Hoffman's "proof" of the possibility of spectrum inversion.

Philosophers have devoted a great deal of discussion to the question of whether an inverted spectrum thought experiment refutes functionalism. (For a review of the inverted spectrum and its many philosophical applications, see Byrne, 2004.) If Hoffman is correct the matter can be swiftly and conclusively settled, without appeal to any empirical data about color vision (or anything else). Assuming only that color experiences and functional relations can be mathematically represented, a simple mathematical result—the Scrambling Theorem—shows that color experiences can be permuted while keeping functional relations constant, thus contradicting functionalism. It will be helpful to illustrate Hoffman’s argument with a very simple example. Suppose that there are two color experiences—experiences of red (state R) and experiences of green (state G), and that the functionalist identifies R with functional state FR, and G with functional state FG. A person can only be in one of these functional states at any particular time. According to the functionalist, then, it is impossible for R to obtain without FR, and vice versa. First, let us put the functionalist theory into the format of the Scrambling Theorem, using Hoffman’s notation, and ‘R,’ ‘FR’, etc., to stand for mathematical surrogates for the experiences and functional states. Let X (the set of ‘‘color experiences of one person’’), and Y (the set of ‘‘color experiences of a second person’’) both be {R, G}. As Hoffman emphasizes, the argument does not depend on restricting the choice of the other functions and sets mentioned in the Scrambling Theorem, so we may take them to be as simple as possible. Let W be {FR,FG}, and define f: X ! W as f (R) = FR and f (G) = FG. (Hence n = 1.) Define the bijection b: X! Y as: b (R) = G and b (G) = R. Then, in this simple case, the Scrambling Theorem tells us that there is a function g: Y! W such that f = gb. And obviously there is: g (G) = FR, and g (R) = FG. 1