refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true.) Lakatos proposed an account of mathematical knowledge based on the idea of heuristics. In Proofs and Refutations the concept of "heuristic" was not well developed, although Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical "thought experiments" are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy "quasi-empiricism". However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which mathematical proofs are valid and which are not. Therefore, he fundamentally disagreed with the "formalist" conception of proof prevailed in Frege's and Russell's logicism, which defines proof simply in terms of formal validity. On its first publication as an article in the British Journal for the Philosophy of Science in 1963–64, Proofs and Refutations became highly influential on new work in the philosophy of mathematics, although few agreed with Lakatos' strong disapproval of formal proof. Before his death he had been planning to return to the philosophy of mathematics and apply his theory of research programmes to it. Lakatos, Worrall and Zahar use Poincaré (1893) to answer one of the major problems perceived by critics, namely that the pattern of mathematical research depicted in Proofs and Refutations does not faithfully represent most of the actual activity of contemporary mathematicians. Cauchy and uniform convergence In a 1966 text Cauchy and the continuum, Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysis. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views. Research programmes Lakatos's second major contribution to the philosophy of science was his model of the "research programme", which he formulated in an attempt to resolve the perceived conflict between Popper's falsificationism and the revolutionary