Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy

11.4.PHILOSOPHY OF MATHEMATICS

At the beginning of the twentieth century, philosophers, logicians, and mathematicians tried to formulate the proper foundations of mathematics. This resulted in the so-called foundational crisis of mathematics, from which several schools emerged, fiercely opposing each other, each with a radically different view of the right approach. During the first half of the twentieth century, the three most influential schools were known as logicism, formalism, and intuitionism. As numbers form an essential element of mathematics, the various philosophical schools also had different approaches to the concept of number.

Logicism, for example, whose most famous members were German mathematician Gottlob Frege (1848–1925) and British mathematician Bertrand Russell (1872–1970), tried to base all of mathematics on pure logic. In particular, they believed numbers should be identified with basic entities from set theory and their arithmetic should be derived from first logical principles. This was an important goal because all traditional pure mathematics can in fact be derived from the properties of natural numbers together with the propositions of pure logic. This idea already appeared in the work of German mathematician Richard Dedekind (1831–1916), who, in 1889, wrote “I consider the number-concept entirely independent of the notions or intuitions of space and time…. I rather consider it an immediate product of the pure laws of thought.”⁸ In 1903, Russell wrote that the object of logicism was “the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles.”⁹ The program of logicism was to reduce the notion of number to elementary ideas founded in pure logic, to “establish the whole theory of cardinal integers as a special branch of logic.”¹⁰ In this way, Russell hoped to give the notion of numbers a definite meaning.

Formalism, on the other hand, did not attempt to give meaning to mathematical objects. In the formalist approach—whose main proponent was German mathematician David Hilbert (1862–1943)—the goal was to define a mathematical theory in terms of a small set of axioms, mathematical propositions that are simply assumed to be true. From these axioms, mathematical theorems were derived by the rules of logical inference. In formalism, one is not interested in the nature of numbers, or in the question of whether numbers have a meaning. Rather one is only interested in the formal properties of numbers; that is, in the rules that govern their relations. Any set of objects that follows these rules could then serve as numbers. The formalist's view is best expressed in a famous statement usually attributed to David Hilbert: “Mathematics is a game played according to certain simple rules with meaningless marks on paper.”¹¹

Intuitionism, originating in the work of L. E. J. Brouwer (1881–1966), is non-Platonic because its philosophy is based on the idea that mathematics is a creation of the human mind. Because mathematical statements are mental constructions, the validity of a statement is ultimately a subjective claim asserted by the intuition of the mathematician. The mathematical formalism is just a means of communication. By restricting the allowed methods of logical reasoning (denying the validity of the principle of the excluded middle—that any proposition must either be true or its negation must be true), intuitionism strongly deviates from classical mathematics and the other philosophical schools, in particular in the accepted methods of proof. For an intuitionist, a mathematical object (e.g., the solution of an equation) would exist if it could be constructed explicitly. This is in contrast to classical mathematics, where the existence of an object can be proved indirectly by deriving a contradiction from the assumption of its nonexistence. The main differences occur, however, in how intuitionism deals with infinity, but that subject is beyond the scope of this book. Statements concerning the arithmetic of finite numbers generally remain true, and, in this context, intuitionism and classical mathematics have a lot in common.

Logicism, formalism, and intuitionism all made valuable contributions to the foundations of mathematics, but they all ran into unexpected difficulties of a rather technical nature. These difficulties finally prevented any of these programs from being fully realized.