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?