Jamesian Philosophy Refreshed

Yesterday's post as gotten me into one of my rare stern-logician frames of mind, so I think I'll stick with that today. I wonder if I've ever discussed Cantor's "diagonal" proof in my blogging. If not, I'll make good on that now. Cantor refuted the common idea that "infinity" is a single all-inclusive category. Once we get beyond enumeration we've gotten to "infinity" and there is nothing more to be said. Or, at least, so we're inclined to think. Yes there is, though, Cantor replies, much more to be said. Because for example not every infinity is equal to every other. Some "trans-finite" numbers are bigger than others. The number of real numbers is a larger transfinite number than the number of integers. Why? Well, diagonals! The question is -- could we in principle create a one-to-one correspondence between every real number and every integer that would allow room for every one of each to show up, eventually, ...

Continuing yesterday's thoughts. Back in 1989, two chemists claimed to have created a simple table top apparatus that would work at room temperatures and that would create excess energy in amounts that, the chemists contended, had no explanation other than the incredible one: atoms were fusing. The scientists were: Stanley Pons and Martin Fleischmann. They were both sufficiently distinguished that the claim could not be dismissed lightly. Pons had a Ph.D. from the University of Southampton; Fleischmann from the Imperial College, London. That's Fleischmann in the photo that goes with this text, holding part of the apparatus, which was a little more complicated than Dogbert's. Their claims set off a frenzy of speculation about what this would mean. Some thought it would be bad news -- that the human race could not be trusted with the prospect of unlimited energy this implied. The more usual view was that this was great good news -- an apparently insatiable hunger ...

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...