What is "ultrafinitism" and why do people believe it? I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist.  Could someone make this a little bit more precise?  Are there good reasons for taking this point of view?  Can you actually get math done from that perspective?

Greg Egan has some fun with this idea in one of his best short stories, "Luminous" (published in the collection of the same name).  A pair of researchers are exploring an apparent "defect" in mathematics:

"You still don't get it, do you,
  Bruno? You're still thinking like a
  Platonist. The universe has only been
  around for fifteen billion years. It
  hasn't had time to create infinities.
  The far side can't go on
  forever-because somewhere beyond the
  defect, there are theorems that don't
  belong to any system. Theorems that
  have never been touched, never been
  tested, never been rendered true or
  false."

Terrific stuff!
