“…the most important of [Ramanujan’s methods] were completely original. His intuition worked in analogies, sometimes, remote, and to an astonishing extent by empirical induction from particular numerical cases. …his most important weapon seems to have been a highly elaborate technique of transformation by means of divergent series and integrals … He had no strict logical justification for his operations. He was not interested in rigour, which for that matter is not of first-rate importance in analysis beyond the undergraduate stage, and can be supplied, given a real idea, by any competent professional. The clear-cut idea of what is *meant* by a proof, nowadays so familiar as to be taken for granted, he perhaps did not possess at all. If a significant piece of reasoning occurred somewhere, and the total mixture of evidence and intuition gave him certainty, he looked no further. It is a minor indication of his quality that he can never have *missed* Cauchy’s theorem. With it he could have arrived more rapidly and conveniently at certain of his results, but his own methods enabled him to survey the field with an equal comprehensiveness and as sure a grasp.”

–from a review of “The Collected Papers of Srinivsa Ramanujan” appearing in the Mathematical Gazette, April 1929, Vol. XIV, No. 200; reprinted in “Littlewood’s Miscellany,” pp 94-99.

]]>One is the distinction between art and craft. So the final rigour in a proof is the craftmanship of a mathematician, making sure that everything works as claimed. No patches needed (though they sometimes are).

Another is that proofs are like describing a route, and this is in a certain landscape. How much detail should you give when describing a walk to the station? You do not want to describe all the cracks in the pavement, but you do want to warn of dangerous manholes.

One of the jobs of mathematicians is t build a landscape in which proofs, routes, can be found. I heard a comment of Raoul Bott on Grothendieck in 1958, that:”Grothendieck was prepared to work very hard to make proofs tautological.” There is a good aim to make it clear **why** something is true; that may need new concepts.

I once had a student criticism of may first year analysis course: “Professor Brown gives too many proofs.” So I decided next year there would be no theorems and no proofs; what they will get were “facts” and “explanations”.

However there is a kind of obligation that an “explanation” should actually explain something!

A good test of a future mathematician is not necessarily current level of performance, but do they actually want to know why something is true.

]]>On the other hand, I think I’m good on language. I can do abstract reasoning when interpreting and using linguistic tools like metaphors and metonymy. I think language is underrated as a math tool. It’s as abstract as greek characters, but its mapping to _some_ intuition is more natural, at least to me. Every time I see greek letters in an explanation I translate it to language and then to some intuition, if needed, but I always do the translation step. If some concept from pure math is explained by a textual description, I tend to capture it easier and I can instantly map to a bunch of real-world applications.

I know, someone will say that natural language is ambiguous, but so is the actual academic math. How much papers do you read where there isn’t a single sentence written in plain natural language? In this case, ambiguity is not language’s fault, since the author decides doing a textual explanation because math isn’t enough to cover the entire concept. There isn’t any notation capable of motivating the reader for reading a math paper. Thus natural language is an advertising tool, but also is employed for covering gaps when pure math notation isn’t enough.

In short, I think pure math (i.e., math heavily relied on notation) is overrated as an abstract reasoning tool. You can do similar reasoning with plain natural language, so you could reach a broader audience and allow more intuitions to emerge — even pure math intuitions. Proving my hypothesis is the harder part, since I should use pure math, but I think I could contribute with some empirical results for motivating a mathematician to try something on this path. There is room for all kind of abstract thinking mathematicians.

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