The Analysis of Matter

a postulate something not very different from a working hypothesis, except that it is more general: it is something which we assume without sufficient evidence, in the hope that, by its help, we shall be able to construct a theory which the facts will confirm. It is by no means essential to science to assume that its postulates are true always or necessarily; it is enough if they are often true. They ought to be so used that, when they are true, they yield verifiable theories, but, when they are not true, no theory can be framed which will fit the facts—until we find a way of working with different postulates. [Pg 168] The most important postulate of science is induction. This may be formulated in various ways, but, however formulated, it must yield the result that a correlation which has been found true in a number of cases, and has never been found false, has at least a certain assignable degree of probability of being always true. I propose to assume the validity of induction, not because I know of any conclusive grounds in its favour, but because it seems, in some form, essential to science and not deducible from anything very different from itself. I do not propose to discuss it, because the problem concerns empirical knowledge in general, not physics in particular; also because the subject is so complicated that a discussion is useless unless it is very lengthy. For the moment I must refer the reader to Mr Keynes and his critics.[39] The other postulates which were at one time thought necessary have gradually been found to be superfluous. At one time, the indestructibility of matter would have been regarded as a postulate. Now, though electrons and protons are supposed to persist as a rule, it is seriously suggested that an electron and a proton may sometimes combine so as to annihilate each other; Eddington has advanced this as an important possible source of stellar energy.[40] It is true that, in this process, energy is supposed to be not destroyed; but the conservation of energy is no more than an empirical generalization, and is not thought to be strictly true. Spatio-temporal continuity was, until lately, a postulate of science, but the quantum theory has called it in question without intellectual disaster. It may be true, but we cannot say that it must be. The existence of causal laws perhaps deserves to rank as a postulate, or may perhaps be proved probable, on the existing evidence, if induction is assumed. Here our proviso is relevant, that a postulate need not be supposed to hold universally. We shall assume that there are causal laws, and try to discover them; but if none are found in a given region, that merely means that science cannot conquer that region. There are at present important regions of this kind. We do not know why a radio-active atom disintegrates at one moment rather than another, or why a planetary electron changes its orbit at one moment rather than another. We cannot be sure that these occurrences severally are governed by laws; but if they are not, science cannot deal with them individually, and is confined to statistical averages. Whether this will prove to be the case, we cannot yet say. FOOTNOTES: [37] Jeans, op. cit., p. 29. [38] To show the depth of Dover cliff, Shakespeare says: "The crows and choughs that wing the midway air Show scarce so gross as beetles." [39] A Treatise on Probability. By John Maynard Keynes. Macmillan, 1920. Le Problème Logique de l'Induction. Par Jean Nicod. Paris, Alcan, 1924. Review of the above by Braithwaite, Mind, 1925. The Foundations of Probability. By R. H. Nisbet. Mind, January, 1926. [40] Nature, May I, 1926, supplement. [Pg 169] CHAPTER XVII WHAT IS AN EMPIRICAL SCIENCE? 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[Pg 170] 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[Pg 171] 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[Pg 172] 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[Pg 173] 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