Will it be a golden age or a dark age for mathematics (and science)?
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I can imagine a future where these companies become the new Zaibatsus[2] holding patents across a wide variety of subject and with mutually re-inforcing silos of proprietary technology created by in-house AI that the rest of the world financed but does not have access to.
[1] For instance: https://www.reuters.com/world/anthropic-quietly-sets-up-biol...
Who's "intellectual property" are we protecting, exactly, when a machine generates the knowledge?
I hope we can update before it's too late to matter.
AI and AI companies have a similar pathway open to them where they will take a lot of information already out there and then gatekeep the synthesis based on that information (obtained without compensation in most cases) as well as to harvest all of the latent knowledge underneath. This is a bad development in my opinion but I'm sure that their shareholders believe it is great.
The same misinformation exists about music agencies, who organize concerts and take a hefty fee.
But those were welcomed by musicians. In the early days (18th century), musicians had to do advertising themselves when traveling to a new city. The innovation was to have an impresario (literally someone who travels ahead of them) to do the organizing.
What musicians don't want is Internet middlemen who pretend to act in their interests and take an even higher cut.
https://en.wikipedia.org/wiki/Monkey_selfie_copyright_disput...
I suppose it's worth distinguishing the type of machine though.
If a genetic algorithm comes up with a unique design and I patent it, that's okay, right?
I think the main issue with LLMs is what went into them. Like we saw recently with Anthropic's revolutionary source control breakthroughs, which turned out to already be in the training data:
That depends.
To what degree was your input decisive?
Did you come up with the genetic algorithm?
Is your patent written in a way that it tries to stop others from generating antennas using genetic algorithms?
I mean, a society devoted to enriching all humans in multiple ways will want mathematicians. A society that aims to just suck all the value it can out of a given human wouldn't need and so wouldn't have mathematicians, if the answers they give can be provided by a machine.
In other words, the crisis in mathematics (as I read it) is that AI can solve without teaching. The result may be impressive, but if the only beings that can move math forward in future are machines grinding away on billions of scenarios that no human would have the patience to undertake, this leaves those in the profession wondering about their place in it.
No one from OpenAI is willing to give a talk on any of their 300+ results.
There is a fixed landscape of logically provable mathematical statements. Normally, the process of exploring that landscape simultaneously improves human understanding. Now, formal proofs and actual knowledge have become decoupled.
If you care about results, and AI doesn't cap out soon, the best humans can do is stand back and do nothing. Useless lumps that aren't able to keep up. Mathematicians wouldn't insist on teaching your dog their new theorem before calling their job done; why should an AI insist on teaching you?
If humans care about human creativity, and don't want to be mere passive observers, they need to make a choice.
AI used in a framework of human augmentation generally gets that for free. Used under the mistaken notion that humans can be easily replaced by AI gets you to the big complaints here.
(These are just my personal read on the topic -- the market is in AI frenzy where AI valuations are disconnected from productive capacity -- heavy on my mind and looking to get some writings out on the topic soon).
I argue this also applies with beauty. We got rid of that, also saying it was unnecessary. The result is sad.
Frontier models can easily one shot what I ask for, and make adjustments as ask to quickly make an app that meets what I wanted.
But often when I programmed outside of work (and at work too) I learnt something.
Quite often that my favourite pattern is an antipattern in some use cases.
That might never happen again at the rate of development.
In contrast, the value of a program is typically just the "black box" service it provides. Indeed, the program itself has always been opaque to most business stakeholders. Before, they paid human programmers and got back a program they didn't understand but could use; now, they pay an LLM provider and get back a program they don't understand but can use. In the pre-LLM era, the economic value of "understanding" and "shared knowledge" about the programs themselves was only that it made maintenance and changes cheaper. It never had intrinsic value, except as intellectual satisfaction to programmers (I was... am... one of those who found that value).
Learning is not an output-efficient process but is a fantastic outcome. AI lowers some portions of the development workflow but not yet every part, and, efficiently produced outputs aren't always the major outcome desired. In the real world / outside the benchmark lab, you care about robustness, reusability, composability, security, and more. Targeting speed to acceptable average output inferred from a prompt or similar benchmark metrics miss a big part of the game.
But paraphrasing
> I can produce in a month what took years for teams of hundreds of people
Is completely insane.
If you can really do that Anthropic would like a word with you.
Professional mathematics is whatever gets you funding so you can keep doing it. Today this is spamming papers and applying for project grants. Turns out the machine can do that quite well, oops.
A lot of progress comes from taking assumptions and breaking them. There was a time when everyone assumed that locked checkouts were absolutely necessary for version control, for instance. You could argue that AI could learn how to do this well but humans have context that machines won't. This is just an extension of the fact that things with independent existence will have different impressions of the world. If they didn't they would functionally be one thing.
> The result may be impressive, but if the only beings that can move math forward in future are machines grinding away on billions of scenarios that no human would have the patience to undertake
Seems like the same thing as neural networks. You have something like a few thousand parameters for computer that works well, but no underlying algorithms and formulas, like the shaders.
I don't think technology will fizzle out in either scenario. In the 'dark age' highly competitive scenario, you will still have those who 'understand what is produced' seeking progress and advancement. They will cohort with whoever/whichever organization is producing the best, most validated results.
I don't think knowledge would be become alien to human understanding, it would just become highly concentrated again as it was in the past. In the past, only the wealthy and noble could afford to spend time on education.
And usually, those who went deep into any field and provided contributions to science were those who had time to devote entire lives to the study or pursuit of knowledge. We still have these issues today, and these issues will always be present.
There is a supply chain of economy, and we provide different rewards based on the value of what is produced. The jobs never go away, the desire for prestige, greed, etc., never goes away. It's why economies exist, and it's why everyone is considering the value of LLMs, and their place in economies, and academies (presumably which exist to advance human understanding/appreciation, but also to prepare younger generations for future work).
What will the ethics of math research be like in the AI era? Will it be like the apparent ethics of the tech oligarchs?
As in: does competition or cooperation produce greater economic gains in terms of cost of living, quality of life, etc?
Subjectively, I've been forced to live and work under a competitive capitalist system that goes against most of my instincts for my entire adult life, which has resulted in a profoundly negative economic outcome for myself.
Others I know have done better, either finding ways to tread water as independent freelancers/artisans or managing to save modest nest eggs (although only by finding themselves on the winning side of the stock market, rarely/never through work at a job).
Vanishingly few have reached economic independence by winning the internet lottery for example, and I don't know them personally.
So based on the evidence before me, competition is a failed model. Yes yes, I'm writing this on an electronic device that cost billions of dollars of R&D to produce. I have food in my stomach and a roof over my head. But have I worked on important problems that would advance the state of the art in my chosen field? Of course not. And neither have most of you reading this.
So I would rephrase the question: will AI allow us to explore cooperation as an alternative economic model to the one we have?
I believe that it will, but not from the top down. If we wait for the wealthy and powerful to share, we'll be waiting forever. No, progress now must come from the most downtrodden among us creating alternative ways to provide time/money/resources to the underemployed and indigent so that they can finally self-actualize outside of the status quo.
Specifically, that means automating necessity. How do I acquire enough money to pay for my food and shelter today, even if I don't have a job? How can I acquire enough money to provide for myself, someone else, and some children (my personal definition of success) without being beholden to an employer? How can I remain insulated from inflation and the tendency for the owners of private property (not personal property) to devalue currency to make the means of production they own more profitable? How can I pay this forward to the next generation so that they can become educated and someday enjoy the commons which I myself have enjoyed? And so on and so forth.
If AI can't do those things, then frankly change may remain forever out of reach. Which makes the question of whether specific fields like math and science will benefit rather moot.
From a historical eye looking back the 'Dark' ages in europe coincides with the Islamic 'Golden' age and the eventual convergence led to the Enlightenment , when tose ideas that had been verboten in europe worked on it the rest of the world as far as China were reintroduced and sparked a new cycle of learning.
Similarly i see especially in the US dominated block netering into a partial Dark - with researchers taking extra pains to hold their research/data closer lest it be scooped and credit/funds/accolades go a brute force LLM run by a competitor.The few that would still publish openly will be the candles holding the ideals of open source collaboration AND communication.
At this point in time it seems to me that the China affiliated labs and orgs are more open in diffusing the knowhow , and coincidentally the society does not appear as anxiety ridden regarding the advent of LL tools.Maybe due to the established Shenzen mindset where everyone copies/learns from everyone else and the only thing that clears is the speed of execution.
In any case probably a convergence back to a equilibrium and a second Enlightenment after a generation.
I’m not bitter about this anymore — it’s been a long time. But the story holds something.
Back in the day, I knew I was talented at mathematics, and I wanted to pursue a research career. For a freshman research class, I selected a problem that was intentionally outside my comfort zone, and came up with some cool ideas. It wasn’t publication quality, though the course wasn’t really expecting publications.
Framing the problem in a way that it could be solved with calculus turned out to be a key insight. I turned to Wolfram, since the integrals were beyond what I as a freshman could have done myself, and I understood that the insight that produced the solution was my original work. I don’t think anyone in the class could have done those integrals. The instructor himself couldn’t tell me what was wrong with them if anything. Yet, he was unimpressed and subtly dismissive or critical.
Today, I might have been in the wave of researchers developing AI for mathematics work. The instructor could have been proud of shaping mathematics through his mentorship. The university might have invited me for talks. Instead, I ended up leaving mathematics as a field altogether. Now, I write software for a bank.
The world is as rich as we make it. Don’t let your opportunities and potential expire. Embrace the materials that are available to work with, or someone else will and they will out compete you and you will have missed the moment.
I get that there’s a romantic idea of what mathematicians do, and the field changing creates a sort of identity crisis. I would argue that the field has entered a new era. Exploring and shaping this era of mathematics is the opportunity of a lifetime. Embrace it, drive it, and make your mark.
Maybe it doesn’t matter and maybe that’s just the new flavour of golden.
Completely spitballing, I wonder what sort of sophisticated, collaborative computing effort might produce a paper analogous to the Higgs boson one, with 5154 authors.
Maybe AI can actually drastically improve this situation by bridging the gap between those "in the club" and those who want to do something useful with mathematical ideas.
I like the UK's archaic system of measures for the similar reasons. It has the stamp of humanity. For those who think about such things it reminds of who we were. Without historical context we think less of ourselves.
> They are a way of celebrating and honoring the human reach for excellence. If you like humans, that's a good thing.
I think progress is inevitable, ideas build on each other, and those with access to capital get to slap their name on their moment in time (yes, even if they did the hard work of converting that capital to progress). If AI takes over math, I really don't think we'll see a slow down in technological progress. Quite likely the opposite.
> I'm glad we don't call them Turning machines for instance.
Are you referring to a Turing machine; the abstract model of a computer, actually named after someone? https://en.wikipedia.org/wiki/Turing_machine
Your bizarre position seems to be that not knowing history somehow makes you more effective? This is honestly the dumbest take I have read all week.
One in particular, about sharing secrets, that had not crossed my mind (I have found that virtually no one ticks the "do not train on my data" box...):
> Similarly, it seems that mathematicians already concerned and guarded about what they write in research statements and research proposals. After all, an administrator or a reviewer may upload those documents into an LLM to get help in writing a report, thus potentially compromising the confidentiality of unpublished research.
When said button is in the basement, at the bottom of a broken set of stairs, locked behind a door that says "beware of the leopard", inside a filing cabinet, and implemented as
dontTrain() { //NYI
}
In some sense, it’s a simulacrum of the Internet’s forum experience - broadening one’s own audience necessitates debasing one’s content. But such is life - he now probably has for more readers than before and likely finds that pull gratifyingly attractive.
Seems contradictory to me, referring to math as a field as “human mathematics” in one sentence, then deriding “anthropocentric visions”.
It'll not necessarily be that great for every mathematician or scientist.
You literally need maths in just about every science, even in biology - think of statistics or any quantitative data set. Or when you need precision or microscopic analysis. cryo-EM is not possible without particle analysis.
Maths will not go away. It is more important than it used to be and it used to be very important already. Albert Einstein without maths wouldn't have been able to conclude much at all.
By all means, please read the article if it's interesting to you and ignore it if it isn't. But I believe that it's mostly directed to those already within the research math community, with a far more important purpose than to be interesting.
As the post itself explains, we (research mathematicians, as a profession) need to coordinate on deciding what a bright future will look like in a world with powerful AI, and in plotting a course to get there. And we need to do it fast, or circumstances might decide for us.
Historically, we've rather sucked at coordination. We need leadership -- we need someone to step forward, and begin by floating ideas and getting the conversation rolling. Personally I am very grateful for what Tao is doing here.
It strikes me that their field previously was quite heavily gatekept, and there was quite a lot of intellectual snobbery.
Seeing them taken down a peg or a hundred is somewhat gratifying.
How is that Bloom filter written in Rust using Agile coming along?
The author spends a lot of time trying to suggest that LLMs are not the end all be all and humans are still needed for understanding etc. Fine. I understand.
If it is so the case that LLMs don't help that much, surely it doesn't matter if LLMs are privately controlled?
How come LLMs are important enough that private control matters but also not enough for actually advancing mathematics significantly on its own?
In other words, you are almost always better off with LLMs even if it is privately controlled. The mathematics community can just ignore LLMs for the base case or use ChatGPT if they want. In either case, it is strange to suggest that things will get worse for them. Its as if they don't have the coordination or agency to adapt.
One big problem here is that OpenAI would be hoarding this ability by 1) not releasing their top models for public use,as we have already seen them do, and 2) limiting the capability of the public API, which we have also seem them do.
This not only kills incentive for math, it means that OpenAI becomes the arbiter of math and research for the world. The research that will be done is the research OpenAI decides to do.
This seems very dangerous to me.
Another path forward is distributed local models. Professors can do research on their HPC machines, individuals can have full access to their models. It's a much more democratic approach.
But coming to final answer is not important and that’s what the author spends time arguing. If they are right, surely whatever slop OpenAI is producing is worthless to human understanding which is the main goal.
The math community has to change prestige markers to promote people who contribute to understanding and not just solving theorems and that’s on mathematicians to do.
If solving problems is the end goal then I don’t mind OpenAI doing it in which case furthering math is already solved by someone.
The only failure mode is if mathematicians can’t get funding for just “understanding”. But it shouldn’t be that hard to do if their claim is right that their main contribution to society has always been understanding.
I’d argue that getting funding purely for understanding may be even easier to justify because the costs go down dramatically per unit understanding.
The question is, if OpenAI is in charge of all math, what will they research? Ideally they would do things to better humanity, but in the worst case scenario they would research war strategies, weapons development, voter suppression algorithms, etc.
OpenAI, and other similar companies seem to be so profit driven that their idea of research could so easily be perverted to cause net harm to not only the maths communities, but well beyond that.
The maths community can continue doing what they want and not be bothered..
Once we've essentially found an algorithm that can solve any math problem we've essentially reached the end of mathematical progress. Checking current solvers against unsolved problems is how we can check if we've reached the goal or not.
Of course humans can still study math, like how we teach people arithmetic, algebra, calculus, etc. But there is no more progress to be had. As a society we can allocate resources elsewhere.
This is just incoherent. Even if we "agree" (we don't but...) not to "abandon" math, how does it follow that "we need a lot more mathematicians" exactly?
Even if you reject the idea that AI can displace the current top quartile/percentile of human mathematicians, wouldn't it be reasonable to assume that it will at least displace some parts of math so that there'll be less demand for mathematicians overall, not more? The mental gymnastics are astounding here.
So the pie just gets larger? Large enough to outpace the displacement? I am gonna wait for this to generalize then, looking forward to AI giving us more software developers, mathematicians, etc. The economic laws have been broken. It's an infinite money glitch, as they used to call it.
I wonder how many mathematicians even truly understand any of the latest research which is only going to increase in output. It will be impossible to keep up unless you slow AI down (not happening).
Edit: Anthropic blog https://www.anthropic.com/research/employment-at-anthropic
The goals of transhumanists go quite a bit further than just replacing people in companies.
2) TFA argues golden age, provided LLM infrastructure is independent.
3) Tao just opened a new SAIR AI platform.
4) Tao just opened a journal for papers with Lean proofs.
5) Academics are known to play both sides of the fence.
It is absolutely bewildering how some people miss the finer points.
It would be cool if some prestigious such program revamped itself to be AI-focused, maybe in collaboration with a CS department.
> Universities, professional societies, funding agencies, and scientific foundations should coordinate the development and governance of competitive LLMs. Scientists will need access to frontier capabilities, transparency about their operation and use of research data, strong confidentiality protections, and the ability to test and reproduce research claims independently.
> The massive computing resources, specialized personnel, data infrastructure, and ongoing investments needed to support this kind of initiative are not within the reach of a single institution, and a huge leadership effort will be necessary to conjure the combination of independent model development, guaranteed access to commercial systems, public computing infrastructure, and open research resources that could do the job.
the post is calling for....an enormous investment in LLM infrastructure.
IMO this is probably not an accurate assessment of Tao in particular at this time.
But in the general case, I think most people have a set of blinders on that will result in a predictable reassessment of AI with the following steps as AI progresses when applied to whatever it is they do:
1) AI is useless and doesn't impact me at all. Look at how bad it is! (Most developers up until late 2025)
2) Ok, now AI is useful but will not replace me. Sucks for the juniors in the field but I am still needed to guide the AI. AI will never be capable of my judgement (or whatever). (Most developers now)
3) Oh shit, I've been replaced in terms of economically valuable utility by AI. (tbd, but likely sooner than is comfortable)
And not everyone will experience #3, there will be a top 1% or so for most fields where #3 remains true for them, but it is likely that for most people #3 comes eventually.