Formalism is the mathematical school of thought which holds that mathematics consists of symbols, rules for combining those symbols, some minimal number of assumptions or axioms, and certain agreed upon rules of inference. Formalism was introduced in the early twentieth century by the great German mathematician, David Hilbert (1862-1943), in response to a certain uneasiness that had arisen among some mathematicians concerning the logical foundations of mathematics. The German logician Gottlob Frege (1848-1925), had made an attempt to derive all of the laws of arithmetic from logic alone, but the young British logician Bertrand Russell (1872-1970) discovered a paradox in Frege's system which doomed it to failure. Russell, with Alfred North Whitehead (1861-1947), made modifications in Frege's work to eliminate Russell's paradox and others that had been discovered by other mathematicians. Russell and Whitehead produced the massive three-volume work Principia Mathematica, which, they claimed, did the job Frege had set out to do. Nevertheless, there remained an undercurrent of opinion among mathematicians that Russell and Whitehead had made too many concessions to eliminate the paradoxes. Rather than arguing the fine points of logic, Hilbert proposed a different way of looking at the foundations of mathematics. Hilbert's formalistic program essentially reduces mathematics to a game, albeit an important game, in which a very strict set of rules must be followed if one is to be allowed to play the game. The equipment needed for playing the game consists of the symbols of the mathematical system, stripped of meaning, except the meaning they are given by the rules of syntax by which they may be combined. The rules for playing the game are the rules of logical inference. The goal of the game is to prove, consistently, i.e., without contradiction, all the theorems that may be stated within the system. The players, the mathematicians, are governed by the very strict rules of the game, the axioms. In this view, Euclidean geometry is just a game and the non-Euclidean geometries of Riemann and Lobachevsky are just different games played using different rules. So long as the rules are applied consistently, the interpretation of the mathematics to any physical reality is irrelevant. In Hilbert's formalism, mathematics is not about symbols which stand for idealized physical objects as in Platonist mathematics.
For the formalist, mathematics is about the symbols themselves, which are devoid of any outside meaning. The symbols are strung together according to very strict syntactical rules to make formulas that may then be used in the proving of theorems. The proving of theorems, by established rules of inference, is the point of mathematics for the formalist. Under Hilbert's plan, theorem-proving would become a mechanical procedure.
Hilbert required that a formal mathematical system should meet three standards: consistency, completeness, and decidability. Consistency means that no contradictions will be found in the system, i.e., a theorem and its negation cannot both be true. Completeness implies that every theorem which is true in the system can be proved within the system. Decidability requires that a finite mechanical procedure exists that determines whether any given claim made by the system may be proved within the system. In 1931, the brilliant logician Kurt Gödel (1906-1978) published a paper showing that any formal system containing the natural numbers cannot be both complete and consistent. If a system is complete, it will contain a contradiction. If it is consistent, then it will not include all possible theorems of the system. Gödel's result doomed Hilbert's program, as well as Russell and Whitehead's, to failure. Nevertheless, the influence of Hibert's formalism did not end with Gödel's publication. Although formalism was unsuited for a philosophy of mathematics, the idea of a mechanical process for computation was adopted by the British logician, Alan Turing (1912-1954), and became the basis for the digital computer which would spur the information revolution of the late 20th century. Turing proposed a machine that could be instructed or programmed to make any computation that could be carried out by a human being with pencil and paper and a sufficient amount of time. The set of instructions given to such a machine would come to be known as a program. Modern computer programs are written in formal languages which can be traced back to Hilbert's desire to formalize the entire discipline of mathematics. Gödel's work showed that this desire was untenable as a foundation upon which to build mathematics, but the position of software developers as among the largest corporations in the world demonstrates that formalism was by no means a dead end.
This is the complete article, containing 752 words
(approx. 3 pages at 300 words per page).
