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PaperclipPerfector  6d ago (text post)   19565 thread views

Culture War Roundup for the week of May 25, 2026

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I still can't make sense of what you are trying to say then. What is the sense in which you claim proofs to be "interchangeable" in classical systems, why do you believe this to be a desirable property, and how do you contend with the fact that constructive proofs are classical proofs?

There may be some HoTT-like sense in which two proofs in a given system are not equivalent (because you introduced some equivalence relation on proofs that does not relate them). If you have a sound embedding from proofs in this system to proofs in another system, and the other system comes with its own equivalence relation on proofs, the images of your two proofs under the embedding might still be equivalent in the other system. Is this not just what would happen here, if you assert that all classical proofs of a given sentence are equivalent (under an equivalence relation you picked) but not all constructive ones are (under an equivalence relation you picked)?

This is doubly-so in cases like cryptography, where almost none of the "incorrectness" of compromised crypto is incorrect at all in the classical mathematical sense.

You will have to elaborate on this statement. What do you take to be an instance of compromised crypto that is not "incorrect in the classical mathematical sense"?

Well, in the classical mathematical sense, crypto doesn’t work at all: just factor the composite number. The entire premise relies on the relatively ill-specified (by mathematical standards) notion of relative computational cost disparity.

What is the sense in which you claim proofs to be "interchangeable" in classical systems

What I mean is when I sit down to write something in a theorem prover, I speak in terms of Peano nats and inductive lists. When I write software anyone would ever want to actually use, I use machine ints and arrays. There is definitely a sense in which I’m doing the same thing in both cases, but nailing this down precisely is… well, non-trivial. (aka, a royal pain in the ass). Like, the discrepancy here is show-stoppingly problematic.

EDIT: basically, the computation is relevant, and more specifically the speed of computation is also relevant, both in terms of practical usability and outright security in the case of crypto. In the classical world, there is no notion of computational relevance at all -- in fact, you outright end up with these counterintuitive weirdo theorems like Zorn's Lemma. The counterintuitiveness depends entirely on this conspicuous lack of computational awareness in the model: as soon as you put that ingredient back in, you're in the construcivist world, and silly results like Zorn's Lemma don't hold anymore unless you postulate them. My contention is the class of problems for which one can say "I don't care about which proof, just that one exists" is relatively narrow and of little practical relevance (in the sense that one can get an incomputable proof, which is, in a fundamental technical sense, useless), while the class of problems about which one does care which proof, in the Curry-Howard sense, is large and of enormous practical consequence. I want to argue that enthusiasm over the former should not translate to enthusiasm about the latter.

I think he means it in the sense that "a nice man on the phone told me he was from BitPanda and asked me to read my password to him so he could check if it was secure" or "SBF stole all my money" can't be detected by enforcing code correctness. All of the badness has happened outside the code, everything inside the code is a perfectly valid transaction.

But this general argument (ChatGPT can claim something in English, then formally prove something completely different) still applies if the proofs are classical.

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