Process and Reality — the Gifford Lectures, 1929

Assumption 30. If two points lie in a given volume, there are linear stretches joining those two points, whose points all lie in that volume.

Assumption 31. If two points lie in a given surface, there are linear stretches joining those two points, whose points all lie in that surface.

Assumption 82. If two points lie in a given linear stretch, there is one, and only one, linear stretch with those points as end-points, whose points lie wholly in the given linear stretch.

It should be noted that the terms ‘volume’ and ‘surface’ are not meant to imply that volumes are threedimensional, or that surfaces are two-dimensional. In the application of this theory of extension to the existing physical world of our epoch, volumes are four-dimensional, and surfaces are three-dimensional. But linear stretches are one-dimensional. A sufficient number of assumptions, some provable and some axiomatic, have now been stated; so as to make clear the sort of development of the theory required for this stage of the definitions. In particular, the notion of the order of points in a linear stretch can now be elaborated from the definition of the notion of ‘between.’ But such investigations will lead us too far into the mathematical principles of geometry.

Assumption 38. A ‘complete locus,’ as defined in Definition 23, consists of an infinite number of points,

CHAPTER III
FLAT LOCI

SECTION I

Mopvern physical science, with its dependence on the exact notions of mathematics, began with the foundation of Greek Geometry. The first definition of Euclid’s Elements runs,

“A point is that of which there is no part.”
The second definition runs,
“A line is breadthless length.”
The fourth definition runs,

“A straight line is any line which les evenly with the points on itself.”

These translations are taken from Euclid In Greek, Book I, edited with notes by Sir Thomas L. Heath, the greatest living authority on Euclid’s Elements. Heath ascribes the second definition ‘to the Platonic school, if not to Plato himself.’ For the Greek phrase translated ‘evenly’ Heath also suggests the alternatives ‘on a footing of equality,’ ‘evenly placed, ‘without bias.’

Kuclid’s first ‘postulate’ is (Heath’s translation) :

“Let the following be postulated: to draw a straight line from any point to any point.”

Heath points out that this postulate was meant to imply, existence and uniqueness.

As these statements occur in Greek science, a muddle arises between ‘forms’ and concrete physical things. Geometry starts with the purpose of investigating cer-

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PUArSLO Ci 461

tain forms of physical things. But in its initial definitions of the ‘point’ and the ‘line,’ it seems immediately to postulate certain ultimate physical things of a very peculiar character. Plato himself appears to have had some suspicion of this confusion when (cf. Heath, loc. cit.) he ‘objected to recognizing points as a separate class of things at all.’ He ought to have gone further, and have made the same objection to all the geometrical entities, namely, points, lines, and surfaces. He wanted ‘forms,’ and he obtained new physical entities.

According to the previous chapter, ‘extension’ should be construed in terms of ‘extensive connection’; that is to say, extension is a form of relationship between the actualities of a nexus. A point is a nexus of actual entities with a certain ‘form’; and so is a ‘segment.’ Thus geometry is the investigation of the morphology of nexus.

SECTION II

The weak point of the Euclidean definition of a straight line is, that nothing has been deduced from it. The notion expressed by the phrases ‘evenly,’ or ‘evenly placed,’ requires definition. The definition should be such that the uniqueness of the straight segment between two points can be deduced from it. Neither of these demands has ever been satisfied, with the result that in modern times the notion of ‘straightness’ has been based on that of measurement. A straight line has, in modern times, been defined as the shortest distance between two points. In the classic geometry, the converse procedure was adopted, and measurement presupposed straight lines. But, with the modern definition, the notion of the ‘shortest distance’ in its turn requires explanation.’ This notion is practically defined to mean the line which is the route of certain physical occurrences.

In this section it will be shown that the gap in the old

1Cf. Part IV, Ch. V, on ‘Measurement.’

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classical theory can be remedied. Straight lines will be defined in terms of the extensive notions, developed in the preceding chapter; and the uniqueness of the straight line joining two points will be proved to follow from the terms of the definition.

A class of ‘oval’ regions must first be defined. Now the only weapon which we have for this definition is the notion of regions which overlap with a unique intersect (cf. Def. 6 of previous chapter). It is evidently a property of a pair of ovals that they can only overlap with a unique intersect. But it is equally evident that some regions which are not ovals also overlap with a unique intersect. However the class of ovals has the property that any region, not a member of it, intersects some ovals with multiple intersects. Also sub-sets of ovals can be found satisfying various conditions.

Thus we proceed to define a class whose regions shall have those relations to each other, and to other regions, which we ascribe to the class of ovals. In other words we cannot define a single oval, but we can define a class of ovals. Such a class will be called ‘ovate.’ The definition of an ovate class proceeds by enumerating all those peculiar properties possessed by individual members of the class, or by sub-sets of members of the class. It will be found in the course of this enumeration that an extensive continuum which possesses an ovate class is dimensional in respect to that class. Thus existence of straight lines in an extensive continuum is bound up with the dimensional character of the continuum; and both characteristics are relative to a particular ovate class of regions in the continuum. It seems probable that an extensive continuum will possess only one ovate class. But I have not succeeded in proving that property; nor is it necessary for the argument.

A preliminary definition is convenient:

Definition 0.1. An ‘ovate abstractive set’ is an abstractive set whose members all belong to the complete ovate class under consideration.

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The characteristics of an ovate class will be divided into two groups: (a) the group of non-abstractive conditions, and (b) the group of abstractive conditions.

Definition 1. A class of regions is called ‘ovate,’ when it satisfies the conditions belonging to the two following groups, (a) and (b):

(a) The Non-Abstractive Group

(1) Any two overlapping regions of the ovate class have a unique intersect which also belongs to that ovate class.

(ii) Any region, not a member of the ovate class, overlaps some members of that class with ‘multiple intersection’ (cf. Def. 6 of previous chapter).

(iii) Any member of the ovate class overlaps some regions, not of that class, with multiple intersection.

(iv) Any pair of members of the ovate class, which are externally connected, have their surfaces touching either in a ‘complete locus’ of points (cf. Ch. III, Def. 23 and Ass. 33), or in a single point.

(v) Any region, not belonging to the ovate class, is externally connected with some member of that class so that their surfaces touch in a set of points which does not form a ‘complete locus.’

(vi) Any member of the ovate class is externally connected with some region not of that class so that their surfaces touch in a set of points which does not form a ‘complete locus.’

(vii) Any finite number of regions are jointly included in some member of the ovate class.