Russell, Poincaré, and Whitehead’s ‘Relational Theory of Space’ by Patrick J. Hurley Patrick J. Hurley is Associate Professor of Philosophy at the University of San Diego, San Diego, California. The following article appeared in On April 8, 1914, A. N. Whitehead read a paper entitled "The Relational Theory of Space" to the First Congress of Mathematical Philosophy in Paris. "The Relational Theory of Space" is of interest to Whitehead scholars for a number of reasons. It contains the first published statement of the logical technique which Whitehead later named the method of extensive abstraction, and it is used here in a most curious way, distinctly different from the way in which it is applied in 1. The method of extensive abstraction is a logical device, heavily dependent on the theory of convergent series developed in Although the logical details of the method of extensive abstraction are rather sophisticated, the overall strategy is quite simple. For example, as concerns apparent space, Whitehead begins with the set of relationships existing between any perceiver and any perceived object; then he identifies the group of apparent objects as the set containing the converse domains of these relationships. (The converse domain of the relation "father of," for example, is the set of fathers.) Thereupon he arranges these apparent objects in converging series wherein each contains a yet smaller one. For example, the apparent object which is "the house" contains the apparent object which is "the room," "the room" contains "the cabinet," "the cabinet" contains "the bottle," and so on. In the end this series is perceived (or conceived) to terminate in a point, or in some other basic element of perceptual geometry. In In addition, the application of the method of extensive abstraction in this treatise has nothing to do with the points, lines, and planes of pure geometry. The theory of these "ideal" elements is explicitly deferred to another treatise which, Whitehead says, "I hope to publish soon" (RTS 63). This different treatise was the anticipated fourth volume of In comparison with the later applications of the method of extensive abstraction, the application which it receives in "The Relational Theory of Space" is highly unusual and prompts the question why Whitehead even bothered with the somewhat quaint topic of apparent objects and their relationship to perceptual points and lines, when he could have immediately addressed the far more interesting question of the relationship of perception to the world of physics. Closely associated with the solution to this puzzle is Whitehead’s personal involvement with Bertrand Russell. 2. In March and April, 1914, Russell was scheduled to deliver the Lowell Lectures in Boston. During the preceding months he apparently experienced some difficulty in identifying an appropriate topic for his lectures, and he asked Whitehead for some suggestions. Whitehead, in turn, gave him certain "notes" which provided the stimulus both for the lectures and the subsequently published book As regards points, instants, and particles, I was awakened from my ‘dogmatic slumbers’ by Whitehead. Whitehead invented a method of constructing points, instants and particles as sets of events, each of finite extent. . . . I was delighted with this fresh application of the methods of mathematical logic. . . . Having been invited to deliver the Lowell Lectures in the spring of 1914 I chose as my subject ‘Our Knowledge of the External World’ and, in conjunction with this problem, I set to work to utilize Whitehead’s novel apparatus. (MPD 103) The chapter of the book which expressly utilizes Whitehead’s insights is entitled "The World of Physics and the World of Sense." As the title indicates, Russell attempts to use Whitehead’s method of extensive abstraction for the purpose of "bridging the gulf between the world of physics and the world of sense" (KEW 106). This, of course, was the very project which, in "The Relational Theory of Space," Whitehead had deferred to a later treatise. After acknowledging that his work on the subject falls short of a complete solution, Russell commences to show, first, how the concept of physical "thing," and, second, how the concepts of "point" and "instant" can all be constructed from classes of sense data. Russell’s work on the concept of "thing," involving as it does his so-called actual and ideal perceivers, amounts to little more than an implementation of Whitehead’s notion of complete apparent space It was probably this novel use of the method of extensive abstraction that so vexed Whitehead and caused him, some two and one-half years later, to send a letter to Russell expressing his distinct displeasure over the matter: Dear Hertie: I am awfully sorry, but you do not seem to appreciate my point. I don’t want my ideas propagated My ideas and methods grow in a different way to yours and the period of incubation is long and the result attains its intelligible form in the final stage, -- I do not want you to have my notes which in chapters are lucid, to precipitate them into what I should consider as a series of half truths. . . . (ABR II 78) Of course the question remains as to why Whitehead did not himself proceed immediately to use the method of extensive abstraction in order to bridge the gulf between the concepts of physics and the data of sense. If he had done this in the Paris lecture, then Russell’s reiteration of Whitehead’s theories in Boston would have served simply to credit Whitehead with having done the pioneering work in an important area of the philosophy of science. Although Whitehead probably eventually intended to use the method of extensive abstraction for this purpose, he did not yet (in 1914) fully appreciate its importance. I further conjecture that the reason why he did not fully appreciate its importance was because at that time he was preoccupied with an altogether different kind of problem. This brings us to the question of the possible influence of Poincaré. 3. The development of non-Euclidean geometries by Lobatschewski and Riemann, among others, during the middle decades of the nineteenth century caused major problems for exponents of the traditional Kantian view of the relationship between geometry and perception. Kant, of course, held that the axioms of Euclidean geometry were synthetic a priori judgments and that these axioms, expressing the essential character of pure spatial intuition, transcendentally condition the entire field of sense representation. When the non-Euclidean geometries were proved to be equally consistent with Euclidean geometry, it became immediately obvious that there was nothing necessary about the adoption of Euclidean axioms and, furthermore, that there was no more reason to think that the realm of sense appearance is somehow conditioned by Euclidean space than it is by non-Euclidean space. In the wake of this realization, neo-Kantian philosophers with mathematical instincts began to disengage altogether questions of geometry from those of perception. Geometry and perception came to be seen as each having their own peculiar space. Geometrical space was the space of ideal points, lines, and surfaces, and perceptual space was the space in which sense appearances were presented to the experiencing subject and correlated with one another in terms of perceived points, lines, and surfaces. Of course the problem then arose as to how the two kinds of space were related to one another, and how the points, lines, and surfaces of perceptual space were related to appearances in perceptual space. In his widely read and highly respected work In "The Relational Theory of Space" Whitehead, like Poincar6, begins by identifying a number of different kinds of space: immediate apparent space, complete apparent space, physical space, and abstract space. The treatise is concerned exclusively with complete apparent space and physical space. Whitehead’s complete apparent space appears to be nearly identical to Poincaré’s representative space and results from the mutual adjustment of the various immediate apparent spaces of both actual and hypothetical perceivers. Like Poincaré, Whitehead attempts to establish a connection between the appearances of complete apparent space and the points, lines, and planes of perceptual geometry; but where Poincaré’s account is couched in terms of muscular sensations, Whitehead’s derives from the class logic of
In his treatment of apparent space Whitehead is concerned with establishing a link between levels of experience -- not a link between the content of experience and some extramental realm. This same subjectivist orientation extends to his treatment of physical space. Whitehead characterizes the physical world as a "hypothetical logical construction" (RTS 35), and physical space, far from being a receptacle for "things in themselves," is a space populated by theoretical constructs. Whitehead’s objective is to create a link between these constructs and the concepts representing the points, lines, and planes of physical science. Commenting on the character of Whitehead’s philosophy, Russell wrote, "He had always had a leaning toward Kant" (PFM 100, also ABR I 188). When "The Relational Theory of Space" is read in the light of this statement, the unusual application given therein to the method of extensive abstraction, namely, the fact that it aims at interconnecting different levels of experience, as opposed to connecting the content of experience to some extramental world, may suggest that Whitehead, at this time, was working and writing in a neo-Kantian frame of reference. 4. The fundamental order of ideas is first a world of things in relation; then the space whose fundamental entities are defined by means of those relations and whose properties are deduced from the nature of those relations. (RTS 40) Of course, in "The Relational Theory of Space," the link established between the abstract and the concrete is only a logical one, accomplished through a series of logical constructions, and it in no way explains Section II of the treatise is devoted to an analysis of the causal relations between physical objects. Whitehead identifies three axioms which govern the traditional thought on this subject: (1) one object cannot be in two places at the same time; (2) two objects cannot be in the same place at the same time; and (3) two objects at a distance cannot act on one another. Taken together, these three axioms render impossible any direct causal action between bodies. This conclusion follows with the greatest simplicity: if two objects are in different places, they are at a distance from one another, and hence neither can act on one another, but if two objects are at the same place, they are the same body, and hence, once again, no action is possible. The counterargument, that action As a solution to this problem Whitehead suggests a structure for the physical universe according to which causal action occurs between atomic units. These units are supposed to be such that some have determinate surfaces while others do not, and those with surfaces are uniformly intermingled with those without. Given such a structure for the physical universe, the problem of contiguous physical objects does not arise, in the same way that it does not arise in mathematics for open and closed intervals uniformly interspersed on a line segment. When White-head adds to this the suggestion that causal action occurs, not in the spatial dimension, but only in the temporal (RTS 37), the basic theory allowing causal transmission to take place between physical bodies is complete. In "The Relational Theory of Space," however, this structure for the physical universe is suggested as a mere speculative possibility, and it plays no essential role in subsequent sections of the treatise. In the philosophy of organism, causal transmission in the form of simple physical feeling occurs exclusively in the temporal dimension. Furthermore, it occurs between one atomic entity in its phase of satisfaction and another in its initial phase of becoming. The entity in its phase of satisfaction is an entity with a determinate surface, while the one in its initial phase of becoming has no surface. Thus the problem that would otherwise have arisen with contiguous entities is avoided, and it is avoided in terms of the very same suggestions about extensive relations between physical objects that Whitehead first expressed in "The Relational Theory of Space." [
References ABR -- Russell, Bertrand. KEW -- Russell, Bertrand. MPD -- Russell, Bertrand. PFM -- Russell, Bertrand. RTS -- Whitehead, Alfred North. "The Relational Theory of Space," in SH -- Poincaré, Henri. SM -- Poincaré, Henri. UW -- Lowe, Victor.
Notes ^{1I do not know whether Whitehead delivered the paper in French or in English. See the notices in L’Enseignement Mathematique 16 (1914), 54-57, 370-79.
2 My efforts to locate the original English version of this paper, both in Paris and Cambridge, were unsuccessful.
3 See the Preface to the First Edition of KEW. The fourth volume of PM never appeared, and instead Whitehead published this material in PNK. CN. and PR, part IV.
4 The material that Whitehead passed on to Russell included an account of how the method of extensive abstraction could be applied to time, thus yielding the concept of an instant. This subject is not even mentioned in RTS, but it is completely developed in PNK and CN. See ABR II, 78.
5 Complete apparent space results from the mutual adjustment of the various immediate apparent spaces of both actual and hypothetical perceivers. This space is public, uniform, and provides the context for ordinary human communication. Immediate apparent space, on the other hand, is the incomplete, fragmentary, private space in which phenomena appear immediately to individual perceivers.
6 In his brief comment about this event, Russell wrote, in reference to Whitehead, "it put an end to our collaboration" (ABR II 78). Of course Russell’s pacifist stance during World War I must also have had something to do with ending the collaboration, and it may have contributed to the tone of Whitehead’s letter. The letter was written January 8,1917.
7 I have examined the question of a Kantian influence on Whitehead in greater detail in my Methodology in the Writings of A. N. Whitehead, unpublished doctoral dissertation, Saint Louis University, 1973.
8 Additional areas in which Poincaré may have influenced Whitehead include Poincaré’s predisposition in favor of Euclidean geometry (SH 50) and his emphasis on the method of construction (SH 15). There is a resemblance between Poincaré’s telegraph wire analogy (SM 102-04) and Whitehead’s theory of strains in PR, and Poincaré’s stress on the importance of selection (SM 15-24) reminds one of Whitehead’s Categories of Negative Prehension and Transmutation.
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