The purpose of this paper is to present a local solution to the Einstein-Rosen-Podolsky (E.P.R.) paradox by way of a mechanical analogue (roulette) and then to note some of its consequences for the foundations of mathematics and probability theory. The detailed mathematical development of the model involves some highly specialized fields in mathematics (set theory, measure theory and group theory). My intention is to avoid these technicalities as much as possible. A complete account that includes proofs and application to physics is in Pitowsky (1982a, 1982b, 1983). Some mathematics is, however, indispensable. I shall denote by x,y,z,w unit vectors in the three dimensional Euclidean Space. If x,y are unit vectors, is the (small) angle between x and y so that 0 ≤ ≤ Π.

The language of thought hypothesis is this: central among the causes of our behavior are inner states with linguistic structure that play roughly the role we pre-scientifically ascribe to our beliefs and desires. A philosophical thesis has been proposed on the basis of that scientific hypothesis- namely, that the way to explicate commonsense notions of the content of beliefs and desires, of their intentionality, is in terms of the meaning of such internal sentences. I am going to compare this explicative strategy with a certain functionalist theory of propositional attitudes, on which propositional attitude ascriptions of the form ‘x believes (desires) that —’ are explicatively more fundamental than anything linguistic and semantic. This is a modern dress version of an old dispute; but the issue is not between the naturalistic proponent of linguistic meaning and the antinaturalistic proponent of irreducible intentionality.

Twenty years have passed since Paul Feyerabend and I first used in print a term we had borrowed from mathematics to describe the relaationship between successive scientific theories. ‘Incommensurability’ was the term; each of us was led to it by problems we had encountered in interpreting scientific texts (Feyerabend 1962; Kuhn 1962). My use of the term was broader than his; his claims for the phenomenon were more sweeping than mine; but our overlap at that time was substantial. Each of us was centrally concerned to show that the meanings of scientific terms and concepts — ‘force’ and ‘mass’, for example, or ‘element’ and ‘compound’ — often changed with the theory in which they were deployed. And each of us claimed that when such changes occurred, it was impossible to define all the terms of one theory in the vocabulary of the other.

In 1962, Thomas Kuhn offered an account of the difficulties which attend the reception of revolutionary scientific ideas: “Communication across the revolutionary divide,” he wrote, “is inevitably partial.” (Kuhn 1962, p. 149). Eight years later, in the postscript to the second edition of The Structure of Scientific Revolutions, he sounded the same theme, characterizing the participants in revolutionary debates as people who attach their terms to nature differently, so that their communication is “inevitably only partial” (Kuhn 1970, p. 198). In his most recent discussion of these issues (Kuhn 1983), Kuhn has explained more clearly than before the nature of the differences in language which he takes to be present in revolutionary debates.

Here is the example that L. J. Savage (1954, p. 21) uses to adumbrate the principle: