In fact, the modern practice (the concept predates the practice of course, but was more of an opinion than a ritual) of mathematics as this ultimate understandable system of truth and elegance seemingly began in Ancient Greece with their practice of proofs and early development of mathematical "frameworks". It didn't reach its current level of rigor and sophistication until 100-150 years ago when Formalism became the dominant school of thought (https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...), spearheaded by a group of mathematicians who held even deeper beliefs that are often referred to as Mathematical Platonism (https://en.wikipedia.org/wiki/Mathematical_Platonism). (Note that these wikipedia articles are not amazing explanations of the concepts, how they relate to realism, or developed historically but they are adequate primers)

Of course, Godel proved that truths exists outside of these formal systems (only a couple decades after mathemticians had started building a secret religion around worshipping Logos. These beliefs were pervasive see eg Einsteins concept of God as a clockmaker or Erdos' references to "The Book"), which leaves us almost back where we started where we might need to consider there may be some empirical results and patterns which "work" but we do not fully understand - we may never understand them. Personally, I think this philosophically justifies not subjecting oneself to the burden of spending excess time understanding or proving things that have never been understood before - it may elude elegance (as the 4-color proof) or even knowability.

We can always look backwards and explain things later, and of course, it's a false dichotomy that some theorems or results must be fully understood and proven (or proven elegantly) before they can be considered true and used as a basis for further results. Perhaps it is unsatisfying to those who wish to truly understand the universe in terms of mathematical elegance, but that asshole used mathematical elegance to disprove mathematical elegance as a perfect tool for understanding the universe already, so take it up with him.

Personally, as someone who at one time heavily considered pursuing a life in mathematics in part because of its ability to answer deep truths, I think Godel set us free: to understand or know things, we cannot rely solely on mathematics. Formal mathematics itself tells us that there are things we can only understand by discovering them, building them, or experimenting with them. There are truths that Cuda Cowboys can uncover that LaTex Liturgy cannot

With AI advisor I do not have this problem. It explains parts I need, in a way I understand. If I study some complicated topic, AI shortens it from months to days.

I was somehow mathematically gifted when younger, sadly I often reinvented my own math, because I did not even know this part of math existed. Watching how Deepseek thinks before answering, is REALLY beneficial. It gives me many hints and references. Human teachers are like black boxes while teaching.

And it's not that AI can't contribute to this effort. I can certainly see how a chatbot research partner could be super valuable for lit review, brainstorming, and even 'talking things through' (much like mathematicians get value from talking aloud). This doesn't even touch on the ability to generate potentially valid proofs, which I do think has a lot of merit. But the idea that we could totally outsource the work to a generative model seems impossible by definition. The point of the labor is develop human understanding, removing the human from the loop changes the nature of the endeavor entirely (basically to algorithm design).

Similar stuff holds about art (at a high level, and glossing over 'craft art'); IMO art is an expressive endeavor. One person communicating a hard-to-express feeling to an audience. GenAI can obviously create really cool pictures, and this can be grist for art, but without some kind of mind-to-mind connection and empathy the picture is ultimately just an artifact. The human context is what turns the artifact into art.

I don't think professional programmers were using notepad in 1985. Here's talk of IDEs from an article from 1985: https://dl.acm.org/doi/10.1145/800225.806843 It mentions Xerox Development Environment, from 1977 https://en.wikipedia.org/wiki/Xerox_Development_Environment

The feedback loop for programming / mathematics / other things I've studied was not a week in the year 2019. In that ancient time the feedback look was maybe 10% slower than with any of these LLMs since you had to look at Google search.

He’s saying go read these books, wrestle with the ideas, go to bed, dream about them, think about them in the shower. Repeat that until you understand enough to understand the answer.

There’s no shortcut here. If you had unlimited time with the advisor he couldn’t just sit you down and make you understand in a few sessions.

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.

I think they also adjust their heuristics, based on looking at thousands of computer moves.

"He [Euclid] lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry."

Source: http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html

However the discussion there is more about math which is unnecessary to a paper, not so much about the problem of math which is unintelligible or, if intelligible, then incorrect. I don't have other papers off the top of my head, although by now it's my default expectation when I see a math-centric AI paper. If you have any such papers in mind, I could tell you my thoughts on it.

However the math in AI papers is indeed different. For example, Kingma and Ba's paper self-presents as having a theorem with a rigorous proof via a couple of lemmas proved by a chain of inequalities. The key thing is that the mathematical details are purportedly all present, but are just wrong.

This isn't at all like what you see in physics papers, which might just openly lack detail, or might use mathematical objects whose existence or definition remain conjectural. There can be some legitimate problems with that, but at least in the best cases it can be very visionary. (Mirror symmetry is a standard example.) By contrast I'm not sure what 'spirit' is even possible in a detailed couple-page 'proof' that its authors probably don't even fully understand. In most cases, the 'theorem' isn't remotely interesting enough as pure mathematics and is also not of any serious relevance to the empirical problem at hand. It just adds an impressive-looking section to the paper.

I do think it's possible that in the future there will be very interesting pure mathematics inspired by AI. But it hasn't been found yet, and I'm very certain it won't come from reconsidering these kinds of badly-written theorems and proofs.

  > People do look

  > Very few are trying to understand why things are working

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 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.

"Special characteristics" is really overstating it. It's just a matter of getting all the prereqs formalized in Lean first. That's a bit of a grind to be sure, but the Mathlib effort for Lean has the bulk of the undergrad curriculum and some grad subjects formalized.

I don't think AI will be all that helpful wrt. this kind of effort, but it might help in some limited ways.

The main bottleneck is having the libraries that define the theorems and objects you need to operate at those levels. Everything is founded on axiomatic foundations and you need to build all of maths on top of that. Projects like mathlib are getting us there but it is a massive undertaking.

It’s not just that it is a lot of maths to go through, it’s also that most maths has not really been proven to this degree of exactitude and there is much gap-filling to do when trying to translate existing proofs, or the reasoning style might be quite distant to how things are expressed in Lean. Some maths fields are also self-consistent islands that haven’t been yet connected to the common axiomatic foundations, and linking them is a serious research endeavor.

Although Lean does allow you to declare theorems as axioms. It is not common practice, but you can skip high up the abstraction ladder and set up a foundation up there if you are confident those theorems are correct. But still defining those mathematical objects can be quite hard on its own, even if you skip the proving.

Anyways, the complexity of the Lean language itself doesn’t help either. The mode of thinking you need to have to operate it is much closer to programming than maths, and for those that think that the Rust borrow-checker is a pain, this is an order of magnitude more complex.

Lean was a significant improvement in ergonomics compared to the previous generation (Coq, Isabelle, Agda…), but still I think there is a lot of work to be done to make it mathematician-friendly.

Most reinforcement-learning AI for maths right now is focused on olympiad problems, hard but quite low in the maths abstraction ladder. Often they don’t even create a proof, they just solve problems that end with an exact result and you just check that. Perhaps the reasoning was incorrect, but if you do it for enough problems you can be confident that it is not just guessing.

On the other side of the spectrum you have mathematicians like Tao just using ChatGPT for brainstorming. It might not be great at complex reasoning, but it has a much wider memory than you do and it can remind you of mathematical tools and techniques that could be useful.

Lean makes the proofs verifiable, and sure to a degree understandable, but normal Lean proofs written by humans are already a challenge to understand without additional documentation, and, as with all code-generation, AI generated proofs are probably even harder to understand. There are many paths to arrive at a proof, the AI may take very messy routes that don't align with how humans create abstractions and assign meaning to break the process down into understandable steps.

In a sense all AI is understandable, nothing is hidden, you can see all the numbers inside a Transformer. Just being able to see the code might not necessarily help mathematicians too much in extracting deeper meaning and lessons from AI-generated proofs, at least not without significant additional work.

I understand that mathematicians are all about understanding why a theorem is true, not just checking that it is, and then using that understanding to push forward other research. Same as most scientists care more about understanding how the world works rather than developing methods to control and manipulate it.

But proven theorems are mathematical tools that can be used to make progress, regardless of your understanding of how they work. I think there is a path forward for maths as an engineering discipline, possibly with much faster progress, just as computer science expanded into software engineering. AI is just the next step of many, software libraries, interfaces and abstractions are also black-boxes in a sense that do stuff for you without knowing how.

A good engineer learns to build on the shoulders of giants, mastering the understanding of what tools do and how they behave, without necessarily needing to know how they are built from scratch. It might be less satisfying to the curious, but you can get a lot more done.

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.

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.

> 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.

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.”

aka optimization-for-its-own-sake, aka pathological optimization.

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.

Yes! This is what frustrates my about the pursuit of AI for the arts too.

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.

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.

Have programmers given up on wanting their mind blown by unbelievable simplicity?

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?

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.

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?