We have reached a position today from which to evaluate judiciously Lakatos' contribution to the philosophy of mathematics. We should applaud unreservedly his decision to take the development of mathematics as a topic worthy of philosophical consideration and admire the first steps he took to elucidate patterns of theory change via his identification of certain mechanisms of concept-stretching. We must recognise, however, that the method of proofs and refutations accounts only for the modification of a concept's definition,¹ omitting more fundamental varieties of theory development, and consequently that much remains to be done. The two tasks before us today are to describe an expanded range of varieties of theory production, especially radically innovative mathematical conceptualisations, and to ponder the philosophical issues relating to such a description. Regarding the former, see Corfield (1998a) and Kvasz (1998). Through this paper I shall largely confine myself to the latter, endeavouring to convey a sense of the vast array of possibilities now open to us.