I don’t think it is strictly true that Grothendieck didn’t rely on examples. Here is a quote from a letter from Luc Illusie, who was a grad student under Grothendieck. Quote:
“In his filing cabinets, located behind his desk, Grothendieck kept many handwritten notes, where he had studied specific examples: he sometimes told me that he was weak on surfaces, but as everybody knows, he was not so weak in local algebra, and he knew enough of curves, abelian varieties and algebraic groups to be able to test his ideas. Also, his familiarity (and constant interest) in analysis and topology was a strong asset. All these examples appeared when you discussed with him.”
Slightly related … there is a recent philosophy of mathematics book (albeit a work of continental philosophy, but it is still interesting). I say slightly related, because firstly it has an entire chapter on Grothendieck and creativity, and secondly the entire book is about the interplay between examples and theory. It goes into the way mathematics moves up and down a ladder of abstraction (from particulars to universals, local to global, specialization to generalization). It focuses on the last 100 years of mathematics, concentrating on algebraic geometry, category theory, model theory, and others. The author splits up the abstractions into:
Eidal mathematics, or “transfusions of form.” This is mathematics that moves up the ladder from the particular to the universal. In this part of the book it focuses on the work of Serres, Langlands, Lawvere, Shelah.
Quiddital mathematics, or “transfusions of reality.” This is abstractions made concrete moving down the ladder. Here the book focuses on Atiyah, Lax, Connes, Kontsevich.
Archeal mathematics, or “decantations of the universal.” This is mathematics that studies the invariants while one moves up and down the ladder. Here it focuses on Freyd, Simpson, Zilber, Gromov.
Now you could say this philosopher is just dressing up stuff already known to mathematicians (Both Kevin Houston’s How to think like a mathematician, and Mason/Burton’s Thinking Mathematically talk about generalization and specialization). But it is still interesting, as I think it may be the only philosophy of mathematics book out there that does an indepth treatment of 20th and 21st Century mathematics that goes beyond Russell and Frege.
I don’t think it is strictly true that Grothendieck didn’t rely on examples. Here is a quote from a letter from Luc Illusie, who was a grad student under Grothendieck. Quote:
Slightly related … there is a recent philosophy of mathematics book (albeit a work of continental philosophy, but it is still interesting). I say slightly related, because firstly it has an entire chapter on Grothendieck and creativity, and secondly the entire book is about the interplay between examples and theory. It goes into the way mathematics moves up and down a ladder of abstraction (from particulars to universals, local to global, specialization to generalization). It focuses on the last 100 years of mathematics, concentrating on algebraic geometry, category theory, model theory, and others. The author splits up the abstractions into:
Eidal mathematics, or “transfusions of form.” This is mathematics that moves up the ladder from the particular to the universal. In this part of the book it focuses on the work of Serres, Langlands, Lawvere, Shelah.
Quiddital mathematics, or “transfusions of reality.” This is abstractions made concrete moving down the ladder. Here the book focuses on Atiyah, Lax, Connes, Kontsevich.
Archeal mathematics, or “decantations of the universal.” This is mathematics that studies the invariants while one moves up and down the ladder. Here it focuses on Freyd, Simpson, Zilber, Gromov.
Now you could say this philosopher is just dressing up stuff already known to mathematicians (Both Kevin Houston’s How to think like a mathematician, and Mason/Burton’s Thinking Mathematically talk about generalization and specialization). But it is still interesting, as I think it may be the only philosophy of mathematics book out there that does an indepth treatment of 20th and 21st Century mathematics that goes beyond Russell and Frege.
Thanks for the piece of counter-data!
I might look into the book, but the naming convention is a big turnoff.