Fooling with infinities

Does infinity exist?

The undergraduate math major might have a minute or two of fun tossing the question around with his buddies at the campus pub. He might point out that, for example, the list of all possible integers is infinite, but so is the list of all even integers.

The philosophy major would take that factoid and spin it into an evening-long discussion of the nature of logical contradictions, or of the ways that language constrains our understanding of reality.

Undergrad philosophers love this sort of thing. That’s why you always find them off in a corner of the pub, talking to each other. No one else will bother with them.

The problem with philosophy is that in its secret heart of hearts it’s essentially a game — a logic game, a word game, or a number game. Internal consistency is the sole criterion for correctness.

Before there was empirical science, logic and words were the only games in town. And even after modern science, philosophers view the numbers of empirical data as just other kinds of logic and words. That’s what numbers are, of course, but philosophy typically ignores that the numbers of science are counters for real things, things whose existence doesn’t require our understanding or our ability to express them without contradiction. To philosophers, real things are just more words, and internal consistency is more important than existence.

I remember a long-ago debate about evolution. My skilled and sober opponent backed me into a linguistic corner out of which I could not readily free myself. When I conceded that he had, indeed, caught me in a careless language error, he smiled broadly and exulted, “See! I told you that evolution wasn’t true!”

Typically of a philosopher, at its simplest his claim was that something in the real world had been disproved because his immediate adversary had failed to win a particular rhetorical point. It was as if he really believed that deflating my argument was in some concrete way materially determinative.

And that’s the problem with philosophy, linguistics, and parts of mathematics. They deal entirely in abstractions, ideations that they confuse with real things. When you see three geese fly over, you’re seeing geese, but the philosopher or mathematician is seeing three.

So when +Plus magazine publishes an article titled “Does infinity exist?” I had two immediate and contradictory reactions. One, this is going to be fun. Two, this is going to be just another number game. Both reactions come from the same source: I was, long ago, an undergraduate philosophy major. I’ve run into this sort of thing before.

The +Plus article starts with Aristotle’s potential and actual infinities, then moves briskly to Cantor’s “countable” (1,2,3,4,5 …) and “uncountable” infinities. From there, it spins off into ever more convoluted but always entertaining complexities.

In fact, my very last undergraduate course, which I completed by mail after emigrating, was a directed studies one-to-one with the prof for whom I had worked grading first year essays. It was an arrogant scheme by someone with scarce mathematics to recast Cantor’s mathematics as a philosophical word game. The idea was inspired by Bertrand Russell’s notion that the finite series defined as “the largest integer ever conceived, plus one” was, being ever unreachable, infinite.

From that, I proposed to explore the idea that there was one unimpeachable absolute in a universe composed entirely of particulars — that being the finite list of everything in the universe. Following Russell, once the list was created, it added itself to the list, and so on and so on, into infinity and beyond. In other words, just another philosophy word game.

Whether or not there are any absolutes was an important question in the 1960’s at a Jesuit university where Anselm and Aquinas were still taken seriously. Among other things, the learnéd gentlemen in black were concerned that, were there no absolutes in a universe composed entirely of particulars, God either couldn’t exist by definition or could exist only entirely outside the universe, which in practical terms amounted to the same thing.

Many of you who were educated at public universities (or at any non-evangelical college in the last 25 years) will have trouble believing that 14th century issues like conceptual arguments for the existence of God could seriously occupy anyone’s mind in the 20th century. Believe it. I couldn’t make up stuff like this.

And although my course proposition makes little sense to me now, it was well enough tricked out in the verbose finery of the day to satisfy the requirements and complete my degree.

The point, such as it is, of this self-indulgent ramble is to repeat the assertion, found fairly frequently here, that the concrete is preferable to the conceptual.

You can’t prove anything with words because you can prove anything with words.

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