The analysis of matter

IT would be generally agreed that physics is an empirical science, as contrasted with logic and pure mathematics. I want, in this chapter, to define in what this difference consists.

We may observe, in the first place, that many philosophers in the past have denied the distinction. Thorough-going rationalists have believed that the facts which we regard as only discoverable by observation could really be deduced from logical and metaphysical principles; thorough-going empiricists have believed that the premisses of pure mathematics are obtained by induction from experience. Both views seem to me false, and are, I think, rarely held in the present day; nevertheless, it will be as well to examine the reasons for thinking that there is an epistemological distinction between pure mathematics and physics, before trying to discover its exact nature.

There is a traditional distinction between necessary and contingent propositions, and another between analytic and synthetic propositions. It was generally held before Kant that necessary propositions were the same as analytic propositions, and contingent propositions were the same as synthetic propositions. But even before Kant the two distinctions were different, even if they effected the same division of propositions. It was held that every proposition is necessary, assertoric, or possible, and that these are ultimate notions, comprised under the head of "modality." I do not think much can be made of modality, the plausibility of which seems to have come from confusing propositions with propositional functions.

Propositions may, it is true, be divided in a way corresponding to what was meant by analytic and synthetic; this will be explained in a moment. But propositions which are not analytic can only be true or false; a true synthetic proposition cannot have a further property of being necessary, and a false synthetic proposition cannot have the property of being possible. Propositional functions, on the contrary, are of three kinds: those which are true for all values of the argument or arguments, those which are false for all values, and those which are true for some arguments and false for others. The first may be called necessary, the second impossible, the third possible. And these terms may be transferred to propositions when they are not known to be true on their own account, but what is known as to their truth or falsehood is deduced from knowledge of propositional functions. E.g. "it is possible that the next man I meet will be called John Smith" is a deduction from the fact that the propositional function " is a man and is called John Smith" is possible—i.e. true for some values of and false for others. Where, as in this instance, it is worth while to say that a proposition is possible, the fact rests upon our ignorance. With more knowledge, we should know who is the next man I shall meet, and then it would be certain that he is John Smith or certain that he is not John Smith. Possibility in this sense thus becomes assimilated to probability, and may count as any degree of probability other than 0 and 1. An "assertoric" proposition, similarly, was, I think, a confused notion applicable to a proposition known to be true but also known to be a value of a propositional function which is sometimes false—e.g. "John Smith is bald."

The distinction of analytic and synthetic is much more relevant to the difference between pure mathematics and physics. Traditionally, an "analytic" proposition was one whose contradictory was self-contradictory, or, what came to the same thing in Aristotelian logic, one which ascribed to a subject a predicate which was part of it—e.g. "white horses are horses." In practice, however, an analytic proposition was one whose truth could be known by means of logic alone. This meaning survives, and is still important, although we can no longer use the definition in terms of subject and predicate or that in terms of the law of contradiction. When Kant argued that "7 + 5= 12" is synthetic, he was using the subject-predicate definition, as his argument shows. But when we define an analytic proposition as one which can be deduced from logic alone, then "7 + 5 = 12" is analytic. On the other hand, the proposition that the sum of the angles of a triangle is two right angles is synthetic. We must ask ourselves, therefore: What is the common quality of the propositions which can be deduced from the premisses of logic?

The answer to this question given by Wittgenstein in his Tractatus Logico-Philosophicus seems to me the right one. Propositions which form part of logic, or can be proved by lope, are all tautologies—i.e. they show that certain different sets of symbols are different ways of saying the same thing, or that one set says part of what the other says. Suppose I say: "If implies , then not- implies not-." Wittgenstein asserts that " implies " and "not- implies not-" are merely different symbols for one proposition: the fact which makes one true (or false) is the same as the fact which makes the other true (or false). Such propositions, therefore, are really concerned with symbols. We can know their truth or falsehood without studying the outside world, because they are only concerned with symbolic manipulations. I should add—though here Wittgenstein might dissent—that all pure mathematics consists of tautologies in the above sense. If this is true, then obviously empiricists such as J. S. Mill are wrong when they say that we believe 2 + 2 = 4 because we have found so many instances of its truth that we can make an induction by simple enumeration which has little chance of being wrong. Every unprejudiced person must agree that such a view feels wrong: our certainty concerning simple mathematical propositions does not seem analogous to our certainty that the sun will rise to-morrow. I do not mean that we feel more sure of the one than of the other, though perhaps we ought to do so; I mean that our assurance seems to have a different source.

I accept the view, therefore, that some propositions are tautologies and some are not, and I regard this as the distinction underlying the old distinction of analytic and synthetic propositions. It is obvious that a proposition which is a tautology is so in virtue of its form, and that any constants which it may contain can be turned into variables without impairing its tautological quality. We may take as a stock example: "If Socrates is a man and all men are mortal, then Socrates is mortal." This is a value of the general logical tautology:

"For all values of , , and , if is an , and all 's are 's, then is a ."

In logic, it is a waste of time to deal with particular examples of general tautologies; therefore constants ought never to occur, except such as are purely formal. The cardinal numbers turn out to be purely formal in this sense; therefore all the constants of pure mathematics are purely formal.

A proposition cannot be a tautology unless it is of a certain complexity, exceeding that of the simplest propositions. It is obvious that there is more complexity in equating two ways of saying the same thing than there is in either way separately. It is obvious also that, whenever it is actually useful to know that two sets of symbols say the same thing, or that one says part of what the other says, that must be because we have some knowledge as to the truth or falsehood of what is expressed by one of the sets. Consequently logical knowledge would be very unimportant if it stood alone; its importance arises through its combination with knowledge of propositions which are not purely logical.

All the propositions which are not tautologies we shall call "synthetic." The simplest kinds of propositions must be synthetic, in virtue of the above argument. And if logic or pure mathematics can ever be employed in a process leading to knowledge that is not tautological, there must be sources of knowledge other than logic and pure mathematics.

The distinctions hitherto considered in this chapter have been logical. In the case of modality, it is true, we found a certain confusion from an admixture of epistemological notions; but modality was intended to be logical, and in one form it was found to be so. We come now to a distinction which is essentially epistemological, that, namely, between a priori and empirical knowledge.

Knowledge is said to be a priori when it can be acquired without requiring any fact of experience as a premiss; in the contrary case, it is said to be empirical. A few words are necessary to make the distinction clear. There is a process by which we acquire knowledge of dated events at times closely contiguous to them; this is the process called "perception" or "introspection"[41] according to the character of the events concerned. There is no doubt need of much discussion as to the nature of this process, and of still more as to the nature of the knowledge to be derived from it; but there can be no doubt of the broad fact that we do acquire knowledge in this way. We wake up and find that it is daylight, or that it is still night; we hear a clock strike; we see a shooting star; we read the newspaper; and so on. In all these cases we acquire knowledge of events, and the time at which we acquire the knowledge is the same, or nearly the same, as that at which the events take place. I shall call this process "perception," and shall, for convenience, include introspection—if this is really different from what is commonly called "perception." A fact of "experience" is one which we could not have known without the help of perception. But this is not quite clear until we have defined what we mean by "could not"; for clearly we may learn from experience that 2 + 2 = 4, though we afterwards realize that the experience was not logically indispensable. In such cases, we see afterwards that the experience did not prove the proposition, but merely suggested it, and led to our finding the real proof. But, in view of the fact that the distinction between empirical and a priori is epistemological, not logical, it is obviously possible for a proposition to change from the one class to the other, since the classification involves reference to the organization of a particular person's knowledge at a particular time. So regarded, the distinction might seem unimportant; but it suggests some less subjective distinctions, which are what we really wish to consider.

Kant's philosophy started from the question: How are synthetic a priori judgments possible? Now we must first of all make a distinction. Kant is concerned with knowledge, not with mere belief. There is no philosophical problem in the fact that a man can have a belief which is synthetic and not based on experience—e.g. that this time the horse on which he has put his money will win. The philosophical problem arises only if there is a class of synthetic a priori beliefs which is always true. Kant considered the propositions of pure mathematics to be of this kind; but in this he was misled by the common opinion of his time, to the effect that geometry, though a branch of pure mathematics, gave information about actual space. Owing to non-Euclidean geometry, particularly as applied in the theory of relativity, we must now distinguish sharply between the geometry applicable to actual space, which is an empirical study forming part of physics, and the geometry of pure mathematics, which gives no information as to actual space. Consequently this instance of synthetic a priori knowledge, upon which Kant relied, is no longer available. Other kinds have been supposed to exist—for example, ethical knowledge, and the law of causality; but it is not necessary for our purposes to decide whether these kinds really exist or not. So far as physics is concerned, we may assume that all real knowledge is either dependent (at least in part) upon perception, or analytic in the sense in which pure mathematics is analytic. The Kantian synthetic a priori knowledge, whether it exists or not, seems not to be found in physics—unless, indeed, the principle of induction were to count as such.

But the principle of induction, as we have already seen, has its origin in physiology, and this suggests a quite different treatment of a priori beliefs from that of Kant. Whether there is a priori knowledge or not, there undoubtedly are, in a certain sense, a priori beliefs. We have reflexes which we intellectualize into beliefs; we blink, and this leads us to the belief that an object touching the eye will hurt it. We may have this belief before we have experience of its truth; if so, it is, in a sense, synthetic a priori knowledge—i.e. it is a belief, not based upon experience, in a true synthetic proposition. Our belief in induction is essentially analogous. But such beliefs, even when true, hardly deserve to be called knowledge, since they are not all true, and therefore all require verification before they ought to be regarded as certain. These beliefs have been useful in generating science, since they supplied hypotheses which were largely true; but they need not survive untested in modern science.

I shall therefore assume that, at any rate in every department relevant to physics, all knowledge is either analytic in the sense in which logic and pure mathematics are analytic, or is, at least in part, derived from perception. And all knowledge which is in any degree necessarily dependent upon perception I shall call "empirical." I shall regard a piece of knowledge as necessarily dependent upon perception when, after a careful analysis of our grounds for believing it, it is found that among these grounds there is the cognition of an event in time, arising at the same time as the event or very shortly after it, and fulfilling certain further criteria which are necessary in order to distinguish perception from certain kinds of error. These criteria will occupy us in the next chapter.

In a science, there are two kinds of empirical propositions. There are those concerned with particular matters of fact, and those concerned with laws induced from matters of fact. The appearances presented by the sun and moon and planets on certain occasions when they have been seen are particular matters of fact. The inference that the sun and moon and planets exist even when no one is observing them—in particular, that the sun exists at night and the planets by day—is an empirical induction. Heraclitus thought the sun was new every day, and there was no logical impossibility in this hypothesis. Thus empirical laws not only depend upon particular matters of fact, but are inferred from these by a process which falls short of logical demonstration. They differ from propositions of pure mathematics both through the nature of their premisses and through the method by which they are inferred from these premisses.

In an advanced science such as physics, the part played by pure mathematics consists in connecting various empirical generalizations with each other, so that the more general laws which replace them are based upon a larger number of matters of fact. The passage from Kepler's laws to the law of gravitation is the stock instance. Each of the three laws was based upon a certain set of facts; all three sets of facts together formed the basis of the law of gravitation. And, as usually happens in such cases, new facts, not belonging to any of the three previous sets, were found to support the new law—for instance, the facts of tides, of lunar motion, and of perturbations. Epistemologically, in such cases, a fact is a premiss for a law; logically, most of the relevant facts are consequences of the law—i.e. all except those required to determine the constants of integration.

In history and geography, the empirical facts are, at present, more important than any generalizations based upon them. In theoretical physics, the opposite is the case: the fact that the sun and moon exist is chiefly interesting as affording evidence of the law of gravitation and the laws of the transmission of light. In a philosophic analysis of physics, we need not consider particular facts except when they form the evidence for a theory. It is of course part of the business of such an analysis to consider what all particular facts have in common, and how they come to be known; but such inquiries are general. We are interested in the concept of topography, but not in the actual topography of the universe; at least, we are not interested in it for its own sake, but only as affording the evidence for general laws.

We have, in view of the above considerations, several different matters to consider, before we can return to actual physics. We have first to consider the nature and validity of the process we have called "perception"; next we have to investigate the general character of the facts known by perception; and lastly we have to examine the inference from facts of perception to empirical laws. After disposing of these topics, we shall resume contact with physics, asking ourselves now, not what physics asserts, but what justification it has for its assertions, and what inessential modifications will increase this justification.