Mathematical methods and human thought in the age of AI

by zaikunzhang • 4 months ago

218 93 4 months ago

Recorded 10 February 2026. Terence Tao of the University of California, Los Angeles, presents "Machine assistance and the future of research mathematics" at IPAM's AI for Science Kickoff.

mchinen3• 4 months ago

He also was on Dwarkesh's podcast last week (https://www.youtube.com/watch?v=Q8Fkpi18QXU ).

I enjoyed the human->depth vs AI->breadth discussion and the waterline rising slowly to fill the 50 lowest hanging Erdos problems but struggling on the next few.

bluecheese4522• 4 months ago

Enough Terence Tao spam.

mchinen3• 4 months ago

I haven't seen any negative sentiment toward Terence Tao before. Coming from outside the academic math sphere, genuinely curious if there's a real issue or if this comment is just spam itself.

tines• 4 months ago

This comment is spam. When Tao says something we should take it seriously.

myhf• 4 months ago

A big trend in AI spam is to take achievements in one field that could be called "AI" and use them as evidence of advancement in other fields that happen to be called "AI".

https://en.wikipedia.org/wiki/Package-deal_fallacy

Tao has been doing a lot of demonstrations of using LLMs for search and translation by experts who already know enough about a field to judge whether generated text is valid or meaningful. Those are valid demonstrations, but they don't justify the LLM-as-intelligent-agent narrative being pushed by most of the reporting on the topic, so the whole situation reeks of payola.

gettingoverit2• 4 months ago

It's actually a strange situation. Tao is the only high-profile mathematician who tries to resolve the reproducibility crisis in math by popularizing proof checkers, and who used LLMs in probably the most legitimate way: to produce formal undeniable proofs with them, that cannot really suffer from LLMs being wrong all the time.

On the other hand, I still find content and arguments he produces to be quite weak, and honestly it's getting annoying to hear them that often. It's the case when he could really get some help of a ghost writer who is more experienced in popularization, otherwise this repetition might cause some serious harm instead.

bluecheese4522• 4 months ago

I have nothing against the guy. What I don’t like is seeing 50 hacker news posts about him every week.

ancillary• 4 months ago

So much of HN is half-baked anecdotes about and by LLMs or philosophizing from VCs who talked to an LLM about Rene Girard for twenty minutes or pop sci articles that appear to be posted so that some bored developer can read the abstract and one experiment and dunk on it. Tao is uniquely positioned as a mathematician who has made enormous contributions to many areas and is old enough to contextualize it all against the past and young enough to be open to its possible futures. More Tao spam sounds good to me!

gradstudent• 4 months ago

I skimmed the paper a couple of times, hoping to find the promised (from the abstract)

pathway to integrating AI into our most challenging and intellectually rigorous fields to the benefit of all humankind.

There's very little insight here though. It seems mostly a retread of conversations we've been having in the academic community for a few years now. In particular, I was hoping to see some discussion of how we might restructure our educational institutions around this technology, when the machines rob students of the opportunity to develop critical thinking skills. Right now our best idea seems to be a retreat to oral and written examinations; an idea which doesn't scale and which ignores the supposed benefits of human+AI reasoning. The alternative suggestion I've seen is to teach prompt engineering, which seems (a) hard for foundational subjects and (b) again, seems to outsource much of the thinking to the AI, instead of extending the reach of human thought.

BDPW• 4 months ago

Physical classrooms don't really scale either, is that really a fundamental problem?

lo_zamoyski2• 4 months ago

Indeed. Education isn't supposed to "scale". We've mucked around with education so much and subjected it to tech fad after tech fad that we hardly have anything resembling education.

Because this has been going on so long, most people's reference point for what constitutes "education" is simply off, mistaking "training" or something like that for it. But the purpose of education is intellectual formation, the ability to reason competently, and the comprehension of basic reality, which enables genuine intellectual freedom (there are moral presuppositions, too; immorality deranges the mind). This is what the classical liberal arts were about.

The very bare minimum criterion (and it is a very bare minimum) for someone to be able to claim to be educated is not only knowledge of their field, but knowledge of the intellectual nature, foundations, and basis of their field in the greater intellectual scope. I would not hold someone with only that bare minimum in especially high esteem vis-a-vis education, but even that bar is higher than what education today provides.

bonoboTP3• 4 months ago

Yes. Tools like Khan Academy help lots of talented kids to progress in the curriculum beyond what's available in physical classrooms available to them.

falcor84• 4 months ago

I don't know if this is what the GP meant, but obviously classrooms do scale, in the sense that almost every kid in the world has access to a classroom. But arguably, for better or worse, this scaling was enabled by "cramming" many of those kids into relatively large classes, with little 1:1 attention from the teachers, and with standardized (mostly multiple-choice) testing, rather than old-school oral testing.

As I see it, it is a very salient question - what would the economics of global schooling look like if we decided that it's imperative for every student to get personal pedagogy and regular individual professional oral examinations for their schoolwork?

nutjob27• 4 months ago

when the machines rob students of the opportunity to develop critical thinking skills

This is a fundamental misunderstanding of human nature. Machines don't rob people of critical thinking skills, people do. Mostly people do it to themselves, often inheriting it from their parents or social environment.

ak_111• 4 months ago

Wait it seems like doing unscalable things - like face-to-face teaching/examination - is exactly the sort of things that humanity can afford to do as it benefits from the surplus free time generated by AI efficiently doing the scalable things.

woolion2• 4 months ago

We assert that artificial intelligence is a natural evolution of human tools developed throughout history to facilitate the creation, organization, and dissemination of ideas, and argue that it is paramount that the development and application of AI remain fundamentally human-centered.

While this is a noble goal, it seems obvious that this isn't how it usually goes. For instance, "free market" is often used as a dogma against companies that are actively harmful to society, as "globalization" might be. An unstoppable force, so any form of opposition is "luddite behavior". Another one is easier transport and remote communication, that generally broke down the social fabric. Or social media wreaking havoc among teen's minds. From there, it's easy to see why the technological system might be seen as an inherent evil. In 1872's Erewhon, Butler already described the technological system as a force that human society could contain as soon as it tolerated it. There are already many companies persecuting their employees for not using AI enough, even when the employee's response is that the quality of its output is not good enough for the work at hand, rather than any ideological reason.

I'm neither optimistic nor pessimistic about the changes that AI might bring, but hoping it to become "human-centered" seems almost as optimistic as hoping for "humane wars".

cowpig2• 4 months ago

"free market" is often used as a dogma against companies that are actively harmful to society

This is a predominantly America-specific piece of propaganda, and it's pretty recent.

Adam Smith's ideas are primarily arguments against mercantilism (e.g. things like using tariffs to wield self-interested state power), something he showed to be against the common good. The "invisible hand" concept is used to show how self-interested action can, under conditions of *competitive markets*, lead to unintentional alignment with the common good.

Obviously that's a significant departure from the way it's commonly used today, where Thiel's book has influenced so many entrepreneurs into believing Monopolies are Good.

But the history of this is very Cold War-influenced, where "free markets" were politically positioned as alternatives to the USSR's "planned economy", and slowly pushed to depart further and further from Adam Smith's original argument about moral philosophy.

abdullahkhalids2• 4 months ago

Economic behavior is inherently game theoretic - agents take various actions and get some positive/negative reward as a result. Whether an agent's reward is positive or negative and of what magnitude, depends on the strategies employed by all agents. If some agents adopt new strategies, the reward calculus for everyone involved can completely change [1].

Over the past few centuries, countless new economic structures and strategies have been discovered and practiced. The rewards for the same action today and in the past can be completely different due to this.

So to me, if someone claimed more than a few decades ago that certain economic strategies and structures are good or bad, its simply not worth listening to them, unless someone reconfirms that the old finding still holds with the latest range of strategies. In that case, the credit and citation goes to that new someone, not the ghosts of the past.

[1] A good interactive demo https://ncase.me/trust/

schmidtleonard• 4 months ago

arguments against mercantilism

It has been funny to watch the rise of "China is beating us" rhetoric against the steady backdrop of "mercantilism is obsolete/bad" dogma, because the elephant in the room is that China has been running a textbook mercantilist playbook.

naasking4• 4 months ago

Thiel's book has influenced so many entrepreneurs into believing Monopolies are Good.

Haven't read his book, but the idea that monopolies are good isn't typically made in a vacuum, it's made relative to alternatives, most often "ham-fisted government intervention". It's easier to take down a badly behaving monopoly than to change government, so believing monopolies are better than the alternatives seems like a decent heuristic.

slopinthebag2• 4 months ago

The government is a rare example of an extremely strong monopoly and not just a duopoly or a company holding significant marketshare. And yet people never seem to criticise it on those grounds despite it suffering from all of the same problems that corporate monopolies are accused of.

Izikiel433• 4 months ago

Globalization was great for poor countries, not so much developed economies.

js8• 4 months ago

No it wasn't. Look at Joseph Stiglitz (Globalization and Its Discontents) and Ha-Joon Chang (Bad Samaritans, Kicking Away the Ladder) for counter-examples.

nutjob27• 4 months ago

This isn't correct. The deal is that the poor countries get development and increased employment, and the rich countries get lower prices. Generally speaking both types of countries get richer.

That some workers lost their jobs is a symptom of any change. I don't know why people always get upset people losing their jobs. It's like death, if no one died relatively few people would be born. If you resist job losses you reduce overall employment and economic development.

intended2• 4 months ago

Oh come now - globalizations was great at the regional level.

It was not that great for sub groups within developed nations.

The original thesis believed that people would be retrained into other equally well paying roles.

Turns out people can’t retrain into new domains, and led to under employment.

zitterbewegung3• 4 months ago

Not even sure if AI was ever "human-centered" . DARPA funded a large amount of AI reseearch from even the 1960s. The DART tool used in the Gulf War made back all of their previous investments. [1]

[1] https://en.wikipedia.org/wiki/Dynamic_Analysis_and_Replannin...

hammock3• 4 months ago

I mean were computers themselves ever human-centered? What’s the difference between AI and a computer?

woolion2• 4 months ago

I just re-read my comment (too late for edits) and there are a number of typos (including missing "not") that significantly degraded the syntax, but the point kind of came across anyway.

sendes3• 4 months ago

We assert that artificial intelligence is a natural evolution of human tools.

While nowhere in the paper this is actually asserted but the abstract, a whiggish narrative of a genuinely unprecedented technology --such that it can replace and supersede human "labour" altogether (one is reminded of The Evolution of Human Science by Ted Chiang)-- sounds naive at best, dangerous at worst.

jebarker• 4 months ago

I don’t see why “natural evolution of human tools” implies “such that it can replace and supersede human labor altogether”. Can you clarify?

sendes3• 4 months ago

A common error in historical thinking tends to see human tools essentially as a positive linear plot between time and progress. But these tools until AI had the common property of being enhancing of human cognition, because they couldn't do the thinking _for you_. AI can do just that, and for all the benefit it brings, seeing it simply as the next step in the "natural evolution of human tools" is alarmingly disarming coming from frontier thinkers.

Zigurd• 4 months ago

I'm glad I can still count on HN to come across the correct use of a lesser known definition of a word.

nutjob27• 4 months ago

supersede human "labour" altogether

For certain types of labor this has always been the case.

The idea that AI will entirely replace all, or most, human labor makes no sense and is just AI hype.

Like all technology before it AI will improve most people's lives.

palmotea4• 4 months ago

Like all technology before it AI will improve most people's lives.

1. Let's be clear: what you're describing is faith.

2. And what are you smoking to assert "all technology before ... AI [improved] most people's lives"?

GodelNumbering• 4 months ago

Today, unlike in the Luddites’ time, we are already seeing skilled workers replaced not with lower-wage human labor, but with AI.

To me this is the weakest claim of the article. This claim been thrown around endlessly without proof.

https://fred.stlouisfed.org/series/IHLIDXUSTPSOFTDEVE

Software Engineer job openings for instance is at 2 year high (still far lower than covid dislocations though), but arguably all Enterprise AI was built or deployed in the last two years. We should have seen a crash in the job openings if the AI job replacement claim was correct.

This is something I've spend some time thinking about (personally written article, not AI slop): https://www.signalbloom.ai/posts/why-task-proficiency-doesnt...

dude250711• 4 months ago

It's not "the age of AI", it's just a Slop Decade.

And the tools did not become "exponentially sophisticated", one thing it's logarithmic, another is that the improvements are questionable. But "pervasive" - yes, granted.

vasco• 4 months ago

Is there a better illustration of the power of UX than the fact that a messaging chat interface was able to set free all of human knowledge from copyright, whereas a bittorrent client couldn't?

gettingoverit2• 4 months ago

In fact, the paper has an error in the argument that AI might find Fermat's theorem to be incorrect due to definition of natural numbers including a zero, because paper's version of a theorem explicitly says that the number should be greater than two, and zero cannot be greater than two.

Surprisingly, this mistake proves the author's point that human can implicitly understand what was said, and that it still has value to it, even if it's incorrect.

areyousure• 4 months ago

The paper explicitly states that n is greater than 2, but the issue is whether a,b,c can be 0.

svat• 4 months ago

The short blog post announcing this paper: https://terrytao.wordpress.com/2026/03/29/mathematical-metho...

In particular:

This is an unabridged version of a solicited article for a forthcoming Blackwell Companion to the Philosophy of Mathematics. […] took over a year to write – which means, at the current pace of development in the field, that some of it is already slightly out of date.

----

Edit: The post at https://mathstodon.xyz/@tao/116319186983426174 mentions also an (entirely unrelated) popular-math presentation titled “What does it mean to think like a mathematician?” https://terrytao.wordpress.com/wp-content/uploads/2026/03/ta... which is interesting too (despite the ChatGPT-generated illustrations and repeating stuff Tao has said before, on his blog etc.)

Mathematical methods and human thought in the age of AI

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