Ard: Yesterday we talked to Marcelo Gleiser and he talked about the idea of knowledge like an island. So as you grow… an island in a sea of ignorance. So as knowledge grows, so does the size of the border that you have of the ignorance that you see. So as you get more and more knowledge, you also see more and more ignorance.
GC: That’s a very nice image. Also people don’t like talking about what they don’t know. They like talking about what they know. I’m the other way around. I prefer thinking about what I don’t know.
Certainty is bad because it’s uncreative. It means you know already – you don’t need to think any more about it. Well it’s also totally uncreative in mathematics. The idea of Hilbert was to ensure certainty He thought it was possible: he thought the possibility of doing this is what it meant to say that mathematics was black or white, that mathematical truth is more solid than any empirical truth. And it’s wonderful that mathematics refuted this.
You know, Gödel’s Incompleteness Theorem is suppressed. The mathematics community doesn’t want to take it into account, because they view it as a tremendously pessimistic, horrible fact that you can’t have a ‘theory of everything’ for mathematics, and that mathematics doesn’t give absolute truth. I think this is absolutely wonderful. The viewpoint is wrong. What Gödel’s Theorem is about… it’s not a negative theorem, it’s a positive theorem. It’s about creativity. It’s the first step in the direction of a mathematical theory of creativity – of saying that math is not a closed system, it’s an open system, just like biology. And this is totally liberating and we should all celebrate…. celebrate this fact rather than bemoaning it, beating our breast, ‘Oh my God. What happened to absolute truth in mathematics?’ Well, what happened was that absolute truth was a closed system. It was a prison: the notion of a formal theory that would give you absolute certainty.
Ard: A theory of everything.
GC: A theory of everything. Yes, a formalisation of all of mathematics in one finite set of axioms. This would have been horrifying.
Let’s say that they have this computer program which can decide if mathematical assertions are true or false.
Well, what good is it to know whether something is true or false? You want to understand what’s happening, right?
David: The why rather than the…
GC: The why, exactly. You want to be convinced emotionally that something is true. That’s why new questions are important, because what counts is not the mathematics we know – the science we know is uninteresting – it’s what we don’t know that’s interesting.
Unfortunately universities spend all their time filling your head with what’s known, but that’s totally trivial. What’s interesting is what we don’t know. That’s what all the courses should be about, so that maybe the students can come up with new ideas before they’ve been brainwashed with the current paradigms. That would be the university I would create, you know, which only would talk about what we don’t know because what we know is really very uninteresting.