My thoughts on a recent post by Gowers
This is a response to a blog post by Timothy Gowers.
I'm not a mathematician. That is of no importance to anything I have to say here.
What I find incredibly frustrating with his post can be summarized in this one quote:
we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.
He's making not one but two big assumptions in his post. First, LLMs will be able to do everything better than humans (hence the "no longer part of their role to find new proofs of theorems"). Second, he uses the word "urgently", because clearly the total replacement of humans is going to happen in a short amount of time.
Professor Gowers is making a weak argument. Once again, we have an AI-promoting argument based on capability rather than profitability. Do LLMs dominate humans in terms of the economics of mathematics research? You can't just extrapolate from low-hanging fruit to the entire field of mathematics. What will it cost to generate a new branch of mathematics? If it costs $100 billion or $1 trillion, should we still give up on human mathematics? Can all of pure mathematics be turned over to LLMs if we'd need models that can work with a billion tokens of context? What would it cost to train a model with that large of a context window if it were possible? And then what would it cost to use it to do proof after proof?
While I'm in no position to calculate these costs, some estimates have been provided for Navier-Stokes, a relatively straightforward problem partially solved by an LLM. Apparently it cost between $10 million and $40 million. This suggests my concerns with these AI-promoting claims are not just theoretical. I can only imagine what it would cost to completely remove humans from the field of mathematics while replacing everything they can do.
I further imagine that the frontier AI labs only care about the removal of humans part and could care less about what small percentage they're actually able to replace. As they tried to do with programmers, once they've scared everyone away from studying mathematics, the frontier AI labs can charge what they want because they're the only game in town. New programmers are the competitive threat in programming and new mathematicians are the competitive threat in mathematics.
The other not insignificant argument by Gowers is that LLMs will be able to completely replace mathematicians quickly. If it takes 50 years to completely replace mathematicians, we'll still need a couple generations of mathematicians before we get there. The alarmist nature of his post is not warranted. At this time, mathematicians are still needed - maybe even more mathematicians than before, given that when paired with an LLM, they're far more productive than they were before.
I want to be clear that I'm not making a prediction here. I don't know the field well enough to accurately predict what will happen even if it were possible. What I'm objecting to is the hyperpromotion of AI that motivates the post by Gowers. (This is not an attack on Gowers, only his blog post.)