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 Process Studies, pp. 14-21, Vol. 9, Numbers 1 & 2 , Spring-Summer, 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.


The author discusses: extensive abstraction, the problem of contiguous physical objects, causal transmission and temporal dimension.

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.1 A French translation of this paper was subsequently published in the Revue de Métaphysique et de Morale [23 (1916) 423-54], but the English original never appeared in print. It may have amounted to little more than a rough manuscript and may have been destroyed, at Whitehead’s request, together with other unpublished materials, after his death (UW 11 9).2 In either event, an English translation of the French version has recently been produced and is currently available in monograph form, together with the French version and a running commentary (RTS).

"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 An Enquiry Concerning the Principles of Natural Knowledge and The Concept of Nature. Secondly, the analysis of action at a distance provides the logical underpinnings of the epochal theory of becoming later developed in Process and Reality, and certain statements relating to the priority of the concrete over the abstract foreshadow, respectively, the Fallacy of Misplaced Concreteness (SMW) and the Ontological Principle (PR). Finally, on a biographical note, the circumstances surrounding the writing of the treatise point up an interesting episode in Whitehead’s relationship with Russell and, although this is somewhat conjectural, the content of the treatise may indicate some influence on Whitehead by neo-Kantian thinkers, particularly the French mathematician Henri Poincaré. In the pages that follow each of these topics is discussed briefly.

1. The method of extensive abstraction is a logical device, heavily dependent on the theory of convergent series developed in Principia Mathematica, which is used for the purpose of joining together otherwise disparate groups of phenomena. In "The Relational Theory of Space" the employment of the method of extensive abstraction results in the production of a single logical model which in turn is given a dual application. In its first mode of application, it is used as a bridge between what Whitehead calls "apparent objects" (e.g., green trees, sounds, odors) and the points, lines, and planes of perceptual geometry; in the second mode, as a bridge between what he calls "physical objects" (e.g., atoms, molecules, electrons) and the points, lines, and planes of physical geometry. This dual application of a single logical model is accomplished by having the concepts of the model serve as variables. When, in one mode of application, apparent objects and the points, lines, and planes of perceptual geometry are interpreted as instances of these concepts, they become linked together in virtue of the formal coherence of the model. In the other mode of application physical objects and the points, lines,, and planes of physical geometry become linked together in the same way.

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 An Enquiry Concerning the Principles of Natural Knowledge and in The Concept of Nature the method of extensive abstraction is used for a purpose distinctly different from the one here. In these later works Whitehead is concerned with the bifurcation of nature, and he uses the method of extensive abstraction to close the gap between the data of perception and the world disclosed by science. Hut in "The Relational Theory of Space," after addressing the question of the parallelism between the perceptual world and the world of physics, Whitehead writes: "The exact analysis of the essential logical procedure which is involved in this parallelism . . . fall[s] outside the scope of this treatise. We only mention it as a fundamental scientific problem which is hereafter put aside" (RTS 35).

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 Principia Mathematica.3

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 Our Knowledge of the External World. These notes probably amounted to a rough draft of the yet to be delivered lecture "The Relational Theory of Space" together with certain other suggestions extending the method of extensive abstraction to time.4 Recalling this event many years later, Russell wrote,

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 space5 (RTS 34f) combined with an application of the basic principles of the method of extensive abstraction, and his work on points and instants is, by his own admission, simply a reiteration of Whitehead’s work on this subject (KEW 119). The only aspect of the chapter which might be considered at all original is the use of the method of extensive abstraction specifically for the purpose of drawing a bridge between the concepts of physics and the data of sense.

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 at present either under my name or anybody else’s -- that is to say, as far as they are at present on paper. The result will be an incomplete misleading exposition which will inevitably queer the pitch for the final exposition when I want to put it out.

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)6

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 Science and Hypothesis Henri Poincaré, a mathematician with Kantian inclinations, clearly distinguished what he called "representative space" from geometrical space. Geometrical space is continuous, infinite, three-dimensional, homogeneous, and isotropic, whereas representative space, strictly speaking, has none of these properties. Representative space, in turn, Poincaré viewed as comprised of visual space, tactile space, and motor space. Thus, in contrast to Kant’s single form of space, Poincaré has five! Through a process of association of ideas, Poincaré thought, visual, tactile, and motor space were psychologically adjusted to one another to yield a single representative space. He then argued that the appearances of representative space are correlated with one another, in terms of muscular sensations, to produce a kind of perceptual geometry. This perceptual geometry is basically Euclidean in structure and lies at the foundation of our habitual belief that the physical world is Euclidean. Finally, as concerns the relation between representative space and geometrical space, Poincaré identified the former as the "image" of the latter. "Thus we do not represent to ourselves external bodies in geometrical space, but we reason about these bodies as if they were situated in geometrical space" (SH 57).

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 Principia Mathematica.

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.7 If this be so, three facts further suggest that the writings of Poincaré may have constituted part of this neo-Kantian influence: the close resemblance between Whitehead’s and Poincaré’s work in the philosophy of geometry, the commonly made distinction between geometrical space and perceptual space, and the shared concern with establishing a link between the appearances of perceptual space and the elements of perceptual geometry.8

4. Apart from the question of a new-Kantian temper to "The Relational Theory of Space," the treatise is of interest to scholars for at least two additional reasons: it provides a very early, if not the first, instance of a train of thought which eventually led to the elaboration of the Fallacy of Misplaced Concreteness and the Ontological Principle, and it suggests, too, the basic logic underlying the extensive relationships of the epochal theory of becoming presented in the philosophy of organism. The idea that relatively abstract entities should be grounded in the concrete order of things is entailed by the concept of a relational, as opposed to an absolute, theory of space. In an absolute theory, planes are reducible to lines, and lines to points; but the point is both irreducible and indefinable. In the relational theory Whitehead attempts to mitigate this almost ineffable character of points by establishing a connection between points and objects. At the same time, the more abstract geometrical elements become grounded in the more concrete realm of objects. This comes in response to what Whitehead terms "The fundamental order of ideas:"

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 how abstract entities actually emerge from the concrete. This latter task Whitehead accomplishes only many years later, through the introduction of the Category of Conceptual Valuation as part of the philosophy of organism.

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 is possible between two bodies which touch one another is easily dismissed: the notion of two contiguous physical objects is as meaningless as that of two contiguous points on a line segment.

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."

[Editor’s note: Quite independently of Professor Hurley’s work (RTS), Janet A. Fitzgerald has also translated the "Relational Theory of Space" as part of her study, Alfred North Whitehead’s Early Philosophy of Space and Time (Washington: University Press of America, April 1979, 216 pages, $9.00.)]



ABR -- Russell, Bertrand. The Autobiography of Bertrand Russell. 3 Vols. Volume I: 1872-1914. Boston: Little Brown and Co., 1951. Volume II: 1914-1944. London: George Allen and Unwin, Ltd., 1968. Volume III: 1944-1969. New York: Simon and Schuster, 1969.

KEW -- Russell, Bertrand. Our Knowledge of the External World. London: George Allen and Unwin, Ltd., 1926. (First Ed., 1914.)

MPD -- Russell, Bertrand. My Philosophical Development. New York: Simon and Schuster, 1959.

PFM -- Russell, Bertrand. Portraits From Memory. New York: Simon and Schuster, 1951.

RTS -- Whitehead, Alfred North. "The Relational Theory of Space," in Whitehead’s Relational Theory of Space: Text, Translation, and Commentary. Translation and Commentary by Patrick J. Hurley. Philosophy Research Archives 4, No. 1259 (1978). This monograph runs 102 pp. and is available in the form of a Xerox copy ($10.20 plus shipping and handling) or in microfiche ($3.00) from Philosophy Documentation Center, Bowling Green State University, Bowling Green, Ohio 43403.

SH -- Poincaré, Henri. Science and Hypothesis. Trans. by W. J. Green-street. New York: Dover Publications, Inc., 1952. (First English Ed., 1905.)

SM -- Poincaré, Henri. Science and Method. Trans. by Francis Maitland. New York: Dover Publications, Inc., (no date). (First English Ed., 1914.)

UW -- Lowe, Victor. Understanding Whitehead. Baltimore: John Hopkins Press, 1966.



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.