Process and Reality — the Gifford Lectures, 1929

The distinction between the three geometrical theories can now be explained by the aid of such a triplet, a point P, a line l not passing through P, and the plane z in which P and l are both incident. Consider all the lines through P and incident in the plane x. Then in the Elliptic Geometrical Theory, all these lines intersect the line /; in the Euclidean Geometrical Theory, all these lines intersect the line J, with the exception of one and only one line— the unique parallel to J through P; in the Hyperbolic Geometrical Theory the lines through P in the plane are divisible into two classes, one class consisting of the lines intersecting /, the other class consisting of the lines not intersecting J, and each class with an infinite number of members. Then it has been shown by Cayley and von Staudt * that the congruence of segments and the numerical measures of the distances involved are definable. The simplest case is that of Euclidean Geometry. In that case the basic fact is that the opposite sides of parallelograms are equal. A further complication is required to define congruence between segments which are not parallel. But it would serve no purpose to enter into the detailed solutions of this mathematical problem.

But the illustration afforded by the particular case of the congruence of the opposite sides of parallelograms, enables the general principle underlying the notion of congruence to be explained. Two segments are congruent when there is a certain analogy between their functions in a systematic pattern of straight lines, which includes both of them.

The definition of this analogy is the definition of con-

1Cf. Cayley’s Sixth Memoir On Quantics, Trans. R. 8. 1859; Von Staudt’s Geometrie der Lage, 1847; and Beitrage zur Geometre der Lage, 1856.

p. 506

gruence in terms of non-metrical geometry. It is possible to discover diverse analogies which give definitions of congruence which are inconsistent with each other. That definition which enters importantly into the internal constitutions of the dominating social entities is the important definition for the cosmic epoch in question.

Measurement is now possible throughout the extensive continuum. This measurement is a systematic procedure dependent on the dominant societies of the cosmic epoch. When one form of measurement has been given, alternative forms with assigned mathematical relations to the initial form can be defined. One such system is as good as any other, so far as mathematical procedure is concerned. The only point to be remembered is that each system of ‘coordinates’ must have its definable relation to the analogy which constitutes congruence.

SECTION VI

Physical measurement is now possible. The modern procedure, introduced by Einstein, is a generalization of the method of ‘least action.’ It consists in considering any continuous line between any two points in the spatiotemporal continuum and seeking to express the physical properties of the field as an integral along it. The measurements which are presupposed are the geometrical measurements constituting the coordinates of the various points involved. Various physical quantities enter as the ‘constants’ involved in the algebraic functions concerned. These constants depend on the actual occasions which atomize the extensive continuum. The physical properties of the medium are expressed by various conditions satisfied by this integral.

It is usual to term an ‘infinitesimal’ element of this integral by the name of an element of distance. But this name, though satisfactory as a technical phraseology, is entirely misleading. There can be no theory of the congruence of different elements of the path. The notion of coincidence does not apply. There is no systematicp. 507 theory possible, since the so-called ‘infinitesimal’ distance depends on the actual entities throughout the environment. The only way of expressing such so-called distance is to make use of the presupposed geometrical measurements. The mistake arises because, unconsciously, the minds of physicists are infected by a presupposition which comes down from Aristotle through Kant. Aristotle placed ‘quantity’ among his categories, and did not distinguish between extensive quantity and intensive quantity. Kant made this distinction, but considered both of them as categoreal notions. It follows from Cayley’s and von Staudt’s work (ef. loc. cit.) that extensive quantity is a construct. The current physical theory presupposes a comparison of so-called lengths among segments without any theory as to the basis on which this comparison is to be made, and in ignoration of the fact that all exact observation belongs to the mode of presentational immediacy. Further, the fact is neglected that there are no infinitesimals, and that a comparison of finite segments is thus required. For this reason, it would be better—so far as explanation is concerned—to abandon the term ‘distance’ for this integral, and to call it by some such name as ‘impetus,’ suggestive of its physical import.”

It is to be noted, however, that the conclusions of this discussion involve no objection to the modern treatment of ultimate physical laws in the guise of a problem in differential geometry. The integral impetus 7s an extensive quantity, a ‘length.’ The differential element of impetus is the differential element of systematic length weighted with the individual peculiarities of its relevant environment. The whole theory of the physical field is the interweaving of the individual peculiarities of actual occasions upon the background of systematic geometry. This systematic geometry expresses the most general ‘substantial form’ inherited throughout the vast cosmic society which

2Cf. my book, The Principle of Relativity, University Press,
Cambridge, 1922.

p. 508

constitutes the primary real potentiality conditioning concrescence.* In this doctrine, the organic philosophy is very near to the philosophy of Descartes.

The whole argument can be summarized thus:

(i) Actual occasions are immovable, so that the doctrine of coincidence is nonsense.

(ii) Extensive quantity is a logical construct, expressing the number of congruent units which are (a) non-overlapping, and (b) exhaustive of the nexus in question.

(iii) Congruence is only definable as a certain definite analogy of function in a systematic complex which embraces both congruent elements.

(iv) That all experimental measurement involves ultimate intuitions of congruence between earlier and later states of the instruments employed.

(v) That all exact observation is made by perception in the mode of presentational immediacy.

(vi) That if such perception merely concerns a private psychological field, science is the daydream of an individual without any public import.

(vii) That perception in the mode of presentational immediacy solely depends upon the ‘withness’ of the ‘body,’ and only exhibits the external contemporary world in respect to its systematic geometrical relationship to the ‘body.’

* This theory of the derivation of the basie uniformity requisite for congruence, and thence for measurement should be compared with that of two deeply interesting articles: (i) “The Theory of Relativity and The First Principles of Science,” and (ii) “The Macroscopic Atomic Theory,” Journal of Philosophy, Vol. XXIV, by Professor F. 8. C. Northrop of Yale. I cannot adjust his doctrine of a ‘macroscopic atom’ to my cosmological outlook. Nor does this notion seem necessary if my doctrine of ‘microscopic atomic occasions’ be accepted. But Professor Northrop’s theory does seem to be the only alternative if this

doctrine be abandoned. I regret that the articles did not come under my notice till this work had been finally revised for publication,