The analysis of matter

APART from pure mathematics, the most advanced of the sciences is physics. Certain parts of theoretical physics have reached the point which makes it possible to exhibit a logical chain from certain assumed premisses to consequences apparently very remote, by means of purely mathematical deductions. This is true especially of everything that belongs to the general theory of relativity. It cannot be said that physics as a whole has yet reached this stage, since quantum phenomena, and the existence of electrons and protons, remain, for the moment, brute facts. But perhaps this state of affairs will not last long; it is not chimerical to hope that a unified treatment of the whole of physics may be possible before many years have passed.

In spite, however, of the extraordinary successes of physics considered as a science, the philosophical outcome is much less dear than it seemed to be when less was known. The purpose of the present chapter is to discuss what is meant by the "philosophical outcome" of physics, and what methods exist for determining its nature.

There are three kinds of questions which we may ask concerning physics or, indeed, concerning any science. The first is: What is its logical structure, considered as a deductive system? What ways exist of defining the entities of physics and deducing the propositions from an initial apparatus of entities and propositions? This is a problem in pure mathematics, for which, in its fundamental portions, mathematical logic is the proper instrument. It is not quite correct to speak, as we did just now, of "initial entities and propositions." What we really have to begin with, in this treatment, is hypotheses containing variables. In geometry, this procedure has become familiar. Instead of "axioms," supposed to be "true," we have the hypothesis that a set of entities (otherwise undefined) has certain enumerated properties. We proceed to prove that such a set of entities has the properties which constitute the propositions of Euclidean geometry, or of whatever other geometry may be occupying our attention. Generally it will be possible to choose many different sets of initial hypotheses which will all yield the same body of propositions; the choice between these sets is logically irrelevant, and can be guided only by æsthetic considerations. There is, however, considerable utility in the discovery of a few simple hypotheses which will yield the whole of some deductive system, since it enables us to know what tests are necessary and sufficient in deciding whether some given set of entities satisfies the deductive system. Moreover, the word "entities," which we have been using, is too narrow if used with any metaphysical implication. The "entities" concerned may, in a given application of a deductive system, be complicated logical structures. Of this we have examples in pure mathematics in the definitions of cardinal numbers, ratios, real numbers, etc. We must be prepared for the possibility of a similar result in physics, in the definition of a "point" of space-time, and even in the definition of an electron or a proton.

The logical analysis of a deductive system is not such a definite and limited undertaking as it appears at first sight. This is due to the circumstance just mentioned—namely, that what we took at first as primitive entities may be replaced by complicated logical structures. As this circumstance has an important bearing upon the philosophy of physics, it will be worth while to illustrate its effect by examples from other fields.

One of the best examples is the theory of finite integers. Weierstrass and others had shown that the whole of analysis was reducible to propositions about finite integers, when Peano showed that these propositions were all deducible from five initial propositions involving three undefined ideas.[1] The five initial propositions might be regarded as assigning certain properties to the group of three undefined ideas, the properties in question being of a logical, not specifically arithmetical, character. What was proved by Peano was this: Given any triad having the five properties in question, every proposition of arithmetic and analysis is true of this triad, provided the interpretation appropriate to this triad is adopted. But it appeared further that there is one such triad corresponding to each infinite series , , , ... , ..., in which there is just one term corresponding to each finite integer. Such series can be defined without mentioning integers. Any such series could be taken, instead of the series of finite integers, as the basis of arithmetic and analysis. Every proposition of arithmetic and analysis will remain true for any such series, but for each series it will be a different proposition from what it is for any other series.

Take, in illustration, some simple proposition of arithmetic, say: "The sum of the first odd numbers is ." Suppose we wish to interpret this proposition as applying to the progression , , ,... , ... In this progression, let be the relation of each term to its successor. Then "odd numbers" will mean "terms having to a relation which is a power of ," where is the relation of an to the next but one.[2] We can now define as meaning that power of which relates to , and we can further define as meaning that to which has the relation . This decides the interpretation of "the sum of the first odd numbers." To define it will be best to define multiplication. We have defined ; consider the relation formed by the relative product of the converse of together with . This relation relates to ; its square relates to ; its cube relates to , etc. Any power of this relation can be shown to be equivalent to a certain power of the converse of multiplied relatively by a certain power of . There is thus one power of this relation which is equivalent to moving backward from to , and then forward; the term to which the forward movement takes us is defined as . Thus we can now interpret . It will be found that the proposition from which we started is true with this interpretation.

It follows from the above that, if we start from Peano's undefined ideas and initial propositions, arithmetic and analysis are not concerned with definite logical objects called numbers, but with the terms of any progression. We may call the terms of any progression 0, 1, 2, 3,..., in which case, with a suitable interpretation of + and , all the propositions of arithmetic will be true of these terms. Thus 0, 1, 2, 3,..., become "variables." To make them constants, we must choose some one definite progression; the natural one to choose is the progression of finite cardinal numbers as defined by Frege. What were, in Peano's methods, primitive terms are thus replaced by logical structures, concerning which it is necessary to prove that they satisfy Peano's five primitive propositions. This process is essential in connecting arithmetic with pure logic. We shall find that a process similar in some respects, though very different in others, is required for connecting physics with perception.

The general process of which the above is an instance will be called the process of "interpretation." It frequently happens that we have a deductive mathematical system, starting from hypotheses concerning undefined objects, and that we have reason to believe that there are objects fulfilling these hypotheses, although, initially, we are unable to point out any such objects with certainty. Usually, in such cases, although many different sets of objects are abstractly available as fulfilling the hypotheses, there is one such set which is much more important than the others. In the above instance, this set was the cardinal numbers. The substitution of such a set for the undefined objects is "interpretation." This process is essential in discovering the philosophical import of physics.

The difference between an important and an unimportant interpretation may be made clear by the case of geometry. Any geometry, Euclidean or non-Euclidean, in which every point has co-ordinates which are real numbers, can be interpreted as applying to a system of sets of real numbers—i.e. a point can be taken to be the series of its co-ordinates. This interpretation is legitimate, and is convenient when we are studying geometry as a branch of pure mathematics. But it is not the important interpretation. Geometry is important, unlike arithmetic and analysis, because it can be interpreted so as to be part of applied mathematics—in fact, so as to be part of physics. It is this interpretation which is the really interesting one, and we cannot therefore rest content with the interpretation which makes geometry part of the study of real numbers, and so, ultimately, part of the study of finite integers. Geometry, as we shall consider it in the present work, will be always treated as part of physics, and will be regarded as dealing with objects which are not either mere variables or definable in purely logical terms. We shall not regard a geometry as satisfactorily interpreted until its initial objects have been defined in terms of entities forming part of the empirical world, as opposed to the world of logical necessity. It is, of course, possible, and even likely, that various different geometries, which would be incompatible if applied to the same set of objects, may all be applicable to the empirical world by means of different interpretations.

So far, we have been considering the logical analysis of physics, which will form the topic of Part I. But in relation to the interpretation of geometry we have already been brought into contact with a very different problem—namely, that of the application of physics to the empirical world. This is, of course, the vital problem: although physics can be pursued as pure mathematics, it is not as pure mathematics that physics is important. What is to be said about the logical analysis of physics is therefore only a necessary preliminary to our main theme. The laws of physics are believed to be at least approximately true, although they are not logically necessary; the evidence for them is empirical. All empirical evidence consists, in the last analysis, of perceptions; thus the world of physics must be, in some sense, continuous with the world of our perceptions, since it is the latter which supplies the evidence for the laws of physics. In the time of Galileo, this fact did not seem to raise any very difficult problems, since the world of physics had not yet become so abstract and remote as subsequent research has made it. But already in the philosophy of Descartes the modern problem is implicit, and with Berkeley it becomes explicit. The problem arises because the world of physics is, prima facie, so different from the world of perception that it is difficult to see how the one can afford evidence for the other; moreover, physics and physiology themselves seem to give grounds for supposing that perception cannot give very accurate information as to the external world, and thus weaken the props upon which they are built.

This difficulty has led, especially in the works of Dr Whitehead, to a new interpretation of physics, which is to make the world of matter less remote from the world of our experience. The principles which inspire Dr Whitehead's work appear to me essential to a right solution of the problem, although in the detail I should sometimes incline to a somewhat more conservative attitude. We may state the problem abstractly as follows:

The evidence for the truth of physics is that perceptions occur as the laws of physics would lead us to expect—e.g. we see an eclipse when the astronomers say there will be an eclipse. But physics itself never says anything about perceptions; it does not say that we shall see an eclipse, but says something about the sun and moon. The passage from what physics asserts to the expected perception is left vague and casual; it has none of the mathematical precision belonging to physics itself. We must therefore find an interpretation of physics which gives a due place to perceptions; if not, we have no right to appeal to the empirical evidence.

This problem has two parts: to assimilate the physical world to the world of perceptions, and to assimilate the world of perceptions to the physical world. Physics must be interpreted in a way which tends towards idealism, and perception in a way which tends towards materialism. I believe that matter is less material, and mind less mental, than is commonly supposed, and that, when this is realized, the difficulties raised by Berkeley largely disappear. Some of the difficulties raised by Hume, it is true, have not yet been disposed of; but they concern scientific method in general, more particularly induction. On these matters I do not propose to say anything in the present volume, which will throughout assume the general validity of scientific method properly conducted.

The problems which arise in attempting to bridge the gulf between physics (as commonly interpreted) and perception are of two kinds. There is first the epistemological problem: what facts and entities do we know of that are relevant to physics, and may serve as its empirical foundation? This demands a discussion of what, exactly, is to be learnt from a perception, and also of the generally assumed physical causation of perceptions—e.g. by light-waves or sound-waves. In connection with this latter question, it is necessary to consider how far, and in what way, a perception can be supposed to resemble its external cause, or, at least, to allow inferences as to characteristics of that cause. This, in turn, demands a careful consideration of causal laws, which, however, is in any case a necessary part of the philosophical analysis of physics. Throughout this inquiry, we are asking ourselves what grounds exist for supposing that physics is "true." But the meaning of this question requires some elucidation in connection with what has already been said about interpretation.

Apart altogether from the general philosophical problem of the meaning of "truth," there is a certain degree of vagueness about the question whether physics is "true." In the narrowest sense, we may say that physics is "true" if we have the perceptions which it leads us to expect. In this sense, a solipsist might say that physics is true; for, although he would suppose that the sun and moon, for instance, are merely certain series of perceptions of his own, yet these perceptions could be foreseen by assuming the generally received laws of astronomy. So, for example, Leibniz says:

A man who, without being a solipsist, believes that whatever is real is mental, need have no difficulty in declaring that physics is "true" in the above sense, and may even go further, and allow the truth of physics in a much wider sense. This wider sense, which I regard as the more important, is as follows: Given physics as a deductive system, derived from certain hypotheses as to undefined terms, do there exist particulars, or logical structures composed of particulars, which satisfy these hypotheses? If the answer is in the affirmative, then physics is completely "true." We shall find, if I am not mistaken, that no conclusive reason can be given for a fully affirmative answer, but that such an answer emerges naturally if we adopt the view that all our perceptions are causally related to antecedents which may not be perceptions. This is the view of common sense, and has always been, at least in practice, the view of physicists. We start, in physics, with a vague mass of common-sense beliefs, which we can subject to progressive refinements without destroying the truth of physics (in our present sense of "truth"); but if we attempt, like Descartes, to doubt all common-sense beliefs, we shall be unable to demonstrate that any absurdity results from the rejection of the above hypothesis as to the causes of perceptions, and we shall therefore be left uncertain as to whether physics is fully "true" or not. In these circumstances, it would seem to be a matter of individual taste whether we adopt or reject what may be called the realist hypothesis.

The epistemological problem, which we have just been stating in outline, will occupy Part II. of the present work. Part III. will be occupied with the outcome for ontology—i.e. with the question: What are the ultimate existents in terms of which physics is true (assuming that there are such)? And what is their general structure? And what are the relations of space-time, causality, and qualitative series respectively? (By "qualitative series" I mean such as are formed by the colours of the rainbow, or by notes of various pitches.) We shall find, if I am not mistaken, that the objects which are mathematically primitive in physics, such as electrons, protons, and points in space-time, are all logically complex structures composed of entities which are metaphysically more primitive, which may be conveniently called "events." It is a matter for mathematical logic to show how to construct, out of these, the objects required by the mathematical physicist. It belongs also to this part of our subject to inquire whether there is anything in the known world that is not part of this metaphysically primitive material of physics. Here we derive great assistance from our earlier epistemological inquiries, since these enable us to see how physics and psychology can be included in one science, more concrete than the former and more comprehensive than the latter. Physics, in itself, is exceedingly abstract, and reveals only certain mathematical characteristics of the material with which it deals. It does not tell us anything as to the intrinsic character of this material. Psychology is preferable in this respect, but is not causally autonomous: if we assume that psychical events are subject, completely, to causal laws, we are compelled to postulate apparently extra-psychical causes for some of them. But by bringing physics and perception together, we are able to include psychical events in the material of physics, and to give to physics the greater concreteness which results from our more intimate acquaintance with the subject-matter of our own experience. To show that the traditional separation between physics and psychology, mind and matter, is not metaphysically defensible, will be one of the purposes of this work; but the two will be brought together, not by subordinating either to the other, but by displaying each as a logical structure composed of what, following Dr H. M. Sheffer,[4] we shall call "neutral stuff." We shall not contend that there are demonstrative grounds in favour of this construction, but only that it is recommended by the usual scientific grounds of economy and comprehensiveness of theoretical explanation.