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The idea for the paper was sparked two years ago by a "toy problem" that Ono presented as a "gift" to entertain Zagier during the lead-up to a math conference c

by twic 7y ago

The idea for the paper was sparked two years ago by a "toy problem" that Ono presented as a "gift" to entertain Zagier during the lead-up to a math conference celebrating his 65th birthday. A toy problem is a scaled-down version of a bigger, more complicated problem that mathematicians are trying to solve. Zagier described the one that Ono gave him as "a cute problem about the asymptotic behavior of certain polynomials involving Euler's partition function, which is an old love of mine and of Ken's—and of about pretty much any classical number theorist." "I found the problem intractable and I didn't really expect Don to get anywhere with it," Ono recalls. "But he thought the challenge was super fun and soon he had crafted a solution." That sounds like a potentially useful problem-solving trick - pull a small but key problem out of your bigger problem, and then nerd-snipe your friends into solving it.

pmiller2 7y ago
Definitely. Two fairly standard techniques in mathematics are solving a cut down version of a problem and lifting that to a full solution, and solving a more general version and specializing to a particular case. Sometimes, amazingly, the more general problem is easier to solve, perhaps because fewer irrelevant details stand in the way. Both methods have yielded a lot of progress in mathematics. For example, solving a cut down version of Fermat’s Last Theorem got us the theory of ideals in algebra. Relatedly, Wiles work on the modularity theorem ultimately led to the actual proof of FLT.
- throwawaymath 7y ago
  > Sometimes, amazingly, the more general problem is easier to solve, perhaps because fewer irrelevant details stand in the way. True, and that's why we like to generalize things. It's usually easier to prove a mathematical object belongs to another class of structure which has certain properties than to prove the object itself has all those properties. Of course as you've noted it's not always like that at all. It's significantly easier to prove many things about finite-dimensional vector spaces over fields versus infinite-dimensional modules over rings. Solving mathematical problems is a lot like working a sponge. It's often useful to alternate between expanding and contracting the scope of your problem.

by twic 7y ago

pmiller2 7y ago

Definitely. Two fairly standard techniques in mathematics are solving a cut down version of a problem and lifting that to a full solution, and solving a more general version and specializing to a particular case. Sometimes, amazingly, the more general problem is easier to solve, perhaps because fewer irrelevant details stand in the way. Both methods have yielded a lot of progress in mathematics. For example, solving a cut down version of Fermat’s Last Theorem got us the theory of ideals in algebra. Relatedly, Wiles work on the modularity theorem ultimately led to the actual proof of FLT.

throwawaymath 7y ago
> Sometimes, amazingly, the more general problem is easier to solve, perhaps because fewer irrelevant details stand in the way. True, and that's why we like to generalize things. It's usually easier to prove a mathematical object belongs to another class of structure which has certain properties than to prove the object itself has all those properties. Of course as you've noted it's not always like that at all. It's significantly easier to prove many things about finite-dimensional vector spaces over fields versus infinite-dimensional modules over rings. Solving mathematical problems is a lot like working a sponge. It's often useful to alternate between expanding and contracting the scope of your problem.