David Peat

40 From Certainty to UncertaintyThe Principia Is PublishedNotwithstanding Brouwer’s objections, Russell and Whitehead pushedahead to publish their research program. The resulting text was so largethat the two philosophers used a wheelbarrow to push the manuscriptto the publisher’s office! With the results now in print, the world’smathematicians had to decide if the two men had truly placed mathematicson a firm logical basis.Some were still worried about Russell’s paradox. Russell himselfclaimed that it was no more than a confusion arising out of mixing updifferent logical types of statement; that is, classes with classes ofclasses. Not everyone was convinced. Had Russell offered a true solutionor was it more a matter of sweeping the problem under the carpet?What’s more, some mathematicians were not happy with the standardsof logical reasoning used by Russell and Whitehead.Gödel’s TheoremMathematicians remained undecided as to whether mathematics hadbeen definitively established as complete and consistent. Finally, in1931, a German paper, “On Formally Undecidable Propositions ofPrincipia Mathematica and Related Systems,” rocked the world ofmathematics and put an end to the program of Hilbert, Russell, andWhitehead. Its author, Kurt Gödel, was 25 and living in Vienna. Hispaper showed once and for all that the internal consistency of the axiomaticmethod, sacred since the time of Euclid, is limited. More precisely,if an axiomatic system is rich enough to produce something likemathematics, then it can never be shown to be consistent. Moreover,such a system will always be inherently incomplete.Gödel’s proof was ingenious in the extreme. To begin with, he wasdetermined to avoid the distinction between mathematics and what isknown as metamathematics. In Hilbert’s program, the goal was to demonstrate,using symbolic logic, that mathematics is both consistent andcomplete. But this meant that mathematics itself was being discussedand analyzed by another symbolic system. The system that talked about