Terence Tao: Math 2.0 [pdf]
268 points - today at 1:16 PM
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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.
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?
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."
"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.
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.
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.
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?
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.
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.
"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.
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?
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)
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).
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.
underrated buried comment based in reality
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.
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...
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.
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.
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