Joel David Hamkins: belief

Transcript context

…So, right. So Hilbert had introduced at his famous address at the turn of the century this list of problems that he thought could guide or were important to consider in the coming century of mathematics. I mean, that’s how people talk about it now, although I’m not sure at all… Of course, I can’t really speak for Hilbert at all, but if you were a very prominent mathematician, I find it a little hard to believe that Hilbert would have conceived of his list in the same way that we now take his lists. I mean, having observed the century unfold, we know that that list of 23 problems did in fact guide whole research programs, and it was extremely important and influential. But at the time, Hilbert would have no reason to think that that would be true, and he was just giving a lecture and had a list of problems that he thought were very important. And so I would find it more reasonable to think that he was just making a list of problems that he thought were extremely interesting and important and fundamental in a way without the heavy burden of guiding this 20th century research. Although it turns out that, in fact, that’s exactly what they did. And we already discussed Hilbert’s views on the nature of set theory and the fundamental character, that quote where he said, “No one will cast us from the paradise that Cantor has created for us.” So I think Hilbert was convinced by Cantor on the importance and the fundamental nature of the continuum hypothesis for the foundations of mathematics, which was a critically important development for the unity of mathematics. I mean, before set theory emerged as a foundation of mathematics, there were… you know, there are different subjects in mathematics. There’s algebra and there’s analysis, real analysis, and topology and geometry, and so on. There’s all these disparate subjects with their own separate axioms, right? And, but sometimes it happens, like when you’re proving, say, the fundamental theorem of algebra, you know, that the complex numbers are an algebraically closed field that you can solve any polynomial equation in. But the proof methods for that theorem come from other parts of mathematics, you know, those topological proofs and so on. And so how does that work? I mean, if you have totally different axiom systems, but you’re using results from one subject in another subject, it’s somehow incoherent unless there’s one underlying subject. So the unity of mathematics was provided by the existence of a mathematical foundation like set theory. And at the time, it was set theory. ’s somehow incoherent unless there’s one underlying subject. So the unity of mathematics was provided by the existence of a mathematical foundation like set theory. And at the time, it was set theory. And so it’s critically important to be able to have a single theory in which one views all of mathematics as taking place to resolve that kind of transfer and borrowing phenomenon that was definitely happening. So that must have been part of Hilbert’s thinking about why it’s so important to have a uniform foundation, and set theory was playing that role at the time. Now, of course, we have other possible foundations coming from category theory or type theory, and there’s univalent foundations now. So there are competing foundations now. There’s no need to just use one set theoretic foundation. Although set theory continues to, in my view, have an extremely successful metamathematical analysis as a foundation, I think is much more successful than set theory for any of those other foundations, but it’s much less amenable to things like computer proof and so on, which is part of the motivation to find these alternative foundations. So, yeah, just to talk about Hilbert, I think he was motivated by the need for unifying foundation of mathematics and set theory was playing that role, and the continuum hypothesis is such a core fundamental question to ask, so it seems quite natural that he would put it on the list. There were other logic-related questions, though, like Hilbert’s tenth problem is also related to logic. This is the question about Diophantine equations, and he asked to provide an algorithm to decide whether a given Diophantine equation has a solution in the integers. So a Diophantine equation is just… I mean, it’s a, maybe a fancy way of talking about something that’s easy to understand, a polynomial equation, except it’s not just one variable, many variables. So you have polynomials in several variables over the integers, and you want to know, can you solve it? So the problem is, as stated by Hilbert, provide an algorithm for answering the question whether a given polynomial equation has a solution in the integers. So he’s sort of presuming that there is an algorithm, but he wants to know what it is. What is the algorithm? But the problem was solved by proving that there is no algorithm. It’s an undecidable problem, like the halting problem. There is no computable procedure that will correctly decide whether a given polynomial equation has a solution in the integers. So that’s quite a remarkable development, I think. So there were also a few other logic-related questions on the list. And so eventually, the continuum hypothesis was shown to be independent from ZFC axioms, as we’ve mentioned. So what… How does that make you feel? What is independence and what does that mean?…

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