It is at once evident that all these tests dependent on a direct intuition of permanence. This ‘permanence’ means ‘permanence in respect to congruence,’ for the vari- ous instruments employed, namely, the yard measure, or the optical instruments, or analogous instruments. For example, the yard measure is assumed to remain con- gruent to its previous self, as it is transferred from one setting to another setting. It is not sufficient to intuit that it remains the same body. Substances that are very deformable preserve that sort of self-identity. The re- quired property is that of self-congruence. Minute varia- tions of physical conditions will make the rod vary slightly; also sense-perception is never absolutely exact.
502 Process and Reality
But unless there be a meaning to ‘exactitude,’ the notions of a ‘slight variation’ and of a ‘slight defect from exacti- tude’ are nonsense. Apart from such a meaning the two occasions of the rod’s existence are incomparable, except by another experiment depending upon the same prin- ciples. There can only be a finite number of such experi- ments; so ultimately we are reduced to these direct judgments.
However far the testing of instruments and the correc- tions for changes of physical factors, such as temperature, are carried, there is always a final dependence upon direct intuitions that relevant circumstances are unchanged. Instruments are used from minute to minute, from hour to hour, and from day to day, with the sole guarantee of antecedent tests and of the appearance of invariability of relevant circumstances.
This ‘appearance’ is always a perception in the mode of presentational immediacy. If such perception be in any sense ‘private’ in contradistinction to a correlative mean- ing for the term ‘public,’ then the perceptions, on which scientific measurement depend, merely throw light upon the private psychology of the particular observer, and have no ‘public’ import.
Such a conclusion is so obviously inconsistent with our beliefs as to the intercommunication of real actualities in a public world, that it may be dismissed as a reductio ad absurdum, having regard to the groundwork of com- mon experience which is the final test of all science and philosophy. A great deal of modern scientific philosophy consists in recurrence to the theory of ‘privacy’ when such statements seem to afford a short cut to simplicity of statement, and—on the other hand—of employment of the notion of observing a public world when that con- cept is essential for expressing the status of science in common experience. Science is either an important state- ment of systematic theory correlating observations of a common world, or is the daydream of a solitary intelli- gence with a taste for the daydream of publication. But
Measurement 503
it is not philosophy to vacillate from one point of view to the other.
Finally, meaning of ‘congruence’ as a relation between two geometrical elements in a strain-locus must be con- sidered. It will be sufficient to consider this meaning in reference to two segments of straight lines, and to treat all other meanings as derivative from this.
A strain-locus is defined by the ‘projectors’ which pene- trate any one finite region within it. Such a locus is a systematic whole, independently of the actualities which may atomize it. In this it is to be distinguished from a ‘duration’ which does depend on its physical content. A strain-locus depends merely upon its geometrical content. This geometrical content is expressed by any adequate set of ‘axioms’ from which the systematic inter-connec- tions of its included straight lines and points can be deduced. This conclusion requires the systematic uni- formity of the geometry of a strain-locus, but refers to further empirical observation for the discovery of the particular character of this uniform system. For example, the question as to whether a complete straight line be a ‘closed’ serial locus of points or an ‘open’ serial locus, is entirely a question for such discovery. The only decision is to be found by comparing the rival theories in respect to their power of elucidating observed facts.
The only relevant properties of straight lines are (i) their completeness, (ii) their inclusion of points, (111) their unique definition by: any pair of included points, (iv) their possibility of mutual intersection in a single point. The additional axioms which express the sys- tematic geometrical theory must not have reference to length or to congruence. For these notions are to be _ derived from the theory. Thus the axioms must have exclusive reference to the intersection of straight lines, and to their inclusion or exclusion of points indicated by the intersections of other lines. Such sets of axioms are
504 Process and Reality
well known to mathematicians. There are many such sets which respectively constitute alternative geometrical the- ories. Also given one set of axioms constituting a definite geometrical theory, different sets of axioms can easily be obtained which are equivalent to each other in the sense that all the other sets can be deduced from any one of them. All such equivalent sets produce the same geo- metrical theory. Equivalent sets have their importance, but not for the present investigation. We can therefore neglect them, and different sets of axioms will mean sets of axioms which constitute incompatible geometrical theories.
There are many such sets, with a great variety of
peculiar properties. There are, however, three such sets
which combine a peculiar simplicity with a very general
conformation to the observed facts. These sets give the
non-metrical properties of the three geometrical theories
respectively known to mathematicians as the theory of
Elliptic Geometry, of Euclidean Geometry, and of Hyper-
bolic Geometry. It will serve no purpose to give the
three sets of axioms. But it is very easy to explain the
main point of difference between the theories, without
being led too far from the philosophical discussion.
In the first place, a definition of a ‘plane’ can be given which is common to all the three theories. The definition already given in Chapter III of this Part will suffice. But an alternative definition can be stated thus: If A, B, C be any three non-collinear points, and AB, BC, CA denote the three complete straight lines containing re- spectively, A and B, B and C, C and A, then the straight lines which respectively intersect both members of any pair of these three lines, not both lines at one of the corners A or B or C, pass through all the points constitut- ing one plane, and all their incident points are incident in the plane.
Thus a plane is defined to be the locus of all the points incident in at least one of such a group of straight lines. The axioms are such that this definition is equivalent to
Measurement 505
the definition in Chapter III. Also the axioms secure
that any straight line passing through two points in a
plane, itself is wholly incident in that plane. Also it
follows from the definition of a plane that a line / and a
point P, not incident in J, are coplanar.
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 Geo- metrical 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 numeri- cal 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 parallelo- grams are equal. A further complication is required to define congruence between segments which are not paral- lel. 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 congru-
ent when there is a certain analogy between their func-
tions 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.
506 Process and Reality
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 con- stitutions of the dominating social entities is the impor- tant 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, alterna-
tive 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 con-
cerned. The only point to be remembered is that each
system of ‘coordinates’ must have its definable relation
to the analogy which constitutes congruence.
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 spatio- temporal continuum and seeking to express the physical properties of the field as an integral along it. The meas- urements which are presupposed are the geometrical measurements constituting the coordinates of the vari- ous points involved. Various physical quantities enter as the ‘constants’ involved in the algebraic functions con- cerned. 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 con-
gruence of different elements of the path. The notion
of coincidence does not apply. There is no systematic
Measurement 507
theory possible, since the so-called ‘infinitesimal’ distance depends on the actual entities throughout the environ- ment. The only way of expressing such so-called distance is to make use of the presupposed geometrical measure- ments. 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 dis- tinguish between extensive quantity and intensive quan- tity. 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 quan- tity is a construct. The current physical theory presup- poses a comparison of so-called lengths among segments without any theory as to the basis on which this com- parison is to be made, and in ignoration of the fact that all exact observation belongs to the mode of presenta- tional 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 dif- ferential 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 environ- ment. The whole theory of the physical field is the inter- weaving of the individual peculiarities of actual occasions upon the background of systematic geometry. This sys- tematic 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.
508 Process and Reality
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 doc- trine of coincidence is nonsense.
(ii) Extensive quantity is a logical construct, express-
ing 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 ulti-
mate intuitions of congruence between earlier and later
states of the instruments employed.