Wednesday, October 03, 2007

To infinity -- and beyond

Cindy asks: "Does the concept of infinity bother you? I'm not kidding....I really am interested....Does it bother you? The thought that stars go on and on and on and on....How do you really get your head around that?"

Good question, Cindy. The greatest philosophical and mathematical minds of all times have wrestled with this question-- and still do.

It is neat that we have invented ways to deal rigorously with the concept of the infinite, within the context of mathematics. What a beautiful thing, for example, is the Fundamental Theorem of Calculus, or, as you mention, the summation of an infinite series.

But no matter how you cut it, the concept of infinity ends in paradox. Here's a little teaser. How many points in any finite line? An infinite number, of course. Now imagine two concentric circles, of radius 1 and radius 2. The outer circle is twice as long as the inner. A radius that passes through any point on the inner circle must intersect exactly one point on the outer circle. And any radius drawn from any point on the outer circle must pass through exactly one point on the inner circle. So you have two lines of different length each with the same number of dimensionless points! Two infinities that are simultaneously different and equal.

The Greeks worried about this little problem, and others like it. So did Galileo. So did such great modern mathematicians as Cantor and Hilbert. So if you and I are befuddled by paradoxes of the infinite, we are not alone.

What about an infinite universe? Do the stars go on forever? It is difficult to imagine space going on forever, but -- as the Greeks observed -- it is impossible to imagine space coming to an end. If space has a boundary, then what lies beyond? The same can be said for time.

General relativity provides a way to mathematically describe a finite three-dimensional universe without a boundary, but only by imagining space curved in a higher dimension -- which pushes infinity into another realm, sort of like the way Plato and Augustine "solved" the paradox by invoking an infinite God.

Are there an infinite number of stars and galaxies? Who knows. Take a look at the Hubble Ultra Deep Field Photo (above) and the difference between a finite universe and infinite universe doesn't make a lot of practical difference. Either way we can say with Pascal: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there...and now instead of then."

Each of us, like Pascal, will respond to the actual or practical infinities in our own way. Some with fear. Some with astonishment. Some with exhilaration. Some with denial.

(Note: Go out tonight and hold up two crossed sewing pins at arms length against the night sky. The area covered by the intersection of the pins is equivalent to what is imaged in the Hubble Ultra Deep Field Photo. In what direction? In any direction. Remember, what you are looking at is not stars, but galaxies -- each galaxy containing hundreds of billions of stars. Infinite? Does it matter?)