
The Ideal Mathematician (1998) [pdf] - anatoly
http://users-cs.au.dk/danvy/the-ideal-mathematician.pdf
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4bpp
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.)

------
loeber
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.

~~~
pathsjs
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

~~~
dkural
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.

~~~
pathsjs
I did not check it myself, so I cannot tell, but this is what I have heard
most often.

Still, it was just to highlight the fact that there is a certain degree of
subjectiveness when evaluating whether something constitutes a proof

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CurtMonash
Some of that's pretty silly.

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.

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a_bonobo
>No discovery of mine has made, or is likely to make, directly or indirectly,
for good or ill, the least difference to the amenity of the world.

\- G. H. Hardy, A Mathematician's Apology

(Yet here we are, applying the Hardy-Weinberg equilibrium in biology, and the
Hardy–Ramanujan asymptotic formula in physics)

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xyproto
The ideal mathematician could be a female.

~~~
walrus
1998\. Language usage has changed. Gender-neutral 'he' was more common then.

