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 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 Whitehead wrote "On Mathematical Concepts" in 1905, at a time when he was two years into writing the 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 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 Given this definition of the material world, Whitehead sets out to analyze different ways of 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 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. 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 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 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 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 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 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 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 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 To maintain some perspective, it perhaps ought to be remarked that no claim is being made that Whitehead’s work in mathematical logic 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
References LA -- B. Russell. "Logical Atomism" in LK -- B. Russell. MG -- A. N. Whitehead. "On Mathematical Concepts of the Material World" in RW -- W. Mays. "The Relevance of ‘On Mathematical Concepts of the Material World’ to Whitehead’s Philosophy" in UW -- V. Lowe.
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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