Monday, April 13, 2020
What do people find enjoyable about mathematics? I don't know all the reasons other people find it enjoyable, and why they do it, but for myself, I think it's a mixture of
* The simple pleasure of discovering patterns hidden from view, and methods that you wouldn't expect to go very far, but do. Of course, what makes something surprising is mostly a reflection of human cognitive limitations and also our current environmental / culural influences. e.g. Humans find the solutions to certain "puzzles" in probability theory surprising, so have a hard time coming up with them; but if we tweaked our brains somehow, maybe they would become "obvious". Also, certain mathematical cultures are trained to think in ways, that make certain solutions easier to produce. The power of culture can be profound -- see, for example, this disucssion of aboriginal peoples in Australia who don't think in "egocentric coordinates":
New York Times article
QUOTE: "But then a remote Australian aboriginal tongue, Guugu Yimithirr, from north Queensland, turned up, and with it came the astounding realization that not all languages conform to what we have always taken as simply “natural.” In fact, Guugu Yimithirr doesn’t make any use of egocentric coordinates at all. The anthropologist John Haviland and later the linguist Stephen Levinson have shown that Guugu Yimithirr does not use words like “left” or “right,” “in front of” or “behind,” to describe the position of objects. Whenever we would use the egocentric system, the Guugu Yimithirr rely on cardinal directions. If they want you to move over on the car seat to make room, they’ll say “move a bit to the east.” To tell you where exactly they left something in your house, they’ll say, “I left it on the southern edge of the western table.” Or they would warn you to “look out for that big ant just north of your foot.” Even when shown a film on television, they gave descriptions of it based on the orientation of the screen. If the television was facing north, and a man on the screen was approaching, they said that he was “coming northward.”"
And they are, apparently, *very* good at keeping track of where objects are placed in space. I can't imagine doing that -- though, who can say what is possible if brought-up for decades in such a culture?
* Some small amount of acclaim, if it leads anywhere surprising.
* I get paid to do it, which pays the bills.
My suspicion is that many other people do mathematics for some additional reasons, that don't apply to me, including:
* They have an almost religious attachment to the work -- almost as though "God" has arranged mathematical structures just-so; and it is the job of humanity to find these "fingerprints of God". There are no accidents; it's all perfect. Ugly mathematics is just people intentionally ignoring this path that is clear for all to see.
I think this view is wrong, and that the beauty is just accidents. And, probably, there are even simpler proofs to most "deep" mathematical results that are just as beautiful; so, which one was "God's" favorite?
* Even though they can't prove it (initially), they think there is some kind of "ultimate explanation" that unites all the special cases in some area; and it is their job to uncover this "more general framework". This is similar to the "religious" view. I think if we could see all possible explanations, we would find that this is mistaken. Some areas will have multiple generalizations, no one any more "correct" than the other; some areas will lead to dead-ends, with no unification possible; and some small number of areas have exactly one explanation that most people believe is the "right way". The areas that will see more successes will likely be ones with more initial structure to work from. The more directions you can branch in before becoming stuck, the deeper you're likely to reach, the greater the chance you'll find the kind of unifications people seek.