[The material of this chapter (with the exception of the last section) can be found in Modes of the Finite, Part 2, Section 2, Chapters 5 and 6.]
8.1. Change as a property
When we think of something as "becoming" different, we ordinarily do not think of the change as instantaneous: you have X one minute, and then suddenly, with no "between," you have Y. Even in substantial changes, there seems to be a gradualness to them so that we can say at some point that the body is not what it used to be, and is not yet what it will be. In the process called "dying," for instance, obviously the body is either alive or dead; but still, when a body is dying, it is going toward being dead from being what you might call "fully alive." It is this changing that we are now going to look at.
In one sense, I suppose, a substantial change has to be instantaneous or immediate: the body is organized either with one form of organization or with some other one; as far as forms are concerned, then as such they don't admit of degrees; their degrees precisely are quantity.
Still, it would seem reasonable to say that any substantial change in a body would always be preceded by or accompanied by an accidental change, where the unifying energy would attempt to cope, as it were, with the efficient cause (the energy disturbing it) until it finally had to assume a different form of internal space of the body; so that it would seem that any change would show up as some property that could be observed.
This may be true in the macroscopic order (i.e. in bodies big enough to be seen with the naked eye); but when you get down to the level of the atom or molecule, things are not so clear. Even accidental changes there are not obviously gradual, because there apparently is no meaning to "between" the different conditions.
For instance, when light falls on a molecule, it is raised to an excited state. Now between the ground state and this excited state, there is a finite energy difference, but according to quantum mechanics, the intermediate degrees have no real meaning. That is, between zero and one, mathematically speaking, there is one-half and two-thirds and so on; but this kind of continuous set of degrees apparently does not describe the real differences in energy. There is just zero or one. Hence, there is no transition in which the energy passes through the one-half point to get from zero to one; and similarly, when it falls back to its ground state, it goes immediately to zero without "passing GO"; one-half is not a concept that refers to anything in this case.
Hence, it would seem that such transitions are of the either/or variety, and in the microscopic order, there is no "act of passing between" A and B.
Nevertheless, there are some cases when the body does have an active transition, and we can consider these.
DEFINITION: PROCESS is a change AS a property of some body.
That is, process is the act of changing, or what the body is doing to get back into equilibrium. Aristotle defined process (what is usually translated as "movement," but his examples show that he was talking about any gradual change) as "the activity of what is in potency insofar as it is in potency." If you remember, his being "in potency" means what we called being unstable; and hence, process is the activity something performs because it is in an unstable condition--or it is the activity by which it gets out of instability into equilibrium again.
Process illustrates, I think, most clearly of all properties, that it is not an "accident," in the sense of something "added to" the body, but is what the whole body is doing as a whole. The body is struggling to adapt somehow to this (in the case of inanimate bodies) excess energy; and what it is doing is observable. This is the property called the "process." Any property is like this; but process shows it best.
Let me use Aristotle's example of the process called "construction" to illustrate what a process is. Clearly, construction of a building is some kind of activity. Is it, Aristotle asks, the activity of the bricks as bricks? No, because when they are acting as bricks, you just have a bunch of bricks. Is it the activity of the bricks as a building? No, because when the bricks are acting as (that system called) a building, the process of construction has stopped. Is construction the bricks acting as partly a building? No, because you could stop construction when the building is half finished, and the bricks would continue acting as part of a building. What construction is is what the bricks are doing (i.e. in this case what is being done to them, but this is an activity on their part also) because they are not yet a building and will be one. That is, it is the act they are performing precisely as in the unstable condition whose purpose is the building. It is what they are doing to get to the purpose from where they are. Note that reactive properties are either processes or, like color, the macroscopic way in which the quantum-mechanical transitions I described above occur. That is, color is the way the molecule reacts to light hitting it, whether that molecule is an element of a more or less tightly knit system (such as the paint on the wall) or a part of a true body (such as you). Each molecule reacts to the light hitting it, gets excited, and falls back to its ground state radiating out the wave length it absorbed. As long as light is striking the body, millions and millions of molecules are going to be doing this, and as each returns to equilibrium, others will be getting excited; and so the large body seems to be "reflecting" a uniform color (which is actually a mixture of the various wave lengths emitted by the different molecules). Of course, as soon as you turn out the light, this reradiation stops, and the body as a whole exists in its ground state. The point is that the property of color is a statistical result of huge numbers of changes going on in the body itself.
There are some things that can be said of all processes just because all of them are the act of something trying to get out of instability to equilibrium. The first of these sounds obvious:
All processes have a DIRECTION, and it is always FROM instability TO equilibrium.
Processes, then, are different from ordinary properties, in that the process is always headed somewhere. And since the unstable condition is (by definition) the self-contradictory condition and the equilibrium the condition in which the body can exist, then the direction of the process has to be the same in all cases. It could not be the case that a process would start from equilibrium and have as its purpose an unstable condition (i.e. start from a condition that could exist and wind up in a condition that couldn't).
This needs some discussion, however. What of so-called "reversible processes"? For instance, you can combine hydrogen and oxygen and get water; and you can put energy into water and get hydrogen and oxygen out of it. Most inanimate processes, in fact, are like this; they can go in either direction. [This does not seem to be the case in living bodies; if you add energy to a corpse of any living body, it seems not possible to bring it back to life.]
Actually, however, even in these "reversible" processes, the process in question is still going from instability to equilibrium. For instance, if you add energy to a mixture of hydrogen and oxygen, you have made it unstable, with a purpose which is water. If you then take the water (which, of course, is now in equilibrium as water) and add energy to it, then you can create a new instability in this body, with now a purpose of being hydrogen and oxygen.
The point is, of course, that you are talking of two different bodies, or systems: the hydrogen-oxygen mixture and the water. It turns out that they are so structured that an instability in one has the other as its purpose; and so you can create a process that goes either way, depending on what you have to start with. But in both cases, the process goes from instability to equilibrium, and not from equilibrium to instability.
There is nothing in the nature of bodies that says that processes have to be reversible.
That is, there is nothing in the nature of a body as such that says that if you set up an imbalance in it with some definite equilibrium as its purpose, you can then set up an imbalance in that resulting body whose purpose will be the bodies you started out with. It often happens; but there is no necessity that it has to happen.
And there are two pieces of evidence that seem to indicate that in fact not all processes are of this "reversible" variety. The first, as I already mentioned, seems to be living bodies. Granted, living bodies can reproduce from inanimate materials; and so it is at least thinkable that either inanimate systems or even corpses could be brought into a condition where an instability could be introduced that would start them living--but it never seems to happen outside a living body itself (I except the initial transition from inanimate to living in evolution, because I think there is reason to say that God is involved in this). The second is that the second law of thermodynamics (that changes go from higher to lower energy-states) seems to indicate that in the long run, there is a kind of irreversibility built in even to the reversible processes.
That is, if you are going to get hydrogen and oxygen back from water, you have to put in more energy than the energy that was needed to get the water out of the hydrogen and oxygen; in any real-world situation, you can't keep flipping back and forth, saving the energy that was given off in the one case and using it to get back the original stuff. In each case, energy will be dissipated into an unusable form, which means that in the long run, there is a given direction to all processes.
In other words, what the second law of thermodynamics states, really, is that the universe as a whole is unstable, and therefore can trade off some of its "pockets" of high energy to raise lower-energy "pockets" to a higher energy state; but that as it does this, some of this excess energy is lost into the universe itself, making it less capable of doing this the next time. The universe is "running down," going from its (unstable) higher-energy state to a (lower-energy) equilibrium, where all processes will stop.
Why spend so much time on this? Because of time itself. We will see later that "time" is a relation among the quantities of processes, and that the "direction" of time reflects the direction of processes: the past is the "more unstable" and the future is toward equilibirum. Hence, "time reversal" doesn't really make any sense. You can't, apparently, run the universe in reverse as if it were a film you could rewind, making it go from its more stable to more unstable condition. The fact that mathematically time could do this simply shows that mathematics is not necessarily an accurate description of what reality is doing.
No, in all probability, time machines are impossible.
Anyone who has opened a physics book and looked at the formulas has seen the mathematical symbols with the little arrows over them. These are vectors, which indicate acts that have not only magnitude but direction.
DEFINITION: A VECTOR is a mathematical description of a process or of a tendency toward a process or of the cause of such a tendency.
That is, vectors describe those forms of energy which are processes (that are going from something to something) because that is the only sort of thing that has a direction. But, of course, since force is the action that causes a process in something, and since the being-affected implies the process in the given direction, then forces and being-affecteds will also have direction, though in a sense they are instantaneous. They have the direction implied by the process they are causing.
That is, the force creates an instability in some body, which, of course, sets up a process whose direction is the purpose: the equilibirum implied by that particular instability. Hence the instability (the being-affected) has the direction that the process has; and since the force is the cause of this instability, then it also has this same direction to it.
Equilibrium, since it has no direction, is expressed mathematically as a SCALAR number.
This, of course, is just a number. Energy itself, in general, is expressed in scalar quantities. This does not mean that processes are not forms of energy, but just that in the language physics uses, processes are singled out as "special," and "energy" as such tends to refer to equilibrium.
8.1.2. The quantities of process
Since process is a form of energy (i.e. a form of activity, but limited in degree), then it will of course be at least in principle measurable, and will have a quantity. Interestingly, however, a given process will actually have two quantities, depending on how you want to look at it.
DEFINITION: The LENGTH of a process is the DIFFERENCE IN ENERGY between the initial instability and the final equilibrium.
DEFINITION: The VELOCITY of the process is the quantity of the process AS SUCH.
The length of the process is actually the process looked at in its relation to the body undergoing the process: how far it went, so to speak, from beginning to end. The velocity is the degree of the process as an act in its own right: how fast it got there.
The two quantities are actually independent of one another. If you go from Boston to San Francisco, the length of the process is the same whether you go by car or by plane; but the main difference between the two modes of transportation is, of course, that these two processes have different velocities, even though the length may be exactly the same in both cases.
Notice that the length of a process is a scalar quantity, while the velocity is always a vector.
The reason for that is that the length is simply a difference in energies, while the velocity is the quantity of the process as such, and the process as a process has a direction from the unstable energy-level to the stable one. Hence, it is that quantity of the process which has the vector attached to it.
Note that I am not necessarily talking of movement when I am talking of "length of process" or "velocity." Any process will have both quantities. The process of blushing, for instance (growing redder) starts from an (unstable) initial paleness and ends with a degree of redness; the velocity of the process is how fast it goes from this paleness to the redness.
In fact, there is a question of whether movement in space can be talked of as a process, strictly speaking. If Newton's first law of motion is correct, then movement at a constant speed in a constant direction does not acquire or give up energy, and so would be in reality a (scalar) form of energy (kinetic energy) and not a real process at all. It turns out that this abstraction can't exist, however, and any real movement is a real process involving differences in energy; but we will discuss this at some length later.
Note that velocities of processes can be directly measured; they need not be arrived at through using time.
That is, the velocity of movement of your car, for instance (what you see as the number on your speedometer), is not arrived at by a little clockwork mechanism. The ghange going on in the wheels as they turn creates a force on the instrument, and the degree of this force determines how far the needle moves. Hence, the velocity (minus the direction, so strictly speaking, the speed) is being directly measured by the speedometer.
And any process, since it is the activity of something changing (and in inanimate bodies would be the getting rid of energy) can at least in principle do work on something-or-other; and hence can cause a force on some measuring instrument, if you have one that can be affected by the energy being got rid of. Hence, processes are always in principle directly measurable.
DEFINITION: ACCELERATION is the velocity of a CHANGE IN VELOCITY.
There is no law that says that a change has to have a constant velocity; and since the velocity itself can be measured (as, say, 0 mph at the beginning, 5 mph a little later, 55 mph for a hour after that, then down to 25, then back to 0 when you reach your destination), then you can see that there is a kind of sub-process by which the velocity goes from 0 to 55, and another when it goes from 55 back to 0. So the two different velocities describe a "length of change of velocity" and there is a definite velocity at which the velocity changed. Suppose you made a jack-rabbit start and got up to cruising speed very fast; but then when you came to the rest area you slowed down very gradually. Your acceleration was much faster than the deceleration (the negative acceleration). Again, the acceleration is independent of the velocities or the "length" of the acceleration as a process.
It is, of course, possible to have variations in acceleration, and to measure those also with a kind of "second order" acceleration emerging as the velocity of that change; and sometimes these "velocities of velocities of velocities" have a use in physics, though they don't have any special name.
We have got into the habit of thinking of velocities as relations of distance (length of the process) and time, even though they can be directly measured. How would you talk about the speed of your car except in miles per hour or kilometers per hour or feet per second or some such relation between the length of the movement and the time you took to travel it? There's no such thing as "vels" or something by which you can measure speed.
But this is a pure historical accident. Because Galileo discovered the law of falling bodies by timing how far the balls he was rolling down a slope got as he reached different parts of a tune he was singing (since clocks didn't measure time in seconds or even minutes in those days, you had to have something different to do it; and the regular pulse of the rhythm of the song was what Galileo actually used), then Newton took this way of measuring velocity, and time became the "independent variable" that velocity was a "function of." You had your stop watch and you measured speed by noting the length of the process and how far the clock went during the time the process took place.
The reason I mention this is that actually, this is going about things backwards; and--not surprisingly--it introduces unnecessary complications into physics that have persisted to the present day.
What is it you are doing when you time some process? Obviously, you are comparing the length of the process with what is going on in your clock, and arriving at the velocity of the process you are interested in. But what is the clock? It could be a sand timer (if you want to measure in multiples of three minutes) or a grandfather clock (if you want to measure in multiples of ticks of the pendulum) or a spring watch (if you want to measure in multiples of ticks of the escapement spring) or a quartz watch (if you want to measure in multiples of the vibrations of the quartz crystal)--and so on.
But what any clock is is some process that has a constant velocity and a measurable length. Within the three-minute time of the egg timer, it is useless as a clock, because you can't measure what part of the sand has fallen through. Within the ticks of the pendulum, the velocity of the pendulum is not regular, and within the ticks of the watch, the velocity of the process is not regular, and so inside these limits, the instrument is useless as a timepiece. Hence, the characteristic of a clock is that it has a regular or constant velocity. You have to establish this first before you can use anything as a clock to measure time.
Therefore, the velocity of the measuring instrument must be known (at least as constant) before time can be measured.
Hence, what you are doing is starting the clock's process at some known point (turning the timer over, noting the position of the hands or the numbers) together with the beginning (the imbalance) of the process to be measured, and stopping both processes together (either actually as with a stopwatch or mentally, noting the position of the hands). You then come up with the following ratio:
Lc/Vc = Lp/Vp
That is the length of the clock's process is to its (constant) velocity as the length of the measured body's process is to its (average) velocity. Since the velocity of the clock's process is constant, the only thing that varies on the left-hand side is the length of the process. Since you know this, and since you know the length of the process on the right-hand side, then the one remaining quantity (the velocity of the body) is now known in relation to the LENGTH of the clock's process.
That is, you don't need to know the actual velocity of the clock, as long as it is constant and as long as you don't use different clocks to measure processes. There are two variables on the right, and only one on the left, and the ratio of the two variables is equal. Hence, if one of the variables on the right is known (the length), then the unknown can be expressed as a function of the length on the right--which is also known.
Timing a process, therefore, is comparing the RATIO of the length to the velocity of the process to the RATIO of these two quantities of some standard process (with a constant velocity).
DEFINITION: The TIME of any process is the RATIO of its length to its velocity.
That is, what is going on in "timing" can now be seen to be comparing the time of the process you are measuring to the time of the standard process. So there are actually two times involved: the clock's and that of whatever you are measuring; and what these times are (we can see from our thought-experiment) in each case is simply the relation between the two quantities of the process.
Now we come to something that seems controversial. Since there is no real connection between the two quantities of any process (i.e. the length does not depend in any way on the velocity or vice versa), this ratio is just a chance number that happens to express what the ratio is in a given case, or in other words,
TIME is NOT something real.
That is, what you are measuring with your clock is not something real. There is a real velocity, and there is a real length to both the process you are measuring and the process you are using to measure it (your clock), but the ratio between these two quantities is just a mathematical trick, describing NO RELATION that actually obtains in the objects.
That is, it makes no difference to the length of the process you are timing that it has a certain velocity; it could have any velocity at all, and the length would be the same. And the same is true of the velocity; the same average velocity is not affected in any way by the fact that the process was this or that length. Hence, there is no connection in reality between the length of the process and the velocity; and so the time of the process as such is not a real relation.
Time is a mental relation which has a foundation in reality; but the relation itself does not exist.
The foundation for our making the comparison exists; but the two numbers (the quantities) are just different--allowing us to divide one by the other. But the number that results from this division is not the quantity of anything at all; it is simply a number that we can arrive at by comparing two quantities.
If this is true of the time of a process, then the time that is "the same" in the clock and the measured process is a fortiori not real. That is, you say the clock took "the same time" to make a revolution of the minute hand as the car took to go 30 miles at its 30 mph constant velocity. The ratios are the same; but these ratios are not the quantity of any form of energy; they are simple numbers. And so "the same time" is a second-order relationship: you compare the ratio (L/V) of one with the ratio (L/V) of the other, and notice that the relation of these two ratios is that the numbers are the same; and so that is supposed to be the "time," as if you actually measured "something." But if you measured something, what is it? Where is it?
This is why "time" in that "the same time" sense appears as "between" things in a kind of "space" that isn't really the space between things. It's a kind of imaginary "line" that stretches from "the past" through "the present" to "the future," along which line "events occur," and which you measure the "length" of by using your clock.
But there's no such line, of course, and you get into all sorts of conundrums, as St. Augustine did, if you try to say that there is. Obviously, "the past" doesn't exist any more, so how could it be measured? And "the future" doesn't exist yet, so you can't measure it either. The only thing that exists is "the present"; but this has to be the present instant, which, of course is only a point along this "line," and so it has no length. So if time is real, it is unmeasurable.
Hence, time in the sense of the comparison of various times (ratios within processes) is not real either.
8.2.1. Complications in physics
This theory of time would predict that it wouldn't just be St. Augustine who got into trouble considering time, but that if physics takes time as the "independent variable" for measuring velocity, then it would be likely that some strange conclusions could result.
And, as the Special and General Theories of Relativity have shown, this does really happen. The two theories, in fact, have a lot to do with just exactly the realization that one person's clock is not measuring "the same time" as another's--unless they are at rest with respect to each other.
Einstein supposed (for reasons that have an experimental foundation in attempts to discover the velocity of light) that the velocity of light in a vacuum was always the same, and would always be observed to be the same (differences due to whether you are moving to or away from the light source show up as different wave lengths of the light, not different velocities).
Now then, with that in mind, suppose you and I are comparing our clocks. We are at rest with respect to each other, and I note that they are synchronized. As my watch's hands come back to the 12, yours also are exactly on the 12.
But now let us let one of us move past the other with a constant velocity. If I am going to observe your clock as it moves past me, I am going to have to do it by means of the light which is traveling (at a constant velocity) between your clock and mine. But since the distance the light travels is changing, then the time the light takes to get from your clock to mine is changing (because the velocity of the light does not change). Hence, a second time is now introduced, which is going to mean that, if our clocks appear to be on the 12 together at the start of the measurement, this will mean that the light that left your clock when it was at the twelve arrived at my clock slightly later, but at the time it arrived, my clock said exactly 12. But now when my clock says 12:05, for instance, then the information from your clock had to travel a different distance to get to my clock; and so I will not now be able to read your clock as saying 12:05. Our clocks cannot now be seen to be synchronized, because I am reading your clock with the interference of a second process whose length is changing.
The result, says Einstein in the Special theory, is that the person who considers himself at rest (and the other one as moving with respect to him) will see the other clock as going slower than his.
But since neither one can claim to be "really" at rest (the only thing that is happening is that the distance between them is changing, not that one is necessarily "there" someplace in "absolute space"), then each observer will observe the other clock as going slower than his. That is, A will say that B's clock is going slower; but B will not say that A's is going faster, but will also observe A's as going slower.
This is really strange if the clocks are actually measuring something; but of course they aren't; they are simply making ratios. But the relation between my ratio and yours is fouled up by the process of information-transfer.
Of course, if the clocks are at rest with respect to each other, then the information going from one to the other always takes the same time (same length, same velocity); and hence, those two clocks can be synchronized. But not if one is moving with respect to the other.
It is also true that if acceleration is taken into account, then this variation in the time of information-transmission becomes extremely complicated; and so the relation between the processes becomes that much more bizarre. This is perhaps not the place to go into this here; but what it amounts to is that these wierd conclusions (that twins, for example, would wind up different ages if one blasted off on a trip and came back again) are due to the fact that you are making comparisons of quantities of processes with processes involved in the transmission of the information necessary to make the comparisons.
Einstein also points out that observations of simultaneity are different depending on motions of the observers. If asteroids hit both ends of a rocket ship moving by another rocket ship, let us say that those people in the middle of the ship struck observe both hits as happening together. But astronauts in the ship going by will have the light transmitted different distances to their eyes from the events, and will see one as happening before the other. Then did they really hit the ship at the same time or not? You can argue this way: If the ship was moving, and on the ship they were observed to hit simultaneously, then the rearmost one would have had to hit the ship slightly before the foremost one; but if the ship was not moving "really," (and only the other one was), then they "really" hit at the same time. But--as we will see more at length--you can only establish movement the way you establish position; with respect to something real, and the only other object to use is the other ship; and so it is a tossup which is "really" moving if the distance between them is changing. Hence, there is no absolute meaning to "simultaneous" any more than there is to any other time-word.
I hasten to say that the Theories of Relativity do not establish that "everything is relative"; they even presuppose that the velocity of light in space is not relative to the observer. Nor is motion as such relative if the distances between objects change; what is relative is which of the objects is considered to be moving and which is "at rest" and what the time of the movement of "the other system" is. The relativity here is analogous to considering whether, when you grew an inch, your feet went an inch farther down from your head or your head went an inch farther up from your feet.
8.2.2. Newton's physics and time
Let me show what happens to Newton's physics because of the introduction of time as an "independent variable," using his force equation.
F = ma
Now acceleration, for Newton, is the time-derivative (tendency to change) of velocity, which is the time-derivative of distance. It looks like this when you put it in the form of the calculus:
F = m d2x/dt2
In order to solve this equation, you have to do some fairly complicated integrating, because you have a "second derivative" (those superscripts aren't squares). But if you note that
v = dx/dt
a = dv/dt
and you solve for dt, you get
dt = dx/v = dv/a
Eliminating dt, and bringing the v's together on one side, you get
a dx = v dv
Solving for a, so you can substitute for it in the force equation, you have
a = v dv/dx
F = m v dv/dx
F dx = m v dv
So a simple substitution for acceleration, once you have eliminated the unnecessary dt from the equation, gives you the force equation directly integrable into the Newtonian form; but this takes several complicated steps in Newton's own way of doing it, precisely because he has used the time, and without realizing it had to perform several operations whose only real function was to eliminate the superfluity.
PREDICTION: This view of time predicts that much of physics could be simplified if time were eliminated as an "independent variable" from the equations of physics.
People might object that you can't observe processes and measure their velocities unless you use clocks. My contention is that velocities can be measured directly, without the use of clocks. For instance, you can measure the velocity of the process of heating by making the increase of heat cause pressure on an instrument (analogously to the speedometer of a car). Perhaps this would involve devising new instruments; but they don't seem to me to be that difficult to invent. It's just that no one has seen the need to do so so far.
And what I suspect, based on this view of time, is that much of the mystery of calculations would drop away; and it is quite conceivable that the elimination of time would show that classical physics and relativistic physics are actually just the same thing at base; and the difference between them depends on what you do with that pesky variable time, which really makes no difference to what you are observing, because what it "measures" is really nothing at all.
You can see that the philosophy of nature is not simply a description of what scientists have discovered about bodies; it is a science in its own right, with consequences that matter to the other sciences that deal with bodies. It might very well be that if philosophers and scientists could learn to cooperate, instead of having scientists look indulgently on philosophers as playing interesting but irrelevant games, both disciplines would be better off.
In any case, my theory is on the line, now. I have made predictions, which should be testable by anyone who wants to make the effort to test them.
8.2.3. The calculus
Let me say a brief word about the differential and integral calculus, which really belongs to the philosophy of mathematics, but is in place here, in a sense.
[For another treatment of this, see Modes of the Finite, Part 4, Section 3, Chapter 3.]
The way the mathematicians explain a derivative like 30 mph (i.e dx/dt = 30 mi/hr) is that the dx is the limit you reach when taking a distance a making it smaller and smaller until it becomes an arbitrarily small delta below any epsilon which is as small as you want to name. As you make the distance smaller and smaller, then (supposing the ratio to be either constant or varying continuously), the "length of time" it takes to go that distance gets smaller and smaller in the proper ratio, so that it too ultimately become arbitrarily small--but the ratio is preserved even here. Then there is the leap to the "limit": that if it keeps getting that way, then "in the limit" (if the distance covered could get to zero, which it can't), then the ratio would be what the "little tiny" one is (or is headed towards).
Where I would differ from this is the notion that the limit can't be reached and the derivative deals with "little tiny" finite quantities. The reason it is said that the limit can't be reached is that the denominator would then actually be zero, and division by zero is forbidden (because there's no inverse operation).
But in one case, division by zero is meaningful: 0/0, because the inverse operation (n times 0 = 0) works in this case. Of course, it's trivial, because the "n" can be any number you want.
But my hypothesis for solving why the calculus works is that the calculus is saying that when there is a function that converges on a definite value as you approach 0/0, then the fraction has a definite meaning: the value that is approached.
This may be just a different way of reading the meaning of "limit"; but it is certainly not what some mathematics professors understand by it. In any case, it makes the derivative an exact number, expressing the TENDENCY to change at some point, and not a ratio between "little tiny" differences in an "infinitesimal change." And it is this "tendential" reality (force or being-affected) that the derivative refers to.
8.3. The path of the process
Before we get into movement as a process, there is one other aspect of processes in general to consider. It does not follow that the length of a process (in the sense of the difference in energy levels between the beginning and end points) is the same as the "distance traveled" to get there.
The length would describe the net work done in the process; but this could involve work being done on the affected object, and work being done by it for a while, and then work being done on it again. What I am talking about can be described in terms of movement most easily. The distance from Boston to Los Angeles is, say, three thousand miles. But if you go from Boston to Los Angeles by way of a trip to Miami, then up to Chicago, and so on, you're going to travel a good deal more than three thousand miles. So the distance traveled and the distance between the end-points need not be the same.
Similarly, if you're heating something, then if you heat it and let it cool and heat it again, then net gain in temperature (the length of the process) is, say, 100 degrees; but it cooled down during part of this process, spilling its heat into the environs; and so the process of heating took a longer path to get between the end-points. Or again, the path by which you get from the beginning balance in your check book to the ending balance is usually considerably longer (because of deposits as well as withdrawals) than the length of the process (the difference between the two balances).
Sometimes, the path of a process is significant in physics, sometimes (because there is no net difference between it and the length) it is not. In any case, what the path amounts to is that the process can be (at least in thought) broken up into smaller processes with different directions (and hence different beginnings and endings) and then these smaller processes can be added up vectorially into the total process.
There remain two topics in this sketch of the philosophy of bodies: movement and evolution. The first is interesting in that there is a question of whether movement is a process at all or not.
The reason is Newton's first law of motion: a body at rest will remain at rest, and a body in motion will remain in motion at a constant velocity (speed and direction) unless acted on by an "unbalanced" force. This implies that there is no change in energy in a moving body (if it is moving with a constant velocity), and hence no real process. It is one of those acts (if this "law" is true) that expresses an apparent process which is really an act.
But the fact that Einstein's General Theory of Relativity precisely denies that this law is true should give one pause. True, Einstein says that movement with a constant acceleration needs no "force" and hence is not a process, and constant velocity would be a constant acceleration of zero; so the first law is a kind of limiting case of Einstein's supposition. But the denial as Newton stated it means that the law is at least worth examining.
Now on the supposition that position "in absolute space" is meaningless, and position in reality is being affected by some body's field, it follows that motion is always motion in the field of some real body. This, of course, what Einstein would say also; there is no meaning to "absolute motion" without reference to some "reference frame," which in the real world is some body with a field.
What we want to find out is whether there could be anything which could meaningfully be observed as motion which would not involve change in energy-level in a field. If everything observed as motion involves changes in the energy-level of the field-interactions of the two objects, then a real change has occurred (there has been work done and a transfer of energy), and motion would be a real process, needing a force of some sort to account for it.
Let us now set up some thought-experiments. First, consider that there are only two objects in the universe, each of which has no size (so that they have no "sides"), but are mass-points; let us for simplicity consider only the gravitational field-interactions of these.
Now if the distance between A and B changes, so that A gets farther from or closer to B, the effect of B's field on A is going to change, and this is a real change in energy. Hence, if there is to be motion without a real change, the distance between A and B would have to be constant.
You would think that A could move in a circle (actually a sphere) around B, and still leave the distance (the radius of the sphere) constant. But the question is this: Could this "movement" be observed by B as a movement? Or by A?
Remember, there are no "sides" to either B or A, and there is no other point of reference in this universe, so that there is nothing in B to establish a "direction" except A. So if A "moved" around B, B would not be able to observe it. Nor could A, because the only direction it could establish would be that toward B; but in the direction toward B, there is no change. Hence, only an ideal observer somewhere in "absolute space" would be able to tell that A is moving; that is, only from a third object could A be said to be moving around B.
We actually are making ourselves a third object when we take this "universal" point of view and suppose that A is "really" moving around B when neither A nor B could say so. But this supposes that motion "in absolute space" means something or that A is moving with respect to us. But then there aren't just two objects in the universe.
If we give a size to B, and assume that the distance between the centers of A and B does not change, we can see that an observer on B's surface (or anywhere on B except the center) would be able to detect A's movement around B (because A would "rise and set" for him).
But in this case the distance between A and the observer is really changing: A gets farther away from or closer to the observer; and hence it looks as if there is a real change going on.
You might argue that there isn't, because the changes in distance on each side of the center cancel each other out, so that there is no net change, really. But there is; and we can see what it might be if we look at the movement from A's point of view.
A has no "sides," so that A can't tell that it is going around B; and the distance between A and B as a whole does not change; and so you woud think that from A's point of view, no change would be observable. But this is not so. A would observe B as rotating, with the observer coming round periodically. There is no real difference in this case between A's going round B and B's rotating.
If there are real field-interactions going on between A and B, then what is going to happen is that (like the real pendulum as opposed to the "perfect" one) either the revolution of A around B (or the rotation of B below A) will tend to slow down (because the changing energy-levels of the parts of B are pulling at A in the direction opposite to the revolution--or are making B rotate more slowly). And the purpose of the process of slowing down is, of course, having A in a synchronous orbit, so to speak: that is, when the revolution/rotation slows so that A is observably at rest above some point on B, the system will be in equilibrium, and any disturbance either way will have a tendency to right itself.
But when A is revolving at exactly the same rate as B is rotating, then no motion can be observed; and supposing there to be no "absolute observer" or third object, no definable movement occurs; and all distances are now fixed.
Of course, if we introduce a third object, then from that point of view A could be observed revolving around B. But a little thought will show that this could only happen if the real distance between A (and/or B) and this third object are changing.
Hence, nothing that could be called "movement" can occur unless there is a real change in energy in a field; and therefore, Newton's First Law is an idealization, and is false as stated.
Furthermore, any movement in a field (in space) is a process, and as such has a purpose; and the purpose is always some equilibrium (such as synchronous orbits) in which the movement as such stops.
8.4.1. Reference frames
This fact of real movement's being an actual change in energy-levels in a field accounts, perhaps, for the odd geometry of Einstein's "paths of movement in space-time." The Einsteinian physics is complicated by the fact that what Einstein was interested in was not only establishing that there was no "absolute space" with reference to which things were "really" moving and no "absolute time" in which they did it; but to make this fit with the fact that distances are real and do not depend on the observer.
The whole of the very complex tensor calculus was developed, not to show that distances are relative, but to have a mathematical system which would leave the distance between two objects the same no matter what reference frame you were on, and no matter how it was moving with respect to any other one.
What is a reference frame? You must have seen those little diagrams with an x-axis, a y-axis, and a z-axis with reference to which you could locate points. This is a Cartesian reference frame, invented by Rene Descartes (Latin: Cartesius). But there are other "coordinate systems," such as polar coordinates, spherical coordinates, and so on, which don't use axes at right angles to each other. We don't need to add confusion by describing them.
What they all are are conveniences for locating objects in the "space" defined by the coordinate system. What Einstein was concerned about is that if you have two observers who are moving with respect to each other, each one's coordinate system is going to measure funny things in the other coordinate system, since the other one is moving and distances between it an the other observer are changing. Then if each of the observers observes two objects, the observations of the two will be different because of the changing distances between the objects and the observers and between the observers and each other.
Einstein's mathematics allows the observers on either reference frame to correct for the motions of the other one and to make observations which will make sense also in the other reference frame (making the proper corrections there). This had to be done, because he assumed that there is no "privileged" reference frame; every one is as good as every other one. That is what is relative about relativity; it does not really deal with the relativity of what is observed.
But why does Einstein talk about space-time and not just space? Because motion of the reference-frames is involved. And, again not to bore you with the mathematics behind this, if you take the "fourth dimension" of Einstein's space-time (x y z and t), the time-dimension, you find that it isn't just plain old t, but involves a negative product of the velocity of light and time (which is a velocity divided by a distance), and so turns out to be an adjustment of distance, not a time at all.
This was necessitated by the observed sameness of the velocity of light in any reference frame; the reference frame would have to have its distances adjusted in the direction of its motion in order to have the measurement of light come out right.
But once again, time has reared its ugly head, complicating things. And so I offer the following
PREDICTION: It should be possible to develop a way of dealing with the real distances between objects in terms of the forces acting on each other, and to describe movements as the changes of these field-interactions without resorting to "reference-frames"; if this were done, many other complications in physics would disappear.
This is a much more tentative proposal than any of the others I have made. My attempts to verify it have run into the difficulty one who has any experience in physics has of describing things without referring to some reference frame; like King George's head in David Copperfield, the reference frames keep intruding themselves.
I would venture to say that if something like this could be done, then we would find macroscopic physics looking remarkably like quantum physics also. I suspect that classical and relativistic physics look the way they look because of the parameters they introduce; and these parameters might very well (especially in the case of time) be unneccessary.
PREDICTION: Instead of taking mass, length, and time as "primitive concepts," and defining everything else with reference to them, this theory would predict a simpler physics if ENERGY, VELOCITY, and FORCE were taken as "primitive," and length, mass, time, and all other concepts were derived from them.
Mathematically, of course, there is no trouble doing this, because if A is equal to B, then you can make B equal to A; it is really just turning the equations inside out (as I did with the force equation, eliminating the time). I suspect that if you did a thorough job of this, you would find the mathematics of physics to simplify itself considerably, because you would be dealing with what is really "out there" (and directly measurable) and not something which is actually at a second remove from what you are observing.
8.4.2. Movement and Zeno
The ancient philosopher Zeno established a number of paradoxes which were intended to prove that movement was simply a way we considered things and was not a real change at all. It is instructive to see if our notion of movement as a real change can survive his critique.
If, says Zeno, you are going to move across the room, you have to move half way before you get to the other side. But in order to get to the half-way point, you first have to move half of that distance; and in order to get there, you first have to move half of that distance, and so on to infinity. No matter how small a distance you move, you first have to cover half of it before you can get there--and half of that, and half of that, and so on. So you can't really move.
Or you can take it the other way. If you get half way, you have to cover half of the remaining distance before you get to the other side, and then half of the now remaining distance, and so on; you never can get there, because you always have half of the remaining distance left.
What is the solution? Zeno is looking at a movement as if it were a series of smaller movements. That is, he is mentally stopping the movement at the half-way point, and then stopping it again at the next half-way point, and so on.
But if movement is a real change, it starts with a definite instability, which implies a definite purpose. So that if you are moving across the room, your body has an instability in it whose purpose is the other side, not half-way. So the instability does not reach a mini-equilibrium halfway across and then resume its trek.
That is, the movement across the room is a single act, not a series of lesser acts; it is a unit, and though you can consider it (because of its path) as a series of smaller movements, it does not exist as a set of acts, but as one single act with two limits and only two: the imbalance and the equilibrium.
Hence, Zeno confuses the path of the movement with the act of moving; it looks paradoxical, but the movement actually has meaning only as a between of the two limits.
And this would add fuel to what I said about the macroscopic world's being like that of quantum mechanics. In quantum mechanics it seems quite clear that changes in energy and movements and so on have to be taken as "between" limits but without all the "intermediate fractions" of a kind of continuous path between the two limits. Perhaps what happens in quantum physics is that the nature of what is being described does not allow of the fiction of reference-frames and continuous spaces and infinitely divisible lines and so on, but deals with acts between definite limits; and the change is describable only in terms of the limits, not something intermediary.
My prediction above probably would have something like this happen in the macroscopic description of things, if you eliminate reference-frames. Possibly the orbits of planets would involve the energy-levels in solar space in which the planets exist, but where the planet is at any moment in this energy-level would not be detectable; it would just be "somewhere on this level."
I don't know. Conceivably, a physics in terms of energy, velocity, and force as primitive concepts, without time and reference frames, would not be able to be developed, or if able to be developed, would not work. It would be interesting to try, however, to see if some of the unsolvable knots in contemporary physics might just turn out to be a tangle in the thread of the investigative process itself--and this would unravel it.
I promised at the very beginning I would say something about evolution. I have a set of notes I hope to work up into decent form some day called "Hypothesis for the Universe" in which I sketch what evolution should look like based on the conclusions I have come to on the nature of bodies, change, process, and life.
[Note: this was subsequently done, and appears as the seventh volume of Modes of the Finite.]
Basically, if evolution is a process, then it started with an instability, and is headed toward an equilibrium. Thus, the universe began in an unstable condition, and the "heat death" is its predictable outcome.
If it began as unstable, then there was something outside the universe that accounted for it. I assume that this is the Infinite Act, God (what else could it be?). But since God is absolutely unchangeable (though He can cause change), then what happens in the universe He creates cannot make a real difference in Him, and therefore the creation and sustaining of the universe is an act of absolute love on His part. As I tried to show in The Finite and the Infinite, God can know the universe He creates without actually being dependent on it or changing (he knows it insofar as He knows Himself as causing it).
I think the evidence supporting evolution is so overwhelming that, even if in details it is wrong, it has to be basically on the right track. To deny it in favor of some literalist interpretation of the Bible is to abdicate reason altogether (in which case, why believe the Bible?).
But if the universe caused by God is evolving, and is in a process, and if God is causing the process, my hypothesis is this:
Evolution is a gradual unfolding of LOVE in two senses: of God's love (respect) for the universe He creates, and of love (unselfishness) IN the universe itself.
That is, this hypothesis predicts that as the process goes on and bodies are more and more capable of doing more and more for themselves, God will manipulate things less and less and leave them more and more free and responsible for what they do; and also, as the process goes on, you will find things acting more and more explicitly not for self-fulfillment, but in imitation of their Divine Creator, who is lavish in giving and not seeking return.
I am not going to try to verify this here; as I say, I have a sketch of it that gives me some hope that it might be on the right track, and it looks as if it can be shown without forcing the data into a kind of Procrustean mold; and it turns out to be beautiful and somewhat hopeful (though not as hopeful as Teilhard de Chardin, I am afraid; I don't see the end, now that humans are free, as inevitable).
But I leave this treatment of bodies here, in this extremely incomplete state. There may be a few things I have said that are true, and they might provoke a further search by those who have more insight than I; and if anything I have said turns out to be productive of light on the subject of bodies, then all the errors and silliness of the rest of what has been put down here can perhaps be forgiven.