Ray Monk on Wittgenstein's Philosophy of Mathematics My feeling is that language is something that happens between people in many different language-communities (including soliloquy). We bring various capacities to bear - our vocal arrangements, our brains, our gestures - to express, try to understand or communicate with each other. These may well all be activities for which we can formulate both (a) rules we find we enact by whether we like it or not, and (b) human-imposed rules, or modifications of a-rules, that make the process work more to our satisfaction.
Then grammarians' rules, dictionaries, meta-languages, T-sentence concepts are all type (b) rules. But Chomskyan generative grammar and certain rules of dialogue would be (a) type rules.
To me exchange of 'truths' is just one of the things such language-communities might regard as what they're doing through language.
And in all this we are then, as you say, largely improvising most of the time.
This could all be bosh, of course, I'm just thinking about how I see it.
I can see how mathematical language could fit my schema. We find we just do count, subtract, divide, multiply. Elaborations make this work better. But I may be on the wrong tack for math. Witt for instance thinks that regarding sets as fundamental is just silly, whereas that's so much part of my psyche now I can't dislodge it to look at it afresh. (I wanted to refer to 'truth-telling' as a subset of language a little earlier!)