Process and Reality — the Gifford Lectures, 1929

These mathematical relations belong to the systematic order of extensiveness which characterizes the cosmic epoch in which we live. The societies of enduring objects—electrons, protons, molecules, material bodies—at once sustain that order and arise out of it. The mathematical relations involved in presentational immediacy thus belong equally to the world perceived and to the natures of the percipient. They are, at the same time, public fact and private experience.

The perceptive mode of presentational immediacy is in one sense barren. So far as—apart from symbolic transference—it discloses the contemporary world, that world, thus objectified, is devoid of all elements constitutive of subjective form, elements emotional, appreciative, purposive. The bonds of the objectified nexus only exhibit the definiteness of mathematical relations.

But in another sense this perceptive mode has overwhelming significance. It exhibits that complex of systematic mathematical relations which participate in all the next of our cosmic epoch, in the widest meaning of that term. These relations only characterize the epoch by reason of their foundation in the immediate experience of the society of occasions dominating that epoch. Thus we find a special application of the doctrine of the interaction between societies of occasions and the laws of nature. The perceptive mode in presentational immediacy is one of the defining characteristics of the societies which constitute the nextis termed material bodies. Also in some fainter intensity it belongs to the electromagnetic occasions in empty space. From the point of view of ap. 499 single experient, that mode discloses systematic relations which dominate the environment. But the environment is dominated by these relationships by reason of the experiences of the individual occasions constituting the societies.

It is by reason of this disclosure of ultimate system that an intellectual comprehension of the physical universe is possible. There is a systematic framework permeating all relevant fact. By reference to this framework the variant, various, vagrant, evanescent details of the abundant world can have their mutual relations exhibited by their correlation to the common terms of a universal system. Sounds differ qualitatively among themselves, sounds differ qualitatively from colours, colours differ qualitatively from the rhythmic throbs of emotion and of pain; yet all alike are periodic and have their spatial relations and their wave-lengths. The discovery of the true relevance of the mathematical relations disclosed in presentational immediacy was the first step in the intellectual conquest of nature. Accurate science was then born. Apart from these relations as facts in nature, such science is meaningless, a tale told by an idiot and credited by fools. For example, the conjecture by an eminent astronomer, based on measurements of photographic plates, that the period of the revolution of our galaxy of stars is about three hundred million years can only derive its meaning from the systematic geometrical relations which permeate the epoch. But he would have required the same reference to system, if he had made an analogous statement about the period of revolution of a child’s top. Also the two periods are comparable in terms of the system.

SECTION IV

Measurement depends upon counting and upon permanence. The question is, what is counted, and what is permanent? The things that are counted are the inches on a straight metal rod, a yard-measure. Also the thingp. 500 that is permanent is this yard-measure in respect both to its internal relations and in respect to some of its extensive relations to the geometry of the world. In the first place, the rod is straight. Thus the measurement depends on the straightness and not the straightness upon the measurement. The modern answer to this statement is that the measurement is a comparison of infinitesimals, or of an approximation to infinitesimals. The answer to this answer is that there are no infinitesimals, and that therefore there can be no approximation to them. Inmathematics, all phraseology about infinitesimals is merely disguised statement about a class of finites. This doctrine has been conclusive mathematical theory since the time of Weierstrass in the middle of the nineteenth century. Also all the contortions of curvature are possible for a segment between any end-points.

Of course, in all measurement there is approximation in our suppositions as to the yard measure. But it is approximation to straightness. Also having regard to the systematic geometry of straight lines, and to the type of approximation exhibited by the rod, the smaller the portion used, the more negligible are the percentage errors introduced by the defects from straightness. But unless the notion of straightness has a definite meaning in reference to the extensive relations, this whole procedure in practical measurement is meaningless. There is nothing to distinguish one contorted segment between end-points from another contorted segment between those end-points. One is no straighter than another. Also any percentage differences between their lengths. can exist.

Again, the inches are counted because they are congruent and are end-on along the straight rod. No one counts coincident inches. The counting essentially is concerned with non-coincident straight segments. The numerical measure of length is the indication of the fact that the yard measure is a straight rod divisible into. thirty-six congruent inch-long segments.

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There is a modern doctrine that ‘congruence’ means the possibility of coincidence. If this be the case, then the importance of congruence would arise when the possibility is realized. Alternatively, the possibility could be of importance as a lure entering into the subjective aim. If the latter alternative were true, congruence would play its part in the form of a tendency of congruent bodies to coalesce, or to resist. coalescence. In fact, there would be adversion to, or aversion from coalescence. Of course the suggestion is fantastic. Recurring to the former alternative, the importance of the thirtysix inches along the yard-measure depends on the fact that they are not coincident and, until the destruction of the rod, never will be coincident. There is a realized property of the rod that it is thirty-six inches in length. Thus although ‘coincidence’ is used as a test of congruence, it is not the meaning of congruence.

We must now consider the use of ‘coincidence’ as a test. Congruence is tested either by the transference of a steel yard-measure from coincidence with one body to coincidence with another body, or by some optical means dependent on the use of an optical instrument and on the congruence of successive wave lengths in a train of waves, or by some other vibratory device dependent on analogous principles.