The axiomatic method providing an easy target for polemical attack by empiricists such as Lakatos (1976). It is nevertheless true that pure mathematicians at any rate regard its use as routine. How then should we account for it? Responses to this question fall into two camps which mathematicians have for some time been wont to call realist and formalist. This was not an alto- gether happy choice of terminology since philosophers had already used both words for more specific positions in the philosophy of mathematics, but I shall follow the mathematicians’ usage here. At the core of attitudes to the axiomatic method that may be called realist is the view that ‘undefined’ does not entail ‘meaningless’ and so it may be possible to provide a meaning for the primitive terms of our theory in advance of laying down the axioms: perhaps they are previously understood terms of ordinary language; or, if not, we may be able to establish the intended meanings by means of what Frege calls elucidations — informal explanations which suffice to indicate the intended meanings of terms. But elucidation, Frege says, is inessential. It merely serves the purpose of mutual understanding among investigators, as well as of the communication of science to others. We may relegate it to a propaedeutic. It has no place in the system of a science; no conclusions are based on it. Someone who pursued research only by himself would not need it. (1906, p. 302) If the primitive terms of our theory are words, such as ‘point’ or ‘line’, which can be given meanings in this manner, then by asserting the axioms of the theory we commit ourselves to their truth. Realism is thus committed to the notion that the words mathematicians use already have a meaning independ- ent of the system of axioms in which the words occur. It is for this reason that such views are described as realist. If the axioms make existential claims (which typically they do), then by taking them to be true we commit ourselves to the existence of the requisite objects. Nevertheless, realism remains a broad church, since it says nothing yet about the nature of the objects thus appealed to