This post is going to be varying slightly from the standard
format of these posts. Instead of focusing on a specific problem, the focus will be shifted toward a number of problems and paradoxes concerning infinity.
Infinity is not a number; it is a concept. So when infinity gets involved in
problem, it sometimes causes the problem to break. This post will
cover four of these problems with Infinity.
The first problem
concerns the Hilbert Hotel. This paradox was proposed by David Hilbert. First,
imagine a hotel with an infinite number of rooms. In every room there is a
mathematician. The problem starts out with a new mathematician walking in, and
wanting a room. Well obviously the hotel is full, but the manager is smart
enough to come up with a solution. He asks every mathematician to move to the
room next to them, so room one moves to room two, and two moves to room three,
and so on. So now room number is empty. Since there is an infinite number of
rooms, there is never not enough room. In fact, even if a infinite number of
mathematicians walked into the hotel they could all find a room. This goes
against the initial understanding that the hotel is full, but at the same time
there is always room.
The second
infinite paradox is called Gabriel’s Trumpet. The trumpet part comes from the
general shape of the trumpet. A hollow cone like shape that tapers off, except
this cone shape tapers off to infinitesimally small. A volume of this cone can
be found, but yet the cone would have an infinite surface area. If there was a
mathematical paint that had no thickness between molecules an infinite amount
of that paint could be poured into the cone, but at the same time the volume of the cone
can be calculated with calculus. This is the paradox, how can something take up a
given space, but yet have infinite surface area?
The third
paradox concerns a dart board. Imagine a dart with a sharp point, that is in
fact a single point thick. The problem arises when the dart is thrown at the
dart board. The question is, if there is a hundred percent chance that the dart
hits the board, what is the probability of hitting a certain singular point on
the board? Since there are an infinite number of point in the surface of a dart
board, the probability of hitting a certain point cannot be great than zero of
the sum of probabilities would be infinite. Also the probability must be more
than zero because if it was zero then the dart would not hit the board. The paradox
is that a probability of hitting the board cannot be determined even if there
is a hundred percent chance the dart will hit the board.
The final
paradox concerning infinity in this post, is about doubling money. Suppose
there is a new casino game that has just arrived. The game is centered around a
coin, and a pot of money. The pot starts at one dollar, and if a heads is
flipped the money in the pot doubles. If a tails is flipped then the game ends, and the pot is collected.
To the average individual it would make sense to put in a few pounds at a time,
but mathematics has a different answer. Mathematics says to put as much money
in as possible. This is because the sum of every outcome
tends towards infinity. The paradox is that like any other casino game there is
a chance to lose it all, but at the same there is a mathematical guaranteed chance of walking
away with an infinite amount of money.
As pointed
out in these four problems, the concept of infinity creates some problems with
mathematics. One of the biggest problems is getting around the idea of
infinity. In the universe there is not anything that is infinite, except maybe
the space of the universe itself, but that is another paradox in itself. In all
of these problem there was something that could not exist in the real world.
There are not any infinite hotels, infinite surface areas, and people hit
dart boards. The answer is not that infinite needs to be removed, but rather
that mathematics as a whole needs to work with infinity in another way. When
some new idea is introduced to mathematics there is always a fight to hold on
to what is known, and not to change. That will simply limit math as a whole. If
no one ever stepped outside the box there would not be the complex number, or
zeta functions, or countless other aspects. The point is infinity is not
something in mathematics that cannot be dealt with.
No comments:
Post a Comment