This is one of the most insightful things I’ve read about the field of mathematics in a long time. (And I read it on tumblr!)
Content includes:
Quotes from top mathematicians/physicists explaining how other top mathematicians systematically try to make their work seem more like a work of genius (as opposed to something *you* could do)
Positive argument for mathematical insight being hard yet doable through hard work as opposed to only doable via magic/genius
Then a separate person follows up with arguments that totally surprised my assumptions. I find that the pendulum in math lectures is swung way too far towards ‘formalism only, no intuition’, but the person (a math tutor) explains how if you swing the pendulum too far the other way and a lecture is all intuition and no messy details, this causes the students also to lose understanding.
Since linkposts are utterly broken, here’s a link to the tumblr post in question.
I agree that there’s an important insight here, but I’m not sure it quite shows that mathematics requires less magical ability than advertised. Perhaps navigating your way through the detailed scutwork as effectively as Feynman or Tao requires genius to just the same extent as spotting the beautiful streamlined arguments would have. Perhaps sometimes it requires some bit of your brain already to have spotted something a bit like the beautiful streamlined arguments, in order to figure out which way to swing the machete when hacking through the details.
I’m also not quite sure I buy “systematically try to make their work seem more like a work of genius”. Other candidate explanations: (1) hoping, perhaps misguidedly, to make it more comprehensible by removing a lot of intimidating grinding, (2) trying to make it more beautiful because aesthetics matter, (3) trying to get to the heart of why a thing is true, so as to provide more enlightenment for all, (4) trying to reduce the page count so that journals won’t refuse to publish. I expect there often is an element of trying to look clever too, but I don’t think that’s the whole story and I think the other reasons are more respectable.
I’ll add another supporting quote. Mathematician Niels Henrik Abel says of Gauss : “He is like the fox, who effaces his tracks in the sand with his tail.”
A counter-example to this sleight-of-hand behavior is Euler. As related by Polya here: https://archive.org/stream/InductionAnd AnalogyIn Mathematics1 #page/n109/mode/2up
(markdown bugs up the underscores in my link)
I think you can get around the messed-up-link problem in at least two ways. First, write some link text, select it and use the make-a-link button like this. Second, use Markdown: link text in square brackets immediately followed by link URL in parens like this. If I’ve guessed correctly, those should both be links to the Polya thing you were hoping to link to.
[EDITED to add: hmm, first one works (once I fixed a wrong guess at what the URL was meant to be) but second one doesn’t function as a link at all, which I suspect may be a Markdown bug. … And on further investigation, my further attempts at making links with Markdown syntax all fail completely—they don’t even try to turn into links and I have no idea why. This happens not only in this comment but also elsewhere, so it surely isn’t a side-effect of the messed-up link above.]