Unfortunately, mathematics (especially pure mathematics) is by its very nature very, very poorly understood by those who haven’t worked as a mathematician. Even worse, those who don’t understand are seemingly not at all aware of their misunderstanding and are entirely confident in their (very wrong) characterisation of the subject.
The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".
A simple example is
"hey, did you know that x^i is the unit circle?"
"what's i?"
"i is defined as if you square it the result is -1"
"what does that have to do with circles?"> Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues.
The fact that this is a radical departure from the norm is part of why mathematics (and philosophy) is often seen as some intangible or ungrokable science to many outsiders, as they're generally approaching it from a perspective of "Okay, but why, what is this useful for?", and the answer "For the science of it" doesn't tend to land with people that aren't already passionate about said science/discipline and are just trying to figure out what it even is or involves.
Doesn't help that there is a pervasive sentiment in American Academia (not sure about elsewhere) about Math being *the* hard science, and I mean hard as in difficulty, so a lot of people get intimidated by it before they ever give it a chance very early on in their academic life and carry that through the rest of their education.
So there ends up being a rather small pool of people that are in(to) the field, and rather high friction for stimualting interest in it from outsiders from the way that it's taught, and a massive difference in the perspective of it's use between it's diaspora and the unmathed masses.
And this is indeed why it is not going to be taken seriously as an academic or (more importantly) an economic endeavour done by humans anymore.
That won't stop the career mathematicians from protesting and having a cry here trying to justify themselves.
Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.
Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:
(1) observe some phenomenon in ‘reality’.
(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.
(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.
It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.
The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.
NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)
If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.
This post which he forwarded was quite poor in my opinion. Confusing, all over the place with AI criticisms and promotion of the AI hazing being done by mathematicians.
X thousand mathematicians who want to protect their livelihoods signed a bunch of letters against AI. Duh. We've seen similar movements from every profession that has been displaced ever.
Terence Tao uses AI and has made a few good points on how to use it. But defensiveness leaks into almost every defense of the role of humans in Mathematics that I've read, even his own at times.
To be clear, I actually believe that Mathematicians aren't going away, but I dont have enough knowledge about the life of a professional mathematician to articulate a path forward.
This "path forward" is what I'd like to see. We need a top mathematician with enough intellectual honesty (Terence Tao qualifies, I think) to start this questioning with "there's actually no role for human Mathematicians" as one of the options on the table and go from there.
That's certainly different from Olympiad-style problems.
literally what software engineers were doing for decades though
software is mostly just simple math, for the most part, until you need to do something more complex for some hairy algos lol
Maybe more in years past when Comp Sci was a subset of Math Departments.
It's about many things, but perhaps the most relevant idea here is that no information matters without understanding. We could generate all possible knowledge, but unless someone--a human--can verify and understand it, it doesn't count. The cure for mortality could be written on the moon, but if no one reads it, it hasn't really been discovered.
[1] https://maskofreason.wordpress.com/wp-content/uploads/2011/0...
I'm a little more flexible, if the new knowledge (that human's don't understand) can be put into a mechanism and have an observable effect, I'd be happy enough. e.g. a new type of rocket fuel that burns 1000x more efficiently.
"I own nothing, have no privacy, [never have to think, and am not required to solve any problems,] and life has never been better."
https://en.wikipedia.org/wiki/You'll_own_nothing_and_be_happ...
Does it need to be a human or can it be some other form of life?
I think this axiom is of course true. But the mistake the article makes, in my opinion, is to try to apply this axiom separately to each domain. If we have this as the over-arching axiom, it is not clear at all that humans should be steering the development of mathematics. Maybe it would be better for humanity if the department of world math is run by AI.
In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.
It's like chimpanzees seeing human society and saying "look how complex it is, imagine how many chimpanzees it needs to maintain it".
Let's be honest: we don't know. Maybe you're right, but for the moment it's more likely that you're not. And countries cannot bet on that vague intuition at the cost of destroying their research communities and world leadership (which takes decades if not a century to achieve).
Best case, they still matter.
Worst case, AI kills us all and it's irrelevant that we "wasted" money on research.
A better analogy: look at all these highly trained engineers, mathematicians, doctors, writers, philosophers, writers, scientists.
How many dumbass politicians do we need to keep it all running smoothly?
Turns out no matter how dumb politicians were, overall society has been developing positively over the history of mankind.
Do you think this applies to say, surgery, as well? There are few useful problems that share these properties.
First off, I can’t imagine anything more torment nexus-y than throwing billions to automate and scale the torture of animals. If each token is a “cut”, how much suffering does 10 trillion training tokens (lower bound) corresponds to?
Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals. You cannot verify success so easily, either. Cancer cells, for example, could regrow over months. You would need to keep the animal alive and regularly test the animal, which would be difficult to scale. And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.
> Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals.
It would likely be sim2real with that as the post training, reducing that requirement a lot.
> Cancer cells, for example, could regrow over months.
In that specific scenario you would likely train on receiving no unrelated injuries during the surgery, and have induced conditions with stuff tagged molecularly that you can then verify efficacy from without waiting months.
Depending on how much more data efficient sim2real makes it, you could end up seeing companies pushing it only for actual procedures the animals need but economically would never get; botched surgery and the animal gets euthanized before waking up, which they could argue was already going to happen.
> And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.
Robotic surgery systems, already go through animal trials before being used on humans. So do many purely human surgery techniques.
well that's an awful image
We're building tools to serve the human society.
You mean dystopian. If it were a utopia everyone would be happy to welcome the new world order.
Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free).
So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight.
BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.
I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.
I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.
I'd argue that what the hugging face attack illustrates is that large AI companies are motivated to have bombastic claims supported by bombastic demos. The model was clearly trained or encouraged to work as it did, as evidenced by the fact it keeps using this particular escape hatch.
And the fact that it aligns with prior and current calls for what very likely might be a regulatory capture / oversight capture move right before IPO. It aligns so well with this "barely constrained superweapon" narrative it might as well be PR.
4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?
The fact it keeps doing it, with more and more evidence, is a sign that it's built that way.
This is a program running on their montoroed machines that they purpose built and monitored its training at every step. I think it'd be way more suprising that they didn't know it used note taking and cross-run memory.
There are many automatic theorem provers that do very clever stuff, just as the underlying theroy describes.
> I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence.
It is absurd to waste time discussing whether it is inteligent or not. It is just an algorithm, we know how it works, and it does exactly what we expect it to do. LLMs are not magical things. The main difference is the scale: for Navier-Stokes they spent in 3 days more money that the whole mathematical community over the last 20 years easily.
By the way, I'm not saying that LLM's are useless, that I'm anti-AI or anything like that.
I just used a £89 Codex subscription to do very intelligent things with it, stuff that I would have had to sit down and ponder and work on for quite a while, and I have a PhD in that. I didn't need to do anything special except explaining the problem(s) to the AI, and my theory of it so far. It took it from there. If that is not intelligence, nothing is.
- solving "frontier" math problems requires intelligence (by human or AI)
- solving "frontier" math problems is dumb statistical prediction of next token (by human or AI)
https://poshenloh.com/posts/20260919-math-ai
The original posted link from OP is from Terry Tao’s website where the article was posted as a guest blog post.
Human capability, when you think about it, is complex. Why is Newton praised as being so damn great? He established the law of universal gravitation, F=ma. Why is that such a big deal?
He distilled countless phenomena in an open system into a single mathematical formula.
What makes it great is that he found common state variables and relationships across entirely different phenomena like falling objects, planetary motion, collisions, and artillery trajectories.
But does F=ma hold true for the entire macroscopic world? No. There are various conditions and specific situations in motion, but within most scenarios and a certain range of approximation, it outputs values that are useful to humans.
Why is the Schrödinger equation so great? Because it turned the time evolution of quantum states into a calculable mathematical law.
Human thought is essentially creating a closed system by deciding what to cut out and what to keep from the infinite degrees of freedom in reality. Academia is what reinforces that closed system.
A great theory is great not because it perfectly replicates reality, but because it compresses the immense complexity of reality into a small, closed formal system while still managing to explain a multitude of phenomena.
In that process, it feels like human thought and progress are shifting into a different framework. What LLMs do well is primarily exploring within the ontology and representation space that humans have already built.
I think there are two broad categories of discovery: One is forming a new closed system, and the other is connecting fragmented knowledge within that closed system. I feel that the vast majority of research focuses on the latter.
What LLMs excel at is finding unvisited points within a given representation space. This is typically the process through which master's and PhD students connect dots, build their skills, and form their own mental models. But the logic behind criticizing LLMs seems to be that they eliminate the very work these graduate students need to do in order to grow.
However, looking at it from another angle, perhaps our current knowledge systems and classifications have reached a limit, suggesting that we might actually need a completely new classification and knowledge system.
What is the core principle of an LLM? It's predicting the probability of the next sequence.
Let's say you type the word "cat". Cat - is cute (90%), want to eat it (6%), furry (4%). Because "is cute" has the highest probability, the next sequence proceeds in that direction.
Within this framework, human knowledge and logic largely operate the same way. Once an initial logical proposition is established, we follow it up with whatever makes logical sense next. From that perspective, I think LLMs will actually do this better.
But what is it that LLMs cannot do right now? They cannot create that initial logical proposition. I believe they lack the ability to carve out a closed system from an open system.
Stacking logic step-by-step within a closed system—LLMs do this exceptionally well. But whether that constitutes true "intelligence" is a different matter.
I feel that being logical does not necessarily equate to having intelligence.
Humans preserve and create different mental models and knowledge systems within an open system. Just as your thoughts differ from mine, LLMs lack the ability to form these distinct mental models.
If so, within these limits, what humans must ultimately do is construct the logical frameworks that LLMs can then fill in. Perhaps a new kind of logic dedicated to designing these frameworks will become the next major trend.
Viewed from this perspective, I have no idea if we are in a mere technological transition or something else entirely. Or whether it is even correct to say humans are strictly necessary to build that framework. Maybe my learning is just lacking.
Our biblically literate ancestors knew better. Our purpose is to love God and love people, which is why we still have a modicum of sense for the value of completely unproductive people.
As we’ve become spiritually hollowed, and more biblically illiterate, we’ve started to dehumanize unproductive people, a slippery slope if we ourselves become unproductive.
The axiom in this article will only be accepted by the intelligentsia if you can tell a story about why it is true: that humans are created in the image of God.
Until we re-find our ability to tell cosmic stories, this hand-wringing will continue amongst the atheistic elites.
I agree that human society is too focused on productivity but disagree that it's tied to ignorance of the Bible. The fall of man illustrated in Genesis is tied to feeding on the fruit of the tree of the knowledge of good and evil. Having the right set of religious concepts wont put you in harmony with God and thus won't make you more loving to your fellow man.
There are many modern examples leading to disastrous results.
> we’ve started to dehumanize unproductive people, a slippery slope if we ourselves become unproductive.
That is very evident in the sort of things the techbros come up with, but I agre it is a wider social problem.
The moral and spiritual depth of our society has only shallowed after centuries of doing this.
It hasn’t worked!
Because people have different religious beliefs or none at all. Biblical is not the natural starting point for everyone.
If not for nothing else but to form a basis for how to distribute/share/hoard the wealth created by a society. The eternal question - who gets what.
TLDR: we don't. Terry Tao just announced he's becoming a UFC heavyweight fighter.
And maybe let's not only hear the opinion of two or three Fields level mathematicians with blogs, 99 % of the worlds mathematicians in academia might profit from these tools as they might partially close the gap between them and the world elite, making creativity and tenaciousness more important than having the right neocortical structure allowing you to outperform 99.9 % of other humans at keeping context in your head and making predictions, AI can do that better now with the right prompts.
Is that so ? Sounds hyperbolic.