Tuesday, September 23, 2014

Revisiting Infinity

"To infinity, and beyond!" ~Buzz Lightyear

Every now and then the notion of infinity is an interesting concept to ponder. It's easy to see why mathematicians have had such a troubling time with it. Take, for example, this common conundrum.

You have hotel with an infinite number of rooms. An infinite number of people show up one night because of your Vacancy sign. (Maybe it is Grandma's Marathon weekend.) So, you give them all of your infinite number of even numbered rooms, 2,4,6, ad infinitum. They check in all happy and your clerks are impressed because you still have an infinite number of vacancies, in case you get more business.

Well, obviously, infinity throws a wrinkle in things if you treat it like an absolute number.

Here's one I've toyed with occasionally. Between any two points there are an infinite number of points. If you have a line segment three inches long, and one that is six inches long, why can't you simply say 3 = 6 since both contain the same number of points?

For the Romans, infinity simply meant a very large number. Galileo later suggested that the infinite does not correspond to mathematical rules in the same way as a common integer. Correctamento, Galileo!

When I was young I used to toy with the idea of time being a "duration" like miles of road, and that between two points in time, let's say four o'clock and eight o'clock, there is an infinite amount of time just as in the number of points in a line segment. (I know that waiting can feel that way sometimes, especially when waiting to open Christmas presents.)

And so... if there is an infinity between a few hours, what about a few minutes? Or between two moments? And how long is Now? Is it infinitely small or infinitely something else? Or is Now only an illusion that occurs between the infinite past and infinite future?

This leads naturally into questions about the nature of time itself. Why do some days go fast and others slow when they all have the same number of hours?

What about space? Is outer space infinite? Or does our universe have an "outer wall" or some kind of endpoint? What would the end of the universe look like? How would you know when you got there?

And then, what about inner space? Maybe this is what Blaise Pascal was talking about when he wrote, “There is a God shaped vacuum in the heart of every man which cannot be filled by any created thing, but only by God, the Creator, made known through Jesus.”

Now for two bonus questions about infinity.

1. What did Dylan mean in his "Visions of Johanna" when he sang, "Inside the museums infinity goes up on trial"?

2. How many angels can dance on the head of a pin? This latter is answered here.

Meantime life goes on all around you. Seize it.

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