Definitions of mathematics

Date: 2015-10-07; view: 501.

Aristotle defined mathematics as "the science of quantity," and this definition prevailed until the 18th century. In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Today, there is no consensus on the definition of mathematics, even among professionals. Some emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. There is not even consensus on whether mathematics is an art or a science. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Some say, "Mathematics is what mathematicians do."

Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. All have severe problems, none has widespread acceptance, and no reconciliation seems possible.

Logicist definitions attempt to reduce mathematics partly or wholly to logic, especially symbolic logic. Two examples of logicist definitions are "Mathematics is the science that draws necessary conclusions" (Benjamin Peirce) and "All Mathematics is Symbolic Logic" (Bertrand Russell). Russell and Alfred North Whitehead attempted to prove rigorously that all of mathematics is derivable from symbolic logic in the Principia Mathematica.

Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other." A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that you can mentally construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems." A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiom has a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.