> Keep in mind that the letter was not written for a mathematician, even though Simone could not understand most of it.

I'd love to see a citation here. Simone Weil referenced mathematics a lot in her philosophical writing, and, growing up in the shadow of her brother, had been exposed to mathematics all her life.

> The unwritten laws of modern mathematics forbid writing down such views if they cannot be stated precisely nor, all the more, proven. To tell the truth, if this were not the case, one would be overwhelmed by work that is even more stupid and if not more useless compared to work that is now published in the journals. But one would love it if Hilbert had written down all that he had in mind.

Here we've had some (technical) progress since 1940: modern Hilberts may publish their proven results, arxiv their useless work, and blog their work that is even more stupid?

"however it is beautiful and surprising that the prime numbers p for which m is a residue are precisely those which belong to certain arithmetic progressions of increment 4m; for the others m is a non-residue"

Fascinating. At first I was confused because I thought he was referring to the law of reciprocity. But it's actually a different law:

  m = 3
  = not a square mod 5. (reciprocal)
  = not a square mod 7. (not reciprocal)
  = 5² mod 11.          (not reciprocal)
  = 4² mod 13.          (reciprocal)

  Add 4*3 = 12:

  = not a square mod 17 (reciprocal)
  = not a square mod 19 (not reciprocal)
  = 7² mod 23.          (not reciprocal)

  Add 4*3 = 12:

  = not a square mod 29 (reciprocal)
  = not a square mod 31 (not reciprocal)

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Is an interesting read, but it's striking how condescending he is at the outset. How about let the reader decide what they understand or not? There's no use in saying anything regarding that.