David Peat

On Incompleteness 35colleague A. N. Whitehead, he embarked on a major undertaking: todiscover the logical foundations of mathematics. This vast researchproject would result in two great volumes known as PrincipiaMathematica.Mathematicians may have thought they were being rigorous untilRussell and others pointed out that, within their arguments, mathematicianswere using subtle forms of reasoning, sometimes unconsciously,that had never been properly formulated. Russell’s plan was touse a formal, symbolic notation in which all rules of inference weretotally explicit. It was to have:• A system of signs• A grammar; that is, rules for combining signs into formulae• Transformation rules that allowed mathematicians to go fromone formula to another• Axioms• Proofs, involving a finite sequence of formulas, starting withan axiom and proceeding step-by-step using the rules of transformationThe Notion of ProofRussell’s program involved basing mathematics on a strictly logicalfoundation, an idea that goes back to Euclid. The ancient Greeks haddiscovered a variety of facts about the geometry of the world but it wasleft to Euclid to gather these facts into a single consistent and logicalscheme called Elements of Geometry.Euclid began with definitions about the simplest possible elementsof geometry—points, lines, planes, and so on. To these he added a fewaxioms, which are the logical starting points of his system and were soobvious, he hoped, as to be self-evidently true. For example, one ofthese axioms tells us that parallel lines do not meet no matter how longthey are.From the starting point of his definitions and axioms, Euclidsought to demonstrate the various theorems known to geometry, such