After a discussion with Michael Harris over at the blog about his book Mathematics without apologies, I realized that there is a lot of confusion surrounding the relationship between homotopy type theory and computer formalization — and that moreover, this confusion may be causing people to react negatively to one or the other due to incorrect associations. There are good reasons to be confused, because the relationship is complicated, and various statements by prominent members of both communities about a “revolution” haven’t helped matters. This post and its sequel(s) are my attempt to clear things up.

In this post I will talk mainly about computer formalization of mathematics, independently of homotopy type theory. There are multiple applications of computers to mathematics, and people sometimes confuse computer verification of proofs with computerized automated proof-finding. Homotopy type theory has very little to do with the latter, so henceforth by “formalization” I will mean only the verification of proofs.

This often means the verification of pre-existing proofs, but it is also possible to use a computer to help construct a proof and verify it at the same time. The tools that we use to verify proofs incorporate a small amount of automation, so that this process can sometimes save a small amount of effort over writing a proof by hand first; but at least in the realm of pure mathematics, the automation mainly serves to save us from worrying about details that the computer cares about but that we wouldn’t have worried about in a paper proof anyway.

Computer-verified proof has been going on for decades. Recently it’s garnered a bit more attention, due partly to complete verifications of some “big-name” theorems such as the four-color theorem, the odd-order theorem, and the Kepler conjecture, whose proofs were so long or complicated or automated that reasonable mathematicians might wonder whether there was an error somewhere.

What is the future of computer-verified proof? Is it the future of mathematics? Should we be happy or worried about that prospect? Does it mean that computers will take over mathematics and leave no room for the humans? My personal opinion is that (1) computer-verified proof is only going to get more common and important, but (2) it will be a long time before all mathematics is computer-verified, if indeed that ever happens, and (3) if and when it does happen, it won’t be anything to worry about.

The reason I believe (2) is that my personal experience with computer proof assistants leads me to the conclusion that they are still very far from usable by the average mathematician on a daily basis. Despite all the fancy tools that exist now, verifying a proof with a computer is usually still a lot more work than writing that proof on paper. And that’s after you spend the necessary time and effort learning to use the proof assistant tool, which generally comes with quite a passel of idiosyncracies.

Moreover, in most cases the benefits to verifying a proof with a computer are doubtful. For big theorems that are very long or complicated or automated, so that their authors have a hard time convincing other mathematicians of their correctness by hand, there’s a clear win. (That’s one of the reasons I believe (1), because I believe that proofs of this sort are also going to get more common.) Moreover, a certain kind of mathematician finds proof verification fun and rewarding for its own sake. But for the everyday proof by your average mathematician, which can be read and understood by any other average mathematician, the benefit from sweating long hours to convince a computer of its truth is just not there (yet). That’s why, despite periodic messianic claims from various quarters, you don’t see mathematicians jumping on any bandwagon of proof verification.

Now there would be certain benefits to the mathematical community if all proofs were computer verified. Notably, of course, we could be sure that they were correct. If all submissions to journals were verified first by their authors, then referees could be mostly absolved of checking the correctness of the results and could focus on exposition and interest. (There are still certain things to check, however, such as that the computer formalization does in fact prove what the paper claims that it does.)

I can imagine that once there is a “critical mass” of mathematicians choosing to verify their own results with a computer, journals might start requiring such verification, thereby tipping the balance completely towards all mathematics being verified, in the same way that many journals now require submissions in (La)TeX. However, we’re a long way from that critical mass, because proof assistants have a long way to go before they are as easy to use as TeX. My guess is that it will take about 100 years, if it happens at all. Moreover, it won’t be a “revolution” in the sense of a sudden overthrow of the status quo. Historians might look back on it afterwards and call it a revolution, but at the time when it happens it will feel like a natural and gradual process.

As for (3), for the most part computer verification is just a supplement to ordinary mathematical reasoning. It’s a tool used by human mathematicians. We have to learn to use it correctly, and establish certain conventions, such as the fact that you’ve verified a proof with a computer doesn’t absolve you from explaining it clearly to humans. But I have faith that the mathematical community will be up to that task, and I don’t see any point in worrying right now about things that are so far in the future we can barely imagine them. (And no, I don’t believe in the AI singularity either.)

(Next time: What about HoTT?)