Jamesian Philosophy Refreshed

An odd campaign emerged in the 19 th century – the effort to reduce numbers to sets. This brings us back, though in a rather different light, to the question with which we opened: what is a number? The simplest possible kinds of numbers are the counting numbers, those we tick off from one to ten on our fingers. There can be no doubt that they are numbers, they are the paradigm, they are at heart what we mean by “numbers.” Thus, whether anything else is a “number” is, if you will, a question about the family resemblance between the counting numbers and that other sort. Whatever else is a number: three is a number! “Is pi a number?” means, “does pi have a close family relationship to three?” “Is i a number?” means the same.    But in the 19 th century, for the first time, mathematicians and logicians started wondering, not idly but in all workaday seriousness, what it means to call three a number. They decided that numbers were special sorts of set. The...

The story of another candidate for numberhood, the infinitesimal, is even stranger than the story of the irrationals or the imaginaries. The infinitesimal is the limit of a process, where the method stipulates that the end of that process can never be reached. Consider: can a single mathematical point have a slope? Our intuitive answer, trained by Euclid, is: no. A point is pure position. It isn’t a line, it isn’t even a tiny part of a line, so it can’t have a slope! Can there be a smallest possible line?   How short can a line get and still have a slope! That’s a question that Euclid taught us not to ask. It is akin to suggesting that we can have two adjacent points. If we could have two adjacent points, then they would presumably constitute the smallest possible line and they would have a slope. But we can’t. Between any two points, properly speaking, infinity of other points can fit. In the seventeenth century it was common for the mathematicians at its cuttin...