Posted by auggierose 2 days ago
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.
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.
That said, few 50 (or even 40) years ago would have predicted that completely abstract number theoretical computations about primes, discrete logarithms, and elliptic curves would be the foundation of our monetary system.
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.
Can you provide an explanation for why their claim that (1) and (2) are the hardest parts is false?
Or perhaps provide an alternative definition of Mathematics that is more explanatory than the one they've provided?
If you feed an AI nonsense in its training data, it will generate nonsense
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.
but you're still right. i disregarded his take, as you would with mine re. math.
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.
If we get to a future where all frontier mathematics contributions are by AI. A future where AI displays creativity in ways that expand mathematic exploration similarly to the ways humans have in the past. A future where AI explains frontier mathematics to curious humans. What will have been lost? Perhaps just "The pleasure of finding things out".
I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand.
Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other.
Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.
But surely, if we know that the dead ends of exploring a problem are valuable, we should be able to explore them even if a solution is already known. It just requires that the mathematics community reshapes itself. And it must. Two years from now people might be able to run the computation that solved NS on their Iphone.
I'm sure that whatever has been discarded during the NS exploration as a dead end you would be able to rediscover using purpose built tooling in the near future. The purpose of human mathematicians in the medium term might be to explore dead ends, and to provide human insights as context to attack other problems. But whether this type of work will remain necessary in the long term im not sure.
Any and every capability AI can demonstrate today is the result of us, humans doing it in the first place for a very long time. The transfer method of these abilities is a subject of another comment, but as Microsoft and NVIDIA puts it, it's a theft of unprecedented scale [0].
AI labs dream of recursive self improvement as an escape from that, but until we arrive there, humans have to do something so AI can do it as well.
And, as of fully autonomous self driving which should have arrived 5 years ago, RSI and AGI is just around the corner, a corner with a radius so large that it never ends.
We dream of building utopias with these tools, but it's a path to dystopia paved with stones made from utopic dreams.
I hope you're right about AGI. What we need is time, and we may not have it. https://www.dwarkesh.com/p/noam-brown
It's because they can't.
It's a mathematical theorem that no algorithm that "solves mathematics" can exist.
>In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an input statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.
I think the headline alone makes a reasonable argument based on the AI tools we have today.
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.
But "frontier" mathematics is still a highly advanced, highly specialized field. It can take years of study to be prepared to understand the established theory and results for a given subtopic.
These two observations, taken together, lead to the predictable outcome of a lot of math "enthusiasts" with an incomplete understanding of the field loudly asserting that they have discovered a radical new result. Often they lack the foundation to even understand what they are doing wrong.
The reluctance of mathematicians to engage with amateurs that come off as cranks is a symptom of how accessible the field is.
I haven't had any interaction with mathematicians that I would describe as hostile.
But companies are our gods (don't believe? E.g. companies cannot die from natural causes). They don't need puny humans. They need AI.
My point however is that companies have no agency of their own. Their age doesn't really matter for that. There are guns that are older than all living people too. If somebody used a flintlock pistol to go around robbing people that wouldn't make the pistol a "godlike eternal entity". Or for that matter if somebody used a 200 year old shovel to create a really nice garden (in the case you believe companies are a net positive).
> But shovels are our gods (don't believe? E.g. shovels cannot die from natural causes). They don't need puny humans. They need AI.