A couple of readers confused about my post on definitions. If we change the sense of a turn, we may change its reference as well. (Not always: if we change the sense of X from "the evening star" to "the morning star," X still refers to the same thing!) But we have not changed any of the facts about what X used to refer to. So if we were to change the sense of the term "cat" to "a large, leaping Australian marsupial," it would henceforth refer to what we now call kangaroos. But that does not mean that the non-human mammal currently living in my house will suddenly have a pouch! Similarly, if we define a new mathematical symbolism, call it M new , that is the same as ours (which we can call M old ) for the first use of number, but every subsequent time it is mentioned, its value goes up by one, so that in M new , 2 + 2 = 5, since the second '2' means what '3' means in M old . That 2 + 2 = 4 is always true in ordinary arithm...