New York Diary

How do you write a limit in html?

A couple days ago, I finished reading The Infinite Book, by John D. Barrow. Though I think overall the book was hit-and-miss, some parts were thought-provoking indeed. The two most fun games that Barrow gives us follow.

1. The problem of infinite numbers of actions in a finite amount of time. We know that the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^n) approaches 1 as n approaches infinity. But concieve of it this way (Barrow's way): a demon switches a light on after half a minute, then off 1/4 of a minute later, then on 1/8 of a minute later still, and so on and so forth (literally ad infinitum). After 1 minute has passed, will the demon have done an infinite number of tasks? And will the light be on or off (or both, or neither)?

2. The problem of infinite size in a random universe. If the universe is infinitely big (or, as an alternative statement of roughly the same problem, if it has been around for an infinitely long time), and if we don't presume that some larger intelligence is conspiring to make earth special, then must we presume there to be an infinite number of earths with an infinite number of people who are exactly like ourselves (or, in the formulation instead of the infinite age of the universe, must we presume that those people did and will exist)? This is the same idea as the monkeys on the typewriters: enough random combinations of atoms in space (letters on the page) and you will get other Earths (other copies of Shakespeare's plays). Of course, the probability of any sufficiently large number of atoms combining to form another you on another Earth around another sun in another spiral galaxy is extremely low. But if you have infinite numbers of atoms combining in infinite numbers of random ways, then even an extraordinarily low probability translates into an infinite number of instances.

There are ways around these mind-benders, to be sure. #1 doesn't take into account the relativistic effects of moving very, very fast. Eventually, you've got to be turning the light on and off at speeds that are appreciably close to c (the speed of light). In fact, the way this problem is set up now, you eventually would be turning the light on and off at speeds faster than the speed of light. This is of course impossible; as you approach c, time slows down so you're never able to exceed the speed of light (or, perhaps, reach the end of the minute?). Even if you could transend c theoretically, though (which you can't), it may not be trivial that no person could ever do this practically: you'd run out of energy from all the work and die first (or, if it wasn't a person, then whatever the power source was would run eventually out of energy), and anyhow you'd be going faster than any known body would be able to go. Maybe the universe is set up not to let us dabble in infinities in any practical way, ever. Maybe infinity has no real, non-mathematical meaning.

The ways around # 2 are rather different. We could of course posit some sort of divine selection that makes our world unique, thereby avoiding the problem of infinite earth-worlds (since the infinity-hypothesis only works if we assume that the creation of our earth was not special). Or we could presume that the universe is neither spatially nor temporally infinite. This kind of makes sense, too, even though we don't have a clue what that would mean we'd see if we looked out from the "edges" of the universe: it would not be absurd to posit that time began with the big bang, that space did too and is merely a function of our universe (or our experience thereof), and that the universe is "closed" and so will not continue to expand infinitely long and far. All these things are at least plausible. The big question here, though, is why # 2 is paradoxical or problematic in the first place. To be sure, it's a little weird to think that not only might we not be special cosmologically, but we might not even be unique (instead, living the exact same lives, with the exact same motivations, and in the exact same observable world, as someone somewhere else in an infinitely large universe). But let's say it's true. Is there anything logically wrong with that?

posted by Skay @ 9:07 PM