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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 Reality.

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 me thoughts).

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 exist.

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 another body.

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 velocity.

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 and inevitable.

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 launch pad.

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 pounds).

And that's really all that anyone needs to know about force.


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