> Symbols, by definition, only represent a thing.

This is missing the lesson of the Yoneda Lemma: symbols are uniquely identified by their relationships with other symbols. If those relationships are represented in text, then in principle they can be inferred and navigated by an LLM.

Some relationships are not represented well in text: tacit knowledge like how hard to twist a bottle cap to get it to come off, etc. We aren't capturing those relationships between all your individual muscles and your brain well in language, so an LLM will miss them or have very approximate versions of them, but... that's always been the problem with tacit knowledge: it's the exact kind of knowledge that's hard to communicate!

> Symbols, by definition, only represent a thing. They are not the same as the thing

First of all, the point isn't about the map becoming the territory, but about whether LLMs can form a map that's similar to the map in our brains.

But to your philosophical point, assuming there are only a finite number of things and places in the universe - or at least the part of which we care about - why wouldn't they be representable with a finite set of symbols?

What you're rejecting is the Church-Turing thesis [1] (essentially, that all mechanical processes, including that of nature, can be simulated with symbolic computation, although there are weaker and stronger variants). It's okay to reject it, but you should know that not many people do (even some non-orthodox thoughts by Penrose about the brain not being simulatable by an ordinary digital computer still accept that some physical machine - the brain - is able to represent what we're interested in).

> If we accept the incompleteness theorem

There is no if there. It's a theorem. But it's completely irrelevant. It means that there are mathematical propositions that can't be proven or disproven by some system of logic, i.e. by some mechanical means. But if something is in the universe, then it's already been proven by some mechanical process: the mechanics of nature. That means that if some finite set of symbols could represent the laws of nature, then anything in nature can be proven in that logical system. Which brings us back to the first point: the only way the mechanics of nature cannot be represented by symbols is if they are somehow infinite, i.e. they don't follow some finite set of laws. In other words - there is no physics. Now, that may be true, but if that's the case, then AI is the least of our worries.

Of course, if physics does exist - i.e. the universe is governed by a finite set of laws - that doesn't mean that we can predict the future, as that would entail both measuring things precisely and simulating them faster than their operation in nature, and both of these things are... difficult.

[1]: https://plato.stanford.edu/entries/church-turing/

First: true propositions (that are not provable) can definitely be expressed, if they couldn't, the incompleteness theorem would not be true ;-)

It would be interesting to know what the percentage of people is, who invoke the incompleteness theorem, and have no clue what it actually says.

Most people don't even know what a proof is, so that cannot be a hindrance on the path to AGI ...

Second: ANY world model that can be digitally represented would be subject to the same argument (if stated correctly), not only LLMs.

> Language models aren't world models for the same reason languages aren't world models. Symbols, by definition, only represent a thing. They are not the same as the thing. The map is not the territory, the description is not the described, you can't get wet in the word "water".

Symbols, maps, descriptions, and words are useful precisely because they are NOT what they represent. Representation is not identity. What else could a “world model” be other than a representation? Aren’t all models representations, by definition? What exactly do you think a world model is, if not something expressible in language?

There is an important implication of learning and indexing being equivalent problems. A number of important data models and data domains exist for which we do not know how to build scalable indexing algorithms and data structures.

It has been noted for several years in US national labs and elsewhere that there is an almost perfect overlap between data models LLMs are poor at learning and data models that we struggle to index at scale. If LLMs were actually good at these things then there would be a straightforward path to addressing these longstanding non-AI computer science problems.

The incompleteness is that the LLM tech literally can't represent elementary things that are important enough that we spend a lot of money trying to represent them on computers for non-AI purposes. A super-intelligent AGI being right around the corner implies that we've solved these problems that we clearly haven't solved.

Perhaps more interesting, it also implies that AGI tech may look significantly different than the current LLM tech stack.

Gödel’s incompleteness theorems aren’t particularly relevant here. Given how often people attempt to apply them to situations where they don’t say anything of note, I think the default should generally be to not publicly appeal to them unless one either has worked out semi-carefully how to derive the thing one wants to show from them, or at least have a sketch that one is confident, from prior experience working with it, that one could make into a rigorous argument. Absent these, the most one should say, I think, is “Perhaps one can use Gödel’s incompleteness theorems to show [thing one wants to show].” .

Now, given a program that is supposed to output text that encodes true statements (in some language), one can probably define some sort of inference system that corresponds to the program such that the inference system is considered to “prove” any sentence that the program outputs (and maybe also some others based on some logical principles, to ensure that the inference system satisfies some good properties), and upon defining this, one could (assuming the language allows making the right kinds of statements about arithmetic) show that this inference system is, by Gödel’s theorems, either inconsistent or incomplete.

This wouldn’t mean that the language was unable to express those statements. It would mean that the program either wouldn’t output those statements, or that the system constructed from the program was inconsistent (and, depending on how the inference system is obtained from the program, the inference system being inconsistent would likely imply that the program sometimes outputs false or contradictory statements).

But, this has basically nothing to do with the “placeholders” thing you said. Gödel’s theorem doesn’t say that some propositions are inexpressible in a given language, but that some propositions can’t be proven in certain axiom+inference systems.

Rather than the incompleteness theorems, the “undefinability of truth” result seems more relevant to the kind of point I think you are trying to make.

Still, I don’t think it will show what you want it to, even if the thing you are trying to show is true. Like, perhaps it is impossible to capture qualia with language, sure, makes sense. But logic cannot show that there are things which language cannot in any way (even collectively) refer to, because to show that there is a thing it has to refer to it.

————

“Can you write a test suite for it?”

Hm, might depend on what you count as a “suite”, but a test protocol, sure. The one I have in mind would probably be a bit expensive to run if it fails the test though (because it involves offering prize money).

Everything is just a low resolution representation of a thing. The so-called reality we supposedly have access to is at best a small number of sound waves and photons hitting our face. So I don't buy this argument that symbols are categorically different. It's a gradient and symbols are more sparse and less rich of a data source, yes. But who are we to say where that hypothetical line exists, beyond which further compression of concepts into smaller numbers of buckets becomes a non-starter for intelligence and world modelling. And then there's multi modal LLMs which have access to data of a similar richness that humans have access to.

> If we accept the incompleteness theorem

And, by various universality theorems, a sufficiently large AGI could approximate any sequence of human neuron firings to an arbitrary precision. So if the incompleteness theorem means that neural nets can never find truth, it also means that the human brain can never find truth.

Human neuron firing patterns, after all, only represent a thing; they are not the same as the thing. Your experience of seeing something isn't recreating the physical universe in your head.

Reminds me of this [1] article. If us humans, after all these years we've been around, can't relay our thoughts exactly as we perceive them in our heads, what makes us think that we can make a model that does it better than us?

[1]: https://www.experimental-history.com/p/you-cant-reach-the-br...

I’m not a math guy but the incompleteness theorem applies to formal systems, right? I’ve never thought about LLMs as formal systems, but I guess they are?

I don't think you can apply the incompleteness theorem like that, LLMs aren't constrained to formal systems

> Language models aren't world models for the same reason languages aren't world models. > Symbols, by definition, only represent a thing. They are not the same as the thing. The map is not the territory, the description is not the described, you can't get wet in the word "water".

There is a lot of negatives in there, but I feel like it boils down to a model of a thing is not the thing. Well duh. It's a model. A map is a model.