[1] for those unfamiliar with math lingo, classical logic has a lot of powerful features. One of those is the law of the excluded middle, which says something can't be true and false at the same time. That means not not true is true, which you can't say in intuitionistic logic. Another feature is proof by contradiction, where you can prove something by showing that the alternative is unsound. There's quite a few results that depend on these techniques, so trying to do everything in intuitionistic logic has run into a lot of roadblocks.
That would be the law of non-contradiction (LNC). The law of the excluded middle (LEM) says that for every proposition it is true or its negation is true.
LEM: For all p, p or not p.
LNC: For all p, not (p and not p).
Classical logic satisfies both, intuitionistic logic only satisfies LNC.
The thing is it can be quite useful to always know what a value is, and there's some cool things you can do when you know how to get a value (such as create an algorithm to get said value). I'm still learning this stuff myself, but inuitionistic logic gets you a lot of interesting properties.
Classical logic was rejected in computer science because the non-constructive nature made it inappropriate for an ostensibly constructive domain. While you can implement its evaluation with a machine, it's extremely unwieldy and inefficient, and allows for a level of non-rigor that proves to be a massive footgun.
As far as lean is concerned, this isn't an example of classical logic. It's just the definition of "not" - to say that some proposition implies a contradiction, and to say that that proposition is untrue, are the same statement.
Most "something"s that you'd want to prove this way will require a step from classical logic, but not all of them. (¬p ⟶ F) ⟶ p is classical; (p ⟶ F) ⟶ ¬p is constructive.
For every "well of course, just...X, that's what everybody does" group-think argument there's a cogent case to be made for at least considering the alternatives. Even if you ultimately reject the alternatives and go with the crowd, you will be better off knowing the landscape.
Every time you go off the beaten path, you're locking yourself into less documentation, more bugs (since there's less exploration of the dark corners), and fewer people you can go to for help. If you've got 20+ choices to make, picking the standard option is the right choice on average, so you can just do it and move on. You have finite attention, so doing a research report on every dependency means you're never actually working on the core problem.
The exceptions to this are when a) it becomes apparent that the standard tool doesn't actually fit your use case, or b) the standard tool significantly overlaps the core problem you're trying to solve.
I of course fully support reinstating logicism, but the same dogmatics love putting up a fight over that as well.
What you've done here is an overgeneralization. "People who like math" and "people who like computers" are massive demographics with considerable overlap.
also, https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon... i.e. there's no reason it should be as you say.
For mathematicians a proof is a means to an end, or a medium of expression - they care about what they say and why.
The correspondence isn’t about C programs corresponding to proofs in math papers. It’s a very a specific claim about kinds of formal systems which don’t resemble how math or programming is done.
I am learning Lean myself so forgive me as I have an overly rosy picture of it as a beginner. My current goal is to write and prove the kind of code normal programmers would write, such as in the recent lean-zip example: https://github.com/kiranandcode/lean-zip/blob/master/Zip/Nat...
Then, I foresee 2 other obstacles, 1 of which may disappear:
1. Actually knowing what the software is supposed to do is hard. Understanding what the users actually want to do and what the customers are paying to do --which aren't necessarily the same thing--is complex
2. The quality of the work of many software developers is abysmal and I don't know why they would be better at a truth language than they are at Java or some other language.
Objection 2 may disappear to be replaced with AI systems with the attention to do what needs to be done. So perhaps things will change in that to make it worthwhile...