So claims the title of a recent Nature article, “Why mathematics is set to be revolutionized by AI.” That title brings you in, but the content is much more circumspect.

We learn, for example, that computational methods of sophisticated mathematics (not ChatGPT!) have shown promise in a few areas. The author, Thomas Fink, director of the London Institute for Mathematical Sciences in the UK, emphasizes the fact that conjectures only need a single counterexample to be disproven.

How would AI revolutionize math?

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What Fink seems to envision is this: AI systems spitting out conjecture after conjecture, and rapidly falsifying all the wrong ones with counterexamples. Then, all the human mathematicians need do is sit back and observe the resulting candidate theorems, selecting the wheat from the chaff. Almost a holiday by the seaside, picking pearls off the beach.

What could go wrong? First of all, if you’ve ever used ChatGPT, it is no easy job to sort the wheat from the chaff. As with the sea, there are deeper problems at play. Mathematical questions of novelty, decidability and mere scalability lurk in the depths, nibbling at our theory’s toes. One can almost hear the tense tuba notes warning us of unseen danger…

Well, what does a mathematician do all day, after all? Does she sit there, randomly pinning one mathematical tail on another mathematical donkey until there are enough to suffice for a paper and then punch the clock? As Fink notes, citing A Mathematician’s Apology (1940) by G. H. Hardy (1877–1947), this is hardly the case. A good theorem, one worth a mathematicians salt, opens the door to greater discoveries. “Further up and further in,” as they say. Such theorems are a rarity, requiring great insight, experience and tenacity — though every so often a genius like Euler will just roll them out like a virtuoso.

Bottom line, it is not enough to randomly assemble “mathenese” and hope for the best. There is not just a bit of chaff among the wheat, but an exponential plethora of chaff to be sifted. A good theorem requires true novelty, and novelty just cannot be created by computers or by any machine for that matter. The mathematicians still have to come up with the good ideas. That AI reality shark nibbling at the toes of the AI math demon has just turned into a chomp!

Still, can’t those workhorse computers at least weed out the bad ideas?

In Fink’s model, just one counterexample is needed, after all…

Yes, only one counterexample is needed. But how vast is the sea in which it is found? Infinitely vast, in fact. It takes a very long time to search an infinite sea for one capsized counterexample. It could take an infinite amount of time if there is no counterexample. Even with a gazillion processors. Even with quantum-fu. We’ll never know, even within the heat death of the universe. This is known as undecidability: “ The undecidability of a problem means that an algorithm is impossible in principle — not only that no algorithm is presently known.” It’s a problem. Chomp! Chomp!

Maybe AI applied to math will be less of a revolution, more like a wobble in a useful direction. Can AI at least wobble the world of mathematics? Here we hit the sorrowful slope of scalability.

The computer world and the scaling problem

Sadly, not a lot of things scale very well in the computer world. This problem is known as NP-Completeness. That is, as we slightly increase a problem’s difficulty, it becomes exponentially, not incrementally, harder.

The Soviets were way ahead of us here because Karl Marx wanted a machine to run the world:

… the means of labour passes through different metamorphoses, whose culmination is the machine, or rather, an automatic system of machinery (system of machinery: the automatic one is merely its most complete, most adequate form, and alone transforms machinery into a system), set in motion by an automaton, a moving power that moves itself; this automaton consisting of numerous mechanical and intellectual organs, so that the workers themselves are cast merely as its conscious linkages

Karl Marx, Grundrisse: Notebook VI / VII – The Chapter on Capital, 1857

Sounds frighteningly familiar. Silicon valley, anyone?

Leonid Levin, Soviet mathematician and discoverer of NP-Completeness, escaped to the United States and let us know that communism faced a technical problem similar to AI: Communism, like the computer world, does not scale. The effort to control all of society, suppressing private innovation, leads to large numbers of undecideable problems which cripple productivity.

Mathematics is not immune to the scaling problem

Mathematics is even more impacted by scalability issues because, while economies consist of things that can be counted by a computer as specific numerical quantities, mathematics consists of concepts that span infinite things. So, if computers cannot count mere things quickly enough for our purposes, they certainly cannot count infinities of things quickly enough.

Chomp! Chomp! Chomp! Burp…

As the red trail of AI’s mathematical ambitions seeps into the sunset amid vigorous thrashing in the water, it is at least a small consolation that a mathematician’s true job of discovering novel theorems is safe forever more.