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.
Interestingly, the main article mentions Bill Thurston's paper "On Proof and Progress in Mathematics" (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?
The Four Color Theorem is a great example! I think this story is often misrepresented as one where mathematicians didn't believe the computer-aided proof. Thurston gets the story right: I think basically everyone in the field took it as resolving the truth of the Four Color Theorem --- although I don't think this was really in serious doubt --- but in an incredibly unsatisfying way. They wanted to know what underlying pattern in planar graphs forces them all to be 4-colorable, and "well, we reduced the question to these tens of thousands of possible counterexamples and they all turned out to be 4-colorable" leaves a lot to be desired as an answer to that question. (This is especially true because the Five Color Theorem does have a very beautiful proof. I reach at a math enrichment program for high schoolers on weekends, and the result was simple enough that we could get all the way through it in class.)
I think you're missing the core component. We care __WHY__ the theorem is true. To be honest, the __IF__ part matters a lot less.
The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important.
I'll give an example from physics. (If you're unsatisfied with this one, pick another physics fact and I'll do my best. _Any_ will do.) Here's a "fact"[0]: The atoms with even number of electrons are more stable than those with an odd number. We knew this in the 1910's, and this is a fact that directly led to the Pauli Exclusion Principle, which led us to better understand chemical bonds. Asking why Pauli Exclusion happens furthers our understanding and leading us to a better understanding of the atomic model. It goes on and on like this.
It has always been about the why. The why is what leads us to new information. The why is what leads to generalization. The why is what leads to causality and predictive models. THe why is what makes the fact useful in the first place.
I do think the why that the Four Colour Theorem is true is captured my statement. The reason why it is true is because there exists some finite unavoidable and reducible set of configurations.
I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason. I suspect that if the size of this finite set were 2, instead of 633, and you could draw these unavoidable configuration on the chalk board, and easily illustrate that both of them are reducible, everyone would be saying "ah yes, the four colour theorem, such an elegant proof!"
Yet, whether the finite set were of size 2 or size 633, the fundamental insight would be identical: there exists some finite unavoidable and reducible set of configurations.
> I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason.
I think that is exactly correct, except for the "no good reason" part. There aren't many (any?) practical situations where the 4-colour theory's provability matters. So the major reason for studying it is coming up with a pattern that can be used in future work.
Having a pattern with a small set (single digit numbers) means that it can be stored in the human brain. 633 objects can't be. That limits the proof.
To me, the four-color theorem is a very interesting proof of concept, perhaps the most interesting mathematical proof of the past 50 years. Perhaps the "pattern that can be used in future work" is the idea of having computers enumerate a large number of cases, which they then solve in individually straightforward ways.
But, I can understand if pure mathematicians don't feel this way. This might be only really an intriguing and beautiful concept to someone who is interested in scaling up algorithms and AI.
> So the major reason for studying it is coming up with a pattern that can be used in future work.
Surely, reducing the infinite way in which polygons can be placed on a plane to a finite set, no matter how large, must involve some pattern useful for future work?
> The reason why it is true is because there exists some finite unavoidable and reducible set of configurations.
OK but respectfully that's just restating the problem in an alternative form. We don't get any insight from it. Why does there exist this limit? What is it about this problem that makes this particular structure happen?
Perhaps the key insight is that there is no concise explanation that underlies this particular structure. Many mathematical statements are true for no concise reason. If you want to discover if these things are true or not, perhaps you need a computer-assisted search, and that's the intellectual lesson to be drawn here.
I really doubt this. I mean mathematicians spent decades trying to answer if the number 2 exists. People spend a lot of time on what seems fairly mundane and frankly, the results are quite beneficial. What's incredible or mind blowing is really just about your perspective, it is really just about your choice to wonder more. https://www.youtube.com/shorts/lcQAWEqPmeg
> The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED.
You just summarised (nearly) everything a mathematician can get out of that computerised proof. That's unsatisfying. It doesn't give you any insight into any other areas of math, nor does it suggest interesting corollaries, nor does it tell you which pre-condition of the statement does what work.
That's rather underwhelming. That's less than you can get out of the 100th proof of Pythagoras.
Another example akin to the proof of the 4-color map theorem was the proof of the Kepler conjecture [1], i.e. "Grocers stack their oranges in the densest-possible way."
We "know" it's true, but only because a machine ground mechanically through lots of tedious cases. I'm sure most mathematicians would appreciate a simpler and more elegant proof.
I understand the emotion and the sadness you mention from a different situation I experienced about a dozen years ago. At that time I was entering Kaggle machine learning competitions, and I did well enough to reach 59th on the global leaderboard. But the way I did it was by trying to understand the problem domain and make and test models based on that understanding.
However by the end of that period, it seemed to transition to a situation where the most important skill in achieving a good score was manipulating statistical machine learning tools (Random Forests was a popular one, I recall), rather than gaining deep understanding of the physics or sociology of the problem, and I started doing worse and I lost interest in Kaggle.
So be it. If you want to win, you use the best tools. But the part that brought joy to me was not fighting for the opportunity to win a few hundred bucks (which I never did), but for the intellectual pleasure and excitement of learning about an interesting problem in a new field that was amenable to mathematical analysis.
I witnessed the same bitter lesson on Kaggle, too. Late stage competitions were almost always won by a mixture of experts using the most recently successful models on that problem. Or a random forest of the same. It was a little frustrating/disappointing to the part of me that wanted to see insights, not just high scores.
Many years ago I heard a mathematician speaking about some open problem and he said, "Sure, it's possible that there is a simple solution to the problem using basic techniques that everyone has just missed so far. And if you find that solution, mathematics will pat you on the head and tell you to run off and play.
"Mathematics advances by solving problems using new techniques because those techniques open up new areas of mathematics."
A proof of a long-open conjecture that uses only elementary techniques is typically long and convoluted.
Think of the problem as requiring spending a certain amount of complexity to solve. If you don't spend it on developing a new way of thinking then you spend it on long and tedious calculations that nobody can keep in working memory.
It's similar to how you can write an AI model in Pytorch or you can write down the logic gates that execute on the GPU. The logic gate representation uses only elementary techniques. But nobody wants to read or check it by hand.
TBH, I think you're worrying about a future that is likely to become much more fun than boring.
For actual research mathematics, there is no reason why an AI (maybe not current entirely statistical models) shouldn't be able to guide you through it exactly the way you prefer to. Then it's just a matter of becoming honest with your own desires.
But it'll also vastly blow up the field of recreational mathematics. Have the AI toss a problem your way you can solve in about a month. A problem involving some recent discoveries. A problem Franklin could have come up with. During a brothel visit. If he was on LSD.
I'm not a mathematician so please feel free to correct me...but wouldn't there still be an opportunity for humans to try to understand why a proof solved by a machine is true? Or are you afraid that the culture of mathematics will shift towards being impatient about this sorts of questions?
Well, it depends on exactly what future you were imagining. In a world where the model just spits out a totally impenetrable but formally verifiable Lean proof, then yes, absolutely, there's a lot for human mathematicians to do. But I don't see any particular reason things would have to stop there: why couldn't some model also spit out nice, beautiful explanations of why the result is true? We're certainly not there yet, but if we do get there, human mathematicians might not really be producing much of anything. What reason would there be to keep employing them all?
Like I said, I don't have any idea what's going to happen. The thing that makes me sad about these conversations is that the people I talk to sometimes don't seem to have any appreciation for the thing they say they want to dismantle. It might even be better for humanity on the whole to arrive in this future; I'm not arguing that one way or the other! Just that I think there's a chance it would involve losing something I really love, and that makes me sad.
> The thing that makes me sad about these conversations is that the people I talk to sometimes don't seem to have any appreciation for the thing they say they want to dismantle
Yes! This is what frustrates my about the pursuit of AI for the arts too.
This seems obviously untrue: why would they be replicating it if they didn’t want it?
I see both cases as people who aren’t well served by the artisanal version attempting to acquire a better-than-commoditized version because they want more of that thing to exist. We regularly have both things in furniture and don’t have any great moral crisis that chairs are produced mechanistically by machines. To me, both things sound like “how dare you buy IKEA furniture — you have no appreciation of woodwork!”
Maybe artisanal math proofs are more beautiful or some other aesthetic concern — but what I’d like is proofs that business models are stable and not full of holes constructed each time a new ML pipeline deploys; which is the sort of boring, rote work that most mathematicians are “too good” to work on. But they’re what’s needed to prevent, eg, the Amazon 2018 hiring freeze.
That’s the need that, eg, automated theorem proving truly solves — and mathematicians are being ignored (much like artist) by people they turn up their noses at.
> why would they be replicating it if they didn’t want it?
Who is "they"?
Most AI for math work is being done by AI researchers that are not themselves academic mathematicians (obviously there exceptions). Similarly, most AI for music and AI for visual art is being done by AI researchers that themselves are not professional musicians or artists (again, there are exceptions). This model can work fine if the AI researchers collaborate with mathematicians or artists to understand that the use of AI is actually useful in the workflow of those fields, but often that doesn't happen and there is a savior-like arrogance where AI researchers think they'll just automate those fields. Same thing happens in AI for medicine. So the reason many of those AI researchers want to do this is for the usual incentives - money and publications.
Clearly, there are commercial use cases for AI in all these fields and those may involve removing humans entirely. But in the case of art, and I (and Hardy) would argue academic math, there's a human aspect that can't be removed. Both of those approaches can exist in the world and have value but AI can't replace Van Gogh entirely. It'll automate the process of creating mass produced artwork or become a tool that human artists can use. Both of those require understanding the application domain intimately, so my point stands I think.
> This model can work fine if the AI researchers collaborate with mathematicians or artists to understand that the use of AI is actually useful in the workflow of those fields, but often that doesn't happen and there is a savior-like arrogance where AI researchers think they'll just automate those fields.
In my experience, the vast majority is people who are hobbyists or amateurs in those fields, who are looking to innovate in approaches — eg, the overwhelming majority of AI music is hobbyists using models to experiment. Similarly, the overwhelming majority of people using AI graphics tools are making memes or pictures to share with friends.
Those people are poorly served by the artisanal approach and are looking to create more art — they’re not engaging in “savior-like arrogance” but trying to satisfy unmet desire for new music and art. You’re merely being snooty.
> But in the case of art, and I (and Hardy) would argue academic math, there's a human aspect that can't be removed.
This is the pretentiousness I called out (and you completely failed to address):
> That’s the need that, eg, automated theorem proving truly solves — and mathematicians are being ignored (much like artist) by people they turn up their noses at.
Nobody is stopping you from your artisanal proofs — have at it. You’re refusing to do the ugly work people actually want, so they’re solving their problems with a tool that doesn’t involve you.
I think we're actually in agreement with respect to the needs of AI in music and graphics. When I say "AI researchers" I'm talking about people in industrial and academic labs. I consider the hobbyists you describe to be the users (who obviously can also be researchers). It's the desires/needs of the latter that should drive the research agenda of the former.
I actually don't understand your position here or what you think I'm arguing for. My point is that the real musicians, artists and mathematicians (whether they're hobbyists, academics or professionals in industry) are not well served by detached AI researchers just trying to automate their work for them. They need AI researchers to understand their workflows and build tools that elevate them, i.e. bicycles for the mind (or hands?).
Again, I do recognize there may be new fully automated workflows that can come out of AI research too but I maintain that the actual artists, musicians and mathematicians today have a valuable role in guiding that development too.
I don’t think the advent of superintelligence will lead to increased leisure time and increased well-being / easier lives. However, if it did I wouldn’t mind redundantly learning the mathematics with the help of the AI. It’s intrinsically interesting and ultimately I don’t care to impress anybody, except to the extent it’s necessary to be employable.
I would love that too. In fact, I already spend a good amount of my free time redundantly learning the mathematics that was produced by humans, and I have fun doing it. The thing that makes me sad to imagine --- and again, this is not a prediction --- is the loss of the community of human mathematicians that we have right now.
>But I don't see any particular reason things would have to stop there: why couldn't some model also spit out nice, beautiful explanations of why the result is true?
Oh… I didnt anticipate this would bother you. Would it be fair to say that its not that you like understanding why its true, because you have that here, but that you like process of discovering why?
Perhaps thats what you meant originally. But my understanding was that you were primarily just concerned with understanding why, not being the one to discover why.
This is an interesting question! You're giving me a chance to reflect a little more than I did when I wrote that last comment.
I can only speak for myself, but it's not that I care a lot about me personally being the first one to discover some new piece of mathematics. (If I did, I'd probably still be doing research, which I'm not.) There is something very satisfying about solving a problem for yourself rather than being handed the answer, though, even if it's not an original problem. It's the same reason some people like doing sudokus, and why those people wouldn't respond well to being told that they could save a lot of time if they just used a sudoku solver or looked up the answer in the back of the book.
But that's not really what I'm getting at in the sentence you're quoting --- people are still free to solve sudokus even though sudoku solvers exist, and the same would presumably be true of proving theorems in the world we're considering. The thing I'd be most worried about is the destruction of the community of mathematicians. If math were just a fun but useless hobby, like, I don't know, whittling or something, I think there would be way fewer people doing it. And there would be even fewer people doing it as deeply and intensely as they are now when it's their full-time job. And as someone who likes math a lot, I don't love the idea of that happening.
CNCs and other technology haven’t destroyed woodworking. There’s whole communities on YouTube — with a spectrum from casual to hobbyist to artisanal to industrial.
Why would mathematics be different than woodworking?
Do you believe there’s a limited demand for mathematics? — my experience is quite the opposite, that we’re limited by the production capacity.
HN has this very unique and strange type of reasoning. You’re actually asking why would mathematics be any different than woodworking because CNC machines? It’s like aby issue can be reduced to the most mundane observations and simplicity because we have to justify all technology. Professional mathematics requires years of intense and usually, i.e. almost always, in graduate schools and the entire machinery of that. You’re comparing something many people do as a hobby to the life’s work and f others. of course you can have wave all this away with some argument but I’m not sure this type of reasoning is going to save the technocrats when it he majority of people realize what this app portends for society.
No Im not — I’m comparing two fields that range from hobby to professional, both of which I’ve worked in professionally. But for which automation occurred at different times.
> You’re comparing something many people do as a hobby to the life’s work and f others.
You’re denigrating the talents and educational efforts of artisanal woodworkers to make a shallow dismissal of my point.
This is actually a metaphor I've used myself. I do think the woodworking community is both smaller and less professionalized than it would be in a world where industrial furniture production didn't exist. (This is a bizarre counterfactual, because it's basically impossible for me to imagine a world where industrial furniture production doesn't exist but YouTube does, but like pretend with me here for a moment.) I don't know that this is necessarily a bad thing, but it's definitely different, and I can imagine that if I were a woodworker who lived through the transition from one world to the other I would find it pretty upsetting! As I said above, I'm not claiming it's not worth making the transition anyway, but it does come with a cost.
One place I think the analogy breaks down, though, is that I think you're pretty severely underestimating the time and effort it takes to be productive at math research. I think my path is pretty typical, so I'll describe it. I went to college for four years and took math classes the whole time, after which I was nowhere near prepared to do independent research. Then I went to graduate school, where I received a small stipend to teach calculus to undergrads while I learned even more math, and at the end of four and a half years of that --- including lots of one-on-one mentorship from my advisor --- I just barely able to kinda sorta produce some publishable-but-not-earthshattering research. If I wanted to produce research I was actually proud of, it probably would have taken several more years of putting in reps on less impressive stuff, but I left the field before reaching that point.
Imagine a world where any research I could have produced at the end of those eight and a half years would be inferior to something an LLM could spit out in an afternoon, and where a different LLM is a better calculus instructor than a 22-year-old nicf. (Not a high bar!) How many people are going to spend all those years learning all those skills? More importantly, why would they expect to be paid to do that while producing nothing the whole time?
That’s no different than high-end artisanal woodworking, which I would argue is comparable to self-directed, cutting-edge research:
- apprenticing
- journeyman phase
- only finally achieving mastery
CNC never replaced those people, rather, it scaled the whole field — by creating much higher demand for furniture. People who never made that full journey instead work at factories where their output is scaled. What was displaced was mediocre talent in average homes, eg, building your own table from a magazine design.
You still haven’t answered why you think mathematics will follow a different trajectory — and the only substantial displacement will, eg, be business analysts no longer checking convexity of models and outsourcing that to AI-scaled math experts at the company.
Woodworking is very far from my world, so I don't really have any grounds to judge how comparable the two things actually are. I'll say two things instead.
First, right now presumably the reason a few people still become master woodworkers is that their work is actually better than the mass-produced furniture that you can get for much less money. Imagine a world where instead it was possible to cheaply and automatically produce furniture that is literally indistinguishable from, or maybe even noticeably superior to, anything a human woodworker could ever make. Do you really think the same number of people would still spend years and years developing those skills?
Second, you've talked about business logic and "math experts at the company" a few times now, which makes me wonder if we're just referring to different things with the word "mathematics". I'm talking about a specific subset, what's sometimes called "pure math," the kind of research that mostly only exists within academia and is focused on proving theorems with the goal of improving human understanding of mathematical patterns with no particular eye on solving any practical problems. It sounds like you're focused on the sort of mathematical work that gets done in industry, where you're using mathematical tools, but the goal is to solve a practical problem for a business.
These are actually quite different activities --- the same individuals who are good at one stand a decent chance of being good at the other, but that's most of what they have in common, and even there I know many people who are much more skilled at one than the other. I'm not really asking anyone who doesn't care about pure math to start caring about it, but when I'm talking about the effect of AI on the future of the field, I'm referring specifically to pure math research.
It would be like having the machine code to something amazing but lacking the ability to adequately explain it or modify it - the machine code is too big and complicated to follow, so unless you can express it or understand it in a better way, it can only be used exactly how it is already.
In mathematics it is just as (if not moreso) important to be able to apply techniques used to solve novel proofs as it is to have the knowledge that the theorem itself is true. Not only might those techniques be used to solve similar problems that the theorem alone cannot, but it might even uncover wholly new mathematical concepts that lead you to mathematics that you previously could not even conceive of.
Machine proofs in their current form are basically huge searches/brute forces from some initial statements to the theorem being proved, by way of logical inference. Mathematics is in some ways the opposite of this: it's about understanding why something is true, not solely whether it is true. Machine proofs give you a path from A to B but that path could be understandable-but-not-generalizable (a brute force), not-generalizable-but-understandable (finding some simple application of existing theorems to get the result that mathematicians simply missed), or neither understandable-nor-generalizable (imagine gigabytes of pure propositional logic on variables with names like n098fne09 and awbnkdujai).
Interestingly, some mathematicians like Terry Tao are starting to experiment with combining LLMs with automated theorem proving, because it might help in both guiding the theorem-prover and explaining its results. I find that philosophically fascinating because LLMs rely on some practices which are not fully understood, hence the article, and may validate combining formal logic with informal intuition as a way of understanding the world (both in mathematics, and generally the way our own minds combine logical reasoning with imprecise language and feelings).
That is kind of hard to do. Human reasoning and computer reasoning is very different, enough so that we can't really grasp it. Take chess, for example. Humans tend to reason in terms of positions and tactics, but computers just brute force it (I'm ignoring stuff like Alpha Zero because computers were way better than us even before that). There isn't much to learn there, so GMs just memorize the computer moves for so and so situation and then go back to their past heuristics after n moves
"I can imagine a future where some future model is better at proving theorems than any human mathematician" Please do not overestimate the power of the algorithm that is predicting next "token" (e.g. word) in a sequence of previously passed words (tokens).
This algorithm will happily predict whatever it was fed with, just ask Chat GPT to write the review of non-existing camera, car or washing machine, you will receive nicely written list of advantages of such item, so what it does not exist.
I can also write you a review of a non-existent camera or washing machine. Or anything else you want a fake review of! Does that mean I’m not capable of reasoning?
If you are not capable of distinguishing between truth and lie, and not capable of reflection which is the drive behind learning from past mistakes - then yes.
> 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.
I really wish that had been my experience taking undergrad math courses.
Instead, I remember linear algebra where the professor would prove a result by introducing an equation pulled out of thin air, plugging it in, showing that the result was true, and that was that. OK sure, the symbol manipulation proved it was true, but zero understanding of why. And when I'd ask professors about the why, I'd encounter outright hostility -- all that mattered was whether it was proven, and asking "why" was positively amateurish and unserious. It was irrelevant to the truth of a result. The same attitude prevailed when it got to quantum mechanics -- "shut up and calculate".
I know there are mathematicians who care deeply about the why, and I have to assume it's what motivates many of them. But my actual experience studying math was the polar opposite. And so I find it very surprising to hear the idea of math being described as being more interested in why than what. The way I was taught didn't just not care about the why, but seemed actively contemptuous of it.
Math-for-engineering and math-for-math courses are very different in emphasis and enthusiasm. Many engineering students tend to not care too much about proofs, so the "get it working" approach might be fine for them.
Also, the math profs teaching the "math-for-engineering" courses tend to view them as a chore (which it kind of is; pure math doesn't get a lot of funding, so they have to pick up these engineering-oriented courses to grab a slice of that engineering money).
I guess, what university and what level of math was that?
I majored in math at MIT, and even at the undergraduate level it was more like what OP is describing and less like what you're saying. I actually took linear algebra twice since my first major was Economics before deciding to add on a math major, and the version of linear algebra for your average engineer or economist (i.e.: a bunch of plug and chug matrices-type stuff), which is what I assume you're referring to, was very different. Linear algebra for mathematicians was all about vector spaces and bases and such, and was very interesting and full of proofs. I don't think actually concretely multiplying matrices was even a topic!
So I guess linear algebra is one of those topics where the math side is interesting and very much what all the mathematicians here are describing, but where it turned out to be so useful for everything, that there's a non-mathematician version of it which is more like what it sounds like you experienced.
in poker AI solvers tell you what the optimal play is and it's your job to reverse engineer the principles behind it. It cuts a lot of the guess work out but there's still plenty of hard work left in understanding the why and ultimately that's where the skill comes in. I wonder if we'll see the same in math
As Heidegger pointed out in _"The question concerning technology"_ the driving mindset behind industrial technology is to turn everything into a (fungible) standing resource -- instrumentalizing it and robbing it of any intrinsic meaning.
Maybe because CS is more engineering than science (at least as far as what drives the sociology), a lot of people approach AI from the same industrial perspective -- be it applications to math, science, art, coding, and whatever else. Ideas like _the bitter lesson_ only reinforce the zeitgeist.
If the shortest proof for some theorem is several thousand pages long and beyond the ability of any biological mind to comprehend, would mathematicians not care about it?
Which is to say, if you only concern yourself with theorems which have short, understandable proofs, aren't you cutting yourself off from vast swathes of math space?
Hm, good question. It depends on what you mean. If you're asking about restricting which theorems we try to prove, then we definitely are cutting ourselves off from vast swathes of math space, and we're doing it on purpose! The article we're responding to talks about mathematicians developing "taste" and "intuition", and this is what I think the author meant --- different people have different tastes, of course, but most conceivable true mathematical statements are ones that everyone would agree are completely uninteresting; they're things like "if you construct these 55 totally unmotivated mathematical objects that no one has ever cared about according to these 18 random made-up rules, then none of the following 301 arrangements are possible."
If you're talking about questions that are well-motivated but whose answers are ugly and incomprehensible, then a milder version of this actually happens fairly often --- some major conjecture gets solved by a proof that everyone agrees is right but which also doesn't shed much light on why the thing is true. In this situation, I think it's fair to describe the usual reaction as, like, I'm definitely happy to have the confirmation that the thing is true, but I would much rather have a nicer argument. Whoever proved the thing in the ugly way definitely earns themselves lots of math points, but if someone else comes along later and proves it in a clearer way then they've done something worth celebrating too.
So Godel proved that there are true theorems that are unprovable. My hunch is that there is a fine grained version of this result <-- that there is a some distribution on the length of the proof for any given conjecture. If true that would mean that we better get used to dealing with long nasty proofs because they are a necessary part of mathematics...perhaps even, in some kind of Kolmogorov complexity-esque fashion, the almost-always bulk of it
Agree that something like this does seem likely. And this line of thought also highlights the work of Chaitin, and the fact that the current discussion around AI is just the latest version of early-2000s quasi-empiricism[1] stuff that never really got resolved. Things like the 4-color theorem would seem to be just the puny top of really big iceberg, and it's probably not going away.
The new spin on these older unresolved issues IHMO is really the black-box aspect of our statistical approaches. Lots of mathematicians that are fine with proof systems like Lean and some million-step process that can in principle be followed are also happy with more open-ended automated search and exploration of model spaces, proof spaces, etc. But can they ever be really be happy with a million gigabyte network of weighted nodes masquerading as some kind of "explanation" though? Not a mathematician but I sympathize. Given the difficulty of building/writing/running it, that looks more like a product than like "knowledge" to me (compare this to how Lean can prove Godel on your laptop).
Maybe it's easier to swallow the bitter pill of poor quality explanations though after the technology itself is a little easier to actually handle. People hate ugly things less if they are practical, and actually something you can build pretty stuff on top on.
I’m not sure that’s quite true. Say the proof of proposition P requires a minimum of N symbols. You could prove it in one paper that’s N symbols long and nobody can read, or you can publish k readable papers, with an average length on the order of N/k symbols, and develop a theory that people can use.
I think even if N is quite large, that just means it may take decades or millennia to publish and understand all k necessary papers, but maybe it’s still worth the effort even if we can get the length-N paper right away. What are you going to do with a mathematical proof that no one can understand anyway?
> If the shortest proof for some theorem is several thousand pages long and beyond the ability of any biological mind to comprehend, would mathematicians not care about it?
Care or not, what are they supposed to do with it?
Sure, they can now assume the theorem to be true, but nothing stopped them from doing that before.
> the primary aim isn't really to find out whether a result is true but why it's true.
I'm honestly surprised that there are mathematicians that think differently (my background[0]). There are so many famous mathematicians stating this through the years. Some more subtle like Poincare stating that math is not the study of numbers but the relationship between them, while others far more explicit. This sounds more like what I hear from the common public who think mathematics is discovered and not invented (how does anyone think anything different after taking Abstract Algebra?).
But being over in the AI/ML world now, this is my NUMBER ONE gripe. Very few are trying to understand why things are working. I'd argue that the biggest reason machines are black boxes are because no one is bothering to look inside of them. You can't solve things like hallucinations and errors without understanding these machines (and there's a lot we already do understand). There's a strong pushback against mathematics and I really don't understand why. It has so many tools that can help us move forward, but yes, it takes a lot of work. It's bad enough I know people who have gotten PhDs from top CS schools (top 3!) and don't understand things like probability distributions.
Unfortunately doing great things takes great work and great effort. I really do want to see the birth of AI, I wouldn't be doing this if I didn't, but I think it'd be naive to believe that this grand challenge can entirely be solved by one field and something so simple as throwing more compute (data, hardware, parameters, or however you want to reframe the Bitter Lesson this year).
Maybe I'm biased because I come from physics where we only care about causal relationships. The "_why_" is the damn Chimichanga. And I should mention, we're very comfortable in physics working with non-deterministic systems and that doesn't mean you can't form causal relationships. That's what the last hundred and some odd years have been all about.[1]
[0] Undergrad in physics, moved to work as engineer, then went to grad school to do CS because I was interested in AI and specifically in the mathematics of it. Boy did I become disappointment years later...
[1] I think there is a bias in CS. I notice there is a lot of test driven development, despite that being well known to be full of pitfalls. You unfortunately can't test your way into a proof. Any mathematician or physicist can tell you. Just because your thing does well on some tests doesn't mean there is proof of anything. Evidence, yes, but that's far from proof. Don't make the mistake Dyson did: https://www.youtube.com/watch?v=hV41QEKiMlM
> I'd argue that the biggest reason machines are black boxes are because no one is bothering to look inside of them.
People do look, but it's extremely hard. Take a look at how hard the mechanistic interpretability people have to work for even small insights. Neel Nanda[1] has some very nice writeups if you haven't already seen them.
> Very few are trying to understand why things are working
What is in question is why this is given so little attention. You can hear Neel talk about this himself. It is the reason he is trying to rally people and get more into Mech Interp. Which frankly, this side of ML is as old as ML itself.
Personal, I believe that if you aren't trying to interpret results and ask the why then you're not actually doing science. Which is fine. There's plenty of good things that come from outside science. I just think it's weird to call something science if you aren't going to do hypothesis testing and finding out why things are the way they are
The problem is that mechanistic interpretability is a lot like neuroscience or molecular biology, i.e. you're trying to understand huge complexity from relatively crude point measurements (no offense intended to neuroscientists and biologists). But AI wants publishable results yesterday. I often wonder whether the current AI systems will stay around long enough for anyone to remain interested in understanding why they ever worked.
People will always be interested in why things work. At least one will as long as I'm alive, but I really don't think I'm that special. Wondering why things are the way they are is really at the core of science. Sure, there are plenty of physicists, mathematicians, neuroscientists, biologists, and others who just want answers, but this is a very narrow part of science.
I would really encourage others to read works that go through the history of the topic they are studying. If you're interested in quantum mechanics, the one I'd recommend is "The Quantum Physicists" by William Cropper[0]. It won't replace Griffiths[1] but it is a good addition.
The reason that getting information like this is VERY helpful is that it teaches you how to solve problems and actually go into the unknown. It is easy to learn things from a book because someone is there telling you all the answers, but texts like these instead put yourself in the shoes of the people in those times, and focus on seeing what and why certain questions are being asked. This is the hard thing when you're at the "end". When you can't just read new knowledge from a book, because there is no one that knows! Or the issue Thomas Wolf describes here[2] and why he struggled.
True. Taking a helicopter is way more impressive. The Everest was climbed in 1953 and the first helicopter to go there was in 2005. It is way harder thing to do.
No, in your analogy building a helicopter capable of going there is impressive. (Though I dispute the idea that it’s more impressive simply because helicopters were invented more recently than mountain climbing.) In any case, riding in a helicopter remains passive and in no sense impressive.
I’ve worked in tech my entire adult life and boy do I feel this deep in my soul. I have slowly withdrawn from the higher-level tech designs and decision making. I usually disagree with all of it. Useless pursuits made only for resume fodder. Tech decisions made based on the bonus the CTO gets from the vendors (Superbowl tickets anyone?) not based on the suitability of the tech.
But absolutely worst of all is the arrogance. The hubris. The thinking that because some human somewhere has figured a thing out that its then just implicitly known by these types. The casual disregard for their fellow humans. The lack of true care for anything and anyone they touch.
Move fast and break things!! Even when its the society you live in.
That arrogance and/or hubris is just another type of stupidity.
I'm sure that many of us sympathize, but can you please express your views without fulminating? It makes a big difference to discussion quality, which is why this is in the site guidelines: https://news.ycombinator.com/newsguidelines.html.
It's not just that comments that vent denunciatory feelings are lower-quality themselves, though usually they are. It's that they exert a degrading influence on the rest of the thread, for a couple reasons: (1) people tend to respond in kind, and (2) these comments always veer towards the generic (e.g. "lack of true care for anything and anyone", "just another type of stupidity"), which is bad for curious conversation. Generic stuff is repetitive, and indignant-generic stuff doubly so.
By the time we get further downthread, the original topic is completely gone and we're into "glorification of management over ICs" (https://news.ycombinator.com/item?id=43346257). Veering offtopic can be ok when the tangent is even more interesting (or whimsical) than the starting point, but most tangents aren't like that—mostly what they do is replace a more-interesting-and-in-the-key-of-curiosity thing with a more-repetitive-and-in-the-key-of-indignation thing, which is a losing trade for HN.
> Move fast and break things!! Even when its the society you live in.
This is the part I don't get honestly
Are people just very shortsighted and don't see how these changes are potentially going to cause upheaval?
Do they think the upheaval is simply going to be worth it?
Do they think they will simply be wealthy enough that it won't affect them much, they will be insulated from it?
Do they just never think about consequences at all?
I am trying not to be extremely negative about all of this, but the speed of which things are moving makes me think we'll hit the cliff before even realizing it is in front of us
> Do they think they will simply be wealthy enough that it won't affect them much, they will be insulated from it?
Yes, partly that. Mostly they only care about their rank. Many people would burn down the country if it meant they could be king of the ashes. Even purely self-interested people should welcome a better society for all, because a rising tide lifts all boats. But not only are they selfish, they're also very stupid, at least in this aspect. They can't see the world as anything but zero sum, and themselves as either winning or losing, so they must win at all costs. And those costs are huge.
Reminds me of the Paradise Lost quote, "Better to rule in Hell, than serve in Heaven", such an insightful book for understanding a certain type of person from Milton. Beautiful imagery throughout too, highly recommend.
> Do they just never think about consequences at all?
Yes, I think this is it. Frequently using social media and being “online” leads to less critical thought, less thinking overall, smaller window that you allow yourself to think in, thoughts that are merely sound bites not fully fleshed out thoughts, and so on. Ones thoughts can easily become a milieu of memes and falsehoods. A person whose mind is in the state will do whatever anyone suggests for that next dopamine hit!
I am guilty of it all myself which is how I can make this claim. I too fear for humanity’s future.
> Are people just very shortsighted and don't see how these changes are potentially going to cause upheaval?
> Do they think the upheaval is simply going to be worth it?
All technology causes upheaval. We've benefited from many of these upheavals to the point where it's impossible for most to imagine a world without the proliferation of movable type, the internal combustion engine, the computer, or the internet. All of your criticisms could have easily been made word for word by the Catholic Church during the medieval era. The "society" of today is no more of a sacred cow than its antecedent incarnations were half a millenium ago. As history has shown, it must either adapt, disperse, or die.
> The "society" of today is no more of a sacred cow than its antecedent incarnations were half a millenium ago. As history has shown, it must either adapt, disperse, or die
I am not concerned about some kind of "sacred cow" that I want to preserve
I am concerned about a future where those with power no longer need 90% of the population so they deploy autonomous weaponry that grinds most of the population into fertilizer
And I'm concerned there are a bunch of short sighted idiots gleefully building autonomous weaponry for them, thinking they will either be spared from mulching, or be the mulchers
Edit: The thing about appealing to history is that it also shows that when upper classes get too powerful they start to lose touch with everyone else, and this often leads to turmoil that affects the common folk most
As one of the common folk, I'm pretty against that
There exists in such a case a certain institution or law; let us say, for the sake of simplicity, a fence or gate erected across a road. The more modern type of reformer goes gaily up to it and says, “I don’t see the use of this; let us clear it away.” To which the more intelligent type of reformer will do well to answer: “If you don’t see the use of it, I certainly won’t let you clear it away. Go away and think. Then, when you can come back and tell me that you do see the use of it, I may allow you to destroy it.”
It's basically meatspace internalizing and adopting the paperclip problem as a "good thing" to pursue, screw externalities and consequences.
And, lo-and-behold, my read for why it gets downvoted here is that a lot of folks on HN ascribe to this mentality, as it is part of the HN ethos to optimize , often pathologically.
Humans like to solve problems and be at the top of the heap. Such is life, survival of the fittest after all. AI is a problem to solve, whoever gets to AGI first will be at the top of the heap. It's a hard drive to turn off.
> But absolutely worst of all is the arrogance. The hubris. The thinking that because some human somewhere has figured a thing out that its then just implicitly known by these types.
I worked in an organization afflicted by this and still have friends there. In the case of that organization, it was caused by an exaggerated glorification of management over ICs. Managers truly did act according to the belief, and show every evidence of sincerely believing in it, that their understanding of every problem was superior to the sum of the knowledge and intelligence of every engineer under them in the org chart, not because they respected their engineers and worked to collect and understand information from them, but because managers are a higher form of humanity than ICs, and org chart hierarchy reflects natural superiority. Every conversation had to be couched in terms that didn't contradict those assumptions, so the culture had an extremely high tolerance for hand-waving and BS. Naturally this created cover for all kinds of selfish decisions based on politics, bonuses, and vendor perks. I'm very glad I got out of there.
I wouldn't paint all of tech with the same brush, though. There are many companies that are better, much better. Not because they serve higher ideals, but because they can't afford to get so detached from reality, because they'd fail if they didn't respect technical considerations and respect their ICs.
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.