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AI in mathematics is forcing big questions

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Re: AI in mathematics is forcing big questions

#101

Post by drdaeman »

VBprogrammer wrote: post #86
sdenton4 wrote: post #80

Earlier quoted context omitted.

How many people drive cars without knowing how an engine works? Or make a phone call without knowing how voice compression for a cellular network does it's thing? Or eats food without knowing how it came together from the supply chain?

This feels like a stretch. It would be impossible for someone who didn't know how an engine worked to repair or improve the design of it.

Why not? One can surely use math even if they have no clue about how to prove theorems. I suck at math, but I use it every day, without knowing how to advance it.

I think it might be fair to say that a proof cannot be without value if it proves something meaningful to a human, that a human can use somehow? But such proof probably doesn’t belong in a library seemingly explicitly dedicated to human-graspable proofs. Just because it violates the intent.

It’s not like such proofs mustn’t exist at all.

Re: AI in mathematics is forcing big questions

#102

Post by crote »

crystal_revenge wrote: post #53
fiforpg wrote: post #11

The use of computers in mathematics has been somewhat controversial from the very start. There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understandin…

> but I would like to understand the problem, too But why should it be the case that this is always possible? It's entirely reasonable that the set of useful mathematical proofs is a proper superset of human intelligible useful proofs. In fact, to argue the contrary would imply there is something incredibly remarkable about human cognition.

> It's entirely reasonable that the set of useful mathematical proofs is a proper superset of human intelligible useful proofs.

If you can't explain something in a way that a child could understands it, you don't fully understand it either.

Re: AI in mathematics is forcing big questions

#103

Post by fdupress »

_zoltan_ wrote: post #91
mittensc wrote: post #90

Earlier quoted context omitted.

There's a difference in math between giving just the answer to a problem and doing it properly/elegantly. So yeah, generated machine-valid proof can be denied if it's incomprehensible, same as human machine-valid proof can be denied for same reasons.

there is a difference but it's overrated. if a theorem is proven, then, as OP said, the theorem is the interface, no matter where the proof is. just as we don't re-prove Fermat's little theorem every time I use it in a proof, because well, it's a theorem.

The theorem doesn't exist in a vacuum. It talks about objects that must be formally defined. And if that formal definition (which is part of the API) is not immediately compatible with those others use, and every single theorem comes with its own definitions of the objects they're working on, you're going to be reinventing the wheel over and over again.

Replacing thought and curation with repeated automation is tech debt, pushed down to fundamental knowledge and understanding.

Re: AI in mathematics is forcing big questions

#105

Post by bananaflag »

6thbit wrote: post #104

I always struggled with my lack of intuition vs that of my peers that had had a more comprehensive ‘math upbringing’. Is direct experience and struggle really the only driver for developing intuition?

I am a mathematician, and I was never the kind to like to struggle by working on problems, but I developed a lot of intuition by 1) thinking deeply about definitions and proofs and why are they this way and not another 2) reading a lot of blogs and expository papers by great mathematicians, even (the more philosophically minded) mathoverflow q&a's (so I absorbed their way of thinking unconsciously). For example, I tell my students to read all 300 of John Baez's This Week's Finds posts [1] and they will deeply understand more math than 99% of their peers.

This is not the "standard" advice that usually gets peddled but for me it "worked".

[1] https://math.ucr.edu/home/baez/twf.html

Re: AI in mathematics is forcing big questions

#106

Post by robrain »

sdenton4 wrote: post #80
zmgsabst wrote: post #70

Earlier quoted context omitted.

No — people don’t successfully use things they don’t understand every day. They approximately use them with varying degrees of success, but also mistakes, broken inferences, etc. My exact point is that your view reduces our ability to do mathematics to that broken, flawed usage and thereby undermines its utility for logical precision: mathematics is only useful because we cleanly understand it. When you try to use ma…

How many people drive cars without knowing how an engine works? Or make a phone call without knowing how voice compression for a cellular network does it's thing? Or eats food without knowing how it came together from the supply chain?

The mechanic who repairs the cars knows how the engine works.

The telco that manages loads and allocates networks knows how voice compression works.

The farmers and supermarkets know how the supply chain works.

None of your questions show why mathematics should include blobs of incomprehensible gloop, where no mathematician, no logician, no philosopher, no man on the street can make sense of said gloop, or use it in any way to further human knowledge.

When it's been decomposed down we can discuss this further, but now it's like saying red is red, just because.

Re: AI in mathematics is forcing big questions

#107

Post by subscribed »

therobots927 wrote: post #24
avaer wrote: post #19

Earlier quoted context omitted.

As a programmer I definitely get annoyed when I see code and I don't understand what it does. But I also definitely don't understand the problem if I can't get the computer to understand it, with tests. In some sense I always considered programming to be more trustworthy than maths arguments without the certainty of a solver proof. With all of these questions in the air, epistemology might be making a comeback.

Tests only work for a limited set of programming verification. In many cases you don’t actually know what the output for any given input should be, so there’s no way of verifying the AI-generated code. You just kind of have to trust it. The only exception I can think of is robotics and quantitative trading. Which have already been extensively utilizing AI.

That's a very handwavy way of saying no.

I disagree, software engineering is a mature discipline now, and at this point we have so many testing frameworks (unit testing, syntax testing, regression testing, fuzzing, testing end to end, live, with a subset of known good and incorrect inputs, chaos monkeys, etc, etc, that to say "there's no way of verifying the AI-generated code" is frankly incorrect.

Or, if you insist, defend the "there's no way of verifying the code, at all", and not only AI-generated.

(if it helps I work in the company where before the code even starts being written, several extensive tests for it must be ready first. It's hard to even commit a broken code, and later in the pipeline it's very easy to catch the subtly broken or incorrect code)

Re: AI in mathematics is forcing big questions

#108

Post by auggierose »

frabcus wrote: post #82
morpheos137 wrote: post #38

Much can be resolved when it is understood math is discovered not created. AI is a tool. if it makes discovery or proof easier that is still mathematics. A proof stands on its own logic regardless how it is derived. The root concern is how ai may provide uplift for mathematical discovery outside of socially expected channels.

I only did undergraduate level in Maths, and to me there is a key aesthetic element which makes it created. The choice of axioms to use, the choice of with theorems are interesting. Yes the "truth" (doesn't exist, see Gödels theorem) is discovered in a vast, wild landscape that Mathematicians explore. But which areas are worth exploring is a critical question. Partly driven by application, partly aesthetic. It's a qu…

The truth very much exists, see Gödel's theorems.

People get confused by this created/discovered thing. Of course it is discovered. It was there even before you created it. ;-)

Re: AI in mathematics is forcing big questions

#109

Post by cornholio »

zmgsabst wrote: post #70
crystal_revenge wrote: post #58

Earlier quoted context omitted.

> Things that aren’t human intelligible aren’t human usable This is objectively false, people use things every single day they don't understand. We still have plenty of things about the world we don't understand but still find useful. You are saying anything we know to be the case, but cannot understand why cannot be used? Can we just stop sleeping because we haven't reasoned why sleep is necessary even though we kno…

No — people don’t successfully use things they don’t understand every day. They approximately use them with varying degrees of success, but also mistakes, broken inferences, etc. My exact point is that your view reduces our ability to do mathematics to that broken, flawed usage and thereby undermines its utility for logical precision: mathematics is only useful because we cleanly understand it. When you try to use ma…

It's easy to find counterexamples: the entire science of pharmacology is based on macroscopic effects that often lack a fundamental understanding of the underlying mechanisms of action. Psychopharmacology is the extreme example. Often, the fact that a drug worked made scientists investigate and discover the mechanism behind it, but for many drugs used every day by billions it's still a mystery, or it's understood only in very broad terms.

So what will you do if the doctor prescribes you an LLM-vibecoded drug that nobody understands how it works, yet it cures some deadly affliction with close to 100% efficacy?

What if, say, these incomprehensible math results lead to a revolution in quantum physics which unlocks chip topologies that are orders of magnitude faster than human comprehensible designs?

Would the high priestess of human reason pass her divining rod over such chips or life-saving drugs and reject it as the work of the AI devil?

Re: AI in mathematics is forcing big questions

#110

Post by hiAndrewQuinn »

crote wrote: post #99
hiAndrewQuinn wrote: post #95

Earlier quoted context omitted.

I don't think that's the point. I think the point is to prove the statement. The techniques and abstractions are a means to an end; making them the point is being seduced by the beauty of the weapon.

> I think the point is to prove the statement. I couldn't disagree more. A lot of mathematical "problems" are almost entirely pointless. Nobody genuinely cares about the moving sofa problem, or about square packing, or about the minimum number of colors needed to draw a map - it is the math that is developed during the solving process that is valuable! An answer to a question like "what is the exact area of a unit ci…

"Mathematics isn't about proving the statement! It's about the collection of substatements which lead to the proof of the statement."

"Okay, so what determines what is and is not allowed in the collection?"

"Whether the given substatement is true or not, of course."

Like this is obviously silly, right. In your view you could have two guys both trying to prove or disprove that the area of the unit circle is 3, and yet only the guy doing it with some vision of nobility where he's building up to this grand theory of approximations is the one actually doing "real" mathematics. The guy who's doing it just because he thinks it's neat and would like an answer to the problem itself doesn't count, and you suspect he couldn't even exist.

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