A few days ago I read this post on dvzer’s blog, about the recent AI advances in mathematics.
He describes three ways of solving a mathematical problem: brute-forcing it, developing just enough understanding to find a solution, or understanding it deeply enough that the solution becomes almost obvious. All three may produce the correct answer, but only the last one really makes you a better mathematician.
That stuck with me in some amorphous form, although I didn’t have anything specific to say at the time.
I sometimes find myself in a similar situation at work. There’s a problem to solve, not nearly enough time, and I manage to get something working without fully understanding all the details. Then a colleague asks me to explain something about the solution, and I feel a bit like an impostor. Or a plagiarist, even though I haven’t actually plagiarized anything (other than the man pages) 🙂
Of course, I’d prefer to understand everything thoroughly, be able to generalize what I’ve learned, reflect on it, perhaps even come up with some improvements. But the business reality is that there simply isn’t enough time. The thing works, there are other things to do, and I move on.
Yesterday I stumbled upon [this article in The Decoder](https://the-decoder.com/mathematicians-say-openai-doesnt-understand-its-own-navier-stokes-proof/), about OpenAI’s mathematical proofs and the criticism from mathematicians who question the methodology and how much actual understanding went into them.
These are people much smarter than me, at least when it comes to mathematics, and they may well be right.
But I keep thinking we’re making a mistake if we judge AI only by what it can do today, and implicitly assume that’s approximately what it will always be able to do.
The GPT models of 2023 couldn’t have produced these proofs. Or done many of the rather impressive programming and other tasks that current models and agents routinely handle. Apparently, “never” can sometimes be as short as three years.
There’s also a somewhat different question: what if AI eventually discovers things that humans simply cannot understand?
Why should we assume that the human brain is capable of understanding everything, given enough education and effort? We readily accept our physical limitations. No amount of training is going to make me an Olympic sprinter. And probably no amount of studying will make me a world-class mathematician either.
We’re also quite comfortable with machines being better than us at various things. Tractors, industrial machinery, calculators, computers… The mechanization of agriculture helped bring its share of employment down to 2% over the 20th century, while producing much more food.
But those machines were mostly specialized tools. The tractor doesn’t decide what to grow. The calculator doesn’t invent new mathematics.
AI is potentially different. It’s much broader, and perhaps eventually much deeper. What happens if it becomes better than us not just at solving problems, but also at deciding which problems are worth solving, developing new theories, and making discoveries?
And what if some of those discoveries are beyond the cognitive capabilities of even the smartest humans?
I can certainly see the problem: how do we verify something that nobody understands? What happens to scientific research, and to the people doing it? How do we retain control over something that may eventually understand the world much better than we do?
I don’t have answers to those questions.
But I also don’t see why we should expect human intelligence to remain the upper limit of what can be understood (is that even the right word?). Or why the shortcomings of today’s AI should tell us very much about what it will be capable of three years from now.
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