Epistemology, reliable inquiry and topology

From one perspective, the fundamental notions of point-set topology have to do with sequences (of points or of numbers) and their limits. A broad class of epistemological questions also appear to be concerned with sequences and their limits. For example, problems of empirical underdetermination–which of a collection of alternative theories is true–have to do with logical properties of sequences of evidence. Underdetermination by evidence is the central problem of Plato’s Meno [Glymour and Kelly 1992], of one of Sextus Empiricus’ many skeptical doubts, and arguably it is the idea in Kant’s antinomies, for example of the infinite divisibility of matter [Kelly 1995, Ch.3]. Many questions of methodology, or of the logic of discovery, have to do with sequences and their limits, for example under what conditions Bayesian procedures, which put a prior probability distribution over alternative hypotheses and possible evidence and form conditional probabilities as new evidence is obtained, converge to the truth ([Savage 1954], [Hesse 1970], [Osherson and Weinstein 1988], [Juhl 1993]). Some analyses of “S knows that p” seem to appeal to properties of actual and possible sequences of something–for example Nozick’s proposal that knowledge of p is belief in p produced by a method that would not produce belief in p if p were false and would produce belief in p if p were true. Even questions about finding the truth under a quite radical relativism, in which truth 1 depends on conceptual scheme and conceptual schemes can be altered, have been analyzed as a kind of limiting property of sequences [Kelly et al. 1994].
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