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The difference in syntax between classical and constructivist mathematics is, as far as I've understood, not because constructivists think a well-formed proposition may be untrue and unfalse at the same time. The difference lies in semantics; a constructivist says that mathematical theorems are not proofs, but rather proofs of provability. A constructivist agrees that if you assume a theorem is a proof, then it either proves the statement true, or false. I, a classical mathematician, agree with the constructivist view, IF we view theorems as proofs of provability. I find not accepting LEM as absurd if one views theorems as proofs.

Now, if one is working within a consistent framework, a classical and constructivist theorem that derives a non-negated proposition is, in effect, the same thing. The classical theorem proves the statement is true, and thus not false. The constructivist theorem proves the statement is provable (it can be proven true), and since the framework is consistent, it cannot thus cannot be proven false (LNC).

If one is working within an incomplete framework (which mathematics is, see Gödel), a classical and constructivist theorem deriving a negation does not amount to the same thing. A classical theorem deriving a negated statement proves it; it proves the negation, and thus proves the non-negated statement is false (LEM). A constructivist theorem deriving a negation merely proves that the non-negated statement is not provable. Just because one cannot prove a statement within an incomplete framework, does not mean that statement is false. There are true, yet unprovable statements.

So, assuming my account of this difference between classical and constructivist theorems is true, my following question arises:

If mathematical theorems are merely proofs of provability in constructivist mathematics, what then are proofs? I suspect it has something to do with computation and construction.

user110391
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    Your question shows some misunderstandings. First of all, constructive mathematics does not reject the LEM. Not accepting some principle is different from rejecting it. It is the difference between taking a neutral position and taking the opposite position. Secondly, constructive mathematics is not some definite thing that everybody agrees upon. There are different systems, some of which are inconsistent with classical logic and some which are not. Thirdly, the "explanation" that constructive logic is about provability is a kind of lie-to-children commonly repeated but not strictly correct. – Zhen Lin Sep 22 '22 at 03:21
  • @ZhenLin I was being imprecise with my reject. I simply meant they reject LEM as a necessary truth, but they don't claim that it is false. I have edited my question now to be more precise. Also, how may it not be correct that the difference is that of proof vs provability? – user110391 Sep 22 '22 at 03:25
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    A proof is a proof is a proof. This is the same in any logic. Your claims about negation are particularly confused because of this. It is difficult to have a discussion in the comment box here, let alone in the answer box. I suggest you read some introductory texts in mathematical logic. Even texts not dealing with constructive logic will probably clear up some of your confusion. – Zhen Lin Sep 22 '22 at 03:33
  • @ZhenLin I have read Stanford Encyclopedia of Philosophy's article on constructivist logic. From what I gathered there, and from what other people have said about constructivist, and specifically intuitionistic, logic, is that it deals with provability and not proof. – user110391 Sep 22 '22 at 03:45
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    That is not correct. At least, that is not how mathematicians now think of intuitionistic logic. You can have a look at any number of textbooks or articles dealing with the internal logic of toposes. – Zhen Lin Sep 22 '22 at 03:50
  • @ZhenLin I will check that out, thanks! – user110391 Sep 22 '22 at 03:57

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The difference lies in semantics; a constructivist says that mathematical theorems are not proofs, but rather proofs of provability.

This is not accurate. As far as I know, no one believes that theorems literally are proofs - a theorem is a statement for which there exists a proof, but the theorem is not identical to the proof, and theorems can have many proofs.

Perhaps what you meant is that constructivists view a proof of a theorem merely as a demonstration that the theorem is provable and nothing more. This position is closer to the idea of formalism than constructivism, though the two can overlap. But I don’t want to speculate too much on what you meant since what you said doesn’t make much sense.

A classical theorem deriving a negated statement proves it; it proves the negation, and thus proves the non-negated statement is false (LEM).

LEM has nothing to do with it. In constructive mathematics, any statement which is not true is false. So a constructive proof of a negated statement disproves the non-negated statement. For more elaboration, see my answer here.

A constructivist theorem deriving a negation merely proves that the non-negated statement is not provable.

It is inaccurate, both in constructive and in classical mathematics, to say that a statement is false iff it is unprovable. Gödel’s incompleteness theorems and other related notions like Tarski’s undefinability of truth apply to both constructive and classical theories. Furthermore, any constructive proof of a proposition is also automatically a classical proof of the same proposition. So the idea that a constructive proof “tells us less” about a proposition than a classical proof is wrong; it’s the other way around.

Edit: some systems are classified as “constructive” which are not compatible with LEM. When I say “constructive”, I’m referring to something like constructive first-order logic or IZF, where if we add in LEM, we get the equivalent classical theory. Constructive logic is also compatible with statements that contradict LEM, like the claim that all functions $\mathbb{R} \to \mathbb{R}$ are continuous. A proof using this principle would obviously not be classically valid.

Mark Saving
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    Strictly speaking there are systems falling under the heading of "constructive mathematics" that prove things that are inconsistent with classical mathematics. For example, Brouwer's infamous theorem that all functions are continuous. – Zhen Lin Sep 23 '22 at 02:03
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    @ZhenLin I wish there were better terminology for this. When I say “constructive”, I generally mean some logical system which is consistent with LEM but also satisfies properties like the number existence property and the disjunction property. So constructive mathematics can be extended in several incompatible ways; with Brouwer’s bar induction + continuity principles, with LEM, or with Church’s thesis. I think dividing up these worlds into classical and non-classical is like dividing food into bananas and non-bananas. It makes bananas sound way more important than everything else. – Mark Saving Sep 23 '22 at 03:45
  • Yes, I agree completely. But unfortunately there is no terminological consensus (yet) and expository articles do not make this clear, so outsiders end up confused by apparently contradictory claims about constructive mathematics. – Zhen Lin Sep 23 '22 at 05:29
  • I have used the word theorem incorrectly. I thought it was the proof, and not the proven statement. I will edit my Q so it is correct. "Perhaps what you meant is that constructivists view a proof of a theorem merely as a demonstration that the theorem is provable and nothing more. " Yes, that exactly what I mean. "It is inaccurate, both in constructive and in classical mathematics, to say that a statement is false iff it is unprovable." I know, and I mentioned this in my Q, as an essential detail of how I (mis)understand the difference. – user110391 Sep 23 '22 at 06:12