Chapter 6

Theorems about effects and causes

There are several conclusions we may draw that follow from the nature of effects and causes as we have so abstractly defined them; and since they are tautologically true because of the way the terms are defined, we might as well state them as theorems:

Theorem I: The cause is outside the effect.

Proof: The effect, by definition, is that which does not make sense, or is contradictory, taken by itself. The cause, by definition, is that which makes sense out of it. Since the effect is an abstraction (just a pair or set of facts), then if the cause is "contained within it," it must be one of the facts that make up the effect. But if the cause is one of the facts that make up the effect, then what resolves the contradiction would be part of this "effect", and the "effect" would then make sense, and so not be an effect. Q. E. D⁽¹⁾ .

When you put the proof like that, it is obvious; but still it is counter-intuitive, as when, for instance, I remember something I have forgotten. The fact was not conscious and is now conscious. Nothing external reminded me of it. So my mind is the cause of its becoming conscious. But my mind is also what was not conscious of it and what is now conscious of it.

But my mind is what is affected, not the effect. The effect is the emergence of something into consciousness without any external stimulus. Whatever it is about the mind that is capable of reawakening past experiences without any external stimulus is the cause of the effect in question, and that, clearly is not part of the problem. The mind and the problem are both within what is affected (me), but the solution is not part of the problem.

So perhaps the theorem can be best understood if you think of the effect as just exactly the problem and nothing else, and the cause as just exactly the fact which solves it, and nothing else. So Spinoza's "cause of itself" which he applies to God is simple nonsense if "cause" is taken in the sense I have defined it. Anything which is self-intelligible ("self-explanatory") is not the cause of itself, but simply not an effect at all.

"Self-explanatory" actually has two senses. In the first, it means that there is nothing to explain; whatever it is makes sense without demanding something else for its intelligibility. Hence, it is neither effect nor cause, but existing as just a fact. Remember, I said that in order to have an effect (or need a cause) there had to be facts in conflict.

In a second sense, a complex object can be "self-explanatory" in that one aspect of the object can be the cause and another aspect be the effect. In this case, the causer contains the effect within it, as was the case in my remembering a fact which I knew but had forgotten. In the sense that I needed nothing else besides myself (in its complexity) to remember, my remembering was self-explanatory; even though using the previous and more accurate sense of the term, my remembering was not (as an effect) self-explanatory, because it needed a power or act of my mind which could bring about the remembering (and which, for example, sometimes is disabled, not allowing the recall).

So something can be the causer of some effect in itself; but nothing can be the cause of itself.

Theorem II: The cause is not altered or different in any way by its having an effect.

Proof: The effect is a set of facts which contradict each other. The cause is simply the fact which, when understood along with the effect, makes the whole intelligible. That is, the effect is an effect because this fact (the cause) is either unknown or ignored; the accident of its not being known (or being deliberately left out of consideration) does not alter what it is in the least. Hence, the cause (as a fact) is what it is, whether or not it is considered as cause of a given effect. Q. E. D.

This again is counter-intuitive, and in fact seems to be disproved by physics. One of the basic laws of motion is Newton's Third Law: "Every action produces an equal and opposite reaction." This seems to say that if A is acting on B, then ipso facto B is acting on A; and hence the fact that A causes something in B necessarily entails A's being affected by B. And this seems to refute the theorem.

But again there is the confusion between effect and what is affected and cause and causer. Let us say that one billiard ball strikes another making it move; the second billiard ball reacts on the first, altering its movement, such that it is impossible in fact for the motion of the first billiard ball to be totally unaffected if it makes something else move.

Let us straighten this out. The effect in question is the fact that the second ball began to move, or if you want to put it in terms of physics, its momentum (and total energy) is greater than before, and obviously it can't give itself more energy than it has.

Now the cause of this is the momentum which is added to it, or if you will, the amount of energy received from outside the ball, which accounts for the excess. You could even say, depending on how you formulated the effect (which is abstract, remember) the amount of energy received from the first ball. But this amount is simply what it is; it will not be altered by the fact that it is making the first ball move, or is making the second ball move, or is heating the second ball, or is dissipated into space or whatever; an amount of energy is just an amount of energy. Add this amount of energy and the problem is solved.

The first ball, then, is the causer, not the cause. It contained the cause (which is why it would be called the "cause" in the ordinary sense); but it contains all sorts of other properties. One of the properties it contains is, of course, the fact that, if it is going to impart this energy to the second ball, it is going to lose it; and if there is to be a transfer of energy, the elastic collision will now make the second ball a causer of a different effect on the first, which will impart an impulse also to it, changing its direction, for instance.

But all that means is that the two billiard balls are complex objects, and are both what is affected and causers of (at least) two different effects. Each of these effects has its own cause, and that cause is simply a fact and is not altered in the least by its being the cause of some effect. In each case, the cause of the action or the reaction was a certain impulse in a certain direction; and that impulse is all that is needed in each case to explain the motions, and is what it is, irrespective of the motions which it explains.

I cannot stress too much that it is absolutely vital to everything that follows that this be clearly understood. If you think "concretely" here, so that it seems to you that the cause is affected by its being a cause, then you must go over what was said and realize that effect and cause are simply ways of talking or thinking about abstract aspects of a concrete situation--valid ways of talking about them, but ways of talking about them nonetheless--and the concrete situation is what it is, irrespective of whether we consider part of it as incomplete and therefore "dependent" on the other part as "completing" it.

It sounds as if I am playing games, and there is no reality to what I am saying at all. Not so. If you consider the billiard balls again, the fact is that the second billiard ball cannot add to its own energy, and it increased its own energy--obviously because of something-or-other the first ball did to it. This "something-or-other," however you want to name it (impulse, energy-transfer, momentum), is the cause. The point I am making is that the "getting up and moving" of the second billiard ball is not divorced from the concrete situation, which is the collision of the two balls. That's what happened, and it makes sense, of course. But it's true that the abstract aspect of that situation, the beginning-to-move of the second ball,wouldn't have made sense without the collision.

To put this another way, the "dependence" of the effect is a hypothetical, not a fact. The effect wouldn't make sense if the cause weren't there. But of course, the cause is always there, or the effect would really be a contradiction; hence, the effect as such prescinds from the whole situation and considers only one aspect of it. It is a real aspect, but it is only a real aspect; and the actual concrete situation contains the cause, without which the particular aspect called the "effect" would be impossible.

The only reason for bothering with distinguishing aspects of a situation in this way is that not all aspects of the situation may be known, and so if you find a situation in which all the aspects you know at the moment are impossible, then you know that the real, concrete situation contains this other aspect (the cause) which makes sense out of the aspect that you happen to have in your grasp.

Thus, even though effect and cause are simply points of view from which to look at a real, concrete situation (which always, as such, makes sense or of course it couldn't exist), it does not follow that considering effects and causes is a silly game, except for anyone who is omniscient. We do get confronted with effects, and have to go outside them to get at their causes; and in this way we increase our knowledge. We don't increase "reality"; we simply know more of it by the fact that the effect tells us precisely that we have (at the moment) got hold of no more than an incomplete view of it.

Theorem III: Identical effects have identical causes.

Proof: An effect is defined as an incomplete view of a situation, such that a missing element makes this incomplete view appear as a contradiction. The fact that one fact (the cause) is removed from the concrete situation (or is unknown), therefore, is what makes the remainder to be an effect.

It follows from this that what makes an effect a given effect (as distinguished from any other effect) is not the language you would use to couch the contradiction, but in what it is that is removed from the concrete situation, which makes the remainder unintelligible. This is another way of saying that what the cause is makes the effect the effect which it is.

And it follows from this that if two effects are identical, then the causes have to be identical. Q. E. D.

There is actually another way to prove this which, for those of a mathematical turn of mind, might be clearer:

Suppose that there are two identical effects, called B₁ and B₂. Now the cause of B₁ has all (and only) the properties necessary to explain B₁. But since the two effects are only abstractions (pairs or sets of facts, not objects), then B₁ can be replaced by B₂ without altering the effect in any way at all, since the two effects as effects are identical. It follows from the fact that no alteration has occurred in the effect that the cause of B₁ has all (and only) the properties necessary to cause the "new" effect. Therefore, the cause of B₁ is also the cause of B₂. Q. E. D.

Once again, the counter-intuitiveness of this is explained by the abstractness of effect and cause as we have defined them, and the fact that we tend to think, when we use these terms, of what is affected and the causer.

For instance, if you see a branch moving outside your window, you might very well notice a squirrel jumping on it, and realize that the "cause" of the movement is the squirrel. Another time, however, you might look out and see the branch moving in just the same way, but without a squirrel making it move; and you go outside and feel the breeze and realize that it's the wind that made the branch move. Now if the theorem above is correct, this sounds as if we are saying that the squirrel is identical with the wind.

But of course, the squirrel and the wind are causers, and contain the causes in question; they are not really the causes. Let us look at the situation more closely. The effect is the movement of an object which has no internal source of movement. Now what is necessary to account for this effect is (a) whatever it is about something that can impart motion to an object that has no internal source of movement (what in physics is called "energy"), and (b) external to the object. Obviously, both the squirrel and the breeze are external to the branch, and both have energy that can be applied to the branch (or they couldn't move it). So in the respect in which they solve the problem, (i.e. as the cause of it) they are identical.

Even if you define the effect you are interested in more precisely, by noticing the amplitude and direction of the movement of the branch--supposing that the two effects so defined are still identical--then you find that the causes in the squirrel and the breeze are still identical: they now are known to have the same amount of energy applied to the same place on the branch (or the movements would be greater or lesser or in different directions).

Well yes, but what is this "energy" that they have? This is one of the interesting things about science, which is actually using our notion of effect and cause very often. You don't know what it is "in itself," so to speak; the name "energy" is simply a term which means something like "Whatever it is about an object that allows it to alter the motion-state of another object." We know that any object that accounts for another object's movement has to have "what it takes" to do this; and (by the theorem above, actually) we know that all of the objects that do this job have the same "whatever it takes"; and so we then give a name to this identical "whatever it is" that's in all the objects that are causers of this particular effect.

So the result is that we don't know what it is that is in all these "energetic" objects; but whatever it is, it has to be there every time; and so, even though we can't point it out directly, we can know that it's there by its effect.

Hence, this theorem, which seems so trivial, turns out in practice to be extremely useful. Most of physics, in fact, is talking about causes in just this abstract way and describing what they are in terms of their being "whatever it is that accounts for" this or that aspect of some effect. And physics has come a long way following Newton's "I make no guesses" as to what the cause actually is "in itself" and being very careful to say no more than what can be said on its being just exactly what is necessary to account for some effect.

Theorem IV: Different effects have different causes.

Proof: Since, as we saw in the theorem on identical effects, a given effect is defined as this one and no other by what is removed (or missing) from its intelligibility (and this missing fact is the cause), it immediately follows that if Effect A is different from Effect B, the facts missing from the intelligibility of A are different from those which are missing from B; which is to say that Effect A has a different cause from the cause of Effect B. Q. E. D.

The second proof goes this way: If different effects were to have the same cause, then the difference between them is irrelevant to their unintelligibility (since they are made intelligible in exactly the same way--which is what "having the same cause" means). But what is "irrelevant to their unintelligibility" means "not part of them as effects," since the effect is nothing but the unintelligibility of the objects in question.

Therefore, in this case, the difference between the effects is irrelevant to their unintelligibility, which means that the effects are not different as effects, but only as affected objects. So if different effects have identical causes, they are not different as effects, which means that they aren't different effects. Therefore, different effects have to have different causes. Q. E. D.

Once again this theorem has widespread application in the sciences, and in that most precise of the sciences, physics. If the branch you saw moving because of the squirrel and because of the breeze were moving differently each time--say with different amplitude of its swing--then the energy applied to it must be different. An application of the same energy to the same part of the same object cannot result in different movements (unless, of course, there is something different about the object's mass the second time to account for the difference--but then this would be part of the cause, making the second cause different). How does physics know this? Because then "the same effect" doesn't make any sense, when you are at this abstract level of thinking.

This abstract level of thinking, by the way, is the cause of many people's difficulties with physics. People want to see what they're referring to, and all of these mysterious things like "energy" and "entropy" which can be "seen" only as derivatives and triple integrals and so on and aren't actually some part of the situation that you can look at and "see what it's like" give them the heebie-jeebies. If you happen to be that kind of person, I pity you if for some reason you are required to read this book; because the level we're going to be operating on is even more abstract than physics.

As long as we are engaged at the moment, however, in the quasi-mathematical procedure of proving theorems, let me now state two corollaries of the theorems we have just proved:

Corollary I: Identical causes have identical effects.

Proof: Suppose there were two identical causes that had different effects. That would mean that in this case there would be different effects that had identical causes, and we just proved that different effects always have different causes. Q. E. D.

Corollary II: Different causes have different effects.

Proof: Suppose there were two different causes that had identical effects. That would mean that in this case there would be identical effects that had different causes, and we proved in Theorem III that identical effects always have identical causes. Q. E. D.

So once both theorems above have been established, you find that you can go both ways with respect to identity and difference; and if you know that a given cause is operating, you can predict what its effect will be, based on what the effect of this cause was in other cases.

But you must be very, very careful here, because it is so easy to confuse the cause and the causer. It does not follow that if you have the same causer, it will produce an identical effect, because different aspects of it may be operating, or there may be a difference in what is affected which actually enters into the causality. For instance, identically the same force (energy-as-applied; I want to use Newton's equation and need this more technical term) applied to different masses will produce different movements, because the movement as an effect has as its cause not only the energy-as-applied but the mass of the affected object. And this can be seen from Newton's equation:

F = m a

(Force equals mass times acceleration.) The movement itself, however, is the acceleration; and if we want to isolate that as an effect, the equation becomes:

a = F/m,

which clearly shows that the acceleration "depends on" (is the effect of) both the force in the causer and the mass of what is affected.

So reasoning from causes to effects is very tricky, because the causes in this abstract sense are in general unobservable, and a given causer can produce all sorts of different effects. Nevertheless, it can be done if you are careful enough; and, in fact, it is what allows sciences to predict.

Notes

1. Quod Erat Demonstrandum: "[Which is] what had to be proved," the traditional ending for the proof of a theorem in Euclidean geometry.