when matematicians solve some problem that has’nt ever been solved how do they know they got the right answer and they did the right steps ?
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Other mathematicians check their work.
By “check their work” what I mean is that other mathematicians read through the proof and make sure that each step follows logically from the last. At the frontiers of math, subfields get so specific that there are only a handful of people in the world who can actually peer review the proof. This means the peer review process can take years.
Still, mistakes do occasionally slip through the cracks. This doesn’t really happen with well established results that have been proven in multiple ways, and it doesn’t happen with big results that a lot of people care about, but it can happen.
Edit: I should probably expand on what “each step follows logically from the last” means. Say I’m trying to prove that 1=2 step by step:
1) let x = y be nonzero
2) then x^2 = xy
3) and so x^2 – y^2 = xy – y^2
4) this implies (x+y)(x-y) = y(x-y)
5) divide through by (x-y) to get x+y = y
6) but x=y, so that means 2x = x
7) therefore 2=1
Now, my result is obviously incorrect. That means one of those steps was wrong, or in other words, there’s one step which does not imply the truth of the next step. Which is it?
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