Process and Reality — the Gifford Lectures, 1929

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 relations 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 member 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 ‘equivalence.’ The connection of this special sense of ‘equivalence’ to physical properties is explained more particularly in Chapter IV of the Concept of Nature.

Assumption 21. An abstractive set is equivalent to itself. This assumption is merely a convenient arrangement of nomenclature. An abstractive set obviously satisfies the conditions for such reflexive equivalence.

Definition 18. A geometrical element is a completep. 456 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 transitive 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 geometrical 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 elements, 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 member 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.’

p. 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 satisfied those conditions; (11) if any abstractive set satisfies those conditions, every member of its associated geometrical 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 ‘segment 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 segment between two points, those points are called the ‘endpoints’ 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, however, 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’ geometrical elements, to be considered in the next chapter. A particular instance of such ‘flat’ elements is affordedp. 458 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 Einsteinian reconstruction of physics is best conceived as the discovery of the interweaving in nature of different types of such conditions.

SECTION IV

Definition 21. A point is said to be ‘situated’ in a region, when the region is a member of one of the punctual 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 ‘surface’ 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 consists of all the points incident in a segment between endp. 459points, the locus is called a ‘linear stretch’ between those end-points.

Assumption 29. There is a one-to-one correlation between 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.