What is a number? Is it some Platonic entity floating outside of space and time that we somehow come into communion with? We’ll be following up our foray into analytical philosophy with Frege with some Bertrand Russell: specifically his Introduction to Mathematical Philosophy (1919), which is the much shortened, non-technical version of his famous Principia Mathematica(written with Whitehead). Frege and Russell agree that numbers and other mathematical notions are reducible to logical operations. Russell, beyond this, sees logical truths as a matter of derivation from definitions: not self-evident truths, and not all from the law of non-contradiction, but by basics that we have to discover through logical analysis, and we try to push the analysis back as far as possible, and wherever possible make mathematics into specific cases of more general principles, so, e.g. properties of sequences of numbers are seen as special cases of sequences of objects.
We’ll focus on chapters 1-3, where he recounts Frege’s derivation of the concept of number (he says these pick out sets of things in the actual world: the number 3 is identical to the set of all trios, for instance, where “trios” are defined without explicit use of the number 3 or any other number), and then chapters 13-18, where he deals with some potential problems with this definition (e.g. ch. 13 asks what happens if there are a finite number of things in the world: then some high number would end up equaling the empty set), giving a crash course in symbolic logic (in ch. 14 and 15), giving a quick account of his theory of descriptions (as discussed in our Frege and Wittgenstein episodes) reducing (in ch. 17) the notion of a class or set itself to more fundamental logical notions (i.e. propositional functions), and (in ch. 18) giving a summary account on the relation between mathematics and logic (i.e. that there’s no line to be drawn between the two).