 Proofs and Refutations is a critique of “formalist” philosophies of mathematics (including formalism proper, logicism and intuitionism), which, in Lakatos’s view, radically misrepresent the nature of mathematics as an intellectual enterprise. For Lakatos, the development of mathematics should not be construed as series of Euclidean deductions where the contents of the relevant concepts has been carefully specified in advance so as to preclude equivocation. Rather, these water-tight deductions from well-defined premises are the (perhaps temporary) end-points of an evolutionary, and indeed a dialectical, process in which the constituent concepts are initially ill-defined, open-ended or ambiguous but become sharper and more precise in the context of a protracted debate. The proofs are refined in conjunction with the concepts (hence “proof-generated concepts”) whilst “refutations” in the form of counterexamples play a prominent part in the process. [One might almost say, paraphrasing Hegel, that in Lakatos’s view “when Euclidean demonstrations paint their grey in grey, then has a shape of mathematical life grown old…The owl of the formalist Minerva begins its flight only with the falling of dusk” (Hegel 2008 [1820/21]: 16).]