## Wednesday, December 28, 2011

### 106: Pre-Newton Nonsense

Before we start, I'd like to mention that I have finally created a
Facebook fan page for this podcast.  You can reach it through a link
at mathmutation.com, or just search for it.  If you're listening and
get a solid majority on the site who are not related to me.  Anyway,
now on to today's topic.
We're all familiar with the basic concepts of classical physics
that they teach us in high school:  the law of intertia, where objects
in motion stay in motion unless acted upon by a force; the law of
acceleration, where acceleration of an object is proportional to a
force divided by its mass; and the law of reciprocal actions, where
every action has an equal and opposite reaction.  These laws lead to
nice,  simple equations that describe the actions of motions and
forces on objects, such as "f=ma", force = mass times acceleration,
and "s=1/2a(t^2)+Vit", describing the distance travelled by an object
under acceleration.  Now these laws are known to be not quite right,
as proven by Einstein early in the last century, but they are still
really good approximations of the real world, and mathematically
correct under some basic assumptions about the universe.
What did people believe before the scientific revolution led by
Galileo and Netwon?  Since the 4th century BC, the dominant theories
of physics in the Western world were those written by the Greek
philosopher Aristotle.  The "law of natural places" said that every
object had a natural altitude where it wanted to rest-- a special-case
rule since gravity was not understood as a general force.  The "law of
rectilinear motion" said that any force would cause motion at a
constant speed.  The "velocity density relation" said an object's
velocity was inversely proportional to the density of a medium.
Vaccums were thought to be impossible, as nature abhors a vaccum;
objects would immediately move into any vaccum temporarily created,
and a true vaccum would be impossible because all objects would move
into it at infinite speed.  Thus the space between planets must be
filled with a special substance known as an ether.
"continuum principle" stated that matter could not be made up of small
atoms, because then by definition there would have to be a vaccum
between the atoms, and his other theories showed that this could not
be possible.  But he did believe that all objects were made up of four
elements in proportion:  earth, air, fire, and water.  Those of you
who play Dungeons and Dragons, or are at least geeky enough to have
played it at one point, are probably familiar with this set of
elements.  The elements were intimately connected with his
gravity-like law of natural places: Fire would seek its "natural
place" in the air, while the other elements wanted to sink downward.
We are lucky enough to have an atmosphere at our altitude because of
its mix of air and fire elements.
These theories led to a few problems.  One of the most obvious
ones was how to explain the use of a bow and arrow.  Once the arrow
leaves the bow, there is no more force on it, so how does it remain in
motion?  Under Newton, we know that its natural tendency is to
continue its motion until an external force slows it down.  But
Aristotle had to come up with a convoluted explanation:  the rapid
motion forward of the arrow created a temporary vaccum, and air
rushing into it pushed the arrow forward further.  A similar
explanation showed how a cat can leap into the air.  Now I know my
late cat Rocky could jump pretty fast when trying to destroy our
ceiling light fixtures, but I don't think he created a vaccum.
So, what were the set of equations that Aristotle's laws led to?
Well, back in Aristotle's day, equations were not seen as an integral
part of a physical theory.  Some of his principles might seem to lead
to equations:  for example, when he states that velocity is inversely
proportional to the density of a medium, one can write it as v=k/d,
where v is velocity, k is some constant, and d is the density.  But
the primary reason why Aristotle's theories weren't usually expressed
as equations was probably that they were so far off from reality, that
any attempt to precisely assign values based on real-world
observations and plug them into equations based on his theories would
inevitably fail.
This didn't really bother Aristotle though:  as typical for his
day, he was notoriously unconcerned with confirming his theories
through actual experiment.  Observations were just a basic starting
point, and then he derived the rest of his theories through pure
philosophy.  In some areas, this led to pretty ridiculous conclusions:
for example,  another of this theories said that women have fewer
teeth than men.  This probably tells us something about the type of
women that Aristotle liked to associate with.
Now we shouldn't be too harsh on Aristotle-- it's not like other
people in his day were any more scientifically rigorous, and the
proposition that we should evaluate and document the laws of nature
was itself a major contribution.  He even got a few things almost
right, like his desciptions of buoyancy in fluids or of air resistance
for falling objects.  But we should also be grateful that future
scientists like Galileo and Newton realized that we needed more
careful observations to create truly valid physical theories, and that
later developments of modern mathematics enabled us to describe these
theories well enough to reproduce and to understand their
implications.
And this has been your math mutation for today.

• Newton's Laws at Wikipedia
• Aristotle's Laws at Wikipedia
• Article on Aristotle's Physics
• Another article on Aristotle's Physics