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Shortly after 1633, when the Catholic
Church placed Galileo's "Dialogue on the Two Chief World Systems" on its Index
of Prohibited Books, Rene Descartes (1596-1650) wrote his famous "Discours de la
Methode", in which he laid out, step by step, his method of reasoning out
problems in philosophy, in natural philosophy (as Science was called then) in
particular. He intended to transform philosophy from the analysis of ancient
texts to the application of reason to the solution of problems: he wanted to
restore the practice of philosophy from an overweening fascination with the
Ancient Greeks to a renewal of what the Greeks had actually done.. He acted to
change the basic assumption behind philosophy from "The Ancients knew
everything", to, as we Americans like to say, "There's a reason for everything."
Descartes took the application of
doubt as the first part of his method, refusing to accept statements about
Reality purely on trust. So radically did Descartes doubt that he doubted not
only other people's reports of what's real, but he also doubted the evidence of
his own senses (this was, of course, a purely philosophical exercise; Descartes
did not allow his method of doubt to interfere with his daily life). At the end
of that exercise Descartes had in mind only one fact about Reality that he could
not doubt -- his own existence; after all, he reasoned, he must exist in order
to doubt his existence. We usually express that conclusion in the Latin
statement, "Cogito; ergo, sum." (I think; therefore, I exist), though Descartes
wrote the original in French.
Using that conclusion as a
self-evident axiom, Descartes attempted to deduce the laws of Nature.
Unfortunately, he did not apply the rules of reason as strictly as he should
have done (in some ways he still reasoned as the Schoolmen had done, giving
words more power than they actually have), so his results did not match Reality.
In fact, Isaac Newton showed that they did not mimic the workings of the real
Universe. In his Principia, in which he laid out his three laws of motion and
then worked out the law of gravity, Newton showed that Descartes' vortex theory
of planetary motion contradicted Johannes Kepler's third law of planetary
motion. But Johannes Kepler had obtained his laws by induction from actual
observations of the planets' motions interpreted through Copernicus' new model
of the solar system. Because, as Bacon required, observation trumps speculation
every time, natural philosophers discarded Descartes' theory and, with it, his
Rationalist approach to science.
However, over three centuries have
elapsed since that time and scientists have learned a great deal more about the
nature of Reality. Perhaps the time has come to us to try again to create a
Rationalist physics of the kind that Descartes sought. I will confess here that
the reasoning in this essay seems weak in places, so don't take it too seriously
until you have more evidence to support it.
I begin where Descartes began: I
exist. I now expand that axiom via the concepts of set theory: I am a member of
the set of all things that exist (and we have defined that particular set to
comprise what we have named "The Universe"). Am I the only member of that set,
as a solipsist would claim? I firmly believe that I am not, but I can't see any
proof that would verify that belief. However, I don't need such a proof, because
my belief is irrelevant to what I am trying to accomplish. What follows will be
valid whether I alone comprise the entire Universe or whether I am merely one
small piece of a greater whole. Let's begin with Descartes' cogito. I know that
I exist because I know that I think. But what do I think? What passes through me
that I call thought?
My thinking comprises a continuous (so
far as I can tell) series of images, most of which appear to give me information
about a world that exists outside me. Can I say for certain that such a world,
called Reality, truly exists? How can I know the answer to that question?
My thinking has an aspect that may
help us. As each thought passes through me I pass a judgment upon it. I decide
whether the thought feels good or bad to me. Hearing Bedrich Smetana's "Vltava"
(The Moldau) gives me great pleasure and makes me feel good. Seeing pictures of
Nazi concentration camps makes me feel bad. To the greatest extent possible to
me I listen to Vltava far more than I look at pictures of Nazi death camps and I
make such an effort in order to satisfy what I call my desires.
In order to satisfy my desires I seek
to maximize the good thoughts that pass through me and to minimize the bad
thoughts that pass through me. But if I exist alone, if I comprise all that
exists, then I must create all of my thoughts. And if I create all of my
thoughts, then all of those thoughts must conform to my desires. Thus, I can
infer that if I comprise all that exists, then I will have only good thoughts.
But I do have bad thoughts. I have
thoughts that conflict with my desires. Because they do not conform to my
desires, these thoughts do not come from me; therefore, they must have come from
something not-me. But in order to provide me with thoughts that something not-me
must exist, so I infer the existence of a thing or things other than me.
To sum up that proof by contraposition
we have the following syllogism: P implies Q. We have not-Q; therefore, we must
have not-P. If I am all that exists, then I create everything. I do not create
everything; therefore, I am not all that exists.
I now assert that I cannot be separate
from that something not-me. If I were separate from it, then it could not affect
me. Thus I must be part of that something that gives me thoughts (percepts). I
call that something Reality. And I call the thing reflected in the thoughts that
Reality gives me the Universe. Now I want to know what I can say about that
Universe, what I can assert about it, put to a proof, and verify as true to
I can assert that the Universe
comprises more than one object. As proof I offer my knowledge that the Universe
comprises at least two objects - me (who has thoughts) and not-me (which gives
If more than one thing exists, something must exist to distinguish those things; otherwise, they would exist as the same object. What can I say about that distinguishing something?
1. It must clearly distinguish not-me from me.
2. It must allow not-me to give thoughts to me.
3. It must not give me thoughts itself. It must exist purely as a medium for the conveyance of thoughts as percepts.
4. It must accommodate all objects and all changes in distinction that actually exist.
We must thus have a void continuum
with one simple property - extent. Thus we must have space, in which all things
My analysis of my thoughts tells me
one more thing about the nature of Reality. I know that I have a thought and
another thought and another thought and so on. I can call having a thought an
event, something that happens. But I can distinguish thoughts, one from another,
and I can thus distinguish events, one from another. We can call what
distinguishes events from each other time. As one wag put it, time prevents
everything from happening all at once.
We thus infer that we exist in a
Universe that comprises objects that exist in space and act out events in time.
Further, we can infer that an object can occupy one position in space at one
instant of time and then occupy a different position in space at another instant
of time. If Reality did not allow that statement to come true, then events would
not occur. So Existence has so structured Reality that objects can have motion.
After a bit of introspection, I can
state as self-evident fact that two of the properties of my existence are extent
and duration; that is, my existence is associated with a span of space and an
elapse of time. Objects and events, either out in what appears to me to be a
world external to me or in my own thoughts, require those properties to display
the differences that distinguish them one from another, one of the more
important of those differences being motion. I now have two possibilities:
either space, time, and motion exist entirely within me or they exist at least
partly outside me. If space, time, and motion exist entirely within me, then I
am the Universe, all that exists, and what follows applies to me in that role.
If, on the other hand, space, time, and motion exist at least partly outside me,
then that part comprises an element of the set of all things that exist separate
from me: what follows applies to that element and applies to me only insofar as
I exist as a body within that element.
It's important to notice that space,
time, and motion exist entirely within the Universe and that absolutely nothing
exists outside the Universe; indeed, there is no "outside" outside the Universe
(though we may continue to use it as a convenient fiction) because there is no
space outside the Universe within which such a relationship can be manifested.
As Gertrude Stein once said of Oakland, "There's no there there." We may thus
say, speaking purely metaphorically, that the Universe manifests its existence
within a frame of absolute nothingness. If we try to imagine viewing the
Universe from outside, much as we imagine seeing Earth from space, we are thus
obliged to fail. In such an attempt to see the Universe as a whole, then, we
come to understand that it can have no position or extent, no instant or
duration, and no motion. The very point of view that we are attempting to
imagine has no existence.
Now we seem to have a contradiction with our self-evident axiom that space, time, and motion exist within the Universe. We want to figure out how Reality is structured to resolve that contradiction. Let's begin with motion. I will assume here what I will verify shortly, that we can describe motion with numbers. We know immediately which number we would associate with no motion, so we know how to resolve the contradiction: we declare that existence must so structure Reality that all of whatever motions exist within the Universe will at all times add up to a net zero. The syllogism looks something like this:
1. All motions always add up to the net motion of the Universe as a whole;
2. The net motion of the Universe as a whole equals zero; therefore,
3. All motions always add up to zero.
Can we use that theorem to say anything useful about motion? Indeed we can and we are not obliged to know and to add up all of the motions in the Universe before we can do so. We merely ask what rules must apply when one body changes its motion. Those rules must tell us how a body can change its motion while also maintaining the net motion of the Universe at zero. Those rules are two and they are simple:
1. A body suffers no change in its motion unless another body forces it to do so, and
2. The change in one body's motion
always necessitates an equal and oppositely directed change in the motion of
Those rules resemble the first and
third of Isaac Newton's laws of motion, the conceptual foundation upon which
modern physics has been built. But "resemble" is not the same as "are identical
to". Those rules may seem unambiguous enough for our understanding, but they are
stated with a vague concept of motion; in particular, although we assumed that
we can assign numerical values to motions, our two rules differ from Newton's
laws in that they do not refer to anything that's measurable. We correct that
deficiency by relating the vague "motion" in our two rules to velocity, which is
a quantity that we associate with moving bodies and which we calculate as a
ratio from measurements of distance crossed and time elapsed.
Toward that end let's exploit our
second rule to define bodies that are dynamically equivalent and see whither
that definition leads us. When I say that two bodies are dynamically equivalent,
what I mean is this: if one body has the same amount of motion as has the other,
then both bodies are moving with the same velocity. (Please note that velocity
is not the same as speed; velocity is speed that's oriented in a specific
direction.) Thus, if we have two dynamically equivalent bodies moving at the
same speed in opposite directions, then we know that the quantity of motion in
one of the bodies is equal to the negative of the quantity of motion in the
other body. It follows, then, that the sum of the two bodies' motions is equal
to zero, so we expect that if the two bodies were to collide and stick to each
other, the resulting two-body cluster would remain motionless at the point of
impact. The cluster's velocity would be equal to zero, which is just what our
two rules lead us to expect.
Suppose that a team of observers
occupies the reference frame in which one of those dynamically equivalent bodies
(call it Able for convenience) is initially motionless. As those observers see
the situation, we would be moving past them at the speed that we attribute to
Able (albeit in the opposite direction) and the second body (called Baker) would
be moving toward Able at twice that speed. Those observers would then see Baker
approach Able at some speed, collide with Able, and then move with Able at half
its original speed. Able and Baker are traveling with the same velocity, so they
each have the same amount of motion, half the total motion of the cluster that
they comprise. That is in accordance with Rule #2: in the collision Able gained
as much motion as Baker lost (from Rule #2) and Baker kept as much motion as
Able gained (from the fact that both bodies are moving together). If Baker keeps
as much as it loses, then Baker keeps half of what it originally had and gives
the other half to Able.
What that other team of observers has
discovered is that half of Baker's original quantity of motion is equivalent to
half of Baker's original velocity. Because that original velocity is somewhat
arbitrary, we are led to infer that any body's quantity of motion is directly
proportional to that body's velocity. That's one piece of the puzzle, but we may
suspect that there is more to quantity of motion than simply a proportionality
to velocity and we can test that suspicion easily.
In the laboratory of your mind you
still have the Able-Baker cluster floating motionless in our reference frame, so
let's take a cue from that other team of observers and use it in an experiment.
Imagine that a third body (called Charlie) that is dynamically equivalent to
either Able or Baker moves toward the cluster at some arbitrarily chosen
velocity. Charlie collides with Able-Baker, sticks to it, and the resulting
three-body cluster moves away with some new velocity. Can we relate that new
velocity to Charlie's original velocity? We know that after the collision Able,
Baker, and Charlie all have the same velocity and thus all possess the same
quantity of motion. We also know that the amount of motion that Charlie lost in
the collision equals the amount that Able and Baker both gained, so we know that
Charlie lost twice as much motion as it kept, which means that Charlie ends up
with one third of its original motion and, therefore, one third of its original
We could go on performing similar
imaginary experiments with progressively more dynamically equivalent bodies in
our clusters, but we can already see whither those experiments will lead us.
They will lead us to infer the following description of the quantity of motion:
in any isolated array of bodies the quantity that remains unchanged by any
collisions among those bodies (and only those bodies) is calculated by adding up
the products obtained by multiplying the number of standard dynamically
equivalent bodies that comprise each body by the velocity of that body.
The phrase "number of standard
dynamically equivalent bodies that comprise each body" will quickly become
tedious if we are obliged to write it often and in physics we will be obliged to
write it often because it names a fundamental property of matter. For
convenience, then, we replace that clumsy phrase with the word "mass" and note
that our usage of that word conforms to Isaac Newton's description of "quantity
of matter". But where Newton presented the existence of mass as a postulate, we
deduced it as an emergent property; that is, it emerged from our analysis of the
fundamental laws of motion as something new and unexpected but also necessary
In the light of that replacement we
can see that our description of the quantity of motion can be rewritten as: in
any isolated array of bodies the quantity that remains unchanged by any
collisions among those bodies (and only those bodies) is calculated by adding up
the products obtained by multiplying the mass of each body by the velocity of
that body. The product of a body's mass and that body's velocity is called the
linear momentum of that body and if we now replace the word "motion" in our
rules by "linear momentum", then those rules do, indeed, become identical to
Newton's first and third laws of motion. Because it specifies, in essence, that
every linear momentum credit gained by one body must be balanced by an equal
linear momentum debit taken from another body, Newton's third law of motion is
also called the law of conservation of linear momentum.
Conservation laws are of fundamental
importance in physics because they are close to the basic structure of Reality.
As an indication of how close, you may note that we just deduced one of those
laws from the basic facts of existence. Another indicator of how close that
relation is was provided by Amalie Emmy Noether (1882-1935), a German
mathematician who demonstrated that conservation laws necessitate (or are
necessitated by) symmetries of space and time. Thus, for example, the
conservation of linear momentum is correlated with the homogeneity of space:
what that means is that the laws of physics must be the same regardless of the
position of the objects to which they apply, so that identical experiments
performed at different places will yield identical results. Indeed, the
conservation laws are so fundamental that we may think of them as comprising the
Constitution of the Universe, an analogy made all the more apropos by the fact
that they constrain the form that other laws of Nature may have.
Consider a simple example of how that
works. If we have some body floating in space, we assign to it a mass in
accordance with how many dynamically equivalent standard bodies will add up to
the same mass: if we are using the metric system, the standard dynamically
equivalent body is called "kilogram" and if we are using the English system,
it's called "slug". Let's say that the mass of the body is two kilograms and
that the body moves past us at two meters per second. The body's linear momentum
is just the product of those two numbers and the units that go with them; that
is, four kilogram-meters per second. Now suppose that the parts of the body
undergo a spontaneous rearrangement within the body. Could the body's mass
change as a consequence of that rearrangement? We are tempted to say that it can
with the proviso that the body's velocity change in order to keep the linear
momentum unchanged. Thus, if the body's mass goes from two kilograms to one
kilogram, we would expect the velocity to go from two meters per second to four
meters per second. But how would that look to an observer passing us at one
meter per second? The body initially moves at one meter per second in their
frame and thus has a linear momentum of two kilogram-meters per second. After
the change the body would ponder one kilogram and would be traveling at three
meters per second: its linear momentum would have increased spontaneously to
three kilogram-meters per second, in violation of the conservation law. In fact,
there is no way of changing velocities that conserves for all possible observers
the linear momentum of a body undergoing a spontaneous change of mass. We are
thus compelled to assert another conservation law, the conservation of mass. In
its usual form it states that mass can be neither created nor destroyed
spontaneously, but it can only be transferred from one body to another. And of
course such transfers must be made in a way that conserves linear momentum.
In discussing Newton's first and third
laws of motion, I have implied the existence of a second law. That second law
differs from the other two in being not so much a description of Reality as it
is a definition of force in mathematical terms. The second law states simply
that a force applied to a body is equal to the rate at which that body's linear
momentum changes in consequence. Because linear momentum is described by the
product of two factors, mass and velocity, there are two ways to describe a
change in linear momentum.
The first and more familiar
description is the product of a mass and the rate at which a velocity changes;
that is, we say that an applied force is equal to the mass of the forced body
multiplied by the body's acceleration. Thus, a locomotive pulls upon a train of
carriages that's standing in a station and the train accelerates, taking a
certain amount of time to go from zero to fifty miles per hour; for a given
amount of force, a more massive train will take longer to achieve that change in
speed than will a lighter train.
The second and less familiar
description of force is the product of a velocity and the rate at which a mass
is changing. The best example that I can provide of this aspect of force is the
thrust of a rocket motor. When the motor is running, propellant flows at low
speed into the combustion chamber and then, expanding rapidly as it burns and
heats itself, spews out the motor's nozzle at high speed; the difference between
those speeds is the velocity used in the product. The rate at which the
propellant flows through the motor provides the rate of mass change for the
calculation. Each of the Space Shuttle's three main engines, for example, takes
in liquid oxygen and liquid hydrogen at the rate of 400 kilograms per second at
effectively zero speed (the propellants enter the motor from the side) and spews
forth the resulting superhot steam at 4165 meters per second, thereby generating
1,668,100 newtons (375,000 pounds) of thrust as the rocketship rises from its
Another way of interpreting the
formula for that second aspect of force is the one used by Einstein to derive
his famous equation relating mass and energy. He described the force exerted by
two rays of light, which force is proportional to the rate at which energy flows
in the rays, and related that to the product of the speed of light and the rate
at which a body's mass was changing. From that application of the formula he
deduced that mass, which we have derived as an inherent property of matter, is a
kind of potential energy.
Finally, to bring us back to Earth
(literally), I want to point out that the first aspect of force that I described
above includes the force with which we are most familiar - weight. That force is
what a body exerts upon the floor or the ground as a result of Earth's gravity
acting to accelerate the body's mass. At sea level the acceleration of gravity
is 9.8 meters per second per second (32.2 feet per second per second). My mass
ponders 100 kilograms (6.8 slugs), so I weigh 980 newtons (220 pounds). If the
acceleration of gravity, the rate at which a dropped body gains speed, were
different, I would ponder the same mass but I would have a different weight. For
example, if I were sitting on the moon, I would weigh merely 163.3 newtons (36.7
And that's really all that anyone
needs to know about force.
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