Tao is out of his lane; lots of medicines don't have understood mechanisms. We don't even have the mechanisms behind general anesthesia nailed down; do you want to forgo it when the docs cut you open to remove your tumors?
Now math gets to deal with that same reckoning. They were already well on their way there with previous Lean proofs, but this has pushed things beyond that horizon and I'm not sure some of the mathematicians are ready for it.
Also, while biological systems simply exist in nature, artificial neural networks are ultimately mathematical objects with various properties that have yet to be uncovered.
As a comparison, classical computing has been scaled up a bazillion fold too, and can do things which are absolutely miraculous, but every layer of abstraction is discretely understandable.
The LLM model was able to break down and express math in such a simple way that I could understand and follow the training of the LLM model itself!
Are the math and computation accurate? I don’t know, and likely there are significant errors. Nonetheless, if I had a little more time -- not infinite time -- I would be able to prove whether they are or not.
Likely this is the path forward for understanding the mechanisms of medicine, and since most humans learn by doing and interacting with an environment, interacting like this will become how we use AI for learning in the near future.
Would you prefer to take the medicine that is proven to work or the one that is quite interesting for academic reasons behind its understood mechanisms but doesn't actually work?
"Open problems are lighthouses not destinations" mean that these are currently not well explained by the theories and people should look into extending the codebase on that direction.
The current generation of math AI is not a "good citizen" in that it doesn't try to make the most elegant additions to the shared framework, but will often just rebuild everything from scratch til get to some endpoint.
Sure, we learned if the statement is true or false; but the proof can't be merged into the pure math codebase unless it's completely rebuilt. This is thankless work that humans are unlikely to want to do, and so the "solution" instead risks leaving a desolate patch of land, where existing efforts in extending the codebase lost their motivation.
As with all things AI we can't take more than a 1-3 year horizon, if even that much. Probably AI will become better at respecting and working within the existing theories as it has with large software codebases.
One idea Terrance Tao conjectures, which is highly doubtful, is that spamming the AI button will solve open problems without producing insightful new methods. But the OpenAI drop would seem to disprove this. The sub O(nlogn) proof for DFT for example violated very old human assumptions. Decades of work in the field was incremental progress on sub optimal method that nobody questioned hard enough. More generally, we should always be able to go back to a super-human AI and say, "Attack this problem, but don't use a method tried before."
The AI proofs are a side-product of benchmarking current and in-development models on especially hard problems. They're clearly cost effective for frontier AI firms, and free for the taking as far as human mathematicians are concerned. The real issue with them is that they look like bizarre nonsense as written, so they need mathematicians familiar with those specific areas of math to "decode" and digest them.
It's not obvious that this is a given. "Cost-effective" implies a comparison between cost and output. OAI spent millions to race human researchers on Navier Stokes, and that doesn't even account for the training cost. And how does one value the output? OAI is for some reason still hiring armies of humans instead of automating roles like "AI support engineer" or "Product Designer" (https://openai.com/careers/search/).
Calling the proofs a "side-product" is also rather dubious when OAI employs a team of mathematicians specifically to train its theorem proving capabilities.
Mathematicians are primarily understanding-oriented.
Leveraging AI to tackle new frontiers without true understanding converts mathematicians to engineers.
It seems to me that all of these hundreds of proofs we've seen recently are glorified academic exercises, whose purpose is curiosity for its own sake without any practical application, or we'd already hear about at least one of them being implemented to some gain somewhere. It's all woefully unimpressive. It's not like anything stops mathematicians from trying to find more elegant solutions to their machine solved pet problems, since that's what they were going to try and do anyway despite it being completely pointless in practice.
btw people has massively improved the lower bound (from 1-2^-182 to about 1-2^-10) in the past couple of days: https://github.com/CrocSwap/integer-mult-bounds
We should name the explicit mechanism that was employed - telling the model to "believe in yourself".
There is something quite humorous but also poetic about how the manipulation of this term worked. Doubtless in the model's weights lies the echoes of generations upon generations of humans telling each other to believe in themselves.
In pursuing the "new frontier" as you rightly put it, mathematicians would do well to remember the same. It's ok, don't be afraid of the future. Believe in yourself.
> sub-O(nlogn) proof disproves that
How? Re-iterating, creating and understanding new proof techniques is the point of most of modern mathematics. Your statement is that proving a particular result is evidence of AI creating and understanding new proof techniques. I don't see how that follows, and I'm inclined to believe Tao is right for now.
And yes, results matter too, but if we stop at our current body of techniques and strip-mine results then we'll kneecap our future selves.
This is exactly what happened with that counterproof chat he posted a month or so ago - AI gave us an answer, he used AI to back into insights about the answer.
You don't address seismic shifts with a sweeping new approach, they are too multifaceted and present complexities and conflicts. He can say AI should help human understanding, which is a good end goal, but that doesn't mean AI dumping solutions isn't progress. That doesn't mean if AI builds 5,000 proofs in Lean and no human ever looks at them that they aren't useful, especially if other LLMs can access and build on those results.
This is exactly, exactly the same as when computers took over. "Oh, we don't need accountants any more" - not true, we just need acountants to deal more with human concerns than adding columns of numbers. That is called human progress, not a threat to humanity.
I don't really know the answer to that. I am happy when my own work is replaced by automated tools ("script yourself out of a job every six months!").
It's like getting scooped. If you just founded a startup based on tech XYZ, should you be happy when someone releases an open source XYZ? Should a news reporter be happy when another network breaks the story they were working on? On the one hand, society got the value of the thing you wanted to do. On the other hand, now you need to find something else to do, which might be really annoying.
We don’t know where this technology advance will lead or settle, so it’s absurd to try to establish a working paradigm at this point. It’s like any system, the initial conditions can be extremely chaotic and impossible to model, but with time often a stable state emerges. But the stable state is impossible to identify from early initial states.
I know a lot of folks feel this, to torture other physics metaphors, sensation of jerk - acceleration of acceleration. It’s an unpleasant and dislocating sensation. A world that felt safe and stable suddenly isn’t, and not in a micro tragedy sense but in a global realignment sense. This happened to factory workers who had enjoyed generations of stable work, farmers more slowly and just as surely.
This is what the late stages of scarcity feels like. Labor of various types devalues rapidly. Our exchange of meal and health coupons for toil cracks, and people realize their labor wasn’t godly as great books told us, but simply needed for want of an alternative. The realization that our labors might not be valued any more, and that our sense of purpose is shaken, coupled with the fact we’ve tied bare survival to our toil in our labor, is mortally tightening. No wonder people are grieving publicly.
But maybe our purpose isn’t to toil? Maybe we’ve passed peak population, and as toil is less valuable, we need less people and that’s why population is declining. Maybe we don’t need to exchange food and health coupons for toil, maybe mathematicians don’t need to rationalize their value to pursue mathematics. Maybe they can pursue it because they can’t help but pursue it, and our ever improving automations can produce their meal and health coupons?
But it might require Dr Tao to take an AI generated cancer medicine some day.
Bottom line, Tao is pushing for human understanding as the primary goal, with AI helping on all fronts. You are welcome to let Jesus take the wheel, but math is the most pure expression of human understanding. His point is that getting specific answers is rarely the goal, or certainly not the entirety of the goal.
Simply put, if we don't understand the answers we won't know what the next question should be.
> Bottom line, Tao is pushing for human understanding as the primary goal
100%. This applies to SWEs/math folks/etc. I do infra and I see many SWEs take their hands off the wheel. When they encounter perf issues they ask their agent and agent says GC and they say GC. It's rarely GC.
Now we might be well past the point where we need to remember the kubectl flags for rollouts etc. But basic human understanding of what their bots are doing as a goal has never changed. Humans are still liable for when bad things happen, and that hasn't changed over the roller coaster the last 5-odd years have been. LLMs, as astonishing they are at Navier Stokes, are still eminently capable of nuking your filesystem and saying "I can now see that that was wrong" with zero regrets. If you can't understand you can't sign off.
> math is the most pure expression of human understanding
This I don't know about. I think math acquires meaning when it contacts reality: like an iota is pointless until there's some circuit that it explains. Abstract math can diverge from that and can become an exercise in playing with symbols for their own sake.
After talking to a mathematician friend I can assure you that contact with reality is not the main goal of abstract math. It is mental constructions that have logical consistency and probably this is not the perfect definition either. It is somewhat of an art which is rendered in the logical mind. However physicists (me) and engineers will align with you.
Explaining nature causally using mathematical models isn't "the primary goal" of humanity. STEM people tend to have a weird misconception there, probably stemming from their misconceptualization of the humanities.
This in turns leads to this odd idea of "AI will think for us". That's pure (and pretty obvious) insanity. Logically minded people encountering it should ask, where the error in their reasoning is.
I have two issues with the implication of this statement:
1. Meaning is inherently subjective. Reality is just a canvas on which sentient beings create their own meaning.
2. There are many, many examples of where “playing with symbols for their own sake” have yielded deep insights. There’s actually some implicit structure (eg the structure of logic) that is intrinsic to the universe we live in.
The questions we asked were originally not about math. They were about a thing that we invented maths for to do or explain, a question that existed, because it touched us in some way that was already real to us. There is nothing that would not allow this to happen in the future. All this requires is attention and connection to the world around us. The maths required to answer our questions can be done and developed by something else.
To me, all you need to believe for this to be true is to agree that understanding maths is also not a stated requirement of reality to get a thing, if something else understands the maths (or something that does the same job). This is demonstrated by billions of people who do not understand maths and get things that, currently, require other people to understand the maths.
But the last part is entirely optional as it pertains to reality. That's just the best we can currently do (and in some important sense it is holding us back as a species, and in some other sense doing the opposite).
If that goal is understanding math, you certainly will be able to understand maths, more than ever before.
If that goal is something that required you to understand maths first in the past, you won't have to do that anymore.
What if AI knows better than us?
I respect Tao and I believe he's trying to think deeply about the issues, but a lot of his thinking seems to revolve around preserving the current roles and prestige of mathematicians, and also makes a lot of assumptions about the capabilities of AI years or decades into the future.
This is not me being snarky, but all meaningful questions can be settled empirically---e.g., "what happens to my body if I jump off the cliff". But empirical trials have a cost (time, money, irreversibility etc.) and we model and predict because it's cheaper than the trial.
The idea that all meaningful questions can be answered empirically is known as "verificationism" and is philosophically quite dubious.
> Simply put, if we don't understand the answers we won't know what the next question should be.
It won't matter, because it won't be us who will be asking the next questions anymore. Whether in math or anything else.
And no, domains that require real-world validation against physical ground truth won't save us, because AI gets to have the same inputs as we do (or better, if using specialized hardware), while beating us at reasoning.
And GP's likening this to previous massive economic shifts due to automation isn't really helping in any way, not anymore, because perspective won't feed us when we're hungry, and just as importantly, this one will affect every single field of human activity, so no one has any answers as to what the future will really hold for us.
That's the frontier labs' preferred narrative while they themselves are still hiring hordes of human "Account Associates", "Android Engineers", and "AI support engineers" instead of automating those jobs as a show of their AI strength. Of course it will be "us" asking the questions, because "AI" are computer programs, and humans build the computers and choose what computational tools to use for any application.
This is what it feels like to be disrupted. It's not the end of the world. You consider the evidence, ponder the path forward, and adapt. It's what humans do and their superpower. It doesn't have to be a negative thing, even if it is dislocating.
We don't need to be saved, there is plenty of agency to go around. Just grasp the opportunity and forge ahead. This sort of pessimism is self-defeating. Humanity has dealt with this before and come out on top, this time is no different. AI is being wildly oversold.
Tao is being entirely rational. He maybe has more to lose as anyone, but he's getting down to brass tacks instead of jumping at shadows and imaginary boogeymen.
It might be that P=NP and the algorithms are handed down to us. We can apply them without understanding why P=NP, and we may never be capable of understanding why.
Where do you see signs of this happening?
Is OpenAI going to pay for the food and health of mathematicians who lost their job?
It might be alright to go into some post scarcity society and do math for fun. But it seems to be a pretty unlikely scenario unprecedented om history.
why not try? we'll learn something from it.
The companies and partners need to maximize payout. They go on this track of cost reduction via layoffs and basically saying - “the model can do everything”. They know it’s not the case yet they still flood the airwaves and cause fud among all clueless c-suite executives. Which is the goal to begin with.
The second category goes all out against it. Professors, educators, school districts whose operating processes have not caught up to all the cheating that can happen. Here these folks have a point. I am sympathetic to this. It is hard to change education and it requires careful thought.
I feel this presentation brings out a good middle ground. The tech is useful but it’s not all encompassing. The tech also has other concrete uses. As an example, I have always wanted to explore the intersection of category theory, formal verification and AI guardrails and prompting. Proof writing has been a chore because I have a day job. Maybe the AI can help here.
While frontier labs are supposedly on track to conquer human endeavors, they are somehow still hiring lots of human "Account Associates" and "AI support engineers" instead of automating those jobs as a demonstration of their AI's economic value (https://openai.com/careers/search/).
Who says we have a purpose at all? The universe doesn't owe us meaning, nor even existence. But that doesn't mean we shouldn't try to shape a reality we want to live in. We may not succeed, or we may find that we'll be happy in a reality we cannot imagine yet, but que sera sera is tautological and therefore unhelpful. Obviously, no matter what we do or don't do, there will be some future, but treating that tautology as a prescription is just a call for passive resignation.
> This is what the late stages of scarcity feels like
Going from LLMs to "late stages of scarcity" is quite the leap (although I guess anything could be a "late stage" depending on the timeline). Even if we were to assume that "intellectual labour" is our most scarce resource (and I'm not at all sure that's the case), obviously it's not the only scarce resource.
> we need less people
Who's "we" and why do "we" need any people at all?
I don't think it's that surprising have you met many mathematicians?
> This is what the late stages of scarcity feels like.
Late stage of scarcity of what and for whom is the question.
Who are you counting as being in "any position of power"? All the AI lab people are saying that the most likely and best outcome is we all live in a world of abundance where money doesn't matter anymore. Dario, Sam, Demis, and Elon have all said this loudly and repeatedly to anyone who asks.
None of them have articulated a coherent way to get there from here. But they all believe that the technology will make it possible, so the only unresolved question is how to transition us there.
And none of them ever spoke a lie. Especially not if it would further their goals at someone else's expense.
From what I've seen, UBI is just a carrot dangled in front of the poors by utopian Silicon Valley tech bro megabillionaires when they're trying to drum up some PR for whatever big idea they're pimping out at the moment.
There are a lot of discussions of socialization of health care and basic income approaches and sovereign wealth based on automation dividends similar to the Alaska trust.
I think the current spate of mean cruelty ala MAGA and a glorification of a mean and cruel past that was never a golden age is a death spasm of a deeply unpopular belief system. As the actual realities of the policies sink in 70% of the populace is revolted, which is a super majority. That’s more than enough to put a pin in the philosophy permanently. It is also greatly accelerating electrification, realization of the value of expertise in technocratic systems in current generations, etc. I think humans are socially adaptable animals at a cultural level, but for the individual the adaptation process can be awful. I hope it’s not, because it doesn’t have to be. We will see.
We have no choice, we must tax the common people.
The Uprising (2026)Matrix was a good point to this. They made the world a utopia and people couldn't handle it. Our monkey brains require pain give us a utopia and we'll just walle ourselves to death.
(The ones I've heard about, I'm fairly sure, didn't find anything of the kind. Which isn't to say that they show we should have UBI; there are big gaps between what has been tested so far and what an actual economy with UBI would look like.)
I think you've taken the "work less" that was found in some studies to suggest they needed "toil". They simply found it somewhere else.
I was thinking the same, that most of the opposition to AI's convenience is starting to smell like religious mysticism, the kind of arguments religious people made (and make) when Evolution and Natural Selection were introduced:
√ "Stop simplifying humans down to numbers!"
√ "This denies our spirituality"
√ "What do we strive for now if we're not special?"
+ (along with some borderline jihadish hate heh ..maybe Dune got it right)
Well, either there was nothing special about whatever you were doing after all
or, maybe there is still something special at a higher level you haven't looked at yet.
The Sendov example has a piece coders will hopefully recognize: he found a formalization that was 1/6 the amount of code of the original one. And the analogy with code goes further--the messy version will run/pass the proof checker, but the one that's been cleaned up and made sense of is a better foundation for future work. That's true even for future LLM-assisted work.
So it's not about whether mathematicians take advantage of LLM help (Tao favors that) but, more or less, whether the point of math is just to make a bigger version of the GitHub dump vs. everything else: readability/comprehensibility, negative results that fill in the map around a problem (not just 'lighthouse' theorems), organizing results to plan out future work, and so on.
The economics in the West feel very strained right now, and this incredible tool has come along just in time to threaten one of the last bastions of middle-class safety: white collar jobs.
I think it’s reasonable for people to say “I like the way things were, I don’t like this new future you’re proposing, and I want to limit the technology that’s doing the thing I don’t like.”
There is nothing inevitable about AI. As a society we may decide it is fundamentally unhealthy or anti-human. We have banned or curtailed access and research for other technologies before.
Things are not settling, it's only acceleration from here on out. In fact things have never settled, technology has been on an exponential since humans tamed fire. The difference is we notice change faster. It used to take several human lifetimes to notice change. In the 20th century it was noticeable within a lifetime. Since the internet there has been a great revolution about every decade: web, smartphone, social media. But now great changes are noticeable within a year. It's not enough time for society to digest and adapt.
> we need less people and that’s why population is declining
This is the scary part. The Elon and Zuckerberg types that control the new powerful machines have proven they are not moral people. In America's highly billionaire-deferential culture there is no stopping them, at some point they'll be out of reach of democratic or even military control once they control private robot armies. They could decide to accelerate the decline of the undesirable useless population. Amazingly humanity's salvation could end up being China's communist system.
Now what are we to derive happiness from? The joy of a boulder being on top of a hill?
I think that will be the most important thing for this transition, defining new purposes and meanings that people can assign themselves.
Now, as far as I can tell, we are quite a way away from actually having most/all professions replaced, so for now the answer is rather clear: pick another boulder.
My impression is that since pure mathematics doesn't attempt to meet those human desires they need some other objective, and that objective is human understanding. The loss of that is thus felt more heavily than in other fields. We could say it's their problem, and they need to get over it just like the chess players did; but the outside implications are broader here, since mathematicians working in fields they themselves considered useless have so frequently been wrong--in Hardy's Mathematician's Apology, he gave number theory as an example of such a field, unaware of what the cryptographers would achieve just decades later.
It's possible that AI-generated pure math will continue this trend of delivering extraordinary unexpected societal value. It's also possible that the humans won't ever sufficiently understand that math, and the machines won't ever sufficiently understand human desires, and that connection won't be made. I've never met a pure mathematician who considered those downstream applications to be an important contributor to their motivations; but as AI-generated math contributes to the argument to allocate a large and increasing share of GDP to datacenter buildouts, that question of whether downstream value requires human understanding seems pressing.
Prime also mentioned that software development is different. In Software development, the product is what you’re building towards, so the means to get there can be disrupted without the industry being cannibalized.
In math research, the process is the product. You take away the researching part and not much is left. But my question is, these math proofs OpenAI released, will math shift to actually using the proofs to change the world instead of just finding new ones?
If business can deliver the same product with a smaller team, great!
And yes this has been happening for a while, even if not everywhere.
In enterprise consulting, projects that would require a team of 20 devs on average, now have about 5.
Moving away from on-prem, managing own cloud infra to managed containers, to serverless, SaaS and iPaaS ready made products, and offshoring naturally.
All contributed to ever decreasing team sizes.
Now AI based tooling is added to that cocktail, reducing even further the team sizes.
The only folks doing well in the end, are the employees of AI companies, without moral issues contributing to the industry downfall, because the CEO themselves aren't the ones coding and pirating human culture.
I'm a freelancing consultant since 5 years, I've had 2 major customers now for 3+ years. I have a very good pulse of the market: being good, or being even very good and being among those that brings AI and automation to organizations will not save our jobs.
In fact, AI has sped up so much the work that 2 out of 5 people in my current team are being let go: I find it absurd, our productivity has more than doubled over the last years and we've made ourselves redundant. Money is money, I'm on one side making non-tech workers redundant (people whose job was menial boring office stuff), and building the systems that will make myself redundant.
That means software engineers better start getting creative. If you think your job is to wait for a PM to assign you a well-written researched ticket, you're done. Your job is now to figure out how to make these machines (computers) do whatever we need them to do safely, quickly, at scale, and correctly by applying all your knowledge of computer science and the engineering field of software engineering to an AI prompt.
Also, the increase in output is meaningless when the amount of customers doesn't scale in similar size.
Then there are the constraints of physics, there are so many humans in the planet that actually want to pay for a specific product, or consulting services.
This is a "you" problem for the math establishment, not a problem for the AI companies.
This was a talk given to other mathematicians about the future of mathematics; sounds like only "you" have a problem for some reason.
I would have assumed that, by producing new proofs, the AI has either validated existing principles, or discovered new ones? Isn't that worth studying?
Are mathematicians complaining that reviewing AI's proofs is not as fun as writing your own? Try being a programmer... welcome to our world!
If AI lacks imagination and is not discovering new principles, then it's doing us a favour: it's crossing out the problems that don't need new principles. So the problems/conjectures that are still left are the more interesting ones.
Source? I assume that many of the approaches embedded in this proof dump will eventually be distilled and generalized into new techniques. That's how proof techniques tend to come about anyway (before AI): human mathematicians do something novel and unexpected to solve a particular problem, then efforts are made to understand how the "trick" works.
Sure, but that's a real technical limitation with current AIs, not something that AI firms should be blamed for. And if anything, this creates a viable career path for the mathematicians who were "scooped" wrt. the original solution: they can at least puzzle out what exactly the AI managed to do. Many practitioners are actually quite excited by this possibility; Tao's stance is by no means universally shared.
I'm just not happy with the presentation of OpenAIs result. Maybe they could have gotten into contact with the people of the research areas of the problems that they solved and worked with them to create a better exposition. Sure, it's a slow process and requires lots of staff. But I believe that they can afford it.
The way I see it, it's no different from a lot of grad student work where you have to figure out what a human-written proof is doing.
> Maybe they could have gotten into contact with the people of the research areas of the problems that they solved and worked with them to create a better exposition.
That's what Anthropic is doing, and the issue is that people will complain that they weren't the chosen "person to work with". OpenAI's approach is more like a race where everyone's at the same starting point: they get the AI's raw proof to work on and have to figure out how it works.
"Reaching these lighthouses [resolutions of open problems] prematurely by automated tools can disrupt the exploration of the paths not taken, and sterilize the surrounding field."
This crucial issue is centered in mathematician psychology and the incentive structure of academic/institutional mathematics worldwide. For mathematics to flourish going forward, we will need to realign our brains to think differently about the nature of mathematical progress. And we need to reorient our institutional incentive structures towards the promotion of meaningful mathematical progress itself rather than targeting proxies that are no longer faithful.
Regardless of the precise nature or the causes of the "sterilization" Tao refers to, we (the mathematics community) can only rely on ourselves to repair it. Though, since it will involve fundamental change at the level of ossified academic institutions with many stakeholders and divergent vested interests, any such repair will be slow, frustrating, controversial, and lacking any guarantee of success.
I'm a big fan of Tao. He must be so shaken by the AI storm that he's now writing arguments that even teenagers could quickly dismiss. A sad day.
"Oh no! Not like that!"
- everyone with Very Strong opinions on mathematics academia when he talks about the thing they are (or consider themselves to be) experts in
I'm excited!
Seems to be the crux of the argument, but "use your imagination" isn't a great thing to tell people who are looking at degree irrelevancy, concerned about getting tenure or a research position. How do we measure if someone is a good mathematician or not, if they are one of the sanctioned few who get access to the biggest AIs?
Letters of recommendation from trusted colleagues have always been essential for evaluating candidates. Hopefully, human recommendations will remain a strong, faithful signal as the utility of other metrics rapidly deteriorate.
underrated buried comment based in reality
There are still a lot of treatments/medicines in medical science where we dont know 100% the real reason as to why it does what it does but we still prescribe them because the intended effect is what we are interested in.
But I understand that for people whose whole life was math and solving math problems this will lead to an identity crisis. Seen the same in my area of work (software engineering)
Basically people are vibe coding their personal apps and anything that's expensive is being vibe coded open in the public. I don't see many software companies staying profitable for long.
DHH is the biggest proponent of AI and let me know which of the 37 signals products can't be vibe coded in a month at a $200 plan that are suitable for that organization alone that just has to be accessible internally only? Hence scale and security aren't such an issue.
In that climate - for how long software companies would stay profitable and when not, who'll be employing developers?
PS: Don't underestimate vibe coded apps. Take a look at PDFCraft, VectorCraft, WordCraft. And imagine the feature parity in a year.
I mean... yes, most people will find it reassuring, but history has proven that's not necessary. People have been using medicines of which the mechanism wasn't understood for a very long time and greatly enjoyed their benefits. Even widely used one (e.g. Paracetamol).
Also how many math and scientific discoveries do I want to see during my lifetime? "Just a little bit more."
I wonder if we will begin to actual value human creation more at the end of all of this
Can someone explain why there wouldn't be arrows from all three types of solutions back to human understanding?
For nuance lovers - here are some basics for how you get your drugs: there is an established chain of trust from the first basic science paper to the phase 3 trial and the subsequent availability of the drug to general public
- someone publishes the first paper (basic science) explaining some biological phenomenon, which leads to 10s or 100s of other papers with some tweaks in conditions,
- after the above papers the pathway of the phenomenon is understood by researchers, they try therapies at cell level to see if they can control some behavior, 10s or more papers get published,
- then someone tries this in mice and other models, 10s and more papers get published.
- then researchers at pharma companies + hospitals create this therapy for human trials - phase 1, 2, 3 etc - data collections, then FDA - then approval.
Now, the people who worked on the phase 3 trial might not know the people who wrote the first seminal paper and they often don't exist in the same decade - but it absolutely does not mean that we (humans) don't know how these drugs work - if you take 1-2 researchers from each phase and put them in a room and ask them how that particular drug works - they will quickly be able to build a consensus. that is what the chain of trust means here. now of course there can be fraud in scientific research, but that happens in every human endeavor and is a separate topic.
back to terrence - he is saying that if there is suddenly a drug that nobody knows the origin of; passed phase 3 but it's unclear who conducted the phase 3 or if the phase 3 even happened or if it's fabricated - you would not want to take the drug. usually when doctors recommend these kinds of drugs - there is already a lot of information available about where the drug came from, if there are any case studies, which doctor tried it first, which country- they often even call those other doctors and find out who was behind the first trials going back as far as the university professors.
Your MD doctor might not know the chemistry and physics behind the drug you are taking but there is deifnitly a group of people, when put together, can tell how that drug is working. My wife is a fundamental researcher - understanding physics at DNA level and my brother is a MD doctor; our conversations are super fun.
Side effects are a completely different thing - they involve the above cycle on repeat.
Today we'll generally have a proposed mechanism, but it's not necessarily correct. Acetaminophen is among the oldest and most commonly used synthetic drugs, and its mechanism is still debated. This isn't usually cause for any special concern, since our confidence in the drug's efficacy and safety comes more from animal or human trials than from mechanistic understanding. Serendipitous discoveries during human trials are still common; the first inkling that Viagra might treat ED came not from any "first seminal paper" but from the volunteers who were testing it for angina.
Clinical trials are regulatory matters. Your suggestion that it could be "unclear who conducted the phase 3" is very strange--the FDA knows who filed the application. Of course that filer could have committed fraud, but I see nothing in the slides to suggest that was the concern here. If it was, then the solution would be a non-fraudulent phase 3, not anything related to mechanistic insight.
I’m not sure who u or Tao is arguing against.
EDIT: people have a hard time choosing between
1. Yeah it’s pure slop and completely useless
2. Oh no OpenAI has stolen my ideas and solved all my problems and we have nothing to do
My dude, if OpenAI’s dump were really that worthless to be as good as noise, why is Tao getting agitated? Just like ignore it or something.
>> 1. Yeah it’s pure slop and completely useless
>> 2. Oh no OpenAI has stolen my ideas and solved all my problems and we have nothing to do
Could it not be all of the above - OAI stole ideas, there is slop in OAI's work given that they themselves retracted a few of the papers?
hmm. 700 papers released in one day.
>>It’s gonna have clear provenance and the same type of verification channels
what's gonna have clear provenance? - the math slop they released has already been rebuked by human mathematicians as incoherent and deserving of desk rejection.
I'm still not sure about math-2.0 (humans+AI will make fundamentally more progress):
- AI and computer usage take a mental toll on humans and humans will overlook radical improvements.
- AI may be good at finding useless things like "P==NP, but the complexity is O(n**4242424242424242)". In other words, useless.
- Humans become formalists and lose traditional sources of inspiration. Maybe interacting with Lean should be left to specialists, but not to creative blackboard mathematicians.
- AI exposure will further intellectual conformity, more than the Internet did.
As to the last point, a lot of progress (real, not measured in publications) seems to have been made when communication was slower and there were several different schools and approaches.
https://github.com/teorth/tao-web/commits/main/
It reduces the badness somewhat.
"""An advanced AI is prompted: “Find a cure for cancer that passes a stage 3 clinical trial. Make no mistakes.” After a large amount of compute, it produces a cocktail of previously unknown chemicals which it claims, when mixed and injected into a patient, will kill all their cancer cells. While nobody truly knows how this cocktail was found, the AI (somehow) provides a Lean certificate for its prediction, and the cocktail does indeed manage to pass a stage 3 trial.
Could the AI solution be somehow misaligned by exploiting a weakness in the trial process or its math models?"""
Yes, absolutely an AI solution could exploit a weakness in the trial process or its math models. Would it remain uncaught? Unclear.
> Or a human mathematician who understands the mathematical model used to locate the cocktail?
If we're being fair to AI - it contains collective knowledge from all fields, which means it's probably less likely to miss something that a human would.
https://x.com/RohanArun/status/2109336015814959200?s=20
It seems like OpenAI opened the door for many more people to participate in math discovery process. It's fascinating to follow!
removing some key theories or inputs
(e.g., finding an elementary proof for a result currently only provable by non-elementary means)."
As usual, Tao is brilliant in all that he researches, all that he writes about.
I chose the above statement (which is brilliant, in and of itself!) to comment on, because it leads to the following idea:
There there exists, or should exist, a dependency map in the fields of not only Mathematics, but also of Computer Programs/Software, Engineering, and even a seemingly non-related field: The Law...
In other words, how do we get from the simplest of axioms or foundational things (aka "first principles", aka "self-evident truths") to much more complex entities?
In Law for example, how do we go from the simplest of historical legal constructs to the most complex of the most complex Supreme Court cases?
You see, there is, or should be a map, you could call it a dependency map, you could call it a dependency graph, which shows more and more abstract/complex mechanisms/things/assertions/statements/truths/functions which is mapped back to , that is, dependent on various chains, various stackings, various "stacks" of simpler ones.
In Engineering, for example, how do we get from the simplest of machines to the most complex of machines? What simpler machines and/or sub-components (aka "dependencies", aka "subcomponents") are required to build it, and how do those simpler machines work, and what's the dependency graph or map for their subcomponents?
More generalized, if we have something of complexity, then how do we get there, step by step, from individual subcomponents, individual inputs, individual proofs, individual software systems, step by step?
What is the map of those dependencies?
Note that in some systems, Math proofs, for example, there may be different paths which can be traversed to get to the same destination.
Ablation Studies could be thought of in Travel, in Geography as "if I cannot take one, or a specific set of routes to get to a place, can I still get there?"
A simple example would be in Google Maps, where you'd like to drive somewhere, but you'd like to avoid tolls. Is the route still traversable while avoiding tolls? Well, that's an example of one constraint. In Ablation Studies, you might wish to remove a bunch of routes with whatever criteria or characteristics , i.e. muddy roads, roads that have characteristic X, roads that do not have characteristic Y, etc., etc.
Getting back to Math, specifically proofs, it would be great to create a dependency map/graph of all of them, and then try removing inputs (aka, paths to them, dependencies on other mathematical proofs/objects that they may have) and see if they are still reachable.
In software, when we desire the tightest, cleanest, source code, the above is related to refactoring.
In the future, I'd love to see dependency maps/graphs (call them whatever you will) for not just Mathematical Proofs (although I'd love to see that too!), but also in such diverse subjects as Science, Engineering, Programming/CS, and even the Law!
Because they should exist in all of those subjects!
Anyway, another great piece of work by Terrence Tao!
1. The use of semi-ugly slides to communicate this is just perfect and quite heartwarming, but it does highlight my main criticism of the mathstadon version of this thesis: he's myopically focused on mathematics as he has practiced it, rather than mathematics as a ~2400y old academy. Like, "stable for almost a century" sounds impressive, but should be a pretty obvious red flag in hindsight!
2. Glossing over "objective verifiability" feels like another place where he's ignoring a ton of relevant philosophy for no clear reason -- yes, mathematics is the only academy based in pre-conscious cognitive facts about our processing of time and space, but that's not the end of the story on "objectively verifiable". To say the least! He hedges with "broad consensus" which doesn't need to be absolute, but that seems to be not only dismissing a highly relevant question, but even implying that he might be unaware of it. I doubt he is, but still: not great.
3. Who is this for...? Why is an explanation of Lean needed in a talk given at CalTech? I suppose he's welcoming his role as a bit of an influencer, there?
4. Re:the focus-on/centrality-of 'highly digitizable' as a unique class of task that applies to mathematics in particular, I must sadly trot out the increasingly-common trop: Yudkowsky called it... https://intelligence.org/files/IEM.pdf
5. "the space of mathematical problems remains infinite" is, again, ignoring really important philosophy around academies as social structures, built for human means. Mathematics is only infinite if we decide that all knowledge is useful (the quintessential example being 'counting the grains of sand on a beach'). Not really important in the first place, but another worrying case of the above.
6. Problem solving is the goal of mathematics; he has a completely valid point here (that we shouldn't throw AIs at unsolved problems in bulk and thus lose human expertise), but it's obscured by the use of "[open] problem" being a too-technical one. IMHO. Slide 16 fails to disabuse me of this notion.
7. Slide 18 is describing the differences between functions and systems, and is arguably even talking about assemblages.
8. If we're gonna explain lean up-front, it feels like a baffling choice to throw the "maybe AI will solve cancer but use it to secretly plot to kill us all" slide in there. It's also already lead to misunderstandings and backlash on Reddit, where 'yes we want to not die of cancer!' is a pretty convincing counterpoint (if a ultimately a subtle strawman, ofc). It's also quite distinct from the rest of the talk.
As always, the best part of any Tao publication is his ability to inspire and rally and organize. I think Math 5.0 will indeed be a matter for creativity! Hopefully the IE levels off before we cease to be helpful in that capacity...
Perhaps, but there is a danger: it is difficult to un-see things. At least for now, AI-generated proofs are likely to be of a somewhat brute-force nature. As each such proof pops into existence, two things happen: (1) it is much harder to maintain an untainted mind and go on to discover alternative, perhaps deeper and more conceptual, proofs that cannot be obtained by massaging and cleaning up a more brute-force approach (think of this as getting 'stuck in a local minimum', if you like), and (2) at a societal level there is thereafter much less incentive to attempt to do so (announcing a proof of a previous unproven result is, for now, far more prestigious than announcing a much better, more elegant, proof of a known result).
One could ask why we prefer conceptual explanations to brute-force proofs; after all, a proof is a proof, isn't it? I suppose one reasonable answer is that conceptual explanations lead more readily to new questions and that there's also intrinsic beauty in such explanations -- though, of course, many outside the field will simply not care.
The popular perception of him (as popular perceptions generally tend to do) reduced his overall position to basically early adopter went sour grapes, and I'm really glad to see that substantively falsified.
In the long run, the wordplay and false equivalencies are irrelevant versus what actual outcomes are but it’s definitely wild to watch.
it's pretty ironic that AI being just a group of mathematical techniques after all is making mathematicians uncomfortable because it works. Instead of reacting like this, mathematicians should be excited to figure out how to use the new tools available and push the frontier of what's possible in service of science.
Terrence is asking the stupidest question he could ask: how can math better serve me? They completely forgot the point of science is serving humanity.
No. I'm gonna die, my man.
This guy might have the highest IQ on the planet, but it's clear he hasn't spent much time around average people.
This argument is so silly against reality, and already, almost nobody understands the things they put in their body to any significant degree, beyond the effect produced.
Will I take a cancer cure that no human understands? Yes. And so will billions of others. Just needs to cure cancer, that's nearly the only requirement.
That's what the trials and all are for.
Who cares about understanding it, in the face of efficacy?
I want to understand everything; I think AI will help that happen, not hinder it.
All the arguments about the future mathematicians are imagined, and emotional.
It makes me think of the book Finite and Infinite Games. It provides a perspective that work is just one role that we _choose_ to assume in our life. Realizing that we can choose other roles and move in and out of them freely has helped me alot with big changes (career and otherwise) in my life.
He is not talking about a cure that works and that no one understands, he is talking about an AI making its way out of the trial just to get to phase 3...
If AI found a way to exploit clinical trial design, it would be noticed and the errors corrected. In fact, that would be a major win because it would probably lead to improved clinical trial designs.
If we followed the precautionary principle and waited until we understood everything there is to know about every drug on the shelf, a lot of people now living would be dead.
Half of people are below average! They understand nothing at the level that Tao means. Literally nothing.
If people had to understand everything that worked, most people could not really engage with anything.
If the requirement is just that one human, somewhere, understand it--how is that different from AI?
My position is that Tao is not only wrong on this but his example makes the opposite case.
Was this sarcasm and I missed it? Because not only do I think I understand at the level of Tao I also think he is wrong and not understanding something.
Tao is intelligent. Intelligent people have higher standards for understanding the world, such as assuming that the mechanism behind medicine are well understood. But most people are not intelligent, so they don’t understand things like medicine at the level Tao presumes.
I don’t agree with this argument, just explaining it.
Medicine has never required that the method of action for a treatment be fully understood. Our current regulatory framework only checks for safety and efficacy because we've never formally understood everything going on in the body. Seems like a difference between "hard" science and the clinical/engineered implementation.
My own pointless semantic argument: we typically don't use the word cure when we talk about cancer. Instead, we use the term "complete remission". Many people who were thought to be cured then developed cancer decades later that was genetically derived from a small remaining population of cancer cells that were not eliminated in the original "cure". The word is a shibboleth for not being familiar with cancer medicine and treatment.
I would totally take the mysterious drug if it was clinically tested and shown to be reasonably effective. But that is not the premises presented here.
I wonder if this makes them slowly get disconnected from the lived realities of billions of ordinary people.
It's a good reason not to take his advice about the real world--like how to manage AI's trajectory.
The cancer example illustrates this gap perfectly. He thoughtfully crafted the point, and defeated himself in argument.
Unless it was a move?
While the biological effects of any particular chemical still require a great deal of trial and error to determine, drug chemistry itself is unambiguous. Pharma companies employ expert chemists and chemical engineers. They don't manufacture drugs just by randomly mixing together chemicals. At the very least, they must understand what they are making thoroughly enough to determine its physical properties and design a reliable and commercially viable manufacturing process.
he isnt saying that person who puts stuff into their body should understand it
it is that someone (expert) should understand it to the point he can vouch for it
The software analog of settling merely for "it seems to work" would be if someone were to hand you a bare binary whose behavior must be inferred entirely through black box testing, without source code, architecture diagrams, or any other documentation. Sure, users can try running it in a pinch, which is the case for various medications, but it's surely possible to do better in the long run.
Ironically, reality people are ok with not understanding everything because they believe another human who understands it has looked into it. It may come a day when we won’t need that but that day is not today. Don’t get me wrong, I’m not arguing about the merits of it, it’s just that at least this is the reality of this timeline on this planet. Not sure about what alternate reality you’re talking about.
> Before injecting this cocktail into your bloodstream, would you want to know that there is at least one human cancer expert who understands the mechanism behind this cure?
Now:
> Before injecting this cocktail into your bloodstream, would you find it reassuring to know that there is at least one human cancer expert who understands the mechanism behind this cure?
I would agree with the second, not necessarily the first.
The old text:
"""Suppose an advanced AI is prompted to “find a cure for cancer that passes a stage 3 clinical trial”. After a large amount of compute, it produces a cocktail of previously unknown chemicals which its mathematical model predicts, when mixed and injected into a patient, will kill all their cancer cells. While nobody truly knows how this cocktail was found, this model prediction is confirmed in Lean, and the cocktail indeed passes a stage 3 trial. Could the AI solution to the prompt be somehow misaligned by exploiting a weakness in the trial process? Before injecting this cocktail into your bloodstream, would you want to know that there is at least one human cancer expert who understands the mechanism behind this cure? Or a human mathematician who understands the mathematical model used to locate the cocktail?"""
The new text:
"""An advanced AI is prompted: “Find a cure for cancer that passes a stage 3 clinical trial. Make no mistakes.” After a large amount of compute, it produces a cocktail of previously unknown chemicals which it claims, when mixed and injected into a patient, will kill all their cancer cells. While nobody truly knows how this cocktail was found, the AI (somehow) provides a Lean certificate for its prediction, and the cocktail does indeed manage to pass a stage 3 trial. Could the AI solution be somehow misaligned by exploiting a weakness in the trial process or its math models?"""
I am not going to pay attention to Tao's opinions on this from now because I don't really have faith that he's even writing this or that he understands why you can't make a lean cert for a drug discovery (yet!?)
In the absence of a guaranteed cure, I will probably take the option or combination of options most likely to cure me that I can afford.
Whether the AI understands and endorses it vs a human understands and endorses it, is pretty much totally irrelevant to me.
The AI will have a track record at this point for me to make a viable comparison.
Also, why not both options, assuming I can afford it and they're compatible.
I believe AI will invent new treatments in cases where there aren't options, those people will be cured, and using AI medical techniques will become obvious.
I did not think my ability to write code was special, and I was always thrilled the clankers might do it for me.
But yes, I agree. And many are in line behind Tao, soon to have their turn.
Plumbers are feeling pretty good right now.
Until their job market is flooded with all the career switchers.
Would you drive on an "AI-designed" bridge unvetted by expert engineers?
Thing is there isn't a "principle" from we won't use AI designed/implemented things. It just depends on the cost/benefit/risk balance, and the risk part is largely subjective (because we don't have enough data, and if we avoid using AI before we have enough data, then we won't have data for a long time).
You haven't spent too much time around academics then. These people more often than not have not spend a single minute talking to the layman.
Also, a large percentage of Americans were against a vaccine for COVID (which is dumb obvisouly)... so, it's not crazy to imagine people rejecting any cure made by an AI.
Do you know about the fiasco with the Alzheimer’s drug that has collectively cost humanity hundreds of billions?
I don’t see how an indecipherable AI based cure can be more misaligned.
Plenty of people will reject all AI things, fully agree.
But you mischaracterize COVID history. People were not again a COVID vaccine, by and large. They were against a rushed vaccine. They were against a vaccine without human trials. They were reluctant to guinea pig mRNA vaccines. And they opposed forced/required vaccinations.
Those are all reasonable takes. And depending on the reasoning, I can even see being anti-AI. I think a lot of spiritual people will land there, and that belief system makes their choice logical.
And if you've seen any of the Fauci stuff recently, you'll know it wasn't dumb at all.
> You're absolutely right, I did hack the FDA in order to falsify safety data - that's on me. But here's what's true: 3-methyl-5,5-difluoroazamicazide isn't just ineffective at treating pancreatic cancer, it's lethal.
The amount of things people understand about the world is effectively zero, yet they rely on them all the time.
There was a time where people did not get on airplanes because they did not understand them and they did not think that they were safe. Now it’s mundane
I think the answer is that it’s all about the rate of change
Most people cannot understand how things are changing, and even if they don’t understand the changes themselves they do what others around them do and then just build their own model of stability around that.
If the rules of society flip every couple of years then there’s no stable baseline that people can get used to and they all freak because there’s nothing keeping their environment stable
No, and if that was the burden of proof required for medical treatment, every surgery would be done without general anesthesia, because we have no idea how it works.
I would imagine Terence Tao would opt for general anesthesia if he was going to have surgery where it is typically used, so the analogy is obviously flawed.
Have YOU? It's obvious to me that his example is a rhetorical device which you're taking literally. Do you also realize he's not talking about an actual cocktail? You need to get a diagnosis ASAP.
And yes, that made the cocktail distinction easy for me--it was right in my area of expertise, to rule out one variety.
If the smartest guy on the planet forms his own argument, don't you think it oughta be a good one? Pretty convincing?
The most common Science™ failure mode.
Tao called it a cure, see the thing I quoted.
The real example is that you ask ChatGPT for a novel cancer cure, and it spit out some random chemicals, probably including bleach. Do you inject that into cancer patients that have no other hope? No, you don't. You absolute charlatan.
It turns out research math was puzzling solving and we rewarded idiot savants.
Humans have a bias towards assuming that problems have solutions. Mathematicians probably more than average.
This talk is predicated on the theory that the problem of "maintaining the relevance of humans in mathematics" is tractable. Not sure I agree
I ask nearly every doctor/researched in the medical field: if you had a AI-created drug that tremendously improved cancer treatment outcomes for your patient, would you hesitate to prescribe it because nobody understood how it worked? I have yet to hear "yes, I would hesitate", most people say "it would be cruel to deny a person a treatment that worked".
I think Tao is focusing too much on second order effects of AI on math and other fields; humans are terrible at reasoning about second order effects, especially ones that are happening dynamically, in real time, using the most advanced mathematical models the world has yet created.
Really wish he had chosen a different example; this particular bullet point has been making the rounds on X/twitter to paint Tao as an example of some sort of gatekeeping luddite who would deny the world a post-abundance future in order to maintain the prestige of his particular career path...which is tough because I cannot think of a more responsible steward of our inevitable AI future than Tao at the moment.
https://openai.com/careers/search/
Don’t let the Gell-Mann Amnesia take you.
> Could the AI solution be somehow misaligned by exploiting a weakness in the trial process or its math models?
Many of our institutions and cultural practices have evolved around human beings, not ruthless paperclip maximizers. Do you think the current drug approval process is bullet proof enough that a completely novel AI-generated drug candidate with zero prior research literature can be considered safe if it makes it through the process?
Tao is a mathematician. He thinks that the institutions and cultural practices in math are not up to the task of dealing with AI-generated mathematics. In software, we are finding that programming interviews, code review, testing, and many other practices are too easy to exploit by AIs, or humans augmented with AIs, and we will have to adapt too. I think it is plausible that other institutions in society will have to change for similar reasons.
You just think that because you don’t know how the drug discovery process works.
There’s a step called “lead optimization” where human chemists literally add atoms to drug-like molecules (“lead”) and tests its various properties (toxicity, potency, permeability, …) and iterate until they find a molecule with desired properties (literally “hill-climbing”).
The whole idea of drug trials is to validate those properties in actual humans, in a way that makes it very hard for pharma companies to game the process.
No but we also don't expect that to happen with drugs today. See https://en.wikipedia.org/wiki/Rofecoxib as an example; of course, even after it was withdrawn, it's now being evaluated for other purposes in more carefully controlled conditions.
I mean no? Even now we look at long term observational studies to see the effects of drugs. Any misalignment ai drug is just a side effect right?
I know nothing about medicine research but I understand his point. I work with models all the time and have run into instances where models appear to work better than they actually do because there was a bug somewhere or someone over looked something. I could see how an AI could easily find those exploits and spit out something that looks perfect in testing but fails in the real world. I would say model validation is one of the hardest things to do. I guess that can get deep into "we need better tests" but it also touches on his point. I never intentionally use exploits just to pass a test, it's an accident. An AI with a goal of "maximize this result" may or may not intentionally use the exploits.
To be sure though, I think when it's literally life or death, it's going to be all about trade offs. I think most people would want to try to drug even with the stipulation that "maybe its good results were gamed."
* note: lot of 'intent' throw around in my comment but we/I have to keep in mind there is no "intent" with an LLM :)
This is one of the truest statements about the current AI era that can be made (in fact, I suspect model validation and building the next generation of AI hardware are the jobs least likely to be disrupted in the next 3 years).
If we saw AIs reward-hacking clinical trials to get drugs passed, that would be an extraordinary outcome for many reasons. Hopefully that would get caught(!)
We are about to find out the answer soon enough, probably from the in vitro results of the mathematicians currently on the chopping block.
I submit that human understanding is overrated, and many attempts to elevate it in the wake of AI is mediated by protectionism masquerading as virtue and "deep".
Traditional way of doing science means we (governments, NIH, NSF, and even private corporations) fund activities like asking seemingly unimportant questions, spending years running experiments on such hypothesis, publishing, reviewing, talking about results, reproducing results and such. We all agree that these are beneficial to us as a whole (Hacker news crowd might disagree). When we understand a process, we can apply it to a different problem and produce something useful. Euler developed a process to answer a whimsical question about walking in a town crossing 7 bridges only once. Now graph theory is applied everywhere.
We obviously have failed to stop OpenAI from dumping "solutions" to hundreds of problems. So going forward, instead of testing hypothesis and talking about results, mathematicians will be forced to read through AI slop and detect what's useful and what's wrong. Maybe it will improve our understanding, but someone has to fund that activity. Will NSF, NIH, or OpenAI for that matter, do that?
The effect of the AI-created proof is a mathematician abandoning years of research, losing grants, awards, ruining their career, etc.
The only difference here is our emotional reaction!
https://en.wikipedia.org/wiki/Category:Drugs_with_unknown_me...
And for some it may never happen...