"We’re gonna need a lot more mathematicians."
What the article really says is that we're going to need much smarter mathematicians.
That is not possible for puny meat-brain humans. Humans are close to their ceiling. AIs are just getting started.
In practice, we're probably going to hit that limit first in IC design. I once went to a talk by the Intel engineering manager who headed the Pentium Pro effort. That was the first superscalar x86 CPU, and it took about 5,000 engineers at peak to design it.
Getting that many people coordinated on one thing was a real achievement.
Then Intel stayed with minor tweaks on that design for years.
We're soon going to be seeing designs of even greater complexity cranked out by AIs. No human will understand them at the gate level. Reading AI-written programming language code is bad enough. Reading AI-written Verilog may be beyond human comprehension, except in small sections.
> Humans are close to their ceiling. AIs are just getting started.
The whole point of mathematics is to vastly exceed that natural ceiling by gradually building a framework for understanding. In fact it’s wrong to speak of a ceiling altogether. If there were a ceiling, we’d have hit it long ago.
AIs have already swallowed the entire history of human thought, but apparently they’re ’just getting started’. I can only assume you don’t know what mathematicians actually do.
Abstractions can certainly enable this kind of "telescoping" effect of understanding, but not everything can be compressed with clever abstractions. Some things are inherently incompressible and there is no neat and insightful short explanation for why it is true, just a massive proof, but it may still have provably good properties for building a chip or power plant.
We have naturally only explored the mathematical universe in the parts where telescoping via clever abstractions can get us. But there is much more. Being able to juggle more things in your mind at the same time can have qualitatively massive benefits.
Information theory and proof theory, algorithmic information theory etc has of course explored this.
> Abstractions can certainly enable this kind of "telescoping" effect of understanding, but not everything can be compressed with clever abstractions. Some things are inherently incompressible and there is no neat and insightful short explanation for why it is true, just a massive proof...
Yes. The four-color theorem was the first proof like that. There have been more since, including Fermat's last theorem. Now AI systems are grinding out more.
The proof(s) of FLT aren’t like that of the four-colour map theorem; despite being heavy on prerequisites, they’re conceptual. In fact FLT is a very nice example of this ‘telescoping’ effect.
Ultimately, you could give a few sentences to summarise the proof of FLT as long as one accepts various concepts and their (also deep and conceptual) proofs. In the case of the four-colour theorem this isn’t true because rather than being amenable to summary in words and fancy concepts it’s just a loooooad of cases to be tediously checked.
I can imagine that AI might be able to produce reams and reams of extremely tedious ‘proofs by a million special cases’ that we’d never be able (or even have thought) to do by hand, but that’s not to say there doesn’t exist an alternate, more conceptual explanation that is compressed.
In fact one might argue that facts that are ‘inherently incompressible’ are by definition uninteresting. The profound statements in mathematics are those that appear to be incompressible, yet turn out to be provable in an elegant way by a change in perspective. In this sense, noting the use of the word ‘appear’, it becomes clear that what is and isn’t interesting (and therefore what is and isn’t to be considered trivial) is entirely subjective and hence mathematics is an inescapably human pursuit. But we’ve known this for a long time.
Animats · · focus · HN ↗
What the article really says is that we're going to need much smarter mathematicians. That is not possible for puny meat-brain humans. Humans are close to their ceiling. AIs are just getting started.
In practice, we're probably going to hit that limit first in IC design. I once went to a talk by the Intel engineering manager who headed the Pentium Pro effort. That was the first superscalar x86 CPU, and it took about 5,000 engineers at peak to design it. Getting that many people coordinated on one thing was a real achievement. Then Intel stayed with minor tweaks on that design for years.
We're soon going to be seeing designs of even greater complexity cranked out by AIs. No human will understand them at the gate level. Reading AI-written programming language code is bad enough. Reading AI-written Verilog may be beyond human comprehension, except in small sections.
xanderlewis · · focus · HN ↗
The whole point of mathematics is to vastly exceed that natural ceiling by gradually building a framework for understanding. In fact it’s wrong to speak of a ceiling altogether. If there were a ceiling, we’d have hit it long ago.
AIs have already swallowed the entire history of human thought, but apparently they’re ’just getting started’. I can only assume you don’t know what mathematicians actually do.
bonoboTP · · focus · HN ↗
We have naturally only explored the mathematical universe in the parts where telescoping via clever abstractions can get us. But there is much more. Being able to juggle more things in your mind at the same time can have qualitatively massive benefits.
Information theory and proof theory, algorithmic information theory etc has of course explored this.
Animats · · focus · HN ↗
Yes. The four-color theorem was the first proof like that. There have been more since, including Fermat's last theorem. Now AI systems are grinding out more.
xanderlewis · · focus · HN ↗
Ultimately, you could give a few sentences to summarise the proof of FLT as long as one accepts various concepts and their (also deep and conceptual) proofs. In the case of the four-colour theorem this isn’t true because rather than being amenable to summary in words and fancy concepts it’s just a loooooad of cases to be tediously checked.
I can imagine that AI might be able to produce reams and reams of extremely tedious ‘proofs by a million special cases’ that we’d never be able (or even have thought) to do by hand, but that’s not to say there doesn’t exist an alternate, more conceptual explanation that is compressed.
In fact one might argue that facts that are ‘inherently incompressible’ are by definition uninteresting. The profound statements in mathematics are those that appear to be incompressible, yet turn out to be provable in an elegant way by a change in perspective. In this sense, noting the use of the word ‘appear’, it becomes clear that what is and isn’t interesting (and therefore what is and isn’t to be considered trivial) is entirely subjective and hence mathematics is an inescapably human pursuit. But we’ve known this for a long time.