Terence Tao: Math 2.0 [pdf]
teorth.github.io105 points by Anon84 3 hours ago
105 points by Anon84 3 hours ago
> 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?
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.
And to be fair - that how medicine works today. We don’t really understand what goes into our body, so that’s why we do clinical trials. And sometimes we discover adverse side effects years and decades after a drug has been released.
Yes, and that isn't the model's fault any more than it would be for a human.
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.
Yep, I think a lot of them are grieving the loss of their identity. Quite understandable given how much of their life force they've poured into math, especially at the level of that Tao is at.
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.
Ironically this is one of the worst counterexamples he could have selected.
If you read all the questions asked by Terrance you would have understood what he is thought behind that question "Could the AI solution to the prompt be somehow misaligned by exploiting a weakness in the trial process?"
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...
He didn't do us any favors by writing that slide in a convoluted and inaccurate way.
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.
Not only that, but also those clinical trials often have low sample sizes. Trials involving only a few hundred patients, or even fewer, are pretty common.
not just clinical trials- there is an enormous amount of in-vivo model study before the molecules go into people.
In vivo is already molecules in body. Do you mean in vitro?
No. I mean in vivo in non-humans, or in human-derived cell lines. https://pmc.ncbi.nlm.nih.gov/articles/PMC6061782/
Agreed. IIRC the mechanism for anesthesia in unknown to this day. In fact, medicine as a whole is very pragmatic and time and time again prefers using techniques with unknown mechanisms but proven results than waiting for a reasonable scientific explanation of what’s going on.
As logic would dictate--particularly for people who aren't the highest IQ guy on the planet.
Half of people are below average! They understand nothing at the level that Tao means. Literally nothing.
Perhaps I'm in the lower half but I do not understand what point you're trying to make. Are you saying that as some higher plane of understanding Tao is actually correct? My thinking is that a simple counterexample would be sufficient - i.e. are there medications that people use for which the mechanism is not understood? There are many such examples as others have given in this thread.
Logic would dictate that medicine should be pragmatic and accept working cures that aren't understood.
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?
Now I’m even more confused, perhaps we are in agreement. My position is that there already exists many such medications that no one knows how it works. It also used to be the default way medicine worked. Lots of things that work were found though observation of trial and error.
My position is that Tao is not only wrong on this but his example makes the opposite case.
Plenty of medicine... Not a single human on earth knows how they work. Now what?
I don't expect Tao to be well versed in medicine, but yeah, that slide is detached from clinical practice.
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.
>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.
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
Nobody understands the mechanism behind general anesthesia, yet there’s a medical speciality dedicated to practicing it and it’s done every single day for a wide variety of procedures. We have figured out how to use general anesthesia in a relatively safe way, but nobody understands why it works.
False. Experts don’t understand many drugs and still prescribe it. And it’s a good thing.
The problem is you won't be faced with a binary choice of certain death or a single unknown AI cure: You will have a choice between several established protocols AND several unknown AI drugs. You doing a dice roll or do you want to make an educated decision?
I don't know how to word this in a way that won't get me into a pointless semantics argument, but the word Cure in the slides is clearly used to mean "substance intended to be (but not necessarily confirmed to be) a cure". If you think it's badly worded that's fine, but ostensibly the point here's not to "win" the argument game, but to get his intended point and engage with that.
No, it doesn't mean "intended to be". "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".
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.
The above commenter has the same allegory- most “users” of math don’t care about the body of work behind it; only the consequence of it being proved is the fact that matters.
But the slide says it passes a stage 3 trial. Surely many dying patients would kill for the opportunity to take the drug.
Many people would, many wouldn’t. The comment I’m responding to casts the decision as so uncontroversial that even asking the question is inhererently damning.
You're glossing over the false statement in your original argument that the cure was not confirmed. The slide says it passed stage 3.
Beat me to it. It is sad to see that a lot of the people commenting lack good reading comprehension. Or maybe it is that they just want to be dismissive of something they don't like, or just for the sake of it.
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.
Wrong: the slide says "the cocktail indeed passes a stage 3 trial." Be careful before you accuse others of bad reading comprehension.
He’s making the same argument I’ve heard software developers make for the past 3 years. Lawyers are saying similar types of things. Knowledge work isn’t as special as we all thought it was. Now, it’s Terence Tao’s turn to go through the same motions we’ve had to go through over the past few years.
I think it is more disconnected in thinking people want to understand the mechanism. As a stage IV cancer patient, I’ve met many people who decide not to continue treatment with approved drugs due to side effects. The notion that every cancer patient will do everything possible to stay alive isn’t true. Most people have a breaking point.
We don’t even know how acitaminophen works, thousands get their livers destroyed every year from it when alternatives exist, and we are yet to ban it. So how is all of the new reality any different?
Funny enough, it is a good argument with just a sprinkle of reality for context: Assume effective alternatives to this cocktail exist. Assume you had access to these effective alternatives. Can this be combined with other drugs? Surgery? You say you just need to cure it. You must consider the deeper mechanisms at play to consider all the possible options for living.
Mathematicians inhibit and explore a world that is completely made up and exists only in abstract.
I wonder if this makes them slowly get disconnected from the lived realities of billions of ordinary people.
How could a theoretical mathematician not be massively disconnected from the reality of life for the ordinary person? We usually celebrate their quirks until they come into conflict like this.
I agree, how could he --also being so smart.
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?
When thinking about math, Tao is in another world unfamiliar to us. But, he grew up in our world and is a professor in our world and if you listen to him, seems like a pretty normal guy. I think he just made a bad analogy here.
It doesn't even need to cure cancer, just prevent it to some unknown degree with unknown side-effects. As long as they want people to take it, they will.
> t's clear he hasn't spent much time around average people.
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.
I think his argument is more "Could the AI solution to the prompt be somehow misaligned by exploiting a weakness in the trial process?" So an AI agent could game (lack of a better word) the process and produce something that may cure cancer, but misses something else.
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 think the current clinical trial process has not been gamed close to the edge of uselessness ?
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.
> 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.
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 right on -- the point is an alignment point in a talk that only uses "alignment" in a completely different sense than it usually is, which is incongruent at best. It's gonna lead to a way bigger backlash among laypeople/policy makers/non-expert stakeholders than the rest of the talk will lead to new growth, combined :(
The cancer thing is just a contrived analogy. His real concern is having himself and his mathematician friends replaced by AI. Appealing to someone's health is an easier sell and then it can normalize the relationship of medical priests, and math priests, etc.
> 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?
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.
The problem with his argument is that he put the cart before the horse. He presupposes we've found and validated a cancer cure, but at that point any rational person would accept said cure. The moral quandary is how you are going to validate it against all the other possible candidates, when you don't have a coherent rationale for it to work.
What makes that quandary the moral variety?
Not understanding a mechanism doesn't make me, or the mechanism, good or evil.
We just treat it like all the other cures right?
> but it's clear he hasn't spent much time around average people.
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.
I own a bar, so...yes, I have. More than my fair share.
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?
It's science (understanding) before engineering (efficacy).
The most common Science™ failure mode.
I agree, if you mean he's prioritizing that.
But since when do we call for a stop to figuring out new things? We don't want a cancer cure unless we understand it?
N people should die because...math is cool for humans? Or?
What if it takes your five years to live down to two in horrible agony?
Then it won't be a cure for cancer
Tao called it a cure, see the thing I quoted.
I think this is just an embarrasingly bad faith / low effort way of engaging with the argument.
"its model predicts [...] will kill cancer cells"
> 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?
It's a cure
...his point is we don't know if it is a cure or not, and it would be good if humans had an idea instead of just having to take the AI's word for it.
I work in drug discovery and AI and his slide about cancer medicine isn't very well informed. We have models and narratives about how drugs work, but they are woefully incomplete.
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.
You are picking on 1 bullet point out of 7. I think the following bullet point is the crux of his argument:
> 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.
> 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?
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 think everyone agrees that it's totally fine if a drug came about due to AI.
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 :)
> I would say model validation is one of the hardest things to do
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(!)
In medicine, or drug discovery more generally, there's FAR too little empirical data for training of ML generally. See OpenAdmet. The kickback to "oh but yeah cancer" is currently just hype. DeepMind has pivoted to this realm but has no demonstrable improvements. For the typical phase 1-2-3 pipeline of drugs, stretching many years, there's not yet any demonstrable improvements.
I think your perspective is limited- in cancer, we have copious genomic information that informs treatment (including clinical trials where treatment is determined by AI).
The effect of the AI-created drug is saving someone's life.
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!
Slide 12 and 13 sum up my conclusions from the Oct 8 100+ reactions to 100+ solutions. Several reported OpenAI drop included answers put a wrap on problems they had been working for years. Several said that they have to completely rewrite grant requests that they had just submitted. Others described learning of the solutions problems they had been working on like losing an old friend or a lover. The general sentiment was to bemoan the loss of a field, as if it would have been better be born in the early 20th century and conclude their career arc before reaching this point. My take away is that the field needs to get it through their heads that their old problems are no longer ambitious, and that their job in the short term is to find the new frontier.
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."
I don’t think your second paragraph is a fair representation of what Tao is saying or contradicts his argument. He is arguing that AI can be useful in the service of human understanding in math and the examples you gave are exactly that. Whilst OpenAI just spammed the AI button the mathematicians that engaged with the output were able to progress their understanding about the problems in some ways (some of which they don’t really like). You need both parts for this to be useful to the field. What’s not clear is whether the juice is worth the squeeze: will all the money spent on spamming the AI and human mathematician time spent studying the outputs progress the field “better” than without the AI?
> solve open problems without producing insightful new methods
> 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.
The DFT paper is an example against AI 'strip-mining'. The paper introduces a new method, and researchers are already trying to improve on it. If anything, the OpenAI dump re-vitalized that branch of study.
Software engineers are primarily outcome-oriented.
Mathematicians are primarily understanding-oriented.
Leveraging AI to tackle new frontiers without true understanding converts mathematicians to engineers.
Actually both are outcome oriented and both can use AI to compress decades of progress. One camp accepts this naturally. Other camp is making their profession to be mysterious and spiritual to run away from the implications of AI
Should we feel sorry that a mathematician had to write a grant request they submitted because an automated tool did what they wanted to do?
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!").
> Before injecting this cocktail into your bloodstream, would you want to know that there is at least one human cancer researcher who understands the mechanism behind this cure.
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?
AI has been turning computer science into biology for the past decade or so. By which I mean that things like neural networks need to be investigated empirically, constructing methodologies and instruments that more closely resemble how fields like biology and medicine have to probe the very complex and messy reality that is beyond our current capacity to fully express in symbolic precision.
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.
Indeed, aren't medical trials based on its effects, rather than how it works?
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?
Some mechanism of particular substrates being effective on a condition (commonly when a medicine is repurposed) not being fully understood is not the same as just guessing with a medical compound. AI boosters keep coming out with this line but it's basically wordplay to conflate the clinical/biomedical version of "not fully understood" with the LLM industry version of "not fully understood"
> Using AI to find and highlight new principles, methods, or insights, rather than merely new proofs.
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.
Absolutely not. New proofs aren't the same thing as new proof techniques, AI is not generating new techniques (yet), and while the existence of more mechanical proofs is interesting those same problems if left to human mathematicians would have been much more likely to actually generate new techniques. Much like how tech has a "juniors" problem we're pushing on the future (no reason to hire juniors, so where are tomorrow's staff engineers going to come from), OpenAI's approach generated a "questions" problem where math and AI could happily coexist if we designed that correctly, but instead nobody's going to be generating or working on the right questions anymore.
I just watched Primeagen’s video on this and Tao’s point is that juniors no longer have the path of solving a proof to earn Field’s medals. He also argues that the community part is being hurt by AI discovering proofs because in the past people used to get invite to talk and collaborate. Now all that is being taken away. The community must adapt because Pandora’s box cannot be closed.
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?
> Tao’s point is that juniors no longer have the path of solving a proof to earn Field’s medals.
This is a "you" problem for the math establishment, not a problem for the AI companies.
Entirely correct. The math establishment needs fundamentally overhaul its incentive structure-irretrievably broken-to function under the assumption that AI involvement in research is completely ubiquitous.
I'd go beyond that and say they need to overhaul their culture and mode of operation. Math needs to be even more collective than it is today, without focus on ego reward and priority. They were already steps in this direction before this year's AI detonation: net-enabled collaboration, first informally and then with Lean formalization. Perhaps there should be a de-emphasis on naming things after people.
Our industry is equally cannibalised, anyone that thinks otherwise is either having too many tokens or in a privileged position.
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.
For now, a skilled person using AI is still miles better than an autonomous AI building something. I’ve been trying to do the latter for months to build open source alternatives and the end products still lack polish and that last 20%. Maybe this changes, but I still think there will be people who can use that AI to be better than AI alone.
Read the second to last slide. What we need now is *imagination*. You assume the need for new software is fixed and that AI is going to satisfy that need with fewer humans (lower cost), but by lowering the cost we can increase the supply of software!
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.
Primeagen is a youtube reaction guy. You may as well tell me what hbomberguy, Jay Leno, or Ben Shapiro had to say about it.
I mean this is funny ad hominiem but OP has a point. Academia has always massively prioritised understanding over outcomes, injecting startup culture into it is basically injecting antimatter.
Putting aside the cancer question: I still don't understand how TT seems to be fixated on what models can do today instead of tomorrow. It's realistic and even conceivable that the models will also become better at explaining and presenting proofs too. Maybe he's not emotionally ready to accept that there may not be a future where his (and to some extent, my) skills are relevant and valued. It breaks my heart. I hope I'm wrong, but it feels like we've run out of higher ground to run to.
This crisis exposes a conflation between A: The broader concept of [abstract] Mathematics and B: The contemporary Mathematics culture and community. This crisis is directly in B only. B will adapt: In how it attributes value, status, hierarchy, and career. There will be a death (Or something close to it), and rebirth. Through this, A will advance in a Kuhnian leap - habits will be broken as incentives changed, and paths ignored will be explored. Insights will flow to the sciences.
I'm excited!
From the slide Beyond Problem Solving:
"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.
> This is in stark contrast to current AI performance on tasks which are subjective, dependent on real world interactions, or for which data is scarce. AI performance is thus extremely jagged: astounding in some directions, while inadequate in others. This is true both within mathematics, and more broadly.
underrated buried comment based in reality
In a nutshell, Math 2.0 is not fully compatible with Math 1.0 and the forced upgrade is breaking features, plug-ins, and we're tracking several new bugs, but this is still the fastest, most secure, and best version of Math ever released, with powerful new features and unrivaled privacy.
It’s funny to see people in the STEM field focus more on the human aspect of creation. It used to be that the result mattered more than your feelings. Now we are moving the goal posts about how things should be done.
I wonder if we will begin to actual value human creation more at the end of all of this
> The future of mathematics — “Math 2.0” — will require both expanding the research frontier, while simultaneously decentering the traditional role of problem solving.
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?
How do we measure if someone is a good mathematician or not?
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.
This consolidation of power is precisely why there is a need for open model development, and exactly why frontier labs have been lobbying hard to abolish them.
I think a better analogy would be conducting a marathon in a fog. If you can't see the path of the proof, how do you know it is completely true in all scenarios? If you can't see whether the AI runner ran through every part of the race, how do you know it didn't draw hallucinated shortcuts in the parts where humans can't see? Or worse, create obscurity and blow smoke to hide the shortcut section? If the same AI was to guide the last living humans to a star, because it found a path clear of danger, could you trust it to get in that ship? Or did it just forgot mentioning an asteroid belt the ship is not built to navigate? Truth is verifiable truth that multiple parties can agree upon. Can you trust with your life something you can't verify?
I'm learning so much about specific vs. generalized intelligence through this entire ordeal.
Exactly. I've never thought as much about metacognition nearly as much as I have up until this point, for better or worse.
Took me far to long to understand that one person can be an expert in one field and be absolutely clueless in another (not trying to throw shade to tao with this post)
page 20 ("A thought experiment on alignment and understanding") is perhaps not likely to quite induce the reaction in most people that Mr. Tao expected.
You described my point much better -- totally agree. It also seems he maybe isn't aware of the nature of experimental drugs, which are often given to patients with serious/terminal conditions before they'd otherwise be approved...
The cancer arguement on slide 20 is pretty weak. I first need to be alive in the long term to worry about the long term effects. If I had terminal cancer I'd gladly take an AI developed 'cure'.
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.
I'm currently on two fairly common medicines that have, in the first paragraph when reading about them, "doctors are unsure of the specific action, but it is thought that [medicine does x to y]"
If I'm terminal with cancer, I'll inject whatever if it can cure that.
Exactly, we only get to run risky clinical trials because patients are 100% desperate and know they are likely to die otherwise.
It is nice to see some sanity back in the conversation!
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.
There is definitely a kind of monocrop problem in some fields, where it seems like having a very diverse spread of academic investigation is needed to have enough diverse traces through the search space. And globalization has been flattening that.
I'm heartened to see that a decent number of slides in this isn't the run of the mill doom and gloom but some actually interesting and potentially productive offshoot ideas.
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.
I think people are missing the point that terrence is trying to make, especially on the cancer drug.
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.
This is a strawman. No one’s saying it’s going to be sudden or anything. It’s gonna have clear provenance and the same type of verification channels
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.
>>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
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?
>>No one’s saying it’s going to be sudden or anything
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.
It's quite noticeable how when Terence tao was sounding pro-AI the sentiment was much more positive and he was being held up as an authority to listen to. Then he puts out some limitations of AI in a presentation about how to work with it as an expert and suddenly on HN he is just some out of touch killjoy trying to hold back progress etc.
> 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?
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.
Just a ~~few~~ ton of things (sorry!), with the upfront caveat that Tao is a hero who's trying his damndest:
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...
Regarding 6., that we shouldn't throw AIs at unsolved problems in bulk and thus lose human expertise. I think that ship has well and truly sailed. AI will continue to be thown at all manner of unsolved problems, and at an increasing rate - by commercial labs, by individual researchers, and by academic research teams. The mathematical community needs to establish paths to meaning, progress, and growth of human expertise in a world where AIs will by default be thrown at every unsolved problem.
>" “Ablation studies”: taking an already proved theorem and seeing whether it can still be proved after
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!
Another example that people should not use medical analogies to illustrate a side point: Discussion boards will focus on the completely irrelevant side point to drown out the renewed AI caution that they do not want to hear.
"our work has brought about enormous advancements to the field of mathematics and we don't like it"
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.
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Has the world gone mad? In which universe would you get a mathematical model to produce a tonic of random chemicals that optimize said mathematical model, put that through stage 3 testing, and only THEN ask if you want to inject this into yourself. My Guy, you just did clinical testing on fucking humans for phase 3, it's a little late to consider the moral ramifications of the analysis-experiment dichotomy.
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.
Man with a cozy and soon to be irrelevant sinecure grasping at straws.
It turns out research math was puzzling solving and we rewarded idiot savants.
Are you going to say that of every other white collar job, since most are likely to be automated? He was problem solving with Erdos at age eight. What were you doing? What do you do now?