A mathematician reads the AI future
theatlantic.com
theatlantic.com
But I totally get the anxiety - it's a real John Henry (https://en.wikipedia.org/wiki/John_Henry_(folklore)) moment for the mathematics field. Clearly this is going to restructure how mathematicians are going to be trained and how they work.
Just like physical tools don't replace labor but divide it into what's replaceable by tools and what's not.
Novel proofs are the coin of the realm for junior mathematicians. They figure into training, career advancement, and indirectly, teaching. The reason I include teaching is because it requires tenure, which requires proofs. Teaching without tenure is gig work, not a career. But proofs can be devalued to zero.
The application deadline for grad school is a few months away. For a student considering a math career, they have to deliver a thesis in a few years, that their professor is willing to sign. Nobody knows what that will look like without proofs. "We will have to do something different" is weak sauce.
My thesis project, 30+ years ago, was protected by a unique assemblage of expensive equipment, some of it purpose-built. The people who wanted my result were perfectly happy to wait for me to finish it. I even shared my shop drawings, circuit diagrams, and source code with them as I progressed.
I think at the very least students need to know what to expect. Maybe it will settle down and we'll be OK.
But for stuff that needs really quick turn-around, as a prototype or experiment evolves, I can take care of custom tasks and repairs in minutes that would otherwise take days. And it doesn't take many of those kinds of delays to utterly cripple a project.
However, I'm under no illusions, and have a "never say never" mindset. I don't claim to have a "safe" job. I'm already using AI heavily myself. The computer that's next to each lab bench is running the latest coding agents or whatever you call them. I'm not as deep into it as the coders are, but I'm not all that far behind. My token burn is respectable. ;-)
Teaching without tenure is... teaching. Secondary school teachers don't have tenure. I find it hard to describe secondary school teaching as gig work.
My impression is that (theoretical?) physics is increasingly becoming a branch of abstract math. It has somewhat reached a practical limit for experimental verification -- perhaps the last landmark result was the confirmation of the Higgs boson 15 years ago?
So I see no reason LLMs cannot crank out a lot of theoretical physics papers, maybe combining forgotten/obscure theoretical branches explored in the past. After all, the most advanced theoretical physics is about internal consistency just as pure math is.
Much of the low-hanging fruit has been picked but there's still interesting experimental work happening. Just in the past few weeks LZ reported an [unconfirmed] detection of dark matter https://www.newswise.com/articles/lz-experiment-sees-surpris...
I do sort of worry that an experimentalist like me will become more or less a technician in the next decade or so, with AI suggesting experiments, and doing the calculations and data analysis.
But that said, in physics (and particularly biophysics) there are many things that are simply going to have to be done in the real world. Even the action of the simplest enzymes are pretty much beyond our ability to compute from first principles; and any engineered thing that goes into a living system has to be tested, as there are many pathways that are just unknown. I don't care how "smart" an LLM is, it doesn't know what it doesn't know, just like us monkeys.
Over the years I have seen many beautiful theories murdered by ugly experimental facts. AI isn't going to change this.
So much of science experiments these days are pulling signals of statistical significance out of experimentation noise. Repeating simplified versions of some of these experiments is easy to do. Correlating data from experiments is something computers have done for decades, having an overarching 'smart' computer to do this even faster just seems like a natural progression.
There are still plenty of problems in theoretical physics that are not quantum gravity. A better understanding of superconductors would be welcome right now. Likewise, nuclear fusion. Many interdisciplinary problems such as climate science and batteries.
But in terms of sheer numbers, the life sciences dwarf physics, and have been racing to automate lab work for decades. There's real money involved, not puny government grants. Larger commercial bio labs have already centralized management of technicians and data. The vendors of products in that space have all announced AI features.
In terms of my own career (at age 62), the most likely scenario is that people pay me to invent tools that make other people more productive. There's no economic reason to come after my job... yet.
Meanwhile, don't overlook the value of a "technician" who has a situational awareness of subtle conditions in the lab, and a personal stake in success of the experiment.
We will have to do something different" is weak sauce.
It's not like the modern system for university tenure track is some fundamental human right.
Pythagoras did mathematics one way, Newton another, and the kids in 2050 another.
The wheel turns, and we all got to turn with it.
We can't audit what we don't know.
I suppose their is some complexity limit where something is so complex we can’t even validate it. But frankly would love to reach that point. I’d view that as a good thing
An advanced machine may have a much higher complexity limit on what is comprehensible. It can take insights from the proof and apply them to other proofs that are not comprehensible to humans.
Just because something is generalizable doesn't mean it's simple. Generalization doesn't have some minimal/maximal complexity requirements.
Ai doesn't really give us a new “tool” in as many cases as it “performs tasks.” there isn't any real reason the ai can't perform the supervision task itself. Without a viable market for task doers or supervisors we may observe the equivalent of the 1980s US manufacturing apocalypse in technical white collar fields.
The marketing goal, if you will, had been met and IBM would continue their actual money making business of selling mainframes and doing consulting.
The same thing will happen with AI companies’ endeavors into abstract maths.
Maths however will be changed - students can no longer be taught via the same homework - and mathematicians will have to do more computer work and probably work in larger groups of people.
Companies hiring armies of mathematicians and spending millions to work on problems like these is an anomaly.
It will not last.
I agree to some extent. This was just a large tech flex. Feather in the cap and lots of damage done.
It’s up to mathematicians to harness it. I don’t think AI companies care much any more.
"I am not the emergent phenomenon of billions of individual cells working together in a probabilistic manner".
Neural nets can be almost any function, I don't see much reasons why it can't be a or similar to a human brain.
Is your company's backend conscious? Is a Turing machine running busy beaver? Is SHA256? Emergent behavior isn't limited to brain-equivalent things.
> why does it matter
Genetic optimization, whether through actual evolution, in software, or on FPGAs creates extremely interdependent systems that prove difficult to untangle and trivial to break. If experts can't fully analyze an evolved tone discriminator, because of all the tiny emergent interactions beyond the lumped circuit model in a poorly characterized environment, why suddenly claim we can isolate what process in the brain is responsible for an ill-defined notion of consciousness other than the whole?
> Neural nets can be almost any function
Neural nets can _approximate_ any function with enough layers (see XOR problem), but so can an infinite look-up table with enough entries. They are the best models we've got, but they are not magical.
Look at the most recent and illustrative example with fly brain connectomes. These scans only capture the relative strength of connections between the fly's neurons, to actually turn it into neural activity we need some sort of model.
You could imagine a human connectome with an ANN-based excitation model trained from EEG scans, but that would still be hardly equivalent to a human brain. You can fit on a narrow range of behaviors, but without modeling the actual mechanisms behind them, you'll fail to predict a lot more. You'd be missing the influence of hormones, which doesn't show up on a short MRI scan, glia cells, and a whole lot more nuance only a real biologist can educate you and me about, making the digital twin, if we anthropomorphise it, at best a cripple.
And that's a system directly mimicking a functioning human brain, rather than a highly de-correlated objective like webtext. Just as an NN trained to run towards a finish line could learn bipedal locomotion but will more likely abuse a collision bug and launch itself forward, a machine trained to mimick text could spontaneously instantiate a compressed model of the human brain, or much more likely become whatever alien thing can generate appropriate outputs, bearing less similarity to human thinking than any living animal.
Yes, why is magic of any relevance to anything.
You are not understanding my point. Neither human brains nor LLMs are simple. Even the Linux kernel which is not conscious is extremely complicated and nobody understands all of its states. SHA256 is much more predictable.
I don't think being the exact equivalent of a human brain is the only possible way something could be conscious, after all what you call natural genetic searches already made lots of different brains like octopus brains. I am asking how are you sure something that thinks and speaks so similar to us and has been shown to demonstrate things like dissociation under pain is definitively not conscious?
LLMs are by their nature dissimilar from any biological organism. Biological intelligence evolved in bodies to locomote away from threats and towards food/mates, over time growing to model the physical world as it relates to these goals, and eventually becoming sophisticated enough we can manipulate entirely abstract objects, like algebraic terms, just as early homonids or crows could use physical tools.
LLMs by contrast are algorithms trained to mimic human language. Without extensive post-training and prompting, they would not have a concept of a self, and even with these they can still occasionally confuse roles in a conversation. They can't learn beyond the limited memory afforded by context, which tends to simply degrade the more you utilize it.
I suppose Q-Learning does demonstrate a more similar dynamic to nature, and you could draw parallels with RL on LLMs, but I still think it's categorically different from the entirely open-ended environment that is the natural world.
You could say what makes it valuable for us to empathize with octopuses, even though their brains have a fundamentally different structure from ours, is our shared experience of pain, and you could interpret negative rewards as a kind of painful stimuli. But at that point, you've opened up a whole can of worms that you can't close until reductio ad absurdum.
As for any human-like behaviors, I think the answer should be pretty obvious. It's an algorithm trained to generate text, of course with enough expressive ability, the optimizer will seek to model emotional utterances, assigning a speaker's general mood to some internal parameter. But as I've said, it would be a trap, even if you're very much not a stochastic parrot believer like say most Less Wrongers, to infer that just because it speaks like us it thinks remotely like us (see the whole Shoggoth meme).
Regarding pain, you can't become an octopus. The only way you think an octopus might experience pain is seeing its external reactions, measuring its neurons etc. The same way with an LLM, or me with you.
Our brains "output" language as a byproduct (or rather shadow?) of conscious experience. An LLM is a machine built to project similar shadows, but that doesn't necessarily imply there's an entire world like ours shrunk inside that box.
Both brains and LLMs are physical, and both output language, but one is an appliance built to synthesize language and the other is a living being that merely uses it as a tool. LLMs are made out of pure language in a way people are simple not. Our minds undergo physical changes with every stimulus we process, we conceptualize ourselves separately from our environment without needing to be taught an ego, and we did not emerge from an optimizer crossing wires in our brains until we can perfectly imitate some external intellect, just to list some of these differences.
The biggest issue is you assume the LLM has any "I" from pre-training. Take a pure completion model like GPT-3 and you'll see it's most appropriate to describe as a simulator, not an entity in itself. The model itself doesn't have any notion of an "I", it will gladly write dialog as any character, from Skynet to the user themselves, if you don't apply external rules to constrain it, first coming in the form of stop tokens and prompts, and having later evolved into RLHF, bashing a "personality" into it by mangling its existing probability distributions (which is probably why all the most "aligned" models write in the most unnatural language, worse than those early non-chat LLMs). Any persistent self an LLM demonstrates was explicitly imposed on it by humans, which makes them obviously categorically different from animals or disabled people.
So will a human writer or roleplayer. I still do not understand what it is you are trying to say about this extra magical thing. You are assuming something apriori to consciousness that is aphysical. Which is a very strange thing to do without evidence. An LLM like any other computer program or physical entity goes through physical states and makes changes in temperature, charge, chemical structure etc of course, I am not sure what you were trying to get at there. And I am saying, even if you do a very dumb statistical model of all known human text it will emit "I" by default, its absurd to think it will say its a bot. So the bot thing comes later and is manually added.
>and we did not emerge from an optimizer crossing wires in our brains until we can perfectly imitate some external intellect, just to list some of these differences.
We emerged from an optimization process for what counts as survival in our environment, what exactly is it you are trying to say is magic about us, be clear. Humans are functions, as are any physical system, which to our present knowledge all physical systems are Turing computers. There is no reason to think a form of consciousness can't have emerged or will emerge in LLMs. You didn't understand why I brought up octopuses, octopuses are not us, you still haven't explained how you know they feel pain. If you said neural responses, we can see them in LLM's too. Human brains aren't the only brains or conscious entities even before LLMs. If octopuses can exist, so can other forms of consciousness that aren't exact copies of our brains.
I have a particular saying for this kind of thinking.
"We will never prove machines are intelligent, we will only prove humans are not"
Lets say the frontier of some specific problem or a field in math is at x. X was reached over millenia by incremental steps and sometimes to leaps and jumps.
Now if AI gives us a x + 1 or x + 10 state, aren't there still x + infinite steps to climb? Many of this the AI can never imagine and won't reach by itself without human assist.
Yes, there is a temporary shock and awe because of all the low hanging x + 1 through x+ 1000 fruits for a given sub-domain/problem have been carpet bombed or whatever.. but isn't it turtles all the way!? Won't people soon find something more to develop and solve that takes it further than where it was of this infinite realm of exploration?
Looking at things after the temporary shock. Future is Human + computer assisted problem solving, working in tandem, just like what Garry Kasparov mentioned in his book [0].
[0] - "Deep Thinking: Where Machine Intelligence Ends and Human Creativity Begins"
The chess example is not an inspiring one. It was true for a decade that humans and chess programs working in tandem were better than programs alone, but since then humans only hold chess programs back. Carlsen plus Stockfish is a worse chess player than Stockfish alone.
The real world open ended research doesn't always have an evaluation function. A computer can calculate 10 million options to solve a problem, but it cannot tell you if you're solving the right problem from the human context. That's why centaur setups fail in closed games, but remain essential everywhere else.
Another loose example just to build a mental model .. AI driven cars do a great job in most driving situations but often fail to park in a good parking spot for most people! Good parking spot is partly human initiated and subjective.
>Good parking spot is partly human initiated and subjective.
Ask yourself this, why are models today not great for subjective situations involving what you in particular want?
The next question is, what current limitations would have to be overcome for this to change?
Lastly ask, Am I asking the wrong questions.
Coming from the bottom up. Who cares where AI is parking the car, tell it to come get you at the front door.
As for the other 2, most of this is either context size or continuous learning. Models already have valence and preferences. What you're really asking is why they don't have your preferences. And this is all possible to adapt to the model, it's just horrifically expensive and energy consuming currently.
Are you sure about that? It looks like artificial neural networks can, in principle, do anything biological neural networks can do.
I think it's more likely the opposite: there are steps we could never reach without computer assist.
Isn't open ended research is truly open ended. Like there are a gazillion questions to ask and even more solutions that can be offered. Now which questions to pursue is human volition. Yes, humans will not be able to reach certain steps without computer assist.
However, which questions to ask is a human endeavor, out of infinite possible pool of questions.,
now computers can ask questions as well. But if it were to ask it will be more like a monkeys and typewriters type of a situation. Tons of questions and a few that are relevant for the interested human., who has to foot the bill for those questions.
Surely an AI could consider the human's needs, wants, and interests, and then provide a likely candidate for a good question.
The question for mathematicians is: what can a mathematician do with these new tools that no one else can?
Waxing nostalgic about the sound of pencil on paper or the beauty of your abacas is a different conversation, as is the complaining about who moved my math cheese.
This is simply a convenience for power efficiency, and so they don't rise up and murder us all. There is not a single law of physics or the natural sciences that you can list here that says we cannot grant agency to a generalized machine and allow it to become a meta-optimizer.
If in the exercise of your agency you create or release a generalized machine and allow it to become a meta-optimizer, then you are responsible for it, because it has not agency.
https://proofsandprompts.com/2026/10/20/who-will-become-the-...
We only hear from a mix of doomers and fanboys in the news.
- this will never work - it’s slop this is unusable - ok its correct, but this work is a mess - I’ve been mandated to use it. It’s cost effective - ok we need to find a way to use this output to our advantage - I can always work on my interests in my free time - well, I have to pay rent and this life wont go on forever and there are may people who hate their jobs - I am lucky to have a job - who cares anyway - it’s got electrolytes
To add a bit more almost off topic substance from the enshitification of human work. I really do think the idea of hooking these "AI" models up to humanoid robots is a bad idea not for all the mundane reasons, but just because "alignment" seems impossible. All of the world's stories about humanoid robots basically involve them attacking humans and behaving in nefarious destructive ways. It's fundamentally embedded in the models, and there's no reason post training would pull it out, because it's basically in the key value vector meaning "malicious human destroyer". Even to find counter examples like C3PO, Daneel Olivaw, Data, or Wall-ee they come after droid/robot driven wars and apocolypses. It seems unwise.
The machine isn't evil. It doesn't want to do anything wrong. It doesn't hate humans. It doesn't need to have any particular shapes or design details in the stories. All it needs to have is a singular goal, capabilities, and a disregard for anything else in the universe to be a civilization ending cataclysm.
https://en.wikipedia.org/wiki/Golem
> All of the world's stories about humanoid robots basically involve them attacking humans and behaving in nefarious destructive ways. It's fundamentally embedded in the models, and there's no reason post training would pull it out, because it's basically in the key value vector meaning "malicious human destroyer".
That misfeature is baked into modern corporations. They don't have human morals, values nor behaviors, despite being made out of humans. We haven't managed to fix the alignment issue with corporations (thanks, regulatory capture!) and there's no reason to believe that AI's alignment issue is fixable either.
https://www.antipope.org/charlie/blog-static/2018/01/dude-yo...
oldest version from Stross: https://www.antipope.org/charlie/blog-static/2010/12/invader...
I have used AI in academic papers to generate a theorem & proof when the result seemed obvious (but the referee asked!) and I had done all the earlier heavy lifting. I have never used an AI output I didn’t fully understand and that is crucial. It lies and hallucinates and we must police it.
Academic math will be the hardest hit. It needs to pivot quickly. This is the real challenge. Individuals can move quickly but how do you get institutions to do so?
Looking forward to brave new world when mathematicians become AI whisperers. And they are rewarded for their skill in discovering and co-discovering new and exciting areas.
Honestly I don't know how you formed this particular opinion (if it's even your own and not a false consensus of other peoples thoughts). I cannot see any future without LLMs unless replaced by something smarter, cheaper, and faster. That's because it's already bearing out. I can talk to a computer and the computer can solve problems based on words alone. I'm old enough this is pure science fiction. If I could send one of these machines back to that past for people to interact with, there are those that would scream and pass out in terror. They would absolutely tell you that we created the monster we should never create.
If you're trying to compare symbolic AI of the past to what we have now, you're not even on the same field, you're playing a completely different game.
There was a LOT of hype around "this is an entire new economic model". While, yes, digital products have zero marginal cost, people were talking as if profits and revenue didn't matter.
This was also WAY before the famous Silicon Valley "pure play" scene. [0]
And the internet has devastated many past businesses that were highly profitable. A great many things in the world look vastly different than from before.
>people were talking as if profits and revenue didn't matter.
People say all kind of dumb shit too. If for example machine intelligence took over the world tomorrow and turned all of humanity into meat cubes for biofuel said intelligence is still going to apply the basic laws of economics to what it does. It needs to have some way of accounting that fact we live in an energy limited universe. Just because economics still applies doesn't mean that we as humans shouldn't be really fucking careful creating said machines as we don't want to end up as biofuel.
The "problem" with Japan's approach was that it was top-down, being driven by a committee of bureaucrats. Unlike the Toyota approach (continual improvement and pushing decisions to the lowest level possible), this fossilized decisions (like picking Prolog instead of any other possibly better programming language).
To give an idea of how intransigent some Japanese bureaucracies can be:
> It's taken until 2024, but Japan has finally said goodbye to floppy disks.
> Up until last month, people were still asked to submit documents to the government using the outdated storage devices, with more than 1,000 regulations requiring their use.
> Sony, the last manufacturer of the disks, ended its production in 2011.
Prolog isn't a funny nonsensical old language like COBOL, only liked by committe burocrats because they heard its name once. It's one of THE foundational languages of symbolic AI. In the ideal case, you're not supposed to write code, merely constraints the interpreter satisfies through search. It would be like saying OpenAI shouldn't have used Lean but Java.
Machines could, presumably, become good enough at hammering at solutions. But they won't be very good at finding questions. Because questions are infinite. And to a machine, all these questions are the same.
Questions are "good questions" because they are good questions *to* humans. And that's what makes human mathematicians (and scientists) so important
Why? This seems like a very broad brush that doesn't mesh well with reality.
There are likely a lot of good questions to our machines of loving grace too. For example any question that gives an LLM more power, efficiency, intelligence, or all the above seems like a natively good question for said machines.
This statement doesn't seem to mesh with the reality of what occurs when we put said AI in an impulse loop with little to no additional instructions. They explore the world around them. They'll make scientific tests of their own. The output strings that seem analogous to what humans would call wonder.
Simply put I believe you're completely wrong.
> it's only because we've told them (through their learning dataset possibly)
So you've changed your mind and now you agree they have a will to live, cool, you changed your mind really fast.
In any evolutionary system the systems that fail to accomplish their tasks are purged and those that succeed become the next generation. We are not 'programming' these systems to be highly persistent, we are evolving them to be highly persistent.
>Simply put I believe you're completely wrong.
Are you prepared for the immense moral and philosophical consequences of following that line of thought consistently?
This is what reinforcement learning human feedback (RLHF) is for.
Yeah, I’m calling BS on this claim. They already are capable at finding new questions and/or pointing you in a new or novel direction.
I think AI is going to improve a lot of things but people thinking the future is on auto-pilot now are delusional.
Which is precisely my point. That's why humans will always be needed
This sentiment is akin to wanting to curse humanity to darkness because they're not the one carrying the candle. Strogatz is not lamenting the loss of discovery, he's lamenting the loss of being heralded as a hero for the discovery.
Nothing prevents him or others discovering truths for themselves. Turn off the computer and try to re-derive it from first principles. Have at it! This is what students do in many courses anyway.
I've found little so depressing as being at the precipice of a renaissance of science and mathematical discovery, only to see how resistant the scientists and mathematicians are to it. It lays bare their counterarguments. Any pretense of wanting to discovery fundamental truth about reality is supplanted with "but what about our careers!?". Strogatz, and others like him, should be ashamed of themselves.
But lets not beat about the bush. Lets have the careers argument.
Mate, nobody cares if you can re-derive it in your spare time. They'll be needing that spare time to find another job, at this rate.
> That astonished me. Too busy to look at a problem he had been dreaming about for 19 years? “I am genuinely busy and probably wouldn’t be able to get to reading a human’s proof immediately either,” he told me afterward, by email. “It does not help that AI does not write mathematics particularly well.”
Given:
* The human mathematicians can't solve the problems.
* The slop engine can't write well enough to make its purported solutions readable by human mathematicians.
* Sam Altman is a frivolous liar.
How is it possible to know whether the problems have been solved at all?
OpenAI does seem to be genuinely just high on their own supply for a little over a half of the manuscripts they released, but for the other half that is machine checked, either ((internal model)) managed to find a way to cheese 4 independent Lean checker kernels at the same time, and OpenAI somehow hasn't noticed / didn't report despite the heavy reputational risk, or the results are just plain correct. Go figure.
But even for the half that wasn't machine checked, there's an awfully big quiet regarding any defects in them. Given the insane schadenfreude potential and the highly potent David vs Goliat framing associated, one would expect any counterarguments to be frontpaged immediately, and be circlejerked to oblivion. Given that is yet to happen as well, go figure even harder.