Date: Nov 30, 2012 6:52 AM
Author: Robert Hansen
Subject: Re: Some important demonstrations on negative numbers
On Nov 29, 2012, at 9:38 AM, Joe Niederberger <email@example.com> wrote:
> I've said a few times now that historically, formal considerations led to the rules around negative numbers and so-called imaginary numbers. That's giving your point a lot of acknowledgement -- as it justly deserves. But, what you are not acknowledging is that those same rules, used as they were without sensible mappings and conceptual underpinnings, led to a protracted period of confusion even among careful thinkers and mathematicians. Read the book, or some of the condensed histories I've referenced.
I did write this several posts ago...
"I am not suggesting that it is mere coincidence that the concrete world is held accountable to the same mathematics and logic we strive to understand so deeply. If the world operated according to some other set of principles then I am sure that those would be the principles that we would strive to understand so deeply. "
I am certainly not without the notion of the real world's role in all of this.
> There is evidenced by the historical record, a continued desire, among general population and mathematicians themselves, for "models" to map the new abstract operation to. So they were developed. Let's use them and not make everyone go through the same old confusions. But Peter has hit on one of the nagging lingerers: (-) x (-) = (+). Just look around the web, maybe starting here:
I like your phrase, "the same old confusions", and I hope, even though you do not agree with my position, you understand my position.
If you don't go through the same "old confusions" then you are not developing mathematical thought, just like if you do not rush home after someone has explained to you how to play the violin, and start trying to play the violin, you are not understanding how to play the violin.
As an analogy, consider climbing Mt Everest. You have three choices. First, with no guidance whatsoever, you can try to find a route up the mountain. That would be like mathematics prior to the discovery/invention of negative numbers (as a small example). Second, a passable route is described to you and you follow that, but that route is still fraught with peril. That would be like mathematics after the discovery/invention of negative numbers. And third, you can stand at the base of Mt Everest and talk about climbing Mt Everest.
I am not preaching that we should let everyone lose with no guidance, that would be for the mavericks. But even though we know the path the student must still make the journey and the path is still as fraught with peril as it ever was. It is still full of "old confusions" and the student must develop formal thinking in order to overcome those confusions. And along the journey, they are going to get lost at times and there will have to be some maverick in them to find their way back to the path.
For Clyde, there is no mountain and thus no such thing as mountain climbing.
For Crabtree, there is a mountain, but he prefers instead to spend his time constructing a toy mountain at ground level, that he claims is even better than "the mountain".
Peter is on the mountain, somewhere.
You tend to enjoy reading books about people that have climbed the mountain.
Lou has laid claim to a section of the mountain and has erected no-traspassing signs and installed a security system.
Dave has surveyed the whole damn mountain, he even knows how to get by Lou's security system, although there are other mountains.
Kirby has heard many tales of mountains and he entertains travelers with those tales, in his little curio shop in the town at the base of the mountain.
I used to play on the mountain a lot, and an adjoining mountain (physics), chose to live and work on yet another mountain (CS), but still spend some summers and weekends at "the mountain".