Logic and the Metaphysics of Hegel and Whitehead
by Charles Nussbaum
Charles Nussbaum is a graduate student in philosophy at Emory University and a bassoonist with the Atlanta Symphony Orchestra. The following article appeared in Process Studies, pp. 32-52, Vol. 15, Number 1, Spring, 1986. 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 aim of this essay is to raise some questions concerning the relation between logic and metaphysics in the philosophies of Hegel and Whitehead. Of paramount importance to the Hegelian perspective on this relation is the well-known distinction between understanding and reason as two levels of thinking, for involved in this distinction is the view that logic, as it has been traditionally conceived, is merely a logic of the understanding, and that reason, or speculative thinking, employs a higher, more inclusive logic, one that is "dialectical" in nature. For obvious reasons "logic of the understanding" means for Hegel the logic of the Aristotelian and Medieval traditions, and does not include logic in its modern mathematical development. But we may, I think, conclude with Errol Harris (AT 74) that from the Hegelian perspective, the philosophical shortcomings of classical logic extend to mathematical logic as well, and that as logic of the understanding, both deal with the "abstract concept of class or aggregate," and are both inextricably connected with a metaphysics of externally related particulars that lose themselves in a "spurious infinite," and with a concomitant mechanical cosmology. Mathematical logic is, consequently, according to this view, an inadequate instrument for the expression of a metaphysics of organically related particulars.
I will argue in what follows that these metaphysical characters do not necessarily follow upon the adoption of a logic that is not dialectical, i.e., modern mathematical logic. I will also try to establish, or at least render plausible, the view that while the distinction between a logic of reason and a logic of the understanding may have been one that was necessary and legitimate for Hegel to maintain, it has, given developments in modern logic, as well as changes in the modern view of the nature of metaphysical thinking, become obsolescent. I will further try to show that Whitehead the metaphysician displays interests in important respects not unlike those of Hegel, but that in his solutions to certain metaphysical problems, Whitehead not only evidences a conception of metaphysics quite different, and considerably more modest than that of Hegel, but also (and this will be my main point) develops his position in terms of a logic that is entirely non-dialectical. This is, however, not at all to say that "dialectic" finds no place whatever in Whitehead’s system. As we shall see, this is not the case. It is only to say that such dialectic as there is no dialectical logic.
To assure ourselves of this, if only in a preliminary way, we need look no further than the first page of Chapter One in Process and Reality, where Whitehead states quite clearly that in the metaphysical system to be presented "the term ‘logical’ has its ordinary meaning, including ‘logical’ consistency, or lack of contradiction, the definition of constructs in logical terms, the exemplification of general logical notions in specific instances, and the principles of inference" (emphasis added). In addition, it may be useful to point out that in his valuable book Understanding Whitehead, Victor Lowe states (UW50) that Whitehead’s "conception of growth has points of similarity with Hegel’s, but differs in having no use for ‘contradiction,’ and in presenting a hierarchy of categories of feeling rather than a hierarchy of categories of thought." We should emphasize that Lowe is here referring to Whitehead’s categoreal scheme, not to the system of eternal objects which, as I shall argue, is a hierarchy of thought, or, more accurately, a logical hierarchy.
Let us begin by looking at certain aspects of Hegel’s dialectical logic and attempting to make some determinations as to its nature and genesis. Specifically let us ask with regard to this logic two questions, viz.: (I) What is its purpose, and what problems did Hegel hope to solve by means of it? (II) What is the nature of the solution arrived at?
The answer to our first question can be seen to be twofold. That is, Hegel’s purposes (among others) may well have been: (A) to provide an alternative scheme to Aristotelian logical abstraction; and (B) to provide an alternative to the concomitant mechanism and the psychological atomism of his own day, and to the concomitant logical scheme and the Newtonian mathematical model of externally related particulars, as well as to the dogmatic insistence upon the subject-predicate form of this logic and to the substance-attribute ontology that was its metaphysical correlate. Let us say something about each of these in turn.
(A) While it is evident that Hegel was greatly inspired by Aristotle, perhaps more so than by any other philosopher, it is equally evident that he believed Aristotle’s philosophy to be in need of reform in the light of intervening developments. One major reform that Hegel seems to have taken upon himself to effect is the production of a logical hierarchy of being that in a sense reverses the direction of abstraction of the Aristotelian logical hierarchy, i.e., that becomes more differentiated and "concrete" as it rises in generality and inclusiveness, rather than more empty and abstract. That is, Hegel seems to have wished to achieve an articulation of the levels, or "moments," of that hierarchy such that the less general is explicitly shown by means of logical structure to be contained in the more general, rather than being enclosed there merely implicitly, as in the Aristotelian method of definition by genus and differentia. "Each new stage of forthgoing, that is, of further determination," Hegel says (SL 840f.), "is also a withdrawal inwards, and the greater extension is equally a higher intensity. The richest is therefore the most concrete and most subjective, and that which withdraws itself into the simplest depth is the mightiest and most all embracing,"
Aristotelian abstraction is, on the other hand, "where the false path branches off and abstraction strays from the highway of the Notion [Begriff] and forsakes the truth. Its higher and highest universal to which it raises itself is only the surface, which becomes ever more destitute of content (SL 619). While the less inclusive is logically a part of, and contained in, the more inclusive, the definition, or formula, presents the more inclusive as part of and dependent upon the less inclusive. According to this logical structure, the less general or inclusive an element, the more fully articulated it is. This is very much in accord with Aristotle’s view that species is to genus as form is to matter.
But why did Hegel think such a reversal of abstraction so important? Sympathetic as he obviously was to Aristotle’s conception of system, there are within this system certain troublesome anomalies and inconsistencies. Among these would, I think, be found the following. (1) There are certain difficulties with Aristotle’s primary ousia. If it is the sensed individual, then, since only form is knowable, that which is ontologically most basic is not knowable. If, on the other hand, primary ousia is, as Father Owens suggests, neither the sensed individual nor the universal but the individual form, the formal cause or the act within the thing which is prior to both the sensed individual and the universal, we must ask, as Hegel may have done, is any meaningful distinction between "individual form" and universal possible within Aristotelian logic? To this Hegel may well have answered no.
(2) The formal cause of a thing is for Aristotle its essence as expressed by its formula or definition. But the definition, expressed as genus and last differentia, traces the individual thing to an abstract universal that obviously contains less in it, and is in a sense less "real" or fullblooded. How can such a cause be truly productive?
(3) God, or the prime mover, the crown of the hierarchy of being, moves all else in the way a desired object moves, by inspiring everything to realize its proper end as defined by its formal cause. For Aristotle formal cause is prior to final cause because act precedes potency; potency involves incompleteness. Therefore God as prime mover, object of desire, and ultimate final cause must include within himself the system of formal causes, the logical structure of the cosmos, particularly if he is to be "thought thinking itself."
But again there is no adequate way to express this via Aristotelian logical classifications. If God is to be definite at all, he must be defined as the sole member of an infima species. Yet this hardly captures his essence as the universal source of all movement, the most highly formed being, indeed as that being which is pure form itself. As pure form God is, from the perspective of the metaphysical matter-form hierarchy, cut off from the universe as from any "matter" (NMMS 110f.). This should alert us, as it most likely alerted Hegel, to the possibility that something is seriously amiss, and that there is a radical inconsistency between the logical and metaphysical aspects of the system. Indeed logical universality and metaphysical comprehensiveness appear to be working at cross purposes.
(B) For Aristotle, causal efficacy in the cosmos, and the structure of the syllogism are related. The connection, by means of a middle term, between elements of the major and minor premises, gives a reason why something is what it is; the syllogism expresses a cause or ground. But with the rejection, in the post-Renaissance development, of formal and final causality, and with the increasing importance of efficient causation, moreover an efficient causation that bore little resemblance to Aristotle’s efficient causation, the relationship between logic (which was still largely syllogistic) and the world had to break down. The logical connection between concepts was interpreted as connection between subjective "ideas" in the mind.
John Dewey makes this point succinctly when he suggests (EN 229) that "Locke’s simple idea is the classic Idea, Form, or Species dislodged from nature and compelled to take refuge in mind." Not only are these ideas subjective, but as "simple," they are conceived according to the analytic method of Descartes and Galileo, i.e., as atomic constituents externally related. Mathematics, then exclusively the science of quantity and externally related particulars, became the formal tool for the investigation of nature. Both mind and matter have become mere aggregates, heaps of atomic constituents which can be separated and examined in isolation.
The logical model in terms of which the mind with its ideas, and substance with its attributes were conceived as unified wholes was that of the subject with predicates, which also had become something quite other than what it had been for Aristotle, since universals were no longer thought of as "forms." But Aristotelian logic allowed itself to be detached from Aristotelian metaphysics and reinterpreted in the way already indicated. A suitable reformation of logic involving a rejection of the subject-predicate paradigm could, Hegel seems to have thought, overcome the problem of externally related aggregates, while at the same time retrieving logic from the subjectivity into which it had fallen, and restoring it to its rightful place as the formal science of being.
This brings us to our second main question (II): In what way does Hegel’s dialectical logic provide an alternative scheme, and enable him to carry out the twofold project (A, B) which we have just sketched? With regard to the first project (A) of dealing with the logical anomalies in the Aristotelian metaphysics the following observations are, I think, relevant. If we see Hegel’s philosophy, at least in part, as an attempt to rethink the Aristotelian metaphysics in the light of the Kantian "Copernican Revolution," the way in which the Kantian development enabled Hegel to resolve the Aristotelian difficulties emerges.
A clue to this solution can be found in Aristotle’s contention that the particular is known actually, the universal potentially. Aristotle means that in perception the individual form is actualized in the mind of the perceiver and in the object as perceived, and that the universal is then potentially reachable by application of the actualized form to other similar particulars. Hegel reinterprets the potential universal as implicit in the particular an sich. But rather than reaching the universal by a kind of inductive procedure, the implicit becomes explicit as a mental or spiritual component required for intelligibility of the individual and implicitly included within it. Once this implicit component has become explicit and then reintegrated with the individual previously an sich, the new synthesis becomes a concrete universal or Begriff (Concept), a genuine, though limited existent. Why is this now a universal? If Hegel has reinterpreted Aristotle’s potential universal as the an sich, he has supplied a criterion of "actualized" universality that is derived from Kant: universality is determined by the necessary being-for-mind of the individual, such necessity arising out of the structure of the mind itself. The way a given individual must be for mind is its universality, its public nature.
We see this process very clearly exemplified in the development from "Sense-Certainty" to "Perception, and thence to "Force and the Understanding" in the large section entitled "Consciousness" in the Phenomenology of Spirit. In "Sense-Certainty" the mental component emerges when that which is implicitly "meant" (das Meinen), but not said in such designations as this,’ ‘here,’ ‘now,’ becomes explicitly recognized. In "Perception" mind is called upon to make good by taking upon itself (as deception, Täuschung) the onus of inconsistencies inherent in the thing and its properties. Finally, in the justly famous, but very obscure section of "Force and the Understanding" known as the "Inverted World," the metaphysical distinction inherent in all designations such as inner-outer, intelligible-sensible, noumenal-phenomenal collapses, and with it the attempt of substance or "essence" metaphysics to evade contradiction by locating "contradictories" (or contraries) in ontologically disparate realms. Indeed the metaphysical opposition of mind and inert, inanimate thing-as-object itself thereby collapses.
No longer is it assumed out of hand that inconsistency signals the "deception," the inadequacy of mind to the object, its incapacity to apprehend the object as it "truly" is in itself. Rather an "inversion" has taken place: the contradictory object is that which is inadequate; it is only a partial determination of spirit, which is the "true something," das wahrhafte Etwas (SL 118). As the true something spirit is the whole (the universal) as well as the properly "grasped" individual, the moment or determination of spirit. But the universal, if it is to be more than an empty abstraction, is no more intelligible without the individual than is the individual without the universal. (I pass over here the distinction between individual and particular.)
This universal-individual, or individual-universal is the Hegelian "concrete universal." The Kantian heritage in this conception can be discerned in the following, obviously approving, account Hegel gives of the nature of the Kantian object of experience:
In point of fact, the comprehension of an object consists in nothing else than that the ego makes it its own, pervades it and brings it into its own form, that is, into the universality that is immediately a determinateness, or a determinateness that is immediately universality. As intuited or even in ordinary conception, the object is still something external and alien. (SL 584f.)
In the Science of Logic, the process that gives rise to the concrete universal is somewhat different, and it is here that Hegel’s solution most clearly shows its Aristotelian, as well as its Kantian characters. Rather than explicitly invoking consciousness to effect the reconciliation of opposites as he does in the Phenomenology, Hegel here interprets Aufhebung (sublation) as a purely logical movement, in which the "contradiction," or opposed or "dirempted" elements, are as "matter" to the "form" of the resolution on a higher level. As inconsistency is then uncovered in this "form," it in turn becomes "matter" to a yet higher reconciliation. Hegel explicitly describes the process in these terms in at least one place (SL 838). In this way he achieves a form-matter hierarchy reminiscent of Aristotle. It is, however, one not founded on the Aristotelian doctrine of definition and substantial form, but on the dialectical process of "diremption" and "sublation" which has become the "source of all activity, of all animate and spiritual self-movement" (SL 835).
But despite the fact that in the Science of Logic consciousness is not explicitly invoked until the "Doctrine of the Notion" (Be griff), or the "Subjective Logic," and perhaps not even until the "Idea," the Logic is nonetheless an account of the development of consciousness, and for Hegel, as for Kant, logic means rules governing the activity of thought. For one thing, Hegel says as much: "As science, truth is pure self-consciousness in its self-development and has the shape of the self, so that the absolute truth of being is the known Notion and the Notion as such is the absolute truth of being (das an und für sich seiende). This objective thinking, then, is the content of pure science" (SL 49). For another, he equates the "unity which constitutes the nature of the Notion" with Kant’s "original synthetic unity of apperception" (SL 584). Only the activity of thought involves not, as in Kant, the discursive synthetic connection of particulars (representations), but the gradual evolution of the Idea into its full articulation and determinateness, the stipulation within itself of its levels and divisions through a process that is logically necessary. In a crucial shift, Hegel interprets the "transcendental unity of apperception," the "unity of consciousness as "the unity of the ego with itself" (SL 584). This means that, boldly walking where Kant feared to tread, Hegel has identified the content of consciousness with consciousness itself, with self-consciousness, and is well on his way to giving the Kantian transcendental ego, as spirit, the "true something," an ontological significance that Kant did not intend.
But be this as it may, Hegel believes himself to have improved considerably on Kant. On the one hand, he claims to have deduced all the categories of thought, rather than merely accepting them uncritically as paralleling the "judgments" of the "traditional" logic. On the other hand, he appears to have resolved the enigma of the Kantian given sensuous content of experience with its inexplicable provenance. Spirit is content to itself, from the perspective of a more formed, or more fully determinate level. The intent here is Aristotelian, but the execution thoroughly Kantian.
If one accepts Hegel’s moves one can see that he has provided a solution to the three Aristotelian problems. (1) The concrete universal as the individual seen in conjunction with the thought structure implied in it, that which makes it an intelligible unity, is neither a naively conceived sense-object, nor an abstract logical universal. (2) As the Concept (Begriff) of the "object," as its ground of definiteness and intelligibility, it is its formal cause, but now including more explicitly within itself than did the original thing an sich. Formal cause, following Kant, is reinterpreted as necessary "formal" structure of consciousness. (3) As the same process is repeated over and over again, a more and more inclusive structure is uncovered, a structure that, as it appears, shows itself to have been present in the process from the start, and at the same time to have been complete prior to the process, and to include the whole process within itself. In other words a divine principle appears that is both immanent and transcendent, rather than merely transcendent, as Aristotle’s prime mover had finally shown itself to be.
In order to deal with the second project (B) of overcoming the mathematical and aggregative model of the science and metaphysics of his day, of providing an alternative to the subject-predicate paradigm, and of retrieving logic from mere subjective or mental abstraction from "ideas," Hegel developed what has been termed "determinate negation." This doctrine arose out of Hegel’s rethinking of two axioms of traditional logic, the principle of identity, and the principle of noncontradiction.
In a way consistent with these axioms, Hegel’s predecessors regarded any definite thing as identical with itself, and as determined as what it was by being other than, by not being something else. But for Hegel, such identity, and such otherness are self-defeating, since two disparate things, in order to be successfully compared, must have something in common, must be held together by some third thing. For Hegel, as for Kant, this third thing is conceived as mind, or spirit. It is significant, I think, that Hegel characterizes Kant’s "notion of synthetic a priori judgments" as the "notion of something differentiated which equally is inseparable, of an identity which is in its own self an inseparable difference" (SL 209). He also terms the first negation of the triadic movement characteristic of the Logic "analytic," and the second negation (the "negation of the negation") "synthetic." This second "immediate" (i.e., the reintegration of the element "analyzed" out) he calls the "third term" (SL 836).
The apparent diversity of disparate things is really identity, i.e., mind with itself; but in order for such identity, or any expression of identity to have sense, i.e., not to collapse into undifferentiated oneness, it must be articulated into diversity. Thus we have the celebrated "identity in diversity," as a necessary combination of the two. Now since the mathematical and mechanical cosmological model of Hegel’s day contains parts which are diverse, but is incapable of holding itself in a unity or identity, its very diversity is inconceivable. The conception is self-defeating.
The classical expression, e.g., in Aristotle (PA 77a10) and Kant (CPR A151=B190), of the principle of noncontradiction is that a logical subject cannot contain contradictory predicates together. But for Hegel such a subject can and must contain such contradictory predicates; indeed the subject-predicate paradigm is nothing other than an attempt to ignore such contradiction which affords mind its motion, and to rigidify, calcify, and compartmentalize a reality that is fluid and interconnected. The recurring image of the caput mortuum, the death’s head, that, for example, appears with such sarcasm and irony in the treatment of phrenology in the Phenomenology of Spirit, is Hegel’s metaphor for the lifeless rigidity of subject-predicate thinking.
For Hegel, the appearance of contradiction must not be evaded by means of more and more sophisticated theories of "essence," of inner and outer, substratum and manifestation, but should be recognized for what it is: the means by which the dialectical process carries itself forward, thereby showing the inadequacy in itself of each partial determination of the whole. Along with the logical subject-predicate paradigm, the related metaphysical doctrine of enduring substratum with changing attributes is also to be rejected in favor of a conception of process evolving towards more and more interconnection between apparently separate existents.
Let us pause here for a moment in order to get our bearings. We have succeeded in outlining two possible aims that may have spurred Hegel to initiate his radical rethinking of the traditional logic, and to develop his dialectical scheme. We have also indicated the nature of the solutions arrived at by Hegel in terms of that scheme. Our purpose in what follows will be to ask ourselves three questions with regard to the dialectical scheme, viz.: (1) What are its presuppositions? (2) Can these presuppositions be called into question? (3) To what extent is Hegel’s reform of logic really revolutionary?
Although it is, as we have already observed, Hegel’s aim to question the subject-predicate paradigm of traditional logic, the way in which he questions subject-predicate thinking is conceived within the ambit of subject-predicate thinking itself: his dialectical method consists in discovering contradiction between predicates that break down the apparent self-sufficiency, the stable "whatness" of the element under consideration. But he does not question (as did no one else at his time) the foundations of the logic of predication itself.
Let us look first at the project of reversing the abstraction of the Aristotelian logical hierarchy. If such a reversal is one s aim but if at the same time one accepts the traditional estimation of apophantic statements, or propositions, as essentially of the subject-predicate form, one will be driven to a dialectical logic like Hegel’s. Why? The level of greater generality is to have more explicit in it than does the level of less generality. But as we saw, the Aristotelian categorization by abstract concepts produces a higher level of abstraction with less explicit in it. Now if one is to remain with the notion of division as to "whatness,’ i.e., remain within a strictly intentional standpoint, a standpoint which Hegel, despite his revolutionary tendencies, seems to have accepted from classical logic, then one will be led, if one’s aim is like Hegel’s, to emphasize the negative of the ‘what,’ that against which the ‘what’ defines itself, as internalized within the higher level of generality, thereby including the ‘what’ and its negation.
Similarly, if one accepts the view current at Hegel’s time that mathematical abstraction, and axiomatic method in general, is limited to the quantitative, one will be driven, as was Hegel, and Kant before him, to posit a mental element to hold such disparate units together in a unity. As Hegel says in Science of Logic, "As absolute negativity the negative movement of absolute mediation is the unity which is subjectivity and soul [Seele]" (SL 836). Kant’s transcendental unity of apperception and Hegel’s identity in diversity are, I think, both engendered in this way. But for both Kant and Hegel, as highly evolved as their respective versions of idealism are, mind serves as a substitute for the rejected traditional substratum, linking elements otherwise disparate. The substance-attribute conception, although greatly modified, continues in some sense to hold sway.
In terms of logic, the subject-predicate paradigm is stronger yet, and is not even seriously questioned by Kant. It had not yet been seen that the subject-predicate form can be viewed as a special case of a more general account of logical terms, functions, and relations: a true "logic of manifolds," as Cassirer calls it (SF 72), eschewing any psychological elements, was yet to emerge. That is to say, the subject-predicate form is symbolized in mathematical logic as monadic relation, consisting of a functional term and an "argument," either a name or a variable. The traditional formulation of subject and predicate as connected by means of a copula has been entirely eliminated. (The importance of this will emerge in the sequel when negation is discussed.) A dyadic, or any polyadic function, differs from a monadic (predicational) function only in the number of arguments terms present.
This rethinking of logical forms in terms of mathematical functionality involves an important emphasis of extensionality over the traditional intentional view. In the place of the concept abstracted from a group of particulars, the function defines a group or class of individuals, each of which, when substituted as a value of the variable in a function satisfies, or renders the propositional function true, and so generates a proposition. The class so defined extensionally is more basic than is a concept subsequently applied to it, which defines it in an intentional sense (Russell, PM 80f.). Functionality, and the extensional view of classes, afford modern logic a greatly increased flexibility not available to traditional logic, and throw a whole new light on both the problems we have isolated, and against which Hegel directed his dialectical logic.
Let us, again, look first at the problem of Aristotelian abstraction. At this point it is my intention to bring Whitehead into the discussion, for his system of eternal objects involves a reversal of abstraction of the requisite kind, but is expressed in the language of a logic which is not dialectical. At the same time, the system of eternal objects is a conception of a logical hierarchy which could never have arisen before the revolution in logic initiated by Boole, Peano, Frege, Russell, and Whitehead himself. As Lowe puts it, "It is in the highest degree doubtful if the Whiteheadian type of nationalistic method in the field of metaphysics would have developed at all, had not the traditional conception of the scope of mathematics first been transcended" (UW 130f.).
Eternal objects are, in Whitehead’s terminology, what had been called universals, but as he himself is quick to point out (SMW 169), the conception is quite different. He means that these objects are not intentional types or essences, but are class concepts expressed by means of mathematical functions. According to Whitehead’s obviously simplified, but still recondite description of the system of eternal objects in the chapter "Abstraction" in Science and the Modern World, one eternal object contains a lower grade object extensionally as a member of a class or series rather than being contained intentionally in a subject-concept from which it can be abstracted in the traditional way. This means that any object can be treated both as a logical individual and as a function delimiting a class of other objects. The "relational essence of an eternal object, Whitehead tells us,
is determinable by reference to that object alone, and does not require reference to any other objects, except those which are specifically involved in its individual essence when that essence is complex. . . . The meaning of the words ‘any’ and ‘some’ springs from this principle -- that is to say, the meaning of the variable’ in logic. (SMW 164)
Now how does all this bear on reversal of abstraction and Hegel’s concrete universal? The fact that an eternal object can be treated as a logical individual and included as a term in another higher grade object means that the latter includes it in its "essence." But the lower grade object also, unless it is simple, and so at the bottom of the logical hierarchy, includes yet lower grade objects in its essence. Hence we have a hierarchy of increasing complexity, higher levels of which include, in an explicit and articulated way, lower levels in their "essences." The complete system of eternal objects is included within what Whitehead calls the "primordial nature" of God.
We have here a reversal of Aristotelian abstraction not unlike that sought by Hegel, but one accomplished by means of a different, nondialectical reformulation of logic. Increasing abstraction produces increasing complexity and "richness" or articulation. Such abstraction Whitehead terms "abstraction from possibility," as opposed to "abstraction from actuality," with which he says it should not be confused (SMW 170). In Atheism and Theism Errol Harris points to the expression of curves in terms of algebraic formulas in analytic geometry as an example of the Hegelian concrete universal, or at least as an illustration of this conception. But this kind of mathematical abstraction, in Whitehead’s terms an abstraction from possibility, is an equally suitable example of the Whiteheadian eternal object. "The algebraic function," as Harris quite rightly says, "is expressed by and immanent in a spatial figure" and is "universal to its particular manifestations" (AT 74). The algebraic function defines a class of "points" that are mutually disposed in a certain way, and classes of these classes. What is most significant is that greater generality and abstraction does not entail greater vacuity and loss of articulation.
Cassirer makes this point very nicely as follows:
Here the more universal concept shows itself also the more rich in content; whoever has it can deduce from it all the mathematical relations which concern the special problems, while on the other hand, he takes these problems not as isolated but as in continuous connection with each other, thus in their deeper systematic connections. The individual case is not excluded from consideration, but is fixed and retained as a perfectly determinate step in a general process of change. . . . Modern expositions of logic have attempted to take account of this circumstance by opposing, -- in accordance with a well-known distinction of Hegel’s -- the abstract universality of the concept to the concrete universality of the mathematical formula. (SF 20, emphasis added)
Furthermore, algebraic form itself can be made an object of study on an even higher level of abstraction, and given various nonquantitative interpretations. That algebra can be given a logical interpretation was Boole’s insight, and Boole’s Laws of Thought in turn inspired Whitehead’s first major work, the Universal Algebra, which was an attempt to treat algebraic patterns at the highest possible level of generality, and "deals with abstract ideas in hierarchical patterns" (UW 139). The connection between the conception of hierarchy in the Universal Algebra and the metaphysical doctrine of eternal objects developed thirty years later is unmistakable, and is remarked upon by Lowe (UW 139).
Let us now pass over to the second Hegelian project of dealing with the problems of externally related particulars in the mathematical cosmology of his day, and of the inadequacy of the subject-predicate paradigm. An extremely important aspect of the system of eternal objects is that they are internally related. As Whitehead says in Science and the Modern World (p. 160), "Since the relationships of A [an eternal object] to other eternal objects stand determinately in the essence of A, it follows that they are internal relations."
But what does this assertion of internal relatedness mean, particularly in light of the fact that Whitehead tells us in Adventures of Ideas (p. 157) that the traditional doctrine of "internal relations is distorted by reason of its description in terms of language adapted to presuppositions of the Newtonian type"? Certainly the "particulars" in a class are not internally related. But these are particulars of the same grade of complexity. However, the class, and the eternal object defining it, are, as we saw, included within a higher grade object as a term, and as part of its "essence. For example, a simple eternal object, say a "particular shade of green," can be incorporated in "another eternal object of the lowest complex grade," say "three definite colors with the spatio-temporal relatedness to each other of three faces of a regular tetrahedron, anywhere at any time" (SMW 166). Although the simple eternal object defines a class of particulars that as values of the variable are that definite shade of green, and although these individuals are externally related, the simple eternal object itself is internally related to the higher grade object, in that it partially constitutes its essence.
But it is far from evident that this solves the problem of external and internal relatedness and that it really produces a viable model of internal relatedness via mathematical logic. Yes, eternal objects as definitive of classes, and classes of classes, can be interpreted as interconnected and internally related in this way, but what about the simple objects, the final result of analysis? They must be externally related; and if they are, the whole system is ultimately a mere aggregate. And then there is the problem of individual existent "things," actualities as opposed to the formal abstract possibilities which are the eternal objects. How are they individual, yet internally related? The answers to these questions will be seen to be connected.
The Whiteheadian actuality, the occasion, represents a complete and thoroughgoing rejection of the substance-attribute conception, and is entirely relational in its essence. But what does this mean? If we think for a moment of the Leibnizian monad, with which the occasion has much in common, we can recall that each monad represents the entire universe from a particular perspective. But as conceived by Leibniz as a qualified substance, such perspectives are attributed to each monad internally, so that the interconnection between monads is only apparent; they are in actuality "windowless" and externally related, and in concord only by virtue of pre-established harmony. Although Leibniz did, in a most modern way, conceive of space as inherently relational, as an "order of co-existence," rather than as a self-subsistent "container," his prejudice in favor of the subject-predicate form for propositions led him to interpret relational forms as ultimately reducible to predicational forms, and to assume that the latter had primary metaphysical significance. According to Leibniz’s conception, it is by means of the particular perspective "built in" to the monad that its individuality is established according to the identity of the indiscernible. The monads are not individuated in space, but space is constructed out of their implied logical relations: they are not in space and time, but space and time are in them.
The Whiteheadian actual occasion is conceived along similar lines, but instead of being a substance persisting through time in apparent relation with other substances, the occasion is a process of actualization of a set of real relations. It is constituted by a selection from the scheme of general relatedness, i.e., from the system of eternal objects. Like the Leibnizian monad, the occasion is individuated by its individual essence, its particular perspective; but unlike the Leibnizian monad this essence is not predicated of the occasion as a substantial substratum, but enters into the inner constitution of the occasion as "a vector transmission of emotional feeling" or, in the language of physics, "the transmission of a form of energy" from past occasions via the eternal objects that communicate the emotional form and make possible the subsequent reenactment by the prehending occasion (PR 315/ 479f.). "The actual entity, in virtue of being what it is, is also where it is" (PR 59/ 93).
We have here an updated version of Kant’s criticism of the Leibnizian monad in the "Amphiboly" in the Critique of Pure Reason: the spatially situated existent is indeed made up of relations rather than being a substance containing its "inhering" attributes internally as predicates which are part of its concept; only the relations are no longer those of the synthetically connected manifold, but the relational connections between the particulars of modern functional or mathematical logic, expanded to include within itself the spatial and temporal relations which Kant could only account for by means of the synthetic a priori. In Whitehead’s view "the extensiveness of space is really the spatialization of extension; and the extensiveness of time is really the temporalization of extension (PR 289/ 442). And as the Kneales have it, "when ‘space’ and words of similar origin occur in pure mathematics, they refer to abstract patterns of ordering which may conceivably be exemplified by widely differing systems of objects" (DL 386). In this way logic resumes its place as the formal science of being, but not as a theory of mental activity, or as a morphology of mind.
We are now in a position to begin to solve the problem of the nature of internal relations between actual occasions, and also to make manifest the affinities between the actual occasion and the Hegelian concrete universal, a goal towards which we have been moving all along. We have already recognized the sense in which eternal objects are internally related: the more general or abstract function includes the less general as a constituent or term. This sense of internality in mathematical functions is not particularly strange or unusual, and something like it is recognized by so different a mathematical philosopher as Wittgenstein: "The internal relation by which a series is ordered is equivalent to the operation that produces one term from another" (TLP 5.232). Now this conception of internal relatedness of forms is on the one hand the means by which Whitehead makes intelligible causal relatedness between prehending and prehended occasions via his theory of propositions. But on the other hand, the concatenation of functional forms, or eternal objects, affords him the means to establish relations, indirect, but nevertheless internal, between contemporary occasions, where causal relations proper are not in question at all. Let us now see how this is achieved.
In the language of mathematical logic, at least as it was current in Whitehead’s day, a proposition is produced from a propositional function by substituting a name as value for the variable in the argument position of the function, or by quantifying over a range of such values. Now for Whitehead, a proposition is "a hybrid between pure possibilities and actualities" (PR 185f./ 282). In a proposition, "The definite set of actual entities involved are called the ‘logical subjects of the proposition’; and the definite set of eternal objects involved are called the ‘predicates of the proposition.’ The predicates define a potentiality of relatedness for the subjects" (PR 186/ 283).
This all seems rather straightforward. The concrescing occasion entertains a "proposal": can the prehended occasions as logical subjects be integrated with certain universals or eternal objects thereby producing that hybrid entity termed by Whitehead a ‘proposition’? In this way "a proposition is an element in the objective lure proposed for feeling" (PR 187/ 284). One can see here the connection between the internal relation between eternal objects, and the fact that from the standpoint of the prehending occasion, its datum, as a logical subject placed within a functional context, is internally related to it. As it is a term in a proposition, so is it an element in the constitution of the prehending occasion entertaining the proposal. This is the relation that Whitehead terms ‘efficient causation,’ and there is an asymmetry about it: from the standpoint of the prehended occasion, or the datum, the relation is an external one. In this strictly causal sense, contemporary occasions are externally related as well.
But perhaps more interesting, and certainly more relevant to our Hegelian themes, is the question of internal relations between contemporary occasions that are not directly related via efficient causation. The possibility of such indirect relatedness appears to be indicated by Whitehead in the following passage:
. . . the objectifications of the presented duration represent a recovery by its contemporaries of a very real efficacy in the determination of M [an actual occasion]. It is true that the eternal objects which effect this objectification belong to the feeling-tones which M derives from the past. But it is a past which is largely common to M and to the presented duration. Thus by the intermediacy of the past, the presented duration has its efficacy in the production of M. (PR 3211 489)
This sharing of a common past is, I suggest, very significant. For it means that all neighboring contemporary actual occasions share to some extent the same content, though with different degrees of comprehensiveness, different valuative emphases, different "perspectives. But especially significant is the fact that this sharing of the same content cannot mean that they "represent" in a similar way the same past world: this is sheer Leibnizianism. It means that the past is literally present in them. As Whitehead says:
The principle of universal relativity directly traverses Aristotle’s dictum, ‘A substance is not present in a subject.’ On the contrary, according to this principle an actual entity is present in other actual entities. In fact if we allow for degrees of relevance, and for negligible relevance, we must say that every actual entity is present in every other actual entity. (PR 50/ 79)
This is also the burden of Whitehead’s "reformed subjectivist principle": all togetherness is experiential togetherness, or an abstraction from experiential togetherness. There is no "objective" togetherness to which experiential togetherness must "correspond" (PR 189f./ 288). But on the other hand, occasions are not, like monads, self-enclosed substances. Again, their perspectives are not representations: concrescence is "the cumulation of the universe and not a stage-play about it" (PR 237/ 363).
Now to say that the past is present in actual occasions is to say that they prehend the forms bequeathed them by past occasions. It is the form that is transmitted, not the original creativity, for the latter has perished. As Whitehead says, "‘Change’ is the description of the adventures of eternal objects in the evolving universe of actual things" (PR 59/ 92). While the occasions are indeed individual "creatures," their distinct individuality depends upon eternal objects as forms of definiteness. But since these forms are not predicated of the occasion as of a substratum, contemporary occasions are genuinely intersecting perspectives on a concatenated past order, and so indirectly contain something of each other. This seems to me to be the most natural construction to place upon Whitehead’s statement just quoted, that "every actual entity is present in every other actual entity." To be sure, future entities are not present in present entities, nor are present entities present in past entities. But neither future entities nor past entities are actual. With this stipulation, I would urge that the view for which I have argued is one that takes Whitehead at his word.
This is not to deny that the final outcome of the concrescing activity of contemporaries is not available to one another. Yet by means of "presentational immediacy" occasions project the concrescence of contemporaries as possible "atomizations of the potential extensive continuum according to available knowledge of their past. Admittedly presentational immediacy does not constitute symmetrical prehensions between contemporary occasions. But some internal relatedness between contemporary occasions seems indicated by Whitehead when he says:
The Cartesian subjectivism in its application to physical science became Newton’s assumption of individually existent physical bodies, with merely external relationships. We diverge from Descartes by holding that what he has described as primary attributes of physical bodies are really the forms of internal relationship between actual occasions, and within actual occasions. (PR 309/ 471).
The terms in which Whitehead describes his divergence from Descartes are extremely suggestive: for it is his mathematical or functional conception of form that enables him to effect his "reform" of the Cartesian subjectivism. Unlike the logic of subject-predicate, mathematics is entirely unconcerned with what something is "in itself," or in isolation. What something is is determined by its function or role. As Cassirer tells us, "We have in pure mathematics a field of knowledge, in which things or their properties are disregarded in principle" (SF 18). Since what something is is determined by its function, the same datum can be implicated by different prehending occasions with different eternal objects. The same datum can come to serve different functions in different concrescences. Yet the datum, though not the original creativity, is literally "present in" both concrescences, though it is (or may be) functioning in different ways. It is in virtue of sharing this common datum, a state of affairs that is only possible given the tendency of mathematical organization simultaneously to place different interpretations on the "same" element, vis-avis its relations, that the concrescing occasions are internally related. This is the central point to be driven home.
We can now deal with the question, earlier deferred, of how it is possible to insist upon internal relations between occasions while at the same time recognizing that from a logical point of view the members of a class of particulars defined by a function would have to be externally related. Since each actual occasion prehends a multiplicity of data, moreover a multiplicity which persists on various levels of logical abstractness and generality, each occasion is ex hypothesi a complex entity. Because every occasion must include some very general and abstract eternal objects as forms of definiteness (i.e., spatial, temporal, "epochal" ones) as well as less pervasive forms, no occasion is constituted as a heap of unileveled components, one as important or unimportant as another. The forms that provide definiteness can be analyzed to logical simples; but this is a mere abstraction, an "abstraction from actuality," to be exact. To assume, as many seem to have done, that the logically simple is per se the metaphysically primary is to fall prey in a particularly egregious way to the fallacy of misplaced concreteness.
At this point the Hegelian parallel begins to emerge. The definiteness of the actual occasion is due to its form, its perspective of pattern, that aspect of the whole included within it. It is individual; but its distinct individuality arises out of the universal manifested in it. On the other hand, the universal as such is merely pure possibility, completely indefinite and unrealized apart from the concrescence of actual occasions and implications with prehended data. But this is the same polarity that we found operative in the concrete universal, though the conception of logical form, and thus the logical basis upon which the rejection of separately persisting externally related individuals rests, is entirely different. We also have in Whitehead, as we do in Hegel, a graded hierarchy of existents, from God to "the most trivial puff of existence in far off empty space" (PR 18/ 28). As the reservoir of possibility, the source of the subjective aim of every occasion, God in his primordial nature is the ultimate formal and final cause of the universe, analogous to Aristotle’s prime mover. Yet God in his consequent nature is fully actual, and is just one, albeit a supreme existent amongst all others. It is in terms of this dual nature, the mental and physical poles, a dual nature shared by all existents, that Whitehead provides his own solution to the Aristotelian transcendence-immanence problem.
Also important for the character of the occasion are those objects not included in its choice of pattern, and this question of negative "prehensions" bears interestingly upon Hegelian determinate negation. If we look at the table of judgments in the Critique of Pure Reason (A70=B95), we find under the heading of judgments as to quality the threefold division: affirmative, negative, infinite. That is, Kant, following Aristotle (DI 19b19), distinguishes between propositions like ‘A is not just’ and ‘A is not-just,’ these being respectively negative and infinite. Taking account of the fact that according to Kant the third category of each of the four groups arises somehow from a combination of the first two (CPR B110), there can be little doubt that we have here in Kant’s divisions a proximate source of Hegelian determinate negation.
Now it is of great interest that the distinction between negative and infinite propositions has not been preserved in symbolic logic. Both propositions referred to above would be symbolized as J(a). As we observed earlier, the conception of subject and predicate as connected by means of a copula has been eliminated. Because of this there is no way to distinguish symbolically ‘A is not just from ‘A is not-just’ as Aristotle does in De Interpretatione. This is not the result of clumsiness or oversight; there is a reason for it, namely the truth-functional stance of modern logic: 1 (a) simply means that the individual named ‘A’ does not satisfy the propositional function in question, i.e., 1 (x), or a true proposition does not emerge when that name is substituted in the argument position. This is important because until the work of Frege the relation between Aristotelian syllogistic of class inclusion and the Stoic propositional calculus developed by Chrysippus and others remained shrouded in darkness. Before the advent of quantification theory it was not realized that the logic of classes presupposed the propositional calculus, and so the notion of truth functionality. As Russell has said (IMP 166), the propositions thought to be the most basic in traditional logic are not the most basic at all.
But once classes are defined by satisfaction or non-satisfaction of a function, the infinite judgment and determinate negation are no longer tenable constructions in logic. Thus when Whitehead says that every occasion is a synthesis of being and not-being" (SMW 163), this sounds like Hegel, and indeed it is similar in intent to determinate negation. But from a logical point of view it is essentially quite different. For what Whitehead means by this is that an "eternal object in all its determinate relationships is excluded from that occasion." But eternal objects, we have established, are mathematical or functional forms. As one final bit of evidence for this let us note that Whitehead, in Science and the Modern World (p. 172), says that in his "account of an actual occasion in terms of its connection, with the realm of eternal objects, we have gone back to the train of thought . . . where mathematics was discussed," and to "the ideas ascribed to Pythagoras. . . .
The exclusion of an eternal object from an occasion simply means that no aspect of the constitution of that occasion satisfies that function, or that the class defined by that function does not enter into the constitution of that occasion. But the important point is that the negation of a function (predicate) does not produce a "negative function," for there can be none such. It merely indicates that no value, or no value from a restricted range can satisfy that function, or that a particular value does not satisfy it. As we have seen, predication is one kind of functional expression. The functional form has thus undercut the basis for determinate negation, i.e., the presence in one substantial thing of contradictory predicates. As the Kneales have it, "the ambiguous and confusing terminology of ‘subject’ and ‘predicate’ will now give place in logic to a more satisfactory distinction of propositional forms according to the doctrine of functions" (DL 436). A thing either satisfies a function or it does not.
But we should realize that a "thing," the value of a variable, can be anything conceivable, simple or complex, and can simultaneously satisfy many different functions. It can be a member of many different classes, and can then enter into many different relations. Furthermore, anything is the sole member of some class (PM 63). In this way modern logic, if judiciously employed, has the flexibility to express something not unlike the Hegelian fluid and multileveled connection between things, and in its own way is able to "destroy," in the manner of Hegel’s "speculative proposition" (PS 38), the rigid subject-predicate paradigm.
This is, however, not at all to say that there is not a dialectical element to be found in Whitehead’s philosophy. There very definitely is such an element, only it is not logical, but emotional, or having to do with feeling. By identifying the dialectical element that he recognizes in the process of concrescence with feeling rather than with logic, Whitehead is able to preserve the integrity of logic, and to avoid what I take to be the Hegelian dual fallacy of both overestimating the power of so-called "reason," and at the same time encumbering logic with psychological, epistemological, and metaphysical elements that have no proper place in it. Whitehead’s dialectic emerges in his striking separation of propositions and judgments, for the most part identified by both Kant and Hegel, and his rather daring recognition of the applicability of both correspondence and coherence criteria in his epistemology. Propositions are for Whitehead either true or false; judgments can be correct, incorrect, or suspended. To the former, he says, applies the correspondence criterion; to the latter, the coherence criterion (PR 1911 291).
This means that whereas a proposition must properly conform to the nexus to which it refers independently of the experience of any particular occasion, and is a logical type, a judgment is concerned with the conformity of components within one particular occasion, and is an emotional type. The judging subject feels the appropriateness, or inappropriateness, vis-a-vis its subjective aim, of a proposed qualification of its datum. For this reason the negative judgment, not the false proposition, is the "peak of mentality" (PR 5/7), the "triumph of consciousness" (PR 273/ 417): the ability to judge negatively, to withhold assent, to develop this way or that, is greater in the higher grade entity. It is the source of error and evil; but as the source of freedom it makes novelty possible. It is in this sense that the occasion is a synthesis of being and not-being. There is a dialectical opposition, but one that is not at all logical, but emotional, having to do with feeling.
Let us now try to recapitulate. We began by isolating two problems in the logic and mathematics of Hegel’s day which he attempted to deal with by means of his dialectical logic. These two problems were: (1) the empty abstraction of the Aristotelian logical hierarchy; (2) the interconnected problems of a mathematical cosmology of externally related particulars, the subjective significance accorded the subject matter of logic, and the inadequacy of the subject-predicate paradigm along with the concomitant substance-attribute dogma of metaphysics. We found in Hegel’s dialectical logic an imaginative attempt at a solution, but one still very much mired in the presuppositions of the logic he wished to replace. We then tried to suggest that mathematical logic involved a more thoroughgoing critique of these presuppositions, and adduced the metaphysics of Whitehead as an example of a proposed solution conceived in terms of mathematical, rather than dialectical logic.
But in addition to the thoroughgoing logical differences between Hegel and Whitehead to which we have largely directed our attention, there is an important difference in their respective conceptions of the nature of the metaphysical argument itself that should at least be touched upon here. Despite Hegel’s rejection of pre-Kantian dogmatic metaphysics, there is reason to see in the Hegelian method a new dogmatism, to see it as perhaps the final attempt to produce a metaphysics in the traditional style, namely the last attempt to directly establish a metaphysical position. Since then, this conception of metaphysics has given way to one of metaphysics as the study of most basic or general presuppositions, and of the metaphysical argument as hypothetical in the manner of a scientific theory, but on a level of higher generality. This certainly seems to be the sense of Whitehead s conception of "imaginative rationalization" or "generalization" (PR 5/ 7).
This point can perhaps be made clearer by noting that Hegel’s metaphysical enterprise can be seen as a highly sophisticated reworking of the cosmological-ontological argument for the existence of God. (I follow Kant here in regarding the former as presupposing the latter.) The Absolute as fully determinate both contains within itself all possible determinations, and is the original ground of these determinations. While recognizing the manifest unsatisfactoriness of the traditional versions of these proofs, Hegel attributes their lack of success to the fact that they move on the level of understanding rather than on that of reason. "A reason-derived knowledge of God," the "highest problem of philosophy" (EL Sec. 36, Zusatz) does indeed set the understanding an impossible task; but for reason such knowledge is not only possible, it is necessary. In order to achieve self-consistency, reason must uncover the divine principle within itself and in the world. It can rest satisfied only when it has, as spirit, appropriated as its own, indeed as itself, the whole which alone is truly infinite and so self-sufficient, which is the true one substance of Spinoza, which is God. It is towards this goal that the dialectical process ineluctably drives thought; immediately it begins truly to think, rather than merely to calculate.
For Whitehead, given his implicit rejection of the Hegelian distinction between a logic of the understanding and a logic of reason, and given his conception of the nature of the metaphysical argument, God is not, and cannot be the inevitable culmination of such a logical progression. In Whitehead’s system God is a necessary metaphysical presupposition. By the "ontological principle," the system of eternal objects, or possibles, must be grounded in some actual existent, in this case God’s mental pole, or God’s primordial nature.
But beyond the requirements of Whitehead’s ontological principle, we can, even if we doubt the legitimacy of dialectical logic, be sympathetic to the Hegelian view that God is in some sense a necessity for rational thought. We may hold that the existence of God cannot be directly established by any logical argument, dialectical or otherwise; but we can insist that some objective principle of order and value is immanent in rational thought in particular, and in the cosmos as a whole. The fact that such is not easily proved means little. One judges a metaphysical scheme by coherence and illuminative power, and such a scheme does not gain adherents in a way a victor carries off the trophies in a debating contest. For as Whitehead has said, "proof," in the strict sense of the term, is "a feeble second-rate procedure" (quoted in UW 367).
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