Gregory Landini is talking at our Logic Seminar next week about Frege and Russell on cardinal numbers. Since our students tend to know a lot more about Frege than Russell, we had a preparatory session on Russell last week, in which I got a chance to show off my stunning historical ignorance. But it was fun to re-read (after a long time) the opening chapters of the Introduction to Mathematical Philosophy. These were, as much as anything, the pages that got me interested in philosophy and the foundations of mathematics when I was a maths student.
Fun to re-read, but also oddly very disappointing. Chapter II starts with stirring words which I well remembered: 'The question "What is a number?" is one which has been often asked, but has only been correctly answered in our own time' (meaning, of course, in 1884 in the Grundlagen). But I'd quite forgotten this passage, later in the same chapter, where Russell writes
We naturally think that the class of all couples is different from the number 2. But there is no doubt about the class of all couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematic number 2 which must always remain elusive.So Russell's stirring words are misleading: he isn't after all claiming to have located, thanks to Frege, the one true metaphysical story about numbers (as classes of classes). It's rather that here we have one way of replacing a problematic entity with something clear and sharply defined that can do the job. And then, of course, Russell is cheating. There isn't such a thing as the class of all couples. Far from there being no doubt about it, he doubts it himself: his official story has a type hierarchy, with classes of couples at each level of the hierarchy above the bottom two. The seductive clarity of the opening chapters IMP is sadly only superficial!