For two thousand years geometry was the paradigm of rigorous reasoning. To argue more geometrico was an ideal many philosophers set themselves. Spinoza set out his philosophy as a series of propositions, each following from its predecessors. Locke, in spite of his criticisms of Descartes, thought he could establish morality as a sequence of deductions from self-evident truths. But, attractive though the ideal of rigorous reasoning undoubtedly is, when we come to reflect on it, we find ourselves impelled towards Formalism. It is not only that axioms are easier to postulate than to justify,1 but the principles of inference themselves, under scrutiny, require a formal exegesis. Some formalist tendency is implicit in Plato’s Theory of Forms, but often Formalists define their position in opposition to straightforward Platonism. Formalists see themselves as radicals-though often, like other radicals, they turn out to be extremely conservative.2 They reject the Platonic Establishment, and are not prepared to swallow guff, or bow down before the great tradition of classical mathematics. They will not make a great act of faith, and believe in a world of Forms, or any other abstract or supersensible entities. Everything must be reauthenticated to their own satisfaction at the bar of reason. And they will be difficult to satisfy. Only the most rigorous proofs will
do. No handwaving, no appeals to intuition will be countenanced. Every assumption must be stated clearly, every rule of inference made quite explicit. Only the most rigorous arguments are worthy of mathematics. Mathematics is not just Plato playing pleasantly with yesful youths by the banks of the Ilyssus, and getting them to say that they see lovely patterns laid up in heaven, but a tough game played against a tough opponent who is as sceptical as it is possible to be and will not let anything pass unless he absolutely has to. It is much harder playing that way, but at least we can be sure that anything we succeed in proving is a real achievement and is something we can be absolutely sure of; having eliminated the sloppy, we can be proud of what remains.