In some sense, mathematical structure is simply analogy at a very high level. One tries to fill in details in a way that is likely to pay off. (E.g. looking for a natural way to make a semigroup you are looking at into a group may just pay off, simply because groups are ubiquitous and useful.) This may be the reason why an eye toward mathematical structure is a good thing to cultivate. This is usually how I a decent way to meet algebraic problems that need attention, when a "picture" needs to be filled in. Ultimately, this "picture" should provide some unification or better understanding of diverse phenomena, or the solution of a reticent problem. Looking for or working on mathematical (or simply algebraic) structure is just another strategy for building a better conceptual picture of the mathematical landscape.
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In some sense, mathematical structure is simply analogy at a very high level. One tries to fill in details in a way that is likely to pay off. (E.g. looking for a natural way to make a semigroup you are looking at into a group may just pay off, simply because groups are ubiquitous and useful.) This may be the reason why an eye toward mathematical structure is a good thing to cultivate. This is usually how I meet algebraic problems that need attention, when a "picture" needs to be filled in. |
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