The analysis of matter

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.

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 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 , there must be no constants involved—except such purely formal constants as "incompatibility" and "for all values of " which are not constituents of the propositions in whose verbal or symbolic expression they occur. It will be seen that transitiveness, e.g., is a logical property of a relation; so is asymmetry or symmetry; so is having terms in its field; so is, in the case of a three-term relation (between), the property of generating a Euclidean space; so is, in the case of a four-term relation (separation of couples), the property of generating a projective space; and so on. We can now state the proposition on account of which structure is important.

When two relations have the same structure (or relation-number), all their logical properties are identical.

Logical properties include all those which can be expressed in mathematical terms. Moreover, the inferences from perceptions to their causes, assuming such inferences to be valid, are concerned mainly, if not exclusively, with logical properties. This latter proposition is one which we must now examine.

Take first the relation between the space of physics and the space of perception. Within the private space of one percipient, there is a distinction between perceived space-relations and inferred ones. There is a space into which all the percepts of one person fit, but this is a constructed space, the construction being achieved during the first months of life. But there are also perceived space-relations, most obviously among visual percepts. These space-relations are not identical with those which physics assumes among the corresponding physical objects, but they have a certain kind of correspondence with those relations. If we represent the position, for physics, of visible objects by polar co-ordinates, taking the percipient as origin, the two angular co-ordinates correspond to perceived relations among visual percepts, while the radius vector (except possibly for very small distances) is inferred by means of causal laws. Let us confine ourselves to the angular co-ordinates. My point is that the relations which physics assumes in assigning angular co-ordinates are not identical with those which we perceive in the visual field, but merely correspond with them in a manner which preserves their logical (mathematical) properties. This follows from the assumption that any difference between two simultaneous percepts implies a correlative difference in their stimuli. Consequently, assuming that light travels in straight lines, two objects which produce percepts which differ in perceived direction must differ in some respect which corresponds with perceived direction. But we need not assume that physical direction has anything in common with visual direction except the logical properties implied by the above assumption. I shall, in Part III., attempt a construction of physical space which will supply some of the detail of the correspondence; for the present, I am concerned to point out that we can only infer the logical (or mathematical) properties of physical space, and must not suppose that it is identical with the space of our perceptions. Indeed, as I shall try to prove later, the whole of a man's visual space is, for physics, inside his head; this will follow from causal considerations.

The same sort of considerations apply to colours and sounds. Colours and sounds can be arranged in an order with respect to several characteristics; we have a right to assume that their stimuli can be arranged in an order with respect to corresponding characteristics, but this, by itself, determines only certain logical properties of the stimuli. This applies to all varieties of percepts, and accounts for the fact that our knowledge of physics is mathematical: it is mathematical because no non-mathematical properties of the physical world can be inferred from perception.

There is, however, one exception to this limitation, at least apparently. The exception I mean is time. We always assume that the time between percepts is the same as the time in the physical world. I do not know whether this view is correct or not; but I will try to set forth the arguments on either side.

In the first place, we must adapt our language to the theory of relativity. I shall assume (what I shall argue in Part III.) that, when we are speaking of physical space, all our percepts are in our head. Consequently psychological time is the same as time measured by our watches, assuming that we carry them on our person. Our head moves along a world-line, and our psychological time-intervals are measured physically by integrating ds along this world-line. Thus there is no difficulty in adapting the statement that psychological and physical time are identical to the requirements of the theory of relativity. In this respect, time differs from space, because physically all our simultaneous percepts are in one place.

I think, however, that the time-intervals between percepts are only to be obtained by means of inferences of the same sort as those which lead us to the physical world. Perceived relations are not between events at different times, but between a percept and a recollection, both of which occur at the same time; or again, where very short times are concerned, between a sensation of maximum vividness and a fading (akoluthic) sensation. Sensations do not decay suddenly, but fade gradually, though very quickly. That is why a quick movement can be apprehended as a whole: the sensations belonging to earlier parts are still present, though less vivid, when the sensations belonging to later parts arise. Thus our knowledge of time seems to be inferred from perceived relations which are not strictly temporal. These relations are, I think, of three sorts. Two sorts have been mentioned: the relation of a vivid to a fading sensation, and the relation of a percept to a recollection. But in addition to these there is an order within recollections: we can recollect a process in the right order. Here, also, however, all that we perceive is in the present, and the time-order of the original events is inferred from relations among the simultaneous events which constitute our present recollection. Thus the conclusion seems to be: Psychological time may be identified with physical time, because neither is a datum, but each is derived from data by inferences of the sort we have found elsewhere, namely, inferences which allow us to know only the logical or mathematical properties of what we infer.

Thus it would seem that, wherever we infer from perceptions, it is only structure that we can validly infer; and structure is what can be expressed by mathematical logic, which includes mathematics.

Before concluding this discussion, we must consider an extension of the notion of similarity which has considerable importance in relation to the inferences leading to the physical world. In defining similarity, we used a one-one relation . But we may substitute a many-one relation, and still obtain something useful. The importance of this is that, as we have seen, if we take a group of events constituting a physical object, the relation of the events which are nearer the object to those which are further from it is many-one, not one-one. If we are observing a man half a mile away, his appearance is not changed if he frowns, whereas it is changed for a man observing him from a distance of three feet. Considerable events may happen in the sun without being perceptible to us even with the best telescopes; but near the sun they may have effects which would be important to a percipient situated where these effects occur. It is obvious as a matter of logic that, if our correlating relation is many-one, not one-one, logical inference in the sense in which goes is just as feasible as before, but logical inference in the opposite sense is more difficult. That is why we assume that differing percepts have differing stimuli, but indistinguishable percepts need not have exactly similar stimuli. If we have and , where is many-one, and if and differ, we can infer that and differ; but if and do not differ, we cannot infer that and do not differ. We find often that indistinguishable percepts are followed by different effects—e.g. one glass of water causes typhoid and another does not. In such cases we assume imperceptible differences—which the microscope may render perceptible. But where there is no discoverable difference in the effects, we can still not be sure there is not a difference in the stimuli which may become relevant at some later stage.

When the relation is many-one, we shall say that the two systems which it correlates are "semi-similar."

This consideration makes all physical inference more or less precarious. We can construct theories which fit the known facts, but we can never be sure that other theories would not fit them equally well. This is an essential limitation on scientific inference, which is generally recognized by men of science: no prudent man of science would maintain that such-and-such a theory is so firmly established that it will never call for modification. Newtonian gravitation came nearer to this certainty than any other theory has ever done; yet Newtonian gravitation has had to be modified. The fundamental reason for this uncertainty, which remains even when we assume all the canons of scientific inference, is the fact that our relation , which connects the physical object with the percept, is many-one and not one-one.