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Although Lakatos"Proofs and Refutations (henceforth PR) is one of the classics of twentieth century philosophy of mathematics, few attempts have been made to build on its ideas.¹ Rather, quite often PR has become an object of respectful reference and detailed exegetical efforts. Of course, Lakatos is credited for having taught philosophers that philosophy of mathematics has to take into account the history and practice of mathematics. This orientation towards the history and practice of the discipline certainly represents progress beyond the ahistorical traditions of logicism, formalism, platonism, and intuitionism. It should help to get rid of the most pernicious vice with which the philosophy of mathematics is plagued; to wit, elementarism, which considers elementary arithmetics of natural numbers or Euclidean geometry as typical for all of mathematics. I think, however, that there is more to learn from Lakatos than just a general orientation towards the history and practice of real-life mathematics. As I want to show in this paper, these insights concern the role of inventing and varying mathematical concepts. More precisely, I want to use Lakatos' ideas of "concept-formation" and "concept-stretching" to sketch an evolutionary theory of mathematical knowledge, which takes axiomatic variation of concepts as the fundamental driving force of the ongoing evolution of mathematics (PR, 83ff.).²

DOI: 10.1007/978-94-017-0769-5_9

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