If a proposition has only intractably large counterexamples, is it effectively true?

Question

Consider something like Goldbach's Conjecture. Suppose we determine that it's semi-decidable (I think that's the term?), such that we can't prove it either way short of computing for an unknown amount of time until we find a counterexample, or forever if no counterexample exists.

Suppose a counterexample exists, but for whatever reason, it's beyond astronomically large. It's in like Busy Beaver territory, utterly impossible to find with any amount of physically possible work.

Suppose after a good effort, we decide GC is almost certainly true and that no counterexample exists. Perhaps assuming it to be true allows for advances in other mathematical fields in unexpected ways (unlikely for GC, but this is illustrative only.)

Which brings me to my question: for absolutely every purpose worth mentioning, couldn't you argue it is true in this case? (The people in the scenario couldn't call it that, as they would have no way of knowing a counterexample wasn't just around the corner, but let's focus on our meta-viewpoint.) It can never be disproven and the counterexample can never be found. This also implies that any unrelated theoretical work relying on its being true cannot run into any contradictions, as that would imply the existence of the counterexample, which we've asserted is impossible to find with any feasible amount of work.

It seems to me like in this case, since it must behave as though it were true for all practical purposes, you might as well just call it true at that point, even if extra-dimensional aliens told you about the counterexample. I ask this both because I'm curious if anyone else shares my point of view and because it seems probable to me that mathematics and computation are trending increasingly towards trying to cope with undecidable or extremely complex problems, and semantic issues like this could be relevant, with some traditionally pure math fields forced to become partially empirical. Indeed, barring a logical blunder on my part, I'm certain there are actual conjectures which meet the criteria I've laid out.

I'll consider this answered if someone can point me towards related materials, points out a blunder, or otherwise contributes insight on this issue.

Answer 1

There is a conjecture: There is no sequence of n consecutive positive integers which contains more primes than the sequence of integers from 2 to n. That's the second Hardy-Littlewood conjecture. And it seems intuitively true, since primes get more and more rare as numbers get larger.

But a lengthy calculation (I mean hours but not days of CPU time) shows that constellations with more primes are possible for some n > 2,200 or so, and the first Hardy-Littlewood conjecture implies that the second one is false. (The existence of such constellations would have been impossible to prove in their time. And the first Hardy-Littlewood conjecture is basically the twin prime theorem on steroids).

Now the problem is that finding a suitable set of primes is basically impossible. I can't remember the details, but you'd have to find over 2,200 primes of several hundred digits at exactly the right distances. It's funny that their first conjecture suggests there is an infinite number of solutions to this problem, when it is absolutely impossible to find just one.

Now I would say that the 2nd conjecture is effectively false.

Answer 2

Definitely not, you could not do this without some sort of extra special bit of reasoning. Just because a counter-example might be incomprehensibly large, and algorithmically impossible to produce, the very fact a counter-example exists out there in the abstract world of Mathematics will have very dire and huge consequences.

For example, the majority of real numbers are incomputable. We have no nice finite algorithmic way of producing them to within arbitrary precision, yet they exist, and assuming they effectively don't exist gives us a very different set (the set of computable numbers). Look here: https://en.wikipedia.org/wiki/Computable_number#Other_properties. You see it says: "The computable real numbers do not share all the properties of the real numbers used in analysis". So even though we cannot algorithmically produce the incomputable numbers, we can easily see the effects of their existence. So the statement

This also implies that any unrelated theoretical work relying on its being true cannot run into any contradictions, as that would imply the existence of the counterexample, which we've asserted is impossible to find with any feasible amount of work.

is definitely completely wrong. We can still run into contradictions even if we cannot feasibly produce the counter-example(s).

EDIT: a ridiculously easier example: there are infinitely many natural numbers. You cannot feasibly write them all down; does that mean for all practical purposes you can assume they are finite? Not at all. It would break Mathematics if you did this.

If you only accept the existence of Mathematical objects within your comprehension and reach, you end up with a Mathematics that looks completely different.

Answer 3

If we interpret "(basic) mathematics" as being a human model of various things about "the real world", there is scant reason to believe that taking that basic math to any logical extreme is necessarily a correct indicator of anything at all.

Nevertheless, "of course", (some) human beings are interested in how their mental models of things play out at ridiculously extreme scales. :)