You might be out of the loop a bit.

https://dspace.mit.edu/handle/1721.1/132379.2

Is a well known project for an AI Physicist. There are plenty of other groups working on similar projects

>I don't see the latest round of ML (or any ML, really) as a step towards it, but I'd love to be proven wrong.

LLM models have been able to do basic math for quite a while now and some have been trained to solve differential equations, calculus problems, etc. Well on their way to more impressive capabilities.

I said "in its own right discovers/invents and proves significant new theorems, advancing some area of research mathematics".

Neither of the things you mention are of this nature, or working towards it. "Finding a symbolic expression that matches data from an unknown function" (Feynman) and "solv[ing] differential equations, calculus problems, etc" are not descriptions of what a research mathematician does.

>Neither of the things you mention are of this nature, or working towards it. "Finding a symbolic expression that matches data from an unknown function" (Feynman) and "solv[ing] differential equations, calculus problems, etc" are not descriptions of what a research mathematician does.

Never said they were but you said:

>It hasn't been done because nobody knows how to make a machine be a mathematician, and I've seen little evidence of any progress towards it."

Which I showed is not accurate. Certainly people have ideas on how to do it and are actively making progress towards that goal.

>Finding a symbolic expression that matches data from an unknown function" (Feynman) and "solving differential equations, calculus problems, etc" are not descriptions of what a research mathematician does.

All research mathematicians started out solving calculus problems and differential equations.

Why do you expect an AI to sprint before it's learned to crawl?

"Discovering/inventing and proving new theorems" is qualitatively different to the things you list. Computers have been able to calculate since they were invented, and there has certainly been plenty of progress on getting them to solve problems, but calculating and solving problems isn't what mathematics research is.

Ever since computers were invented there has been a hope that you could set up a system that would just churn out interesting new theorems. Indeed it was one of the primary motivations for the invention of the computer, but it hasn't materialised yet.

You clearly consider the progress on solving problems to be progress towards being able to do mathematical research. I don't think it is, any more than progress in, say, graphics is. But maybe I will turn out to be wrong and you will turn out to be right. We won't have the answer until the problem is solved and we have our wonderful machine churning out theorems.

But I think you will probably be able to agree that since mathematical research is something human minds are capable of it's something that an AGI should be capable of, i.e. if an AI approach is inherently incapable of it, it's not AGI. You may consider it an unnecessarily stringent requirement, in that there may be other, easier challenges that AIs can perform that will convince you that they are AGI. That's fine - you think about the problem differently to me, so you find different things persuasive. If you are convinced that a given AI is AGI, though, you shouldn't be too concerned about my particular goalpost given that your AGI should be able to achieve it (and convince me) pretty soon.

We'll see what happens. I'm just explaining what I would find convincing, and pointing out that contrary to the oft-repeated accusation that started this discussion, I for one have never once "moved the goalposts".

>But I think you will probably be able to agree that since mathematical research is something human minds are capable of it's something that an AGI should be capable of, i.e. if an AI approach is inherently incapable of it, it's not AGI.

Indeed. To be clear, I'm not saying I think any current system is remotely close to AGI. I just think that saying that no one is thinking about or making progress on a math research AI is inaccurate.

Feels to me like your test is really "do something on your own right", which is the hard fluffy sentient part, and then some additional guard rails that it needs to be math for some reason

The "do it on your own right" is actually the weaker part of it. It's somewhat ill-defined, and I could imagine some future instance where it's highly debatable whether the AI was working on its own or being used as a tool by a human. There aren't yet any cases where that's in question, though, so it's a hypothetical future debate. In any case, it's certainly not the meat of the test.

It has to be maths for a specific reason. I think it's in some sense the purest form of an ability distinctive to human minds and pervasive in how they work. As I mentioned, it's an ability that can be demonstrated in the absence of any particular physical capability, and yet despite it being perhaps the oldest goal of AI it may be the one we have made least progress towards.

Anyway that's my goalpost, and it's not moving. AGI, being "general", surely should be capable of this hitherto uniquely human activity. If our attempts so far are not capable of it, then clearly they are not "general". If you know of any evidence that my goalpost has been achieved, please let me know. I'm very eager to see it happen.

The reason is simple - in math, it means solving open problems.

That’s a different thing, since solving an existing open problem doesn’t mean inventing a new theorem.

Most humans are not capable of discovering or prove new significant theorems.

Most humans aren't capable of playing chess to grandmaster level either, or producing artwork in arbitrary styles, or remembering and distinguishing between millions of faces.

I'd settle for a demonstration that a computer has truly independently discovered/invented and proved some significant part of our existing mathematical edifice. This hasn't been achieved yet, either. However, I suspect that once we've figured out how to do this at all, surpassing human capabilities will be inevitable in a relatively short time. So I don't see much value in softening the test unless/until there's some actual candidate available that would pass the softer test.

The value in requiring genuinely new maths is that it makes it unlikely that knowledge of the result has been encoded in the algorithm or training set. Certainly, if GPT-3 were to output Euler's formula that wouldn't be at all convincing as a "discovery".

Not an example of complex mathematical reasoning, but aren't AlphaZero and its cousins evidence that ML can independently rediscover principles that humans have found, as well as discover new principles of its own? For example, LC0 plays for advantages that humans hadn't considered before in Chess.

Very good point. Apparently human players stared studying AIs playing go and Chess to learn new techniques.

I'm not an expert in those algorithms, so... maybe? If so, maybe someone will successfully apply those ideas to the challenge I've described. I'd love to see it happen.