The Untenability of Whitehead’s Theory of Extensive Connection by Lee F. Werth Lee F. Werth teaches philosophy at Cleveland State University, Cleveland, Ohio. The following article appeared in Whitehead’s account of the perceptive mode of presentational immediacy, as he explicitly states, is dependent upon a definition of straight lines in terms which make no appeal to measurement (unlike the definition, ‘the shortest distance between two points’): "It is to be noted that this doctrine of presentational immediacy and of the strain-locus entirely depends upon a definition of straight lines in terms of mere extensiveness" (PR 493; cf. also 495-96). Accordingly, straight lines are defined in terms of a scheme of extensive connection. The definition of a straight line requires that some geometrical elements be In my demonstration of these claims, the relevant theorems will be derived from Whitehead’s definitions and assumptions in the chapter on "Extensive Connection," particularly definitions 2, 7, 9-13, 15-17, and assumptions 6-9, 23-26. The reader will want to have Whitehead’s text in front of him in examining this argument. 1. Every abstractive set must be composed of an infinite number of members. This follows from Assump. 9, Assump. 6, Def. 10, Assump. 7, and Assump. 8. An included region I shall not argue for the importance of this inconsistency since it is easily resolved by a terminological revision, i.e., by refusing to call included regions ‘parts’. 2. Any two members of the infinite number of members of an abstractive set are such that one of them includes the other nontangentially. This follows from criterion (i) 3. Of any two member regions 4. Any two members of an abstractive set themselves determine an abstractive set. This follows from the satisfaction of the two criteria determining an abstractive set, i.e., nontangential inclusion and there being no region included in every member of the set (Def. 10), and from 2 and 3. 5. Member regions 6. Regions 7. 8. Every region nontangentially included by 9. Every member region of an abstraetive set a included by region 10. Every member of an abstractive set /3 includes some members of . This follows: from the transitivity of inclusion (Assump. 6) such that if 11. Every member of an abstractive set a includes some members of the set . This follows from 8, 9, and 10. 12. Abstractive set a covers abstractive set . This follows from Def. 11 and from 11. 13. Every member of an abstractive set 14. Abstractive set 15. Abstractive set a and abstractive set 16. Abstractive set a and abstractive set 17. Abstractive sets a and 18. Abstractive sets 19. Abstractive sets It is worth noting that were the paragraph following Definition 11 in If all the member regions of equivalent abstractive sets are required to include I have not chosen to hold Whitehead to the full ramifications of the paragraph following Definition 11, but rather to construct my argument within the scope of the more liberal Definition 11 without the unfortunate paragraph immediately following it. 20. If every member of an abstractive set includes some members of another abstractive set, it will always be the case that every member of that latter abstractive set includes some members of the former abstractive set. This follows from 11 and 13. We will not be troubled by the assumption of asymmetry of inclusion (Assump. 8) since we are not requiring every member of either abstractive set to include all of the members of the convergent tail of the other. All we require is that some members of one set, If we conceive of the Chinese nest-of-boxes toy as an analogue of an abstractive set (PR 454; cf. also CN 79), remembering that there is no last box, it The above is by way of illustration through an analogue and nothing has been asserted that is not a consequence of the proof wherein it is demonstrated that to cover is to be covered. 21. The two abstractive sets of 20 cover each other. This follows from Def. 11 and from 20. 22. The relation of covering, if it is to hold between any two abstractive sets always holds symmetrically. This follows from 20 and 21. 23. The two abstractive sets of 21 are equivalent. This follows from Def. 12 and from 21. 24. The two abstractive sets will be members of the same geometrical element. This follows from Def. 13 and from 23. 25. The two abstractive sets will be equivalent to every member of the geometrical element of which they are members. This follows from an argument analogous to 16 through 19, i.e., Assump. 23, Assump. 24, Def. 12, and from 22. 26. When every member of an arbitrary geometrical element 27. The members of geometrical element 28. The members of geometrical element 29. Geometrical element 30.
31. A further, less formal, and more general proof is included to meet the objection that my argument depends upon deriving the abstractive set 1. If set a covers set , every member of the abstractive set 2. 3. If 4. If 5. If 6. To cover an abstractive set is to be covered by that set, i.e., symmetry of coverage. The rest of the argument that no geometrical element I must confess that I do not like to reason on the basis of the largest region of an abstractive set. Whitehead remarks that "though an abstractive set must start with some region at its big end, these initial large-sized regions never enter into our reasoning" (PB 455). Moreover, although Whitehead does speak of ‘big end’ and ‘far enough down’, I believe such terms to be logically improper in the context of a theory which is logically prior to measurement. These terms may be used metaphorically, but should not themselves play a formal role. Therefore, I have constructed my argument upon the longer and more formal proof. Rejoinder A Reader, justifiably unhappy with the conclusion that all geometrical elements are points, would probably attack the mainstay of the demonstration, that mainstay being the symmetrical character of the relation of covering (22). Looking back at the original definition of demonstration, that mainstay being the symmetrical character of the covering (Def. 11), one reads: "an abstractive set The reader gladly asserts that I have been overly restrictive in my interpretation of Def. 11. Nowhere in that definition of ‘cover’ does it state that ‘includes’ is to be read as 1. Let us begin with an example where an abstractive set 2. The sets will not be equivalent since equivalence requires that each set cover the other. 3. The sets will belong to different geometrical elements since only equivalent sets belong to the same geometrical element. 4. Any member set of the one geometrical element 5. Geometrical element
6. The geometrical element 7. Geometrical element It would seem that I am now forced to renounce the whole of my argument based upon the theorem of the symmetry of coverage which was itself derived from an apparently overly restrictive interpretation of ‘includes’ in Def. 11. Actually ‘includes’ as opposed to ‘nontangentially includes’ is a pseudo-issue. Irrespective of which way ‘includes’ is to be interpreted, Whitehead’s attempt to define straight lines remains unsuccessful. If ‘includes’ is restricted to nontangential inclusion, then coverage is symmetrical and we are able to derive the theorem that all geometrical elements are points. However, if ‘includes’ is broadened to allow tangential inclusion, we are then across the other horn of the dilemma. It can then be shown that Were we to restrict ourselves to the sort of two-dimensional circular regions found diagrammed in If it It remains to specify the nature of the We conclude the formal discussion by noticing that either (My use of the expressions ‘shape,’ circular,’ ‘ovate,’ ‘concentric, ‘big,’ and ‘small’ are intended solely as conceptual aids. They are to help the reader identify the relevant abstractive sets, a sets become possible when the meaning of ‘include’ is liberalized and not because they are somehow logically dependent upon concentricity, measurement, or any logically prior definitions of particular shapes.) If no geometrical element is a point, then Whitehead’s geometrical account of the perceptive mode of presentational immediacy in terms of mere extensiveness is untenable, as it is when all geometrical elements are points. Def. 5 of a straight line requires that there be a "locus of To refute my argument requires the satisfaction of two criteria. Asymmetrical coverage between abstractive sets must be possible, i.e., the refutation of theorem 22. It must also be established that it is not always possible to construct a covered set Viewed 6922 times. |