[shalmanese]

> But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.

AI is going to both help and hinder this process though. At the end of the day, mathematics is mostly a social process at this point. The goal is not raw number of theorems proven, it’s how proving theorems affects the working operational models of mathematicians. Only a rare few new theorems in mathematics nowadays have direct real world applicability.

If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.

[xyzzy123]

Weird question, do you think AIs might prove a lot of theorems that are mainly useful to other AIs (i.e, make nearly no impact on the human culture of working mathematicians), which then get used to prove results that humans do actually care about?

It seems like if AIs can prove and index a huge number of (largely uninteresting to humans) things there might be sort of "parallel cultures"? Big results are most valuable to humans and AIs both (most context efficient!), but a very large number of less general but still non-obvious results might be an effective approach to solving problems?

[dyauspitr]

> Only a rare few new theorems in mathematics nowadays have direct real world applicability.

I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe.

> If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.

I think the idea here is that all mathematicians will just be using AI for their future work so they don’t really have to absorb it as long as it’s in the training data.

[mkl]

> > Only a rare few new theorems in mathematics nowadays have direct real world applicability.

> I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe.

I am a mathematician. It is true. The key is we're talking about new theorems, and direct, current real world applicability. Some theorems that have no applicability now may in the future, as theory often precedes applications by a long way and the usefulness is likely to come from other things built on top of the new maths, and a lot of pure maths will never have direct real world applications but contributes to our overall understanding.

[adastra22]

The key word in that sentence is “new.” New math is typically explored without expectation of practical use. There are exceptions, but it is generally true.

On the other hand, there are many applied mathematicians and theorists from other fields that mine new maths for applications to their fields. But they are almost always not the ones that come up with the new math.

Historically, of course, mathematics was always driven by the need to explain things. Many of the mathematicians from the 17th and 18th centuries were physicists (or, less commonly, engineers). But for the last hundred years or so that really hasn’t been the case.

[anonymousDan]

Out of interest, what would you estimate the proportion of new maths that is used by other fields to be? Do you think much of this new maths is potentially underutilised as it were?

[teiferer]

> Only a rare few new theorems in mathematics nowadays have direct real world applicability.

Has this ever been different?

Math is abstract, rightfully so. It does not have to have direct applicability. Understanding builds over time and applications eventually follow. Number theory used to be a fringe "pure" theory field without applications for the longest time. If we'd only be interested in (and thus fund) what has direct applicability then society would be much worse off.

Side note: I recall my high school class mates rolling their eyes in every math class with "when will I ever need this in my life?" never asking the same question about PE or history or art classes. Now they struggle with their tax return and are routinely getting screwed over by loan sharks. But make no mistake, they can be proud of their A for hitting the goal 5 out of 5 times during soccer in PE class.