of Deuring's second paper on the subject only the first few pages were ever printed, apparently also because Hasse thought that the method of proof of its main result was “unfair”, and therefore not fit to print.

From these two examples, and several other remarks and actions by Hasse, we shall launch into a general discussion of how to pinpoint Hasse’s idea of the purity of method.

2) Kati Kish Bar-On: Mathematics and Society Reunited: The Social Aspects of Brouwer’s Intuitionism

Brouwer’s philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, which also underlies his mathematical intuitionism, as he strived to create mathematics that develops out of something inner and a‑linguistic. Thus, points of connection between Brouwer’s views and the social world seem less probable and are rarely mentioned in the literature. In this lecture, I examine Brouwer’s views on the construction, use, and practice of mathematics through a socially oriented prism. I highlight the social character of mathematical practice as Brouwer addressed it in the Significs Dialogues - documented dialogues between Brouwer and other members of the Signific Circle, a social movement focused on the connection between language, mathematics, and society centered in the Netherlands. After fleshing out the connection between society, people, and mathematical knowledge in Brouwer’s thought, I pose two critical questions: (1) How do social, personal, and political events have shaped the development of intuitionism, and (2) How does Brouwer’s social perspective affected the content of his intuitionism.

A link to the paper can be found here

3) Tim Lork: Actors of Hyperpoliticized Purity – Prosopographical Perspectives on Aryan Mathematics

'Aryan Mathematics' is a prime example of 'hyperpoliticized fringe doctrines' (Michael Gordin): Intellectual positions with scientific aspirations, instrumentalized by or directly derived from political ideology – in this case, from national socialism and its racist idea of ‘purity’. Traditionally analyzed in philosophy, fringe doctrines have also become a subject of empirical research in sociology and the history of science in the last years. Along these lines, the talk demonstrates potentials and limits of a prosopographical and bibliographical approach to 'Aryan Mathematics', using data collected in the DFG project Political crises and disciplinary development. Mathematics in Germany, 1920-1960 at the University of Wuppertal ('Promath').

Accordingly, the talk provides a historical sketch of 'Aryan Mathematics' and a short introduction to 'Promath', followed by prosopographical and bibliographical comparisons of the contributors to the journal 'Deutsche Mathematik', the central publication organ of ‘Aryan Mathematics’, with the mathematical elite during the Nazi regime. Who supported 'Aryan Mathematics' in public? How did this group differ from the established mathematical elite? Had the engagement for 'Aryan Mathematics' any impact on academic careers? By addressing questions like these, the talk presents empirically substantiated insights concerning the character of 'Aryan Mathematics'.

4) Tabea Rohr: Pure Thought and National Pride. Frege on Logic and Politics

Frege first presents his logic in Concept Script. A formula language, modeled on that of arithmetic, for pure thought from 1879. Throughout his life, Frege stresses the universality of logical laws, which are acknowledged by all rational beings. On the other hand, Frege expresses nationalist and even fascist views in his diary. In this talk, we try to highlight this contrast and discuss, if there are any links between the logical and political thinker Gottlob Frege.

5) Stefan Dollinger: The conceptually long reach of the early 19th century: the field-defining One Standard German Axiom (OSGA) and the lack of adequate descriptions of Austrian German today

Germanistik - German Studies - was institutionalized in the early 19th century as precisely that: German-istik, which focuses on all things German; not Austrian, not Swiss, not South Tyrolean, not Luxembourgish or Liechtensteinish. While after WWII, German literature denazified fully and is today quite avant-garde, German linguistics and dialectology have largely gotten away with re-hiring WWII-linguists after the war. Combined with a naïve belief in the objectivity of data, neither data nor theory seems to match social realities today. Reading present-day accounts of Austrian German (now called “German in Austria”), one wonders how the German “Postivismusstreit” has gone by unnoticed in that field. The present paper scrutinizes this new wave of German linguists in Austria, since about 2012, for its bias against Standard Austrian German. This bias is never openly addressed, but easily discernible in their writings. While the current generation of scholars (unlike their teachers and, especially teachers’ teachers) has nothing to do with Nazism, their views, I claim, are compromised by a One Standard German Axiom (OSGA), which has, with occasional critique, been operative in German linguistics since the 1820s.

6) Christophe Eckes: Teaching mathematics at the École polytechnique under the Vichy regime: between modernization and counter-modernity?

While a first major study of the École polytechnique under the Vichy regime was coordinated by historians Marc-Olivier Baruch and Vincent Guigueno in the 1990s, more specific research on the mathematical sciences practiced and taught there during this period remains largely to be done.

With this talk, we would like to look at how the teaching of mathematical sciences was viewed during the Occupation by members of the so-called “Conseil de perfectionnement de l'École polytechnique”, a body responsible for coordinating teaching at the École polytechnique. We will also attempt to determine the extent to which the Vichy regime's propaganda was exerted there, before analyzing the interventions and reflections specifically dedicated to the teaching of mathematical sciences within this “Conseil de perfectionnement”.

Through this case study, we aim to show whether and to what extent the teaching of mathematics as reformed by the governing bodies of the École Polytechnique during the Occupation bears the hallmark of a desire to modernize the machinery of the State which, as the American historian Robert Paxton rightly pointed out in the early 1970s, was one of the characteristic features of the Vichy regime. We will also explore whether, and to what extent, the duality between modernity and counter-modernity as elaborated by the historian of mathematics Herbert Mehrtens can be employed in such a context.

7) Rémi Blondel: French mathematics, a symbol of purity and universality?

The aim of my presentation is to examine the relationships that French mathematicians maintain, on the one hand, with mathematics and, on the other, with France. More specifically, I will study how mathematicians express these relationships through the concept of purity. My study will focus on the first half of the 20th century, a period during which mathematical exchanges between France and Germany flourished.

I will first examine which mathematical objects, fields, and practices are considered ‘pure’ by mathematicians. I will then examine how the notion of purity allows mathematicians to glorify the individuality of an author, but also to emphasize the superiority of the nation and the universality of a thought. Finally, I will show how the questions raised by this notion are both mathematical and linguistic and that the ideology underlying these questions is not specific to mathematics and mathematicians but that, on the contrary, mathematical purity is part of a much more global vision of the world.