  Maddy claims that the correct model of mathematical justification will vindicate the hypothesis that traditionally scientific or philosophical arguments do not enter into the justification of mathematical statements. For example, the French analysts Baire, Borel and Lebesgue criticised the Axiom of Choice on the basis of a definabilist methodology, according to which the existence of an object depends on its definability: functions should be definable rules, membership of a set should given in a definable way, etc. But definabilism ended up having no influence on the practice of mathematics, and its sympathisers were silenced or gave up on it when its mathematically undesirable features became clear. For example, the Axiom of Choice allows the taking of maximal ideals in rings and other structures; it entails the kinds of maximality principles that even the analysts were already using; it simplifies transfinite arithmetic; and despite its suspicious abstractness it turns out to be equivalent to ‘concrete’ and ‘mathematical’ statements such as the claim that every vector space has a basis. The reasons for accepting the Axiom of Choice were in the end purely mathematical.