14 ms·

The cultural divide between mathematics and AI

300 points by rfurmani 2y ago

mistrial9 2y ago
> Throughout the conference, I noticed a subtle pressure on presenters to incorporate AI themes into their talks, regardless of relevance. This is well-studied and not unique to AI, the USA in English, or even Western traditions. Here is what I mean: a book called Diffusion of Innovations by Rogers explains a history of technology introduction.. if the results are tallied in population, money or other prosperity, the civilizations and their language groups that have systematic ways to explore and apply new technology are "winners" in the global context. AI is a powerful lever. The meta-conversation here might be around concepts of cancer, imbalance and chairs on the deck of the Titanic.. but this is getting off-topic for maths.
- golol 2y ago
  I think another way to think about this is that subtly trying to consider AI in your AI-unrelated research is just respecting the bitter lesson. You need to at least consider how a data-driven approach might work for your problem. It could totally wipe you out - make your approach pointless. That's the bitter lesson.
golol 2y ago
Nice article. I didn't read every section in detail but I think it makes a good point that AI researchers maybe focus too much on the thought of creating new mathematics while being able to repdroduce, index or formalize existing mathematics is really they key goal imo. This will then also lead to new mathematics. I think the more you advance in mathematical maturity the bigger the "brush" becomes with which you make your strokes. As an undergrad a stroke can be a single argument in a proof, or a simple Lemma. As a professor it can be a good guess for a well-posedness strategy for a PDE. I think AI will help humans find new mathematics with much bigger brush strokes. If you need to generalize a specific inequality on the whole space to Lipschitz domains, perhaps AI will give you a dozen pages, perhaps even of formalized Lean, in a single stroke. If you are a scientist and consider an ODE model, perhaps AI can give you formally verified error and convergence bounds using your specific constants. You switch to a probabilistic setting? Do not worry. All of these are examples of not very deep but tedious and non-trivial mathematical busywork that can take days or weeks. The mathematical ability necessary to do this has in my opinion already been demonstrated by o3 in rare cases. It can not piece things together yet though. But GPT-4 could not piece together proofs to undergrad homework problems while o3 now can. So I believe improvement is quite possible.
esafak 2y ago
AI is young, and at the center of the industry spotlight, so it attracts a lot of people who are not in it to understand anything. It's like when the whole world got on the Internet, and the culture suddenly shifted. It's a good thing; you just have to dress up your work in the right language, and you can get funding, like when Richard Bellman coined the term "dynamic programming" to make it palatable to the Secretary of Defense, Charles Wilson.
- deadbabe 2y ago
  AI has been around since at least the 1970s.
  - tromp 2y ago
    Or 1949 if you consider the Turing Test, or 1912 if you consider Torres Quevedo's machine El Ajedrecista that plays rook endings. The illusion of AI dates back to 1770's The Turk.
    - abstractbill 2y ago
      Yes, and all of these dates would be considered "young" by most mathematicians!
    - int_19h 2y ago
      "[The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine...Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent." - Ada Lovelace, 1842
  - bluefirebrand 2y ago
    Not in any way that is relevant to the conversation about AI that has exploded this decade
invalidOrTaken 2y ago
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- deleted 2y ago
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nicf 2y ago
I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of this divide don't seem to quite get what it is that mathematicians want from math: that the primary aim isn't really to find out whether a result is true but why it's true. To be honest, it's hard for me not to get kind of emotional about this. Obviously I don't know what's going to happen, but I can imagine a future where some future model is better at proving theorems than any human mathematician, like the situation, say, chess has been in for some time now. In that future, I would still care a lot about learning why theorems are true --- the process of answering those questions is one of the things I find the most beautiful and fulfilling in the world --- and it makes me really sad to hear people talk about math being "solved", as though all we're doing is checking theorems off of a to-do list. I often find the conversation pretty demoralizing, especially because I think a lot of the people I have it with would probably really enjoy the thing mathematics actually is much more than the thing they seem to think it is.
- SwtCyber 2y ago
  Understanding why something is true - that's the beauty of it
- jasonhong 2y ago
  Interestingly, the main article mentions Bill Thurston's paper "On Proof and Progress in Mathematics" (https://www.math.toronto.edu/mccann/199/thurston.pdf https://www.math.toronto.edu/mccann/199/thurston.pdf), but doesn't mention a quote from that paper that captures the essence of what you wrote: > "The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." Incidentally, I've also a similar problem when reviewing HCI and computer systems papers. Ok sure, this deep learning neural net worked better, but what did we as a community actually learn that others can build on?
meroes 2y ago
My take is a bit different. I only have a math undergrad and only worked as an AI trainer so I’m quite “low” on the totem pole. I have listened to colin Mclarty talk about philosophy of math and there was a contingent of mathematicians who solely cared about solving problems via “algorithms”. The time period was just preceding the modern math since the late 1800s roughly, where the algorithmists, intuitivists, and logical oriented mathematicians coalesced into a combination that includes intuitive, algorithmic, and importance of logic, leading to the modern way we do proofs and focus on proofs. These algorithmists didn’t care about the so called “meaningless” operations that got an answer, they just cared they got useful results. I think the article mitigates this side of math, and is the side AI will be best or most useful at. Having read AI proofs, they are terrible in my opinion. But if AI can prove something useful even if the proof is grossly unappealing to the modern mathematician, there should be nothing to clamor about. This is the talk I have in mind https://m.youtube.com/watch?v=-r-qNE0L-yI&pp=ygUlQ29saW4gbWNjbGFydHkgMyBwaGlsb3NvcGhpZXMgb2YgbWF0aA%3D%3D https://m.youtube.com/watch?v=-r-qNE0L-yI&pp=ygUlQ29saW4gbWN...
throw8404948k 2y ago
> This quest for deep understanding also explains a common experience for mathematics graduate students: asking an advisor a question, only to be told, "Read these books and come back in a few months." With AI advisor I do not have this problem. It explains parts I need, in a way I understand. If I study some complicated topic, AI shortens it from months to days. I was somehow mathematically gifted when younger, sadly I often reinvented my own math, because I did not even know this part of math existed. Watching how Deepseek thinks before answering, is REALLY beneficial. It gives me many hints and references. Human teachers are like black boxes while teaching.
- sarchertech 2y ago
  I think you’re missing the point of what the advisor is saying.
  - throw8404948k 2y ago
    No, I get it. My point is human advisor does not have enough time, to answer questions and correctly explain the subject. I may get like 4 hours a week, if lucky. Books are just a cheap substitute for real dialog and reasoning with a teacher. Most ancient philosophy papers were in form of dialog. It is much faster to explain things. AI is a game changer. It shortens feedback loop from a week to hour! It makes mistakes (as humans do), but it is faster to find them. And it also develops cognitive skills while finding them. It is like programming in low level C in notepad 40 years ago. Versus high level language with IDE, VCS, unit tests... Or like farming resources in Rust. Booring repetitive grind...
    - WhyOhWhyQ 2y ago
      Books aren't just a lower quality version of dialog with a person though. They operate entirely differently. With very few people can you think quietly for 30 minutes straight without talking, but with a book you can put it down and come back to it at will. I don't think professional programmers were using notepad in 1985. Here's talk of IDEs from an article from 1985: https://dl.acm.org/doi/10.1145/800225.806843 https://dl.acm.org/doi/10.1145/800225.806843 It mentions Xerox Development Environment, from 1977 https://en.wikipedia.org/wiki/Xerox_Development_Environment https://en.wikipedia.org/wiki/Xerox_Development_Environment The feedback loop for programming / mathematics / other things I've studied was not a week in the year 2019. In that ancient time the feedback look was maybe 10% slower than with any of these LLMs since you had to look at Google search.
m0llusk 2y ago
> The last mathematicians considered to have a comprehensive view of the field were Hilbert and Poincaré, over a century ago. Henri Cartan of the Bourbaki had not only a more comprehensive view, but a greater scope of the potential of mathematical modeling and description
- coffeeaddict1 2y ago
  I would also add Grothendieck to that list.
  - pathsjs 2y ago
    Grothendieck did not have a comprehensive view of mathematics, nor he ever claimed to. There are vast swathes of mathematics (e.g. PDE or probability) that never fell under Grothendieck's radar
woah 2y ago
> Perhaps most telling was the sadness expressed by several mathematicians regarding the increasing secrecy in AI research. Mathematics has long prided itself on openness and transparency, with results freely shared and discussed. The closing off of research at major AI labs—and the inability of collaborating mathematicians to discuss their work—represents a significant cultural clash with mathematical traditions. This tension recalls Michael Atiyah's warning against secrecy in research: "Mathematics thrives on openness; secrecy is anathema to its progress" (Atiyah, 1984). Engineering has always involved large amounts of both math and secrecy, what's different now?
- anon291 2y ago
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  - feoren 2y ago
    I get the feeling you've never really talked to many academics.
    - nicf 2y ago
      Especially not mathematicians! No one goes into math academia for the money, and people with math Ph.D.'s are often very employable at much higher salaries if they jump ship to industry. The reason mathematicians stay in the field --- and I say this as someone who didn't stay, for a variety of reasons --- is because they love math and want to spend their time researching and teaching it.
    - anon291 2y ago
      I work with the ones that made the jump to industry, so no, I'm confronted with the divide day in and day out. The academics that either switch to industry or maintain close industry ties, typically do not seem to share these concerns, or at least, can contextualize them.
      - daveguy 2y ago
xg15 2y ago
> One question generated particular concern: what would happen if an AI system produced a proof of a major conjecture like the Riemann Hypothesis, but the proof was too complex for humans to understand? Would such a result be satisfying? Would it advance mathematical understanding? The consensus seemed to be that while such a proof might technically resolve the conjecture, it would fail to deliver the deeper understanding that mathematicians truly seek. I think this is an interesting question. In a hypothetical SciFi world where we somehow provably know that AI is infallible and the results are always correct, you could imagine mathematicians grudgingly accepting some conjecture as "proven by AI" even without understanding the why. But for real-world AI, we know it can produce hallucinations and its reasoning chains can have massive logical errors. So if it came up with a proof that no one understands, how would we even be able to verify that the proof is indeed correct and not just gibberish? Or more generally, how do you verify a proof that you don't understand?
- tech_ken 2y ago
  > Or more generally, how do you verify a proof that you don't understand? This is the big question! Computer-aided proof has been around forever. AI seems like just another tool from that box. Albeit one that has the potential to provide 'human-friendly' answers, rather than just a bunch of symbolic manipulation that must be interpreted.
- oersted 2y ago
  Serious theorem-proving AIs always write the proof in a formal syntax where it is possible to check that the proof is correct without issue. The most popular such formal language is Lean, but there are many others. It's just like having a coding AI, it may write some function and you check if it compiles. If the AI writes a program/proof in Lean, it will only compile if the proof is correct. Checking the correctness of proofs is a much easier problem than coming up with the proof in the first place.
  - nybsjytm 2y ago
    > Checking the correctness of proofs is a much easier problem than coming up with the proof in the first place. Just so this isn't misunderstood, not so much cutting-edge math is presently possible to code in lean. The famous exceptions (such as the results by Clausen-Scholze and Gowers-Green-Manners-Tao) have special characteristics which make them much more ground-level and easier to code in lean. What's true is that it's very easy to check if a lean-coded proof is correct. But it's hard and time-consuming to formulate most math as lean code. It's something many AI research groups are working on.
kkylin 2y ago
As Feynman once said [0]: "Physics is like sex. Sure, it may give some practical results, but that's not why we do it." I don't think it's any different for mathematics, programming, a lot of engineering, etc. I can see a day might come when we (research mathematicians, math professors, etc) might not exist as a profession anymore, but there will continue to be mathematicians. What we'll do to make a living when that day comes, I have no idea. I suspect many others will also have to figure that out soon. [0] I've seen this attributed to the Character of Physical Law but haven't confirmed it
- maroonblazer 2y ago
  I'd not heard that Feynman quote before, so thanks for sharing; I love it. I'd include writing, art-, and music-making in that category.
- csomar 2y ago
  Back to gambling? Mathematics is a relatively new career. My understanding is that these guys used to gamble about solving proofs for a living.
- SwtCyber 2y ago
  In a way people don't do math just for its utility, they do it because it's beautiful, challenging, and deeply fulfilling
tech_ken 2y ago
Mathematics is, IMO, not the axioms, proofs, or theorems. It's the human process of organizing these things into conceptual taxonomies that appeal to what is ultimately an aesthetic sensibility (what "makes sense"), updating those taxonomies as human understanding and aesthetic preferences evolve, as well as practical considerations ('application'). Generating proofs of a statement is like a biologist identifying a new species, critical but also just the start of the work. It's the macropatterns connecting the organisms that lead to the really important science, not just the individual units of study alone. And it's not that AI can't contribute to this effort. I can certainly see how a chatbot research partner could be super valuable for lit review, brainstorming, and even 'talking things through' (much like mathematicians get value from talking aloud). This doesn't even touch on the ability to generate potentially valid proofs, which I do think has a lot of merit. But the idea that we could totally outsource the work to a generative model seems impossible by definition. The point of the labor is develop human understanding, removing the human from the loop changes the nature of the endeavor entirely (basically to algorithm design). Similar stuff holds about art (at a high level, and glossing over 'craft art'); IMO art is an expressive endeavor. One person communicating a hard-to-express feeling to an audience. GenAI can obviously create really cool pictures, and this can be grist for art, but without some kind of mind-to-mind connection and empathy the picture is ultimately just an artifact. The human context is what turns the artifact into art.
- deleted 2y ago
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EigenLord 2y ago
Is it really a culture divide or is it an economic incentives divide? Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. What differs are the incentives. Math rewards openness because there's no real concept of a "competitive edge", you're incentivized to freely publish and share your results as that is how you get recognition and hopefully a chance to climb the academic ladder. (Maybe there might be a competitive spirit between individual mathematicians working on the same problems, but this is different than systemic market competition.) AI is split between being a scientific and capitalist pursuit; sharing advances can mean the difference between making a fortune or being outmaneuvered by competitors. It contaminates the motives. This is where the AI researcher's typical desire for "novel results" comes from as well, they are inheriting the values of industry to produce economic innovations. It's a tidier explanation to tie the culture differences to material motive.
- nybsjytm 2y ago
  > Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. There is a major caveat here. Most 'serious math' in AI papers is wrong and/or irrelevant! It's even the case for famous papers. Each lemma in Kingma and Ba's ADAM optimization paper is wrong, the geometry in McInnes and Healy's UMAP paper is mostly gibberish, etc... I think it's pretty clear that AI researchers (albeit surely with some exceptions) just don't know how to construct or evaluate a mathematical argument. Moreover the AI community (at large, again surely with individual exceptions) seems to just have pretty much no interest in promoting high intellectual standards.
  - zipy124 2y ago
    I'd be interested to read about the gibberish in UMAP, I know the paper "An improvement of the convergence proof of the ADAM-Optimizer" for the lemma problem in the original ADAM but hadn't heard of the second one. Do you have any further info on it?
  - 2y ago
mcguire 2y ago
Fundamentally, mathematics is about understanding why something is true or false. Modern AI is about "well, it looks like it works, so we're golden".
nothrowaways 2y ago
You can't fake influence
Sniffnoy 2y ago
> As Gauss famously said, there is "no royal road" to mathematical mastery. This is not the point, but the saying "there is no royal road to geometry" is far older than Gauss! It goes back at least to Proclus, who attributes it to Euclid.
- troymc 2y ago
  I never understood that quote until recently. The story goes that the (royal) pharaoh of Egypt wanted to learn geometry, but didn't want to have to read Euclid. He wanted a faster route. But, "there is no royal road to geometry."
  - FilosofumRex 2y ago
    The last Egyptian pharaoh was Nectanebo II, who ruled from 358 to approximately 340 BC. Alexander founded Alexandria in 331 BC as the crown jewel of his empire where Euclid wrote his magnum opus, The Elements in 300 BC! Unless the royal pharaoh of Egypt, refers to Ptolemy I Soter, Macedonian general who was the first Ptolemaic Kingdom ruler of Egypt after Alexander's death.
    - troymc 2y ago
      Yep, exactly. Here's a translation of Proclus: "He [Euclid] lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry." Source: http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html
NooneAtAll3 2y ago
I feel like this rumbling can be summarized as "Ai is engineering, not math" - and suddenly a lot of things make sense Why Ai field is so secretive? Because it's all trade secrets - and maybe soon to become patents. You don't give away precisely how semiconductor fabs work, only base research level of "this direction is promising" Why everyone is pushed to add Ai in? Because that's where the money is, that's where the product is. Why Ai needs results fast? Because it's production line, and you create and design stuff Even the core distinction mentioned - that Ai is about "speculation and possibility" - that's all about tool experimenting and prototyping. It's all about building and constructing. Aka Engineering/Technology letters of STEM I guess next step is to ask "what to do next?". IMO, math and Ai fields should realise the divide and slowly diverge, leaving each other alone on an arm's length. Just as engineers and programmers (not computer scientists) already do
umutisik 2y ago
If AI can prove major theorems, it will likely by employing similar heuristics as the mathematical community employs when searching for proofs and understanding. Studying AI-generated proofs, with the help of AI to decipher contents will help humans build that 'understanding' if that is desired. An issue in these discussions is that mathematics is both an art, a sport, and a science. And the development of AI that can build 'useful' libraries of proven theorems means different things for each. The sport of mathematics will be basically over. The art of mathematics will thrive as it becomes easier to explore the mathematical world. For the science of mathematics, it's hard to say, it's been kind of shaky for ~50 years anyway, but it can only help.
tylerneylon 2y ago
I agree with the overt message of the post — AI-first folks tend to think about getting things working, whereas math-first people enjoy deeply understood theory. But I also think there's something missing. In math, there's an urban legend that the first Greek who proved sqrt(2) is irrational (sometimes credited to Hippasus of Metapontum) was thrown overboard to drown at sea for his discovery. This is almost certainly false, but it does capture the spirit of a mission in pure math. The unspoken dream is this: ~ "Every beautiful question will one day have a beautiful answer." At the same time, ever since the pure and abstract nature of Euclid's Elements, mathematics has gradually become a more diverse culture. We've accepted more and more kinds of "numbers:" negative, irrational, transcendental, complex, surreal, hyperreal, and beyond those into group theory and category theory. Math was once focused on measurement of shapes or distances, and went beyond that into things like graph theory and probabilities and algorithms. In each of these evolutions, people are implicitly asking the question: "What is math?" Imagine the work of introducing the sqrt() symbol into ancient mathematics. It's strange because you're defining a symbol as answering a previously hard question (what x has x^2=something?). The same might be said of integration as the opposite of a derivative, or of sine defined in terms of geometric questions. Over and over again, new methods become part of the canon by proving to be both useful, and in having properties beyond their definition. AI may one day fall into this broader scope of math (or may already be there, depending on your view). If an LLM can give you a verified but unreadable proof of a conjecture, it's still true. If it can give you a crazy counterexample, it's still false. I'm not saying math should change, but that there's already a nature of change and diversity within what math is, and that AI seems likely to feel like a branch of this in the future; or a close cousin the way computer science already is.
- tylerneylon 2y ago
  PS After I wrote my comment, I realized: of course, AI could one day get better at the things that make it not-perfect in pure math today: * AI could get better at thinking intuitively about math concepts. * AI could get better at looking for solutions people can understand. * AI could get better at teaching people about ideas that at first seem abstruse. * AI could get better at understanding its own thought, so that progress is not only a result, but also a method for future progress.
lmpdev 2y ago
I did a fair bit of applied mathematics at uni What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career. They repeatedly and successfully applied these strategies into unsolved problems I argue would not take a tremendous jump in performance for an AI to begin their own journey similar in kind to the greats, the only thing standing in their way (as with all contemporary mathematicians) is the extreme specialisation required to reach the boundary of unsolved problems AI need not be Euler to be an important tool and figure within mathematics
- joe_the_user 2y ago
  What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career. I know this claim is often made but it seems obvious that in this discussion, trick means something far wider and more subtle than any set computer program. In a lot of ways, "he just uses a few tricks" is akin to the way a mathematician will say "and the rest of the proof is elementary" (when it's still quite long and hard for anyone not versed in a given specialty). I mean, before category theory was formalized, the proofs that now are possible with it might classified as "all done with this trick" but grasping said trick was far from elementary matter. I argue would not take a tremendous jump in performance for an AI to begin their own journey similar in kind to the greats, the only thing standing in their way (as with all contemporary mathematicians) is the extreme specialisation required to reach the boundary of unsolved problems. Not that LLMs can't do some impressive things but your narrative seems to anthropomorphize them in a less than useful way.
lairv 2y ago
> A revealing anecdote shared at one panel highlighted the cultural divide: when AI systems reproduced known mathematical results, mathematicians were excited, while AI researchers were disappointed This seems very caricatural, one thing I've often heard in the AI community is that it'd be interesting to train models with an old data cutoff date (say 1900) and see whether the model is able to reinvent modern science
j2kun 2y ago
This is written in the first person, but there is no listed author and the website does not suggest an author...
- mkl 2y ago
  It's in the usual location at the top of the page: "By Ralph Furman".
wanderingmind 2y ago
Terence Tao recently gave a lecture on Machine Assisted Proofs that helped even common folk like me to understand on the upcoming massive changes to Math within the next decade. Especially, its fascinating to see how AI and especially Lean might provide an avenue for large scale collaboration in Math Research, to bring them on par with how research is done in other sciences https://www.youtube.com/watch?v=5ZIIGLiQWNM https://www.youtube.com/watch?v=5ZIIGLiQWNM
FilosofumRex 2y ago
I find this cultural divide exists predominantly among mathematicians who consider existence proofs as real mathematics. Mathematicians who practice constructive math and view existence proofs as mere intellectual artifacts tend to embrace AI, physics, engineering and even automated provers as worthy subjects.
weitendorf 2y ago
If you look closely at the history of mathematics you can see that it worked similarly to current AI in many respects (not so much the secrecy) - people were oftentimes just concerned with whether something worked rather than why it worked (eg so that they could build a building or compute something), and the full theoretical understanding of something sometimes came significantly later than the knowledge of whether something was true or useful. In fact, the modern practice (the concept predates the practice of course, but was more of an opinion than a ritual) of mathematics as this ultimate understandable system of truth and elegance seemingly began in Ancient Greece with their practice of proofs and early development of mathematical "frameworks". It didn't reach its current level of rigor and sophistication until 100-150 years ago when Formalism became the dominant school of thought (https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...), spearheaded by a group of mathematicians who held even deeper beliefs that are often referred to as Mathematical Platonism (https://en.wikipedia.org/wiki/Mathematical_Platonism https://en.wikipedia.org/wiki/Mathematical_Platonism). (Note that these wikipedia articles are not amazing explanations of the concepts, how they relate to realism, or developed historically but they are adequate primers) Of course, Godel proved that truths exists outside of these formal systems (only a couple decades after mathemticians had started building a secret religion around worshipping Logos. These beliefs were pervasive see eg Einsteins concept of God as a clockmaker or Erdos' references to "The Book"), which leaves us almost back where we started where we might need to consider there may be some empirical results and patterns which "work" but we do not fully understand - we may never understand them. Personally, I think this philosophically justifies not subjecting oneself to the burden of spending excess time understanding or proving things that have never been understood before - it may elude elegance (as the 4-color proof) or even knowability. We can always look backwards and explain things later, and of course, it's a false dichotomy that some theorems or results must be fully understood and proven (or proven elegantly) before they can be considered true and used as a basis for further results. Perhaps it is unsatisfying to those who wish to truly understand the universe in terms of mathematical elegance, but that asshole used mathematical elegance to disprove mathematical elegance as a perfect tool for understanding the universe already, so take it up with him. Personally, as someone who at one time heavily considered pursuing a life in mathematics in part because of its ability to answer deep truths, I think Godel set us free: to understand or know things, we cannot rely solely on mathematics. Formal mathematics itself tells us that there are things we can only understand by discovering them, building them, or experimenting with them. There are truths that Cuda Cowboys can uncover that LaTex Liturgy cannot
krnsll 2y ago
As a mathematician, I can't help but simmer each time I find the profession's insistence on grasping the how's and why's of matters to be dismissed as pedantry. Actionable results are important but absent understanding, we will never have any grasp on downstream impact of such progress. I fear AI is just going to lower our general epistemic standards as a society, and we forget essential truth verifying techniques in the technical (and other) realms all together. Needless to say the impact this has on our society's ethical and effectively legal foundations, because ultimately without clarity on how's and why's it will be near impossible to justly assign damages.
FabHK 2y ago
> One striking feature of mathematical culture that came up was the norm of alphabetical authorship. […] There are some exceptions, like Adleman insisting on being last in the RSA paper. lol, took me a second to get the plausible reason for that
SwtCyber 2y ago
If AI-generated proofs become incomprehensible to humans, do they still count as -math- in the traditional sense?
- sigmoid10 2y ago
  We already have proofs by exhaustion that could only ever be verified using computers. Some people would argue they are not "elegant" but I don't think anyone would argue they are not math.
  - SwtCyber 2y ago
    But I wonder if there's a distinction between a proof that is merely computationally intensive and one that is conceptually inaccessible
hjhack 2y ago
[dead]
randomNumber7 2y ago
Imo mathematicians want to be very smart, when a lot of ai is actually easy to undertand with good abstract and logic thinking and linear algebra.
- bwfan123 2y ago
  whaa ? have you read the article ?
- recursive 2y ago
  I've known a few mathematicians, and none of them cared about being perceived as smart. They wanted to figure stuff out, which, incidentally, tended to actually make them smart.
trostaft 2y ago
> Unlike many scientific fields, mathematics has no concept of "first author" or "senior author"; contributors are simply listed alphabetically. I don't think this is (generally) true? Speaking as a math postdoc right now, at least in my field of computational mathematics there's definitely a notion of first author. Though, a note of individual contributions at the bottom of the paper is becoming more common.
- nicf 2y ago
  I was an algebraic geometer when I was still doing research in the field, and it was definitely true in that corner of the world. Authors are alphabetical, and you usually cite the paper by listing them all, no "et al"'s. I think I didn't even know there was such a thing as "first author" until I worked in ML.
- pathsjs 2y ago
  Computational mathematics may have adopted conventions from computer science, but I assure you that the alphabetical order convention is definitely the standard in most pure mathematics (with some exceptions, such as the Ascoli-Arzelà theorem)
bwfan123 2y ago
I had an aha moment recently. An excited AI researcher claimed, wow: claude could solve this IMO problem. Then, a mathematician pointed out a flaw which the AI researcher overlooked. The AI researcher then prompted the AI with the error and then the AI produced another proof he thought worked, but again was flawed. The AI played on the researcher's naivete. Long story short, current AI is doing cargo-cult math - ie, going through the motions with mimicry. Experts can see through it, but excited AI hypesters are blind, and lap it up. Even alpha-geometry (with built-in theorem prover) is largely doing brute-force search of a limited axiomatized domain. This is not to say AI is not useful, just that the hype exceeds the actual.
- xanderlewis 2y ago
  Everything you’ve said is obvious, and yet really needs saying and isn’t said enough.
calibas 2y ago
> While mathematicians traditionally pursue understanding for its own sake, industry researchers must ultimately deliver products, features, or capabilities that create value for their organizations. This really isn't about mathematics or AI, this is about the gap between academia and business. The academic wants to pursue knowledge for the sake of knowledge, while a business wants to make money. Compare to computer science or engineering, where business has near completely pervaded the fields. I've never heard anybody lamenting their inability to "pursue understanding for its own sake" and when someone does advance the theory, there's also a conversation about how to make it profitable. The academic aspect isn't gone, but it's found a way to coexist with the business aspect, for better or worse. Honestly it sounds like mathematicians have had things pretty good if this is one of their biggest complaints.
BrenBarn 2y ago
I think a lot of this is not so much "math vs. AI" as "anyone who cares about anything other than making as much money as possible vs. anyone who only cares about making as much money as possible".

300 points by rfurmani 2y ago

> Throughout the conference, I noticed a subtle pressure on presenters to incorporate AI themes into their talks, regardless of relevance. This is well-studied and not unique to AI, the USA in English, or even Western traditions. Here is what I mean: a book called Diffusion of Innovations by Rogers explains a history of technology introduction.. if the results are tallied in population, money or other prosperity, the civilizations and their language groups that have systematic ways to explore and apply new technology are "winners" in the global context. AI is a powerful lever. The meta-conversation here might be around concepts of cancer, imbalance and chairs on the deck of the Titanic.. but this is getting off-topic for maths.

golol 2y ago
I think another way to think about this is that subtly trying to consider AI in your AI-unrelated research is just respecting the bitter lesson. You need to at least consider how a data-driven approach might work for your problem. It could totally wipe you out - make your approach pointless. That's the bitter lesson.

golol 2y ago

Nice article. I didn't read every section in detail but I think it makes a good point that AI researchers maybe focus too much on the thought of creating new mathematics while being able to repdroduce, index or formalize existing mathematics is really they key goal imo. This will then also lead to new mathematics. I think the more you advance in mathematical maturity the bigger the "brush" becomes with which you make your strokes. As an undergrad a stroke can be a single argument in a proof, or a simple Lemma. As a professor it can be a good guess for a well-posedness strategy for a PDE. I think AI will help humans find new mathematics with much bigger brush strokes. If you need to generalize a specific inequality on the whole space to Lipschitz domains, perhaps AI will give you a dozen pages, perhaps even of formalized Lean, in a single stroke. If you are a scientist and consider an ODE model, perhaps AI can give you formally verified error and convergence bounds using your specific constants. You switch to a probabilistic setting? Do not worry. All of these are examples of not very deep but tedious and non-trivial mathematical busywork that can take days or weeks. The mathematical ability necessary to do this has in my opinion already been demonstrated by o3 in rare cases. It can not piece things together yet though. But GPT-4 could not piece together proofs to undergrad homework problems while o3 now can. So I believe improvement is quite possible.

esafak 2y ago

AI is young, and at the center of the industry spotlight, so it attracts a lot of people who are not in it to understand anything. It's like when the whole world got on the Internet, and the culture suddenly shifted. It's a good thing; you just have to dress up your work in the right language, and you can get funding, like when Richard Bellman coined the term "dynamic programming" to make it palatable to the Secretary of Defense, Charles Wilson.

deadbabe 2y ago
AI has been around since at least the 1970s.
- tromp 2y ago
  Or 1949 if you consider the Turing Test, or 1912 if you consider Torres Quevedo's machine El Ajedrecista that plays rook endings. The illusion of AI dates back to 1770's The Turk.
  - abstractbill 2y ago
    Yes, and all of these dates would be considered "young" by most mathematicians!
  - int_19h 2y ago
    "[The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine...Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent." - Ada Lovelace, 1842
- bluefirebrand 2y ago
  Not in any way that is relevant to the conversation about AI that has exploded this decade

invalidOrTaken 2y ago

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nicf 2y ago

I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of this divide don't seem to quite get what it is that mathematicians want from math: that the primary aim isn't really to find out whether a result is true but why it's true. To be honest, it's hard for me not to get kind of emotional about this. Obviously I don't know what's going to happen, but I can imagine a future where some future model is better at proving theorems than any human mathematician, like the situation, say, chess has been in for some time now. In that future, I would still care a lot about learning why theorems are true --- the process of answering those questions is one of the things I find the most beautiful and fulfilling in the world --- and it makes me really sad to hear people talk about math being "solved", as though all we're doing is checking theorems off of a to-do list. I often find the conversation pretty demoralizing, especially because I think a lot of the people I have it with would probably really enjoy the thing mathematics actually is much more than the thing they seem to think it is.

SwtCyber 2y ago
Understanding why something is true - that's the beauty of it
jasonhong 2y ago
Interestingly, the main article mentions Bill Thurston's paper "On Proof and Progress in Mathematics" (https://www.math.toronto.edu/mccann/199/thurston.pdf https://www.math.toronto.edu/mccann/199/thurston.pdf), but doesn't mention a quote from that paper that captures the essence of what you wrote: > "The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." Incidentally, I've also a similar problem when reviewing HCI and computer systems papers. Ok sure, this deep learning neural net worked better, but what did we as a community actually learn that others can build on?

meroes 2y ago

My take is a bit different. I only have a math undergrad and only worked as an AI trainer so I’m quite “low” on the totem pole. I have listened to colin Mclarty talk about philosophy of math and there was a contingent of mathematicians who solely cared about solving problems via “algorithms”. The time period was just preceding the modern math since the late 1800s roughly, where the algorithmists, intuitivists, and logical oriented mathematicians coalesced into a combination that includes intuitive, algorithmic, and importance of logic, leading to the modern way we do proofs and focus on proofs. These algorithmists didn’t care about the so called “meaningless” operations that got an answer, they just cared they got useful results. I think the article mitigates this side of math, and is the side AI will be best or most useful at. Having read AI proofs, they are terrible in my opinion. But if AI can prove something useful even if the proof is grossly unappealing to the modern mathematician, there should be nothing to clamor about. This is the talk I have in mind https://m.youtube.com/watch?v=-r-qNE0L-yI&pp=ygUlQ29saW4gbWNjbGFydHkgMyBwaGlsb3NvcGhpZXMgb2YgbWF0aA%3D%3D https://m.youtube.com/watch?v=-r-qNE0L-yI&pp=ygUlQ29saW4gbWN...

throw8404948k 2y ago

> This quest for deep understanding also explains a common experience for mathematics graduate students: asking an advisor a question, only to be told, "Read these books and come back in a few months." With AI advisor I do not have this problem. It explains parts I need, in a way I understand. If I study some complicated topic, AI shortens it from months to days. I was somehow mathematically gifted when younger, sadly I often reinvented my own math, because I did not even know this part of math existed. Watching how Deepseek thinks before answering, is REALLY beneficial. It gives me many hints and references. Human teachers are like black boxes while teaching.

sarchertech 2y ago
I think you’re missing the point of what the advisor is saying.
- throw8404948k 2y ago
  No, I get it. My point is human advisor does not have enough time, to answer questions and correctly explain the subject. I may get like 4 hours a week, if lucky. Books are just a cheap substitute for real dialog and reasoning with a teacher. Most ancient philosophy papers were in form of dialog. It is much faster to explain things. AI is a game changer. It shortens feedback loop from a week to hour! It makes mistakes (as humans do), but it is faster to find them. And it also develops cognitive skills while finding them. It is like programming in low level C in notepad 40 years ago. Versus high level language with IDE, VCS, unit tests... Or like farming resources in Rust. Booring repetitive grind...
  - WhyOhWhyQ 2y ago
    Books aren't just a lower quality version of dialog with a person though. They operate entirely differently. With very few people can you think quietly for 30 minutes straight without talking, but with a book you can put it down and come back to it at will. I don't think professional programmers were using notepad in 1985. Here's talk of IDEs from an article from 1985: https://dl.acm.org/doi/10.1145/800225.806843 https://dl.acm.org/doi/10.1145/800225.806843 It mentions Xerox Development Environment, from 1977 https://en.wikipedia.org/wiki/Xerox_Development_Environment https://en.wikipedia.org/wiki/Xerox_Development_Environment The feedback loop for programming / mathematics / other things I've studied was not a week in the year 2019. In that ancient time the feedback look was maybe 10% slower than with any of these LLMs since you had to look at Google search.

m0llusk 2y ago

> The last mathematicians considered to have a comprehensive view of the field were Hilbert and Poincaré, over a century ago. Henri Cartan of the Bourbaki had not only a more comprehensive view, but a greater scope of the potential of mathematical modeling and description

coffeeaddict1 2y ago
I would also add Grothendieck to that list.
- pathsjs 2y ago
  Grothendieck did not have a comprehensive view of mathematics, nor he ever claimed to. There are vast swathes of mathematics (e.g. PDE or probability) that never fell under Grothendieck's radar

woah 2y ago

> Perhaps most telling was the sadness expressed by several mathematicians regarding the increasing secrecy in AI research. Mathematics has long prided itself on openness and transparency, with results freely shared and discussed. The closing off of research at major AI labs—and the inability of collaborating mathematicians to discuss their work—represents a significant cultural clash with mathematical traditions. This tension recalls Michael Atiyah's warning against secrecy in research: "Mathematics thrives on openness; secrecy is anathema to its progress" (Atiyah, 1984). Engineering has always involved large amounts of both math and secrecy, what's different now?

anon291 2y ago
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- feoren 2y ago
  I get the feeling you've never really talked to many academics.
  - nicf 2y ago
    Especially not mathematicians! No one goes into math academia for the money, and people with math Ph.D.'s are often very employable at much higher salaries if they jump ship to industry. The reason mathematicians stay in the field --- and I say this as someone who didn't stay, for a variety of reasons --- is because they love math and want to spend their time researching and teaching it.
  - anon291 2y ago
    I work with the ones that made the jump to industry, so no, I'm confronted with the divide day in and day out. The academics that either switch to industry or maintain close industry ties, typically do not seem to share these concerns, or at least, can contextualize them.
    - daveguy 2y ago

xg15 2y ago

> One question generated particular concern: what would happen if an AI system produced a proof of a major conjecture like the Riemann Hypothesis, but the proof was too complex for humans to understand? Would such a result be satisfying? Would it advance mathematical understanding? The consensus seemed to be that while such a proof might technically resolve the conjecture, it would fail to deliver the deeper understanding that mathematicians truly seek. I think this is an interesting question. In a hypothetical SciFi world where we somehow provably know that AI is infallible and the results are always correct, you could imagine mathematicians grudgingly accepting some conjecture as "proven by AI" even without understanding the why. But for real-world AI, we know it can produce hallucinations and its reasoning chains can have massive logical errors. So if it came up with a proof that no one understands, how would we even be able to verify that the proof is indeed correct and not just gibberish? Or more generally, how do you verify a proof that you don't understand?

tech_ken 2y ago
> Or more generally, how do you verify a proof that you don't understand? This is the big question! Computer-aided proof has been around forever. AI seems like just another tool from that box. Albeit one that has the potential to provide 'human-friendly' answers, rather than just a bunch of symbolic manipulation that must be interpreted.
oersted 2y ago
Serious theorem-proving AIs always write the proof in a formal syntax where it is possible to check that the proof is correct without issue. The most popular such formal language is Lean, but there are many others. It's just like having a coding AI, it may write some function and you check if it compiles. If the AI writes a program/proof in Lean, it will only compile if the proof is correct. Checking the correctness of proofs is a much easier problem than coming up with the proof in the first place.
- nybsjytm 2y ago
  > Checking the correctness of proofs is a much easier problem than coming up with the proof in the first place. Just so this isn't misunderstood, not so much cutting-edge math is presently possible to code in lean. The famous exceptions (such as the results by Clausen-Scholze and Gowers-Green-Manners-Tao) have special characteristics which make them much more ground-level and easier to code in lean. What's true is that it's very easy to check if a lean-coded proof is correct. But it's hard and time-consuming to formulate most math as lean code. It's something many AI research groups are working on.

kkylin 2y ago

As Feynman once said [0]: "Physics is like sex. Sure, it may give some practical results, but that's not why we do it." I don't think it's any different for mathematics, programming, a lot of engineering, etc. I can see a day might come when we (research mathematicians, math professors, etc) might not exist as a profession anymore, but there will continue to be mathematicians. What we'll do to make a living when that day comes, I have no idea. I suspect many others will also have to figure that out soon. [0] I've seen this attributed to the Character of Physical Law but haven't confirmed it

maroonblazer 2y ago
I'd not heard that Feynman quote before, so thanks for sharing; I love it. I'd include writing, art-, and music-making in that category.
csomar 2y ago
Back to gambling? Mathematics is a relatively new career. My understanding is that these guys used to gamble about solving proofs for a living.
SwtCyber 2y ago
In a way people don't do math just for its utility, they do it because it's beautiful, challenging, and deeply fulfilling

tech_ken 2y ago

Mathematics is, IMO, not the axioms, proofs, or theorems. It's the human process of organizing these things into conceptual taxonomies that appeal to what is ultimately an aesthetic sensibility (what "makes sense"), updating those taxonomies as human understanding and aesthetic preferences evolve, as well as practical considerations ('application'). Generating proofs of a statement is like a biologist identifying a new species, critical but also just the start of the work. It's the macropatterns connecting the organisms that lead to the really important science, not just the individual units of study alone. And it's not that AI can't contribute to this effort. I can certainly see how a chatbot research partner could be super valuable for lit review, brainstorming, and even 'talking things through' (much like mathematicians get value from talking aloud). This doesn't even touch on the ability to generate potentially valid proofs, which I do think has a lot of merit. But the idea that we could totally outsource the work to a generative model seems impossible by definition. The point of the labor is develop human understanding, removing the human from the loop changes the nature of the endeavor entirely (basically to algorithm design). Similar stuff holds about art (at a high level, and glossing over 'craft art'); IMO art is an expressive endeavor. One person communicating a hard-to-express feeling to an audience. GenAI can obviously create really cool pictures, and this can be grist for art, but without some kind of mind-to-mind connection and empathy the picture is ultimately just an artifact. The human context is what turns the artifact into art.

deleted 2y ago
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EigenLord 2y ago

Is it really a culture divide or is it an economic incentives divide? Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. What differs are the incentives. Math rewards openness because there's no real concept of a "competitive edge", you're incentivized to freely publish and share your results as that is how you get recognition and hopefully a chance to climb the academic ladder. (Maybe there might be a competitive spirit between individual mathematicians working on the same problems, but this is different than systemic market competition.) AI is split between being a scientific and capitalist pursuit; sharing advances can mean the difference between making a fortune or being outmaneuvered by competitors. It contaminates the motives. This is where the AI researcher's typical desire for "novel results" comes from as well, they are inheriting the values of industry to produce economic innovations. It's a tidier explanation to tie the culture differences to material motive.

nybsjytm 2y ago
> Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. There is a major caveat here. Most 'serious math' in AI papers is wrong and/or irrelevant! It's even the case for famous papers. Each lemma in Kingma and Ba's ADAM optimization paper is wrong, the geometry in McInnes and Healy's UMAP paper is mostly gibberish, etc... I think it's pretty clear that AI researchers (albeit surely with some exceptions) just don't know how to construct or evaluate a mathematical argument. Moreover the AI community (at large, again surely with individual exceptions) seems to just have pretty much no interest in promoting high intellectual standards.
- zipy124 2y ago
  I'd be interested to read about the gibberish in UMAP, I know the paper "An improvement of the convergence proof of the ADAM-Optimizer" for the lemma problem in the original ADAM but hadn't heard of the second one. Do you have any further info on it?
- 2y ago

mcguire 2y ago

Fundamentally, mathematics is about understanding why something is true or false. Modern AI is about "well, it looks like it works, so we're golden".

nothrowaways 2y ago

You can't fake influence

Sniffnoy 2y ago

> As Gauss famously said, there is "no royal road" to mathematical mastery. This is not the point, but the saying "there is no royal road to geometry" is far older than Gauss! It goes back at least to Proclus, who attributes it to Euclid.

troymc 2y ago
I never understood that quote until recently. The story goes that the (royal) pharaoh of Egypt wanted to learn geometry, but didn't want to have to read Euclid. He wanted a faster route. But, "there is no royal road to geometry."
- FilosofumRex 2y ago
  The last Egyptian pharaoh was Nectanebo II, who ruled from 358 to approximately 340 BC. Alexander founded Alexandria in 331 BC as the crown jewel of his empire where Euclid wrote his magnum opus, The Elements in 300 BC! Unless the royal pharaoh of Egypt, refers to Ptolemy I Soter, Macedonian general who was the first Ptolemaic Kingdom ruler of Egypt after Alexander's death.
  - troymc 2y ago
    Yep, exactly. Here's a translation of Proclus: "He [Euclid] lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry." Source: http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html

NooneAtAll3 2y ago

I feel like this rumbling can be summarized as "Ai is engineering, not math" - and suddenly a lot of things make sense Why Ai field is so secretive? Because it's all trade secrets - and maybe soon to become patents. You don't give away precisely how semiconductor fabs work, only base research level of "this direction is promising" Why everyone is pushed to add Ai in? Because that's where the money is, that's where the product is. Why Ai needs results fast? Because it's production line, and you create and design stuff Even the core distinction mentioned - that Ai is about "speculation and possibility" - that's all about tool experimenting and prototyping. It's all about building and constructing. Aka Engineering/Technology letters of STEM I guess next step is to ask "what to do next?". IMO, math and Ai fields should realise the divide and slowly diverge, leaving each other alone on an arm's length. Just as engineers and programmers (not computer scientists) already do

umutisik 2y ago

If AI can prove major theorems, it will likely by employing similar heuristics as the mathematical community employs when searching for proofs and understanding. Studying AI-generated proofs, with the help of AI to decipher contents will help humans build that 'understanding' if that is desired. An issue in these discussions is that mathematics is both an art, a sport, and a science. And the development of AI that can build 'useful' libraries of proven theorems means different things for each. The sport of mathematics will be basically over. The art of mathematics will thrive as it becomes easier to explore the mathematical world. For the science of mathematics, it's hard to say, it's been kind of shaky for ~50 years anyway, but it can only help.

tylerneylon 2y ago

I agree with the overt message of the post — AI-first folks tend to think about getting things working, whereas math-first people enjoy deeply understood theory. But I also think there's something missing. In math, there's an urban legend that the first Greek who proved sqrt(2) is irrational (sometimes credited to Hippasus of Metapontum) was thrown overboard to drown at sea for his discovery. This is almost certainly false, but it does capture the spirit of a mission in pure math. The unspoken dream is this: ~ "Every beautiful question will one day have a beautiful answer." At the same time, ever since the pure and abstract nature of Euclid's Elements, mathematics has gradually become a more diverse culture. We've accepted more and more kinds of "numbers:" negative, irrational, transcendental, complex, surreal, hyperreal, and beyond those into group theory and category theory. Math was once focused on measurement of shapes or distances, and went beyond that into things like graph theory and probabilities and algorithms. In each of these evolutions, people are implicitly asking the question: "What is math?" Imagine the work of introducing the sqrt() symbol into ancient mathematics. It's strange because you're defining a symbol as answering a previously hard question (what x has x^2=something?). The same might be said of integration as the opposite of a derivative, or of sine defined in terms of geometric questions. Over and over again, new methods become part of the canon by proving to be both useful, and in having properties beyond their definition. AI may one day fall into this broader scope of math (or may already be there, depending on your view). If an LLM can give you a verified but unreadable proof of a conjecture, it's still true. If it can give you a crazy counterexample, it's still false. I'm not saying math should change, but that there's already a nature of change and diversity within what math is, and that AI seems likely to feel like a branch of this in the future; or a close cousin the way computer science already is.

tylerneylon 2y ago
PS After I wrote my comment, I realized: of course, AI could one day get better at the things that make it not-perfect in pure math today: * AI could get better at thinking intuitively about math concepts. * AI could get better at looking for solutions people can understand. * AI could get better at teaching people about ideas that at first seem abstruse. * AI could get better at understanding its own thought, so that progress is not only a result, but also a method for future progress.

lmpdev 2y ago

I did a fair bit of applied mathematics at uni What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career. They repeatedly and successfully applied these strategies into unsolved problems I argue would not take a tremendous jump in performance for an AI to begin their own journey similar in kind to the greats, the only thing standing in their way (as with all contemporary mathematicians) is the extreme specialisation required to reach the boundary of unsolved problems AI need not be Euler to be an important tool and figure within mathematics

joe_the_user 2y ago
What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career. I know this claim is often made but it seems obvious that in this discussion, trick means something far wider and more subtle than any set computer program. In a lot of ways, "he just uses a few tricks" is akin to the way a mathematician will say "and the rest of the proof is elementary" (when it's still quite long and hard for anyone not versed in a given specialty). I mean, before category theory was formalized, the proofs that now are possible with it might classified as "all done with this trick" but grasping said trick was far from elementary matter. I argue would not take a tremendous jump in performance for an AI to begin their own journey similar in kind to the greats, the only thing standing in their way (as with all contemporary mathematicians) is the extreme specialisation required to reach the boundary of unsolved problems. Not that LLMs can't do some impressive things but your narrative seems to anthropomorphize them in a less than useful way.

lairv 2y ago

> A revealing anecdote shared at one panel highlighted the cultural divide: when AI systems reproduced known mathematical results, mathematicians were excited, while AI researchers were disappointed This seems very caricatural, one thing I've often heard in the AI community is that it'd be interesting to train models with an old data cutoff date (say 1900) and see whether the model is able to reinvent modern science

j2kun 2y ago

This is written in the first person, but there is no listed author and the website does not suggest an author...

mkl 2y ago
It's in the usual location at the top of the page: "By Ralph Furman".

wanderingmind 2y ago

Terence Tao recently gave a lecture on Machine Assisted Proofs that helped even common folk like me to understand on the upcoming massive changes to Math within the next decade. Especially, its fascinating to see how AI and especially Lean might provide an avenue for large scale collaboration in Math Research, to bring them on par with how research is done in other sciences https://www.youtube.com/watch?v=5ZIIGLiQWNM https://www.youtube.com/watch?v=5ZIIGLiQWNM

FilosofumRex 2y ago

I find this cultural divide exists predominantly among mathematicians who consider existence proofs as real mathematics. Mathematicians who practice constructive math and view existence proofs as mere intellectual artifacts tend to embrace AI, physics, engineering and even automated provers as worthy subjects.

weitendorf 2y ago

If you look closely at the history of mathematics you can see that it worked similarly to current AI in many respects (not so much the secrecy) - people were oftentimes just concerned with whether something worked rather than why it worked (eg so that they could build a building or compute something), and the full theoretical understanding of something sometimes came significantly later than the knowledge of whether something was true or useful. In fact, the modern practice (the concept predates the practice of course, but was more of an opinion than a ritual) of mathematics as this ultimate understandable system of truth and elegance seemingly began in Ancient Greece with their practice of proofs and early development of mathematical "frameworks". It didn't reach its current level of rigor and sophistication until 100-150 years ago when Formalism became the dominant school of thought (https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...), spearheaded by a group of mathematicians who held even deeper beliefs that are often referred to as Mathematical Platonism (https://en.wikipedia.org/wiki/Mathematical_Platonism https://en.wikipedia.org/wiki/Mathematical_Platonism). (Note that these wikipedia articles are not amazing explanations of the concepts, how they relate to realism, or developed historically but they are adequate primers) Of course, Godel proved that truths exists outside of these formal systems (only a couple decades after mathemticians had started building a secret religion around worshipping Logos. These beliefs were pervasive see eg Einsteins concept of God as a clockmaker or Erdos' references to "The Book"), which leaves us almost back where we started where we might need to consider there may be some empirical results and patterns which "work" but we do not fully understand - we may never understand them. Personally, I think this philosophically justifies not subjecting oneself to the burden of spending excess time understanding or proving things that have never been understood before - it may elude elegance (as the 4-color proof) or even knowability. We can always look backwards and explain things later, and of course, it's a false dichotomy that some theorems or results must be fully understood and proven (or proven elegantly) before they can be considered true and used as a basis for further results. Perhaps it is unsatisfying to those who wish to truly understand the universe in terms of mathematical elegance, but that asshole used mathematical elegance to disprove mathematical elegance as a perfect tool for understanding the universe already, so take it up with him. Personally, as someone who at one time heavily considered pursuing a life in mathematics in part because of its ability to answer deep truths, I think Godel set us free: to understand or know things, we cannot rely solely on mathematics. Formal mathematics itself tells us that there are things we can only understand by discovering them, building them, or experimenting with them. There are truths that Cuda Cowboys can uncover that LaTex Liturgy cannot

krnsll 2y ago

As a mathematician, I can't help but simmer each time I find the profession's insistence on grasping the how's and why's of matters to be dismissed as pedantry. Actionable results are important but absent understanding, we will never have any grasp on downstream impact of such progress. I fear AI is just going to lower our general epistemic standards as a society, and we forget essential truth verifying techniques in the technical (and other) realms all together. Needless to say the impact this has on our society's ethical and effectively legal foundations, because ultimately without clarity on how's and why's it will be near impossible to justly assign damages.

FabHK 2y ago

> One striking feature of mathematical culture that came up was the norm of alphabetical authorship. […] There are some exceptions, like Adleman insisting on being last in the RSA paper. lol, took me a second to get the plausible reason for that

SwtCyber 2y ago

If AI-generated proofs become incomprehensible to humans, do they still count as -math- in the traditional sense?

sigmoid10 2y ago
We already have proofs by exhaustion that could only ever be verified using computers. Some people would argue they are not "elegant" but I don't think anyone would argue they are not math.
- SwtCyber 2y ago
  But I wonder if there's a distinction between a proof that is merely computationally intensive and one that is conceptually inaccessible

hjhack 2y ago

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randomNumber7 2y ago

Imo mathematicians want to be very smart, when a lot of ai is actually easy to undertand with good abstract and logic thinking and linear algebra.

bwfan123 2y ago
whaa ? have you read the article ?
recursive 2y ago
I've known a few mathematicians, and none of them cared about being perceived as smart. They wanted to figure stuff out, which, incidentally, tended to actually make them smart.

trostaft 2y ago

> Unlike many scientific fields, mathematics has no concept of "first author" or "senior author"; contributors are simply listed alphabetically. I don't think this is (generally) true? Speaking as a math postdoc right now, at least in my field of computational mathematics there's definitely a notion of first author. Though, a note of individual contributions at the bottom of the paper is becoming more common.

nicf 2y ago
I was an algebraic geometer when I was still doing research in the field, and it was definitely true in that corner of the world. Authors are alphabetical, and you usually cite the paper by listing them all, no "et al"'s. I think I didn't even know there was such a thing as "first author" until I worked in ML.
pathsjs 2y ago
Computational mathematics may have adopted conventions from computer science, but I assure you that the alphabetical order convention is definitely the standard in most pure mathematics (with some exceptions, such as the Ascoli-Arzelà theorem)

bwfan123 2y ago

I had an aha moment recently. An excited AI researcher claimed, wow: claude could solve this IMO problem. Then, a mathematician pointed out a flaw which the AI researcher overlooked. The AI researcher then prompted the AI with the error and then the AI produced another proof he thought worked, but again was flawed. The AI played on the researcher's naivete. Long story short, current AI is doing cargo-cult math - ie, going through the motions with mimicry. Experts can see through it, but excited AI hypesters are blind, and lap it up. Even alpha-geometry (with built-in theorem prover) is largely doing brute-force search of a limited axiomatized domain. This is not to say AI is not useful, just that the hype exceeds the actual.

xanderlewis 2y ago
Everything you’ve said is obvious, and yet really needs saying and isn’t said enough.

calibas 2y ago

> While mathematicians traditionally pursue understanding for its own sake, industry researchers must ultimately deliver products, features, or capabilities that create value for their organizations. This really isn't about mathematics or AI, this is about the gap between academia and business. The academic wants to pursue knowledge for the sake of knowledge, while a business wants to make money. Compare to computer science or engineering, where business has near completely pervaded the fields. I've never heard anybody lamenting their inability to "pursue understanding for its own sake" and when someone does advance the theory, there's also a conversation about how to make it profitable. The academic aspect isn't gone, but it's found a way to coexist with the business aspect, for better or worse. Honestly it sounds like mathematicians have had things pretty good if this is one of their biggest complaints.

BrenBarn 2y ago

I think a lot of this is not so much "math vs. AI" as "anyone who cares about anything other than making as much money as possible vs. anyone who only cares about making as much money as possible".