Human minds have a bad habit of polishing their work until the seams are invisible. It’s akin to movie magic or stage illusions.

A mathematician publishes a three-page proof that took months of time and reams of paper to work out, but we never get to see the iterative process, the naive methods becoming refined, the chronological order that yielded the logical order, the false trails and dead ends.

There are practical reasons for this: brevity is often an aid to clarity, and few mathematicians wish proofs were any longer than they are already. But I wonder if part of the motivation for succinct, laconic proofs is that the authors look smarter when you can’t see how they did the trick. It’s as if the spirits of Tartaglia and Fermat linger still, recalling a time when mathematics was as secretive as alchemy. (Speaking of alchemy, the physics, chemistry, and other empirical disciplines are not immune from this disease.)

Never before have we had access to so much information, and never before has it been so hard to find the information we need. Sure, computers can make efficient indexes, but if the semantics are unrecognizable outside of the field, the words are so much rubbish to non-specialists. I hope that someday standardized descriptions of mathematical structure or canonical forms of equations will help alleviate this, so that equations in one paper can be automatically compared to equations in another.

Think of it as disassembling software to see if it uses a certain algorithm, except for sharing knowledge instead of suppressing it.

Addendum 2016-07-29:

There’s one thing that comes through in a lot of this history, by the way: notation, like ordinary language, is a dramatic divider of people. I mean, there’s somehow a big division between those who read a particular notation and those who don’t. It ends up seeming rather mystical. It’s like the alchemists and the occultists: mathematical notation is full of signs and symbols that people don’t normally use, and that most people don’t understand.

— Stephen Wolfram, “Mathematical Notation: Past and Future”, October 20, 2000

http://www.stephenwolfram.com/publications/mathematical-notation-past-future/

https://web.archive.org/web/20010813153029/http://www.mathmlconference.org:80/schedule.html