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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 ge
by 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.