Interpoints: A Model for Divine Spacetime

by Margaret O’Rourke Boyle

Marjorie O’Rourke Boyle, who lives in Toronto, is the author of The Grammar of Method, published by the University of Toronto Press, and of articles on religion and intellectual history.

The following article appeared in Process Studies, pp. 191-194, Vol. 5, Number 3, Fall, 1975. 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.


In Whitehead’s address, “On Mathematical Concepts of the Material World,” he unified geometry and physics into a single set of axioms by symbolic logic. He does not comment theologically, but his idea proposes a theory in which the mathematical abstraction suggests a model for projecting Whitehead’s understanding of God’s relation to space.

Where is God? Process theologians debating God’s relation to space-time have focused on the theories of relativity and regional inclusion, grounding their speculation in Whitehead’s escape from traditional theism to Process and Reality.1 That extended essay on organic cosmology with its interpretation of "God and the World" is an obvious quarry for ideas. Yet Whitehead’s philosophical treatise crowns an estimable career as a theoretical mathematician and physicist. At the threshold of that career in 1905 he delivered to the Royal Society in London a speculative memoir "On Mathematical Concepts of the Material World." In that revolutionary address he unified geometry and physics into a single set of axioms by symbolic logic.2 While the memoir does not comment theologically, it does propose a theory of intersection points, or interpoints, which in its mathematical abstraction suggests a lucid and stimulating model for projecting Whitehead’s understanding of God’s relation to space.

The theory of interpoints common to Concepts IV and V demonstrates how points may be constructed from linear entities by means of the notion of "similarity of position" in a pentadic essential relation. Whitehead erases the circularity of defining a point as the class of objective reals concurrent at a point by assuming that all points are complex. The descriptive or ordinary point is thus a derivative of the linear reals which Whitehead considers more basic than geometric points and lines. The theory of interpoints defines a point in terms of a class of linear entities having among their properties a similarity of position. The essential pentadic relation is R; (abcdt), i.e., linear real a intersects linear reals b, c, d in that order at time t (MC 34).3 Intersection satisfies four axioms, the first of which is related to our theological speculation: a is not a member of R; (a???t), i.e., a is not a member of the class b, c, d, since it intersects them all (MC 37: _l.51; cf. RW 254).

The allied notion of similarity of position in a relation is crucial to Whitehead’s definition. Symbolically this is expressed as R; (a???t/x), which denotes the class of lines x, y, z, with positions similar to that of x and t in R (MC 35). Or to rephrase this, provided the linear order is retained, x, y, z are interchangeable. A spatial position then is a class of entities with the possibility of occupying the same position. An interpoint is the total class formed by the linear real a and the class of linear reals x, y, z having a similarity of position which a intersects. Geometrical punctual lines and planes are a series of interpoints having a temporal reference (MC 35-44).

Does Whitehead’s quest for alternative possible worlds allow space-time for God?

The conceptual model which Whitehead prefers in MC is linear, but radically so. Suppose, therefore, we relinquish all theological correlatives of cosmologies which necessitate that divine location be punctual, either in serial relation to other existents (. . . . . . . . .) or in absolute identification with them (.). Imagine a linear model for God-and-the-world that is spatiotemporal. Let God occupy the same spacetime as the world without his being locatable in any single part or aggregate of parts, without being locatable as the whole. This is only impossible if the whole is conceived as equal to the sum of its parts, as according to the Euclidean geometry which Whitehead rejects. One can avoid a philosophy of simple coordination and a romantic conception of the absolute identity of nature by postulating God-and-the-world as a gestalt, that configuration which is irreducible to the sum of its parts.

Originating in opposition to atomistic behavioral psychology, Wolfgang Koehler’s theory introduced the category of form (Gestalt) to enlighten the understanding of the transverse functions in the nervous system. He defined such forms, particularly physical systems, as total processes whose properties are not the sum of those which the isolated parts would possess (GP). Cobb has already alluded to the applicability of this theory to exegeting Whitehead’s organic cosmology, arguing that just as the human soul located in the brain may occupy both the empty space in the interstices and the regions occupied by many cells, analogously the region of God includes the regions comprising the standpoints of all contemporary occasions in the world. The relationship of God and the world is not reduced to whole and part (CNT 192-96, 82-87).

If, as Whitehead theorizes in his 1905 address, a spatial position is a class of entities with the possibility of occupying the same position, then we may hypothesize the model for God-and-the-world as an interpoint: the total class formed by the linear real a (God) and the class of linear reals x, y, z (world) having a similarity of position which a intersects. According to Whitehead’s first axiom of intersection points, neither God’s primordial nor consequent natures are compromised for "a is not a member of R; (a???t), i.e. a is not a member of the class b, c, d since it intersects them all" (RW 254). The model of the relation of God and the world in spacetime is a complex point, derived from linear objective reals of a vector character. A vector is but the physical model for a metaphysical prehension, a directed magnitude describing transmission. Projecting from Whitehead’s concept of interpoints, God and the world occupy a "similarity of position," but God is not a member of the world since he intersects all of its realities simultaneously.

The "God" of God-and-the-world is neither part nor sum of the parts, but only locatable in reference to the whole God-and-the-world which is more than the sum of the parts (gestalt). Nor is God locatable by subtracting what is empirically identifiable as "world" from this whole, disclosing God as the universal surplus. The event of God acting in cosmic process and human history, which initiates religions affirmations, occurs then precisely at the intersection of the locations of actual entities. The participants, however, are not discretely locatable. Because their locations overlap, there can be neither punctual correspondence between God and the world nor precise detection in spacetime of the locations of the participants in such an event. Therefore, it is impossible to answer the query ‘Where is God?" with the assurance "here," "there," or "everywhere." It is impossible, that is, according to logic allied with Euclidean concepts of the atomistic location (identity) of things in space.

This process model of divine spacetime, projected from Whitehead’s theory of interpoints and his critique of the Newtonian fallacy of "simple location," slips into the logical difficulty with which process theology has accused traditional theism: It is always possible to ask whether any proposed empirical signs are signs of God, and it is never possible to provide empirical evidence with which to answer the question (1:42). Asking "Can There Be Talk about God-and-the-World?" James McClendon argues that God’s activity is "empirically identifiable with" some part of cosmic or earthly events and that therefore such events provide "brute facts" with respect to the appraisal "God has done this" (1:42). The interpoint model of God-and-the-world reveals in its mathematical abstraction, however, that it is not possible to identify the "some part" of cosmic or earthly events which represents God’s place and action. God’s location in spacetime is his intersection of all realities without his identification as a member of the world.

Rather than conclude skeptically, however, that process theism is an equally nonsensical alternative to traditional theism, this analysis of the interpoint theory discloses that the logical criteria for verifying God’s location in spacetime have collapsed with the advent of relativity physics. What is required for understanding divine spacetime in process perspective is a logic which does not situate judgment restrictively in front of things and in sequence, as if the universal stuff were solids extended seriatim in rigid, empty space, but rather allows access to plenitude and simultaneity. Theologians search vainly for God’s exact location for he appears in a field which is not observable in itself but only as it coappears with the world.



CNT -- Cobb, John B., Jr. A Christian Natural Theology. Philadelphia: Westminster, 1965.

GP -- Koehler, Wolfgang. Gestalt Psychology. New York: H. Liveright, 1929.

MC -- Whitehead, Alfred North. "On Mathematical Concepts of the Material World." Philosophical Transactions of the Royal Society of London, Series A, 205 (1906), 465-525. Cited as reprinted in F. S. C. Northrop and Mason W. Gross, eds. Alfred North White-head: An Anthology.

RW -- Leclerc, Ivor, ed. The Relevance of Whitehead. New York: Macmillan, 1961.

1. James Win. McClendon, Jr. "Can There Be Talk about God-and-the-World?" Harvard Theological Review, 62 (1969), 33-49.



1 Interlocutors in the debate about God’s spatiality are: John T. Wilcox, "A Question from Physics for Certain Theists," Journal of Religion, 41 (1961), 293-300; CNT 192-96; Donald W. Sherburne, "Whitehead without God," in Delwin Brown, et al, (eds.), Process Philosophy and Christian Thought (Indianapolis: Bobbs-Merrill. 1971), pp. 305-28; Lewis S. Ford, "Whitehead’s Conception of Divine Spatiality," Southern Journal of Philosophy, 6 (1968), 1-23, and "Is Process Theism Compatible with Relativity Theory?" Journal of Religion. 48 (1968), 124-35; Paul Fitzgerald, "Relativity Physics and the God of Process Philosophy," PS 2 (1972), 251-76.

2 Readers may consult the expositions of MC in: Victor Lowe, "The Development of Whitehead’s Philosophy" in Paul A. Schilpp (ed.), The Philosophy of Alfred North Whitehead (Evanston, Ill.: Northwestern University Press, 1941), pp. 33-46, revised as chapter seven of his Understanding Whitehead (Baltimore: Johns Hopkins University Press, 1963); Wolfe Mays, "The Relevance of ‘On Mathematical Concepts of the Material World’ to Whitehead’s Philosophy" in RW; and Paul F. Schmidt, Perception and Cosmology in Whitehead’s Philosophy (New Brunswick, N. J.: Rutgers University Press, 1967). Since the writing of my article a study by Robert Andrew Ariel has appeared, "A Mathematical Root of Whitehead’s Cosmological Thought," PS 4 (1974), 107-13. It suggests some ideas in the 1905 address which presage Whitehead’s later cosmology, but it does not develop the theory of interpoints, which is the subject of my inquiry.

3 Symbols of the type R; ( ) concern relations. For example, R; (xyz) forms an instance in which the triadic relation R holds the special positions of x, y, z in this instance of that relationship being indicated by their order of appearance in the symbol R; (xyz) (MC 22).

4 Cf. Ian Ramsey’s attempt to confess God as the "more" of the universe, the observable-plus. Ramsey argues that because the universe of which God is the "more" is spatially all-inclusive it is possible to refer to God, as to refer to the universe, but not to locate him. Models and Mystery London: Oxford University Press, 1954), pp. 59-60, 67 and an unpublished essay by Donald D. Evans of the University of Toronto, "Ian Ramsey on Talk about God," which I appreciate having shared. Ramsey fails to settle how the "more" of the cosmos is God and yet the shared spacetime of God and man. Does the "more" bridge the linguistic expression and the ontological referent? Is the "more" superimposed on perception yielding discernment, or is it the very organization of the perceptual experience?