• GPT4 (and other LLMs) is some kind of generalized homotopy engine. You can give it any input, ask it to apply any "translation". Language translation, style translation, or even keeping the style but talking about another subject, or translating from one program to the other – and it gives you something different, yet identical. Write something like ... but ... There is some deep understanding of what identity is here, in particular with respect to the the messy expectations of our human sign systems: you can throw any kind of equivalence path, and GPT4 will handle them just fine. It seems the limit is not in its ability to generalize to any kind of identity schema we throw at it, but in the complexity of these schemas.

• I'm not saying GPT has an explicit understanding of these schemas/homotopies. My point is that even though GPT doesn't know much about homotopy type theory, I think it knows them in a latent way: GPT would perform much better at translating a piece of code in one language to another than it'd be at explaining what it just did in sound terms what through the lens of homotopy type theory. That knowledge about identity/equivalence is implicit.

• "Category Theory is indebted to [structuralism] in mathematics, particularly the idea of a homomophism which is a structure preserving map which first appeared in algebra. There was a parallel movement of structuralism in the humanities which can be traced back to the structuralist linguistics of Saussure and semiotics." (https://philosophy.stackexchange.com/a/7208)

Here's what Saussure, the founder of structuralism in linguistics said in manuscripts we found in the 1990s (source: http://www.revue-texto.net/docannexe/file/116/saussure255_6.pdf)

"The notion of identity will be, in all these orders, the necessary basis, the one that serves as an absolute basis: it is only through it and in relation to it that we can then determine the entities of each order, the primary terms that the linguist can legitimately believe to have before them."

"A certain vocal being is thus constituted and recognized in the name of an identity that we establish, and then thousands of others are obtained using the same principle, we can begin to classify these identity patterns of all sorts that we take, and are obliged to take, for the primary and specific and concrete facts, although they are each in their infinite diversity only the result of a vast prior operation of generalization.
Couldn't we limit ourselves to implying this great fundamental operation? Isn't it obvious from the outset that as soon as we talk about a group, for example, we mean the generality of cases where a group exists, so there is little subtle interest in recalling that this entity is fundamentally and primarily based on an identity?
We will immediately see that it is not allowed to substitute abstract entities for the fact of the identity of certain concrete facts with impunity because we will deal with other abstract entities, and the only pole in the middle of this will be identity or non-identity."

• Human Language as a Boson Gas of Entangled Words (https://link.springer.com/article/10.1007/s10699-019-09633-4):

Wherever there is meaning, identity/indistinguishability arises as words/phrases become substitutable as they refer to the same concept under the lens of the text they appear in, which acts as an "indentity schema" or homotopy context for these words. Because this entice words entangled around the same concept to display the similar occurrence frequencies, they behave in a fashion analogous to bose enstein condensates which allows to account for the Zipf distribution of word frequencies in the text. Meaning <-> Indistinguishability via the identity patterns set up by the text <-> Zipf distributions.

• The Runabout Inference-Ticket http://www.thatmarcusfamily.org/philosophy/Course_Websites/Logic_F09/Readings/Priori_RIT.pdf.

This paper discusses the idea of analytically valid inferences, which arise solely from the meanings of certain expressions. It uses the example of the word "and" to illustrate this point, arguing that the meaning of "and" is entirely determined by the inferences it allows.


• Categorical harmony and path induction: https://philpapers.org/archive/WALCHA-2.pdf

"This paper responds to recent work in the philosophy of Homotopy Type Theory by James
Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along ‘pre-mathematical’ lines. I give an alternate justification based on the philosophical framework of inferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. This categorical harmony is stated in terms of adjoints and says that any concept definable by iterated adjoints from general categorical operations is harmonious."

• The role of grammar in transition-probabilities of subsequent words in English text (https://deepai.org/publication/the-role-of-grammar-in-transition-probabilities-of-subsequent-words-in-english-text)

"Sentence formation is a highly structured, history-dependent, and sample-space reducing (SSR) process. While the first word in a sentence can be chosen from the entire vocabulary, typically, the freedom of choosing subsequent words gets more and more constrained by grammar and context, as the sentence progresses. This sample-space reducing property offers a natural explanation of Zipf's law in word frequencies, however, it fails to capture the structure of the word-to-word transition probability matrices of English text. Here we adopt the view that grammatical constraints (such as subject--predicate--object) locally re-order the word order in sentences that are sampled with a SSR word generation process. We demonstrate that superimposing grammatical structure -- as a local word re-ordering (permutation) process -- on a sample-space reducing process is sufficient to explain both, word frequencies and word-to-word transition probabilities."

Put in other words, if the authors of the Reversal Curse paper were to fix the weights of the LLM in an ad-hoc way, by directly adjusting the weights to "locally" satisfy the reversal tasks the LLM fails at, their fixes would also be "neuronally local". As a consequence, the LLM's conception of the meaning of the words would degrade in its hability to make a general use of them through the dataset. It would be a more fragmented understanding, concepts would be more specific (multiple local accounts of what the word "is" mean), Less words would be indistinguishable when within the same text, and, as their bose-einsteiness fades away, so would the Zipf distribution.