 Let us assume, then, that an idealized deductive proof provides one kind of security: the transparency of each step ensures the validity of the argument as a whole, and hence guarantees that if the premises are all true then the conclusion must be true. But what of the axioms that are brought in at the beginning of the proof process? The traditional answer to this question is to claim that the truth of the axioms is secure because the axioms are “self-evident”. This certainly seems to have been the generally accepted view of the axioms of Euclidean geometry, for example. However, this attitude is much less prevalent in contemporary mathematics, for various reasons. Firstly, the discovery of non-Euclidean geometry in the early 19th Century showed that apparent self-evidence, at least in the case of the Parallel Postulate, is no guarantee of necessary truth. Secondly, the increasing range and complexity of mathematical theories—and their axiomatizations—made it much less plausible to claim that each individual axiom was transparently true. Thirdly, many mathematical subfields have become abstracted to a considerable degree from any concrete models, and this has gone hand-in-hand with the tendency for at least some mathematicians to adopt a formalist attitude to the theories they develop. Rather than expressing fundamental truths, on this view axioms serve simply to provide the starting position for a formal game.