Joel David Hamkins: belief

Transcript context

…And if I may, going to Perplexity’s definition of Hilbert’s program, it was David Hilbert’s early 20th-century project to give all of classical mathematics a completely secure finitary foundation. In essence, the goal was to formalize all of mathematics in precise axiomatic systems and then prove using only very elementary finitary reasoning about symbols that these systems are free of contradiction. Right, exactly right. Let’s imagine what it would be like if he had been right. So we would have this finitary theory, and it would prove that the strong theory was free of contradiction. So we could start enumerating proofs from the strong theory. I mean, right now, we can write a computer program that would systematically generate all possible proofs from a given theory. And so we could have this theorem enumeration machine that just spit out theorems all day long in such a manner that every single theorem would eventually be produced by this device. And so if you had a mathematical question of any kind, you could answer it by just waiting for either the answer to come out yes or the answer to come out no. So the nature of mathematical investigation in Hilbert’s world is one of just turning the crank of the theorem enumeration machine. Devoid of creative thinking or imagination, it’s just getting the answer by rote procedure. So Hilbert, in effect, is telling us, with his program, that the fundamental nature of mathematics is rote computation. I mean, the way I think about the Hilbert program seems extremely attractive in the historical context of being worried about the antinomies, the inconsistencies, and so how can we kind of block them. And so- It seems natural, first of all, to have a strong theory that’s going to answer all the questions, because the idea of logical independence and pervasiveness that we now know exists just wasn’t, you know, there was no known… They didn’t know anything like that happening ever. And so it’s natural to think that it wouldn’t happen, and also that they would be able to guard against this inconsistency. So it seems like the goals of the Hilbert program are quite natural in that historical context. But, you know, when you think a little more about what the nature of it would be like, it shows you this kind of rote procedure. And now you’re saying, well, that doesn’t seem so unlikely maybe, in the light of increasing computer power and so on, it’s actually maybe turning into our everyday experience, where the machines are calculating more and more for us and in a way that could be alarming. Okay, but to talk about the alternative to the Hilbert point of view, I mean, if he’s wrong, then what is the nature of mathematical reality? Well, it would mean that we couldn’t ever maybe, for the first goal, we couldn’t ever write down a theory that answered all the questions. So we would always be in a situation where our best theory, even the infinitary theories, would have questions that they stumble with and are unable to answer. Independence would occur. the questions. So we would always be in a situation where our best theory, even the infinitary theories, would have questions that they stumble with and are unable to answer. Independence would occur. But then also, because of the failure of the second goal, we would also have to be constantly worrying about whether our theories were consistent or not, and we wouldn’t have any truly convincing means of saying that they were free from contradiction. And the fact of Gödel’s Incompleteness Theorem shows that that is exactly the nature of mathematical reality, actually. Those are the two incompleteness theorems. So the first incompleteness theorem says you cannot write down a computably axiomatizable theory that answers all the questions. Every such theory will be incomplete, assuming it includes a certain amount of arithmetic. And secondly, no such theory can ever prove its own consistency. So not only is it the case that the finitary theory can’t prove the consistency of the strong infinitary theory, but even the infinitary theory can’t prove its own consistency, right? That’s the second incompleteness theorem. And so it’s, in that sense, a decisive takedown of the Hilbert program, which is really quite remarkable, the extent to which his theorem just really answered that whole puzzle. It’s quite amazing. There’s another aspect, kind of easy to think about. I mean, if you’re wondering about theories that prove their own consistency, then would you trust a theory that proves of itself that it’s consistent? I mean, that’s like… It’s like the used car salesman telling you, “Oh, I’m trustworthy.” I mean, it’s not a reason to trust the used car salesman, is it? Just because he says that. So similarly, if you have a theory that proves its own consistency, well, I mean, even an inconsistent theory would prove its own consistency. And so it doesn’t seem to be a logical reason to believe in the consistency, if you have a theory that proves itself consistent.…

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