Message-Id: <l03102803b37250fa7fc9@[209.209.18.94]>
Date: Tue, 1 Jun 1999 02:21:38 -0700
To: pinhole@exploratorium.edu
From: Ronald Wong <ronwong@inreach.com>
Subject: math and science (inverse square)
Sarah Wise asked:
>Why do so many natural phenomena seem to follow a pattern of squares or
>inverse squares? It seems oddly coincidental that the physical data fit the
>mathematics so perfectly...
>....Examples I've thought of so far that fit this pattern are Coulomb's law,
>the Law of Universal Gravitation...
Good question Sarah and you offer some interesting examples.
The important thing to keep in mind is that this is NOT a scientific
question. It can't be answered within the framework of science. What
hypothesis could possibly be proposed in light of this question that could
then become the subject of a scientific experiment?
Although any answer to the question would be purely speculative, there is
nothing wrong in asking it. Philosophers do this all the time and so do
students. It is when they ask us such questions that we, as teachers, have
an opportunity to bring them to an awareness of the fact that there are
questions which just cannot be asked of science - giving them examples of
what are and are not scientific questions and what it is that accounts for
the difference.
Other questions that you may have been asked that fall into the same
category are:
Why do bodies have this property, called inertia, that will allow them to
move with uniform velocity, in an inertial frame of reference, whenever
there are no unbalanced forces acting on them?
Why does light travel in straight lines?
How is it that one object can, by it's very existence, reach out over vast
distances and exert a force of attraction on another (never mind
that it's strength depends on the inverse square of the distance)?
Not only is this last question unscientific but it brings to mind another
reason why some answers won't be found as a result of scientific effort.
There are many people today who believe that heavenly bodies move the way
they do because there are forces of attraction acting between them due to
something called gravity. Every mass has a gravitational field associated
with it and any mass placed in this field will experience a gravitational
force acting on it, drawing it to the source of the field.
Interestingly enough, you would be hard pressed to find a physicist today
who would agree with this.
They probably WOULD agree that for most people today using the Newtonian
concept of gravitational force to explain to them why the heavenly bodies
move as they do would be sufficient. But only in the same sense that the
concept involving the basic elements of earth, water, air, and fire - and
it's attendant cosmology and element, the quintessence - was sufficient
for explaining nature to those who lived three hundred years ago during a
time when Newton's concept of gravity was being expounded and verified.
It's been over three-quarters of a century since the deficiencies of
Newton's Universal Law of Gravity were resolved. In the process, a whole
new way of looking at things came into being. It involved something called
space-time and the notion that, by nature, all things - objects as well as
energy - follow the shortest possible path through space-time.
>From this standpoint, the planets move the way they do because, by nature,
they are following the shortest path through space-time. The presence of a
massive body like our sun distorts space-time in such a way that the
shortest path for the planets ends up being one where the path is
elliptical in shape, where a line connecting the planet to the sun sweeps
out equal areas in equal intervals of time, and where the square of the
planet's period is proportional to the cube of its mean distance from the
sun. Apparently, Kepler's Laws are a direct result of our planets pursuing
their natural paths in the curvature of space-time brought about by our
sun's presence.
Because the path is their "natural" path through space-time, no force is
required to "explain" their motion (This is the relativistic equivalent of
Newton's first law where no unbalanced force is required to explain the
uniform motion of a body in an inertial frame of reference or the earlier
Aristotelian principle that objects seeking their natural place require no
force to account for their motion).
When Einstein proposed the General Law of Relativity he effectively did
away with the force of gravity. There is no such force. It isn't required
to explain heavenly motion. The planets move the way they do "naturally" -
without any unbalanced force acting on them.
In other words, with time, the results of scientific efforts have made any
questions regarding the gravitational force or the comparison between it
and the electrical force moot. There is no force of gravity to begin with
and thus no basis for a comparison.
After 2 000 years of success, the concept involving the four basic elements
(and the geocentric universe it supported) gave way to the universe of
Kepler, Newton's laws of mechanics, and Newton's concept of gravitational
force. In less than 300 years Newton's concept of universal gravitation
with its inverse square relationship has given way to the principles of
relativity.
In a similar manner we've also found that light doesn't necessarily travel
in straight lines as it travels through space and that the concept of
inertia may involve something more than just the object itself.
Who knows what the future holds but, if the past is any guide, we can
expect our understanding of nature to continue to change in significant and
profound ways.
So, here's another point we as teachers can make when the opportunity
arises - the idea that our understanding of what nature is is constantly
changing - sometimes in profound ways.
Which leads to one last point:
There is a very strong sense of finality in the phrase, "...because that's
the way it is.". As was mentioned in an earlier post, there are occasions
when we are tempted to use it but, in light of the history of science, we
should really refrain from doing so. By it's very nature, scientific
activity continuously changes our understanding of what "is" is - making
what "IT is" even more sublime in the process.
Science courses have an important advantage over other courses of study. In
addition to lecturing to our students and performing
demonstrations/presentations for them - elements common to other courses of
study - we can add hands-on class activities and lab investigations that
differ significantly in terms of goals and procedures from other studies.
The proper mix and implementation of these elements creates an environment
which help students to remember the lessons they have learned long after
they have passed through our classrooms.
As a result, we have an opportunity to teach some very important lessons in
a way that students will remember for a very long time - namely, the kind
of questions one can ask of a given field of study are limited, the
fundamental concepts of a body of knowledge are subject to change and the
continued efforts on our part to address the questions that we are
confronted with will lead to a deeper and more profound understanding of
ourselves and the world we live in.
That's what makes science teaching so enjoyable, rewarding and well worth
our while.
Speaking of enjoyable and rewarding, have a good summer ya'll.
ron