Finitists say that it is impossible to actually have an infinite number of things. Now, either it is logically contradictory that there is an infinite number of things or not. If it is not logically contradictory, then the finitist's absurdity-based arguments are seriously weakened. If it is logically contradictory, on the other hand, then standard mathematics is contradictory, since standard mathematics can prove that there are infinitely many things (e.g., primes). But from the contradictory everything follows ("explosion", logicians call it). So in standard mathematics everything is true, and standard mathematics breaks down.
I suppose the finitist's best bet is to say that an infinite number of things is not logically contradictory, but metaphysically impossible. That requires a careful toning down of some of the arguments for finitism, though.
9 comments:
Can't a finitist be some sort of non-realist about mathematical objects? Perhaps a Pretense Theorist or a Nominalist of some sort? That's William Lane Craig's preferred way of dealing with this problem, as he mentioned at that Baylor conference they had for Alvin Plantinga.
Pretending that a contradiction is true is quite problematic, given explosion.
We are pretending that there are infinite things (though we know there can't be), and extrapolating what would follow (including obvious contradictions like Hilbert's Hotel).
But everything follows from a contradiction, and following is transitive. So there is no task of extrapolating what would follow: we know what would follow--everything.
I don't see how it's a problem if we're making it clear that this is just pretend, and that the mathematical statements are not actually true.... In any case, there are at least a dozen other anti-realist views out there. It would require a rebuttal of all those views to then come back around and say mathematics defeats finitism.
It may be problematic that I take "metaphysical impossibility" in the Swinburnean sense of being "fully-informed logical impossibility", but, given that caveat, I think the stories about Time Travel are also logically contradictory, but that doesn't keep us from pretending. It doesn't keep us from talking about things that would be true if I could travel back in time.
Also, relevance logic is relevant. :-)
Good point, but it would be an unhappy result if relevance logic were needed for doing mathematics.
One quick note on pretense theory: I think it's altogether too quick to say we would immediately entail everything. After all, people often live under the pretense that God does not exist. That may very well be a logical contradiction. Their worldview is, at best, incomplete; at worst, inconsistent/incoherent. But it's still possible to make rational statements about the state of the world sans God, and how to deal with the existential problems inherent in such a worldview, etc etc. The pretense can be upheld, and logical/rational consequences can be spelled out.
Besides, we already know from Gödel that mathematics is either incomplete or inconsistent, right? Not sure if that's totally relevant, but it at least indicates we should confuse doing mathematics with doing philosophy of mathematics.
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