Definitions in Mathematics
The field of mathematics is much more than just the applications. In fact, the applications are a result of actual mathematics which comes in the form of proofs and theorems. For example, in ring theory, mathematicians needed to prove that
a * 0 = 0 for all values of
a. Below is the proof:
a * 0 = a(0 * 0) = a * 0 + a * 0. (1)
Then we add -(a * 0) to both sides to get
(a * 0) + -(a * 0) = (a * 0 + a * 0) + -(a * 0) (2)
This gives us
0 = a * 0. (3)
The applications of this benefit many individuals when used to show
5 * 0 = 0, but this is merely the result of a more broad result that has been proven.
How are these proofs constructed? Through definitions. To prove the above result, we could not assume that
a(0 * 0) = a * 0 + a * 0; instead, we needed to use the definition of a "ring" which, by definition, allows for line (1). Similarly, we needed to use the definition of a "ring" in order to know that we were allowed to use
-(a * 0) in line (2).
Definitions in Economics
Economics, however, does not use definitions in the same capacity. Here, definitions are used strictly for the "defining of terms" rather than the "relation of terms." In Economics, one cannot prove that, in the short-run, expanding the money supply (which causes inflation) will decrease unemployment. The definitions in economics are not set up to do that; even more so, they cannot do that.
Part of the reason that the definitions in economics cannot do this is because of the definitions. Think of the terms "good," "market," and "demand." All of these terms have sloppy definitions. They do not really relate to anything else. On the other hand, we have terms such as "currency" and "GDP" that have extensive, precise definitions. These definitions have been chosen purposefully, and measurements of "currency" and "GDP" are precise because of this.
Another part of why Economics has "poor" definitions is due to the study of economics itself. Economics relies heavily on the demand of individuals. This demand cannot be quantified nor is there any guarantee that it will remain the same from one moment to the next. Thus, no real way to construct a proof that will be true beyond any particular moment. Because of this, economics does not need rigorous definitions. In Mathematics, however, we can construct proofs regardless of the numbers we use and, thus, transcend the limitations up to a very broad context. In the proof above, we used
a instead of a number so that we did not have to rely on using that number and only that number. By using
a, we know that multiplying any number by
0 will give us
Response to Edesess
Edesess is mostly (probably 95%) correct. In truth, most of the definitions of economics are not "precisely defined" to the same level that mathematical definitions are required to be. In Mathematics, the definitions are carefully considered and decided upon by the Mathematical Community as a whole (not to say that Economic definitions aren't, but that's outside my knowledge). Also, by the nature of Economics, the use of the definitions cannot be used to prove anything.
In response to Edesess, however, economics should not be treated as Mathematics because of the fundamental differences in how they make discoveries. Economics is advanced through polls, market data, supply and demand graphs; Mathematics is furthered by research, proofs, and theorems.