What is creativity? By Gregory Chaitin

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The notion of creativity is central in all your work. What do you understand by it?

Well, I don’t know! I think it’s very interesting to try to understand creativity better. Part of what I understand by it is something that cannot be done mechanically, something that cannot be done automatically or routinely. So, in a way, for me it’s necessarily an incomputable function. So, for example, when Turing talks about the halting problem, he proves there is no general procedure, no general-purpose algorithm, that is, no mechanical way to solve that problem, which means it’s a problem that requires an unlimited amount of creativity. We could describe it using Feyerabend language -Paul Feyerabend never talks about these things, but he has some colorful, strong ways of referring to this- and the title of his book, Against method. That is a theorem of computer science, Turing’s 1936 theorem that there is no algorithm to solve the halting problem. So that means that you need to use different methods. There is no single method that will solve all cases. Similarly, Gödel’s incompleteness theorem shows that, even in elementary arithmetic, there is no axiomatic theory which will answer all possible questions. If there had been, that would give you a mechanical procedure to answer questions in elementary number theory, because you could just run mechanically through all possible proofs in a formal axiomatic theory. So both of these theorems state that there are no general-purpose methods in pure mathematics, that mathematics is rich. My version of incompleteness, the way I put it is pure mathematics has infinite complexity, in a sense that algorithmic information theory defines more precisely. Essentially, I can prove that pure mathematics is infinitely complicated. Any formal axiomatic theory only has finite complexity and, therefore, is incomplete, it cannot encompass all of pure mathematics. Another way to put it is solving the halting problem is infinitely complicated: no single algorithm of finite complexity will work.

You can take this two results by Turing and Gödel pessimistically and say they are a slap in the face of pure mathematics and even of pure thought. But I think the right way to take them is optimistically, the way Emil Post initially took them, and say these results are opening a door in pure mathematics to the very important issue of creativity. They’re saying creativity is essential in fundamental mathematics, it plays a fundamental role, and they’re starting to give us hints about how to understand creativity. Turing has a paper where he talks about oracles, using an oracle for the halting problem. Using an oracle is a little bit like divine inspiration: getting a yes/no answer from an oracle is like one bit of creativity, because the oracle can answer questions that you cannot answer mechanically. I’ve always been fascinated by this question, though I wasn’t working directly on this. But now, with this theory of evolution, I’m actually using incompleteness and incomputability in order to force evolution to go on forever. I need to face my organisms with a challenge that requires an infinite amount of creativity, and pure mathematics, with the work of Gödel, Turing and my own, gives us a mathematical problem that, if used to challenge organisms with it, makes evolution go on indefinitely. It’s a first step, but creativity is a very deep, important question. Certainly, if you look at mathematicians like Euler or Ramanujan, whose creativity seems really breathtaking -especially Euler- it’s hard to think of a rational explanation. Somehow, Euler seems to go straight to the source of new ideas. This may sound a little mystical, but creativity is mysterious. Some mathematicians say that thinking about things which are incomputable is mysticism, but I don’t think so. I disagree: I think you need to go beyond incompleteness and think about creativity. And prove whatever we can prove.

So, in a way, what you’re studying is the emergence of creativity through biological evolution.

Yes, that’s what I’m studying. I’m forcing my organisms to be creative by the fitness measure I take in my metabiological model, which is very simple, it’s just a random walk in software space. It’s a hill-climbing, random walk in fitness space. Another key intellectual issue if you want to come up with a theoretical biology is: what is the space of organisms? What kind of mathematics should we use for the space of all possible organisms? And I think the only space which is rich enough would be the space of all possible algorithms, of all possible computer programs. In population genetics, all we’re looking at is a fixed gene pool and the frequency of each gene in the population, and that’s not a very rich space of possibilities to model evolution and creativity. It models in a very detailed way some aspects of evolution that are very interesting, but it does not deal with creativity and where new genes come from. You have a finite set of genes in that model.

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