From the Fall of the Berlin Wall until about a decade ago my animating intellectual question was pretty much, ‘how did a small number of natural sciences become qualitatively more impressive than the social sciences?’ Because I met the late George Smith relatively early in my thinking about this (as an undergraduate), I often formulated this question in terms of evidence; he liked his version of my question as ‘how to turn data into high-quality evidence?’ And his striking answer was (to simplify) that deviations from theoretically significant expectations through theory-mediated measurement were made informative in an iterative process over time in the sciences that succeeded in achieving lift-off. While I was much more interested in the social organization of the sciences than George, I never did much better than he did in answering my own question. But I did make sure his version was published in the best possible form and with full detail (see his ‘Closing the Loop.’)
Along the way, I became an expert on Newton and his reception and on developments in twentieth century economics, and I found a home in professional philosophy (as a ‘historian of philosophy,’ a ‘philosopher of social science,’ and ‘HPS-type’). At least this is a way one can tell my professional story before I switched (or returned) to political theory. Once in political theory I recognized that Machiavelli’s great contribution to the art of government — that in the right conditions, which includes considerable layers of social opacity, social disorder can itself be informative — anticipates George’s insight into how successive approximation can be fruitful.
Be that as it may, the way professional philosophy is organized, there is no list of ‘open’ questions that are treated as ‘important.’ In fact, there are ways of posing my animating question in terms that made it sound vaguely Popperian (viz. ‘is there a scientific method;’ ‘a logic of discovery?;’ ‘could concepts be stress tested?’)—and so discarded. Sometimes I was directed to others — Alison Wylie, Nancy Cartwright, and Hasok Chang back in the day — whose projects had a family resemblance with my own.
There are some subfields in philosophy where through the force of personality and/or institutional entrenchment (or donor/foundation persistence) some questions do feel significant and open. But more often than not these are treated (after the fact) as evidence of ‘fashion’ in philosophy. Because there is no consensus on what really matters philosophically, and great jobs are scarce in the profession, lots of professional philosophers are in the business of artificially producing prestige (by keeping acceptance rates for the really important journals ridiculously low).
The previous was not really meant to be polemical.
Professional philosophers often think of themselves as constituting a field that’s not very far from professional mathematics. One of our core areas — mathematical logic — even has some overlap with foundational practices in mathematics. And I am just old enough to remember the time when some wise folk thought that having mathematical logicians around in philosophy was a thing thought good, perhaps standards-protecting (even standards-raising), for a department. And I was taught by and mentored for a long time by a philosopher, Jody Azzouni who himself studied mathematical practice, and found that useful in debating what other professional philosophers argued about philosophically important topics.
But judging by the reactions collected at ‘Proofs and Prompts’ to the Open AI math dump last week, the way enquiry is organized in professional mathematics is quite distinct from the relative lack of organization in philosophy. Or to put it differently, professional mathematics has much more of a closed guild structure than we do in philosophy. (Some evidence for this can be gleaned from the way the mathematicians use ‘community’ in ‘mathematical/mathematics community’—no philosopher would speak of our profession as such as a community unless it were in order to be satirical.) There who you study with and what ‘open’ problems you work on have a rather tight connection and, these in turn, have a rather tight connection to one’s professional identity and trajectory. To put it in terms of the economists: barriers to entry and, once in, path-dependent endowment effects are rather steep.
Having a list of such open problems, and an implied sense of their significance to the community, allows for much coordination, including the distribution of honors, jobs, and grant proposals. While there is, undoubtedly, a political economy to the institutions that fund all of this, it has allowed for a non-trivial amount of bottom-up intellectual and collective self-governance as well as exploration. It is, for example, pretty clear that quite a few leading mathematicians have been early adopters of LLMs in their own research and, thereby, have indirectly helped train the next generation of these.
The situation described in the previous paragraph is fairly unique in human history. Few civilizations have invested in pure mathematics to the degree ours has with the rise of the modern research university and dedicated research institutions since (say) the late nineteenth century. (By ‘civilization’ I am happy to be very inclusive.) But by making its official problems explicitly legible (to adopt social science jargon), the ‘mathematical community’ became sitting ducks when the barriers to entry were huge for individual researchers but a relatively easy technological hurdle for a few for-profit corporations with no hard budget constraint (relative to the mathematical community). These companies now control a vital resource that enables one to be on the leading edge of the mathematical frontier. My use of ‘one’ is, of course, a weasel word—because it is wholly unclear if any human can have any leading role on that frontier henceforth.
Perhaps the implied guild structure explains the unseemly schadenfreude with which the demise of professional mathematics has been welcomed by so many commentators. Cheering on the misfortune of those who harm nobody, and who are engaged in a noble activity, is itself evidence of the advanced corruption (recall) of our age.
One sometimes hears from economic historians and sociologists of technology that the technological disruption to the mathematical community, if it is one, is no big deal all things considered. The mathematicians turn out to be no different from the hand-weavers, the loom operators, and the human (mostly female) ‘computers’ that got displaced by machines in the context of progress. But while there is more than a kernel of truth in this (and there is quite a bit of intellectual labor embodied in modern machinery, etc.), this is one of the first times since the rise of the modern research university that a ‘pure’ field risks being overtaken and displaced by one of the (applied) technologies it partially birthed. Philosophers are somewhat used to being told by the practitioners in the fields it helped birth that philosophy is useless and pointless; but it may hit the mathematicians harder. Many mathematicians are in a state of “shock,” “loss,” and depression (as well as anger), but a guild that gets undermined also opens up new opportunities for those impatient or sidelined under the old regime.
As regular readers know (recall here), I argue that the structural (Humphreys) opacity that is inherent in LLMs enhances genuine expertise. But it doesn’t follow from this that the remaining place for human mathematical expertise is on the research frontier of commonly defined open questions. My own scholarly work (with Russo and Wagemans here) suggests that in such conditions there will be more work for certification and less for production. For, opaque systems, when used to produce results, systematically secrete new layers of opacity that need to be checked and monitored.
I increasingly suspect professional mathematics will turn into practiced and idiosyncratic virtuosity, where math slides back into modern art.1 (Danny Calegari’s examples of animations are worth pondering.) Since modern universities find ways to hire, evaluate, and promote all kinds of artists, this may require less administrative adjustment than one might fear.
Of course, if the cutting edge LLMs become more like a utility, then, alternatively, maybe credentialling within mathematics disappears entirely. Automated iterative feedback from deviations then mops it all up.
What about professional philosophy and the scholarship about it (i.e. history of philosophy)? Prestige producers and the journals they control are fighting a rearguard action against the use of LLMs. I expect more and more ‘professional philosophy’ to be co-produced by professional philosophers and LLMs, until we reach the point where LLMs ‘edit,’ referee, and summarize all such papers and are effectively the only ones actually reading them. Given the way it is organized the good news is professional philosophy is not so vulnerable to a hostile takeover; but the bad news is that it is vulnerable to increasing (ahh) hollowing out of its mode of production because that mode is not itself intrinsic to philosophy. (Okay that was polemical.)
That last claim is too strong undoubtedly. Such legibility to outsiders is, of course, a matter of degree and intensity, and one wonders if the philosophical contributions of (say) Jorge Luis Borges will be re-evaluated in light of such considerations (not to mention the fortunes of bits of continental philosophy). By emphasizing Borges here I do not mean to focus on his brilliant thought experiments, but rather on how he indicates new ways of discernment. That means, but this is self-serving, that philosophers will increasingly value and gravitate toward judgment and voice—but I would say that.
Let me rephrase that. You may think that if we take Liam Kofi Bright’s famous distinction between basically pleasant bureaucrats and Sexy Murder Poets seriously, I am predicting the demise of the former and the renewed attraction of the latter within professional philosophy. But while that result would greatly amuse me, it’s not what I am predicting. Rather I am claiming (recall my treatment of synthetic philosophy) that philosophy will rediscover that its vocation is in the tending, nurturing, and monitoring of the machinery of governance of the bureaucratic structure (including the LLMs) that is opaque to society.
- An earlier version of this post was published at Digressions&Impressions.
1 In some ways this would be a return to Hardy’s perspective, but also with a different conception of art and beauty underwriting it. That is, mathematics will become more like painting in response to the development of photography and the newer digital technologies. (The second half of this footnote was suggested by a token of Opus 5.5.)
{ 68 comments }
D. S. Battistoli 10.15.26 at 9:16 pm
It strikes me that this post may become more descriptively accurate by appending the word “analytic” before most instances of the word “philosopher”:
I mentioned in a comment to Speranta‘s most recent post here at CT that “I’ve been reading a lot of Henri Bergson lately, in preparation for some writing about his influence on African philosophy.” And since, I’ve gotten into the actual reading of African and Asian philosophers who build with Bergson.
Why is this relevant? Because part of the Analytic/Continental split was Russell, the co-author of Principia Mathematica, calling Bergson, the most influential philosopher of their day, a thickwitted mumblefunk. Russell also didn’t have much time for the Neoplatonists, to whom Bergson looked with increasing care as his own career went on. Bergson also insisted, contra Russell though not in argument with him, that philosophy should situate itself in a theoretical space significantly poised upon and in conversation with the high theory generated by psychology, sociology, and biology (not, as per the analytics, as a descriptive annex to mathematics).
Furthermore, Bergson believed that what he calls intuition, still important to Plato and Plotinus, but which was treated with ever-decreasing respect by philosophers from Aristotle to Descartes and onward, played an essential role, alongside reason, in animating the space described by philosophy.
Now, nothing that you say about mathematics is wrong, but it seems this is the place to note that part of the reason so many philosophers from non-Western traditions have not followed analytic philosophy into its centralization of mathematic logic is that they have continued to value so much of the non-mathy stuff Russell was trying to excise from philosophy as anglophones and their friends practiced it.
In short, philosophy today is less in danger from LLMs than is mathematics because Russell failed in his herculean effort to make philosophy an appendage to maths and to discredit anyone who wasn’t dancing to his beat.
Which is why it is useful to make a distinction between “philosophy” (a field practiced all around the world, including by non-analytics) and “analytic philosophy.” Or, to put it in the polemic terms of a 2025 philosophy-substack brouhaha: go ask Judith Butler if mathematical logic is core to the work she and aligned philosophers practice. (That is a joke, not an invitation to polemics or a Butlerian debate).
Lee A. Arnold 10.15.26 at 9:26 pm
It will be interesting to hear from the mathematicians who go over these OpenAI proofs whether A.) the machine has invented genuinely new mathematical concepts that did not exist before, or else B.) it has only combined wildly disparate techniques and elements of existing mathematics, to solve these problems.
I suspect the answer is B.
However, the role of the humans will not simply be relegated to verification and governance of the results and of the resulting intellectual industry. Instead, in verifying these results, humans may occasionally find that they are led to think, on their own, of genuinely new concepts that the machine could never find. Because these concepts did not exist anywhere in the language of mathematics before.
I expect the same thing will be true of using natural language LLMs, such as to investigate philosophical subjects: their creativity will be of the lesser, “combinatory” kind, and rarely (if ever) will the machines come up with a genuinely new concept that does not already exist in the corpus of human language.
In other words, the really interesting question is whether there will always be rational concepts which we humans are capable of creating, which are always beyond the contemporaneous output of machines.
As Warren McCulloch (one of the people who started all of this) put it in an essay titled, Mysterium Iniquitatus of Sinful Man (1955):
“To the theoretical question, ‘Can you design a machine to do whatever a brain will do?’ the answer is this: ‘If you will specify in a finite and unambiguous way what you think a brain does do with information, then we can design a machine to do it.’ Pitts and I have proved this constructively. But can you say what you think brains do?”
M Caswell 10.15.26 at 9:37 pm
Philosophy maybe requires that there be questions, and not just problems.
Alex SL 10.15.26 at 9:54 pm
I know too little about institutionalised mathematics and philosophy to have a strong opinion on most aspects of the post. However,
The areas where current ‘AI’ (LLMs and their ilk) can be used with some kind of reliability, and therefore the areas that are experiencing a demise of the value of expertise, are those where a solution can be brute-forced and then easily checked for correctness. As everybody knows, coding is the area the tech bros latched onto first: throw out lots of lines of code, test run it, and you see if it works; if it doesn’t, try again until you hit your token limit. Similarly, to my limited understanding the big AI breakthroughs in math are often brute-force throwing millions of dollars worth of tokens at the problem, while confirming the correctness of one of the attempts is relatively easy. This means that subfields or problems of math that don’t work like that should be less affected (if there are any; my ignorance again).
It also means that, from what I am reading, the other aspects of the AI ‘solutions’ aren’t so great. Apparently, the explanations of the proofs that the AI spat out leave a lot to be desired, and mathematicians may spend the next few years trying to make sense of them and clean up after the mess left behind by the AI bros, who have in the meantime wandered off to destroy the next area of human creativity to chase media attention that inflates their share prices a bit more yet. (But also, the AI bubble is starting to make little pfffff noises this week. That doesn’t mean the hype will end within the month, but it is quite possible that the two major AI labs will have gone bankrupt by the time the mathematical community has fully assessed the proofs their bots vomited out. It is also well possible that nobody will have the appetite to burn ten million dollars to solve another open research problem two years from now because they are too busy paying down their bad debts after 80% of the data center projects went broke. We will see.)
At any rate, what this implies, and what the second half of your post describes, is bleak. Assume for a moment that technological progress e.g. in robotics continues forever, and that both increasingly larger AI ‘models’ and humanoid robots will be cheaper than exploiting human employees (I have doubts, but let’s assume). What does that world look like? The goal of the tech billionaires is clearly to have everything that matters, everything that is creative or enjoyable, done by computers and by robots that they control. Replace not only cutting-edge mathematicians, but artists, philosophers, biologists, engineers, economists, programmers, accountants, etc., with a very complex chatbot. You need to know something, you don’t try to get a degree in it or talk to an expert, no, you ask the all-knowing mystery box oracle run by Google or, gods help us, Xai. Replace nurses, plumbers, delivery drivers, electricians, cleaners, and waiters with robots. Pretty much all of humanity can just sit on its couch being unemployed or elbow each other in the face to become one of the trillionaires’ influencers or enforcers keeping the rest of us in our place. I somehow doubt that that could be a stable system, but if it were, it would be a dystopia worse than 1984, worse than Soylent Green.
somebody who spoke to a mathematician about this 10.15.26 at 11:37 pm
The working mathematicians I know (a weirdly large number for my present situation, but understandable given my undergraduate interests) tell me that the AI slop dumped on their profession is interesting “directionally” but the value in mathematical work is in the ability to connect it to other parts of their work, to take an answer or proof and extend the methods, ideas and understanding developed there to new areas, or to old areas that could benefit from the fresh tools. The shit garbage that The Computer Boys slopped out may technically add up but there’s a reason nobody who barfed it out even attempted to get it published with any kind of peer review because nobody involved understood it in the slightest. Nobody, including The Computer Boys, can connect any of it to anything whatsoever. Further, you can’t even feed this shit back into the Computer and ask the lying/asskissing machine to do more because there’s too much of it and the lying/asskissing machine can’t remember all of the slop garbage it barfed out. Despite these deficiencies, the belief of my interlocutors, relatively universal, is that of course AI will be used to eliminate all mathematics in all universities permanently, then eliminate all of it in high school and elementary school. After all, the slop machine can get close enough that only a small number of people will have their airplanes crash, and probably none of them will be rich enough to matter.
This prediction lines up with how AI is being used in every other field – forced into it at knifepoint by the ultra-wealthy mega-elite super-brain eugenicist Computer Boys no matter the damage to their lessers – so I believe them. Eventually mathematics will return, but not until the super-brain mega-IQ elite men explode from injecting themselves with too many horse steroids and/or ketamine.
David Duffy 10.16.26 at 1:23 am
I think most mathematicians, like everyone else in academic research, really want to know more. If a rapid path to new knowledge is via automation, it will be taken. Yesterday I wrote a quick computer simulation to look at what a statistical analysis does, because I am unable to mentally visualize it as a good mathematician would. Human insecurity about loss of intellectual superiority to computers was always a non-problem for us 99% below the highest levels of individual achievement.
As to the current whirlwind pace of development of AI, I am reminded of what has gone on in human genomics. You might remember that the public funding for the Human Genome Project was greatly accelerated because of the threat that Celera Genomics offered of completing the work 10 (or was it 100) times faster and then “patenting the genome”. Subsequent to that, technological advances in large-scale genotyping and sequencing have predominantly been in the private sector – public research now largely depends on the products of those corporations.
In the case of the AI boom, there has not been a similar significant public science response to my knowledge, though as usual, the basics came out of academia. That comes back to the original topic, in that it was my expectation, at least, that the advances due to such technology would be at the leading edge of human knowledge, eg AlphaFold or mathematical proofs, rather than being immediately rolled out en masse as chatbots or replacements for the white collar professional.
engels 10.16.26 at 9:04 am
professional mathematics has much more of a closed guild structure than we do in philosophy. (Some evidence for this can be gleaned from the way the mathematicians use ‘community’ in ‘mathematical/mathematics community’—no philosopher would speak of our profession as such as a community unless it were in order to be satirical.)
Given that professions are (arguably almost by definition) closed quasi-guilds, whereas to me at least “communities” can be informal and porous I find this contrast strange. In fsct I think the term “community” is often just bureaucratese for a group, as in the “international community” or the “disabled community”. Calling philosophy a “professsion” was a long-running source of ire on CT I believe, and subsequently on Twitter, which caused some “professional philosophers” to abandon the self-description I think.
Petter Sjölund 10.16.26 at 12:13 pm
Lots of parallels to the reaction in 1976 when Appel and Haken proved the Four Color Theorem on an IBM 370.
People like Paul Halmos and Herbert Wilf called it “not real mathematics”, complaining that it offered a raw answer but no conceptual breakthrough and no insight.
The maths community balked at the prospect of vetting the IBM 370’s output. The New York Times initially refused to report on it because editors were worried there might be some hidden computer error in the proof. Researchers despaired that their life’s work had been wiped out by the machine, wondering what roles were left for human intuition.
The Four Color Theorem proof wasn’t generally accepted until it could be verified with independent software such as the Coq proof assistant decades later,
I guess this is that times 372.
D. S. Battistoli 10.16.26 at 1:05 pm
engels @7:
Thank you for the historical context on attitudes toward the historically contingent concept of a “philosophy profession” here at CT. I’d not been aware of that. Eric‘s “the way the mathematicians use [. . .] ‘mathematical/mathematics community’—no philosopher would speak of our profession as such as a community” struck me as well.
I tend to think back to Weber’s famous “Staat ist diejenige menschliche Gemeinschaft” (“A state is a human community”), itself basically dependent on his separate formulation:
In English:
Weber’s definition of “communal [social relationship]” has held up fantastically well in the century since he formulated it, with “language communities” “online communities,” and the like all able to fit under and be productively analyzed in line with it.
At the same time, I think get what Eric is trying to say in the original post: Mathematicians think of themselves like sweep rowers all in one crew shell, pulling together, if at different specific places and with slightly different angles of attack, to a common destination. Meanwhile, philosophers think of themselves like an online book club that requires everyone to have read a few of the same books, but doesn’t care if they read the same or different things this month, or if they skip a bunch of meetings. From a Weberian perspective, both are communities, but it might be off to the side of Eric’s point to try to enforce that definition.
JBL 10.16.26 at 2:55 pm
@Petter Sjölund 8: Speaking as a professional mathematician (albeit who was not born when 4CT was proved), your statement
“The Four Color Theorem proof wasn’t generally accepted until it could be verified with independent software such as the Coq proof assistant decades later”
does not ring true at all to me. I do not think that by the late 1990s anyone doubted the correctness of the proof of 4CT. There had been errors caught and corrected in the *human* part of the proof, but no problems were ever found in the computer part; and it had been independently verified by an independent group using a different computer-aided strategy in the mid-1990s. (The Coq formalization didn’t come until 2005.) As a colleague once said to me while I was counting midterm exams that had come out of the copy machine: “If you get a different answer from what you expected, it will be because you miscounted.”
The situation with AI does not seem particularly similar to the 4CT at all.
somebody who actually has used theorem proving software 10.16.26 at 3:29 pm
Theorem proving software is not what the Computer Boys are barfing out garbage from. The four color theorem was proved with the assistance of a machine because it could be reducible (at the time) to around eighteen hundred configurations (later reduced to 633 by….a human mathematician). The software checked each configuration, but the configurations themselves were calculated by a human (indeed, a human identified the method by which particular maps could be reduced to these configurations). The computer didn’t prove the four-color theorem. Instead, humans programmed it with an equivalent to the four-color theorem and the computer did the checking of it. (Indeed, a human mathematician found a flaw in the proof, but the human mathematicians who had advanced the proof were able to show that the flaw wasn’t significant.) You can get Rocq to output garbage if you want, just program it to! It doesn’t mean that Rocq is bad theorem proving software.
I urge the interested party to read: about Rocq and consider the difference between the use of theorem proving software (common in mathematics since the 90s) and grok producing a racist anime girl who just shrugs and says a theorem is right and gives you 9,000 pages of unreadable, often meaningless garbage when you ask why.
LFC 10.16.26 at 4:55 pm
I haven’t yet quite gotten through the entire OP but w.r.t. the opening — From the Fall of the Berlin Wall until about a decade ago my animating intellectual question was pretty much, ‘how did a small number of natural sciences become qualitatively more impressive than the social sciences?’ — it seems strange that the OP appears not even to gesture, unless I missed something, at the very long-running debates about the social sciences v. the natural sciences.
If you think, as some do, that the social sciences are, generally, immature sciences not basically different in kind from the natural sciences, then the question why the social sciences have not become as impressive in various respects as the natural sciences is going to present itself w/ perhaps some urgency. OTOH, if you don’t think the social sciences are, in general, simply immature versions of the natural sciences, and you don’t think the question of, say, what causes social revolutions will ever be finally “solved” in the definitive or nearly-definitive sense that physics solved question [fill in the blank] or that biology solved the basic mechanism of DNA, then the question “how did a small number of natural sciences become qualitatively more impressive than the social sciences?” will be less pressing and prob. less puzzling.
J, not that one 10.16.26 at 8:58 pm
I think Alex is saying something similar, but this
Is true, but it’s true in a symmetrical way. The proofs are also supposed to be legible, in particular ways. This is not the case with computer-generated proofs. It isn’t a new problem. I remember around the time I was in college in the 80s the computer-assisted proof of the four-color problem was still not fully accepted. The reasons weren’t exactly the same, but as I recall, they were similar, having to do with the difference between a proof that proceeds by deduction and a proof that proceeds by generating combinatorial possibilities and coming to a conclusion by exhaustion.
But it’s interesting to think that certain fields are vulnerable to appropriation because of a certain kind of legibility. The discussions of AI are interesting in part because non-technical pundits seem to feel that software systems should be legible to them, since they’re somehow not really engineering products, but I don’t know, almost humanistic objects that software people ought to be able to explain in humanistic terms. We don’t ask that bridges be legible in the same way.
It’s even more interesting that many people seem to be much more comfortable with the idea that an AI is illegible to humans than with the idea that an engineering artifact of a certain kind is illegible to them.
engels 10.16.26 at 9:07 pm
DSB, thanks for bringing Weber to bear on the topic. I think it has been observed (by Stanley Cavell among others) that one of the features distinguishing the so-called continental philosophers from their analytic cousins is the conscious preoccupation with a shared canon of texts you refer to (by contrast, analytic philosophy has arguably had maths envy for much of its existence and two of its founding trinity were mathematicians).
JBL 10.17.26 at 12:20 am
I just want to repeat (speaking as a mathematician) that the objections to the 4CT proof are of a completely different kind from the objections to the LLM proofs. The 4CT proof is a very large finite calculation; it is uncheckable in principle by hand because of the size of the computation, but also large calculations done by computer are vastly more reliable than large calculations done by hand. If someone started checking cases by hand of the 4CT argument and proposed that there was an error, everyone would tell that person to (1) stop wasting their time, and (2) go look for their own error.
The kind of illegibility of LLM proofs is not remotely similar to that. But also, the advance in the quality of LLM proofs over the last six months makes it clear that their illegibility is a transient problem: if the people making these things cared about mathematics (they don’t, as far as I can tell, except as a PR thing or as part of the weird AI cult stuff that I haven’t wrapped my head around), they would be able to get them to produce proofs that were human-friendly. Anthropic is not operating at the same scale as OpenAI but has been bringing human mathematicians on board to help make their proofs more human-readable, but the point is that “hard to read” is primarily a stylistic rather than substantive objection.
To be clear, I do not use LLMs, do not like what they’re doing to my field or the world more broadly, and have been part of a group of people trying to keep hyped-up claims about these not-yet-verified proofs out of Wikipedia; but unfortunately mathematics as a discipline is not going to be saved by aesthetic arguments about the style or quality of the proofs, almost all of which are almost certainly correct.
FWIW a thing that no one here has said but that is clear to all the mathematicians I’ve talked with about it: the reason for the progression
(1) coding
(2) mathematics
is that both coding and mathematics have the feature that you can tell whether something is right or not just by examining its text. (But coding came first because coding is easier than math.) This distinguishes mathematics from most other realms of argumentation: if I give you two philosophical or legal arguments about some point, you can sit down and point out their strengths and weaknesses, and you can argue maybe that one of them is better than the other; but if I sit you down with a mathematical proof of a statement P and a mathematical proof of the statement not-P, you can conclusively and definitively explain why one of them is right and the other is wrong.
engels 10.17.26 at 10:49 am
The conclusion of #15 removed me of this famous taxonomy (I can’t seem to find the definitive version):
* Proof by vigorous hand waving: Works well in a classroom.
* Proof by seduction: “Convince yourself that this is true!”
* Proof by accumulated evidence: “Long and diligent search has not revealed a counterexample.”
* Proof by Divine Intervention: “Then a miracle occurs …”
* Proof by forward reference: Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.
* Proof by funding: How could three different government agencies be wrong?
* Proof by example: The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.
* Proof by omission: “The reader may easily supply the details” or “The other 253 cases are analogous”
* Proof by deferral: “We’ll prove this later in the course”.
* Proof by picture: A more convincing form of proof by example. Combines well with proof by omission.
* Proof by intimidation: “Trivial.”
* Proof by adverb: “As is quite clear, the elementary aforementioned statement is obviously valid.”
* Proof by cumbersome notation: Best done with access to at least four alphabets and special symbols.
* Proof by exhaustion: An issue or two of a journal devoted to your proof is useful.
* Proof by obfuscation: A long plotless sequence of true and/or meaningless syntactically related statements.
* Proof by wishful citation: The author cites the negation, converse, or generalization of a theorem from the literature to support his claims.
* Proof by eminent authority: “I saw Karp in the elevator and he said it was probably NP- complete.”
* Proof by personal communication: “Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication].”
* Proof by reduction to the wrong problem: “To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem.”
* Proof by reference to inaccessible literature: The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.
* Proof by importance: A large body of useful consequences all follow from the proposition in question.
* Proof by mutual reference: In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to follow from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.
* Proof by metaproof: A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.
* Proof by vehement assertion: It is useful to have some kind of authority relation to the audience.
* Proof by ghost reference: Nothing even remotely resembling the cited theorem appears in the reference given.
* Proof by semantic shift: Some of the standard but inconvenient definitions are changed for the statement of the result.
* Proof by appeal to intuition: Cloud-shaped drawings frequently help here.
Petter Sjölund 10.17.26 at 1:12 pm
My point wasn’t so much that the objections to the 1978 proof were the same, or of the same kind, as the present objections to the OpenAI dump, but more that some of sentiments expressed by members of the mathematics community back then that are similar to the current ones. This is described in Robin Wilson’s book Four Colors Suffice: How the Map Problem Was Solved.
In the book, Bill Tutte describes the despair of a purist mathematician watching the machine take over:
“I imagine one of them outgribing in despair, crying ‘What shall I do now?'”
J, not that one 10.17.26 at 3:06 pm
JBL @ 15
Thanks for the corrections on the four color problem, but I’d take exception to the following two sentences:
This treats the software as magic that maybe the corporate owners aren’t allowing to do everything it can do. I’d like to be able to go into a store and immediately find a non-stretchy top that fits me perfectly, but I don’t imagine the manufacturers are denying me this deliberately. Or, I’d like to be able to run faster, but telling me if I cared, I could run a 5k in 10 minutes is just silly.
and
This is like saying writing prose is easier than playing chess. (The sentence previous to that where you say code can be proved correct by inspection is also not quite right, but maybe you mean not really “looking at it” but “trying to run it.”)
engels 10.17.26 at 3:48 pm
”I imagine one of them outgribing in despair, crying ‘What shall I do now?’”
Granted Tutte and Wilson are mathematicians and I’m not but this really trivialises the epidemic issues involved imho.
Alex SL 10.17.26 at 9:12 pm
David Duffy,
“Subsequent to that, technological advances in large-scale genotyping and sequencing have predominantly been in the private sector – public research now largely depends on the products of those corporations.”
Maybe that is how it is in human medicine, or maybe you mean that the sequencing service providers are sometimes private companies (?), but when I need large-scale genotyping or sequencing data for invasive weeds or rare/threatened species, the money is coming from publicly funded research grants or appropriations. Corporations don’t have a profit incentive to generate the data.
Similarly, as I wrote further up, I expect corporate interest in sinking multiple millions into math problems to disappear once they cannot use it as a publicity stunt to boost share prices. There is just no personalised, short-term profit in it otherwise, no matter the social benefits.
JBL 10.17.26 at 11:03 pm
@Petter Sjölund: fair enough; I can’t say anything interesting about how mathematicians broadly felt about the 4CT proof. I am a bit skeptical that the view the Tutte quote jokes about could really have been that widespread, but that may just be the bias of coming of age in an era when the Appel—Haken strategy (do something interesting to reduce your infinite problem to a large-but-finite problem that can be checked on a computer) had become commonplace (even if generally viewed as inelegant).
JBL 10.18.26 at 12:06 am
@J: “This treats the software as magic that maybe the corporate owners aren’t allowing to do everything it can do.” No, but I’m not sure I can convey convincingly why not. 2 years ago LLM prose was tortured and almost always identifiable immediately, and also LLMs were perhaps B- students in undergraduate mathematics classes. One year ago LLMs were A students in sophomore level proof-based mathematics classes and some people found them interesting to work with in a research setting but they weren’t able to produce anything anyone thought of as interesting new mathematics. Sometime this summer they became capable of solving the kinds of research problems we give REU students or beginning PhD students to work on — unsolved, but mostly because no one had ever tried very hard. And in the last month or so they became capable of solving at least some problems that many experts have devoted considerable time to thinking about over many years. Also over the same period LLM prose has gotten better (though it’s still not great). I would be very happy if there were a reason to expect this obvious acceleration to suddenly come to a complete halt but it seems to me that expecting that is the magical thinking here. (And again, in case it is not clear: I am not happy about this! A bunch of terrorists have set off a nuclear bomb in the middle of my research community.)
“ This is like saying writing prose is easier than playing chess.” No? I mean I hate to be pulling rank (that’s a lie actually this is fun) but do you do either of these activities at any serious level? The easy parts of doing mathematics are extremely similar to writing code, in terms of the underlying structure of the activity. I teach my intro to proofs students a bunch of formal structures (proof by induction, proof by contradiction, proof by division into exhaustive cases, …) and proofs I write in my research are made by assembling these same pieces in complicated but rigid ways. Likewise CS students learn a bunch of formal structures, and the code they write assembles those same basic pieces in complicated but rigid ways. Playing chess is quite like both — and computers have been better at that than humans for 20–30 years now. Imagine the crisis that Deep Blue 2 and then Stockfish would have caused in Departments of Chess, if universities had such things.
“Running code” is one particular way of “looking at code” (which happens to use the aid of a compiler etc.).
JBL 10.18.26 at 12:13 am
BTW all it is super interesting to learn that the phrase “the mathematics community” sounds weird to (some) outsiders, definitely something to ponder for me.
Jimmy 10.18.26 at 3:54 am
Whether the profession is dying or transforming, the operational question underneath is the same one that shows up wherever AI enters a discipline: the scarce resource shifts from producing candidates to verifying them. In mathematics the verification culture (refereeing, replication of proofs, seminars) is the infrastructure that makes machine-generated results usable at all. I run a small independent tracker in a completely different domain – codexreset.today follows AI coding-agent usage-limit reset windows and reset-failure reports – and the common thread is that a discipline’s health becomes visible in its verification records, not its output volume.
Charles Stewart 10.18.26 at 11:31 am
The post How to formalise all of mathematics contains the reaction of many mathematicians to OpenAI’s dump.
David Duffy 10.18.26 at 11:48 pm
AlexSL wrote “the money is coming from publicly funded research grants”
Oh absolutely, but either the equipment and consumables used by the genotyping laboratory you use come from biggish corporations, or you use a commercial service, and the huge improvements in throughput come from those.
Tm 10.19.26 at 2:14 pm
OP: “‘how did a small number of natural sciences become qualitatively more impressive than the social sciences?’”
LFC has already pointed out that this framing is somewhat strange. I think it’s actually a meaningless question.
“Professional philosophers often think of themselves as constituting a field that’s not very far from professional mathematics. One of our core areas — mathematical logic — even has some overlap with foundational practices in mathematics.”
I think that stems from the time when everything was a branch of philosophy. Why is the book called “Philosophiae Naturalis Principia Mathematica”? But for mathematicians, logic is of course a branch of Mathematics. As is Computer Science, btw, at least theoretical Computer Science. It’s wholly a branch of Mathematics, concerned with building mathematical models and proving theorems about them.
“Few civilizations have invested in pure mathematics to the degree ours has with the rise of the modern research university and dedicated research institutions since (say) the late nineteenth century.”
Surely, pure mathematics research is orders of magnitude less resource intensive than almost any other field of research, especially the physical and life sciences but also the social sciences? Running AI models to solve problems in pure Mathematics may be the most expensive Mathematical undertaking ever.
“These companies now control a vital resource that enables one to be on the leading edge of the mathematical frontier.”
The paradox here is that you need trained mathematicians to even make sense of the output that these companies’ AI models have generated. Nobody who isn’t a trained mathematician understands even remotely what they are about, let alone what the proofs mean and whether or not they are correct. And as a trained mathematician myself (albeit not a working mathematician and my training was 30 years ago), I would say that even most trained mathematicians understand only a fraction and only a handful of people are qualified and capable of evaluating this output.
Most AI slop tries to look and sound plausible to the general audience but these papers (and several mathematicians commenting about them have described them as AI slop) are gibberish to most people, including I suspect to the very people who trained the models. This raises some interesting questions I think.
Tm 10.19.26 at 3:23 pm
DSB 1: I think it was pretty normal for philosophers at the time to vigorously disagree. Is that different today? Russell was also famously critical of Henry James, so it’s not that his vitriol was reserved for non-anglophones.
JBL 15: “both coding and mathematics have the feature that you can tell whether something is right or not just by examining its text. (But coding came first because coding is easier than math.)
Coding *is* math, resp. the theory of Programming Languages (Formal Languages, Generative Grammar) and machine models and theoretical Computer Science in general is a branch of Math (at least that’s how theoretical CS was taught in my time). And Russell and Whitehead showed in the early 20th century that Math can *in principle* be reduced purely to symbolic manipulation. Math is that field of inquiry which is independant of empirical observation (although some would dispute this).
Tm 10.19.26 at 3:23 pm
Leibniz’ 17th century vision that all questions might at some point be reducible to calculation has long been thought ridiculous but maybe it’s making a comeback.
(https://publicdomainreview.org/essay/let-us-calculate-leibniz-llull-and-the-computational-imagination/)
Alex SL 10.19.26 at 8:32 pm
Tangentially related opinion piece just out about biology, arguing that because it is now so easy to produce a paper, we have to completely re-think how we evaluate colleagues: https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.3004075
I agree, unsurprisingly, that counting papers and citations was at any rate always a stupid idea, not least because those metrics are so easily manipulated. We should actually look at the value of what people do in their research instead of trying to force everything into one simple number. But the other aspect here is that I disagree with the piece on how easy it has become to get a paper: yes, the chatbot can write a manuscript, but where are the data coming from? The time-determining steps in biology are field work, lab work, glasshouse trials, measurements, etc. The author actually acknowledges that (“the bottlenecks are structural and experimental rather than computational”) but then waves it away because there is an incentive to have more papers, without ever explaining how that solves the bottlenecks. Bit odd, that.
David Duffy,
Okay, if that is what was meant. But that simply describes that we are living in an economy dominated by private for-profit corporations providing goods and services to paying customers. It isn’t saying anything about the effects of automation as such, which I read your previous comment as being about.
JBL at 15: “a thing that no one here has said”,
Sorry, but isn’t that what I said in my second paragraph at 4?
engels 10.19.26 at 9:20 pm
We need to talk about Gödel’s Incompleteness Theorem.
D. S. Battistoli 10.19.26 at 10:28 pm
Tm @27:
Not incidentally, James and Bergson were highly sympathetic to one another, with Bergson citing James’ work throughout his career (Bergson being a non-Comtean positivist who eagerly kept up with advances in psychology) and James being at the least a highly appreciative reader of Bergson. James and Jamesians were a, if not the, primary source of positivist psychological insight on which Bergson leaned throughout his career.
The tradition you mention of vigorous disagreement is as old as philosophy itself, though there was long also a tradition of charitable critique: the idea was that you should demonstrate your capacity to understand your opponent’s philosophizing on their own terms before attacking or departing from it. No one is required to be, say, a Platonist, but you should be able to describe what Plato was doing if he comes up in your argument.
Russell was famously mocking even in formally published pieces toward Bergson’s works. He wrote to one correspondent, “I mean to begin with a really sympathetic account, before criticizing him; but all the phrases that occur to me are turning him into ridicule.”
Though Neil Wilson would later re-formulate, in a clearly analytic context, the philosophical “principle of charity” (“We select as designatum that individual which will make the largest possible number of [our interlocutors]’ statements true.”—1959. “Subjects and Substrata, p. 532), I do trace to Russell the origin of a presumption that philosopher-ego can elect to decline to write for publication in accord with the principle of charity if they feel the context to call for it; in practice, engaging with “continental” philosophers is often just such a sirenic context for analytics.
Which gets me to your point about “non-anglophone”: I rather despise the proposed analytic-continental diairesis of philosophy (as if philosophy was unified and exclusively European for all human history, and then split into a methodology opposed to a geography in the early twentieth century), and so I usually talk about it in different ways, as in this case “anglophones and their friends.” Regardless, the fact that Russell was critical of the majority of his day’s non-anglophone European philosophy does not preclude him from also being critical of anglophones like James.
Mitchell Porter 10.19.26 at 11:17 pm
Skeptics who think the OpenAI proofs are gibberish (#11, #26?) may wish to see the latest post at Terry Tao’s blog, which reports a number of human-authored papers distilling or extending the AI proof of the non-existence of Siegel zeros (a problem adjacent to the Riemann hypothesis).
https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/
Tm 10.20.26 at 12:34 pm
Mitchell 32: “Skeptics who think the OpenAI proofs are gibberish (#11, #26?)”
Several mathematicians commenting about them have described them as “AI slop” and unreadable and are asking whether it’s really their job to ‘clean up tafter the machine’, so to speak, and turn these papers into publishable material (supposing they are correct, which is still an open question).
But you are missing my actual point: I said that they are “gibberish to most people, including I suspect to the very people who trained the models”, because to most people, anyt complicated math is gibberish, because advanced math is really hard and only a small fraction of people has the competence to understand it. The presentation matters of course. The proof of the Pythagorean theorem can, I believe, be presented in a way that at least in principle, anybody familiar with basic arithmetic can understand it if they are willing to invest a few minutes of attention. And I regret that this isn’t happening in most school math classes. But again, most present day advanced math is really beyond what non-specialists make sense of.
engels 30: I second that. I wonder what philosophers of math have to say now about the significance of the incompleteness and undecidability theorems. The thing with incompleteness is that any specific, sufficiently powerful axiomatic system is incomplete but we can always go to the meta level. Somebody (David in Tokyo I think) on the other thread recently claimed that undecidability really was proof that the brain is just a computer, which I found too far-fetched to comment to. It seems to me that AI whatever we think of it is a wide field for philosophers to investigate.
LFC 10.20.26 at 7:01 pm
TM @27 and DSB @31
The references to Henry James should be to William James. (Henry was the novelist, William the philosopher/psychologist.)
Aviva Mar 10.21.26 at 3:31 am
The legibility point is what I’ll keep thinking about. A published list of open problems was a coordination tool for people, and it turned out to be an equally good target list for a company with far more compute than any department. Philosophy’s lack of an agreed list looks almost protective by comparison, though I suspect that protection is temporary. I’d gently push on one thing: the early adopters you mention suggest the guild wasn’t simply caught off guard, since some members were already helping build the tools. Does that change who should decide how dumped results get absorbed?
Tm 10.21.26 at 7:41 am
Oops you are right … thanks LFC
ETB 10.21.26 at 9:35 am
Naturally, I oppose the increasing alienation of labour (for, hopefully, self-evident reasons), which it seems to me is central to this. While class struggle within an embourgeoised society is not exactly a new problem, interestingly here the process impacts work more typically chosen for intrinsic satisfaction rather than the traditional manual and routine labour (which, one suspects, is not entirely accidental). Class reductionism is best avoided, yet broader solidarity might go a long way towards some of the issues AI raises – though solidarity across those divides is, of course, the one thing that can’t be assumed…
SusanC 10.21.26 at 2:29 pm
“Skeptics who think the OpenAI proofs are gibberish”
The main objection I’m hearing comes down to a philosophical position on what constitutes a mathematical proof. (And it’s an argument of the nature of proof that got discussed at length in philosophy books long before we had these AI generated proofs)
Is it
– you start from the axioms, apply the deduction rules, and any sequence of steps that gets you to the conclusion is fine, no matter how opaque or convoluted it is
– proofs are supposed to convey understanding of why it’s true
The current drop of AI proofs gives us the first but not the second, and it turns out that a great many mathematicians think a type a proof is crap, i.e. they are advocates of position b.
Now, quite possibly, given a type (a) proof and doing some work you can get to a type (b) proof that will satisfy adherents of position b. This seems to in fact be happening with the most interesting results. But some considerable amount of work needs to be done by somebody, and the fact that OpenAI didn’t do this work and left things half-done is the core of the objection.
There’s also a concern about the possibility of various forms of cheating.
One of the ways you can cheat is that the formally proven statement is true, but is not the same as the original conjecture. I.e., you solve an easy problem, claim you’ve solved a hard problem, and hope nobody checks. So somebody needs to be able to check.
[ I have personally looked at the formal statements of some the combinatorics results, and the combinatorics ones i looked at aren’t doing that kind of cheating, although they are stylistically annoying. They tend to provide their own versions of definitions that are in Lean’s Mathlib, so you need to check the AI hasn’t cheated by providing a subtly wrong definition ]
ETB 10.21.26 at 3:31 pm
I would also gently suggest that it might be worth giving this a read:
https://maevyn.codes/decoherence
(I wouldn’t dare to venture an opinion on such matters, but perhaps those with greater expertise might?)
engels 10.21.26 at 4:18 pm
My hazy understanding is that Gödel made both logicism (TM) and formalism (JPL?) very unattractive, if not defunct. Yes you can “go meta” but that leads straight to a Sisyphus loop. Gödel himself was a Platonist iirc.
How about: mathematical objects are only created by (intuitionism) or perceived by (Platonism) the human mind, therefore AIs can not produce mathematics or prove theorems, at best they might guide humans to do so.
If I was a guild master bent on protecting sweet, sweet my privilege I’d probably go with something like that.
https://plato.stanford.edu/entries/intuitionism/
https://plato.stanford.edu/archives/fall2014/entries/platonism-mathematics/
D. S. Battistoli 10.21.26 at 4:44 pm
LFC @35:
Quite so, thanks!
engels 10.21.26 at 9:15 pm
Actually thinking about it, Platonism is quite likely consistent with robot Fields Medalists so I should have just recommended a revival of intuitionism as the go-to philosophy for opulent maths guildbros jealously guarding their lucrative closed shop.
engels 10.21.26 at 11:14 pm
Son, we live in a world with theorems, and those theorems have to be proved by men with lemmas. Who’s gonna do it? Etc
JPL 10.22.26 at 12:27 am
Engels @41: “(JPL?)”
“Hey, I haven’t commented on this thread!”, I said (I think without externalizing it); but “JBL” is not “JPL”. I’m not a mathematician. But before Godel there was the Frege-Hilbert controversy, in which, in my opinion, Frege prevailed. (And Hilbert later revised his earlier view.) Still, it’s an interesting question, since LLM-based AI robots behave as if Hilbert had the last word, and that interpretation can’t possibly be true. Thoughts are the activity of the semantic categories and schemata, which are logically prior to the formalism; the formalism only expresses those thoughts for people who have semantic categories and can recreate the thoughts expressed. (Just like with NLs.) (I think I would stand with the Platonists (of some sort).)
Tm 10.22.26 at 11:12 am
SusanC: “The current drop of AI proofs gives us the first but not the second” (you start from the axioms, apply the deduction rules, and any sequence of steps that gets you to the conclusion is fine, no matter how opaque or convoluted it is)
I don’t think this is actually true. Have you had a look at some of the papers? I think they try to mimic the human style (what else? It’s what they’ve been trained on) but they aren’t good at it. As an aside, we know that LLMs hallucinate and it would be surprising if this doesn’t happen in the Math papers. But to find that out, somebody needs to comb through the papers and as I said there aren’t many people really competent to understand the background and find errors. So I really don’t know where this is going.
engels 41: I always found it fascinating that you can mathematically prove that certain things are impossible, cannot be done no matter how hardyou try. That is the opposite of the tech oligarch religion which claims that whatever they want they can make it happen.
Tm 10.22.26 at 11:16 am
Just adding this from wikipedia:
“Verifying and confirming whether the research is valid will take a long time, given some papers are hundreds of pages long and with dense calculations.[1] Upon release, OpenAI researchers themselves had not been able to read through all the papers.[22] Three papers were retracted within a day, and an additional 14 papers were revised.[1][23] About half of the manuscripts were accompanied by machine-verifiable proofs, certified as correct by the automated logic platform Lean;[22][23] however, verifying that the proof corresponds to the paper must be done manually; some of the formalizations appear to prove weaker results than those claimed in the papers.[24] OpenAI disclaimed the validity of unformalized results.”
So the papers include “machine-verifiable proofs” but the question remains what the proofs actually prove.
https://en.wikipedia.org/wiki/OpenAI_bulk_release_of_mathematical_papers
J, not that one 10.22.26 at 7:52 pm
I could be wrong, but I assume the kind of thing Susan C points at @39 wouldn’t in any way be tolerated for a proof sent in by a human, who except in very exceptional circumstances would be told back and put in the effort to get the work into the expected form. It does seem hard to understand why the corporations couldn’t have redone the work with the standard definition. If the proof is really enabled by changing up the definitions (so they weren’t wrong enough to be cheating but different enough so they couldn’t be translated), that seems significant enough that eliding the difference might plausibly feel in some way like “cheating” even beyond not demonstrating understanding (which would be bad enough).
ETB 10.22.26 at 11:02 pm
A level Mathematics
Answer all parts. You must show all your working.
(1) If it takes 1 AI thread 1 hr to produce 1 paper, and 6 mathematicians 3 hrs to review it, then (assuming 1 maths department closes every 6 months) how long until the backlog exceeds what remains of the workforce? Give your answer in months to 3 significant figures. (5 marks)
(2) If NVIDIA’s market capitalisation exceeds $5 trillion, and OpenAI and Anthropic together target a $3 trillion valuation, then determine when the bubble will burst and the global economy collapse (assume growth is exponential). (2 marks)
(3) Using your answers to parts (1) and (2), calculate the probability that this does not end badly. Give your answer to 3 decimal places. (2 marks)
(4) Show that your answers to parts (1) to (3) were not AI generated. (1 mark)
Total for Question 1 is 10 marks
Note: Answers beginning “Certainly!” will receive 0 marks.
engels 10.23.26 at 4:53 pm
Ok I did it (sorry).
Son, we live in a world that has theorems, and those theorems have to be proved by men with lemmas. Whose gonna do it? You? You, Lt. Weinburg? I have a greater responsibility than you could possibly fathom. You weep for Frege’s Basic Law V, and you curse Russell’s paradox. You have that luxury. You have the luxury of not knowing what I know. That the Ramified Theory of Types, while unwieldy, is necessary for logical consistency. And Principia Mathematica, while grotesque and incomprehensible to you, can be used to derive much of modern mathematics from logical principles. You don’t want the truth because deep down in places you don’t talk about at parties, you want me in that lecture hall, you need me in the lecture hall. We use words like atomic proposition, predicative function, Sheffer stroke. We use these words as the backbone of a life spent defending the consistency of mathematics. You use them as a punchline. I have neither the time nor the inclination to explain myself to a man who rises and sleeps under the blanket of the very rigour that I provide, then questions the manner in which I provide it. I would rather you just said thank you, and went on your way, Otherwise, I suggest you pick up a piece of chalk, and provide a counter example to one of my conjectures. Either way, I don’t give a damn what you think you are entitled to assume.
engels 10.23.26 at 6:01 pm
Also, the issue Susan is describing hss some overlap with my concerns about formalism versus realism etc though it predates LLM proofs, at least in theory: eg an automated theorem prover could spew out a series of formally valid inferences that began with Peano arithmetic and ended with Goldbach’s conjecture but which was so long, complicated and unstructured no human could follow it, kind of like the four colour problem on steroids.
JimV 10.23.26 at 7:44 pm
From Dr. Scott Aaronson’s post “The Mathocalypse” https://scottaaronson.blog/?p=10169
…among the 372 huge results released yesterday by OpenAI, on the recommendation of its advisory group of Timothy Gowers, Edward Witten, and other distinguished mathematicians, was a proof of Subhash Khot’s Unique Games Conjecture (UGC), a statement that my wife [Dana] has worked toward proving for the entire time I’ve known her [20 years].
Additional commentary from Dana on the UGC proof:
The Unique Games Conjecture is true, though it wasn’t really needed for the proof of its most famous corollaries (like optimal hardness of approximating Max-Cut and other constraint satisfaction problems). The published proof of the conjecture somehow makes the short code approach* work with a bunch of new ideas (it’ll take time to understand exactly how, the writeup is poorly written). I still wonder whether it’s possible to prove the conjecture via the half-space code.
* Addendum: Upon further inspection, the proof constructs a completely new tree-based code that somehow has a noise test with properties similar to that of the short code.
…Just now, however, Dana tells me that she’s been asking Astra [another AI] all day to explain the new proof of the UGC to her and it’s been doing an amazing job and she’s starting to understand the construction. She’s adapting incredibly quickly to what’s already the new way of “doing” math and CS theory!
…Dana tells me she now mostly understands the proof of the UGC, and is amazed by the new ideas in it, and wants to give talks about it soon.
—-excerpts end —-
When Roger Ailes first claimed to have proved Fermat’s Theorem, that proof was found to be wrong. I guess it was human slop.
engels 10.24.26 at 8:20 am
the papers include “machine-verifiable proofs” but the question remains what the proofs actually prove
I actually don’t understand this. All that needs checking then is that the result stated in human mathematical language matches the final line of the machine readable proof, which is surely not difficult: what have I misunderstood?
engels 10.24.26 at 8:23 am
John Healey: “I understand and share the concerns of working people, so I am setting out a distinctively British approach to AI…”
https://www.theguardian.com/uk-news/2026/oct/24/john-healey-pledges-to-use-budget-to-harness-power-of-ai-to-bolster-britains-security
Be afraid. Be very afraid.
David, a Bostonian in Tokyo 10.25.26 at 11:21 am
Alex SL wrote:
“Similarly, as I wrote further up, I expect corporate interest in sinking multiple millions into math problems to disappear once they cannot use it as a publicity stunt to boost share prices. ”
This is exactly right. But it’s even worse than that. LLM-based AI is horrifically computationally expensive. I haven’t worked the numbers exactly, but Open AI and Anthropic are planning on _multi-trillion-dollar IPOs_ next year, and making those IPOs fly depends on them being able to make claims about the technology. So, if any of you are business types, what percentage of that multi-trillion dollars should Open AI be willing to spend on a publicity stunt this effective? (Assume, for the sake of argument, 1%.) My guess is that that number would be much larger than the total US expenditure on mathematics for not just a year, but for a decade.
(Before you yell that computers are getting faster, allow me to point out that they’re not. Really, they’re not. Improvements per two-year generation have been about 20% for two friggin decades now. For good generations. The RTX5080 is less than 1.5 times faster than the two generation (four years) older RTX3080 (consumer GPUs). )
That is, it wasn’t _millions_ of dollars, it was _billions_ of dollars. Which itself is a tiny percentage of those trillions. One percent of that IPO would be multiple tens of billions. That’s serious money that’s never going to be spent on mathematics again.
engels 10.25.26 at 3:01 pm
Before you yell that computers are getting faster, allow me to point out that they’re not. Really, they’re not. Improvements per two-year generation have been about 20% for two friggin decades now. For good generations.
So why is all my 10-year-old tech unusable?
ETB 10.25.26 at 6:49 pm
As I understand it, Lean guarantees that the formal statement follows from the axioms – not that the formal statement is the theorem the paper claims. Mismatches can arise from weaker substituted statements, vacuous truth, definitions that don’t capture the intended notion, etc. so a human still has to check that the formalisation says what the paper says.
Alex SL 10.25.26 at 8:36 pm
David, a Bostonian in Tokyo,
I am not a business type, but maybe it is somewhere in the middle? Assume an ad budget for inflating the IPO valuation of tens of billions, but only part of that goes into making mathematicians’ lives harder as a publicity stunt, because you still need other billions for making cybersecurity harder as a publicity stunt, making biologists’ lives harder as a publicity stunt, subsidised subscription fees, and actual ads. Still, your point stands, of course.
I for one am glad to know that I live in a sensible free market economy with private enterprises that rationally allocate investment based on customer preferences instead of some unaccountable command economy wasting a trillion dollars on some harebrained ideological scheme that will all come crashing down sometime in the next few years. /s
David, a Bostonian in Tokyo 10.26.26 at 4:52 am
To add another rant here, the LLM programming thing is similarly insane. Using sample code to generate new code is an old idea, and the automatic programming folks have been working on/thinking about how to do this for as long as there’s been code. But they’ve never had the resources the LLM types have. (The LLMs have engorged enormous volumes of code, and have engorged that code into a humongously inneficcient (albeit gloriously general) memory structure that’s insanely expensive (computationally) to access.) It doesn’t matter that the code generated is bad, folks who have an idea for a program often have the problem that just starting a coding project is a lot of work. (That’s the sound of me raising my hand and screaming ME ME ME that you are hearing: I have two simple programs that I really need to write, and haven’t even started. Sigh.)
Anyway, the folks who have been thinking about this for decades must be livid. And everyone else is happily using their wildly subsidized accounts (although the level of subsidy (users get to use three to six times as much compute as their subscription costs) isn’t completely insane, merely large).
Tm 10.26.26 at 7:45 am
engels 53: Again what I quoted from qikipedia: “About half of the manuscripts were accompanied by machine-verifiable proofs, certified as correct by the automated logic platform Lean; however, *verifying that the proof corresponds to the paper must be done manually*; some of the formalizations appear to prove weaker results than those claimed in the papers”
engels 10.26.26 at 1:20 pm
Thanks TM and ETB, but then it sounds like half of it can binned, because the LLM misrepresented the results of its proofs. I was puzzled but the implication that there was ongoing uncertainty.
Tm 10.26.26 at 9:41 pm
In related news:
The Bull Case for AI Armageddon
For investors mulling the Anthropic share offering, a 10% chance of extinction may be a 90% win for your portfolio
https://www.wsj.com/tech/ai/the-bull-case-for-ai-armageddon-673521f4
engels 10.26.26 at 11:15 pm
In other news, this almost makes AI seem benign and carefully thought through:
https://www.theguardian.com/us-news/2026/oct/26/california-sunlight-space-mirrors-satellites
Kester 10.27.26 at 3:27 pm
I came up with a neat idea in maths someone please steal it.
We introduce the Kester function, an operator that modifies a real-valued function according to whether the function is continuous at each point.
Given a function f : X → ℝ, the Kester function K[f] is defined by:
Kf =
2f(x), if f is continuous at x,
f(x), if f is discontinuous at x.
The construction is unusual because the value of K[f] at x depends not merely on f(x), but also on a local property of f at that point.
In detail:
https://mathforums.com/t/the-kester-functiona-function-defined-by-the-continuity-of-its-underlying-function.374705/
engels 10.27.26 at 3:30 pm
Btw, speaking of Henry James, “A Bostonian in Tokyo” sounds like a great Henry James novel… maybe Chat-GPT could write it?
engels 10.28.26 at 11:56 am
Decent effort from GPT-5.4 mini:
At that season when the climate of Tokyo, having for some weeks maintained an air of exquisite urbanity, was observed to lapse, with a delicacy almost moral, into an afternoon of rain, there arrived in the city a young Bostonian whose consciousness, if one might speak so of a quality at once so sensitive and so irreproachably supplied with opinions, had already been formed, in the old manner, by all that was most enviable in the cultivated life of the American North-East. He had come, as such persons do, not because Tokyo had been recommended to him by any vulgar necessity, but because it had been represented, by a succession of minds equally ornamental and equally incapable of certitude, as a place where the very terms of perception underwent, under pressure of beauty and strangeness, a refinement almost too severe for comfort.
His name was Mr. Percival Vane—though there were ladies in Back Bay who, had they been asked, would have said “Perry,” and others, in whose presence he would have accepted no abbreviation whatsoever—and he had the look, in his thinness, his neatness, and that overtraced air of having arrived at conclusions too early in life, of a gentleman who had never in the least been forced to choose between principle and convenience because Providence, with a tenderness almost too considerate, had arranged that both should always attend him. He had in his pocket a letter of introduction, in his portmanteau a collection of linen that seemed to imply a moral faith in civilization, and in his mind, mixed in proportions not easily disentangled, a curiosity about Japan, a distrust of his own curiosity, and the conviction that whatever awaited him there would somehow constitute a judgment upon Boston.
He had been met, at the station, by a young gentleman attached to the American legation, who spoke with a rapidity that suggested either perfect ease or complete impatience, and who had conducted him, through the evening glitter and the dim streets, to a house in the foreign settlement where the paper lanterns without and the Western chairs within appeared to have entered into a compromise neither could altogether respect. There, before he had yet had time to decide whether he was tired or only rendered theatrical by fatigue, he encountered Mrs. Ellery Prowse, a lady of singular composure, who had lived in Tokyo three years and was therefore considered by new arrivals to possess, not information exactly, but an authority of manner that seemed to make information unnecessary.
Mrs. Prowse, who wore her knowledge as some women wear jewels—with a consciousness that it was both becoming and exposed to envy—received him with a kindness that had evidently been taught by experience how to avoid kindness’s more embarrassing forms. She asked after Boston, not as one asks after a place, but as one might inquire into a compromised reputation; and when Mr. Vane, with that admirable caution which is one of the few virtues never wholly appreciated by those who possess it, began to speak of the city’s institutions, its libraries, its clubs, and the singular habit of its summers, she listened in a way that made him feel he was being both understood and revised.
Outside, the rain fell with an oriental patience which, had Mr. Vane possessed the proper imagination for adjectives, he would perhaps have called symbolic; and within, as the lamp-light widened upon the furniture and the tea-trays and the faces that bent, with differing degrees of discipline, toward conversation, he became aware that Tokyo was not to be approached, if one wished to keep one’s self-respect, as a spectacle merely, but as an intelligence. This intelligence, as he presently divined, was everywhere, and nowhere more than in the silence between remarks—those intervals in which, to a Bostonian, so habituated to the consolations of assertion, a great deal of life appeared to be taking place without permission.
“Tokyo,” said Mrs. Prowse, after a pause which had already begun to make Mr. Vane uneasy, “is the kindest place in the world to one who does not insist upon being the center of it.”
He received this with the attention due to a paradox and the discomfort due to a truth; for if he had come, in part, to discover Japan, he had perhaps come still more to discover how much of himself could survive being discovered there. […]
someone who has read a henry james novel 10.28.26 at 5:32 pm
There is no way anyone who has read James could think that engels @ 65’s experiment has produced something that’s even close to a “decent effort” at aping James’ style. Like come on. The last two paragraphs in particular are just flatly revolting. Here’s how a lady character delivers a witty observation in Henry James:
From The Aspern Papers:
She had heard Miss Bordereau was ill and had a suspicion that she was in want; and she had gone to the house to offer assistance, so that if there were suffering (and American suffering), she should at least not have it on her conscience. The “little one” received her in the great cold, tarnished Venetian sala, the central hall of the house, paved with marble and roofed with dim crossbeams, and did not even ask her to sit down. This was not encouraging for me, who wished to sit so fast, and I remarked as much to Mrs. Prest. She however replied with profundity, “Ah, but there’s all the difference: I went to confer a favor and you will go to ask one. If they are proud you will be on the right side.” And she offered to show me their house to begin with—to row me thither in her gondola.There’s no point of view shifting in the middle of the quote, the remark is revelatory of the character of the speaker and the person the speaker is describing (and ironic given the later developments of the chapter). The lying/asskissing machine can’t do any of this because it can only write what vaguely sounds right on the surface, it can’t hold together a thought for more than a few words at a time. I don’t even like James that much but his writing was not like this, at all. He was a meticulous writer always constructing the whole from every word, sentence and paragraph. He’s probably one of the least machine-imitatable stylists out there just for this reason.
Well maybe the fault is in myself. There is something skin-crawling about LLM prose that I just can’t shake. Something is very, very, very wrong in the passage engels quotes at 65, horribly wrong, I read it with a wince of disgust. Does nobody else feel this? Well, I guess there’s a trillion dollars invested in making people tell me “it’s just you, this is actually better than Henry James if you think about it, because you can ask the machine to make it about whatever you want!” and I don’t have a trillion dollars to invest on the other side so the reality will soon be that I’m wrong, and it’s wicked of me to invite others to be in the wrong with me.
Tm 10.28.26 at 6:54 pm
Engels 63: don’t worry, these satellites will absolutely be AI guided, which means that the lunatics running this startup are already planning their IPO valued at 90 trillion Trumpcoins.
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