Chapter 4

Theory and verification

Once the hypothesis has passed the test of the experiment, it is no longer called a "hypothesis" but a "theory." The word "theory" comes from the Greek theoría, which means "a looking at" or in English, "a way of looking at things." So it's not something presupposed any more, but it makes up part of our attitude toward the world.

The idea behind this is that a theory is to be accepted, absent evidence to the contrary. Why? Because something doesn't make sense without it, and makes sense with it. If you didn't have a reason for rejecting it, then you would be accepting the world as unreasonable. The theory may not be true, because the experiment, as I said, hasn't been able to verify it (in the sense of "prove it true"), but just has failed to falsify it; still, the fact that it hasn't been verified is no reason for rejecting it, because if you don't have any reason other than that it hasn't been verified, your rejection means the acceptance of the unexplained effect, or the acceptance of a contradiction for which you have no resolution (or for which you have an untested "explanation," which amounts to the same thing).

But as usual, things are not so simple. There is, for instance, the Ptolmaic theory of the earth-centered universe with the heavenly bodies circling in spheres and epicycles around it, which hasn't really been falsified, since if you wanted to, you could fix it up to fit in with all the observations up to the present. Interestingly, Newton's view of the universe has been falsified, as we will see shortly in discussing predictions. But there is Einstein's finite but unbounded universe, which in itself sounds even more bizarre than Ptolemy's. Why is one accepted and the other rejected?

The answer is that there are three basic checks on a theory, which don't prove it, but allow you to choose between competing ones that fit the facts so far observed; they are simplicity, logic, and comprehensiveness.

The simplicity of a scientific theory obviously does not mean that it is simple to understand. General Relativity is a simple theory of bodies' motions in space, but you have to know esoteric mathematics like the tensor calculus and a good deal of physics to be able to follow it. The Ptolmaic theory of heavenly bodies is much easier to understand, but it's not simple.

Simplicity is just an application of Occam's Razor, which I have mentioned a couple of times in passing in this book. It is time to see why it is a good canon of a theory. First of all, the notion, formulated by William of Occam, says that a theory is better the fewer things not in evidence it assumes to be true; and the ideal, of course, is none at all or just one. (It's called a "razor," of course, because you "shave off" everything from the cause except what's absolutely necessary for the effect to be what it is.)

Now why should this be a criterion for a good theory? Who says that you're more likely to hit at the correct explanation if you pick one that doesn't have many parts to it rather than one that has a lot of them? We see every day events depending on the convergence of a huge number of other events. Why did the tree in my back yard grow there? Let us say that it was because a squirrel picked up the nut from its parent tree and instead of eating it buried it there, at the edge of but not in my lawn, so that the sapling didn't get mowed down, and it was in soil that was rich from the grass clippings and leaves rotting above it; but it was not so far in the woods that it didn't get light--and so on and so on. The cause of a singular event like this is often a chance concatenation of an enormous number of factors, none of which can be left out. Simple explanations (i.e. explanations that reduce everything to one factor) are in cases like this simplistic explanations; and it is the sign of the fanatic that he doesn't recognize this.

But for this very reason these explanations are not theories in the scientific sense of the term. It isn't, as scientists so often say, that they aren't "repeatable." Theories about the evolution of the universe are not testable by "repeating" evolution, and they are theories (and testable, by the way).

No, the reason lies deeper in what you mean by a theory as an explanation. I stressed at the beginning of this section that science wasn't interested in finding out facts so much as it was in making sense out of the otherwise unintelligible facts that confront it. True, the cause that makes sense out of the unintelligibility will also be a fact; but it is sought not because it is another fact to know, but because it is the fact that makes sense out of the effect.

Now if we look at a complex theory, we will see why it is that a theory is better the simpler it is. Take the Ptolmaic theory of the heavenly bodies. It assumes that each body is on a sphere that is centered on the earth as the center of the universe and is rotating around it. The planets, however, are on little spheres on the surface of their main sphere, and as the main sphere rotates, the little one does also, making the planet move erratically as seen against the background of the sphere on which the stars appear (which of course moves with perfect regularity around the earth once every sidereal day). The different speeds of the spheres and the different distances make them appear in the different positions with respect to each other in the course of the years.

All right, but now what connects them all into a system? How are they interrelated? There is no answer to this in the Ptolmaic view; they just happen to be arranged in such a way that the appearances are what they are. This is one of the reasons why it is so easy to fix up this view to fit new observations; if Mercury, for instance, is in a position slightly different from what past and less accurate observations would lead you to expect, you just adjust the size of Mercury's sphere or its epicycle or its distance until the motion fits the observation. If stars are discovered to move with respect to each other, you put them on different epicycles within the sphere of the stars; if astronauts go through these spheres on the way to other planets, then you just make the spheres penetrable--force-spheres, not bodies of crystal. And so on.

So ultimately your explanation of the motions of the heavenly bodies is "they're just arranged this way, that's all." But that's no explanation. As you can see from the discussion of probability, chance cannot be a cause. Insofar as the factors in some complex theory, therefore, are not connected, then they just happen to be together by chance; and insofar as they are together by chance, the explanation is no explanation at all. It's like what the medievals were accused of saying (the serious ones didn't): "it's the will of God" when confronted with strange and anomalous events. Since that was the "explanation" of anything and everything, it is the explanation of nothing.

But then when Newton developed his Theory of Universal Gravitation, all you needed to assume was two things: (a) that bodies were attracted to each other proportionally to the product of their masses and inversely as the square of the distance between the centers, and (b) there was an initial tangential velocity (i.e. one at right angles to the line between the centers) that was great enough to prevent their falling into each other.

As to this second point, you don't even have to assume some "centrifugal force" as a separate force. If you throw a ball parallel to the surface of the earth, it will curve downward in a steadily increasing arc (a parabola, if you're interested). The harder you throw it, the farther it will go (the shallower the arc) before it hits the ground.

Now suppose you are standing on the summit of Mount Everest so that there are no obstructions ahead of you at this height anywhere in the world (another thought-experiment, notice), and you throw the ball very hard straight out. It will curve downward toward the earth. But the earth itself is curved; and so if you throw it hard enough, the arc it is traveling in toward the earth might be shallower than the curvature of the earth, in which case it will miss the earth and continue on around it, and eventually wind up hitting you on the back of the head. Now of course, to make this work, you had better be on the moon where there's no air rather than on the earth, and you had better develop your pitching arm rather thoroughly; but I think you can see the principle. Given an initial tangential motion of the proper speed ("orbital velocity"), then the single force that makes bodies fall down keeps satellites up. Any speed beyond this just changes the shape of the orbit until "escape velocity" is reached. But that's another story.

Now this one force of gravity ties all the planets together into a single solar system around the sun, explains why the orbits aren't circles but ellipses (that's what depends on the initial speed), explains what the sun itself is doing inside the galaxy we call the "milky way," explains the shape of that and other galaxies, and explains systems of stars and galaxies. About the only thing it doesn't explain is why all the galaxies are moving away from each other (except the ones locked into a system like the milky way and our little companion that can only be seen from the southern hemisphere); and for this an initial explosion (the "big bang") had to be added.

Now there is an explanation. Assume just this one fact, that there is a force of attraction between bodies due to their mass, and falling bodies make sense, orbiting satellites make sense, planetary systems (with satellites like the moon around the planets) make sense, galaxies make sense.

Unfortunately, it's the wrong explanation, as we'll see. But you can understand why this is a theory that, absent evidence to the contrary, is to be preferred as an explanation to Ptolemy's. It actually explains; Ptolemy's doesn't.

So the reason a simple theory is preferable to one that assumes more in evidence is not really because it is "truer" by that fact; it is because the more complex one relies on coincidence among its parts, and coincidence precisely doesn't explain.

Now of course, theories can have complex parts if they can show what the relation is among them and don't just have them working together by chance. But of course, in that case what connects them is the true cause; and so even if the theory has complex parts, it's still basically a simple theory, because ultimately it rests on the one fact which connects all the parts.

If you look at this theory of science of mine, you will see that it rests on the one fact that scientists know that contradictions don't really occur, and yet they find evidence of contradictions in what confronts them (effects). Given that one fact, everything else follows: observation, hypothesis, experiment, theory, and (as we will see) verification. So this theory of science is a simple theory of science, even to explaining why simplicity is a criterion of a good scientific theory.

Now of course, the criterion that the theory has to be logical simply means that you should be able to deduce all of the otherwise contradictory effects from the cause by the "p implies q" type of reasoning, where "p" is your statement of what the cause is, and "q" is the event in question. For instance, if theories are supposed to be explanations of events that are otherwise contradictory, then it follows logically that simplicity in the sense discussed above would have to be a criterion of a good theory. And it has been recognized as one, by people who knew it worked, but didn't know why.

The third criterion is actually connected with the second; it is the criterion that the theory has to be comprehensive. What that means is that the theory has to explain all of the aspects of the problem in question, or it fails as a theory. If one tiny part of the effect remains unexplained by the theory, then the theory can't be stating the cause of the effect, because part of the world remains self-contradictory under it, and the theory's whole purpose is to show how the world, assuming it, is not self-contradictory. (Once again, notice that it is effect and cause that explain why this criterion is a criterion of a good theory.)

It does not matter how insignificant this aspect of the world is; if it is such that the theory has to make sense out of it, and the theory doesn't, then the theory is wrong. We have seen any number of examples of this in the course of this book. To take just one that comes to mind, Skinner's supposedly "scientific" theory of why we think we're free when we're not, that we aren't aware of what's forcing us to choose and do something. But, as I mentioned, that would mean that compulsives, who fit the antecedent, would then have to feel free, and they don't. Hence, his theory doesn't explain something that it has to explain, and so it must be rejected.

Or take another famous case, that of the Newton's Theory of Universal Gravitation, which was supposed to explain the orbits of the planets, including their shapes and so on. One of the things his theory does is say that when there is something like the sun that is basically determining the orbit around it of Mercury, say, and there are also the other planets pulling on Mercury from outside, even though they are moving around the sun themselves (and so are in different positions at different times), the orbit of Mercury will "precess" due to these "perturbations." Precession is what a spinning top does as it begins to slow down and the whole top as it tips begins moving around in a circle.

Imagine Mercury's orbit, then, as an oval, with the sun offset toward one end of it. The end nearest the sun is called the "perihelion," because the Greek word means "nearest the sun." Now if you imagine the whole orbit moving in a circle around the sun (i.e. with the perihelion point moving in a circle around it), you get the basic theoretical picture of precession. Of course, the actual motion of Mercury is like that child's toy of many years ago the "Spirograph," where a pen traced intricate patterns by being attached to intermeshing circular gears; but that need not worry us.

Obviously, to figure out what the precession of Mercury would have to be due to the presence of the orbiting earth, and also of Venus, and Mars and Jupiter and so on was no small undertaking; but it was done and it agreed with the observations on Mercury--until the beginning of this century, when more accurate instruments and calculations showed that Newton's view of how much the precession should be was off by a matter of (as I recall) four seconds of arc per century. To make this intelligible, an arc of 90 degrees is an arc which is the part of the circumference of a circle cut off by radii which make an angle of 90 degrees at the center of the circle. An arc, then, of one degree is one three hundred sixtieth of the circumference; and arc of one minute is a sixtieth of this, and an arc of one second is a sixtieth of that. The precession of Mercury's orbit was off by four of those per century. Not, you would say, enough to amount to a hill of beans.

Nevertheless, it was enough to destroy the theory. Newton said that Mercury had to be here today, if his theory was true; but Mercury was over there, a few yards off in the millions of miles of its orbit. People checked and rechecked, and couldn't make the observations agree with the calculated position, and couldn't make the calculations come out different. Something that was supposed to be explained couldn't be explained. "(p implies q) and not q implies not p."

Einstein then came along with a different notion (warping of space-time instead of a force of attraction) and explained all that Newton explained plus the location of Mercury which Newton's theory couldn't explain; and that's why Einstein's view is held and Newton's isn't. Einstein's (as far as we now know) is comprehensive; it explains all that it's supposed to explain. Newton's, for all its simplicity and elegance, isn't; and so it's just wrong.

Now connected with this notion of comprehensiveness is that it almost inevitably results in predictions from the theory, and gives rise to the final step of scientific method, that of verification, which as always is at best "non-falsification."

Actually, the problem that destroyed Newton's theory could be called a falsified prediction. But to see why, we have to see what the basis of predictions is. And once again, the notion of effect and cause gives the explanation.

It is in practice impossible for your initial observation to take in every aspect of the effect in question, especially if it is an effect of any generality at all, such as the effect connected with the fact that bodies fall down at a constant acceleration. Hence, the explanation you come up with in your hypothesis, if it is really the cause of the effect, will, of course explain all the aspects of the true effect, not just the ones you happened to have seen and which piqued your curiosity; and not even just the ones you ran across in your careful observation.

Hence your "p" in the "p implies q" will almost inevitably actually have more logical implications than the ones you happen to have observed; and these will be the predictions from your theory. Newton didn't have the orbit of Mercury before him as he developed his theory, I assume; but the theory, as accounting for all motions of all heavenly bodies, would have to include the orbit of Mercury; and so you could predict the orbit from the theory. Unfortunately, it turned out to be different from what the theory predicted, and this destroyed the theory.

Einstein's theory, in fitting the observations of Mercury, also would, of course, apply to the other planets; and so his theory predicted a similar divergence from Newtonian calculations in the orbit of Venus; and this was checked and found to be as Einstein said it would be.

Further, since his theory said basically that bodies left to themselves fall (move with constant acceleration) in straight lines (shortest distance between points); but that in the presence of massive objects, space gets warped out of Euclidian shape, so that straight lines no longer look like what Euclid thought they did, and are sometimes orbits (I kid you not) in Einstein's geometry, then it follows logically from this that anything that travels through space in straight lines (even massless things like light) will be following the weird-shaped straight lines of the new geometry, and from our Euclidian perspective, will travel a curved path.

The theory therefore predicted that during an eclipse, when the sun is dark enough so that you can see the starts behind it, the stars seen near the sun will appear to be in the wrong places, because the light coming to us from them will be bent along the curve around the sun (i.e. the straight line between them and us will be a Euclidian curve). And observations of the starts in the background of an eclipsed sun showed that they were not in the positions we knew them to be, but appeared to be--just where Einstein said they would appear to be. Another prediction verified. This displacement would have to occur if Einstein's theory is true, because it logically follows from the theory.

But of course, the fact that it occurs doesn't prove the theory true, as I have so often said, because nothing follows from "p implies q and q."

Nevertheless, insofar as the predictions predict events that are very unlikely on any other supposition than the theory, the theory is on that much firmer ground. If light has no mass, it can't be attracted by massive objects, it would seem; so why should it be bent around them? But the General Theory of Relativity doesn't suppose a "force" of gravity at all, but just a warping of the geometry of space.

Note that how space gets warped and what it means to have "nothingness" warped is not something Einstein undertakes to answer, and it is his right not to have to. From the fact that space is warped, the rest follows, and so he can start from this as his explanation of what is implied by it, without having to go back to its own explanation. All that means is that, to the extent that the fact he uses as his cause doesn't make sense by itself, he has not got the ultimate explanation.

That is, as I pointed out when discussing effects and causes in Section 2 of the first part, all you need is some fact which is necessary to account for your effect; you don't need to go behind it to the condition (cause of the cause) for the effect. So there is no need to fault Einstein for not explaining how space-time can be warped in the presence of massive bodies. I tried to give some hint of what might be behind this in the discussion of distance, position, and space in Chapter 5 of Section 1 of the second part 1.1.5.

So not every scientific theory has to be "repeatable"; but it would be rare indeed for any theory not to have implications beyond what were the initially observed factors of the effect; and so it is all but inevitable that theories will predict--and that they will predict things that can be checked.

I have tried to show in the course of this book that this applies to philosophical theories as well as to scientific ones. A major reason why I disagree with the theories I disagree with isn't that I don't find them "congenial" to my Weltanschauung, but that I have discovered predictions from them (like the Skinnerian prediction above) that just don't fit the facts. And since I have something of a scientific turn of mind, I can't accept them under those conditions.

My own theory of thinking and reasoning, by the way, predicts that you ought to be able to take every aspect of human mental activity under its umbrella and show how it follows from trying to know relationships among objects (or relationships among relationships among objects) and how we try to reconnect objects so that we can see new relationships that we haven't seen so far.

And up to this point, I have been able to show why there are the different modes of thought of mysticism, logic, mathematics, and now science; and I hope to show in the next chapter how this basic insight also explains art, and in the following one how it takes evaluation into account.

I don't see any reason why we shouldn't draw out the logical consequences of philosophical theories and test them; and if they predict things that aren't so, reject them. That's what this by now immense tome is partly about; the rest of it is an attempt to develop a theory of the world and our place in it (including our knowledge of it) that will predict things that stand up to the test. If you are reading this and I have been dead for a while, that in itself is a prediction from my theory (because it's my ambition that this should be so, and a prediction from Chapter 4 of Section 4 of the third part 3.4.4 is that legitimate ambitions we carry beyond the grave will be fulfilled).(1)

There are only two brief topics left that I want to discuss in this superficial sketch of scientific thinking: why scientists use models, and what a scientific law is.

First of all, models in scientific theory are looked upon as metaphors, and they are really analogies. Metaphors, as we will see in the next section, are esthetic facts, not analogies. When we say that the meadow is smiling, we are not drawing an analogy between the meadow and a smiling face, or there would be some indirect perceptive similarity between the two. Analogies, if you will refer back to Chapter 7 of Section 2 of the first part 1.2.7, are similarities in causes that are known only by the fact that the effects are similar. That is, because the effects are similar, then the theorem that similar effects have analogous causes comes into play--and so you know the fact that the causes are somehow similar, without knowing the points of similarity. Metaphors like the smiling meadow, however, are simply using the emotions as receiving instruments analogous to sense organs (since the emotions do, as we saw in Chapter 5 of Section 2 of the third part 1.2.5, respond to outside energy as well as the state of the body); and so just as "the meadow is green" means "the meadow has in it the cause of my eyes' reacting the same way they do when I look at emeralds and so on," so "the meadow is smiling" means "the meadow has in it the cause of my reacting emotionally the same way as I do when someone smiles at me." So it is silly to examine meadows to try to find where the lips are.

On the other hand, if you happen to know that the "q" from this particular theory looks a lot like the "q" from some other theory, then the theorem of similar effects takes over, and you can argue to some kind of similarity between the "p's" of the two theories. Hence by examining the effects that are similar to the effects of your theory, you can actually learn something about the cause in relation to the cause of those effects.

So, from noticing that the equation of an electron (which we can't observe directly, because it's too small) looks a lot like the equation of motion of a little speck of dust, with things that seem to resemble the three-dimensional translational motion and also the spin of the particle on each of its three axes, we can say that by analogy the electron is like a little particle, and can then talk about its "spin," referring to whatever about it makes its equation look like the spin of a particle.

This does not mean that an electron is a tiny particle, however; because one thing that particles don't do is interfere with one another like waves; but electrons do. That is, the equation of an electron also has some aspects to it that look very much like what happens if you shake a length of rope and the hump moves down the rope, and then you shake it "out of phase" with your initial one, and parts of the hump get bigger and parts get smaller. That's the kind of interference I mean. I talked about it earlier when discussing position in Chapter 5 of Section 1 of the first part 1.1.5.

But particles aren't waves and waves aren't particles. Right. So electrons aren't similar to particles in the sense that they're particles too small for us to see; they're analogous to particles in the sense that there is something in common between electrons and particles, but we don't know exactly what. Whatever it is, it's what is responsible for the similarity in the equations. By the same token, there's an analogy and not a direct similarity between electrons and waves; and obviously whatever it is about the electron that makes it analogous to a wave is compatible with whatever makes it analogous to a particle, even though in the macroscopic world, waves and particles are incompatible. Well yes; but who says that just because waves and particles are incompatible, what is in some unknown way similar to a wave can't be in some other unknown way similar to a particle?

In any case, scientists use models because they are analogies, not metaphors; they aren't sneaking in a little artistry on the side. You can really learn something by studying a model; you can study meadows until you're blue in the face, and you won't learn anything perceptive about smiling faces.

And once more, it is the theory of effect and cause, as developed back in Section 2 of the first part of this book, that explains why scientists are so enamored of models, in spite of the fact that they just look like poetry.

Finally, what is a "scientific law," and why are theories that have been verified called "laws"?

Let's make a definition of a law first.

A scientific law is a description of some invariant relationship.

The difference, then, between a theory and a law is that a law just states a (constant) fact, and a theory is an explanation of an effect. For instance, the law of falling bodies is that in fact no matter what their weight, they all fall to the earth at the rate of 32 feet per second per second. The theory of gravitation explains this by the force of gravity that is proportional to the masses of the bodies and the earth and the inverse of the distance between their centers. To take another example, Boyle's and Charles's laws of gases say that a gas (to oversimplify) increases in volume or pressure 1/273rd for each degree Celsius over zero Celsius. The kinetic-molecular theory of gases says that heat is the speed of molecular motion, and -273o Celsius is the point at which the speed is zero (motion stops). It therefore explains the expansion in that molecules moving faster hit each other (and also the container's walls) harder, and so increase the pressure or the expansion (if it's something like a balloon). It also explains, of course, why the expansion is 1/273rd for every degree.

So the law simply states a fact, while the theory states as a fact something that is the cause of some other fact. Then why do people say that well-verified theories (i.e. theories that fit the three criteria above and have predictions that are true) "become" laws? It is simply that these unobserved explanations are then taken to be facts.

As I said earlier, any theory is to be accepted as a fact if there is no evidence to the contrary; because even though it might be false (and things like what happened to Newton's Theory of Universal Gravitation are always possible--after all, the discovery of the failure of his theory happened centuries after his death), you have no reason for saying it is false, and you have reason for saying that it is true. Hence, if you refer back to Chapter 5 of Section 1 of the first part 1.1.5, you have physical certainty of its truth.

So just because a theory can't be proved true (in the sense that its falseness would be a contradiction), this is no reason for rejecting it when it doesn't fit your lifestyle or is inconvenient on the grounds that "well, it's just a theory." You reject it under pain of condemning yourself to irrationality; and what possible reason could you have for choosing irrationality over rationality?

I have said this often in the course of this book, but it needs saying (at least during the time I am alive) again and again. As I originally wrote this, I that noontime read a review of two feminist books including articles by philosophers who realized that their positions contradicted themselves; but said, "But we have to hold both horns of the dilemma, and simply use which is more convenient to serve women's interests"--after showing that there couldn't even be "the interests of women as a group." By the time you are reading this, I hope this deconstructive aberration will have sunk into the cesspool of repudiated thought where it belongs.



1. Judging by the lives of people like Richard Wagner, you probably don't even have to be a saint to have them fulfilled, which gives me a good deal of encouragement, even though one of my ambitions happens to be to be a saint. Somehow or other, by the time the end comes, I'm going to stop fighting my Master.