Wednesday, September 26, 2012

Infinity and Atomism

Now this is interesting: one of the ancient arguments for the existence of indivisibles (e.g., "atoms") is that infinite divisibility implies an actual infinity.  Suppose you can infinitely divide both a mustard seed and a mountain.  So they are both composed of the same number (an infinity) of parts.  Therefore, they are both the same size.  Or, by the same reasoning, a part of a mountain is the same size as the whole mountain. 

This isn't considered a problem in mathematics, where a set is infinite just in case one of its subsets (parts) is the same size as the parent.  (For instance, the set of integers and the set of squares.)  But it's an open question in physics whether there is an actual infinity. 

Some evidence seems to imply it.  Inverse square laws (like Newton's equation for gravity or Coulomb's law for electrostatics) can evaluate to infinity.  Some solutions of Einstein's equations (e.g., some black holes) evaluate to infinity.  If the topology of the universe is flat (analysis of the cosmic background radiation thus far suggests it is), then space is an actual, physical infinite.