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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 particu

• jasonhong 2y ago
  Interestingly, the main article mentions Bill Thurston's paper "On Proof and Progress in Mathematics" (https://www.math.toronto.edu/mccann/199/thurston.pdf https://www.math.toronto.edu/mccann/199/thurston.pdf), but doesn't mention a quote from that paper that captures the essence of what you wrote: > "The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." Incidentally, I've also a similar problem when reviewing HCI and computer systems papers. Ok sure, this deep learning neural net worked better, but what did we as a community actually learn that others can build on?
  • nicf 2y ago
    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.)
    • rssoconnor 2y ago
      Is the proof of the Four Colour Theorem really that unsatisfying? The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. To verify this computational fact one uses a (very) glorified pocket calculator.
      • godelski 2y ago
        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. [0] Quotes are because truth is very very hard to derive. https://hermiene.net/essays-trans/relativity_of_wrong.html https://hermiene.net/essays-trans/relativity_of_wrong.html
• Henchman21 2y ago
  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.
  • bluefirebrand 2y ago
    > 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 That's the part I find unnerving
    • feoren 2y ago
      > 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.
      • 2y ago
• lordleft 2y ago
  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?
  • nicf 2y ago
    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.
    • deleted 2y ago
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    • GPerson 2y ago
      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.
      • nicf 2y ago
        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.
• mcguire 2y ago
  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."
  • mb7733 2y ago
    What the hell is that quote? No, a simple proof is the absolute mathematical ideal!
  • psunavy03 2y ago
    That seems like a justification that is right on the knife's edge of being a self-licking ice cream cone.
  • senderista 2y ago
    Really? I've always had the impression that "elementary" proofs of hard problems are highly valued.
    • ants_everywhere 2y ago
      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.
  • lupire 2y ago
    That's the attitude of poor mathematicians who are insecure about their own faults.
• jvans 2y ago
  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
• optimalsolver 2y ago
  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?
  • nicf 2y ago
    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. Does that answer your question?
    • diamondage 2y ago
      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
• godelski 2y ago
  > 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 https://www.youtube.com/watch?v=hV41QEKiMlM
  • solveit 2y ago
• 7402 2y ago
  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.
  • kevindamm 2y ago
    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.
• the_cat_kittles 2y ago
  taking a helicopter to the top of a mountain is not the same thing as climbing it
  • cladopa 2y ago
    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.
    • jlev1 2y ago
      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.
• SwtCyber 2y ago
  Understanding why something is true - that's the beauty of it
• piokoch 2y ago
  "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.
  • whynotminot 2y ago
    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?
    • alternatex 2y ago
      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.
  • Philpax 2y ago
    Luckily, we have ways of verifying mathematical results, and using that to improve our AI systems: https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/ https://deepmind.google/discover/blog/ai-solves-imo-problems...
• crazygringo 2y ago
  > 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.
  • Ar-Curunir 2y ago
    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).
  • losvedir 2y ago
    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.
• ssivark 2y ago
  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.
• DeepSeaTortoise 2y ago
  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.