R. Hansen >But the mathematics isn't the pictures, it is in formal thinking.
Everybody reading this board knows a bit at least what truly "formal mathematics" is. And it is not the entirety of what mathematics as a human endeavor consists of.
So, what do you make of the fact that Hilbert made a few "errors" (i.e. formally unsubstantiated steps,) relying on geometric intuition, while producing proofs based on his famous axioms? If he were only using thinking "formally" that should not have happened.
R Hansen says: >According to your picture theory then, none of us understand a minus times a minus
Absurd - I said that there are no really convincing common sense examples. The history of the subject speaks volumes in this regard. It's not *my theory*. People accept the rule one way or another, and use it. So what? I've constructed no theory. I'm pointing out the obvious. David Tall has a theory, I reference it to show what one looks like. Accept it or not, it speaks to the issues you seem to care about, in an articulate way.
R. Hansen >There is a difference in looking at a painting and wondering what it does for you and looking at a painting and wondering what it did for the painter.
You cannot argue any point at all without lapsing into these analogies. That's rather informal of you.
If you have a problem with the mathematical picture that I was alluding to earlier, say it what it is. Please spare us another art analogy regarding violins, or pianos, or paintings.