Jamesian Philosophy Refreshed

I happened onto a logic blog recently and, before my eyes burnt out with the beauty of the abstractions involved, I learned a new term. ZFC Set Theory. The first two letters, "ZF," abbreviate names, Zemelo and Frankel. Both big names in the history of logic -- I need say no more of them now. The C stands for "choice" as in the "axiom of choice." Now, that sounds important. What choice can there be in logic? Well, the axiom of choice involves mutually disjoint nonempty sets. Take for example two such sets: the set of all teacups and the set of all named bodies of water on the planet. The axiom of choice means that for any set of such sets, it is possible to create a transversal set, containing exactly one element from each. Simple enough in this case: I can simple designate one set as consistent of teacup A plus Lake Erie. Another set consists of teacup B plus the Atlantic Ocean. And so forth. Why is this important? Well, Bertrand Russell in 1904 ...

  To understand this rigorous proof that there is no one-to-one correspondence betwee integers and points on a line, suppose we encounter a ChartMaker (CM) who wants to try to create this one-to-one correspondence.   He presents a chart that looks like this: Integer                                     Point 1                                                                                 ...

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...