I ran across this concept in the Math Forum and thought it interesting.
Positive and negative numbers
One thing that can help is to realize that mathematics is a world of its own that can be used to MODEL things in the real world, but, like any model, is not IDENTICAL to that real world. Negative numbers are part of the "model world", not the real world. So there are some situations where negative numbers make sense (and a negative answer to a problem is valid), and others where they do not (so that a negative answer just means there is no solution to the original problem). In the case of temperatures, the 0 point (except in absolute temperature) is arbitrary, so that SOME negative values are possible, but others are not.
In the case of money, a negative answer may or may not be valid. If you are just spending money from a basic checking account, a negative balance means that you are overdrawn--but it DOES still have meaning, because you now owe that much money to the bank. If you have an account with automatic overdraft protection, the negative balance means that you have borrowed that much and have to pay it back. So how to interpret the negative result depends on the situation; often positive and negative are just two sides of the same coin, each with its own interpretation.
The idea of negative numbers was often considered suspect even into the 1800's. I've read a book by a mathematician of that time trying to present algebra in a way that didn't treat negative numbers as real; he called them fictitious, and presented them as just a shorthand for operations that should properly be done in reverse. But he admitted that using negative numbers made the work easier, always gave the right answer (when interpreted correctly), and unified what would otherwise have required several cases (depending on which number is greater, for example). That is, negative numbers serve as a good, though imperfect, MODEL--you don't have to recognize the existence of the negative numbers themselves as concrete entities in order to make good use of them. (And, by the way, even counting numbers are really an abstract concept too--you never saw a "three" by itself, did you? ALL numbers live in the math world, not the real one.)
What we are doing when we use negative numbers is translating a real problem into a problem about, say, locations on a number line (which correspond to amounts of money you have or owe, say), solving that new problem, and then deciding how to translate the answer back into the real world problem. The negative numbers live in this separate world of math, and may have various meanings or lack of meaning in the real world.
- Doctor P., The Math Forum