Michael Dummett writes (page 349)

Since primality is decidable, the statement that any particular natural number is prime must be determinately either true or false, since the decision procedure, if applied to that number, would have a determinate outcome. If we deny this, we shall be forced to repudiate the relatively liberal intuitionist criterion for decidability and be driven back into strict finitism, a doctrine that involves infinite sequences with finite upper bounds on the number of their terms, and that is very dubiously coherent. [my emphasis]

How do proponents of finitism respond to this claim that their position is "dubiously coherent"?

Dummett, M. The logical basis of metaphysics. (1991) Harvard University Press.

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    A well known response took place in an exchange between Harvey Friedman and the strict finitist Yessenin-Volpin. The question is : If 2 is accepted but not 2^100, then where do you draw the line? Friedman starts by asking if 2^1 is valid and VY responds yes. Next, 2^2 and again yes, but with a perceptible delay. Next, 2^3 and again yes but with more delay. This continues a few more times until Friedman realises how VY is responding to his criticism. Sure he was always prepared to answer yes, but he was going to take 2^100 times a long to say yes to 2^100 as he did to say yes to 2^1. – Nick Jul 23 '19 at 22:11
  • @NickR Thanks! I'll look up that exchange between Harvey Friedman and Yessenin-Volpin. – Frank Hubeny Jul 23 '19 at 23:44
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    I do not think that finitists would accept the "infinite sequences with finite upper bounds on the number of their terms" as a fair description. As far as they are concerned, "infinite" is just a manner of speaking. Nor do they have to pony up the "largest" number. Priest gives it a curious spin in What Could the Least Inconsistent Number Be? His paraconsistent arithmetic with finite models has a least number N such that N=N+1, but it makes not a least bit of difference to any practical uses of arithmetic what it is, as long as it is exorbitantly large. – Conifold Jul 24 '19 at 4:55
  • @Conifold I agree that Dummett's phrase "infinite sequence with finite upper bound" seems to be a misrepresentation of finistism. That is one of the reasons why I asked this question - to get more clarity on what finitism actually is, not what the opponents of finitism think it is. Thanks for the reference. – Frank Hubeny Jul 24 '19 at 12:25

There is no practical difference between denying and assuming infinity since we are unable to carry out any practical experiment that would tell us which of the two options is true.

More generally, falsehoods don't make any difference to any practical use as long as nobody in fact uses them.

This is indeed routinely applied in everything we do including science. False theories are regarded as perfectly scientific as long as nobody finds a way of putting any falsehood in them to any practical use. When somebody does, the theory of course is falsified and scientists work to think up another one which would be without at least this particular falsehood. This seems to have worked well so far.

Human beings seem to be finite at least in the quantity of information they can handle and the speed at which they can compute solutions. The logic of the human mind not only is organised to accommodate this fact but it is indeed itself both the result of, and optimal solution to, this fundamental constraint.

Denying or asserting the reality of the infinite are both self-indulgent metaphysical views. Assuming the reality of the infinite should be good enough. It will always work since no one will ever be contradicted by the facts of the matter.

Claiming infinity doesn't exist is just as futile but you also won't risk being contradicted by the real world as long as you refrain from specifying the boundary of the universe.

The only sort of claims that would be falsifiable is for people who believe the universe is finite to make: They could assert what the limit is. Strangely enough, many people claim infinity doesn't exist, but very few of them want to assert what the limit would be.

Many people in the past, broadly before Copernicus and Galileo, would confidently assert that the universe didn't extend beyond "the outermost sphere of the heavens", that is, broadly, the "firmament", or celestial spheres of the planets and stars.

This claim of course was falsified, several times over as people realised, in stages, that the world extended beyond all visible stars, beyond Andromeda, the only galaxy visible, if barely, in the night sky, with the naked eye, and which is already a staggering 2.5 million light-years away from us, and indeed extends well beyond whatever anyone could possibly imagine in that narrow, indeed limited, space of the mind which is available for people to do their imagining in.

Thus, reasonable people have learnt from experience to limit their claims, if not their intimate conviction. They choose to merely assert that the universe is finite without specifying what the limit might be, and thus without risking being contradicted by the fact of the matter.

Perhaps the fundamental question regarding our notion of the infinite is how come, with our very limited brain and very limited mind, we nonetheless came to conceive of the infinite.

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  • Questions about infinity do seem to be impractical and unfalsifiable and so it may not matter whether one assumes the various infinities exist or not. – Frank Hubeny Jul 24 '19 at 12:33
  • I don't agree with this line of reasoning; mathematics is not some kind of empiricism / scientific theory testing. Rather, you put some axioms in the system - and work from there. These do not demand a justification. (Countable) Infinity is an axiom. If you don't have it (ultrafinitism), fine, you can't make some claims, but nothing that was true before becomes /false/. "Lesser" finitisms are largely degenerate - if you have Infinity and Powerset you nec'y have the reals. If you don't have Powerset it's a world of hurt - you have to really change reasoning itself to do anything useful. – BadZen Aug 30 at 22:42
  • @BadZen If you now "can't make some claim", then presumably some implication that was true before now is false, which is why you now can't make this particular claim that the implication is true. And so it is not true that "nothing that was true before becomes /false/". – Speakpigeon Aug 31 at 10:09
  • @BadZen "If you don't have Powerset it's a world of hurt - you have to really change reasoning itself to do anything useful." This sounds interesting but a bit fluffy as it is. So, may be you could flesh out this idea into a full-length answer, if you can be bothered. – Speakpigeon Aug 31 at 10:11
  • I don't think that should be an answer; it literally doesn't answer what OP is asking. Also it's a whole litany / really a different topic. (See victoriagitman.github.io/files/ZFC-.pdf and en.wikipedia.org/wiki/Pocket_set_theory for example, if interested) Re the previous: there is a huge and important difference between a proposition not being /true/ (roughly: provable) and that proposition being /false/ (it's negation is true!) Deleting an independent axiom of your theory - for example Infinity - does make any propositions false - if it did the axiom would be inconsistent. – BadZen Aug 31 at 14:34

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