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...