Process and Reality — the Gifford Lectures, 1929

For the sake of brevity the terms ‘connection’ and ‘connected’ will be used in the place of ‘extensive connection’ and ‘extensively connected.’ The term ‘region’ will be used for the relata which are involved in the scheme of ‘extensive connection.’ Thus, in the shortened phraseology, regions are the things which are connected.

A set of diagrams will illustrate the type of relationship meant by ‘connection.’ The two areas, A and B, in each diagram exhibit an instance of connection with each other.

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DIAGRAMS I

• (i) (ii) A (CQ)
• (iii) (iv) a Cag
• (v) (vi)

Such diagrams are apt to be misleading: for one reason, because they introduce features as obvious, which it is our business to define in terms of our fundamental notion of ‘connection’; for another reason, because they introduce features which are special to the two-dimensional, spatial extensiveness of a sheet of paper.

In the three diagrams of Set II, the areas, A and B, are not connected; but they are ‘mediately’ connected by the area C.

SECTION II

Definition 1. Two regions are ‘mediately’ connected when they are both connected with a third region.

Assumption 1. Connection and mediate connection are both of them symmetrical relations; that is to say, if region A is connected, or mediately connected, with region B, then region B is connected, or mediately connected, with region A.

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DIAGRAMS II

(i) (ii)

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(iii)

Cc

It is obvious that the part of this assumption which concerns mediate connection can be proved from the terms of the definition. In the subsequent development of definitions and assumptions we shall not draw attention to such instances of the possibility of proof.

Assumption 2. No region is connected with all the other regions; and any two regions are mediately connected.

Assumption 3. Connection is not transitive; that is to say, if A be connected with B, and B with C, it does not thereby follow that A is connected with C; though in certain cases it does happen that A is connected with C.

Assumption 4. No region is connected, or mediately connected, with itself.

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This assumption is merely a convenient arrangement of nomenclature.

Definition 2. Region A is said to ‘include’ region B when every region connected with B is also connected with A. As an alternative nomenclature, region B will be said to be ‘part’ of region A.

This definition of ‘inclusion’ is due to Professor de Laguna; it constitutes an important addition to the theory of extension. In such investigations, as the present one, the definitions are the really vital portion of the subject.

Assumption 5. When one region includes another, the two regions are connected.

Assumption 6. The relation of inclusion is transitive.

Assumption 7. A region does not include itself.

Assumption 8. The relation of inclusion is asymmetrical; that is to say, if A includes B, then B does not include A.

Assumption 9. Every region includes other regions; and a pair of regions thus included in one region are not necessarily connected with each other. Such pairs can always be found, included in any given region.

Definition 3. Two regions are said to ‘overlap,’ when there is a third region which they both include.

Assumption 10. The relation of overlapping is symmetrical.

Assumption 11. If one region includes another region, the two regions overlap.

Assumption 12. Two regions which overlap are connected.

Definition 4. A ‘dissection’ of any given region A, is a set of regions, which is such that (i) all its members are included in A, (ii) no two of its members overlap, (iii) any region included in A, but not a member of the set, either is included in one member of the set, or overlaps more than one member of the set.

Assumption 15. There are many dissections of any given region.

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Assumption 16. A dissection of a region is not a dissection of any other region.

Definition 5. A region is called an ‘intersect’ of two overlapping regions, A and B, when (i) either it is included in both A and B, or it is one of the two regions and is included in the other, and (ii) no region, also included in both A and B, can overlap it without being included in it.

Definition 6. 1. If there be one, and only one, intersect of two regions, A and B, those regions are said to overlap with ‘unique intersection’; if there be more than one intersect, they are said to overlap with ‘multiple intersection.’

Assumption 17. Any region included in both of two overlapping regions, and not itself an intersect, is included in one, and only one, intersect.

Assumption 18. If A includes B, then B is the sole intersect of A and B.

Assumption 19. An intersect of two regions, which is not one of the two regions, is included in both regions.

Assumption 20. Each pair of overlapping regions has at least one intersect.

Definition 7. Two regions are ‘externally’ connected when (i) they are connected, and (ii) they do not overlap. The possibility of this definition is another of the advantages gained from the adoption of Professor de Laguna’s starting point, ‘extensive connection,’ over my original starting point,’ ‘extensive whole and extensive part.’ External connection is illustrated by diagrams (v) and (vi) in Set I of the diagrams. So far, we have not discriminated between the two cases illustrated respectively by these two diagrams. The notion of external connection is a long step towards the elaboration of the notion of a ‘surface,’ which has not yet been touched upon.

Definition 8. A region B is ‘tangentially’ included in a region A, when (i) B is included in A, and (ii) there are

1Cf. my Principles of Natural Knowledge, and Concept of Nature.

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regions which are externally connected with both A and B.

Definition 9. A region B is ‘non-tangentially’ included in a region A when (i) B is included in A, and (11) there is no third region which is externally connected with both A and B.

The possibility, at this stage, of the three definitions 7, 8, and 9, constitutes the advantage to be gained by starting from Professor de Laguna’s notion of ‘extensive connection.’ Non-tangential inclusion is illustrated by diagram (i) of the first set; and the two cases—as yet undiscriminated—of tangential inclusion are illustrated by diagrams (11) and (iil),

SECTION III

Definition 10. A set of regions is called an ‘abstractive set,’ when (i) any two members of the set are such that one of them includes the other non-tangentially, (ii) there is no region included in every member of the set.

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-dimensioned space, we see that abstractive sets can have different 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 reference to the notions, point, line, area.

Definition 11. An abstractive set o is said to ‘cover’ an

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