I only recently learned that Pascual Jordan, a well known physicist, with significant contributions to the development of early quantum mechanics was a paid member of the Nazi Party. He in fact joined a SA unit.
Are there any well-known mathematicians who were fascist or Nazi?
The book Mathematicians Under the Nazis by Segal is a systematic study of, well, what it says in the title. It includes several examples of mathematicians who showed various degrees of sympathy for the Nazis. This includes Gentzen, Vahlen, Bieberbach and Kähler.
Probably the most famous mathematicians who were members of NSDAP are Oswald Teichmuller and Ludwig Bieberbach.
From "Probabilism", Erkenntnis, vol. 31, No. 2/3 (1989), pp. 169-223 (translation of a 1931 article in Italian)
But where my spirit rebelled most ferociously and clashed against the concept of "absolute truth" was in the political field, and I could not say what part, surely very great, this sense of impatient revolt must have had in the development of my ideas. To be confronted by papier-mâché idols and a miserable political class that would have preferred Italy in ruins rather than failing (sacrilege!) to render due homage! Those delicious absolute truths that stuffed the demo-liberal brains! That impeccable rational mechanics of the perfect civilian regime of the peoples, conforming to the rights of man and various other immortal principles! October of '22! It seemed to me I could see them, these Immortal Principles, as filthy corpses in the dust. And with what conscious and ferocious voluptuousness I felt myself trampling them, marching to hymns of triumph, obscure but faithful Blackshirt!
From "La crisi dei principi e l'economia matematica", Acta Seminarii, 2 (1943), pp. 33-68
Facciamo che quando la vita, cessato il turbine della lotta, tornerà a rifiorire, liberata dagli egoismi che pretendevano soffocarla, siamo in grado di indicarle una nuova via meglio concepita, che superi l'assurda artificiale barriera elevata tra l'uomo e la possibilità di benessere largitegli dalla natura o conquistate dal suo ingegno. Soltanto allora la guerra "del sangue contro l'oro" sarà veramente vinta. Come vogliamo, come è necessario.
My translation:
Let us ensure that when life, having ceased the whirlwind of struggle, flourishes again, freed from the selfishness that claimed to stifle it, we are able to point it to a new and better conceived way, one that overcomes the absurdly artificial high barrier between man and the possibility of prosperity bestowed upon him by nature or conquered by his cleverness. Only then will the war "of blood against gold" be truly won. It is what we want, it is what is necessary.
The list of Itaian fascist mathematicians is, unfortunately, quite long, given that fascism lasted for 20 years and it was basically impossible to get a relevant position in accademia without supporting the regime (some professors resigned when it was imposed to swore loyalty to fascism to every University professor).
It is not easy to distinguish between those that were sincerely fascist and those that merely adopted a comfortable neutrality, or simply faked loyalty to fascism being intimately at fight with it.
Surely Francesco Severi was strongly supporting fascism and you can find a reasonably detailed description of his positions in here: paper by Goodstein and - AMS Babbitt, together with references therein.
Surely a devoted fascist was Luigi Fantappié; its famously recorded in André Weil autobiography an episod in which he went to Volterra (who was a Jew) to convince him of how beautiful the Italian new Law against Jews was. The Law eventually determined that both Volterra and Levi Civita were permanently thrown out of Italian University. Also he was appointed for some years as a Professor at University of Sao Paulo, Brasil, and he gave conferences throughout Brasil which were intented to promote fascism in that country.
Enrico Bompiani and Mauro Picone were two others very relevant nathematicians that completely sided on the fascist side from the very beginning, and gained powerful poisition in the academic environment through this.
A brief but informative summary of this development may be found in Benzi's slides and in this work about the Italian Mathematica Union UMI (paper by Giacardi and Tazzioli)
As to the Wikipedia quotes about Gentzen one should probably add that various authors describe him as politically naive and unworldly. It seems he joined the SA because he was told to do so to secure his academic position. He remained in contact with his "esteemed" teacher Paul Bernays, a Jew who had left Germany in 1933, and in 1941 Gentzen even suggested that Kurt Gödel - who had fled to the US in 1940 - should be offered a professorship in Germany.
Van der Waerden in the Third Reich by Reinhard Siegmund-Schultze book review of "The Scholar and the State. In Search of Van der Waerden"
See the comments below
Ernst Witt joined the Nazi party soon after Hitler assumed power, and various stories have circulated about his commitments. From MacTutor History of Mathematics:
On 30 January 1933 Hitler came to power. Witt joined the Nazi Party on 1 May 1933 and also the SA, the military wing of the Party. It has been claimed that Oswald Teichmüller, who was like Witt a student at Göttingen, convinced him to join. Emmy Noether, who was Jewish, was dismissed from her post by the Nazis but continued to give lectures in her home. At one of the lectures Witt turned up wearing his SA (Sturm Abteilung: Storm Section) uniform.
