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The main thing about constructivists is that they do not generally accept proofs where you you assume the negation of what you are want to prove and arrive at a contradiction (Reductio ad absurdum -- though it seems that some cases remain valid in intuitionist/constructivist logic).
As a layman, my gut feeling is that the constructivists would probably accept Euclid's proof of an infinite magnitude of primes, because it gives you a rather concrete algorithm for finding a prime which is not in your list of all primes, but not Hilbert's proof of his Nullstellensatz, because it does not give you a way to determine the algebraic relationship.
WP on constructivism
Moving on further
I am not sure I follow. Suppose you have a program and a claim about that program ("this Turing machine will halt for any input", "This C code will never invoke undefined behavior", etc). In my mind, a proof which is just half a gigabyte of gibberish without rhyme or reason would still be acceptable for such practical problems. If someone formally verified that a given version of the Linux kernel did not suffer from a given class of exploits, I would not complain about them using proof by negation.
By contrast, open problems in mathematics are not things where our main interest is in knowing the answer. Few people are interested in P==NP because they think there is a practical algorithm for solving SAT to be discovered. If an ASI told us what the answer is ("P is equal to NP, but the polynomial has coefficients and exponents so large that your mathematics can't even express them using all the protons in the universe"), that would be of little value. People are interested in these big open questions because their answer sometimes lead to the development of new and exciting branches of mathematics.
The textbook example of a problem which looked promising in that regard until it was proven would be the four color theorem. "So it can just be proven by brute-forcing 1834 configurations with a computer? Seems it was not a nice problem, after all."
With most of our existing software, the problem is even identifying what formal properties we would want, or constructing it so that has the desired properties (and ideally we can easily proof them).
The Hilbert thing is more a sign of the times. Back then, there was a lot of resistance and emotional slapfighting over this, because in ye olden days, people were very concerned about which postulates were True (TM), and viewed any attempt to discuss this as an attack on Truth.
Today, we worry less about this absolute notion of Truth and more about models: is Euclid's parallel postulate True or False? Well, there's actually nothing to fight over: you just get different geometries, but they're all meaningful and useful! So rather than saying "The parallel postulate is true, therefore XYZ", you can just say "When the parallel postulate is true, XYZ holds." Even questioning logical primitives that seem "very true" like "can you appeal to a theorem more than once in the same proof?" turns out to have surprisingly useful implications! For example, if you're modeling a cookie, it makes a lot more sense to say you can only eat it once, rather than you can eat it as many times as you want. It's not wrong; it's just a different thought model. A thought model that, incidentally, is the foundation of a very popular programming language.
One can say "Ok, but what if the postulates imply a contradiction? Surely that's bad, right?" Well... yes, but actually even here there's a lot of subtlety. Not for mathematicians, but for everyone else: see, the trick to defining logical systems that dodge Russell's Paradox is to have a cumulative hierarchy of universes, which is what a lot of dependently-typed proof systems use. But it turns out this is an enormous pain in the ass to work with for writing normal software, to the point that literally nobody does it, and instead we settle for simpler systems that are much more ergonomic to write in yet still, in practice, give a pretty strong (but not rock-solid!) guarantee that you haven't contradicted yourself.
Anyway, constructivism is a more grounded model compared to classical logic, in the sense that it actually computes results, and you can do the legacy thing by just saying "When the law of the excluded middle holds, XYZ."
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