Not too long ago, it was believed that all processes were either deterministic or random.
Some things could not be predicted accurately, like the time that an atom of uranium would
decay, because they were random in their nature. Knowing what time the atom was created
or what temperature it was stored at will not help your prediction.
Other things, like where a home run will land were thought to be deterministic. If you
know how hard it was hit, the direction of the wind, the air density, etc., you can tell
where it will land. The more closely you know the initial parameters, the more accurate
your prediction will be, but a tiny deviation in the initial conditions will only cause a
tiny deviation in the final calculation.
However, during the mid 20th century some mathematicians and scientists came to appreciate
that many processes did not fit into either of these 2 categories neatly. These things
were not truly random, and many simpler systems could be described by a few simple
equations. On the other hand, they seemed to not be predictable, because starting with
the same initial conditions did not always generate the same end result. They created a
third category, called chaos, to describe these processes.
For example, imagine an equilateral triangle made of wood with a strong magnet, positive
pole facing up embedded at each vertex. suspend a magnet with negative pole facing down
by a string exactly over the center of the triangle. If you move the pendulum to any
point and then release it, the pendulum magnet will come to rest over one of the embedded
magnets due to their mutual attraction. This simple physical system can be completely
described by a few equations relating momentum, gravitational force, magnetic force, air
resistance and such, and so you would think if you measured where the pendulum was before
release you could accurately predict which of the 3 vertices it would end up with. But
Two hallmarks of chaotic systems exist in the above example. The equations describing the
system are highly interdependent, and the magnetic forces felt by the pendulum at 5
milliseconds after release are highly dependent on the forces present at 4 milliseconds
after release due to various feedback loops among the variables. This creates the second
hallmark, a system highly dependent on initial conditions. This means that a small change
in where you start the pendulum may have a large effect on which vertex the pendulum stops
at. Since it is impossible to measure all the initial parameters exactly, in some regions
the outcome of the pendulum drop is unpredictable, even though we feel we should be able
to just plug in the numbers to see what happens.
The classic example of this dependence is the weather. This is a complex system with many
interdependent variables, but in theory if you could measure the temperature, direction,
humidity, and density of all the air masses on Earth, figure in the effects of energy
input from the Sun etc., you should be able to plug the numbers into a big supercomputer
and crank out the exact weather for the whole Earth very far into the future. But we
cannot measure perfectly; even if there were a perfect problem in every cubic meter of the
atmosphere, an undetected eddy might happen in between them. And since the system is
highly dependent on initial conditions, making a small error in the inputs can lead to a
totally different outcome, so our predictions will be worthless. This example gave us the
name of the 'butterfly effect' a butterfly flapping its wings could change the initial
state of the Earth enough that it could cause a typhoon to occur or a predicted one to
never materialize. Small changes have huge consequences.
Many environmentalists worry that a lack of knowledge of this so called 'non-linear
dynamics' may cause trouble. People may not realize that small changes in the Earth's
environment now, like a .5% change in the CO2 composition of the atmosphere or a 10%
decease in the number of salamander species, can cause immense differences in what the
Earth looks like in the future.
Rob Landolfi, Science Teacher, Washington, DC
'As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.'