Innumeracy: Mathematical Illiteracy and Its Consequences

by John Allen Paulos

Many people believe that determining the truth of any mathematical statement is merely a matter of mechanically plugging into some algorithm or recipe which will eventually yield a yes or a no answer, and that given a reasonable collection of basic axioms, every mathematical statement is either provable or unprovable. Mathematics in this view is cut-and-dried and calls for nothing so much as mastery of the requisite algorithms, and unlimited patience. The Austrian–American logician Kurt Gödel brilliantly refuted these facile assumptions by demonstrating that any system of mathematics, no matter how elaborate, will of necessity contain statements which can be neither proved nor disproved within the system. This and related results by logicians Alonzo Church, Alan Turing, and others have deepened our understanding of mathematics and its limitations. Given our concerns here, however, it’s sufficient to note that not even theoretically is mathematics mechanical or complete.