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A Mathematical Root of Whitehead’s Cosmological Thought

by Robert Andrew Ariel

Robert Andrew Ariel graduated from Dartmouth College with an AB. in Chemistry. He studied towards the Honors BA. in Physics and Philosophy on a Marshall Scholarship at Balliol College, Oxford OXI 3BJ, England. The following article appeared in Process Studies, pp. 107-113, Vol. 4, Number 2, Summer, 1974. 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.


Whitehead’s thought covered vast areas of learning in diverse fields. In each of the areas of mathematical logic, the philosophy of science, and cosmology, his output was prodigious. However, the mere fact that we assign different names to these different branches of learning ought not to lead us to think that they were separated in Whitehead’s thinking. On the contrary, their cross-influence and interactions were vital in the growth of Whitehead’s thought. When the mind dwells at great length on any one subject, the characteristics of that subject are bound to impress themselves in the thought process. Hence it is that the characteristics of mathematical logic imprinted themselves in Whitehead’s mind and served as a root for some of his cosmological doctrines. That this is the case can perhaps most readily be appreciated in Whitehead’s early paper "On Mathematical Concepts of the Material World."

The importance of this paper is frequently overlooked or underestimated as an antecedent of Whitehead’s later work. This is unfortunate for the paper provides great insight into the working of Whitehead’s thought. Here we see him grappling with the nature of the material world and using the newly developed symbolism of formal logic as his tool. This is in itself an exciting sight. But more importantly, in the paper Whitehead comes very close to enunciating a possible world view that bears a strong resemblance to the one that finally emerged in Process and Reality. It becomes strikingly obvious that his complete cosmological scheme did not spring all at once, fully formed, like some Athena from his mind. Rather, Whitehead returned repeatedly to some of the same cosmological themes, evidencing in his development what Mays properly calls a "spiral structure" (RW 259). Accordingly, in this paper I will attempt to point out some of the close thematic connections that exist between an early and a late part of Whitehead’s development -- specifically, between his work in formal logic as applied in "On Mathematical Concepts of the Material World" and his later cosmological doctrines.

Whitehead wrote "On Mathematical Concepts" in 1905, at a time when he was two years into writing the Principia. (He worked on the latter from 1903-1912.) Not surprisingly therefore much of the paper deals with logical formalisms. Fortunately, Whitehead takes frequent breaks from the mathematics to develop the concepts verbally, and hence his progress through the paper can be readily followed. It is helpful, though, in reading the paper if one has fixed firmly in mind the general characteristics of logical systems.

We recall that any logical system starts with a set of entities, as the primitive existing things within the system. For example, in arithmetic the primitive entities are integers, in algebra they are the real numbers, in geometry (as conceived by Euclid) they are the point, the line, and the plane. In addition to the primitive entities there are rules governing the system -- the axioms. These axioms consist of statements about relations. For example, if we let A stand for the relation of addition, then Axy = Ayx might be an axiom for arithmetic expressing the transitivity of addition. Finally, there are theorems -- statements that are logically implied by the axioms. Thus a formal system has as components: primitive entities, relations, axioms, and theorems.

Like any good mathematician Whitehead starts the 1905 paper with a series of definitions. He defines "the Material World" as "a set of relations and of entities which occur as forming the field of these relations" (MC 13). (Note: A "Field" is the set of primitive entities with which a formal system deals.) This is a most revealing definition. It shows that Whitehead conceives of the world as a logical system and therefore believes that it is possible to view the world in this way. By this one stroke Whitehead has cut through a morass of potential problems. Anyone who quarrels, on whatever grounds, with the possibility of investigating such all-embracing world views is brought up short. By his defining the material world in mathematical terms, his ability to investigate ways of conceiving the material world becomes as certain as his ability to do mathematics. No one challenges the latter. Hence the former must likewise remain unchallenged.

Given this definition of the material world, Whitehead sets out to analyze different ways of conceiving of the material world, i.e., sets out to investigate the different possible primitive entities and relations between these entities. Indeed, it is the choice of one relation, the "essential relation," that defines which concept of the material world we are dealing with. Mays (RW 240) points Out the similarity between every world view’s having an "essential relation" and Whitehead’s opinion expressed in Process and Reality that certain characteristics are shared by every cosmic epoch. Alternatively, it could be argued that there is a connection between which essential relation is chosen and which cosmic epoch is described. The five different essential relations which Whitehead discusses in the 1905 paper all correspond to different ways of picturing the current epoch, it is true, but there is no reason why Whitehead’s method could not be generalized to other cosmic epochs by making several modifications. (See MC 82 for an expression of Whitehead’s belief in the possibility of extending his method, albeit in a different way.)

Whitehead’s method in dealing with each concept of the material world rims as follows: He chooses a set of primitive entities and an essential relation between some of the members of this set. He then asks: What other properties must this essential relation possess in order for the field of the essential relation to be "meaningful," i.e., so that the theorems of Euclidian1 geometry are rendered "true" in the field? These other properties then become the axioms of that concept of the material world.

The above is probably unclear and will likely remain so unless an example is given. Indeed, Whitehead deduces five separate "concepts" of the material world based on the above procedure. The first concept is the "normal" one and will serve as an example. The other four will serve to illustrate the variety of the possible concepts and the power of the procedure.2

Concept I is the "standard" world view of classical physics. The set of primitive entities is defined as the union of the sets of all point of space, all instants of time, and all particles of matter. (These three are, of course, the primitive entities of Newtonian physics.) The fundamental relation is one which defines the nature of "extension." Whitehead shows that from the concept of extension one can derive all of Euclidian geometry (not surprisingly, since Euclid had the concept of extension, e.g., lines, planes, etc., in mind while developing his geometry). Whitehead dryly remarks that this concept I of the material world would be "beautiful . . . if only we limit ourselves to the consideration of an unchanging world of space" (MC 28). The realization that the world is changing, though, upsets the beauty. For to account for change we must introduce an indefinite if not infinite number of "extraneous relations" (i.e., relations other than the fundamental relations) of the form M (p,x,t) which indicate that particle p is at point x at instant t. The laws of mechanics have their origin in (i.e., are axioms of) these extraneous relations. Whitehead criticizes this conceptual scheme by means of Ockham’s razor. He asks if it might not be possible to form a simpler concept of the material world, ideally one that does not involve an indefinite number of extraneous relations.

As a first step toward simplifying the concept of nature, Whitehead introduces concept II. In concept II the fundamental relation is also one of extension. But the extraneous relations involve only relations between points of space and instants of time, whereas concept I involved points, instants, and particles of matter. The particles of matter have been dropped. Whitehead observes that "If we abolish the particles . . . everything will proceed exactly as in the classical concept [i.e., concept I]. The reason for the original introduction of ‘matter’ was, without doubt, to give the senses something to perceive. If a relation can be perceived, this concept II has every advantage over the classical concept" (MC 29). The advantage to which Whitehead refers, of course, is greater simplicity. It is here that we begin to see the power of Whitehead’s method. Because the mathematical formalism does not necessitate the existence of particles, Whitehead dispenses with them. Note that the world we are left with is one wherein there is only time and space (i.e., extension) and relations between these two. This is in some ways a foreshadowing of Whitehead’s later cosmological views, where the relation is taken as primary.

Whitehead derives concept III by "abandoning the prejudice against points moving" (MC 30). Thus moving points, and instants of time, comprise the set of primitive entities. The essential relation is a four-place one, connecting three moving points and an instant of time. But most importantly, in sharp contrast to the indefinite extraneous relations required of concepts I and II, concept III requires only one that defines three mutually perpendicular axes whereby one can define motion, acceleration, etc. This concept "pledges to explain the physical world by the aid of motion only. . . . The ‘corpuscle’ will be a volume in which some peculiarity of the motion of the [primitive entities] exists and persists" (MC 31). Here again we see Whitehead’s tendency to regard things (corpuscles) as derived from something more basic, a tendency that, of course, finally reaches full fruition in his cosmological scheme some twenty-five years later.

Concepts IV and V are very different from concepts I, II, and III. Whereas the first three concepts take points to be primitive, the last two take lines to be primitive. This is a bigger leap than might at first appear, and Whitehead has to go though some lengthy mathematics to show that "points" can be derived from "lines," since the concept of "point" must appear somewhere, either derivatively or primitively, in any scheme purporting to describe a concept of the world.

Concept IV is precisely analogous to concept I, but uses lines instead of points. The essential relation is a five member one, relating the condition for the intersection of four lines at an instant of time. Whitehead is able to show that thirteen axioms can be written in terms of this relation that will define Euclidian geometry and thereby shows that this is a "reasonable" relation on which to build a concept of the material world. However, concept IV suffers from the same flaw as concept I in that an indefinite number of extraneous relations are needed to specify the relationships between the derived points (in contrast to the primitive points of concept I) and instants of time. Hence by the criterion of Ockham’s razor it too is unsatisfactory.

It is the fully developed form of concept V that bears the closest similarity to Whitehead’s later cosmological construction. Concept V, like concept IV, relies on lines as primitive and regards points as derived. Unlike concept IV, concept V enables Whitehead to make do without the indefinite number of extraneous relations, substituting a single extrinsic relation in its place. Furthermore, he can derive the concept of a "corpuscle" as "a volume with some special property in respect to linear objective reals ‘passing through’ it" (MC 43). The development of concept V is the most mathematical of all the concepts considered. Fortunately, we need not consider the details of the derivation, but may instead simply ask: What is the nature of the concept that ultimately results?

First, each "point" is defined in terms of the intersection of the primitive lines. Whitehead observes that these lines could be taken as "the lines of force of the modern physicist" (MC 32), with the one proviso that, unlike line of force, these primitive lines never end. Thus in this concept "action at a distance" is readily explicable, since the point acting and the point "being acted upon" can both share a common component, namely the line they have in common. But most importantly, observe that each point is built up from these lines (i.e., the intersection of several lines can be used to define a point), and, since these lines can be regarded as lines of force or influence, it would be but a small jump to say that each point is composed of the "influences" of other points. Note how very close this is to the scheme outlined in Process and Reality. Whitehead never explicitly says in the 1905 paper that outside influences "create" the point; he regarded the point as a geometrical form. He is, though, so close. Indeed, he gets even closer than this. Who can read his definition of a "corpuscle" in concept V as "a volume with some special property in respect to linear objective reals ‘passing through’ it" without extrapolating the "corpuscle" to "the event," and "linear reals" to "prehensions," and the "special property" to "synthetic ability"? Further, since every point and every other point have a line in common, i.e., every two points have a "part" in common, we can see why Whitehead may later think that every event prehends every other (noncontemporaneous) event. (The matter of contemporaneousness does not arise this early, though. Whitehead has not yet begun to question the notion of time.)3 But perhaps most interesting of all, Whitehead observes that "It is necessary to assume that the points in this concept disintegrate, and do not, in general, persist from instant to instant" (MC 43).4 Since a point is composed of lines and the lines are moving, then, unless all the lines are moving uniformly in the same direction, the lines will be constantly "touching" and "breaking contact with" other lines. Hence "old" points are constantly "dying" and new ones being "born" all the time. Once again we see how very close White-head is to the view that he would ultimately adopt concerning the transience of actual occasions.

In sum, then, in concept V we see that Whitehead has already taken the following crucial steps: (1) He regards points and "corpuscles" as derived, not primitive. (2) He regards them as being derived from lines (which he seems to want to equate to lines of force) passing through other points, lines capable of communicating "influence" from these other points. (3) He regards a point as being a transient entity which exists only for an instant before "dying."

When one realizes that these concepts were brought forward by Whitehead at the peak of his ‘mathematical’ period, one sees how long these ideas were gestating before reaching their mature form in Process and Reality some twenty-five years later. One can also realize how inaccurate the suggestion is that Whitehead "turned into" a philosopher, after having been a mathematician. He was always a philosopher, even in his early writing. Even though he may not have thought of himself in these terms, Whitehead clearly does show an interest in, and a questioning of fundamentals, which is the true mark of the metaphysician (cf. UW 169-71). On the other hand, he was also always a mathematician since, as he clearly indicates in the 1905 article, he seems to conceive of the world as a formal logical system.5 No wonder he regards metaphysics as a possible occupation. Given his outlook he could no more think metaphysics impossible than he could think formal mathematical systems derived from axioms as impossible. Whether he consciously realized what he was doing I do not know; but I am sure that he implicitly thought of metaphysical principles as the axioms of the "logical system" of the world.

To maintain some perspective, it perhaps ought to be remarked that no claim is being made that Whitehead’s work in mathematical logic impelled him to his later cosmological doctrines. On the contrary it is quite certain that his cosmology, though strongly influenced by "On Mathematical Concepts of the Material World," was still a "free creation of the human mind." Indeed to see the divergent effects that a common mathematical background could have, it is instructive to contrast White-head and Russell, since during their extensive collaboration on the Principia they were both exposed to similar mathematical influences.

Russell writes (LA 362f) of a procedural technique which both he and Whitehead adopted:

One very important heuristic maxim which Dr. Whitehead and I found, by experience, to be applicable in mathematical logic, and have since applied in various other fields, is a form of Ockham’s razor. When some set of supposed entities has neat logical properties, it turns out, in a great many instances, that the supposed entities can be replaced by purely logical structures composed of entities which have not such neat properties. In that case, in interpreting a body of propositions hitherto believed to be about the supposed entities, we can substitute the logical structure without altering any of the details of the body of propositions in question.

The techniques which Russell outlines is, of course, precisely the one employed by Whitehead in the 1905 paper. The "supposed entities" are the primitive particles of matter, points of space, and instants of time of the normal world view. The "purely logical structures" which can replace these are the five possible concepts of the material world developed in the paper. And the unchanging "details of the body of propositions in question" are the details of geometry, and, ultimately, Whitehead hoped, physics as well. In essence, Whitehead employed the maxim to forge the somewhat random "normal" world view with its separate concepts of matter and extension into a system, a synthesis within whose framework, he hoped, a place could be found for each of the manifold phenomena of the material world (MC 81f).

In sharp contrast, Russell employed the maxim for the purpose of analysis. He cites it as a guiding force for the dissection of language which he performed (LA 364). Specifically, he uses it as a model for his analysis of definite descriptions into all the smaller units contained in such descriptions, as in his essay "On Denoting" (LK 39).

This basic difference in temperament -- Whitehead’s leaning toward synthesis of systems, Russell’s bending toward detailed analysis -- seems to be a recurrent theme in both of these men’s work. As a final ironic note it is interesting to observe a few connections that exist between the two works cited above: Both "On Denoting" and "On Mathematical Concepts of the Material World" were written in 1905, a time when both men were collaborating on the Principia. Both Whitehead and Russell, looking back later in life, regarded their respective essays as among the finest pieces of work they had produced (UW 466; LK 39). Both employed their common work in mathematical logic, specifically Russell’s maxim quoted above, as a guide in the formulation of their respective papers. Yet because of the difference in outlook between the two men, one essay becomes a penetrating analysis of common language, while the other becomes a synthesis of possible world views and, indeed, a stepping stone to a cosmology.

 

References

LA -- B. Russell. "Logical Atomism" in Contemporary British Philosophy. Ed. J. H. Muirhead. London and New York, 1924.

LK -- B. Russell. Logic and Knowledge, ed. B. C. Marsh. London: Allen and Unwin, 1956.

MG -- A. N. Whitehead. "On Mathematical Concepts of the Material World" in Philosophical Transactions of the Royal Society of London, Series A, 205 (1906). Pp. 465-525. Reprinted in F. S. C. Northrup and M. W. Gross, Alfred North Whitehead: An Anthology. New York, 1953. All page references refer to the reprinted version.

RW -- W. Mays. "The Relevance of ‘On Mathematical Concepts of the Material World’ to Whitehead’s Philosophy" in The Relevance of Whitehead. Ed. Ivor Leclerc. London: Allen and Unwin, 1961. Pp. 235-60.

UW -- V. Lowe. Understanding Whitehead. Johns Hopkins, 1962.

 

Notes

1 Whitehead specifies Euclidian geometry, but observes the other geometries might also be employed, provided some minor changes were made in the system. He employs Euclidian geometry because he in fact believes it to be true of the "real world." (This is hardly surprising when one realizes that Whitehead’s essay was written in the same year that Einstein struck the first major blow against the classical Newtonian-Euclidian world view with his paper on special relativity.) It is significant, though, that even later, in his own Principle of Relativity, Whitehead still retains a "uniform" geometry (of which Euclidian geometry is a subspecies) in sharp distinction to the nonuniform, "warped" geometry of Einstein’s general relativity.

If the method Whitehead employed in the 1905 paper were to be generalized so as to be applicable to other cosmic epochs (as was suggested above might be possible by choosing other essential relations to represent different cosmic epochs), one would desire a criterion of "meaningfulness" for the field of the essential relation other than the satisfaction of Euclidian geometry (or any other geometry, for that matter). Whitehead suggests (PR 103) that even the bare fact of dimensionality, apart from the number of dimensions, may be characteristic of only this epoch. Hence geometry, which presupposes dimensionality, might not be able to be used as a criterion of "trueness" in other epochs.

2 W. Mays (RW 235-60) gives n more detailed account of each of the five potential world views. The more abbreviated accounts here are presented solely to enable parallels to be drawn between aspects of these world views and Whitehead’s later cosmology.

3 See UW 161 for a description of Whitehead’s view of time at this stage.

4 Whitehead is referring to concept IV, but the same wonld hold true for concept V, as Whitehead indicates MC 81.

5 Cf. W, Mays, The Philosophy of Whitehead (London and New York, 1959), chapter 7. Mays seems to be one of the few commentators on Whitehead who adequately appreciates the importance of the formal logical method on the development of Whitehead’s thought. Lowe (UW. ch. 7) is also very useful in this regard.


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