Economics, Mathematical and Empirical

Abstract

Though its influence is a hidden one, mathematics has shaped our world in fundamental ways. What is the practical value of mathematics? What would be lost if we had access only to common sense reasoning? A three-part answer can be attempted: 1. Because of mathematics’ precise, formal character, mathematical arguments can remain sound even if they are long and complex. In contrast, common sense arguments can generally be trusted only if they remain short; even moderately long nonmathematical arguments rapidly become far-fetched and dubious. 2. The precisely defined formalisms of mathematics discipline mathematical reasoning, and thus stake out an area within which patterns of reasoning have a reproducible, objective character. Since mathematics defines the notion of formal proof precisely enough for the correctness of a fully detailed proof to be verified mechanically (for example, by a computing mechanism), doubt as to whether a given theorem has or has not been proved (from a given set of assumptions) can never persist for long. This enables mathematicians to work as a unified international community, whose reasoning transcends national boundaries and survives the rise and fall of religions and of empires. 3. Though founded on a very small set of fundamental formal principles, mathematics allows an immense variety of intellectual structures to be elaborated. Mathematics can be regarded as a grammar defining a language in which a very wide range of themes can be expressed. Thus mathematics provides a disciplined mechanism for devising frameworks into which the facts provided by empirical science will fit and within which originally disparate facts will buttress one another and rise above the primitively empirical. Rather than hampering the reach of imagination, the formal discipline inherent in mathematics often allows one to exploit connections which the untutored thinker would have feared to use. Common sense is imprisoned in a relatively narrow range of intuitions. Mathematics, because it can guarantee the validity of lengthier and more complex reasonings than common sense admits, is able to break out of this confinement, transcending and correcting intuition.