When is it meaningful to say that an undecided conjecture is true or false?

Question

I see that other questions have already been asked about mathematical truth but here I want to ask clarifications on a particular perspective.

One can think the answer to the question could be "when in principle there is a way to prove or disprove". This could be good for very simple mathematical statements, like "217*29=6293" or "77647 is prime". But many mathematical statements are more complicated and a such simple criterion doesn't work. One can think that a statement can be considered true or false if its truth or falsity can somehow affect the world we live in (this definition probably meets the taste of physicists), but I fear things are even more subtle.

There are statements that we don't know if they are true or false, that maybe are not provable by principle (so we include the possibility that truth of falsity cannot affect in any way our world), but that surely are true or false. To be more concrete, let's think to the statement "every even integer greater than 2 can be expressed as the sum of two primes" (Goldbach's conjecture).

If, by principle, there would be no way to know if the Goldbach conjecture is true or is false, then the conjecture would be true: in this case we will never know the true about the conjecture but in this case too we would know for certain that the conjecture is true or is false: a third possibility is not allowed: a mathematical true does exists. We could build a table like that (where X mark a possibility)

In other words, if the conjecture were false it would be provable by principle (at worst, by resorting to brute force of trivial calculations, no matter if maybe impossible in practice). But in some case the possibility that some mathematical statements are not provable, became really critical about the meaning of the very idea of true or false about a mathematical statement (at least I think so, I'm asking about that): think to another conceptual simple but unsolved problem: "there are infinite twin prime" (twin prime conjecture).

To solve this problem we need a creative proof, brute force is useless (maybe it is useful here or there in developing a creative proof, but this is not the point here). Now, who assures us that a creative proof of this problem exists? And if this were not the case, in what sense the true or false sets by which we suppose that well posed mathematical statements are divided, would be applicable to this? I wonder if the statement "the twin prime conjecture is true or it is false", in the current state of knowledge, is meaningless or not. We could be tempted to build a table like that

But this make sense? Wouldn't it be more correct filing in this way?

In other words if we suppose that mathematical statements are true or false, we should suppose too that:

• or they are provable
• or, if they are not provable, only a possibility about truth is open

If we don't know if a mathematical statement is provable, and if the unprovability were compatible with both options about the truth, would it have any meaning to claim "that statement is true or is false"?

Answer 1

Wittgenstein discusses at length the meaning of unproved conjectures, and of the status of their truth value, during his middle period in Philosophical Grammar and Philosophical Remarks, and during the late period in Lectures on the Foundations of Mathematics and Remarks on the Foundations of Mathematics. Although much in his views changes, he basically comes to the conclusion presented in the OP: the talk of truth/falsity of (typical) unproven conjectures lacks sense. At least, unless there is an algorithm that can be used to "work them out", a "decision procedure". Essentially, Wittgenstein rejects the platonist analogy between mathematics and physics, where we pretend to have a God's eye for the infinite and interpret unproven conjectures "by analogy" to hypotheses about physical reality, and mathematical research "by analogy", often invoked, to geographical discovery of new landscapes.

Panjvani calls this the "no-conjecture thesis", and discusses it at length in Wittgenstein and Strong Mathematical Verificationism, as well as how it evolves from the middle to the late period. For the late view see Empirical Regularities in Wittgenstein's Philosophy of Mathematics by Steiner and Dawson's dissertation Leaving Mathematics As It Is: Wittgenstein’s Later Philosophy of Mathematics.

According to Wittgenstein, any "creative" proof (not a routine calculation whose steps are known prior to being actually carried out) changes the mathematical system it is given in, establishes new conceptual connections and hence alters its rules of use. And, in contrast to physics, mathematical sentences are "grammatical", rule-making, not empirical, so it is the system that gives sense to them. Therefore, saying that the conjecture is true or false prior to such a proof being given appeals to a yet non-existent sense, a nonsense.

The sentences like the Goldbach conjecture, where there is a decision procedure in case of falsity only, are special. But even with them the meaning of their truth, as established by a proof, will be different from what truth/falsity could mean before it. And in the case of the twin prime conjecture it is altogether lacking in advance. Here is from Wittgenstein's discussion of the Goldbach conjecture from Philosophical Grammar, ch. 5:

"Only the so-called proof establishes any connection between the hypothesis and the primes as such. And that is shown by the fact that... until then the hypothesis can be construed as one belonging purely to physics. - On the other hand when we have supplied a proof, it doesn't prove what was conjectured at all, since I can't conjecture to infinity. I can only conjecture what can be confirmed, but experience can only confirm a finite number of conjectures, and you can't conjecture the proof until you've got it, and not then either.

[...] Suppose someone was investigating even numbers to see if they confirmed Goldbach's conjecture. Suppose he expressed the conjecture - and it can be expressed - that if he continued with this investigation, he would never meet a counter-example as long as he lived. If a proof of the theorem is then discovered, will it also be a proof of the man's conjecture? How is that possible?"

In other words, what "the man" had in mind when he conjectured that even numbers are sums of two primes was some sort of physical prediction requiring infinitely many checks. And confirming such a thing is not humanly possible. Even if a "creative" proof "by principle" is then discovered it will not be "confirming" this at all. It will instead establish a new connection between the conjecture and the primes, so it will prove something else, something without a sense at the time of making the conjecture.

"a mathematical proof incorporates the mathematical proposition into a new calculus, and alters its position in mathematics. The proposition with its proof doesn't belong to the same category as the proposition without the proof. [...] So if I want to raise a question which won't depend on the truth of the proposition, I have to speak of checking its truth, not of proving or disproving it. The method of checking the truth corresponds to the sense of a mathematical proposition. If it's impossible to speak of such a check, then the analogy between 'mathematical proposition' and other things we call propositions collapses.".

Things are even worse with the twin prime conjecture where no sense of "checking" can be attached in advance. Even if it is false, how would we check (non-"creatively") that the last pair of twins is the last?

"How strange it would be if a geographical expedition were uncertain whether it had a goal, and so whether it had any route whatsoever. We can't imagine such a thing, it's nonsense. But this is precisely what it is like in a mathematical expedition. And so perhaps it is a good idea to drop the comparison altogether."

I will not go into the changes that Wittgenstein's thought underwent in the late period, and how it altered his view of unproven conjectures. In a sentence, he came to associate the sense of mathematical propositions not with a rigid calculus and decision procedures but rather with looser "language games", mathematical practices. But, suffice it to say, the conclusion about the lack or change of meaning of unproven conjectures after a proof remained in place. As well as the anti-platonist idea that their truth is not "determined" in advance by "mathematical facts". Except, perhaps, in an empirical sense of what future mathematicians are likely to accept for compelling, but ultimately empirical and/or pragmatic reasons. Here is again on Goldbach from Lectures on the Foundations of Mathematics (XIV):

"Suppose someone said, "What you, Wittgenstein, say comes to saying we could also extend arithmetic in such a way as to prove I this is not so, or to make it a primitive proposition." I'd say: certainly. Because of course you haven't yet made this extension. The road is not yet actually built. You could if you wished assume it isn't so. You would get into an awful mess.

[...] If we adopt the idea that you can continue the road either in this way or in that way (Goldbach's theorem true or not true) - then a hunch that it will be proved true is a hunch that people will find it the only way of proceeding. Though before anyone has found a proof we could say, "If someone has found a proof we have a perfect right not to acknowledge it"."

Answer 2

We have much more concrete examples than the ones you have in mind. The Continuum Hypothesis, the Determinacy of Infinite Games, and various Large Cardinal Propositions have been proved independent of the remainder of set theory. That means that we can make them true, adding them as axioms to our existing theory, and there would be no more contradictions in our theory than if we make them false. There are proper models of set theory both with and without these propositions as axioms, so they cannot be proved either true or false.

So we do need the category you propose, and we have it. Something that cannot be proved true nor proved false in a mathematical theory is 'undecidable'. It is possible that questions like Goldbach's conjecture or the infinitude of twin primes are undecidable in arithmetic in the same way that these propositions about sets are undecidable in the theory of sets.

From this perspective, the answer to the title question is that a given proposition is not necessarily either true or false. It is true if proven, or false if it leads to a contradiction. But in some cases, it can instead be proved that neither is the case. Since we have actual examples of such statements, you could say (along with Intuitionists and Constructivists) that the classical Law of the Excluded Middle does not hold: that it is not true that for every mathematical statement, it is either true or false.