by Lee F. Werth
Lee F. Werth teaches philosophy at Cleveland State University, Cleveland, Ohio.
The following article appeared in Process Studies, pp. 37-44, Vol. 8, Number 1, Spring, 1978. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.
If no geometrical element is a point, then Whitehead’s geometrical account of the perceptive mode of presentational immediacy in terms of mere extensiveness is untenable, as it is when all geometrical elements are points.
Whitehead’s account of the perceptive mode of presentational immediacy, as he explicitly states, is dependent upon a definition of straight lines in terms which make no appeal to measurement (unlike the definition, ‘the shortest distance between two points’): "It is to be noted that this doctrine of presentational immediacy and of the strain-locus entirely depends upon a definition of straight lines in terms of mere extensiveness" (PR 493; cf. also 495-96). Accordingly, straight lines are defined in terms of a scheme of extensive connection. The definition of a straight line requires that some geometrical elements be incident in another geometrical element: "The locus of points incident in a ‘straight segment’ is called the ‘straight line’ between the endpoints of the segment" (PR 466, def. 5). See also his earlier definitions of ‘geometrical element’ (def. 13), ‘point’ (def. 16), ‘segment’ (def. 18), ‘end-points’ (def. 19), ‘straight’ (def. 3), and ‘incident’ (def. 15). ‘Being incident in’ is defined only as a relation between geometrical elements. But this definition contradicts a derivable theorem that no geometrical element a is ever incident in a geometrical element b. Moreover, from this theorem we can demonstrate that, according to the explicit theory of extensive connection Whitehead proposes, all geometrical elements are points. Since no geometrical element can satisfy the definition of a straight line in terms of mere extensiveness, it follows that Whitehead’s geometrical account of presentational immediacy and of the strain-locus is untenable in its present form.
In my demonstration of these claims, the relevant theorems will be derived from Whitehead’s definitions and assumptions in the chapter on "Extensive Connection," particularly definitions 2, 7, 9-13, 15-17, and assumptions 6-9, 23-26. The reader will want to have Whitehead’s text in front of him in examining this argument.
1. Every abstractive set must be composed of an infinite number of members. This follows from Assump. 9, Assump. 6, Def. 10, Assump. 7, and Assump. 8.
An included region B is said to be ‘part’ of region A (Def. 2). Comparing this use of ‘part’ with that in Def. 16 seems to yield an inconsistency in so far as points have abstractive sets as members (Def. 17) and abstractive sets have parts, i.e., included regions. Therefore, points have parts.
I shall not argue for the importance of this inconsistency since it is easily resolved by a terminological revision, i.e., by refusing to call included regions ‘parts’.
2. Any two members of the infinite number of members of an abstractive set are such that one of them includes the other nontangentially. This follows from criterion (i) of the definition of an abstractive set (Def. 10).
3. Of any two member regions A and B of an abstractive set such that B is nontangentially included in A, there will be an infinite set of regions nontangentially included in B, such that A includes them all and B includes all of them but A and B. This follows from the transitivity of inclusion (Assump. 6), the asymmetry of inclusion (Assump. 8), the irreflexivity of inclusion (Assump. 7), and the fact that every region includes other regions (Assump. 9).
4. Any two members of an abstractive set themselves determine an abstractive set. This follows from the satisfaction of the two criteria determining an abstractive set, i.e., nontangential inclusion and there being no region included in every member of the set (Def. 10), and from 2 and 3.
5. Member regions A and B determine an abstractive set to be called a. This follows from 4.
6. Regions A and B include member regions C and D of a such that A includes B, B includes C, C includes D. This follows from 3.
7. C and D of abstractive set a determine an abstractive set to be called B. This follows from 4.
8. Every region nontangentially included by A or B is a member of a. This follows from 3 and 5.
9. Every member region of an abstraetive set a included by region B is also a member region of an abstractive set . This follows from regions C and D being members of a and also regions determining the set (7).
10. Every member of an abstractive set /3 includes some members of . This follows: from the transitivity of inclusion (Assump. 6) such that if C includes D, C includes D and whatever else D includes; from criterion (i) from the definition of an abstractive set which states that any two members of an abstractive set are such that one of them Includes the other nontangentially (Def. 10); and from 1, which asserts the infinity of members of an abstractive set and thereby assures us that every region including another region necessarily will include more than one region.
11. Every member of an abstractive set a includes some members of the set . This follows from 8, 9, and 10.
12. Abstractive set a covers abstractive set . This follows from Def. 11 and from 11.
13. Every member of an abstractive set includes some members of an abstractive set a. This follows from 9 and 10.
14. Abstractive set covers abstractive set a. This follows from Def. 11 and from 13.
15. Abstractive set a and abstractive set are equivalent. This follows from Def. 12, and from 12 and 14.
16. Abstractive set a and abstractive set are members of the same geometrical element. This follows from Def. 13 and from 15.
17. Abstractive sets a and cover every member of the geometrical element of which they are members. This follows from Assump. 23 and from 16.
18. Abstractive sets a and are covered by every member of the geometrical element of which they are members. This follows from Assump. 24 and from 16.
19. Abstractive sets a and are equivalent to every member of the geometrical element of which they are members. This follows from Def. 12 and from 17 and 18.
It is worth noting that were the paragraph following Definition 11 in Process and Reality included as part of that definition, an inconsistency could be generated. It is asserted that "when the set a covers the set , each member of a includes all the members of the convergent tail of provided that we start far enough down in the serial arrangement of the set " (PR 455; emphasis mine). This need not be inconsistent with Definition 11 which states that a covers "when every member of the set a includes some members of the set ," we merely conceive of the "some members" as those very same members as "all the members of the convergent tail of ." However, we must remember that equivalent abstractive sets cover each other. Thus, each member of , if is to cover a, includes all the members of the convergent tail of a. This seems possible until we remember that the relation of inclusion is asymmetrical (Assump. 8). Therefore, since every member of must include all the regions of a’s tail, some of the including regions will be in ’s tail. A tail region of will include all the regions of a’s tail. Since a tail region of a must also include all the tail regions of ’s tail, it is necessary that at least one of the included regions of ’s tail be a region which includes the a region including it. This is a violation of the assumption of the asymmetry of inclusion (Assump. 8).
If all the member regions of equivalent abstractive sets are required to include all the members of each other’s convergent tails, it is clear that there can be no equivalent sets (Def. 12) insofar as equivalent sets require mutual coverage. Without the possibility of equivalent sets there can be no geometrical elements (Def. 13) and without the possibility of geometrical elements there can be no points (Def. 16).
I have not chosen to hold Whitehead to the full ramifications of the paragraph following Definition 11, but rather to construct my argument within the scope of the more liberal Definition 11 without the unfortunate paragraph immediately following it.
20. If every member of an abstractive set includes some members of another abstractive set, it will always be the case that every member of that latter abstractive set includes some members of the former abstractive set. This follows from 11 and 13.
We will not be troubled by the assumption of asymmetry of inclusion (Assump. 8) since we are not requiring every member of either abstractive set to include all of the members of the convergent tail of the other. All we require is that some members of one set, regardless which ones, be included by every member of the other. Even where regions of one set are also regions of the other, asymmetry of inclusion is not violated as only ‘larger’ less converged regions of either set include ‘smaller’ more converged regions and never the converse. An included region will therefore never include its includer.
If we conceive of the Chinese nest-of-boxes toy as an analogue of an abstractive set (PR 454; cf. also CN 79), remembering that there is no last box, it is clear that unless two abstractive sets are superimposed there will be no inclusion of any member region of one set by a member region of the other. But when the sets are superimposed, it is apparent that there will be member regions common to both. Moreover, when every region of one set includes some regions of the other set, every region of the latter set will include some regions of the former set. Once superimposed, the two sets become indistinguishable. A new observer on the scene might notice that the present largest, least converged box is larger than before, or that there are more boxes than before. But he might also notice no difference whatever from the box situation prior to the superimposition of the additional ‘nest-of-boxes’ abstractive set. Whether or not the difference is noticed depends upon whether or not the additional nest-of-boxes has a largest member larger than the largest member of the initial nest-of-boxes.
The above is by way of illustration through an analogue and nothing has been asserted that is not a consequence of the proof wherein it is demonstrated that to cover is to be covered.
21. The two abstractive sets of 20 cover each other. This follows from Def. 11 and from 20.
22. The relation of covering, if it is to hold between any two abstractive sets always holds symmetrically. This follows from 20 and 21.
23. The two abstractive sets of 21 are equivalent. This follows from Def. 12 and from 21.
24. The two abstractive sets will be members of the same geometrical element. This follows from Def. 13 and from 23.
25. The two abstractive sets will be equivalent to every member of the geometrical element of which they are members. This follows from an argument analogous to 16 through 19, i.e., Assump. 23, Assump. 24, Def. 12, and from 22.
26. When every member of an arbitrary geometrical element a covers every member of an arbitrary geometrical element b, every member of geometrical element b will cover every member of geometrical element a. This follows from the symmetry of coverage (22).
27. The members of geometrical element a and the members of the geometrical element b are equivalent to each other. This follows from Def. 12 and from 26.
28. The members of geometrical element a and the members of the geometrical element b are members of the same geometrical element. This follows from the definition of a geometrical element (Def. 13) and from 27.
29. Geometrical element a is identical to geometrical element b. This follows from 28.
30. No geometrical element a is ever incident in a geometrical element b. This follows from 29 and Def. 15 which asserts that "a is said to be ‘incident’ in the geometrical element b, when every member of b covers every member of a, but a and b are not identical." We have seen that a and b must always be identical when every member of b covers every member of a.
31. All geometrical elements are points and all member sets are punctual sets. This follows from Def. 16 and Def. 17 and from 30.
A further, less formal, and more general proof is included to meet the objection that my argument depends upon deriving the abstractive set a from an abstractive set and then deriving from a. Under these conditions, it might be objected that it is no wonder that a and should both belong to the same geometrical element.
1. If set a covers set , every member of the abstractive set a includes some members of the abstractive set . (From Definition 11).
2. a’s largest, least converged region is either bigger, smaller, or the same size as ’s largest, least converged region.
3. If a’s largest region is the same size as ’s largest region and every member of a includes some members of , it is clear that every member of also includes some members of a.
4. If a’s largest region is smaller than ’s largest region and every member of a includes some members of , then every member of includes some members of a. The inclusion of members of a by ’s largest members, i.e., all those larger than a’s largest region, will require the transitivity of inclusion (Assumption 6).
5. If a’s largest region is larger than ’s largest region and every member of a includes some members of , then every member of includes some members of a. Those members of a included by members of will be those "far enough down in the serial arrangement of the set " (PR 455), i.e., those members of a which are sufficiently converged to allow ’s member regions to include them. The existence of those members "far enough down" requires that a have an infinite number of regions, which a is said to have according to the paragraph following Definition 10. (See Def. 10 and 1 in the preceding argument.)
6. To cover an abstractive set is to be covered by that set, i.e., symmetry of coverage. The rest of the argument that no geometrical element a is ever incident in a geometrical element b, and that all geometrical elements are points, is as before, i.e., that which follows the symmetry of coverage theorem (22).
I must confess that I do not like to reason on the basis of the largest region of an abstractive set. Whitehead remarks that "though an abstractive set must start with some region at its big end, these initial large-sized regions never enter into our reasoning" (PB 455). Moreover, although Whitehead does speak of ‘big end’ and ‘far enough down’, I believe such terms to be logically improper in the context of a theory which is logically prior to measurement. These terms may be used metaphorically, but should not themselves play a formal role. Therefore, I have constructed my argument upon the longer and more formal proof.
A Reader, justifiably unhappy with the conclusion that all geometrical elements are points, would probably attack the mainstay of the demonstration, that mainstay being the symmetrical character of the relation of covering (22). Looking back at the original definition of demonstration, that mainstay being the symmetrical character of the covering (Def. 11), one reads: "an abstractive set a is said to cover an abstractive set when every member of the set a includes some members of the set ." As a measure of caution one rereads the definition of an abstractive set (Def. 10) and attends carefully to criterion (i) of that definition. "Any two members of the set (of regions) are such that one of them includes the other nontangentially" (PR 454, emphasis mine).
The reader gladly asserts that I have been overly restrictive in my interpretation of Def. 11. Nowhere in that definition of ‘cover’ does it state that ‘includes’ is to be read as ‘nontangentially includes’. I am at once accused of deriving my theorem of the symmetry of coverage from a fallacious and overly restrictive interpretation of Whitehead’s original Def. 11. It will be argued: it may well be that symmetrical coverage does occur between or among abstractive sets when the inclusion is nontangential, but inclusion need not be nontangential. If tangential inclusion allows for nonsymmetrical coverage, there will be no theorem 22, and the conclusion that all geometrical elements are points cannot be derived. Some geometrical elements will not be points. My accuser might supply the following proof.
1. Let us begin with an example where an abstractive set covers another abstractive set a but is not covered by a.
2. The sets will not be equivalent since equivalence requires that each set cover the other.
3. The sets will belong to different geometrical elements since only equivalent sets belong to the same geometrical element.
4. Any member set of the one geometrical element a will be covered by any member set of the other geometrical element b, since all the members of any geometrical element are equivalent.
5. Geometrical element a and b are not identical since the equivalent a sets of a are not equivalent to the equivalent sets of b.
6. The geometrical element a is incident in the geometrical element b since 4 and 5 together satisfy the definition of incidence (Def. 15).
7. Geometrical element b is not a point since it has geometrical element a incident in it, and the definition of a point (Def. 16) requires that it have no geometrical element incident in it.
It would seem that I am now forced to renounce the whole of my argument based upon the theorem of the symmetry of coverage which was itself derived from an apparently overly restrictive interpretation of ‘includes’ in Def. 11.
Actually ‘includes’ as opposed to ‘nontangentially includes’ is a pseudo-issue. Irrespective of which way ‘includes’ is to be interpreted, Whitehead’s attempt to define straight lines remains unsuccessful. If ‘includes’ is restricted to nontangential inclusion, then coverage is symmetrical and we are able to derive the theorem that all geometrical elements are points. However, if ‘includes’ is broadened to allow tangential inclusion, we are then across the other horn of the dilemma. It can then be shown that no geometrical element is a point. Any arbitrarily selected abstractive set can cover and not be covered by some other abstractive set a, if the inclusion of the member regions of the covered set a is allowed to be tangential. It follows that a geometrical element b associated with any group of equivalent abstractive sets is such that it always has some other geometrical element a incident in it, the reason being that each of the equivalent abstractive sets which are the members of a geometrical element b is such that each can cover and not be covered by the a equivalent abstractive sets which are the members of the geometrical element a. And if every geometrical element has some other geometrical element incident in it, no geometrical element can be a point.
Were we to restrict ourselves to the sort of two-dimensional circular regions found diagrammed in Process and Reality, we would not be able to construct a counterinstance to the symmetry of coverage even if we allowed inclusion to be tangential. Remembering that Whitehead warns us not to be misled by the limitations imposed upon us by such diagrams (PR 450), let us interpret ‘inclusion’ as allowing tangential inclusion and ‘region’ as allowing any desired shape, in which case asymmetrical coverage can indeed be obtained. For example, it is possible to construct an abstractive set of ovate regions which covers but is not covered by an abstractive set of circular regions.
If it is also possible to construct an abstractive set of circular regions such that: (i) the members are concentric; (ii) every member tangentially includes some members of some other abstractive set a (in which case covers a); (iii) a does not cover , then we may have confidence in the claim that any arbitrarily selected abstractive set can be shown to cover some other abstractive set asymmetrically. If an abstractive set can be constructed which satisfies these very constraining criteria, it ought then to be acknowledged that any abstractive set can cover asymmetrically.
It remains to specify the nature of the a set covered by but not covering . Every member of a includes part but not all of any member; otherwise, we risk symmetrical coverage since inclusion is transitive. It is to be remembered that every abstractive set has an infinite number of members. We can conceptualize but not visualize an infinite abstractive set; we understand that it has no smallest (most converged) member. There is no logical limitation which prohibits every a member region, even the least converged, from having one end which is smaller than any arbitrarily selected member, even members well down the convergent tail of . The small end of any a region cuts into and thus includes part but not all of even the very converged regions. (An a member cannot be circular; at least one end must be elongated and pointed.) By this reasoning every a member escapes including any member; nevertheless, covers a.
We conclude the formal discussion by noticing that either all geometrical elements are points, which follows from the theorem of symmetrical coverage (22), or no geometrical element is a point, which follows from allowing tangential inclusion to satisfy the definition of coverage, a liberalization which enables any abstractive set to asymmetrically cover an a set.
(My use of the expressions ‘shape,’ circular,’ ‘ovate,’ ‘concentric, ‘big,’ and ‘small’ are intended solely as conceptual aids. They are to help the reader identify the relevant abstractive sets, a sets become possible when the meaning of ‘include’ is liberalized and not because they are somehow logically dependent upon concentricity, measurement, or any logically prior definitions of particular shapes.)
If no geometrical element is a point, then Whitehead’s geometrical account of the perceptive mode of presentational immediacy in terms of mere extensiveness is untenable, as it is when all geometrical elements are points. Def. 5 of a straight line requires that there be a "locus of points incident in a straight segment" (PR 466; emphasis mine), and without points we once again have no definition of straight lines.
To refute my argument requires the satisfaction of two criteria. Asymmetrical coverage between abstractive sets must be possible, i.e., the refutation of theorem 22. It must also be established that it is not always possible to construct a covered set a such that a’s associated geometrical element can be said to be incident in ’s associated geometrical element (for if a covered set a can always be constructed where a does not cover its coverer, then all geometrical elements are not points). If these two criteria are satisfied, some geometrical elements will be points and some will not, and straight lines can be definable in terms of mere extensiveness.