“God, as the Pythagoreans said, is a geometer-but not an algebraist”

A remark of Simone Weil, which I learned from Karen Olsson’s book The Weil Conjectures.

In the original, “Pour moi, je pense bien que Dieu, selon la parole pythagoricienne, est un géomètre perpétuel – mais non pas un algébriste.” Here’s the letter to her brother from which this is taken.

I wanted to write more in Shape about the complicated moral weights people assigned to the difference/tension between algebra and geometry. I wrote a little about what Poincaré thought about it, who saw the two subjects as representing two temperaments, both indispensable, though any given mathematician might possess them in different proportions.

But then there is this, from S.I. Segal’s “Topologists in Hitler’s Germany”

“the … Nazi movement saw “truly German” mathematics as intuitive and tied to nature, often geometrical, and certainly not axiomatic. Axiomatics, “logic chopping”, too great abstraction, was Franco-Jewish.”

It is grimly funny to imagine the Nazi ideologues locating in abstract algebra and the insolubility of certain equations in radicals the ultimate origins of rootless cosmopolitanism.

3 thoughts on ““God, as the Pythagoreans said, is a geometer-but not an algebraist”

  1. NDE says:

    I was expecting the Atiyah quote on this topic . . . (copy-and-pasted from https://www.goodreads.com/quotes/9026708: “Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.”)

  2. dratman says:

    That perhaps explains why Hitler kept ordering the Luftwaffe to bomb Hilbert’s Hotel. He dreamed of attacking it an uncountably infinite number of times. What ignorance! Fortunately Whitehall had requisitioned the hotel and renamed it Bletchley Park as soon as war was declared. Hitler’s algorithms to find it never terminated, and on April 30, 1945 the allies pulled the plug on all his computations.

  3. AG says:

    The quote from Hermann Weyl in section 1 of the article “Singular Adventures of Baron Bourgain in the Labyrinth of the Continuum” https://www.ams.org/journals/notices/202011/rnoti-p1716.pdf
    expresses a sentiment akin to the one in the highlighted quote in the post.

    The following opening of section 4.1 (on Hausdorff-Banach-Tarsky paradox) does not appear in the published version.

    The types of creatures on the earth are countless, and on an individual level their self-preservation instinct as well as longing for procreation is always unlimited; however the space on which this entire life process plays itself out is limited. It is the surface area of a precisely measured sphere. Hitlers Zweites Buch

    It is a pity the demented housepainter was not briefed about the Hausdorff-Banach-Tarsky constructive solution of the Lebensraum problem. \footnote{When, as part of the ‘Final Solution,’ Hausdorff, his wife Charlotte, and a sister of hers were ordered to leave their house for a local internment camp in January 1942, they opted for suicide. During the night of July 3, 1941, forty distinguished representatives of Lvov intelligentsia, including S. Ruziewicz, perished at the hands of the S. S. ‘Nachtigall’ batallion. Banach was saved by Rudolf
    Weigel, the inventor of the typhus vaccine, who employed him as a feeder of lice.}

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