alt Hacker NewsI don't think that's the point. I think the point is to prove the statement. The techniques and abstractions are a means to an end; making them the point is being seduced by the beauty of the weapon.
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New techniques and abstractions is how mathematics expand. Mathematics is about studying structures, proving statements is a part of it but it is not all what mathematics is about. If anything, proofs themselves are a means to an end (understanding). Eg Galois developed some techniques and abstractions to prove that there is no general solution to polynomial equations of degree >=5, but these techniques and abstractions gave rise to whole new mathematical fields.
Mathematics has to be also understood from the perspective of theory building, not just problem solving.
> I think the point is to prove the statement.
I couldn't disagree more.
A lot of mathematical "problems" are almost entirely pointless. Nobody genuinely cares about the moving sofa problem, or about square packing, or about the minimum number of colors needed to draw a map - it is the math that is developed during the solving process that is valuable!
An answer to a question like "what is the exact area of a unit circle" is a mere curiosity. Calculating a good-enough approximation is trivial, after all. But wanting an exact answer leads to developing calculus, which leads to most modern physics. Science was able to make a giant leap forwards due to the techniques developed, while the actual answer itself is mostly useless.
Most of the interesting research I’ve ever done started while reading through the intermediate steps in an unrelated paper.
As far as I can tell from colleagues in other domains, it’s the same there. One paper will mention something off-hand and that’ll cause someone else to have a spark of insight, which turns into it’s own valuable research