Written: May 1, 2011
My great guru, Gian-Carlo Rota, once said (in :"Two Turning Points in Invariant Theory", Math. Intell. 21(1), p. 26):
Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day.And indeed, most mathematical articles, especially seminal ones, are written in such a terse and unmotivated style, revealing as little as possible on how the proof was discovered, and only saying the absolute minimum to enable the careful reader to certify the formal correctness of the proof, but without conveying any understanding.
Hopefully, this would change, and mathematicians would become more generous with their ideas, and tell us how in the world they discovered their breakthroughs. Of course, quite a few brilliant mathematicians are bad teachers, and even if they would have been willing to write in a way that would bring understanding and insight, they are unable. After all the world's champion runner has hardly any clue on why he is so fast, not knowing any biochemistry, and the world's fastest driver probably does not any thermodynamics.
Already in the print-era it was possible to do an OK job of explaining, but with electronic communication, and the internet, one can do marvels. So please, next time you discover a mathematical breakthrough (or even just a nice little theorem), please do not write it like Klaus Roth wrote his seminal 1953 article "On Certain Sets of Integers", but make it into a lively and interactive wiki and he created a program that tests, empirically, every single statement made in the article. [Here is his report]
Well done David!, and let's hope that many people would follow in your finger-steps!