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Can AI do maths yet? Thoughts from a mathematician

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Re: Can AI do maths yet? Thoughts from a mathematician

#311

Post by exprofmaddy »

red75prime wrote: post #283
exprofmaddy wrote: post #166

Earlier quoted context omitted.

Some differential equations that model physics admit singularities and multiple solutions. Therefore, functions are not the most general way of describing relations. Functions are a subset of relations. Although "non-deterministic" and "stochastic" are often used interchangeably, they are not equivalent. Probability is applied analysis whose objects are distributions. Analysis is a form of deductive, i.e. mechanical,…

> Some differential equations that model physics admit singularities and multiple solutions. And physicists regard those as unphysical: the theory breaks down, we need better one.

For example, the Euler equations model compressible flow with discontinuities (shocks in the flow field variables) and rarefaction waves. These theories are accepted and used routinely.

Re: Can AI do maths yet? Thoughts from a mathematician

#312

Post by jvvw »

vouaobrasil wrote: post #146
busyant wrote: post #138

As someone who has a 18 yo son who wants to study math, this has me (and him) ... worried ... about becoming obsolete? But I'm wondering what other people think of this analogy. I used to be a bench scientist (molecular genetics). There were world class researchers who were more creative than I was. I even had a Nobel Laureate once tell me that my research was simply "dotting 'i's and crossing 't's". Nevertheless, I…

I was just thinking about this. I already posted a comment here, but I will say that as a mathematician (PhD in number theory), that for me, AI signficantly takes away the beauty of doing mathematics within a realm in which AI is used. The best part of math (again, just for me) is that it was a journey that was done by hand with only the human intellect that computers didn't understand. The beauty of the subject was…

I agree wholeheartedly about the beauty of doing mathematics. I will add though that the author of this article, Kevin Buzzard, doesn't need to do this for his career and from what I know of him is somebody who very much cares about mathematics and the future of the subject. The fact that a mathematician of that calibre is interested in this makes me more interested.

Re: Can AI do maths yet? Thoughts from a mathematician

#313

Post by goalieca »

Syzygies wrote: post #247

"Can AI do math for us" is the canonical wrong question. People want self-driving cars so they can drink and watch TV. We should crave tools that enhance our abilities, as tools have done since prehistoric times. I'm a research mathematician. In the 1980's I'd ask everyone I knew a question, and flip through the hard bound library volumes of Mathematical Reviews, hoping to recognize something. If I was lucky, I'd get…

> People want self-driving cars so they can drink and watch TV. We should crave tools that enhance our abilities, as tools have done since prehistoric times.

Improved tooling and techniques have given humans the free time and resources needed for arts, culture, philosophy, sports, and spending time to enjoy life! Fancy telecom technologies have allowed me to work from home and i love it :)

Re: Can AI do maths yet? Thoughts from a mathematician

#314

Post by sumeno »

xmprt wrote: post #245
phkahler wrote: post #217

Earlier quoted context omitted.

>> It's not about training directly on the test set, it's about people discussing questions in the test set online Don't kid yourself. There are 10's of billions of dollars going into AI. Some of the humans involved would happily cheat on comparative tests to boost investment.

The incentives are definitely there, but even CEOs and VCs know that if they cheat the tests just to get more investment, they're only cheating themselves. No one is liquidating within the next 5 years so either they end up getting caught and lose everything or they spent all this energy trying to cheat while having a subpar model which results in them losing to competitors who actually invested in good technology. H…

Why is this any different from say, Theranos?

CEOs and VCs will happily lie because they are convinced they are smarter than everyone else and will solve the problem before they get caught.

Re: Can AI do maths yet? Thoughts from a mathematician

#315

Post by exprofmaddy »

Quekid5 wrote: post #181
exprofmaddy wrote: post #143

Earlier quoted context omitted.

Agreed. If someone believes the world is purely mechanistic, then it follows that a sufficiently large computing machine can model the world---like Leibniz's Ratiocinator. The intoxication may stem from the potential for predictability and control. The irony is: why would someone want control if they don't have true choice? Unfortunately, such a question rarely pierces the intoxicated mind when this mind is preoccupi…

> If someone believes the world is purely mechanistic, then it follows that a sufficiently large computing machine can model the world Is this controversial in some way? The problem is that to simulate a universe you need a bigger universe -- which doesn't exist (or is certainly out of reach due to information theoretical limits) > ---like Leibniz's Ratiocinator. The intoxication may stem from the potential for predi…

People wish to feel safe. One path to safety is controlling or managing the environment. Lack of sufficient control produces anxiety. But control is only possible if the environment is predictable, i.e., relatively certain knowledge that if I do X then the environment responds with Y. Humans use models for prediction. Loosely speaking, if the universe is truly mechanistic/deterministic, then the goal of modeling is to get the correct model (though notions of "goals" are problematic in determinism without real counterfactuals). However, if we can't know whether the universe is truly deterministic, then modeling is a pragmatic exercise in control (or management).

My comments are not about simulating the universe on a real machine. They're about the validity and value of math/computational modeling in a universe where determinism is scientifically indeterminable.

Re: Can AI do maths yet? Thoughts from a mathematician

#316

Post by vector_spaces »

halayli wrote: post #280
amanda99 wrote: post #266

Earlier quoted context omitted.

> AI is often wrong, never knows when it's wrong, but people are like this too. When talking with various models of ChatGPT about research math, my biggest gripe is that it's either confidently right (10% of my work) or confidently wrong (90%). A human researcher would be right 15% of the time, unsure 50% of the time, and give helpful ideas that are right/helpful (25%) or wrong/a red herring (10%). And only 5% of the…

these numbers are just your perception. The way you ask the question will very much influence the output and certain topics more than others. I get much better results when I share my certainty levels in my questions and say things like "if at all", "if any" etc.

I agree with this approach and use it myself, but these confidence markers can also skew output in undesirable ways. All of these heuristics are especially fragile when the subject matter touches the frontiers of what is known.

In any case my best experiences with LLMs for pure math research have been for exploring the problem space and ideation -- queries along the line of "Here's a problem I'm working on ... . Do any other fields have a version of this problem, but framed differently?" or "Give me some totally left field methods, even if they are from different fields or unlikely to work. Assume I've exhausted all the 'obvious' approaches from field X"

Re: Can AI do maths yet? Thoughts from a mathematician

#317

Post by fosk »

FiberBundle wrote: post #308
whimsicalism wrote: post #305

Earlier quoted context omitted.

Ingredients to a top HN comment on AI include some nominal expert explaining why actually labor won’t be replaced and it will be a collaborative process so you don’t need to worry sprinkled with a little bit of ‘the status quo will stay still even though this tech only appeared in the last 2 years’

It didn't appear in the last two years. We have had deep learning based autoregressive language models (like Word2Vec) for at least 10 years.

Early computer networks appeared in the 1960s and the public internet as we know it in the 1990s.

We are still early in AI.

Re: Can AI do maths yet? Thoughts from a mathematician

#318

Post by vouaobrasil »

daxfohl wrote: post #233
vouaobrasil wrote: post #202

Earlier quoted context omitted.

Just a counterpoint, but I wonder how much you'll really understand if you can't even prove the whole thing yourself. Personally, I learn by proving but I guess everyone is different.

My hunch is it won't be much different, even when we can simply ask a machine that doesn't have a cached proof, "prove riemann hypothesis" and it thinks for ten seconds and spits out a fully correct proof. As Erdos(I think?) said, great math is not about the answers, it's about the questions. Or maybe it was someone else, and maybe "great mathematicians" rather than "great math". But, gist is the same. "What happens…

> I have a hard time imagining computers would ever have a strong enough understanding of the human experience with mathematics to even begin pondering such questions unprompted, let alone answer them and grok the implications.

The key is that the good questions however come from hard-won experience, not lazily questioning an AI.

Re: Can AI do maths yet? Thoughts from a mathematician

#319

Post by red75prime »

exprofmaddy wrote: post #311
red75prime wrote: post #283

Earlier quoted context omitted.

> Some differential equations that model physics admit singularities and multiple solutions. And physicists regard those as unphysical: the theory breaks down, we need better one.

For example, the Euler equations model compressible flow with discontinuities (shocks in the flow field variables) and rarefaction waves. These theories are accepted and used routinely.

Great. A useful approximation of what really happens in the fluid. But I'm sure there are no shocks and rarefactions in physicists' neurons while they are thinking about it.

Switching into a less facetious mode...

Do you understand that in context of this dialogue it's not enough to show some examples of discontinuous or otherwise unrepresentable by NNs functions? You need at least to give a hint why such functions cannot be avoided while approximating functionality of the human brain.

Many things are possible, but I'm not going to keep my mind open to a possibility of a teal Russell's teapot before I get a hint at its existence, so to speak.

Re: Can AI do maths yet? Thoughts from a mathematician

#320

Post by lern_too_spel »

bongodongobob wrote: post #226
lern_too_spel wrote: post #172

Earlier quoted context omitted.

If Pauling's eugenics policies were bad, then the laws against incest that are currently on the books in many states (which are also eugenics policies that use the same mechanism) are also bad. There are different forms of eugenics policies, and Pauling's proposal to restrict the mating choices of people carrying certain recessive genes so their children don't suffer is ethically different from Hitler exterminating p…

From what I remember, he wanted to mark people with tattoos or something.

This is mentioned in my link: "According to Pauling, carriers should have an obvious mark, (i.e. a tattoo on the forehead) denoting their disease, which would allow carriers to identify others with the same affliction and avoid marrying them."

The goal wasn't to mark people for ostracism but to make it easier for people carrying these genes to find mates that won't result in suffering for their offspring.

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