Background on the "Arithmetization of Mathematics"
Russell says that all of mathematics has been reduced to arithmetic, and so the reduction of arithmetic to logic reduces all of mathematics to logic. He is, in referring to the arithmetization of mathematics just echoing a common sentiment of his day, which is why he doesn't go into detail. Here is the basic idea: We start with arithmetic: the theory of 0,1,2,... . We introduce the integers (positive and negative numbers) as pairs of numbers with the following equivalence relation: (a,b)=(c,d) if a+d=b+c (that is, if a-b=c-d, but that won't do, since a-b might be negative). Thus, -1 is (0,1). We now introduce rational numbers as pairs of integers under the equivalence relation a/b=c/d if ad=cb. Finally, we introduce real numbers as infinite sequences of rational numbers. For example,
.
We can introduce not only the arithmetic of the real numbers but also the notions of function on the real numbers, derivative, integral, and so on, base on arithmetic (and some logic).
Thus, Russell thinks that if he can move one step further back, and introduce the natural numbers in terms of logic, he will have succeeded in reducing all of mathematics to basic logical principles. The thesis that all of mathematics is logic is known as logicism. It is generally held that both Frege and Russell are logicists. The defining characteristic of logicism, as it is typically used, is not what I just said above, but that it is the important common feature of the views of Frege, Russell, and their followers. Russell tends to eliminate mathematics and its objects in terms of mathematical principles, while Frege tends to use logical principles to show us what those objects are, without any qualms about their reality. There seems to be an immediate problem with the idea of reducing mathematics to logic. Mathematics appears to be about objects, while logic is supposed to be so general that it doesn't presuppose the existence of any objects of any ordinary sort. Thus, to be a logicist, one needs some kind of special "logical objects." For Frege, the subject matter of logic is concepts, and concepts have extensions, which are special logical objects. He can therefore prove that the numbers, which, inevitably, turn out to be particular extensions, exist. (The term "extension," opposed to "intension," is a term derived from medieval logic, that Frege and Russell use in very different ways.) For Russell, logic is about propositional functions. Just as 2x indicates the doubling function, for Russell, 'x is a philosopher' indicates the function that takes, for example, Socrates to Socrates is a philosopher, that is, to true. For Frege, concepts have extensions, for Russell, propositional functions determine classes. Thus, the propositional function x is a philosopher determines the class of philosophers. What is a class? A class, at base, is determined by its members (extensionally), not by how it is conceived (not intensionally). The class of people is the same as the class of featherless bipeds. A class, Russell says, can be given in two ways: by listing its members or by giving a propositional function that determines it. The list method only works for small (at least finite) classes, and so the propositional function method is more fundamental. Aside: Russell uses class, set, extension, manifold, aggregate, collection and some other terms as if they are interchangeable. They aren't: Frege objected that the class of trees in the forest determines the forest but the extension of trees in the forest determines the trees. Cantor's notion of set is one of a collection determined by enumerating its members, and if there are a lot, that just means that people can't be the ones doing the enumerating. He knows that not every class is coextensive with a set, something that deeply surprised Russell, and has no interest in classes. Riemann, Cantor's teacher, introduced the term manifold, though only in mathematical contexts, but it is more closely allied with sets than with classes. The terms collection and aggregate are generally today reserved as neutral or intuitive terms. Russell never seems to realize that there is any option other than his classes. Now it is a matter of five minutes to transmogrify Frege's numbers into Russell's. A number is a class of classes. The number two, for example, is the class of pairs. In general, a number is a class that contains all and only those classes that are equinumerous with one of its members. Frege's extensions are supposed to be meaning like entities associated with concepts, accessible to anyone who has the concepts. Thus, Frege's two is supposed to be the real two that we ordinarily use, more carefully explained. Russell's two (the class of pairs) is a replacement for the ordinary ill-formed notion of that number, one that recommends itself for purposes of philosophical and mathematical clarity. -- ProfessorShaughanLavine? - 19 Jan 2005