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Comment by baddox | Hacker News Reader

If I can understand the complexity classes themselves as well as the consequences of P = or != NP (which I believe I can), then it seems like I should expect (or at least hope for) a similarly-comprehensible proof.

Contrast that with, say, special relativity. I can't even really understand the consequences of special relativity (length contraction, relativity of simultaneity, time dilation, etc.) beyond accepting that my intuitions are completely errant at relativistic speeds. Because of that, I don't really expect or hope for an intuitive proof or explanation of special relativity.

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I can understand Fermat's Last Theorem but I don't understand its proof.

Granted. I do think problems in the domain of number theory are a bit different. I don't know of any quickly-explainable consequences to Fermat's Last Theorem, and in my first comment I was really thinking about consequences. With Fermat's Last Theorem, to my knowledge, the original problem is itself pretty arbitrary, so I'm not surprised to find its solution and proof to be complicated despite the simplicity of the problem statement.

Also, I'll preemptively mention that Gödel showed us that any formal problem is a problem in number theory. That said, a problem like P =? NP would almost surely be incomprehensible if expressed as an isomorphic problem in number theory. So my original point, clarified and rephrased, is that I would expect (or desire) the proof of a problem to be as intuitive as the statement of the problem combined with its consequences in the domain the problem is stated.

I don't know the history of mathematical theories, but in physics it has often been the case that something took years (or decades) for people to wrap their heads around as they were poking at the edges of understanding. But as they continued to do so, new ideas and concepts emerged which made a better mental framework for understanding them. Hence, for example, the work-energy theorem can now be proved by a high school senior, but in the day there was big debate over the factor of 1/2 in the term for kinetic energy.

So it may be with mathematics. Things like Fermat's Last Theorem and P != NP may be difficult to grasp simply within our current frameworks, but new ideas may give rise to new frameworks that makes them more digestible.

If you can just get it proved somehow then you can proceed with confidence towards these new ideas.

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