Friday, July 29, 2005

Infinite Jest: Physics and Math Grapple with Infinity

This article in today's Wall Street Journal works for me on so many levels, I think I'm going to smile about it all day.

Definition of Infinity Expands for Scientists and Mathematicians

Basically, infinity throws science for a loop. A sampling ...

If thinking of infinities makes your head spin, you're in good company. Georg Cantor, the early-20th-century mathematician who did more than anyone to explore infinities, suffered a nervous breakdown and repeated bouts of depression. In the 1930s, some fed-up mathematicians even argued that infinities should be banned from mathematics. Today, however, infinities aren't just a central part of mathematics. More surprising, says cosmologist John Barrow of the University of Cambridge, England, in his charming new tome, "The Infinite Book," scientists who study the real world are having to take infinities seriously, too.

.. and ...

To mathematicians, "equal" means you can match the elements in one set to the elements in another, one to one, with nothing left over. For instance, there is an infinite number of integers: 1, 2, 3, 4 . . . . There is also an infinite number of squares: 1, 4, 9, 16 . . . . You can match every integer with a square (1 with 1, 2 with 4, and so on), so the two sets are equal, as long as you never stop matching. But wait: Every square also belongs to the set of integers. That suggests that the set of integers is larger, since it contains all the squares and then some. Surely there are more integers than squares, right?

Actually, no. Before his breakdown, Cantor asserted that if the elements in one infinite set match up one to one with the counting numbers, then those infinities are of equal size. The infinity of squares and the infinity of integers (and the infinity of even numbers) are therefore equal, even though the infinity of integers is denser.

... and, my favorite ...

Decimals, however, are different, mathematicians say. There is an infinite number of them, too, but this infinity is larger than the infinity of integers or squares. Even in the tiny space between zero and 1, there's an infinite number of decimals with no certainty as to what comes next. What comes after .1, for example? Is it .11 or .2?

The fact that the Almighty (who is infinite Himself, by the way, in His being, wisdom, power, holiness, justice, goodness and truth) placed us in an infinite universe to noodle on an infinite numbers system (there are an infinite number of decimals between the number 1 and 2!!!!) should shake any atheist to the core.