The Analysis of Matter

can be no reason to think so, since the group of events will produce exactly the same percepts; therefore the substance at the centre, if there is one, is irrelevant to science, and belongs to the realm of mere abstract possibility. If we can reach the same conclusion as regards matter in physics, we have diminished the difficulty involved in building our bridge from perception to physics. The substitution of space-time for space and time has made it much more natural than formerly to conceive a piece of matter as a group of events. Physics starts, nowadays, from a four-dimensional manifold of events, not, as formerly, from a temporal series of three-dimensional manifolds, connected with each other by the conception of matter in motion. Instead of a permanent piece of matter, we have now the conception of a "world-line," which is a series of events connected with each other in a certain way. The parts of one light-ray are connected with each other in a manner which enables us to consider them as forming, together, one light-ray; but we do not conceive a light-ray as a substance moving with the velocity of light. Just the same kind of connection may be held to constitute the unity of an electron. We have[Pg 245] a series of events connected together by causal laws; these may be taken to be the electron, since anything further is a rash inference which is theoretically useless. What is peculiar about a string of events which physics takes as belonging to one electron is a character which is present approximately in the common-sense "thing," a character which I should define as the existence of a first-order differential law connecting successive events along a linear route. That is to say, given an event belonging to an electron at one place in space-time, there will be other events at certain neighbouring regions of space-time, separated from the first and from each other by small time-like intervals, such that, when the intervals are taken small enough, if , , are three such events, and the interval between and is equal to that between and , then the difference between and tends towards equality with the difference between and , in certain measurable respects. This is a way of saying that accelerations are always finite—or, where they are not (as perhaps in quantum phenomena), there are other characteristics involved which are subject to a condition analogous to finite acceleration. Let us take first the common-sense "thing." If I watch a moving object, I have a series of percepts which change gradually, both as regards position and as regards qualities—colour, shape, etc. The gradualness of the change is the criterion by which I am led to regard the percepts as all belonging to one "thing." But on a common-sense basis there are exceptions, such as explosions. Science deals with these as rapid, but not instantaneous, changes, and so removes the exceptions. We thus arrive at the conclusion that, given an event at a time , there will be closely analogous events at neighbouring times. We may symbolize this by saying that, if there is an event at time , there will be, at any neighbouring time an event: [Pg 246] where is a continuous function of the time, while is determined by the second-order differential equations of physics. The string of events so connected is called one piece of matter. In the case of the sudden changes contemplated by the quantum theory, there is still continuity in everything except spatial position, and the spatial position undergoes a change which is one of a small number of possible changes. Thus in this case also the new occurrences can be causally connected with the old, though the laws of the connection are somewhat different from what they are in the usual case. Thus the string of events constituting one material unit is distinguished from others by the existence of an intrinsic causal law, though this law is only differential. A light-wave, in this respect, is analogous to a material unit; it differs in the fact that it spreads spherically instead of travelling along a linear route.[55] It will be seen that, if a piece of matter is a string of events, the distinction between motion and other continuous changes is not so simple as it seemed. We could form continuous series of events which would not all belong to one piece of matter; therefore the change from one to another would not be a "motion." A "motion" is a string of events connected| with each other according to the laws of motion. This might seem like a vicious circle, but in fact it is not. What we assert is: Strings of events exist which are connected with each other according to the laws of motion; one such string is called (me piece of matter, and the transition from one event in the string to another is called a motion. This contains as much as can be verifiable in physics, since every percept is an event. There is no mathematical advantage in asserting more, and to assert more is to go beyond the evidence. Therefore it is prudent, in physics, to regard an electron as a group of events connected[Pg 247] together in a certain way. An electron may be a "thing," but it is absolutely impossible to obtain any evidence for or against this possibility, which is scientifically unimportant, because the group of events has all the requisite properties. The light thrown on the notion of substance by the connection between physics and perception, which was the third branch of our problem, has already been touched upon. We saw in former chapters that the physical object to be inferred from perception is a group of events, rather than a single "thing." Percepts are always events, and common sense is rash when it refers them to "things" with changing states. There is therefore every reason, from the standpoint of perception, to desire an interpretation of physics which dispenses with permanent substance. As we have seen that such an interpretation is possible, we shall henceforth adopt it. There is, however, a view not uncommon in philosophy, and perhaps nearer to common sense than the view which I have adopted. This view is, I think, that of Dr Whitehead. It holds that the different events which constitute a group—whether those which make up a physical object at one time or those which make up the history of a physical object—are not logically self-subsistent, but are mere "aspects," implying other aspects in some sense which is not merely causal or inductively derived from observed correlations. I consider this view impossible on purely logical grounds, and have so argued elsewhere. But at the moment I prefer to argue that it is empirically useless. Given a group of events, the evidence that they are "aspects" of one "thing" must be inductive evidence derived from perception, and must be exactly the same as the evidence upon which we have relied in collecting them into causal groups. The supposed logical implications, if they exist, cannot be discovered by logic, but only by observation; no one, by mere reasoning, could avoid being deceived by the three-card trick. Moreover, in calling two events "aspects" of one "thing," we imply that their likeness is more important than their difference; but for science both are facts, and of exactly the same importance. One may say that the theory of relativity has grown up by paying attention to small differences between "aspects." I conclude, therefore, that the "thing" with "aspects" is as useless as permanent substance, and represents an inference which is as unwarrantable as it is unnecessary. FOOTNOTES: [52] Tractatus Logico-Philosophicus. [53] Cf. Analysis of Mind, chap. X. [54] See Principia Mathematica, vol. I., Introduction to second edition. [55] The non-substantial character of the election emerges even more forcibly from the Heisenberg theory mentioned in Chapter IV. than from the older theory. [Pg 249] CHAPTER XXIV IMPORTANCE OF STRUCTURE IN SCIENTIFIC INFERENCE THE inference from perception to physics, which we have been considering, is one which depends upon certain postulates, the chief of which, apart from induction, is the assumption of a certain similarity of structure between cause and effect where both are complex. I want, in this chapter, to inquire more closely into this postulate, not with a view to establishing its validity, which I shall take for granted, but with a view to discovering what it asserts and what are its consequences. The first point is to be clear as to what we mean by structure. The notion is not applicable to classes, but only to relations or systems of relations. It is fully defined, and made the basis of a general kind of arithmetic, in Principia Mathematica.[56] But as the later parts of that book are not read, I may be excused for repeating, in outline, what is needed for our present purposes. Two relations , are said to be "similar" if there is a one-one relation between the terms of their fields, which is such that, whenever two terms have the relation , their correlates have the relation , and vice versa. The most familiar example is that of series: two series are similar when their terms can be correlated without change of order. But it would be a great mistake to suppose that series are the only important application of the notion of similarity between relations. A map, for example, if accurate, is similar to the region which it maps. A book spelt phonetically is similar to the sounds produced when it is read aloud. A gramophone record is similar to the music which it produces. And so on. [Pg 250] It should be observed that similarity applies not only to two-term relations, but to relations with any number of terms. Suppose we have two relations , each -adic; suppose there is a one-one relation which relates all the terms in the field of to all the terms in the field of ; let , , ... be terms which have the relation and let , , ... be the terms correlated with them by the relation . Then and are similar if there is a one-one relation such that, when the above conditions are fulfilled, , , ... have the relation , and conversely. Two relations which are similar have the same "structure" or "relation-number." The "relation-number" of a relation is the same as its "structure," and is defined as the class of all relations similar to the given relation. Relation-numbers satisfy all the formal laws of arithmetic which are satisfied by transfinite ordinal numbers; ordinal numbers, both finite and transfinite, are a particular kind of relation-numbers—namely, the relation-numbers of relations which generate well-ordered series. The formal laws satisfied by relation-numbers are: They do not in general satisfy the commutative law, nor the other form of the distributive law, viz.: Relation-numbers are important for the following reason. In addition to the propositions which can be proved by logic (considered in Chapter XVII.), there are other propositions which can be enunciated by logic, though they cannot be proved or disproved except by empirical evidence. Such, for example, is the proposition: "There are classes which are not[Pg 251] finite." This is a proposition which is purely logical in content, but there is no a priori way of knowing whether it is true or false. (Many such have been proposed, but they are all fallacious.) Then, again, there are propositions which contain some particular constituent, but would be capable of enunciation in logical terms if that constituent were turned into a variable. Take, e.g.: "Before is a transitive relation." This is not a statement which pure logic can enunciate, because before is an empirical relation. But " is a transitive relation," where is variable, can be enunciated by pure logic. We will say that a proposition containing a certain constituent attributes a "logical property" to if, when is replaced by a variable , the result is a propositional function which can be expressed by logic. The test of a logical property is very simple: apart from the constant ,