On Dec 17, 2012, at 12:38 PM, Joe Niederberger <email@example.com> wrote:
> Everybody learns mathematics with the aid of pictures, even you. Now I'm supposing you pretend that you didn't, or that you needn't have. I hope nobody is convinced of your word games here. I'm not.
Didn't I say that I wouldn't write a math textbook void of illustration? I have had this discussion with others and my distinctions are very reasonable. Yes we learn mathematics with the aid of pictures. But the mathematics isn't the pictures, it is in formal thinking. Think about it. You can't even come up with a convincing picture of minus times a minus. According to your picture theory then, none of us understand a minus times a minus. 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. I am talking about the painter, not you.