We Will Never Prove Most Theorems

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Physics limits what we can prove

Brett: The overwhelming majority of theorems in mathematics are theorems that we cannot possibly prove. This is Godel’s theorem, and it also comes out of Turing’s proof of what is and is not computable.

The things that are not computable vastly outnumber the things that are computable, and what is computable depends entirely upon what computers we can make in this physical universe. The computers that we can make must obey our laws of physics. 

If the laws of physics were different, then we’d be able to prove different sorts of mathematics. This is another part of the mathematician’s misconception: They think they can get outside of the laws of physics. However, their brain is just a physical computer. Their brain must obey the laws of physics.

If they existed in a universe with different laws of physics, then they could prove different theorems. But we exist in the universe that we’re in, so we’re bound by a whole bunch of things, not least of which is the finite speed of light. There could be certain things out there in abstract space that we would be able to come to a more full understanding of if we could get outside of the restrictions of the laws of physics.

Happily, none of those theorems that we cannot prove at the moment are inherently interesting. Some things can be inherently boring—namely, all of these theorems which we cannot possibly prove as true or false.

Those (Read more...)