Joel David Hamkins: commitment

Transcript context

…This minefield of paradoxes. Yeah. Right. A minefield. That’s a really good way of describing the situation. And so Hilbert said, “Well, look, we have to fix this problem, you know. We want to use the set theory foundations, but we want to do it in a way that is trustworthy and reliable. We can’t allow that the foundations of mathematics are in question, you know.” This is a kind of attitude, I think, that underlies Hilbert and the Hilbert program. And so he proposed, “Look, we’re going to have this strong theory, this set theory that we want to be proving our theorems in. But on the one hand, we want it to be as strong as possible. We would like it to answer all the questions. ” There’s another famous quote of Hilbert in his retirement address where he proclaims, “Wir müssen wissen, wir werden wissen,” so, “We must know, we will know,” in which he’s very optimistic about the ability of mathematics to answer all of the questions of mathematics that we have posed. We have all these problems we want to solve, and he is saying, “We’re going to do it. We’re going to solve all these problems.” So we want to propose this strong theory, and one has the sense that he had in mind set theory in which all the questions are going to be answered. Okay? But secondly, we want to combine that with a very weak arithmetic, purely finitistic theory; we want to prove that the reasoning process of the strong theory is safe. Okay? So in order to make sense of that point of view, you basically have to invent the philosophy of formalism where we can look at what a proof is, what is the nature of mathematical reasoning. And on Hilbert’s way of thinking about this, a proof is basically itself a finitistic kind of object. It’s a sequence of… If you think about the nature of what a proof is, it’s a sequence of assertions which can be viewed as sort of sequences of symbols that conform with certain rules of logical reasoning. And this is a formalist way of understanding the nature of proof. So we think about a proof in a kind of syntactic, formal way. Even though the contents of those statements might be referring to infinite uncountable objects, the statements themselves are not infinite uncountable objects. The statements themselves are just finite sequences of symbols. So when you kind of think of proof as… Maybe it’s fair to say almost, like, outside of math? It’s, like, tools operating on math. And then for Hilbert, he thought proof is inside the axiomatic system. Something like this.…

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