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The cultural divide between mathematics and AI

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Re: The cultural divide between mathematics and AI

#111

Post by godelski »

solveit wrote: post #98
godelski wrote: post #93

Earlier quoted context omitted.

> 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…

> 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. [1]: https://www.neelnanda.io/mechanistic-interpretability

  > People do look
This was never in question

  > 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

Re: The cultural divide between mathematics and AI

#112

Post by FilosofumRex »

I find this cultural divide exists predominantly among mathematicians who consider existence proofs as real mathematics.

Mathematicians who practice constructive math and view existence proofs as mere intellectual artifacts tend to embrace AI, physics, engineering and even automated provers as worthy subjects.

Re: The cultural divide between mathematics and AI

#113

Post by eru »

rssoconnor wrote: post #88
nicf wrote: post #17

Earlier quoted context omitted.

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 underlyin…

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.

> 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.

Re: The cultural divide between mathematics and AI

#114

Post by gessha »

lordleft wrote: post #19
nicf wrote: post #8

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 thi…

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?

It’s kind of like knowing the answer to the ultimate question of life and not knowing the question itself ;)

Re: The cultural divide between mathematics and AI

#115

Post by troymc »

FilosofumRex wrote: post #107
troymc wrote: post #79

Earlier quoted context omitted.

I never understood that quote until recently. The story goes that the (royal) pharaoh of Egypt wanted to learn geometry, but didn't want to have to read Euclid. He wanted a faster route. But, "there is no royal road to geometry."

The last Egyptian pharaoh was Nectanebo II, who ruled from 358 to approximately 340 BC. Alexander founded Alexandria in 331 BC as the crown jewel of his empire where Euclid wrote his magnum opus, The Elements in 300 BC! Unless the royal pharaoh of Egypt, refers to Ptolemy I Soter, Macedonian general who was the first Ptolemaic Kingdom ruler of Egypt after Alexander's death.

Yep, exactly. Here's a translation of Proclus:

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

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

Re: The cultural divide between mathematics and AI

#116

Post by dinkumthinkum »

zmgsabst wrote: post #110
nicf wrote: post #70

Earlier quoted context omitted.

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 b…

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.

Re: The cultural divide between mathematics and AI

#117

Post by 7402 »

nicf wrote: post #8

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 thi…

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.

Re: The cultural divide between mathematics and AI

#118

Post by godelski »

jebarker wrote: post #101
solveit wrote: post #98

Earlier quoted context omitted.

> 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. [1]: https://www.neelnanda.io/mechanistic-interpretability

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 the…

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.

[0] https://www.amazon.com/Quantum-Physicists-Introduction-Their...

[1] https://www.amazon.com/Introduction-Quantum-Mechanics-David-...

[2] https://thomwolf.io/blog/scientific-ai.html

Re: The cultural divide between mathematics and AI

#119

Post by nicf »

zmgsabst wrote: post #110
nicf wrote: post #70

Earlier quoted context omitted.

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 b…

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.

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?

Re: The cultural divide between mathematics and AI

#120

Post by roenxi »

rssoconnor wrote: post #99
godelski wrote: post #97

Earlier quoted context omitted.

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…

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 o…

> 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.

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