alt Hacker NewsWill it be a golden age or a dark age for mathematics (and science)?
Comments
I get that Tao is a mathematician first and I respect the fact that he stays 'in his wheelhouse' (he didn't write the post but he did post it on his blog). But I think it would be fair to extrapolate and to wonder about this question not just for mathematics but for all of science. Keep in mind that scientific knowledge can be hoarded or used to patent stuff and that there is plenty of evidence that OpenAI, Anthropic and possibly others are going to go after other sources of income[1] than just selling access to their AI inference engines.
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...
> For those hard theorems where we really want LLMs to help us, these Lean proofs are extremely long, like hundreds of thousands of lines of code long. A seemingly emergent trend is that the longer the Lean version of a proof, the messier and less faithful its machine-generated “natural language” translation seems to be.
This seems to be a common and continuing objection to the OpenAI math repo release. Specifically that at least some, perhaps many of
* the natural language proofs are low quality or otherwise extremely difficult to understand (although possibly correct?)
* the lean proofs are too long, convoluted, or otherwise extremely difficult to understand
And specifically: iterate on the OpenAI repo to produce "better" proofs that prove the same things.
My read on Terence Tao's comments of late is that professional mathematicians are principally upset about AI-submitted math papers lacking the kind of broad teaching content that high quality human math papers offer. While the answer might be correct, AI math papers don't teach the reader in the way a human author would have. Professional mathematics is not just about solving problems; it's just as much about spreading knowledge so that others can program their human brains with the same understanding that the paper's author had when they wrote the paper (and achieved the result).
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.
These are all great points but I'd also point out that the 'dark age' scenario is a highly competitive scenario, while the 'golden age' scenario is a highly collaborative one.
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).
Obviously a golden age.
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.
I think I'm a bit on the "golden age" side of the question.
The math slopocalypse isn't useful without being able to turn it into math that can be taught to actual humans by finding the underlying unifying principles.
All those O(n log(n)^(1-delta)) results suggest that there's probably a unifying "trick" behind all of them and that there's going to be an upper bound on delta for any problem. You should really be able to teach that in a textbook and then have any human student replicate any of the LLM results as worked problems.
For anything that an LLM manages to prove with thousands or millions of lines of Lean code, there's likely a missing branch of mathematics sitting there somewhere.
Actual humans need to beat on the problems and the LLM-generated solutions until they come up with the underlying principles, probably using AI to help them do it.
You probably won't get there by just asking an LLM to invent a new branch of mathematics and "don't make any mistakes"; the mathematicians who get good at directing LLMs to search for unifying principles, and then synthesizing (and double-checking) the results, will likely be very successful in their field.
I bet for a short golden age where AI will reap the low hanging fruit of easy (for a computer) cross-referencing and inference. And then a dark age with none interested in putting new ideas out in the field. Hopefully, I am wrong.
Authorship and credit seem like anti-patterns in math. One of the least accessible parts of math to me is that literally everything is named after people. Navier-Stokes? Who? I don't really want to know to be honest, I'd rather the ideas and information be framed with a prioritization towards understanding, not author's reputation. I feel the same way about all of the Greek notation, it's 2026, can we not use something in regular circulation to broaden the audience?
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.
Some great thoughts, many of which extend beyond mathematics.
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.
Im going to go with it will be both , the Golden aspects and the Dark will co-exist depending on many factors but most especially the society and recption towards the changes it will usher not only in the way scientist and institutions choose to use the tools(basic research and advanced, peer review and citations) but also how the society will engage with it (awards accolades and grants).
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.
It may be relevant that Grant "3blue1brown" Sanderson (at https://youtu.be/LKrwhjOI8Nc ) and in ways Stephen Wolfram (at https://youtu.be/YRFfP1YDoVA , seemingly) have tackled the same question in the past few days. With an optimistic stance.
> I therefore disagree with claims that mathematics is over because machines can prove theorems. Without humans who understand mathematics, there is, quite literally, no human mathematics—a fundamental part of human culture. If we agree that we should not abandon mathematics because poorly understood machines bruise our anthropocentric vision of it, then society will need more mathematicians, not fewer. Like… a lot more.
Seems contradictory to me, referring to math as a field as “human mathematics” in one sentence, then deriding “anthropocentric visions”.
I'll be a golden age for mathematics and science, especially science.
It'll not necessarily be that great for every mathematician or scientist.
"Logistical component A is constrained by abstract concept B." am I being a dick, or is this not extremely interesting to read? TBF I've probably over-consumed this particular category of argument recently.
In the US, we've already seen the decrease in funding for education & research and closing borders down to many foreign researchers. The current admin and mainstream media is science-skeptic. We're already heading for a dark age. AI's contribution is largely because those policies are coupled with the same admin putting all their eggs into the AI basket, and likewise, basically all the major players in AI being on board with the admin's policies. This should be considered a systemic issue. Things like having a "healthy scientific skepticism and the limits of LLMs’ creativity" are important, but won't solve this alone.
I think Golden age, because I'm more inspired than ever to learn more math now.
I will go for "golden age".
Most of the skepticism towards using AI in maths could be applied to using computers in maths in general.
For instance, a computer can find a solution using brute force, like for the 4 color theorem. By doing this, we skip the potentially insightful part where the problem is reduced to something humanly solvable.
And yet, I don't think any mathematician today will tell you that computers are anything but essential tools.
I believe that AI is the same, and that mathematicians and other scientists will embrace it like they embraced computers. In fact scientists have been using AI (aka. machine learning) before it was mainstream, which is totally unsurprising because they invented it in the first place, the industry just gave them more resources, which turned out to be what AI needed.
All this “we will need far more mathematicians” seems like cope. Now that we can generate infinite text on the Bible, we need far more priests to understand it. It rubs me the wrong way.
I hate to be controversial but if it helps more of those brilliant minds move from exploring and proving less practical number theories and into applications of numerical science to real-world problems, with AI as a supporting tool I think this is where the golden age begins.
I don’t really have much sympathy for mathematicians, frankly.
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.
its not just math .. its math, science, engineering, medicine, tech .. ie. everything.
We need to ensure we dont outsource thinking itself - yes its great to have a lot of our knowledge encoded in books, and in the weights of LLMs [easier to access ]
Do we even 'have' a technology if it doesn't reside actively in some human minds ?
Some of math is handed down to apprentice mages, and not encoded well in books [ perhaps books dont encode intuition well, Ive had many cases where someone has explained something not in the textbook, a trick they learned from someone else ]
Apprentice mages only become mages by grappling with hard problems, asking questions, going on long detours, discussing/arguing with master mages, trying lots of tough thought experiments - all valuable "training".
Training using the AI might lead to a golden age where more mages learn wider and deeper, and the subject flourishes and more humans understand the tech.
The danger is using the AI makes us lazy - we dont grapple with programming syntax, draw math diagrams slowly by hand, check edge cases, try to prove things ourselves the slow way and build intuition in the process.
Another issue is ownership and access. Already RAM GPUs and soon CPUs are harder to get hold of due to competition with the hyperscalar datacenter buildout. This is an example of economic inequality effectively stealing resources from the mainstream.
Will math remain open or will it be owned by the frontier models ? With Math and science journals Elselvier almost had a stranglehold, bullying universities to pay massive bulk subscriptions - fortunately Arxiv solved some of that. Can I even own a copy of a music track, tv show or movie these days, when there it is so much more profitable for companies to rent it out pay-per-view or monthly ? There is every financial incentive to slurp up the whole of Arxiv, github and every science textbook into a frontier model and rent out access via pay per token. I no longer need to go to stack overflow to look up a simple algo, google can give me a pithy summary, so its likely that the LLMs will steal traffic from github, or github could be drowned in the volume of vibe code. Is there any point buying a textbook or going to university when the AI covers the material ? If no universities then we dont have a physical space for science and tech to be discussed. That activity might go online .. or may disappear altogether - we already have less socializing in person due to the rise of social media.
We have several "alignment" problems : humans arent aligned with each other, frontier models are aligned with profit-making which may differ from the common good, the race is on and so frontier models are more aligned to short term shareholder / VC value than to longterm benefit to humanity as a whole. These mis-alignments may have always been there, but are multiplied by the power and reach of new AI technology.
Mathematics is needed for numerous things. I do not see it going away.
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.
> Private control of frontier LLMs is potentially dangerous.
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.
>(4) Mathematical progress cannot be measured by checking off boxes on a problem list.
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.
> Without humans who understand mathematics, there is, quite literally, no human mathematics—a fundamental part of human culture. If we agree that we should not abandon mathematics because poorly understood machines bruise our anthropocentric vision of it, then society will need more mathematicians, not fewer. Like… a lot more.
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.
[flagged]
I love how Terrence Tao went from being AI shiller -- being a walking advertisement for openai and anthropic to a Oh nO MaThS iS KilLeD wHaT aBoUT mY JoB in a few months.
Terry remarks in a Nymberphile video that he thinks the Heroic age of mathematics is now over. the age of individual researchers is done, and hhatnlike many other disciplines, mathematics will be done in large teams, big facilities, and in complex ways that no sole individual will understand end to end.
I imagine he means in that video something like modern physics at CERN where no one single person actually understand how the whole thing works, but nonetheless is confident in the results.
I find Terry's hypothesis there somewhat comforting to my own limited cranium. most of the math he does already is far beyond me, nevertheless I trust that he is correct.
it's a different world for math, but one we all readily inhabit already everything we cross a bridge or buy refrigerated food.