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And if we’re going to make a “general theory of mathematics” a first step is to do something like we’d typically do in naturalscience, and try to “drill down” to find a uniform underlying model—or at least representation—for all of them. and at t steps gives a total number of rules equal to: ✕. ✕.
For integers, the obvious notion of equivalence is numerical equality. For example, we know (as I discovered in 2000) that (( b · c ) · a ) · ( b · (( b · a ) · b )) = a is the minimal axiom system for Boolean algebra , because FindEquationalProof finds a path that proves it.
No doubt there’ll at least be some “natural-science-like” characterizations of what’s going on. The same is true of axioms for areas of abstract algebra like group theory—as well as basic Euclidean geometry (at least for integers). Will there still be “human-level descriptions” that involve numbers?
And in fact, to my knowledge, my Boolean algebra axiom is actually the only truly unexpected result thats ever been found for the first time using automated theorem proving. But what about something more like a theory in naturalscience? But, OK, so we know its true. And thats interesting. So, OK, weve got a complicated proof.
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