I thought of a better way of putting what I was trying to say. Communication may be orthogonal to the point of your question, but representation is not. An AI needs to use an internal language to represent the world or the structure of mathematics—this is the crux of Wittgenstein’s famous “private language argument”—whether or not it ever attempts to communicate. You can’t evaluate “syntactic legality” except within a particular language, whose correspondence to the world is not given a matter of logic (although it may be more or less useful pragmatically).
If your point is that it isn’t necessarily useful to try to say in what sense our procedures “correspond,” “represent,” or “are about” what they serve to model, I completely agree. We don’t need to explain why our model works, although some theory may help us to find other useful models.
But then I’m not sure see what is at stake when you talk about what makes a proof correct. Obviously we can have a valuable discussion about what kinds of demonstration we should find convincing. But ultimately the procedure that guides our behavior either gives satisfactory results or it doesn’t; we were either right or wrong to be convinced by an argument.
I thought of a better way of putting what I was trying to say. Communication may be orthogonal to the point of your question, but representation is not. An AI needs to use an internal language to represent the world or the structure of mathematics—this is the crux of Wittgenstein’s famous “private language argument”—whether or not it ever attempts to communicate. You can’t evaluate “syntactic legality” except within a particular language, whose correspondence to the world is not given a matter of logic (although it may be more or less useful pragmatically).
See my reply to Chappell here and the enclosing thread: http://lesswrong.com/lw/f1u/causal_reference/7phu
If your point is that it isn’t necessarily useful to try to say in what sense our procedures “correspond,” “represent,” or “are about” what they serve to model, I completely agree. We don’t need to explain why our model works, although some theory may help us to find other useful models.
But then I’m not sure see what is at stake when you talk about what makes a proof correct. Obviously we can have a valuable discussion about what kinds of demonstration we should find convincing. But ultimately the procedure that guides our behavior either gives satisfactory results or it doesn’t; we were either right or wrong to be convinced by an argument.