I think the "practical" justification is best: Just like defining empty sums as 0 and empty products as 1, and 0! = 1, and 0^0 = 1 sometimes, and primes starting at 2: There's arguable logic for several choices, and mathematicians have settled on the most convenient choice, that leads to the simplest formulas and theorems. There's a modest physics justification, based on charge: There are positive and negative charges, and the electric force between two charged particles behaves like our usual sign convention: flipping the sign of either charge flips the force, and flipping the signs of both charges flips the force back. Note that set theory has some problems, because union and intersection are idempotent, and order of operations sometimes makes a difference: Interpreting A+B as (A union B) and A*B as (A intersect B), the distributive law works; in fact union distributes over both union and intersection, as does intersection. But things get messy when we bring in subtraction. If we follow the usual convention, A - B means A - (A intersect B), or A intersect (B complement). But then (A - B) + C is >= (A + C) - B; and the distributive law is a mess. Is there even a formula for (A-B)*(C-D)? Rich ------- Quoting James Propp <jamespropp@gmail.com>:
I started writing a new draft titled "Going Negative" and would love to get your feedback. I plan on publishing it on the 17th and will continue to tinker with it between now and the weekend.