The Methods of Mathematics Are Fallible

Even if the subject matter is not

Brett: If I compare math to physics: We have this domain called particle physics, and the deepest theory we have in particle physics is called the standard model. This describes all of the fundamental particles that exist and the interactions between them, the forces that exist between them, and the gauge bosons, which mediate the force between particles like electrons, protons and neutrons.

Now, what is matter made of? We would say matter is made of these particles described by the standard model of physics. But does that rule out the fact that these fundamental particles might themselves consist of even smaller particles? We have a possibly deeper theory called string theory. So our knowledge of what the most fundamental particles are and what, in reality, the most fundamental particles are, is different.

So, too in mathematics. Deutsch explains that mathematics is a field where what we’re trying to uncover is necessary truth. The subject matter of mathematics is necessary truth, in the same way that the subject matter of particle physics is the fundamental particles. 

But since the subject matter of fundamental particle physics is the fundamental particles, that doesn’t mean you actually find the fundamental particles. All it means is that you have found the smallest particles that your biggest particle accelerators are able to resolve.

But if you had an even bigger particle accelerator, you might find particles within those particles. 

This has been the history of particle physics. We used to think that atoms were fundamental. Then, of course, we found they contained nuclei and electrons. In the nuclei, we found out that there were protons and neutrons. Inside the protons and neutrons, we found out they were made up of quarks. And that’s where we’re at right now. We’re at the point where we say that quarks are fundamental and electrons and fundamental.

But that doesn’t mean that we’re going to end particle physics right now. What we need are further theories about what might be inside of those really small particles. 

Comparing that to mathematics, if necessary truth is the subject matter of mathematics, mathematicians are engaged in creating knowledge about necessary truth. Because a mathematician has a brain—which is a physical object—and all physical objects are subject to making errors of degradation via the second law of thermodynamics—or simply the usual mental mistakes and errors that any human being makes—a mathematician is just as fallible as anyone else. So what they end up proving could be in error.

Naval: If I understand this point, even mathematics is capable of error because mathematics is a creative act. We’re never quite done. There could have been a mistake in your axiom somewhere.