I don't think any of the research mathematicians I know exhibit a degree of ignorance like the one depicted in the article towards the philosophical underpinnings of their field - the implied attitude towards rigorous proof in particular seems to be an egregious strawman. It is generally accepted that any research mathematician, given an arbitrarily chosen section of a high-level proof, should be able to convert it to a structured step-by-step proof (though possibly at the expense of a prohibitive amount of time), and the same generally holds for any half-decent graduate student by the time they've spent a year or two in their field.
The choice of title makes it even more questionable - imagine a similarly uncharitable piece titled "The ideal [racial category]".
(That being said, the insinuation that mathematicians tend to play the motte-and-bailey game with formalism and platonism does ring true.)
On a whole, I found this article somewhat condescending. It starts out declaring that it will try to create an "impossibly pure" mathematician in order to display paradoxical & problematic aspects of his role, but it really does neither. It shows a (negatively) stereotypical mathematician hopelessly confined to his field, and by this example uncovers nothing paradoxical or self-contradicting, but rather pokes at the perceived societal shortcomings of the mathematician in a way that could be interpreted as flippant.
And this "pure mathematician" is even contrived in an unrealistic fashion: no ideal mathematician would say that a proof is simply "an argument that convinces someone who knows the subject." This entire student-mathematician exchange is facetious at best.
This article really reminds me of a strawman: it pretends to paint a picture of an "ideal mathematician," but at points like the above simply resorts to vaguely offensive stereotypes (in most of the exchanges, the mathematician is stereotypically bad at explaining just about anything).
The final point the authors (who, by the way, are reputable mathematicians) make about mathematicians and their relationship to the outside world is a deserving one. There are many humorous and correct insights in this article (for example, the convention of concealing any sign that the author or intended reader is a human being is one that I find to be quite unfortunate and worthy of address), but I really found the pseudo-platonic dialogues to be nothing but off-putting. I think this might have been much better delivered as an earnest essay.
As a former mathematician, I found it quite entertaining :-)
Plus, not only I would not find absurd the claim that a proof is simply "an argument that convinces someone who knows the subject." I actually agree with it!
For an egregious example, look at Perelman proof of Poincaré conjecture. According to a few experts, it is a complete proof, and it is just a matter of sketching out the details. According to Yau, it is more like a plan of attack, and the actual proof is due to his students.
We are talking about two field medalists talking about one of the most famous problems in mathematics and disagreeing whether something constitutes a proof
Yau is wrong. Perelman proved it just fine. Hamilton made the plan of attack. Perelman executed it. Mere mortals could finally follow the argument, Yau & his students transparently tried to steal credit from a man who doesn't care for formal recognition.
" no ideal mathematician would say that a proof is simply "an argument that convinces someone who knows the subject." This entire student-mathematician exchange is facetious at best."
From your point of view, what would be a possible definition of proof from an ideal mathematician, and what would be a more-correct form of the article's definition?
An unquestionably rigorous proof is indeed "I know it when I see it", give or take a few differences. (E.g., my adviser Andy Gleason told me to take a theorem that had been proved by Tarski-Principle hand-waving and see if I could find a constructive proof for it. I did, by invoking a theorem from Vol. II of van der Waerden. He was pleased.)
A published proof is a claim that if one tries to flesh it out to an unquestionably rigorous one, one will succeed. (Errors are however distressingly common.)
The accusation of Platonism in the article is mainly hogwash.
The choice of title makes it even more questionable - imagine a similarly uncharitable piece titled "The ideal [racial category]".
(That being said, the insinuation that mathematicians tend to play the motte-and-bailey game with formalism and platonism does ring true.)