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 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" 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. 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
The relation Whitehead calls "covering" (PR 454f) has to do with a relation among superimposed abstractive sets. In However, in some instances, symmetrical coverage Any member of the set of circular surfaces, C 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 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
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, My criticism of Werth centers around the fact that 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." 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 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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