Comment by _alternator_
17 hours ago
Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of Navier-Stokes will probably take a similarly long time. In the mean time, it looks like all open problems will be solved (or proved that they can't be solved).
It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.
This reminds me of the time an AI was taught how to play a racing sim game (Gran Turismo if I remember correctly). The AI was able to race its car very well, but it took a lot of risks that a human player probably would not. A human player might be able to copy the approach the AI took, but they would probably crash.
Going back to chess, I think the situation is similar where you can’t expect an amateur player to get better by trying to play like a strong engine. I think even professional chess players mainly use engines to prepare or memorize variations that are counterintuitive for their opponent. In other words, getting into situations that look wild, but that part of one player’s preparation.
I’m not sure how it is in math, but in chess, it seems like top players can play just like engines when they are in “normal” positions, so that is where I get a bit confused as to where the direction of insight is coming from because it’s been my view that AI is able to make leaps that we would never think of taking and I’m not sure that anyone could actually learn how to do that on their own unless they were willing to keep failing over and over.
> The AI was able to race its car very well, but it took a lot of risks that a human player probably would not.
There is a parallel with autonomous vehicles in real life. On northbound 1 in SF going through GG Park, the left turn lane onto Crossover Drive is always backed up. Waymos often do a very late merge into that turn lane in order to jump the queue and save time. With 360 degree sensing they can do this safely in real time but it feels too risky for most humans to attempt.
Well maybe its time to pivot from mathematics, and science as whole from personal attribution to being about progress of the field? Maybe your contribution to humanity as a mathematician is to find the right meaningful question to ask, and not to stamp your name on some fact?
This is the direction of experimental particle physics and observational astronomy, where the budgetary scale at which progress occurs means we now fund these efforts at a societal level. These fields have graduated beyond "tabletop science".
For 3000 years mathematics has only been a "tabletop science". Even big programs like the classification of finite simple groups have been comprised of small teams chipping away at different (publishable) parts of an overall program.
This latest Navier-Stokes advance cost something like $22m in tokens, already well beyond what a mathematician's research grant can fund. As the easy open problems get mined, the cost of frontier progress will continue to climb. Some part of mathematics as a field will need to transition from tabletop science to big science: Coordinated top-down programs addressing high-priority objectives.
TBD is what the role of individual mathematicians will look like in a "big science" paradigm, but we could look to experimental high energy physics for ideas. For all practical purposes, AI converts math from a theoretical field into an experimental/observational one.
Yes. But this is hard for mathematicians to stomach, because like everyone else, deep down in a place where they don't like to talk about at parties, they have egos and a sense of purpose based in part on demonstrating mastery of a technically difficult field, as well as social connections based on their participation in it, and taking all that away from them probably feels like a kind of death.
The situation is not that different from John Henry competing against the machine. The question is really: Do people deserve to be allowed to continue doing what they have always done, when doing it is no longer necessary to advance the greater good?
I totally understand and agree, I just feel like with the progress that we have in automating informational work, you are going to have an exponential amount of these "deaths" as new fields where humans can provide any kind of value get more and more short-lived.
At some point in my suggestion the machine will ask better questions than you, and that will be pointless as well, and you keep doing what you like doing, or you move on to something new. But if you keep tying your value to outcome and recognition instead of process you are going to have some incredibly depressing years ahead, and every time will just be as hard to stomach because of your ego.
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So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before?
Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
If this is V0.5 of AGI/ASI then by V1 the only system that will be understanding any of this is the AI itself. If AI creates a new field of mathematics month 1, then solutions to new problems in month 2, then another field of mathematics on top of that at month 3 there's no human who will ever keep up with that.
Or the alternative is a flattening of abilities, the AI cannot proceed further than the collective intelligence of humans and in that case this is correct. We'd be in a future where nobody wants to work in a field with an AI dominating it and when AI hits the limit of no useful training data input we'd have this giant gap of nobody know wtf it's done for years and nobody willing to figure it out and advance it.
Ooo here's a dytopian story: - AI gets better at everything humans do - humans stop trying - AI cannot improve anymore than its input data + human support - AI slowly degrades itself (model collapse) for decades, it slowly hallucinates little by little until its hallucinating entire scientific fields losing quality over time - there's a mass population of people in the future who never learned to do anything and now have to relearn and figure out the equivalent of 100k years of AI work in order to prevent its slow degredation while all the systems they've come to rely on start failing around them. The AI has solved every problem but every real solution is saturated with 1000 false ones. - humanity starts from scratch?
I love the idea of an archive of every solution to every problem existing but it's impossible to figure out the correct one. Infinite library like!
Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor.
It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even more useful to have mathematicians available who already understand that specific math.
In that case isn't it actually more valuable to have an AI do this? It works faster and solves more math.
The random engineer looking at a funny problem 10 years later now has the literal author of the math to talk to about it and implement it.
I have never even spoken to a world class mathematician and now I can have them design with me?
How is this not better in almost everyway?
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there's a book I read "The Practice Effect" such that technology becomes super advanced based on using something, it gets better and better, but the people regress and become more like a medieval society as they just care that using things improves them.
I think this is how Asgard worked.
The tech got so advanced people kind of used the advanced tech, but looked pretty much like a how medieval society would. Even their politics.
Fun suggestion, thanks!
> Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
Most modern mathematical problems are sufficiently abstract that their proofs or disproofs have no direct application. There's no problem you can fix or invention you can build based solely on OpenAI's construction, because analytic solutions to the Navier-Stokes equations are not used for practical purposes in fluid dynamics. The problems and their proofs are only interesting to the degree that they help us better understand how the math works.
IIUC the Navier-Stokes proof is understandable by human beings, but if it weren't it would be no more useful than a proof that 3 dimensional florg-complete entry seams have no durdle-nodes.
So...why can't an AI do the exact same thing? Make AI so it understands math better for future math to understand more math.
Unless your argument is that mathematicians are effectively useless?
I am assuming that's not your point though.
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