There Is No Settled Mathematics

Proofs are not certainties

Naval:
There are two other scientific thinkers who I like who come to similar conclusions as Deutsch.

One is Nassim Taleb, who 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 a final truth. All you can do is work with the best explanation you have today, which is still far better than ignorance. 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 one I find fascinating is Gregory Chaitin. He is a mathematician very much in the vein of Kurt Gödel because he explores 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; the theorem isn’t 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 bid to find good explanations back at the core of it.

At some deep level, mathematics is still an art. Of course, very useful things come out of mathematics. You’re still building an edifice of knowledge, but there is no such thing as a conclusive, settled truth. There is no settled 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.

Brett:
This is something that we inherit from our schooling more than anything else. It’s part of our academic culture, and it bleeds into the wider culture as well. People have this idea that mathematics is this pristine area of knowledge where what is 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 is 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. Mathematicians have this intuitive way of realizing that their proof—the theorem they have reached by this method of proof—is absolutely, certainly true.

In fact, it’s a confusion between the subject matter and their knowledge of the subject matter.