[resident423]

Remember when people thought solving Erdos problems required intelligence? Is there anything an LLM could ever do that would cound as intelligence? Surely the trend has to break at some point, if so what would be the thing that crosses the line to into real intelligence?

[NitpickLawyer]

> Remember when people thought solving Erdos problems required intelligence? Is there anything an LLM could ever do that would cound as intelligence?

Hah. It reminds me of this great quote, from the '80s:

> There is a related “Theorem” about progress in AI: once some mental function is programmed, people soon cease to consider it as an essential ingredient of “real thinking”. The ineluctable core of intelligence is always in that next thing which hasn’t yet been programmed. This “Theorem” was first proposed to me by Larry Tesler, so I call it Tesler’s Theorem: “AI is whatever hasn’t been done yet.”

We are seeing this right now in the comments. 50 years later, people are still doing this! Oh, this was solved, but it was trivial, of course this isn't real intelligence.

[latexr]

That is a “gotcha” born of either ignorance (nothing wrong with that, we’re all ignorant of something) or bad faith. Definitions shift as we learn more. Darwin’s definition of life is not the same as Descartes’ or Plato’s or anyone in between or since because we learn and evolve our thinking.

Are you also going to argue definitions of life before we even learned of microscopic or single cell organisms are correct and that the definitions we use today are wrong? That they are shifting goal posts? That “centuries later, people are still doing this”? No, that would be absurd.

[NitpickLawyer]

I don't see it as a gotcha. Just an (evergreen, it seems) observation that people will absolutely move the goalposts every time there's something new. And people can be ignorant outsiders or experts in that field as well.

For example, ~2 years ago, an expert in ML publicly made this remark on stage: LLMs can't do math. Today they absolutely and obviously, can. Yet somehow it's not impressive anymore. Or, and this is the key part of the quote, this is somehow not related to "intelligence". Something that 2 years ago was not possible (again, according to a leading expert in this field), is possible today. And yet this is somehow something that they always could do, and since they're doing it today, is suddenly no longer important. On to the next one!

No idea why this is related to darwin or definitions of life. The definitions don't change. What people considered important 2 years ago, is suddenly not important anymore. The only thing that changed is that today we can see that capability. Ergo, the quote holds.

[latexr]

> For example, ~2 years ago, an expert in ML

See, that’s a poor argument already. Anyone could counter that with other experts in ML publicly making remarks that AI would have replaced 80% of the work force or cured multiple diseases by now, which obviously hasn’t happened. That’s about as good an argument as when people countered NFT critics by citing how Clifford Stoll said the internet was a fad.

> made this remark on stage: LLMs can't do math. Today they absolutely and obviously, can.

How exactly are “LLMs can’t” and “do math” defined? As you described it, that sentence does not mean “will never be able to”, so there’s no contradiction. Furthermore, it continues to be true that you cannot trust LLMs on their own for basic arithmetic. They may e.g. call an external tool to do it, but pattern matching on text isn’t sufficient.

> The definitions don't change.

Of course they do, what are you talking about? Definitions change all the time with new information. That’s called science.

[NitpickLawyer]

The definition of "can/cannot do math" didn't change. That's not up for debate. 2 years ago they couldn't solve an erdos problem (people have tried, Tao has tried ~1 year ago). Today they can.

Definitions don't change. The idea that now that they can it's no longer intelligence is changing. And that's literally moving the goalposts. Read the thread here, go to the bottom part. There are zillions of comments saying this.

You are keen to not trying to understand what the quote is saying. This is not good faith discussion, and it's not going anywhere. We're already miles from where we started. The quote is an observation (and an old one at that) about goalposts moving. If you can't or won't see that, there's no reason to continue this thread.

[latexr]

> The definition of "can/cannot do math" didn't change. That's not up for debate.

That is not the argument. The point is that the way you phrased it is ambiguous. “Math” isn’t a single thing, and “cannot” can either mean “cannot yet” or “cannot ever”. I don’t know what the “expert” said since you haven’t provided that information, I’m directly asking you to clarify the meaning of their words (better yet, link to them so we can properly arrive at a consensus).

> Definitions don't change.

Yes they do! All the time!

https://www.merriam-webster.com/wordplay/words-that-used-to-...

> And that's literally moving the goalposts.

Good example. There are no literal goal posts here to be moved. But with the new accepted definition of the words, that’s OK.

> There are zillions of comments saying this.

Saying what, exactly? Please be clear, you keep being ambiguous. The thread barely crossed a couple of hundred comments as of now, there are not “zillions” of comments in agreement of anything.

> You are keen to not trying to understand what the quote is saying. (…) If you can't or won't see that, there's no reason to continue this thread.

Indeed, if you ascribe wrong motivations and put a wall before understanding what someone is arguing, there is indeed no reason to continue the thread. The only wrong part of your assessment is who is doing the thing you’re complaining about.

[noosphr]

I've spend a good chunk of time formalising mathematics.

Doing formalized mathematics is as intelligent as multiplying numbers together.

The only reason why it's so hard now is that the standard notation is the equivalent of Roman numerals.

When you start using a sane metalanguage, and not just augmrnted English, to do proofs you gain the same increase in capabilities as going from word equations to algebra.

[xxs]

>the standard notation is the equivalent of Roman numerals.

But the Roman numerals are easy. I was able to use them before 1st grade and I can't touch any "standard notation" to this day.

[_0ffh]

Well, the famous Turing test was evidently insufficient. All that happened is that the test is dead and nobody ever mentions it anymore. I'm not sure that any other test would fare any better once solved.

[thesmtsolver2]

When will LLM folks realize that automated theorem provers have existed for decades and non-ML theorem provers have solved non-trivial Math problems tougher than this Erdos problem.

Proposing and proving something like Gödel's theorem's definitely requires intelligence.

Solving an already proposed problem is just crunching through a large search space.

[throwaway198846]

Automated theorem provers can't prove this problem. Which non-trivial Math problem you think are thougher than this Erdos problem?

[ogogmad]

> Proposing

I think GIT is a negative answer to a problem originally posed by David Hilbert. It was not proposed by Goedel originally. I think Goedel's main new idea was (i) inventing Goedel numbering (ii) using Goedel numbering to show that provability from a finite FOL signature, and a single FOL formula, is reducible to an equation involving primitive recursive functions (iii) devising a method to translate FOL statements about arbitrary primitive recursive functions into statements about only the two primitive recursive functions + and ×.

Later work establishing the field of computability theory (or "recursive function theory" as it was then known) generalised the insights (i) and (ii). In light of that, Goedel's only now-relevant contribution is (iii).

> When will LLM folks realize that automated theorem provers have existed for decades

This is very misinformed. Automated theorem proving was, sadly, mostly a disappointment until LLMs and other Machine Learning techniques came along. Nothing like the article's result was remotely within reach.

[virgildotcodes]

So the only intelligent people in history are those who invent new fields of mathematics, got it.

You can just about make out those goalposts on the surface of the moon with a good telescope at this point.

[crazylogger]

"Hi ChatGPT, propose and prove something radically new in the genre of Gödel's theorem."

How is this not just another proposed problem (albeit with a search space much larger than an Erdos problem's)?

[dmurray]

I think the point the GP is making is that Gödel's theorem wasn't part of any "genre". Gödel, or somebody, had to invent the whole field, and we haven't seen LLMs invent new fields of mathematics yet.

But this isn't a fair bar to hold it to. There are plenty of intelligent people out there, including 99% of professional mathematicians, who never invent new fields of mathematics.