THROUGHOUT the theory of relativity, there is an application, with increasing stringency, of a principle which begins to make itself felt in physics with Galileo, in spite of the fact that he did not possess the mathematical technique which it demands. The principle I mean is that of "differential laws," as it may be called. This means that any connection which may exist between distant events is the result of integration from a law giving a rate of change at every point of some route from the one to the other. One may give a simple illustration of a differential law from the "curve of pursuit": a man is walking along a straight road, and his dog is in a field beside the road; the man whistles to the dog, and the dog runs towards him. We suppose that at each moment the dog runs exactly towards where his master is at that moment. To discover the curve described by the dog is a problem in integration, which becomes definite given certain further data. The Newtonian law of gravitation gives a very similar type of law, except that it is the acceleration of the planet, not its velocity, that is directed towards the sun at each moment. It has long been a commonplace of physics that its causal laws should have this differential character: they should tell primarily a tendency at each moment, not the outcome after a finite time. In a word, its causal laws take the form of differential equations, usually of the second order.

This view of causal laws is absent from quantum theory, from the ideas of savages and uneducated persons, and from the works of philosophers, including Bergson and J. S. Mill. In quantum theory, we have a discrete series of possible sudden changes, and a certain statistical knowledge of the proportion of cases in which each possibility is realized; but we have no knowledge as to what determines the occurrence of a particular change in a particular case. Moreover, the change is not of the sort that can be expressed by differential equations: it is a change from a state expressed by one integer or set of integers to a state expressed by another. This kind of change may turn out to be physically ultimate, and to mark out at least a part of physics as governed by laws of a new sort. But we are not likely to find science returning to the crude form of causality believed in by Fijians and philosophers, of which the type is "lightning causes thunder." It can never be a law that, given at one time, there is sure to be at another time, because something might intervene to prevent n . We do not derive such laws from quantum phenomena, because we do not, in their case, know that will not continue throughout the time in question. The natural view to take at present is that quantum phenomena have to do with the interchange of energy between matter and the surrounding medium, while continuous change is found in all processes which involve no such interchange. There are, however, difficulties in any view at present, and it is not for a layman to venture an opinion. It seems not improbable that, as Heisenberg suggests, our views of space-time may have to be modified profoundly before harmony is achieved between quantum phenomena and the laws of transmission of light in vacuo. For the moment, however, I wish to confine myself to the standpoint of relativity theory.

Although physics has worked with differential equations ever since the invention of the calculus, geometry was supposed to be able to start with laws applying to finite spaces. If we accept the Einsteinian point of view, there can no longer be any separation between geometry and physics; every proposition of geometry will be to some extent causal. Take first the special theory. Relatively to axes () we can obtain propositions of geometry by keeping constant; but relatively to other axes these propositions will refer to events at different times. It is true that these events, in any system of co-ordinates, will have a space-like interval, and will have no direct causal relations with each other; but they will have indirect causal relations derived from a common ancestry. Let us take some example, say: The sum of the angles of a triangle is two right angles. Our triangle may be composed of rods or of light-rays. In either case, it must preserve a certain constancy while we measure it. Both rods and light-rays are complicated physical structures, and the physical laws of their behaviour are involved in taking them as approximations to ideal straight lines. Nevertheless, so far as the special theory is concerned, all this might be allowed, and yet we might maintain a certain distinction between geometry and physics, the former being a set of laws supposed exact, and approximately verified, for the relations of the , , co-ordinates in any Galilean frame when is kept constant.

But in the general theory the intermixture of geometry and physics is more intimate. We cannot accurately reduce to the form: and therefore we cannot accurately distinguish one co-ordinate as representing the time. We cannot therefore obtain a timeless geometry by putting =constant. With this goes a change in our axioms. We no longer have, as in Euclid, in Lobatchevsky and Bolyai, and in projective geometry, axioms dealing with straight lines of finite length. We have now only, as our initial apparatus, a geometry of the infinitesimal, from which large-scale results must be obtained by integration. From this point of view, Weyl's extension of Einstein appears natural. As we saw in the last chapter, quoting Eddington, the statement that the distances , are equal is the assertion of a relation between the four points. , , , . If all the relations which constitute our initial apparatus are to be confined to the infinitesimal, so must this relation; if so, , , , must all be close together, and Weyl's geometry results.

At this point, however, the pure mathematician is likely to feel a difficulty which does not greatly trouble the physicist. The physicist thinks of his infinitesimals as actual small quantities, which may—e.g. in astronomical problems—be such as would be reckoned large in other problems. For him, therefore, a statement in terms of infinitesimals is quite satisfactory.

But for the pure mathematician there are no infinitesimals, and all statements in which they seem to occur must be expressible as limits of what happens to finite quantities. To take our particular case: We must be able to say of a small finite quadrilateral that it is approximately a parallelogram, if we are to be able to assign a meaning to the statement that an infinitesimal quadrilateral may be accurately a parallelogram. The case is exactly analogous to velocity in elementary kinematics: we can assign a meaning to velocity only because we can measure finite distances and times, and so form the conception of the limit of their quotient. It is not wholly clear how we are to satisfy this requirement in the case of Weyl's theory. I think, however, that there is not the slightest reason to suppose that it cannot be satisfied. Let "" mean " form a parallelogram." We are supposed to have also , , etc., but not etc. Also if we have and , we are to have . But if we take "" to mean ", , , form an approximate parallelogram," we cannot (if there is any way of specifying a degree of approximation) argue from and to . Now if we assume, as Weyl does, that lengths at a given point are comparable, we can perhaps give the necessary definitions. We shall have to take , not , as our fundamental relation, since the distance between any two points is finite, and it is assumed that no finite quadrilateral can be accurately a parallelogram. Or perhaps we shall have to go a step further, and take as fundamental a relation of eight points, say meaning " is more nearly a parallelogram than " We shall then say that, given any four points, , , , , it is possible to find points , nearer to and respectively than and are, such that Further, we can say that, if , , , are sufficiently near together, and then the ratio of to can be made to approach zero as a limit by diminishing the size of in a purely ordinal sense. (Ordinal relations among points, as we saw earlier, are presupposed in the theory of relativity.)

It is highly probable that the above process can be simplified. It is, however, of no importance in itself; its only purpose is to show that the derivatives required can be correctly defined, and that, however the mathematical treatment may confine itself to infinitesimals, relations between points whose distances are finite must be presupposed if the infinitesimal calculus is to be applicable.

This last result, whose generality is obvious from the theory of limits, is of some philosophical importance. Wherever mathematics works in a continuous medium with relations which may be loosely described as next-to-next, there must be other relations, holding between points at finite distances from each other, and having the next-to-next relations as their limits. Thus, when we say that laws have to be expressed by differential equations, we are saying that the finite relations which occur cannot be brought under accurate laws, but only their limits as distances are diminished. We are not saying that these limits are the physical realities; on the contrary, the physical realities continue to be the finite relations. And if our theory is to be adequate, some way must be found of so defining the finite relations as to make the passage to the limit possible.

It is considered a merit in the general theory of relativity, particularly in Weyl's form (or the still more general form suggested by Eddington), that it dispenses with what we may call "integrated" relations as regards its fundamentals. Thus Eddington, after pointing out that he is concerned with structure, not with substance, proceeds (p. 224):

It is obvious that, in the above passage, Eddington is imagining displacements at a small finite distance from each other, not at an infinitesimal distance; he is not thinking of all the apparatus involved in a procedure which replaces infinitesimals by limits. One might suggest that he is supposing, e.g., that a footrule will not change much during the portion of a second required to transfer it from one part of a given page to another. But when we say that it will not change "much," we imply some standard of quantitative comparison other than the footrule; and this leads to the problems we have been considering.

I cannot but think that Eddington's point of view lends itself to development and further analysis by means of mathematical logic; in particular, this applies to the conditions for the possibility of measurement, a subject which will be considered explicitly in the next chapter. But for the present my concern is with "the comparability of proximate relations." In the first place, what is meant by "comparability"? A moment's reflection shows that what is wanted is a symmetrical transitive relation which each of the relations in question has to some others, but not to all. (It is assumed, in the particular case of Eddington's general geometry, that when there is such a relation of the interval to the interval , there is also such a relation of the interval to the interval . But this, as he admits (p. 226), is not essential.) Now why should we suppose that a transitive symmetrical relation of the above sort is more likely to exist between small intervals than between large ones? I.e., if is between and , and between and , is it more likely that the relation in question will hold between and than between and ? I do not see why we should think so. And I think further that, with a correct interpretation of infinitesimals, the whole belief that causation must always be from next-to-next becomes untenable unless continuity is abandoned. Causal laws may all be differential equations, but the grounds for thinking that they are must be empirical, not a priori. They cannot be derived from the impossibility of action at a distance unless distance itself is a derivative from causality, which may well be the case, but does not represent any part of the views of those who are anxious to dispense with action at a distance. It may well be, therefore, that there is one department of physics—that included in the general theory of relativity, as supplemented by Weyl—in which everything proceeds by differential equations, while there is another part—that dealt with by quantum theory—in which this whole apparatus is inapplicable. There is absolutely no a priori reason why everything should go by differential equations, since, even then, causation does not really go from next-to-next: in a continuum there is no "next." It is, at bottom, because "next-to-next" seems natural that we like a procedure of differential equations; but the two are logically incompatible, and our preference for the second on account of the first proceeds only from logical confusion.