Technology

Mathematical language is a language, but mathematics is not just a language. It is a structure with internal rules that are not determined by pure convention (as natural languages are). We could internationally agree from tomorrow to call "blue" whatever it's now called "red" and vice versa, but we couldn't agree to say that "2 + 2 = 5", because that would lead to internal inconsistencies (we could agree to use the symbol "5" for 4, but that's a different matter).

This is also related to a staple of science: that scientific and mathematical truth is not determined by a majority vote, but by internal consistency. Indeed modern science started with this very paradigm shift. Quoting Galilei:

But in the natural sciences, whose conclusions are true and necessary and have nothing to do with human will, one must take care not to place oneself in the defense of error; for here a thousand Demostheneses and a thousand Aristotles would be left in the lurch by every mediocre wit who happened to hit upon the truth for himself.

If we want to train an algorithm to infer rules from language, we need to give samples of language where the rules are obeyed strictly (and yet this may not be enough). Otherwise the algorithm will wrongly generalize that the rules aren't strict (in fact it'll just see a bunch of mutually inconsistent examples). Which is what happens with ChatGPT.

Edit: On top of this, Gödel's theorem and other related theorems have shown that mathematical reasoning cannot be reduced to pure symbol manipulation, Hilbert's unfulfilled dream. So one can't infer mathematical reasoning from language patterns. Children learn reasoning not only through language training, but also through behaviour training (this was pointed out by Turing). This is why large language models have intrinsic limitations in what they can achieve and be used for.

I didn't know they made such claims. They're borderline dangerous claims...