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So I'm part of this math meme group and this was posted

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I'm not an expert in "modal homotopy type theory" but are both claims true? And is this a fair critique of Wittgenstein's insight?

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In broad strokes, yes.

One central feature of Wittgenstein's Philosophical Investigations is the notion of a language game. Just like certain actions do not make sense except in the context of a game, certain words do not make sense except in the "context" of a certain community, ie those language users. This gave rise to Ordinary Language Philosophy, certain sketches of OLP might characterize it as claiming that ordinary languages, as opposed to formal logics, are the best place to do philosophy in.

Homotopy Type theory is a type theory in which the notion of propositional equality is interpreted as homotopy and type isomorphism as homotopy equivalence. More generally, this gives rise to multiple notions of equality, which I suppose the meme (purposefully, for humor's sake) uses as "is". However, we don't need the notion of homotopy to recognize this meme- just note that (arguably) most mathematicians work in a many sorted logic and with many notions of equality: set equality, numerical equality, and equality up to isomorphism.

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No. That meme has little relationship to Wittgenstein's philosophy. Even if a word has only one "intended" meaning, Wittgenstein argues that we have no reliable way to communicate that meaning or to verify that we are indeed understanding the same meaning. See his argument on following a rule.

On the other hand, they may be referring to Wittgenstein's argument that language is a sort of rule-based activity or language game. He says that there are many of these language games (different contexts), and that many philosophical problems are essentially confusions where two different language games are getting mixed up. However, this cannot be addressed by type theory of word meanings because the words may have no meaning at all; they are being used simply for effect.

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