Yup, I imagine within a year or two OpenAI and Anthropic will have in-house versions of pretty much everything anyone would ever use, with in-house agents hardening and improving it at a torrid pace that the human/code-review/github cycle could never keep up with. Their models will default to using these in-house versions, and be trained in all the idiosyncrasies of those tools and libraries. Their coding agents will phone home with feedback about usability that in-house agents will act on immediately, operations agents will do the same re bugs and runtime performance. I don't really see a how or why a human-run project on open source would even bother anymore.
People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.
Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free).
So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight.
BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.
> Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.
I guess Terence's main point has been all the time that if we let AI solve all these existing problems, we don't notice the new ones and then there is stagnation.
Because the bounded universe creates bounds on what the intelligences within it can conceive of as mathematics.
We can conceive of lots of mathematics that our own universe doesn't necessarily support. But only that much and no more: we're still made of stuff inside the universe, and so are our tools, so there are upper limits on our conception.
Mathematics that our universe does not support is still mathematics. Also you're viewing this through a human lens. How do you know it doesn't support what you think it doesn't support? Mathematics after all is a very human pursuit. Some alien species may have started off with ternary or even continuous logic. We have no counterexamples to show that that is beyond our universe.
Bounds do not imply bounds. There are an infinite number of numbers between zero and one. Heck, with our finite minds in finite time we have created uncountable infinities!
> There probably is a finite amount of math to be done
> The universe is bounded by rules, as far as we can tell, and not a lot of them
Respectfully, this is not a useful frame for the discussion. Nobody is expecting to reach the limit you have noted either with or without the assistance of LLMs. So there is always more math that could be done.
The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.
At those levels math and physics are the same bound: the bound of things the universe allows to be conceived of inside it.
It’s possible the math our monkey brains + sand can ever conceive of in this universe is a low and accessible amount.
> The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.
When you have a moment, please reference some proofs supporting this.
There are lots of thoughts you’re not biologically capable of thinking. That means the list of thoughts you’re capable of thinking is finite. That means the amount of math you’re capable of discovering, even if you lived forever, is finite.
That’s true of all humans and all constructs.
So however much math there is to discover, that’s the finite subset you’ll ever have access to.
It’s hard to prove what thoughts no human and no construct is capable of generating, but surely there are some, and it’s possible or even likely some of those are math-related.
Even if we accept that there are "lots of thoughts" we're not capable of thinking (though I wonder how you would define a thought if not as something that you can think), it still does not follow that the "amount of math" (as if it's a definite quantity) that one could discover is necessarily finite, if you lived forever.
By analogy, the "amount of math" we can possibly discover could still be countably infinite even if the space of all possible thoughts would be uncountably infinite. Countably infinite is still plenty big, and it is certainly not finite.
To put what I said another way, math may be infinite in principle, but the creatures in our universe can only conceive of or perceive so much of it: a finite amount.
We (or our constructs) can plausibly mine all there is and then there's no more that's physically possible for us or our constructs to mine within the universe in which we exist.
If an ant can't conceive of or perceive trigonometry that doesn't mean trigonometry doesn't exist. But if neither an ant nor a human nor any creature or construct or technology inside the current universe, now or ever, can conceive of or perceive trigonometry, then we might as well call it non-existent. It may exist: but we'd literally never know it and neither would our constructs or space aliens or inter-dimensional aliens, or their constructs, etc.
Say that there's a level of mathematics at which a blerg is a zorg, but our universe includes neither blergs nor zorgs and no intelligences in our universe can conceive of blergs/zorgs because that would require having evolved outside our universe... in that case we can consider our universe's mathematics solved without it needing to work down to the blerg and zorg level.
The natural numbers are countably infinite. For each n ∈ ℕ, there is a proof by reflexivity that n = n. Hence there are countably infinitely many such proofs, one for each natural number.
I didn’t demand anything. I requested some proofs since this thread is quite literally about a claim that requires proofs. As evidenced by the further comments in the thread. If you disapprove of the conversation, please feel free to use the tools and move on. No hard feelings!
Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.
I think that's a variation on the interesting numbers paradox joke.
Statement: All numbers are interesting.
Proof: Assume by contradiction that there's a non-empty set of uninteresting numbers. Then that set contains the smallest uninteresting number. That property makes it interesting.
Yeah, I've seen this before as well. I guess I've just become old and grumpy and can't appreciate jokes like these anymore. Also taking jokes seriously is peak HN so..
Edit: actually, forget about the above. I just find it very annoying when people dismiss good conversations with not-so-good jokes.
I mean, there's some truth to that joke in this case, so I wouldn't agree that it's dismissing. The point of it is that what matters is how we define "interesting", because by the joke's definition in particular, there can't be uninteresting problems. That seems to be a very subjective concept that can also change with time. Mathematics is so specialized that there can be 3 people in the world who find one specific problem interesting. If they solve it, they'll move on to something else.
I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.
Firstly, I'm very happy we're having this conversation. It's so pointless, yet pedantic it warms my heart in the best way possible.
Secondly, I stand by the statement that there is no merit to this joke. This is because the way it defines interesting is very hand-wavy. There are interesting and non-interesting problems, but by a sleigh of hand you can turn the non-interesting problems interesting, thus proving that basically everything in the universe is interesting. At least in the mathematically describable universe. When everything is interesting, nothing is interesting. So we can dismiss the proof as a silly joke.
What mathematicians find interesting is a different story. However, we can almost certainly say there is only a finite number of problems mathematicians as physical beings can solve. If we have 200 mathematical symbols at our disposal, and we consider all strings of these symbols of length 1000,000, we have captured all the descriptions of problems that fit to 1M symbols. But that's a finite number. Going beyond that starts to be difficult for a human to grasp (if 1M is not too much already), so all mathematical problems that are solvable by a physical mathematician are in that set of strings. And that's not even saying anything about whether or not they're interesting..
I still don't quite agree with the approach to mathematics as enumerating problems of a certain size, but to be honest I'm not prepared or motivated to keep this conversation going without turning to handwavy arguments based on my imperfect idea of what mathematics is and how it works. However, it's certainly given me something to think about, so it hasn't been completely pointless. Thank you for the discussion.
By the way, if I'm not completely mistaken, Gödel's argument to show the incompleteness of mathematics relies on encoding all mathematical statements as numbers. So I'm certainly not being very original here.
An interesting problem must have a description that fits in a brain, at least for now. Your description-length argument assumes arbitrarily large storage.
Sorry, I assumed the inductive construction was implied; you can indeed describe properties of that particular interesting problem (though of course you can’t hold its definition in your head), so it goes in the list. Keep going. At some point you’ll hit problems where the process of constructing the problem doesn’t even fit in a brain, etc. There are at least countably many problems, but finitely many problems which any algorithm-which-fits-in-the-brain can describe given finitely many inputs-which-fit-in-the-brain.
This isn’t an enormously important point - the actual question at issue is an empirical one, “in a steady state, can we produce interesting problems at a rate that exceeds our ability to solve them and integrate our understanding” or something like that - but I did rankle at a “trivial” proof which is invalid due to equivocating between multiple definitions of the word “interesting” (which should really take an object, “interesting to me” vs “interesting to something smarter than me”).
I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.
I would argue that novel and/or valuable results are not necessarily interesting!
I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.
I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.
I find myself less worried about it than at first. I think what we'll see are that some things are low-hanging fruit and can be solved just by tireless search. Maybe half the millennium and other such high-visibility problems will fall this way.
Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.
It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.
There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.
That's exciting, and kind of makes sense in retrospect. Sometimes a "fresh pair of eyes" on a problem can be all you need. Someone who comes in with a different background and can understand the problem in different terms and work on it from a different angle. It doesn't even have to be them doing the work, just a "that kind of reminds me of ... did you think about trying something like that?" that can get a team unstuck after thinking about it in the same way and never making progress.
I know someone—who doesn’t even use LLMs all that much, I don’t think—who has verbally told me numerous times “you are absolutely right”. I never said anything, but the comment always takes me out of the conversation.
Also, find the right manager and the right role. With a manager that expects a highly regimented routine, you probably aren't going to do so well. With a manager that lets you explore and be more creative, you might do a lot better.
I've found one of my own strengths is in finding ways to use existing features, maybe with slight modifications, together to do the things that customers want, allowing the team to avoid several large projects and the resulting maintenance burden entirely. My first manager understood the value of that and we worked really well together for a few years. After a reorg, my subsequent manager considered it lazy and PIP after six months. I don't fault them, and different management styles work for different people. But make sure you find someone you're compatible with.
To be devil's advocate, two things may offer a glimmer of hope:
First, math, generally, is useless. I mean, yes there are of course practical uses of basic thru undergrad-level math, and some beyond that. But for many mathematicians, the sum result of their entire career may lead to exactly zero results that have any real-world value. The entire field they work in may have meaning only to the handful of other individuals on the planet that also work in that field. But to those handful of people, the meaning defines their lives. From a socio-economic perspective, those departments should have been defunded a century ago. Yet they continue. Why? Because it scratches an itch. Not just for those individuals in the field, but also for us as a species. To stop exploring, to eliminate the search for pots of gold that may be buried in some odd corner of sphere packing, or coloring theorems, or Garside categories, and to put a boundary on the limits of our understanding, just because they aren't immediately applicable, is an idea that most humans would not be willing to sacrifice, even if it reduced their tax burden a couple cents. If it was going to happen, it'd have happened already.
The second is, even with AI, it's not free. As the software industry is discovering, far from it. So, given that, who is going to decide what theorems to research and how much it's worth? Congress? Of course not. AI itself? In theory that sounds plausible, but that falls victim to thing 1 above: most math is useless, so AI itself has no value metric it can assign to things, and besides which, without the human element, once the initial curiosity has subsided, there'd be no reason to continue any funding for AI to do it. So no, the only possible owners of this is going to be mathematicians themselves, the ones who care about the field and deeply understand the kwah of their vision.
Combining these, there's a future where, humanistically, "nothing changes". The method changes, the efficiency changes, the scope changes, but the work itself: publishing proofs, remains the domain of professional mathematicians. AI will enable them to be dramatically more daring and broad in their investigations and scope, and will likely write the entirety of the proof. However it will remain the work of the mathematicians to determine, what areas are worth spending limited AI resources on to investigate further, how far to go down rabbit holes, how to prioritize potential connections, and what the ultimate meaning of the findings is. So rather than being an end of mathematics, it could be a dawn of something far greater than anything we've ever seen before.
Microsoft has their own Durable Task framewor[1] for that kind of stuff, and it supports both running as a self-hosted standalone service like temporal, and running serverless on Azure Functions. It actually predated airflow, temporal, etc., IIRC.
This one seems to be more database-specific use case. The advantage is probably that you can track the exact state of the job in the database itself, rather than having to cross-reference the workflow log with the codebase and trace through it line by line to figure out what the state is. Plus I assume it's less overhead and latency, and operationally one less thing to spin up.
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