This definition practically limits abstractive sets to those sets which were termed ‘simple abstractive sets’ in my Principles of Natural Knowledge (paragraph 37.6). Since every region includes other regions, and since the relation of inclusion is transitive, it is evident that every abstractive set must be composed of an infinite number of members.
By reference to the particular case of three-dimen- sioned space, we see that abstractive sets can have differ- ent types of convergence. For in this case, an abstractive set can converge either to a point, or to a line, or to an area. But it is to be noted that we have not defined either points, or lines, or areas; and that we propose to define them in terms of abstractive sets. Thus we must define the various types of abstractive sets without ref- erence to the notions, point, line, area.
Definition 11. An abstractive set o is said to ‘cover’ an
Extensive Connection 455
abstractive set 8, when every member of the set o, includes some members of the set @.
It is to be noticed that each abstractive set is to be conceived with its members in serial order, determined by the relation of inclusion. The series starts with a region of any size, and converges indefinitely towards smaller and smaller regions, without any limiting region. When the set a covers the set 6, each member of a includes all the members of the convergent tail of 8. provided that we start far enough down in the serial arrangement of the set 8. It will be found that, though an abstractive set must start with some region at its big end, these initial large-sized regions never enter into our reasoning. Attention is always fixed on what rela- tions occur when we have proceeded far enough down the series. The only relations which are interesting are those which, if they commence anywhere, continue throughout the remainder of the infinite series.
Definition 12. Two abstractive sets are said to be ‘equivalent’ when each set covers the other.
Thus if o and 6 be the two equivalent abstractive sets, and A, be any member of a, there is some member of 6, B, say, which is included in A,, also there is some mem- ber of gq, Az say, which is included in B,; also there is some member of 8, B, say, which is included in A», and so on indefinitely. Two equivalent abstractive sets are equivalent in respect to their convergence. But, in so far as the two sets are diverse, there will be relationships and characteristics in respect to which those sets are not equivalent, in a more general sense of the term ‘equiva- lence.’ The connection of this special sense of ‘equiva- lence’ to physical properties is explained more particu- larly in Chapter IV of the Concept of Nature.
Assumption 21. An abstractive set is equivalent to
itself. This assumption is merely a convenient arrange-
ment of nomenclature. An abstractive set obviously sat-
isfies the conditions for such reflexive equivalence.
Definition 18. A geometrical element is a complete
456 Process and Reality
eroup of abstractive sets equivalent to each other, and not equivalent to any abstractive set outside the group.
Assumption 22. The relation of equivalence is transi- tive and symmetrical.
Thus any two members of a geometrical element are
equivalent to each other; and an abstractive set, not
belonging to the geometrical element, is not equivalent
to any member of that geometrical element. It is evident
that each abstractive set belongs to one, and only one,
geometrical element.
Definition 14. The geometrical element to which an
abstractive set belongs, is called the geometrical element
‘associated’ with that abstractive set. Thus a geometri-
cal element is ‘associated’ with each of its members.
Assumption 23. Any abstractive set which covers any
member of a geometrical element, also covers every
member of that element.
Assumption 24. An abstractive set which is covered by
any member of a geometrical element, is also covered by
every member of that element.
Assumption 25. If a and b be two geometrical ele-
ments, either every member of a covers every member of
b, or no member of a covers any member of b.
Definition 16. The geometrical element a is said to be
‘meident’ in the geometrical element b, when every mem-
ber of b covers every member of a, but a and b are not
identical.
Assumption 26. A geometrical element is not incident in itself.
This assumption is merely a convenient arrangement of nomenclature.
When the geometrical element a is incident in the
geometrical element b, the members of a will be said to
have a ‘sharper convergence’ than those of b.
Definition 16. A geometrical element is called a ‘point,’
when there is no geometrical element incident in it.
This definition of a ‘point’ is to be compared with
Buclid’s definition: ‘A point is without parts.’
Extensive Connection 457
Definition 16.1. The members of a geometrical element are said to be ‘prime’ in reference to assigned conditions, when (i) every member of that geometrical element sat- isfied those conditions; (11) if any abstractive set satisfies those conditions, every member of its associated geometri- cal element satisfies them; (ili) there is no geometrical element, with members satisfying those conditions, which is also incident in the given geometrical element.
The term ‘prime’ will also be applied to a geometrical
element, when its members are ‘prime’ in the sense
defined above.
It is obvious that a point is, in a sense, an ‘absolute’
prime. This is, in fact, the sense in which the definitions
of a point, given here, conforms to Euclid’s definition.
Defimtion 17. An abstractive set which is a member of a point will be called ‘punctual.’
Defintion 18. A geometrical element is called a ‘seg-
ment between two points P and Q,’ when its members are
prime in reference to the condition that the points P and
Q are incident in it.
Definition 19. When a geometrical element is a seg-
ment between two points, those points are called the ‘end-
points’ of the segment.
Definition 20. An abstractive set which is a member of a segment is called ‘segmental.’
Assumption 27. There are many diverse segments with
the same end-points: but a segment has only one pair of
end-points.
This assumption illustrates the fact that there can be many geometrical elements which are prime in reference to some given conditions. There are, how- ever, conditions such that there is only one geometrical element prime to any one of them. For example, the set of points incident in one geometrical element uniquely defines that geometrical element. Also another instance of uniqueness is to be found in the theory of ‘flat’ geo- metrical elements, to be considered in the next chapter. A particular instance of such ‘flat’ elements is afforded
458 Process and Reality
by straight lines. The whole theory of geometry depends upon the discovery of conditions which correspond to one, and only one, prime geometrical element. The Greeks, with their usual fortunate intuition, chanced upon such conditions in their notions of straight lines and planes. There is every reason, however, to believe that, in other epochs, widely different types of conditions with this property may be important—perhaps even in this epoch. The discovery of them is obviously of the first importance. It is possible that the modern Einstein- ian reconstruction of physics is best conceived as the dis- covery of the interweaving in nature of different types of such conditions.
Definition 21. A point is said to be ‘situated’ in a
region, when the region is a member of one of the punc-
tual abstractive sets which compose that point.
Assumption 28. If a point be situated in a region, the
regions, sufficiently far down the convergent tails of the
various abstractive sets composing that point, are
included in that region non-tangentially.
Definition 22. A point is said to be situated in the ‘sur-
face’ of a region, when all the regions in which it
is situated overlap that region but are not included
in it.
Definition 23. A ‘complete locus’ is a set of points which compose either (i) all the points situated in a region, or (ii) all the points situated in the surface of a region, or (iii) all the points incident in a geometrical element.
A ‘locus’ always means a ‘locus of points.’
Definition 24. When a complete locus consists of all the points situated in a region, it is called the ‘volume’ of that region; when a complete locus consists of all the points in the surface of a region, the locus itself is called the ‘surface’ of that region; when a complete locus con- sists of all the points incident in a segment between end-
Extensive Connection 459
points, the locus is called a ‘linear stretch’ between those end-points.
Assumption 29. There is a one-to-one correlation be-
tween volumes and regions, between surfaces and regions,
and between linear stretches and segments, and between
any geometrical element and the locus of points incident
in it.
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 ‘sur- face’ are not meant to imply that volumes are three- dimensional, 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-dimen- sional, 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 develop- ment of the theory required for this stage of the defini- tions. 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 Defi- nition 23, consists of an infinite number of points,
Flat Loci
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 foot- ing 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-
PUArSLO Ci 461
tain forms of physical things. But in its initial defini- tions 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 geo- metrical entities, namely, points, lines, and surfaces. He wanted ‘forms,’ and he obtained new physical en- tities.
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.