The Analysis of Matter

of the philosopher. Minkowski had the same[Pg 396] quality, but he did not live to see the general theory. For philosophical purposes, therefore, I have allowed myself to be guided almost entirely by Eddington. In the general theory of relativity, we start with a four-dimensional continuum of points, whose properties, to begin with, are purely ordinal. We then assign four co-ordinates to each point on any principle such that the ordinal properties of the co-ordinates are the same as those of the points. We then assume that, if two points are very close together, there is a quadratic function of the co-ordinates which has the same value however the co-ordinates may be assigned, subject to the above ordinal condition. If this function is positive, its square root is called the (time-like) interval; if negative, the square root of the function with its sign changed is called the (space-like) interval. Omitting niceties, we may say that the remainder of the theory turns mainly on geodesics. A geodesic is a route between two space-time points such that the integral of the interval along this route is stationary. In the important routes, it is a maximum. It appears that energy can be divided into parcels which move in geodesics; when these parcels move with a velocity less than that of light, they are regarded as pieces of matter. Weyl, by imposing certain limitations on measurement, succeeds in including electromagnetic phenomena in this scheme. Thus we have a comprehensive theory which may be taken to embrace everything except quantum phenomena. But although there is so much to give pleasure to the logician in this scheme—more especially the method of tensors and Hamiltonian derivatives—yet the philosopher cannot but feel dissatisfaction with the apparently arbitrary assumption about intervals. This assumption seemed less arbitrary than it is, because of its connection, historically, with the theorem of Pythagoras and its modifications in non-Euclidean geometry. But the theorem was believed formerly because[Pg 397] it had been proved; when the proof was found to have no value, it was believed because empirical evidence was thought to show its approximate truth. This empirical evidence, of course, remains, but the theory of relativity has made its value much more problematical than it formerly seemed. And it is customary to carry out measurements carefully, taking trouble to secure bodies that are as nearly rigid as possible, and optical instruments that are accurate. If our co-ordinates are to be arbitrary, as they are in the general theory of relativity, it is doubtful whether we still have a right to expect that they will verify anything analogous to the theorem of Pythagoras. As against these doubts, it may be said that the general theory has justified itself by the correctness of all its verifiable consequences. This is true, and I do not wish to minimize the force of the argument. But I seem to observe that, in obtaining these results, the theory does not make use of the full liberty in assignment of co-ordinates which it claims at the start. In astronomy, its co-ordinates are still assigned by the usual careful methods, and it is not clear that this care is useless. From the method of tensors, it seems to follow that we can employ any co-ordinates subject to the ordinal condition. But the method of tensors, as used, assumes the formula for interval; for this reason, Dr Whitehead found it necessary, in his Principle of Relativity, to give a theory of tensors independent of the formula for interval. There is thus still legitimate room for doubt as to whether the formula for interval is really quite independent of the choice of co-ordinates. And, apart from this question, there is great difficulty in suggesting any non-technical meaning for interval; yet such a meaning ought to exist, if interval is as fundamental as it appears to be in the theory of relativity. There is difficulty also as to what is meant by measurement. And there is the feeling that, perhaps, tensor equations represent purely ordinal properties of the space-time continuum, and could, by a better[Pg 398] technique, be set forth without the use of any co-ordinates at all. The necessary technique does not exist at present, but it is not impossible that it may be created before long. In Part II., we approached a different type of question: the question of the evidence for the truth of physics, i.e. of the relation of physics to perception. For the purposes of this inquiry, it is convenient to use "perception" somewhat more narrowly than it would be used in psychology. Our purpose is epistemological, and therefore perception is only relevant in so far as it is explicit and the percept is observed: percepts which pass unnoticed cannot be made into premisses for physics. The use of percepts for inference as to the physical world rests upon the causal theory of perception, since the naive realism of common sense turns out to be self-contradictory. The serious alternatives to the causal theory of perception are not common sense, but solipsism and phenomenalism. Solipsism, as an epistemologically serious theory, must mean the view that from the events which I experience there is no valid method of inferring the character, or even the existence, of events which I do not experience. If inference is taken in the sense of strict deductive logic, there is, so far as I can see, no escape from the solipsist position. And it should be observed that this position cannot admit unconscious events in me, any more than events outside me: its basis is epistemological, and therefore, for it, the important distinction is between what I experience and what I do not experience, not between what is mine and what is not mine in some metaphysical or physical sense. We cannot escape from the solipsist position without bringing in induction and causality, which are still subject to the doubts resulting from Hume's sceptical criticism. Since, however, all science rests upon induction and causality, it seems justifiable, at least pragmatically, to assume that, when properly employed, they can give at least a probability. In the present work, I have made this assumption baldly,[Pg 399] without attempting to justify it; I have done this because I do not believe that a justification could be much briefer than Mr Keynes's Treatise on Probability, and also because, while I am convinced that a justification is possible, I am not satisfied with those put forward by others or with any that I have been able to invent myself. It seemed best, therefore, to make the assumption as stark as possible, without any attempt at artificial plausibility. Intermediate between solipsism and the ordinary scientific view, there is a half-way house called "phenomenalism." This admits events other than those which I experience, but holds that all of them are percepts or other mental events. Practically, it means, when advocated by scientific men, that they will accept the testimony of other observers as to what they have actually experienced, but that they will not infer thence anything which no observer has experienced. It may be said, in justification of this position, that, while it employs analogy and induction, it refrains from assuming causality. But it may be doubted whether it can really abstain from causality. Phenomenalists appear to take testimony for granted, i.e. to assume that the words which they see and hear express what they themselves would express if they used them. But this involves causality, and involves it in the form in which the cause is in one person and the effect in another. There does not seem, therefore, to be any substantial justification for this half-way house. We therefore assume, though with less than demonstrative certainty, that percepts have causes which may be not percepts, and, in particular, that when a number of people have similar percepts simultaneously, there is what may be called a "field" of causally connected events, which, it is found, have relations that often enable us to arrange them in a spherical order about a centre. We thus arrive at a space-time order of events, which is found to be the same whichever of many[Pg 400] possible methods of arriving at it we adopt; in this order, a percept is located in the head of the percipient. In drawing inferences from percepts to their causes, we assume that the stimulus must possess whatever structure is possessed by the percept, though it may also have structural properties not possessed by the percept. The assumption that the structural properties of the percept must exist in the stimulus follows from the maxim "same cause, same effect" in the inverted form "different effects, different causes," from which it follows that if, e.g., we see red and green side by side, there is some difference between the stimulus to the red percept and the stimulus to the green percept. The structural features possessed by the stimulus but not by the percept, when they can be inferred, are inferred by means of general laws—e.g. when two objects look similar to the naked eye but dissimilar under the microscope, we assume that there are differences in the stimuli to the naked-eye percepts which produce either no differences, or no perceptible differences, in the corresponding percepts. These principles enable us to infer a great deal as to the structure of the physical world, but not as to its intrinsic character. They put percepts in their place as occurrences analogous to and connected with other events in the physical world, and they enable us to regard a dictaphone or a photographic plate as having something which, from the standpoint of physics, is not very dissimilar from perception. We no longer have to contend with what used to seem mysterious in the causal theory of perception: a series of light-waves or sound-waves or what not suddenly producing a mental event apparently totally different from themselves in character. As to intrinsic character, we do not know enough about it in the physical world to have a right to say that it is very different from that of percepts; while as to structure we have reason to hold that it is similar in the stimulus and the percept. This has become possible owing to the facts that "matter" can be[Pg 401] regarded as a system of events, not as part of the stuff of the world, and that space-time, as it occurs in physics, has been found to be much more different from perceptual space than was formerly imagined. This brings us to Part III., in which we endeavour to discover a possible structure of the physical world which shall at once justify physics and take account of the connection with perception demanded by the necessity for an empirical basis for physics. Here we are concerned first with the construction of points as systems of events which overlap, or are "co-punctual," in space-time, and then with the purely ordinal properties of space-time. The method employed is very general, and can be adapted to a discrete or to a continuous order; it is proved that , events are sufficient to generate a continuum of points, given certain laws as to the manner of their overlapping. The whole of this theory, however, aims only at constructing such properties of space-time as belong to analysis situs; everything appertaining to intervals and metrics is omitted at this stage, since causal considerations are required for the theory of intervals. The conception of one unit of matter—say one electron—as a "substance," i.e. a single simple entity persisting through time, is not one which we are justified in adopting, since we have no evidence whatever as to whether it is false or true. We define a single material unit as a "causal line," i.e. as a series of events connected with each other by an intrinsic differential causal law which determines first-order changes, leaving second-order changes to be determined by extrinsic causal laws. (In this we are