Tuesday, January 27, 2009

Big Infinity and Little Infinity

The distinction between what I call "Big Infinity" and "Little Infinity" has been on my mind for a while, so coming back to it now as a starting point for postings seems particularly appropriate. It happens to be pretty central in some of the popular philosophical inquiries these days, which means it's worth getting a clearer understanding of what we mean by "infinity".

What does it mean for something to be infinite? You might answer that it has something to do with endlessness or eternity; perhaps it relates to time or the size of the universe somehow. It seems to me that my very first thought about infinity is "something that goes on forever" or maybe "an amount that cannot be quantified". Either way it seems to have a suggestion of magnitude within it.

"Big Infinity" is the kind of infinity that tries to suggest something that is unbounded on the ends. Infinite space means means space without a boundary at which it no longer continues. If you recall your High School Geometry lesson, you might think of Big Infinity in the definition of a line. Unending in both directions, right? It passes through all of the points that lie on itself.

So it's big. Really big.

In ancient times, the philosopher Zeno is attributed with an interesting paradox, also called the paradox of the arrow. Zeno said that an arrow fired at a target could never actually hit it because it would always cross half of the distance in the approach. By continually halving the distance the arrow would become incrementally closer but never actually hit its target.

This is something like "Little Infinity". Think of all of the numbers between 0 and 1 (0.1, 0.01, etc.) and you have a sense of what Little Infinity represents. The infinite amount of space between two defined boundaries. Tiny differences, minuscule differences, but differences nonetheless.

Start by thinking about bounded and unbounded infinity then. How do you percieve them? Do you believe in them at all? Do you think one makes more sense than the other? What does it mean for something to be infinite?

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