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Notes -
I don't follow your technical aside. Isn't the point of an isomorphism that all the isomorphisms are the same in the sense in which you are isomorphic?
I don't understand this example, because if you have A|B, A+A = A, B+B=B, but if A+B=A then A maps to true and if A+B=B then B maps to true.
But how? Like, what is the failure mode that you might get if you try to prove the correctness of a program for which this disquisition about isomorphisms is relevant?
I'm not sure I follow your confusion. In a strictly mathematical sense, you may have equivalence, but if you publish your paper in the world where electrons have positive charge and everyone else publishes their paper in the world where they have negative charge, well… you can see how this appeal to "Well they’re equivalent either way 🤷♂️" is both technically correct and also asinine.
But it's not just this sense: in the bool example, there's only one bit of information at play, but say you're talking about something more complicated like graph isomorphism or SAT solving. The representation of the problem matters a lot in practice! You can famously write a SAT solver in a single line of Haskell code, but you can also do it in 60,000+ lines of low-level systems code. I'm sure you can guess which is the useless party trick and which is the industrial-grade SAT solver. But even more relevant is that the technical curiosity is, in a strictly mathematical sense, more correct! The industrial-grade SAT solver has a maximum variable count just by the fact that it labels them with 32-bit ints, which is obviously ridiculous by theoretical standards: there’s no magical limit on the number of variables you can have in a SAT problem.
So, let’s say we’re interested in proving the correctness of a SAT solver. I bet there are a fair number of people who’ve played around with Lean and similar systems who would have a fair shot at proving the party trick correct.
There’s not a single person on the planet who stands a shot at proving the industrial SAT solver correct, nor is there likely to be within a thousand years.
And this is how it always is! There is this enormous disconnect between the truly formal representation of a system and "equivalent" (in any colloquial sense) real-world systems anyone cares about, even for systems that are themselves formal fields of study like SAT (much less anything an MBA would consider “real world”, like whether his website is secure).
The reason money has any interest in Lean 4 beyond "the government will give me a tax deduction for my charitable donation" is because we want systems to stop being so broken, and formal verification is supposed to help with that.
My point is just that there are… asterisks. A lot of asterisks.
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