Date: Nov 27, 2012 3:33 PM
Author: Joe Niederberger
Subject: Re: Some important demonstrations on negative numbers

Robert Hansen says
>How to recognize convention and the way it is developed.

I'll add it wasn't developed without a lot of worry and disagreement - even vehement denunciations of what we take for granted today. See:
http://plus.maths.org/content/negative-math

Barry Manzur also discusses (somewhat less satisfactorily in my opinion, but worthwhile) in "Imagining Numbers" --Chapter "Justifying Laws" (particularly Ditsributibve Law, and as James Elander points out, 1 as identity for mult.)

For myself --I like to have a reasonable model (reasonable for me.) The one that comes to mind most readily is a game of poker, with winning (pos.) losing (neg.) and the fantasy of being able to reverse time to "un-lose" N times. If I un-lose a $3 loss 5 times (which I model as -3 x -5), where am I?

Cheers,
Joe N