> One striking characteristic of Grothendieck's mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called "Grothendieck prime". In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."
As a curiosity, the contrast between Grothendieck and Ramanujan is very striking. One famous story about Ramanujan from Wikipedia (https://en.wikipedia.org/wiki/1729_(number)):
"Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but Ramanujan remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways"."
They, of course, were very different personalities, doing very different mathematics with very different impacts on the field. I always found it interesting that Ramanujan seemed to be very comfortable with numbers, their properties, patterns (continued fractions) and Grothendieck was very comfortable with structures and their rhythms without paying attention to concrete examples.
If anyone's interested in Grothendieck's writing, which is primarily in French, I threw his "Séminaires de Géométrie Algébrique" (SGA, Algebraic Geometry Seminars) and "Éléments de Géométrie Algébrique" (EGA, Elements of Algebraic Geometry) into an LLM to translate it to English. It's spotty in some sections, so I intend to do another pass, but it's better than my remedial French.
Field medals are still handed out to people who deign to look upon his prophetic ramblings. I'm half-convinced the religious stuff probably unveils the geometric structure of the universe.
Interestingly, the way in which Grothendieck conceived of equality is nowadays being questioned, especially due to the rise of formalized mathematics and Lean. More concretely, there is this fun paper by Kevin Buzzard which deconstructs it: https://arxiv.org/abs/2405.10387
Money quote:
> In this paper I argue that the first assertion above is false, the second is dan-
gerous, and the third is meaningless.
Super. I always wanted to learn about sheaves and schemes and the like, and this gives a simple introduction that really motivates digging deeper into the details.
It is also immediately clear why this plays a role in semantics for logics: although a ring is not that important in logic (I would think), the idea to study a theory through its syntactical consequences turned into semantics is very natural, and exactly what I do for abstraction logic as well, in particular via "valuation spaces". And it has the same property, once you set up everything the right way, things like completeness just automatically flow out of it.
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[ 3.1 ms ] story [ 37.4 ms ] threadFor more life and times stuff I also suggest Labatut's Cease to Understand the World book and https://theanarchistlibrary.org/library/konstantinos-foutzop...
> One striking characteristic of Grothendieck's mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called "Grothendieck prime". In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."
"Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but Ramanujan remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways"."
They, of course, were very different personalities, doing very different mathematics with very different impacts on the field. I always found it interesting that Ramanujan seemed to be very comfortable with numbers, their properties, patterns (continued fractions) and Grothendieck was very comfortable with structures and their rhythms without paying attention to concrete examples.
EGA: https://github.com/jcreinhold/ega (https://jcreinhold.github.io/ega/)
SGA: https://github.com/jcreinhold/sga (https://jcreinhold.github.io/sga/)
https://mikepierce.github.io/grothendieck-kimchi/translation...
Money quote:
> In this paper I argue that the first assertion above is false, the second is dan- gerous, and the third is meaningless.
Articles on his life: https://www.math.columbia.edu/~woit/wordpress/?p=7335
Two Titans (Grothendieck and Witten) - https://www.math.columbia.edu/~woit/wordpress/?p=12868
AMS Math articles on Grothendieck - https://www.math.columbia.edu/~woit/wordpress/?p=78
It is also immediately clear why this plays a role in semantics for logics: although a ring is not that important in logic (I would think), the idea to study a theory through its syntactical consequences turned into semantics is very natural, and exactly what I do for abstraction logic as well, in particular via "valuation spaces". And it has the same property, once you set up everything the right way, things like completeness just automatically flow out of it.