Role of mathematics in modern economics

Mathematics has become an important tool in modern economics. Almost every field in modern economics uses mathematics and statistics. The mathematical approach to economic analysis is used when economists make use of mathematical symbols in the statement of a problem and also draw upon known mathematical theorems to aid in reasoning. It is not difficult to understand why the mathematical approach has become a dominant approach since finding the boundary of a theory, developing an analytical framework of a theory, establishing reference systems, and providing analytical tools all need mathematics. When discussing an economic issue, it is very important to distinguish between:

Two types of conditions: necessary and sufficient conditions for a statement to be true and
 Two types of statements: positive analysis and normative analysis. It is easy to confuse the distinction between necessary conditions and sufficient conditions, a problem that results often in incorrect conclusions.

For instance, it is often heard that the market institution should not be used based on the fact that some countries are market economies but remain poor. The reason this logic results in a wrong conclusion is that they did not realize the adoption of a market mechanism is just a necessary condition for a country to be rich, but is not a sufficient condition. Becoming a rich country also depends on other factors such as political system, social infrastructures, and culture. Additionally, no example of a country can be found so far that it is rich in the long run, that is not a market economy. 

If we apply a theoretical result to real world without knowing the boundary of a theory, we may get a very bad consequence and hurt an economy seriously.  Some of the advantages of using mathematics are that

the “language” used and the descriptions of assumptions are clearer, more accurate, and more precise,
the logical process of analysis is more rigorous and clearly sets the boundaries and limitations of a statement,
it can give a new result that may not be easily obtained through observation alone, and
It can reduce unnecessary debates and improve or extend existing results. It should be remarked that, although mathematics is of critical importance in modern economics, economics is not mathematics. Economics uses mathematics as a tool in order to model and analyze various economic problems. Statistics and econometrics are used to test or measure the accuracy of our predication, and identify causalities among economic variables. Thus, the production of an economic conclusion usually goes in two stages:

Stage 1: (non-mathematical language stage) Produce preliminary outputs, propose economic ideas, intuitions, and conjectures.

Stage 2: (mathematical language stage) Produce inter-mediate outputs, give a formal and rigorous result through mathematical modeling that specifies the boundary of a theory.

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