Let me tell you a theorem whose proof is so long you couldn't fit it in the observable Universe if you wrote one symbol on every electron, proton and neutron available. In fact.... (to be continued)pic.twitter.com/Uq93Exq27f
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Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
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Let me tell you a theorem whose proof is so long you couldn't fit it in the observable Universe if you wrote one symbol on every electron, proton and neutron available. In fact.... (to be continued)pic.twitter.com/Uq93Exq27f
So, by stepping out of an axiom system, assuming it's consistent and using that to draw conclusions, we can shorten proofs of some results by an arbitrary amount! For some more about the world's longest proofs, try this:https://johncarlosbaez.wordpress.com/2012/10/19/insanely-long-proofs/ …
Of course two questions arise here, one (far) more straightforward than the other. First, are there naturally-occurring statements whose shortest proofs are extremely long in what we all now think are reasonable axiom systems; and, second, what is reasonable, and why?
Hales’ original proof of the Kepler conjecture (on densest packing of space by sphere) took a hundred pages of writing together with about 3 gigabytes of computer calculations. If we wrote out those calculations in a text file, they’d fill about 2 million pages!
Which, of course and alas, leaves completely open the question of whether some shorter proof may exist. Still, it is surely a sobering, cautionary example.
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