Jamesian Philosophy Refreshed

Recently, the physicist Sabine Hossenfelder, who maintains a wonderful blog, posted her thoughts about "infinity," its necessity as a mathematicians' tool, on the one hand, but its unreality from the point of view of the science of physics, on the other. Her point is that a physicist only regards something as real if it is necessary in order to explain an observation. Infinity, she says, will never fill that bill.   She also went a bit into the mathematics of infinity, and the fact that not all infinities are the same. The real numbers are a different sort of infinity from the integers. I used the co mment section under her blog post to ask a question that has long been on my mind.  I remember when I was a kid, many decades ago, some of us nerdy types excitedly read "One, Two, Three, Infinity" by George Gamow. Gamow covered some of the same territory you just did although I'm guessing progress has been made in the last half century or so. If I recall corr...

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, ...