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Wittgenstein is critical of the 'private linguist' and his exclusive use of the ostensive definition, where the definition provided for a given word is an example or a 'pointing out' of what the word might refer to.

But it would seem that the origin of our language is rooted in ostensive definition. It is after all how children with no prior knowledge of a public language begin to learn a language. Their mother or father points to something while stressing a given word.

Wittgenstein doesn't seem to have been one to hold that the entire language game is a farce.

As such, what allowed him to criticize ostensive definitions in the private language argument while not coming to criticize language acquisition through ostensive definition in the case of children?

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But it would seem that the origin of our language is rooted in ostensive definition. It is after all how children with no prior knowledge of a public language begin to learn a language. Their mother or father points to something while stressing a given word.

That would be Augustine's reasoning (or rather, remembrance) that Wittgenstein criticises. But it is hard to agree with Augustine. Children definitely do not learn their first language by the method you suggest, which would give them a lexicon, but no grammar.

I have seen little children babbling, imitating adults in conversation. Their sounds make no sense at all, but what is important is that they have grasped that talking is important, because adults talk. Meaning comes afterward, and it is only secondarily linked to a list of substantive nouns that children learn by heart. Instead, children must first learn to do basic operations such as demand, refuse, accept, call, react to calls, etc. They do not learn to call their mothers "mummy" by someone pointing to their mother and calling her "mummy". They learn it by realizing that their mother reacts immediately and joyfully if they emit sounds like "ma", "mo", "mem", "muma", etc.

As such, what allowed him to criticize ostensive definitions in the private language argument while not coming to criticize language acquisition through ostensive definition in the case of children?

As far as I understand, he opens Philosophical Investigations by exactly criticizing language acquisition by ostensive definition in the case of children. In fact, it seems to me that he strawmans a remark by Augustine that has no greater ambitions than recalling a first childhood that has actually been forgotten into a whole "philosophical theory of language" that doesn't seem to match any actual body of philosophy.

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We can understand one of the central theses of Investigations to be "Meaning is use." For Wittgenstein, coming to understand the meaning of a bit of language is not a matter of forming a mental state in which the bit of language appropriately maps on to the right parts of the world. Much of Investigations works at encouraging skepticism about that account of meaning. Instead, coming to understand the meaning of words and phrases is just learning how to use them in the right ways in the right contexts.

The main processes of language acquisition are therefore imitation, experimentation, and participation in activities involving language.

If we sometimes employ ostensive definition when teaching language, we radically over-estimate its significance for actual learning, on this view. We notice those obvious acts of pointing, and incorrectly infer that that's how learning works. In doing so, we miss the more subtle processes through which a person tries out bits of language and gradually begins to deploy the right ones, or rather ones that work in the various contexts of one's life.

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