Infinity is not quite an
endlessly useful concept, but it is a very useful one. That doesn’t mean that there are actual
infinities. Tom
Hartsfield has some mathematical reasons for rejecting infinity as
something that can describe the real world.
I am very much inclined to agree.
My favorite example is not mathematical but logical.

Suppose that there is a grand
hotel with an infinite number of rooms (presumably serviced by an infinite
number of modestly paid workers). The
hotel is completely booked on Labor Day weekend, when another guest shows up
and requests a room. No problem! The hotel manager has every guest move to the
next highest numbered room, so the guy whose wife kicked him out of the house moves
from room one to room two and the meth addict in two moves to three, etc. ad
infinitum. Given the infinite number of
rooms, each guest will always be able to move up one. So the hotel manager can accommodate the new
guest. Likewise, the manager could have
each guest move to the next highest odd numbered room. That would free up an infinite series of even
numbered rooms. Let’s hope he has plenty
of towels.

So: here we have a hotel that
is completely booked and yet has room for an infinite number of guests. That is a logical contradiction and so no
such hotel can actually exist.

I am pretty sure that that is
right. Infinity has intrigued and
troubled philosophers since before the time of Socrates. The most famous example are the marvelous
paradoxes of Zeno. Zeno was a monist who
wanted to show that actual change (in this case, motion) involved logical
contradictions. Thus reality was one and
unchangeable.

The simplest of his paradoxes
goes like this: Achilles wants to run across the stadium. That looks easy enough. Half way across, however, it occurs to him
that before he can complete his run he must cover half the remaining distance
(one quarter of the stadium field).
After he has done that, he will have cross the next eighth, etc. ad
infinitum. No matter how fast he runs,
he will always have some remaining portion of distance to cover. He collapses in existential despair.

In fact, Achilles has every
reason to believe he can make it. If he
runs at a steady pace, he is covering each portion at half the time. So if it takes him one minute to cover the
first half, what will be true at two minutes?
He will be over the finish line.
One half plus one quarter plus one eighth, etc., is how much shorter
than one? Shorter by an infinite amount,
one might say. Whatever theoretical significance
that might have, it has no practical significance at all. Shorter by an infinite degree is just as
long.

Infinity in either direction,
small or large, isn’t real.

## No comments:

## Post a Comment