There Is No Settled Mathematics

There are two other scientific thinkers that I like who are unrelated to David Deutsch but come to very similar conclusions. One is Nasim Taleb, who's popularized the idea of the black swan, which is that no number of white swans disproves the existence of a black swan. You can never conclusively say all swans are white; you can never establish final truth. All you can do is work with the best explanation you have today, which is still better than ignorance—far better. But at any time, a black swan can show up and disprove your theory, and then you have to go find a better one.

The other fellow who I find fascinating is Gregory Chaitin. He is a mathematician who is very much in the Kurt Gödel vein, where he tries to explore the limits and boundaries of what is possible in mathematics. One of the points that he makes is that Gödel's incompleteness theorem doesn't say that mathematics is junk; it's not a cause for despair. Gödel's incompleteness theorem says that no formal system, including mathematics, can be both complete and correct. Either there are statements that are true that cannot be proven true in the system, or there will be a contradiction somewhere inside the system.

This could be a cause of despair for mathematicians who view mathematics as this abstract, perfect, fully self-contained thing. But Chaitin makes the argument that actually it opens up for creativity in mathematics. It means that even in mathematics, you are always one step away from falsifying something and then finding a better explanation for it. It puts humans and their creativity and their ability to find good explanations back at the core of it.

At some deep level, mathematics is still an art. There are very useful things that come out of mathematics, and you're still building an edifice of knowledge. But there is no such thing as conclusive settled truth; there is no subtle science; there is no settled mathematics. There are good explanations that will be replaced over time with more good explanations that explain more of the world.

This is something that we inherit from our schooling more than anything else. It's part of our academic culture and breeds into the wider culture as well. People have this idea that mathematics is this pristine area of knowledge where what has proved to be true is certainly true. Then you have science, which doesn't give you certain truth, but you can be highly confident in what you discover. You can use experiments to confirm that what you're saying appears to be correct, but you might be wrong.

And then, of course, there's philosophy, which is a mere matter of opinion. This is the hierarchy that some people inherit from school: mathematics are certain, science is almost certain, and the rest of it is more or less a matter of opinion. This is what Deutsch calls the mathematician's misconception; that mathematicians have this intuitive way of realizing that their proof, their theorem, that they've reached by this method of proof is absolutely certainly true. In fact, it's a confusion between the subject matter and our knowledge of the subject matter.