Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

Popper formulates and explains his non-justificationist theory of knowledge. Science--empirical science--aims at true explanatory theories, yet it can never prove, finally establish, or justify any of its theories as true, not even if it is in fact a true theory. Science must continue to question and criticize all its theories, even those which happen to be true.

Much philosophy of science is methodology of science. How should one go about doing and evaluating it? The question is one of the methodology of methodology, i.e. of metamethodology. There is a vague thesis common to Descartes and more recent philosophers such as Quine and Lakatos: that what is good methodology, good evidence, good reason for accepting, rejecting or revising beliefs in mathematics and in the sciences properly so called, does not differ in significant kind from what is good methodology, (...) evidence or reason elsewhere, even in epistemology and ethics; and further, that mathematics and the natural sciences provide good paradigms for methodology generally. Following von Hayek,1 I call this form of epistemological and methodological monism: Scientism. (shrink)

Can we know anything for certain? There are those who think we can (traditionally labeled the "dogmatists") and those who think we cannot (traditionally labeled the "skeptics"). The theory of knowledge, or epistemology, is the great debate between the two. This book is an introductory and historically-based survey of the debate. It sides for the most part with the skeptics. It also develops out of skepticism a third view, fallibilism or critical rationalism, which incorporates an uncompromising realism about perception, science, (...) and the nature of truth. (shrink)

It is argued in this paper that the valid argument forms coming under the general heading of Demonstrative Induction have played a highly significant role in the history of theoretical physics. This situation was thoroughly appreciated by several earlier philosophers of science and deserves to be more widely known and understood.

William Whewell’s philosophy of scientific discovery is applied to the problem of understanding the nature of unification and explanation by the composition of causes in Newtonian mechanics. The essay attempts to demonstrate: the sense in which ”approximate’ laws successfully refer to real physical systems rather than to idealizations of them; why good theoretical constructs are not badly underdetermined by observation; and why, in particular, Newtonian forces are not conventional and how empiricist arguments against the existence of component causes, and against (...) the veracity of the fundamental laws, are flawed. (shrink)

Holton, we acknowledge, has given a good counter-example to a theory, and that theory is interesting and worth refuting. The theory we have in mind is like Smith's, but is more reductionist in spirit. It is a theory that ties value to Reason and to processes of reasoning, or inference - not to the recognition of reasons and acting on reasons. Such a theory overestimates the importance of logic, truth, inference, and thinking things through for yourself independently of any ideas (...) about where you might end up. Now it might well be thought that any Kantian theory of value would need to be tied to just such a conception of Reason. But while the theory behind The Moral Problem is Kantian in some very salient respects, the survival of Smith's analysis of value in the face of Holton's argument is very instructive. It teaches us a memorable moral: that a Kantian theory like Smith' s does not need to be tied - even loosely - to an overly intellectualised, logocentric conception of Reason. (shrink)