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On Behalf of the Unhappy Reader: A Response to Lee F. Werth

by Elizabeth M. Kraus

Elizabeth M. Kraus is Professor of Philosophy at Fordham University, Bronx, N.Y. The following article appeared in Process Studies, pp. 125-133, Vol. 9, Numbers 3&4, Fall and Winter, 1979. 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.


Werth’s attack on the tenability of Whitehead’s theory of extensive connection (PS 8:37-44) constitutes a serious challenge to the coherence of the philosophy of organism and therefore demands serious consideration. At the same time, both the attack and the doctrine attacked are so arcane and abstruse as to render them inaccessible and/or uninteresting to all but a few specialists in the philosophical community, with the end result that both are, in practice, passed over. This article attempts an enterprise as risk-laden as it is necessary: to present a more intuitive version of the relevant issues in PR IV, 2 and, on that basis to evaluate Werth’s arguments. The risk is obvious. No intuitive translation of a purely formal demonstration can capture the rigor central to such an argument, and hence is always romantic. All it can hope to do is to render the major ideas and general contours of the demonstration "interesting" and thereby provoke a more widespread investigation of its content.

I

If PR IV, 2, the locus of the derivation under attack, is to become intelligible, the reader must be aware of the sort of undertaking in which Whitehead is engaged. That the subject matter is geometry is immediately obvious. Not so obvious, however, is the type of geometry within the parameters of which the undertaking is conducted. Throughout his extended discussion of the properties of regions, it is not metric properties which are the focus of Whitehead’s concern: not their dimensionality, not their "shapiness" (e.g., straightness or flatness) as grasped in sense perception, not their size, not, in fact, any property which is visualizable. Hence not only has he moved beyond Euclidean geometry, but beyond the non-Euclidean varieties as well, into the sphere of the more general, nonmetric geometries which have arisen since the Renaissance.

There are three species of metric geometry, species distinguished from each other and ranged in a hierarchy of generality on the basis of their intent: to discover the properties of regions which persist through more and more drastic sorts of transformations.

The least general species, affine geometry, isolates those properties which remain constant when a figure is uniformly stretched or shrunk, For example, parallel lines remain parallel when viewed through a telescope or microscope, yet lose their parallelism in the distortion produced by a fish-eye lens. They are affine invariants.

Projective geometry, the geometry of perspective, allows more dramatic transformations and hence reveals more general invariants. It was this sort of geometry which da Vinci intuitively grasped and illustrated in his sketches of the "bird’s-eye view" (angolo inferiore) and the "worm’s eye view" (angolo superiore) in the Codex Huygens. Anyone who has ever observed the apparent convergence of railroad tracks has had first hand experience that at least one affine invariant -- parallelism -- does not survive a shift of perspective, whereas straightness does. What additional properties (projective invariants) remain constant through perspectival transformations is the question at issue in projective geometry. It is to be noted that it is this type of geometry which dominates PR IV, 3, with its concern for the formal definitions of straightness and flatness. It should also be noted that affine transformations are special cases of projective transformations, ones retaining the same line of sight" but increasing or decreasing the distance separating viewer and object viewed.

When formalized by Arthur Cayley at the turn of the century, projective geometry was taken to be the most general variety of geometry. However, at the same time topology was growing from its embryonic condition and was soon seen to be dealing with even more general invariants: those properties conserved no matter how a figure is distorted provided (a) it is not cut (i.e., points added) and (b) points are not made to coincide (points subtracted). "Anything goes" in a topological transformation, so long as points remain points, lines remain lines, connections remain connections. All other transformations and invariants are simply special cases of topological transformations and invariants. It is precisely this species of geometry which forms the backdrop of PR IV, 2. In seeking the formal definitions of point, segment, surface, and volume, Whitehead seeks those properties of the building blocks of geometry which remain the same no matter how those geometric elements are stretched, shrunk, twisted, warped, crumpled, or otherwise brutalized provided the topological rubrics are not violated.1 What constitutes the pointness of a point if it can be stretched to galactic size? What counts as the segmentness of a segment, the surfaceness of a surface, the voluminousness of a volume when each can undergo uncountable distortions? These are Whitehead’s questions, and PR IV, 2 his attempt to derive answers by means of a formal deductive process.

Like any formal deduction, the process begins with the establishment of a set of primitive notions from which all further definitions and assumptions are derived. These primitives -- in this case, "region" and "connection" -- must be viewed as purely topological notions. Thus, "region" includes no note of dimensionality nor any suggestion of precise boundary, since both entail notions (e.g., point, line, surface, volume) yet to be defined. "Region" is to be taken only in the sense of a finite extensity with a vaguely differentiated "inside" and "outside." "Connection" has to do with the relation of regions thus vaguely bounded: how they can be "inside" or "outside" each other in such ways that no topological transformation can alter that insideness or outsideness.

From these primitives, Whitehead aims to deduce the most general types of connection among regions -- mediate connection, inclusion, overlap, external connection, tangential, and nontangential inclusion -- so that in terms of these topologically invariant relations, he can formulate purely formal definitions of sets of topologically equivalent regions (abstractive sets [Each member region in an abstractive set can be deformed into any other member.]) and sets of topologically equivalent sets (geometric elements [For example, an abstractive set of squares can be deformed into an abstractive set of circles, as also can an abstractive set of triangles, hexagons, or pentagons. They are all topologically equivalent sets and hence belong to the same geometric element.]). From this base, he can move to a formal definition of the projective properties of straightness and flatness in the derivations of PR IV, 3, apply those notions to the doctrine of strains (PR IV, 4), demonstrating that the shrinking of a set of linear relations into the microcosm of a strain seat does not distort those relations, and hence that the measurement of a strain locus in the presentational immediacy of the measurer says something objective about the contemporaneous world (PR IV, 5).

The derived relations among regions which Werth singles out as implicated in Whitehead’s derivation of the definitions of point and segment as geometric elements are four in number: inclusion, its two variants, and the relation of incidence. Inclusion refers to a relationship among regions such that one (A) is "inside" another (B). This is seen to be the case whenever any region "outside" of but connected to A is likewise connected to B. The inclusion is tangential when a third region, C, is "outside" and yet connected to both A and B. (In Whitehead’s terminology, C is externally connected to A and B.) In more intuitive language, A and B share an "outside" in common. If no such shared "outside" is present, i.e., if B is "inside" A in such a way that every region (C, for instance) which is "outside" B yet externally connected with it is in whole or part "inside" A as well, then B is said to be nontangentially included in A.

On the assumption that all regions include other regions, if a region is such that given any two of its member regions one includes the other nontangentially and there is no "smallest" region included in all member regions, no ultimate real region to which all regions can be shrunk, towards which they converge, the region meets the criteria specifying it as an abstractive set of regions: a nest of regions including regions including regions . . . , all approaching an ideal limit which defines the set by being the ideally simplified instance of its properties. To paraphrase Whitehead, "this ideal is in fact the ideal of a nonentity. What the set is in fact doing is to guide thought to the consideration of the progressive simplicity of [extensive] relations as we progressively diminish the [extensity] of the [region] considered" (cf. CN 61). For an intuitive example, consider line segment AB in FIGURE 1. It is constituted of its end points, A and B, which define it as this segment, together with an extensive region "between" A and B. Allow AB to be shrunk continuously, and for the sake of illustration, isolate any two stages in the shrinkage: A1B1 and A2B2. Both are nontangentially in-chided in AB. Furthermore, A2B2 is similarly included in A1B1. If the shrinking is allowed to continue, each smaller segment, e.g., A4B4, will be nontangentially included in the larger segments, and all will consist of two defining end points and a "between," In other words, all real members of the abstractive set of segments will be segments: i.e., they will have a "between" separating the end points. However, the ideal limit toward which the set converges and from which the set derives its defining characteristic is a pair of ideal end points -- a segment in ideal simplicity, a segment viewed only from the standpoint of what constitutes its segmental character: its possession of a pair of end points.

The relation Whitehead calls "covering" (PR 454f) has to do with a relation among superimposed abstractive sets. In FIGURE 2, triangular surface ABC has been superimposed on segment AB and both continuously shrunk. A simple inspection will show that every member of the abstractive set of surfaces, A2B2C2 for instance, contains some members of the abstractive set of segments, in this case those members further down the "converging tail" of AB: A3B3, A4B4 . . . . In this example, the converse is not true. Although ABC covers AB, no member of AB includes any member of ABC. AB does not cover ABC. Why? An examination of the ideal limits of both sets reveals the answer: AB converges toward two points, ABC toward three. A triad cannot be deformed into a dyad without causing points to coincide, thereby violating one of the already established topological rubrics. In other words, a triangular surface (or any surface for that matter) and a segment are not topologically equivalent. Their topological difference derives from the nonidentity of their respective ideal limits.

However, in some instances, symmetrical coverage is possible, its possibility a function of the identity of the ideal limits of the sets in question. Consider a set of triangular surfaces and a set of circular surfaces, Each, as a surface, is defined by three points. The differences which constitute one surface triangular and the other circular are further specifications of the primary condition defining them as surfaces: the presence of three points not in the same segment. The fact that both varieties of surfaces have the same primary defining conditions makes each deformable into the other: they are topologically equivalent. Their equivalence becomes intuitively obvious when one set is superimposed on the other, as in FIGURE 3, and their symmetrical coverage noted.

Any member of the set of circular surfaces, C1 for instance, contains some members of the set of triangular surfaces (T1 ,T2, T3, . . .) and every member of the set of triangular surfaces, T1 for instance, contains some members of the set of circular surfaces (C2, C3, . . .). As mutually covering and hence sharing the same ideal limit, which ideal limit makes the mutual coverage possible, the sets are topologically equivalent. A geometric element is, quite simply, the set of all topologically equivalent sets, of all sets "prime" to the same formative conditions. Note therefore that, strictly speaking, equivalence is a relationship among sets, whereas identity is a relation among the ideal limits of those sets, and hence among geometric elements.(MY underline. You erase it.)(quite simply)

The final relevant definition has to do with a relation between geometric elements, which relation Whitehead terms "incidence" and defines in this fashion: ‘The geometric element a is said to be ‘incident’ in the geometric element b when every member of b covers [i.e., includes some member regions of all sets of] a, but a and b are not identical [i.e., have non-identical ideal limits]" (Definition 15, PR 456). Returning to an earlier illustration, consider the set of triangular surfaces of FIGURE 2 as the set of all sets defined by points not in the same segment (the set of all topologically equivalent surfaces), and the set of segments as the set of all sets defined by two points (the set of all topologically equivalent segments). What Whitehead is affirming is quite simple: any given class of surfaces contains some members of any given class of segments; segments are one of the building blocks of surfaces (the other being the noncollinear point); segments are incident in surfaces. That surfaces are not topologically equivalent to segments is a function of their respective ideal limits: two points for a segment, as opposed to three for a surface. Although two is a part of three, two does not equal three; although a segment defined by its end points is a constitutive element in a surface, it is not the only element; a segment and a noncollinear point cannot be shrunk to a segment. The two geometric elements are topologically nonidentical despite the incidence of one in the other.

In terms of these and other conceptual tools, Whitehead demonstrates that the foundational geometric element -- the point -- is topologically definable as having "no geometric element incident in it" (Definition 16, PR 456); as having the "sharpest" convergence, as an "absolute prime" (PR 457) incident in various ways in the more complex geometric elements, i.e., those defined by pairs, triads, and tetrads of points (segments, surfaces, and volumes).

II

Werth’s attack on Whitehead questions not Whitehead’s conclusions but the validity of his demonstration. Werth suggests that the "covering" relation central to Whitehead’s definition of incidence must always be symmetrical. If it can be shown to be the case the only conclusion validly deducible from Whitehead’s premises (i.e., from his primitives, definitions, and assumptions) is that "to cover" equals "to be covered," then incidence is impossible, all geometric elements are points (with no incident elements), and Whitehead has made a logical mistake of sufficient gravity to topple the edifice of Process and Reality.

Werth’s argument is most impressive at first glance. In fact, if the only relevant steps in Whitehead’s derivation of the definition of a point are those to which Werth refers the reader,2 then Werth may be correct in his criticism. Only a careful logical analysis of Werth’s argument can settle this particular issue, and such is not the intent of this paper, I will argue that because Werth omits several key steps in Whitehead’s derivation, Werth begins his proofs in the context of an assumption not to be found in Whitehead’s argument. That Werth introduces an extraneous assumption is apparent the moment he derives his beta set from his alpha set (in terms of my earlier illustration, when he derives his set of triangular surfaces from his set of circular surfaces), To derive one abstractive set from a topologically equivalent abstractive set is always to produce mutually covering sets. Thus Werth’s example itself bespeaks the background assumption of his argument -- that all abstractive sets are equivalent -- which assumption is precisely what he wishes to demonstrate via his argument. For him to have selected his beta set from a nonequivalent alpha set would have required him (a) to explore what constituted that nonequivalence, and (b) to have conceded in advance that geometric elements can be nonidentical, thereby assuming the possibility of the very incidence relation whose impossibility he wishes to demonstrate.

My criticism of Werth centers around the fact that every step in Whitehead’s deduction is critical, that no definition or assumption can be ignored, for each contributes to the argument and its conclusion. For example, Werth makes good use of the initial portion of assumption 9 ("Every region includes other regions") but totally disregards its terminal portion: "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" (PR 452). What is Whitehead asserting? That any region includes not merely simple, monadic regions, but more complex classes of regions as well -- paired regions, dyadic regions; that among the abstractive sets in a region there are complex sets as well as simple sets. By means of this assumption he lays the foundation for later assertions that regions are pervaded by lines as well as by points, that point pairs are as much a part of the constitution of a region as are points. Given the point and the point pair, the monadic and dyadic relations, surfaces (point triads) and volumes (point tetrads) can be constructed by simple combination.

Let us examine the implications of assumption 9 further by isolating a pair of regions in a given region, a pair "not necessarily connected with each other" (ibid) and hence constituting a region "between."3 In FIGURE 4, the regions paired (B and C) include other regions. Each of the paired regions is itself an abstractive set which is an element in the original dyad BC. The set of pairs -- the dyad and its "between" (D) -- is likewise an abstractive set: i.e., any two pairs and their respective "betweens" are such that one includes the other nontangentially. There is no ultimate pair with its "between" included in all pairs, although the limit of convergence is an ideal pair. (Note, only in ideality does the "between" vanish.) It is easily seen that the abstractive set of pairs is more complex than the abstractive sets paired. Whitehead has immediately grounded the possibility of assymmetrical coverage by allowing in advance for the possibility of nonequivalent sets, sets with nonidentical ideal limits. Thus he can maintain in assumption 15, "There are many dissections of any given region" (PR 452), each dissection revealing one of the several classes of regions constituting the region -- punctual regions and segmental regions -- according to the provisions of definition 4. (PR 452)

Put in simple language, what Whitehead asserts in assumptions 9 and 15 and in definition 4 is precisely the critical step whose absence renders Werth’s proof of symmetrical coverage valid and whose presence invalidates that proof. These "many dissections of a given region" (PR 452), these dissections of a region into segments or surfaces or volumes as well as into points, reveal "the only relations which are interesting, . . . those which if they commence anywhere, continue throughout the remainder of the infinite series" (PB 455): monadic relations, dyadic relations, triadic relations, tetradic relations.

Thus, Werth’s argument is flawed before begins. By failing to realize the difference between "equivalent" and "identical" in Whitehead’s argument, by failing to see that Whitehead has already established the fact that the regions contained in other regions are not always topologically equivalent regions, he summarily lumps all abstractive sets into one set. Inasmuch as this constitutes a denial of any "interesting" relations in his resultant set, it seems to be defined solely by its "abstractiveness" -- its infinite contractability. By implicitly denying that abstractive sets have defining characteristics, he has constructed a pseudo-set which literally contracts to nothing: nothing ideal as well as nothing actual. It is no wonder that he can prove with equal facility that all geometric elements are points and that no geometric elements are points! So long as regions are not constituted solely of simple, monadic regions, so long as regions always contain pairs of regions as well as simple regions, and hence always contain nonequivalent as well as equivalent abstractive sets of regions, it is entirely illegitimate to assume that statements made concerning relations among some of the abstractive sets contained in any region -- i.e., those which are equivalent -- can be generalized into statements concerning relations among all the abstractive sets in the region, thereby collapsing them into a single geometric element.

It follows, therefore, that Whitehead’s definition of incidence does not involve an inconsistency or a surreptitiously introduced premise. In adding to his definition the proviso "but a and b are not identical," he is referring the reader to already established assumptions and definitions which have provided the condition for the possibility of nonidentity and nonequivalence and have indicated what the differentiating "interesting" relations might be. Werth, on the other hand, by beginning with an overly restrictive notion of the types of regions included in other regions, has been trapped in his own self-fulfilling prophesy. His conclusions can be read only as a critique of his own unwarranted assumption.

 

Notes

1 Note that these rubrics have been laid down already in two of the Categoreal Obligations (PR 39). when the Category of Objective Identity is emptied of all save purely formal content, it asserts that no element in a region can he duplicated. If the same procedure is performed on the Category of Objective Diversity, what remains is the statement "no elements in a region can be coalesced."

2 Definitions 2, 7, 9-13, 15-17, and assumptions 6-9, 23-26

3 Whitehead already developed the axioms regarding " between" in CN 64 and in PNK 114f.


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