Mathematics used to be portrayed as a deductive science. Stemming from Polya, however, is a philosophical movement which broadens the concept of mathematical reasoning to include inductive or quasi-empirical methods. Interest in inductive methods is a welcome turn from foundationalism toward a philosophy grounded in mathematical practice. Regrettably, though, the conception of mathematical reasoning embraced by quasi-empiricists is still too narrow to include the sort of thought-experiment which Mueller describes as traditional mathematical proof and which Lakatos examines in Proofs and (...) refutations. This paper extends the concept of mathematical reasoning along two further dimensions to accommodate thought-experiment.Keywords: Thought-experiment; Informal proof; Mathematical reasoning. (shrink)

Lakatos's Proofs and Refutations is usually understood as an attempt to apply Popper's methodology of science to mathematics. This view has been challenged because despite appearances the methodology expounded in it deviates considerably from what would have been a straightforward application of Popperian maxims. I take a closer look at the Popperian roots of Lakatos's philosophy of mathematics, considered not as an application but as an extension of Popper's critical programme, and focus especially on the core ideas of this programme (...) as they appeared where Popper specifically addressed philosophical problems concerning mathematics. (shrink)