Oh, MY! Hansen's: "But the mathematics isn't the pictures, it is in formal thinking."
The height of mathematical naiveté. Spoken like a truly devoted disciple of how mathematics traditionally has been taught in American schools and undergraduate courses ... which is quite unlike how mankind's mathematical knowledge actually is created.
Mathematical knowledge is organized into mathematical theories ... and is created through *personal* theorizing done by its creators ... only subsequently being shared with others. Like many other animal species, the human mind theorizes by working with, and on internal pictures ("schema", if you prefer). Most American adults have, and use personal theories about how to tie their shoes. Putting those theoristic pictures into written words is a very complex challenge. Putting them into mathematical languages is even more demanding ... so much so that mathematical theories necessarily are relatively simplistic.
The mathematics is in the theorizing and in the resulting theories ... regardless of how (or whether) they are expressed. Yes, there IS an area of genuine mathematics that focuses on the processing of formalisms ... and many adults have been myopically reared to equate that smaller part, with the whole enterprise. [Americans are rarely educated in the nature of mathematical theories or mathematical theorizing.] There even is an arena of mathematics that focuses only on formalisms, while ignoring any possibility of conceptual meanings to under-stand those formalisms.
But to claim that all of mathematical theorizing is done through "formal thinking" is to invoke an ultra narrow meaning of "mathematics" ... a meaning that ignores most of what has been done, and is being done, in the field.
"You can't even come up with a convincing picture of minus times a minus." The pictures have been around for centuries ... but they have been "convincing" only to those persons who are willing and able to learn how to newly interpret them. [Even some of the greatest were hampered by their myopic clinging to inappropriate interpretations.]
The 2-space graph of the -3x function yields a very clear picture of what (-3)*(-2) means. But to be "convinced", one must be willing to learn that with signed numbers, the word "times" speaks only about how to operate with the two numerators (the 3 and the 2) ... NOT about combining the signs.
The picture consists of the -3x line (a proportion), and the (1,0)-to-(1,-3) segment of the vertical slope-scale ("tangent" to the unit circle), and the vertical (-2,0)-to-(-2,6) altitude. That picture says that (-3)*(-2) is the function-value, at -2, of the proportion whose per-unit-rate is -3 per 1. When that picture is not "convincing", it is probably because the convincee has yet to learn that such evaluations are called "multiplying" only because of what is done with the numerators.
From: Robert Hansen Sent: Monday, December 17, 2012 6:48 PM To: email@example.com Subject: Re: Some important demonstrations on negative numbers > a MACS syllabus
On Dec 17, 2012, at 12:38 PM, Joe Niederberger <firstname.lastname@example.org> 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.