Degrees of Belief and Knowledge in Mathematics

Belief is very often analysed in terms of degrees, and since most human beings experience belief as a matter of degree, this kind of analysis has strong support. Much more controversial is the claim that knowledge, too, may be a matter of degree. The two questions are linked—after all, knowledge is, according to the traditional analyses, a special kind of belief. Many philosophers have analysed knowledge as belief that does not allow doubt and consequently, no degrees either. This view has been challenged by some authors, among them David Lewis, who gives a contextualist account of knowledge in [Lew96]: “S knows that P iff S’s evidence eliminates every possibility in which not-P—Psst!—except for those possibilities that we are properly ignoring.” The option of “properly ignoring possibilities” yields a spectrum of knowledge contexts from the loose standards of every-day usage (“I know that [my cat] Possum is not in my study without checking the closed drawers”) to the demanding standards of epistemology (Cartesian Doubt). The weaker my criteria for knowledge are (or, equivalently, the more possibilities I am allowed to be properly ignoring in the context), the lower my degree of knowledge is. In this spectrum of knowledge contexts, where does mathematics feature? Mathematics is a deductive science in which the notion of proof plays a crucial rôle. In philosophical contexts, mathematics is often used as an epistemological rôle model: mathematical knowledge is assumed to be absolute. However, we shall argue in this paper that even