Friday, November 27, 2009

Even bigger, even smaller

I have a suspicion that humans are not really built to think about or deal with any kind of infinity.  "Being present everywhere" or "going on forever" seem fairly easy to imagine but I think that is just because we can't do so and our minds just conk out.  Adults know that they weren't really happy "ever after".  They were mortal and eventually ceased to be.  There were holes in the good times where things weren't so happy.
I read a little about Georg Cantor and his work with mathematical infinities.  It would be easy to assume anything that is infinite is just that: endless.  Sometimes, it is fun to talk about the biggest number with a child who is capable of understanding.  "What is the biggest number?"  "A million."  "What about a million plus one?  Would that be bigger than a million?  Ok, a billion.  What about a billion plus one?  That would be bigger than a billion.  Eventually, we all see that given strength and attention, we could conceptually keep adding one to anything, thereby making something big even bigger.

He also showed that the counting numbers 1,2,3... can go on for as long as we are capable of listing or imagining them but that the points between any two numbers in the series make a bigger infinity than the counting numbers.  The idea is related to ideas of the ancient Greek Zeno. Once we start thinking of fractions, we see that we can always conceive of halving any segment however small.  We can see that the halving can go on forever without ever reaching the next number in the counting series.  We can think of a point halfway between 2 and 3 and then a point halfway between 2 and that point, always taking smaller and smaller bits.  There are any number of points between 2 and 3 even though there are no integers there. 

Zeno said that Achilles, a renown warrior, could not possibly beat a tortoise in a race if the tortoise got any head start.  He pointed out the infinities, that Achilles would have to cover half the distance in the tortoise's lead.  Then, he would have to cover half the remaining distance.  While the warrior was covering those distances, the tortoise would be moving on.  Poor hero!  Can't even catch a turtle!

Popular Posts

Follow @olderkirby