David Peat

On Incompleteness 39pure pattern of symbols, each following logically on the next accordingto strict rules of procedure. Rather than puzzle over the meaning ofthese symbols we should be concerned with establishing strict rules formanipulating them in order to go from one line of a proof to the next.This was Hilbert’s great program for the foundation of mathematics—hisroyal road to certainty. Hilbert wanted to list every possibleassumption and logical principle used in mathematics: nothing was tobe hidden; everything had to be up front. Rather than relying on words,every step of a proof should be replaced by rigorous strings of symboliclogic along with rules for going from one step to the next. Ideallythe whole thing could be automated. Provide a computer the axiomsof mathematics and a set of procedural rules, and it would work outevery theorem in mathematics.Hilbert’s axiomatic approach appeared foolproof. There seemedto be no chance of making a mistake in logic. There were no hiddenassumptions, nothing could exist within the system that had not previouslybeen defined, and nothing lay outside the system other than symboliclogic. This was exactly the approach espoused by Russell andWhitehead as they worked on their vast scheme to encompass mathematicswithin a frame of total rigor.IntuitionismNot everyone agreed with Hilbert’s reduction of mathematics to purelogic. The Dutch mathematician L. E. J. Brouwer argued that mathematicscould not be reduced to strings of meaningless symbols alone.The notion of counting, he argued, arises out of our intuitive experienceof time that allows us to distinguish the now from what is notnow. It is at a deep, psychological level, he claimed, that we have theconcept of “two-ness” or difference. Since our ability to count arisesout of this very basic mental experience, Brouwer argued for intuitionism,an investigation of the deep psychological level at which our mathematicalreasoning operates.