Algebra Etc.

Ramanujan claimed to have taught himself math using Carr’s “Synopsis of Results in Pure & Applied Mathematics” which is unique in its emphasis of continued fractions. Do you know what Hardy judged Ramanujan’s “formal command of” to be unmatched by anyone in the world? (hint: they are used exhaustively in much of his most brilliant work) You guessed it! Continued fractions. Similarly, Feynman was given Woods’s “Advanced Calculus” in high school, which taught the unpopular trick of differentiating under the integral sign. he explicitly credited Woods with giving him a “different box of tools” that helped him solve problems that others found difficult (if you’ve ever done math competitively you probably know differentiating under the integral sign as the “Feynman trick”.) I should have mentioned Grothendieck: As a broke stateless teenager with a sick mother and a dead father at the University of Montpelier, Alexander Grothendieck (by his own account) could not find any textbooks that gave satisfactory definitions of length, area, and volume. Unsatisfied, he worked entirely alone for years developing a “general theory of integration”. When he arrived in Paris in 1948 and handed the manuscripts around the (probably very amused) Cartan circle, he learned that he had spent years rediscovering the then half-century old Lebesgue integration, which was not by any means cutting edge, and in fact standard material in Paris. It was, in other words, totally worthless. Interestingly though, his retrospective view of these solitary years (what he called his “années cruciales”) was that they were the most formative of his mathematical life. He writes incessantly about the importance of this kind of solitude in ReS, and gives a beautiful “heirs and builders” analogy, which is very similar to the “bag of tools” analogy of Feynman. In those years, he writes, he “learned to approach the things which [he wanted] to know with [his] own eyes.” Moreso than Feynman and Ramanujan, Grothendieck lived under an extreme kind of intellectual solitude which begat a unique view of and approach to mathematics. It is intuitive that a unique view of mathematics will create unique mathematical work, but it is not exactly intuitive that a unique view of math is an inherently valuable one. But it is! Consider the integral trick: the integral trick was valuable because nobody knew about it. Some tools are valuable because nobody else uses them. In Grothendieck’s words, “It is within this solitary act, above all else, that ‘creation’ lies.” (This insight runs deeper than mathematics but I’d consider it a deep betrayal to the spirit of Grothendieck to write about it)