John W. Lango is Associate Professor of Philosophy, Department of Philosophy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10021. He is the author of Whitehead’s Ontology and various articles on metaphysics and ethics. E-mail jlango@hejira.hunter.cuny.edu.
The following article appeared in Process Studies, pp.16 -42, Vol. 29, Number 1, Spring-Summer, 2000. 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.
SUMMARY
Dr. Lango draws upon the logic of relations in an attempt to understand what nexus are.
Whitehead’s process metaphysics is also a relation metaphysics. For instance, one of the categories in his categoreal scheme in Process and Reality is the "principle of process," and another is the "principle of relativity." Most significantly, prehensions are "Concrete Facts of Relatedness" (22-23). Although it is customary to classify him as a process philosopher, it is reasonable to classify him also as a relation philosopher.1
Because he also is a relation philosopher, my view is that it is sometimes fruitful to interpret aspects of his metaphysics in terms of the logic of relations. One of his categories of existence is that of "Nexus" (Process 22). A nexus consists (roughly speaking, in actual entities interrelated through their prehensions of one another. In this paper, I shall draw upon the logic of relations in an attempt to understand what nexus are. The importance of the category of nexus lies partly in its great generality, a generality that can be fully represented in terms of the logic of relations.
Studies of Whitehead’s metaphysics usually concentrate on the category of actual entities, and consider the category of nexus incidenrally.2 My aim in this paper is to supplement these studies by focusing on the category of nexus.3 Let me indicate why it is of value to investigate his metaphysics in this way. Central to Process and Reality is a discussion of the order of nature that stems from a variety of societies of actual entities (e.g., material bodies and living organisms). This notion of "society" is essential to his metaphysics because it helps to link his speculative conception of actual entities with entities of ordinary experience. Societies are a particular type of nexus. Therefore, to understand how societies bring order to nature, we need to understand the unity of interrelatedness provided by nexus.
Let me mention another reason for focusing on his category of nexus. His metaphysics can be understood as a sort of metaphysics of events. "An actual occasion is," he remarked, "the limiting type of event with only one member." More generally, an event is "a nexus of actual occasions" (Process 73).4 Just as he used his technical term "eternal object" rather than the standard term "universal" (Process 48), so he used his technical term "nexus" rather than the standard term "event." Although Process and Reality is centered on the standpoint of a single subject actual entity, it also is important to grasp how his metaphysics pertains to larger events.
At the end of Whitehead’s Modes of Thought there is the following sentence: "Poetry allies itself to metre, philosophy to mathematic pattern" (238). Hence we should expect the metaphysics of Whitehead the mathematician to embody mathematical patterns. Now it should be realized that the logic of relations is a branch of mathematics.5 In using that logic to interpret the category of nexus, I shall claim that nexus exhibit very general mathematical patterns. Such a claim is compatible with understanding them also as processes.
But the logic of relations cannot by itself provide an adequate analysis of the category of nexus. The metaphysics of nexus needs to be thoroughly discussed before that logic can be utilized adequately Accordingly, this paper is divided into two parts: in the first part, "The Metaphysics of Nexus," the text of Process and Reality is examined in order to try to determine what White-head understood nexus to be; in the second part, "The Mathematics of Nexus," nexus are interpreted in terms of the logic of relations. The discussion in this second part is nontechnical and intuitive.
Note that, in order to have sufficient focus, I shall not examine his other writings. The purpose is to avoid difficult questions about reconciling discrepancies between them and Process and Reality. This paper is not a study in the development of his thought.
Part One: The Metaphysics of Nexus
I. The Question of Ontological Status
What sort of entities are nexus? What is their ontological status? The text of Process and Reality suggests different answers. I shall contrast two answers, which I shall call the realist thesis and the eliminativist thesis. (Later, I shall consider four additional theses about nexus, the subjectivist thesis, the transcendentist thesis, the inclusivist thesis, and the abstractivist thesis.)
That there can be different answers is indicated by the two sentences that immediately follow the list of categories of existence. The first is this: "Among these eight categories of existence, actual entities and eternal objects stand out with a certain extreme finality" (22). In a paper of this brevity, I have to assume that the reader is largely familiar with actual entities and eternal objects, in order to have adequate space for a discussion of nexus. For the sake of simplicity, I shall limit my discussion to actual entities other than God (i.e., to actual occasions). Hence by "actual entity" I shall mean "finite actual entity" (i.e., "actual occasion").
And the second sentence is this: "The other types of existence have a certain intermediate character" (Process 22). It follows that nexus have an intermediate character. But what is that intermediate character? The realist and the eliminativist provide different answers which I want now to contrast.
The eliminativist thesis was summarized well by Dorothy Emmet:
It is however difficult to see why prehensions, nexus, propositions, multiplicities and contrasts should be described as "categories of existence." They are surely rather modes in which actual entities and eternal objects can be together. (70)
Presumably, a mode is not an entity; since nexus are merely modes in which entitles can be together, they are not themselves entities. Note that the category that best supports her claim is Whitehead’s nineteenth category of explanation, with its word "community" (Process 25).
In light of this quotation, the eliminativist thesis can be stated briefly thus: Nexus are not entities. But we can still ask this: What sort of modes of togetherness are they?
In contrast, the realist thesis is briefly this: Nexus are entities. Although intermediate, they still are real, they really are entities. In this paper, I shall defend realism against eliminativism.
II. The Fourteenth Category of Explanation
Among Whitehead’s categories of explanation, there is one that explains nexus:
(xiv) That a nexus is a set of actual entities in the unity of the relatedness constituted by their prehensions of each other, or -- what is the same thing conversely expressed -- constituted by their objectifications in each other. (Process 24)
This category is the focal point of my paper. It expresses a general definition of nexus that I shall discuss in the next part in terms of the logic of relations.
For purposes of illustration, let us have a simple example: A and B are actual entities, and B prehends A. And so there is the nexus -- call it the A-B nexus -- that is constituted by B’s prehension of A.6 The phrase "prehensions of each other" in the category suggests that, in order to have a nexus, A must also prehend B. However, I think that Whitehead used that phrase loosely, and so its meaning can also be expressed as "prehensions of one another." Characterizing exactly how the actual entities in a nexus are linked with one another through prehensions is a main aim of this paper.
The question of ontological status is exemplified as follows: Is the A-B nexus itself an entity? Are there three different entities, namely, A, B, and the A-B nexus? Or, is the A-B nexus merely a mode of togetherness of the two entities A and B? Are there only two entities, namely, A and B?
An eliminativist could answer these questions as follows: The key to interpreting the category is the word "set." Nexus are not entities (i.e., individuals); rather, they are sets. The word "relatedness" refers to the relation of prehension. A nexus is a set of actual entities among which this relation of prehension holds.7 The mode of togetherness of the actual entities in a nexus is this relation of prehension. Consider, for example, the A-B nexus: Let S be the set whose members are just A and B. Because B prehends A, it follows that the set S is a nexus. But the nexus S is not a third entity (i.e., individual) in addition to A and B. The nexus S is only a set.
But what is the ontological status of sets? Why are they not entities? The eliminativist could answer thus: Whitehead’s use of the term "set" in the category is misleading. To conform more with his categories of existence, he should have used the term "multiplicity" Multiplicities, even though included among the categories of existence, are not to be called "entities." A multiplicity consists of many entities; it is not itself one single ("proper") entity (Process 30). What he meant to say in the category is this: A nexus is a multiplicity of actual entities interrelated by prehensions. In brief, nexus are not ("proper") entities; rather, they are multiplicities.
My view is that these answers are mistaken. Nexus are neither sets nor multiplicities (228). Let me explain.
III. The Relation of Prehension
The eliminativist claims that the actual entities in a nexus are interrelated by a relation of prehension. How are we to understand this relation? What is its ontological status? A standard view is that properties and relations are universals. Hence, the relation of prehension could be construed as a sort of abstract entity, namely, a universal. However, instead of the standard term "universal," Whitehead preferred his own term "eternal object." Consequently, it would be better to say this: The relation of prehension is an eternal object.
Accordingly. the eliminativist could enlarge the above interpretation of the fourteenth category of explanation as follows: A nexus is a set of actual entities that are linked together by means of an eternal object, the relation of prehension. The mode of togetherness of the actual entities in a nexus is to be understood in terms of the conception of "eternal object." There indeed is a third entity in the A-B nexus in addition to A and B, namely, an eternal object. But there is not a fourth entity, the nexus itself. Thus do "actual entities and eternal objects stand out with a certain extreme finality" (Process 22).
I think that this eliminativist interpretation is incorrect. In discussing the topic of universals and particulars, Whitehead stated that, in addition to actual entities, "prehensions and subjective forms are also ‘particulars"’ (Process 48). How, then, can the relation of prehension be a universal, when prehensions are particulars? My answer is this: Indeed, properties and relations are universals. For instance, there is the property of being an actual entity. This property is a universal (i.e., an eternal object). Particular actual entities are instances of that property. Similarly, particular prehensions are instances of the relation of prehension. Just as properties have instances, so there are relation instances.8
To have an example, let me enlarge the one above by adding a second simple nexus, the C-D nexus: C and D are actual entities, and D prehends C. The point is that D’s prehension of C is (numerically) different from B’s prehension of A. And both are instances of the relation of prehension.
IV. Simple Physical Feelings
Such prehensions are particulars. This claim should be understood in terms of Whitehead’s conception of "simple physical feeling." A physical feeling is simple when it has just one actual entity as its (initial) datum (Process 236-39). For instance, in the two simple nexus, B has a simple physical feeling of A, and D has a simple physical feeling of C. The two simple physical feelings are (numerically) distinct, and both are instances of the relation of prehension.
Therefore, my view is that, when Whitehead used the term "prehensions" in the fourteenth category of explanation, what he meant was "simple physical feelings." Henceforth, for the sake of brevity, I shall usually abbreviate "simple physical feeling" as "feeling." Briefly, actual entities that feel one another constitute a nexus.
It should be realized that Whitehead’s conception of "simple physical feeling" is more elaborate. It is the feeling of a prehension. In the A-B nexus, B feels A by means of one of A’s own prehensions. B’s feeling has A as its "initial datum" and A’s prehension as its "objective datum." A’s prehension is called the "objectification" of A for B (Process 236), which clarifies the meaning of the term "objectification" in the second part of the category.
Accordingly, I propose to rewrite the category as follows:
(xiv) That a nexus is a multiplicity of actual entities in the unity of the relatedness constituted by their simple physical feelings of one another, or -- what is the same thing conversely expressed -- constituted by their prehensions being (simply physically) felt by one another.
V. The Realist Thesis
In light of this interpretation of the category, the realist thesis about nexus can be stated thus: The nexus itself is a particular entity. Rather than merely being a set of actual entities interrelated by their feelings of one another, a nexus is a particular entity in addition to those actual entities and their feelings. For instance, the A-B nexus involves these five particulars: actual entities A and B, B’s feeling of A, one of A’s prehensions, and the A-B nexus itself.
The realist thesis is expressed in the following passage:
Actual entities involve each other by reason of their prehensions of each other. There are thus real individual facts of the togetherness of actual entities, which are real, individual, and particular, in the same sense in which actual entities and the prehensions are real, individual, and particular. Any such particular fact of togetherness among actual entities is called a "nexus" (plural form is written "nexus"). The ultimate facts of immediate experience are actual entities, prehensions, and nexus. (Whitehead, Process 20)
Thus a nexus is not a multiplicity; it is not many entities. Instead, it is "individual," it is a single entity. And it is not a set; it is not an abstract entity. Instead, it is "real" and "particular." When actual entities are interrelated by feelings, the resultant nexus are themselves "real individual facts."
What, then, is the "intermediate character" of nexus? The realist answer is this: Indeed, actual entities are the "Final Realities" (Whitehead, Process 22). But nexus also are realities, albeit intermediate realities. They are intermediate in this sense: Without actual entities, there would not be nexus. For nexus come into being when actual entities have simple physical feelings of one another. In this sense, nexus are composed of (i.e., have as constituents) actual entities. Just as living cells -- although composed of molecules -- are not just sets of molecules; so nexus -- although composed of actual entities -- are not just sets of actual entities.9
VI. The Process of Becoming of a Nexus
Whitehead was both a process philosopher and a relation philosopher. I have been discussing the relatedness in a nexus in abstraction from the idea of process. But it is important to understand how nexus involve processes. How does a nexus come into being when actual entities have feelings of each other?
An actual entity comes into being through an internal process of concrescence (Whitehead, Process 22). In that process of becoming, there is a succession of phases (Whitehead, Process 26). The first phase of concrescence involves a "multiplicity of simple physical feelings" of antecedent actual entities (Whitehead, Process 236). Later phases involve derivations of conceptual prehensions and integral prehensions (Whitehead, Process 26). In the final phase (i.e., the satisfaction), the actual entity comes into being as "one complex, fully determinate feeling" (26). That one fully determinate prehension is complex insofar as it has as components prehensions from earlier phases.
How does the A-B nexus come into being? First, A comes into being through such a process of concrescence. Then B comes into being, in a process of concrescence that includes the following: In the first phase, there is a feeling of A. In later phases, B integrates that feeling of A with other prehensions. In the final phase, B comes into being as one complex prehension, a prehension that retains as a component the feeling of A. This complex process of becoming is the process of becoming of the A-B nexus. The two final realities A and B come into being, and in that process the intermediate reality the A-B nexus comes into being.
To generalize, the actual entities in a nexus come into being, and in that process the intermediate reality, the nexus of those actual entities, comes into being. Just as each actual entity is an entity that is (numerically) distinct from its component prehensions, so each nexus is an entity that is (numerically) distinct from the actual entities that constitute it.
VII. The Subjectivist Thesis
In discussing the question of the ontological status of nexus, I have been contrasting two theses, realism and eliminativism. I shall now discuss a third: subjectivism.
Whitehead distinguished two uses of the term "nexus" in Process and Reality. According to "the second use of the term nexus," a nexus is "merely relative to the feeling" of it (231). In this sense of the term, nexus fall under the category of contrasts (228). This second use suggests the following subjectivist thesis about the ontological status of nexus: When a subject actual entity prehends a nexus of antecedent actual entities, that nexus only exists as that subject’s "perspective" on those antecedent actual entities (221). The antecedent actual entities only become a nexus from the perspective of the subject. Nexus exist only in the eye of the prehender.
How is a nexus prehended? For an illustration, let me add the following to the description of the two simple nexus: D feels B, and D feels A. And, to repeat, B feels A. Consequently, D has a complex physical prehension of the nexus between A and B. It is vital to understand the process whereby D’s prehension of the A-B nexus comes into being. This complex process involves the following succession of processes: A comes into being. In B’s first phase of concrescence, B feels A. B comes into being. In D’s first phase of concrescence, D has feelings of A and B; in particular, D feels B’s feeling of A. In a later phase, D synthesizes those two feelings into a complex prehension of the nexus between A and B (see Process 226).
The main point is this: The nexus between A and B does not come into being because D prehends it. Instead, B’s feeling of A has already come into being before D begins its process of concrescence. And, therefore, the nexus between A and B has already come into being before D begins its process of concrescence. The subjectivist thesis is not correct.
But suppose that the realist thesis can be refuted, and that subjectivism is correct. The subjectivist could still read the second part of this paper as providing a definition of what nexus are from the standpoint of a subject actual entity. Similarly, the other definitions discussed there could be relativized to subject actual entities. The logic of relations is neutral between subjectivism and realism.
In conclusion, the realist thesis is this: To say that actual entities are the final realities does not mean that nexus are merely multiplicities (or sets), nor does it mean that they are merely subjective. Even though actual entities are the final realities, nexus also are objective realities.
VIII. The Transcendentist Thesis
Whitehead mentioned his "first use of the term nexus" in a paragraph about nexus and God (Process 231). Lewis Ford claims that Whitehead inserted the paragraph to "resolve one problem about the ontological status of the independent nexus" (232). Briefly, Whitehead’s paragraph says this: The ontological principle requires nexus to be "somewhere," and so they have to be in the consequent nature of God (231). Thus the paragraph can be read as containing what I shall call the transcendentist thesis about the ontological status of nexus, namely, that they only exist as real entities in God. (In thus relativizing nexus to a single divine subject actual entity, transcendentism might be construed as a form of subjectivism.)
I am skeptical about the transcendentist thesis. Whitehead’s ontological principle "means that actual entities are the only reasons" (Process 24). For example, the two final realities A and B are the reasons for the intermediate reality the A-B nexus. They come into being, and in that process it comes into being. Hence the A-B nexus is somewhere -- it is where A and B are. Indeed, God prehends A, B, and their nexus. But, just as A and B are final realities that are (numerically) distinct from God, so their nexus is an intermediate reality that is (numerically) distinct from God. Whatever motivated Whitehead to insert the paragraph about nexus and God, I do not think that the transcendentist reading of it is coherent with the metaphysics of nexus found in the bulk of Process and Reality. But I have no space to discuss the role of God in Whitehead’s metaphysics more fully. Those who accept the transcendentist thesis can read the second part of this paper as containing a discussion of nexus as they exist in the consequent nature of God.
In conclusion, the realist thesis is this: To say that actual entities are the final realities does not mean that nexus are merely multiplicities (or sets), or that nexus are merely subjective, or that nexus only exist in the consequent nature of God. Even though actual entities are the final realities, nexus also are objective realities that exist in the universe together with the actual entities that constitute them.
IX. Societies
Let me repeat an analogy: A cell is not a set of molecules, but rather is composed of them. But there is more than an analogy here. According to Whitehead, cells and molecules are societies of actual entities. The derivative notion of "society" is essential to his metaphysics, for it serves to link his speculative conception of actual entities with entities of ordinary experience, such as material bodies and living organisms (including cells and molecules). Note that he called his method of philosophy "speculative philosophy" (Process 3). From the standpoint of common sense, the thesis that the universe is a plenum of actual entities is indeed speculative. Hence the notion of "society" helps to make his metaphysics more comprehensible.
What, then, are societies? What is their ontological status? Societies are nexus. They are nexus of actual entities that are ordered by means of a "defining characteristic" (i.e., an eternal object). This defining characteristic is "inherited" by later actual entities in the society from earlier actual entities in the society. Each actual entity in the society has a conceptual prehension of the defining characteristic which it derives from its feelings of earlier actual entities in the society (Whitehead, Process 34). One of my aims in this paper about nexus is to shed some light on the notion of "society."
There is a problem of terminology. To common sense, there are three stages in the history of a living organism: coming into being, enduring, and perishing. But living organisms are categorized as nexus. And so, to generalize, we could distinguish three stages in the history of any nexus: coming into being, enduring, and perishing. Here the term "coming into being" refers to the beginning of the nexus, the coming into being of its earliest constituent actual entities. Analogously, the history of an actual entity has three stages: its first phase of concrescence, its supplementary phases, and its final phase. But Whitehead called the whole history of an actual entity its process of becoming. Here the term "becoming" refers to the actual entity in its entirety, to all of its phases of concrescence. Analogously, I have been calling the whole history of a nexus its process of becoming. In this use of the term "becoming," I am referring to the nexus in its entirety, to all of its constituent actual entities. For the sake of simplicity, I shall continue to use the term "the becoming of a nexus" in this broader sense.
To common sense, each physical object is one single entity. How does the notion of "society" capture this idea? One answer is that a society is one entity because it has a defining characteristic. But this answer is problematic. Because of its defining characteristic, a society has one nature. But two societies can have the same nature; each can have identically the same eternal object as its defining characteristic. Therefore, that a society has a defining characteristic does not explain why it is one entity. Instead, it is a single entity because it is a nexus. Because nexus are "real, individual, and particular" (Whitehead, Process 20), societies are real, individual, and particular. In opposing eliminativism, I have argued that a nexus is not just a set of actual entities and an eternal object (i.e., the relation of prehension). Similarly. my view is that a society is not just a set of actual entities and an eternal object (i.e., a defining characteristic). Thus the realist thesis about nexus elucidates how Whitehead’s metaphysics can construe material bodies and living organisms as single entities. Although speculative, his metaphysics does conform somewhat to common sense.
X. Contemporaries
Simple physical feelings were also termed by Whitehead "‘causal’ feelings": "A simple physical feeling is an act of causation" (Process 236). Each actual entity feels only those actual entities that have already come into being -- i.e., that are in its "causal past." Hence, it cannot feel actual entities that are in a "unison of becoming" with it -- i.e., that are its causal "contemporaries" (Process 123-25): For "contemporary events happen in causal independence of each other" (Process 61).
Therefore, a set of mutually contemporaneous actual entities does not constitute a nexus. Let me rephrase this essential point in the language of the fourteenth category of explanation: A set of mutually contemporaneous actual entities does not have a "unity of the relatedness constituted by their prehensions [i.e., simple physical feelings] of each other" (Process 24).
For an illustration, let me expand the description of the two simple nexus. To repeat: B feels A, and D feels C. Moreover, C feels A. But C does not feel B, and B does not feel C. Instead, B and C are contemporaries, B and C are in a unison of becoming. Let us call them the B-C unison. The main point is that there is no B-C nexus. B and C do not have a unity of relatedness constituted by a feeling. The B-C unison is only a multiplicity.
Nonetheless, B and C do have a common past. Both feel A. They both are linked to A by means of feelings. Therefore, there is a unity of relatedness among the three actual entities, B and C and A, that is constituted by their feelings of each other. There is a B-A-C nexus.
The main claim illustrated by the B-A-C nexus is that two actual entities in a nexus may be interrelated indirectly by means of simple physical feelings. They may be interrelated through the mediation of other actual entities.
This claim is fundamental to Whitehead’s complex doctrine of perception in the mode of presentational immediacy. This doctrine too is essential to his metaphysics. For it serves to link his speculative conception of actual entities with the common-sense view that we have sensory perceptions of physical objects. Hence, it also helps to make his metaphysics more comprehensible. Briefly, it includes the following ideas: I perceive sense-data. But I do not directly perceive a physical object as qualified by sense-data. Instead, I derive the sense-data that I attribute to the physical object from antecedent states of my own body (i.e., from the causal past). Those past states of my body were causally affected by the physical object (e.g., by means of light waves).10
Accordingly. the claim can be illustrated more complexly as follows: My present subject (actual entity) and contemporaneous actual entities in the physical object do not constitute a nexus. But the following actual entities do constitute a nexus: my present subject (actual entity), contemporaneous actual entities in the physical object, some past actual entities in my body. and some past actual entities in the physical object.
XI. Inclusivism
In summary, my view is that a nexus is composed of actual entities that are interrelated by their simple physical feelings of one another. And so actual entities that are mutually contemporaneous do not comprise a nexus. In contrast, Jorge Nobo contends that "any set of occasions constitutes a nexus." In particular, he contends that "a group of mutually contemporaneous occasions constitute a nexus" (21). I want now to discuss this inclusivist thesis, the thesis that every collection of actual entities constitute a nexus.
Nobo provides as textual support a quotation from Adventures of Ideas (258-59), a quotation that links "the term Nexus" to the idea of "mutual immanence" (Nobo 21). "Contemporary occasions are," he contends, "mutually immanent" (21). His defense of this contention involves Whitehead’s theory of extension, a subject too complex for consideration here. If Nobo is right about Adventures of Ideas, my suspicion is that Whitehead’s conception of nexus evolved. Nevertheless, my concern is with how he understood nexus in Process and Reality. I do not think that the fourteenth category of explanation -- the primary category in which nexus are explained -- supports inclusivism. But the inclusivist thesis is supported by certain passages in Process and Reality. Some examples are these: "the contemporary nexus perceived in the mode of presentational immediacy" (126), and "the nexus of contemporary actual entities" (317).
Whitehead also stated (in the paragraph from which Nobo quoted) that the "mutual immanence of contemporary actual entities is "of the indirect type" (Adventures 259). In the B-A-C nexus, B and C are contemporaries, but they both feel A, and so they are interrelated indirectly. It is important to distinguish the set containing just B and C from the set containing B, C, and A. The former set does not contain actual entities that by themselves constitute (in my sense of the term) a nexus, because it does not contain an intermediary actual entity that interrelates the two contemporaries. In contrast, the latter set does contain actual entities that constitute a nexus, because it does contain such an intermediary. In general, my view is that a set of actual entities cannot constitute a nexus unless it contains sufficient intermediaries.
What, then, did Whitehead mean by "the contemporary nexus perceived in the mode of presentational immediacy" (Process 126)? In light of the difference between the B-C multiplicity and the B-A-C nexus, let me contrast two answers: On the one hand, this passage could be read as referring to mutually contemporaneous actual entities in abstraction from actual entities in their common past. On the other hand, it could be read as referring to mutually contemporaneous actual entities together with actual entities in their common past. (cf. the illustration at the end of the preceding section.)
I am skeptical about inclusivism, but there is no space to dispute Nobo’s contentions further. And so my response to them is this: Let us suppose that the inclusivist thesis is correct. My paper can still be read as a discussion of an important type of nexus, namely, those nexus that are interrelated by means of simple physical feelings. Let us call nexus of this type cohesive nexus. Also, to have a term that contrasts with "inclusivism," I shall call my view about nexus cohesivism.
Clearly, the A-B nexus -- in which B coheres concretely with A by means of a simple physical feeling -- is different from the B-C unison -- in which B and C come into being independently. If the inclusivist thesis is correct, the A-B nexus is to be classified as a cohesive nexus, whereas the B-C unison is to be classified as a noncohesive nexus. The main point is that the inclusivist can read the second part of this paper as an attempt to understand cohesive nexus in terms of the logic of relations.
XII. Abstractivism11
To review, a nexus is constituted by actual entities that are interrelated by their simple physical feelings of one another. Some of the actual entities in a nexus may be interrelated through the mediation of other actual entities in that nexus. Actual entities that are mutually contemporaneous do not constitute a nexus.
But actual entities may also be interrelated through the mediation of entities of other categoreal types. In particular, they may be interrelated indirectly by means of their conceptual feelings of one and the same eternal object. Accordingly. in opposition to my claim that mutually contemporaneous actual entities do not constitute a nexus, there is the following abstractivist thesi:12 When actual entities are thus interrelated through the mediation of an eternal object, they constitute a nexus.13 Let us call such a nexus an abstract nexus.
Utilizing Whitehead’s doctrine of "perception in the mode of presentational immediacy." the abstractivist thesis can be illustrated as follows: Another person and I are presently perceiving one and the same sense-datum. Let S be my present subject (actual entity), let T be a contemporaneous subject (actual entity) of that other person, and let O be the sense-datum. Because a sense-datum is an eternal object (61), S and T have conceptual feelings of one and the same eternal object. Hence S and T, even though mutually contemporaneous, constitute a nexus. Let us call this abstract nexus the S-0-T nexus.
I am skeptical about the abstractivist thesis. Actual entities are "real, individual, and particular," and so "in the same sense" are nexus (Whitehead, Process 20). In the B-A-C nexus, B and C are interrelated indirectly by means of their simple physical feelings of A. Because B and C are interrelated through the mediation of A -- an entity that is real, individual, and particular -- the nexus among B, A, and C is itself real, individual, and particular.
In contrast, in the putative S-O-T nexus, S and T are interrelated indirectly by means of their conceptual feelings of an eternal object. But that eternal object is not an entity that is real, individual, and particular. Instead, it is a universal (Whitehead, Process 48). Because S and T are interrelated through the mediation of a universal -- an entity that is not real, individual, and particular -- the interrelatedness of S, T, and the eternal object does not constitute an entity that is real, individual, and particular. In brief, there really is no S-O-T nexus.
I have voiced an objection to the abstractivist thesis, but there is no space to examine it more fully. In conclusion, my response to it is this: let us suppose that it is correct; let us suppose that there are such abstract nexus. My paper can still be read as a discussion of cohesive nexus. The main point is that the abstractivist can read the second part of this paper as an attempt to understand cohesive nexus in terms of the logic of relations.
Although I have been defending my own interpretation of the category of nexus, one of my aims has been to distinguish alternative interpretations. Since the logic of relations is neutral between these competing views about the metaphysics of nexus, my hope is that those who disagree with my interpretation will still find the second part helpful.
Part Two: The Mathematics of Nexus
I. Introduction
Having explored the metaphysics of nexus with reference to the text of Process and Reality, I shall now interpret the category of nexus in terms of the logic of relations. Although the term "logic" might suggest otherwise, I shall not be concerned with deductive arguments about nexus. Instead, I shall discuss some mathematical patterns exhibited by nexus that are definable in terms of the logic of relations. In addition to mathematical patterns that are evident in Process and Reality, I shall discuss some mathematical patterns that are implicit there.
"Whitehead is," Charles Hartshorne remarked, "the first to embody modern relational logic in a fairly complete metaphysical system." And, he added, "Metaphysics ought to be the study of relational structures as embodied in reality as such" (14). In this second part of my paper, I shall attempt to discern some relational structures that Whitehead’s metaphysics attributes to reality: relational structures involving nexus.
In particular, the logic of relations is used to formulate a definition of nexus. Also, it is used to distinguish various sorts of nexus. Although I have been defending the realist thesis, closely similar definitions could be utilized by the eliminativist, the subjectivist, and the transcendentist. The logic of relations is neutral between these competing views about ontological status. Note also that the inclusivist and the abstractivist can understand me to be defining cohesive nexus.
In addition, I shall discuss the type of order involved in the creative advance of actual entities. The goal is to determine the type of order involved in the creative advance of nexus. Although Whitehead’s idea of "creative advance" does not involve "the notion of a unique seriality" (Process 35). the creative advance of actual entities and nexus is ordered nonserially. In so doing, I shall consider his definition of durations. Even though durations are not nexus, the topic of durations is quite relevant. The nonserial ordering of actual entities induces a nonserial ordering of durations.
My view is that just as the process of becoming of an actual entity involves phases, so does the process of becoming of a nexus. To show that defining nexus in terms of the logic of relations is compatible with understanding them as processes, that logic is used to define the phases in the process of becoming of a nexus. Just as a "duration is a cross-section of the universe" (Whitehead, Process 125); so a phase of a nexus is a cross-section of that nexus. The nonserial ordering of actual entities induces a nonserial ordering of these phases. The creative advance of actual entities involves a creative advance of phases of nexus.
II. The Relation of Simple Physical Feeling
Let me preface this section with a comment by Bertrand Russell about the "calculus of relations": "A careful analysis of mathematical reasoning shows [. . .] that types of relations are the true subject-matter discussed" (23). Similarly, my view is that an analysis of Whitehead’s categoreal scheme shows that (mathematical) types of relations are importantly involved there.
In an earlier section, I discussed the relation of prehension, a relation that has particular prehensions as instances. Subsequently, I rewrote the fourteenth category of explanation in terms of the conception of "simple physical feeling." In this second part of my paper, I shall focus on the relation of simple physical feeling. For the sake of brevity, I shall term this relation "feels." Particular (simple physical) feelings are instances of the relation feels. What type of relation is it? The aim of the present section is to answer this question.
The theory of types of relations is a sort of mathematical theory. For the type of a particular relation is definable in terms of purely formal properties of the relation. Consider, for example, the relation being a (natural) parent of. Your child cannot be your parent. To generalize, a relation R is asymmetric when it has the following formal property: For any entities x and y, if xRy, then it is not the case that yRx.
The relation feels is asymmetric. Each actual entity feels only those actual entities that have already come into being, and so they cannot feel it. Actual entities cannot be both in the causal past and in the causal future. Consider, again, the A-B nexus: When B comes into being, A has already come into being, and so B feels A. But when A comes into being, B has not yet come into being, and so A cannot feel B. A cannot be both in B’s causal past and in B’s causal future.14
You cannot be your own parent. In general, a relation R is irreflexive when the following formal property holds: For any entity x, it is not the case that xRx. The relation feels is irreflexive. When an actual entity is coming into being, it has not already come into being. It is not in its own causal past. Thus it cannot feel itself.
Another relation is being an ancestor of. An ancestor of any of your ancestors is also your ancestor. Generally speaking, a relation is transitive when it has the following formal property: For any entities x, y. and z, if xRy and yRz, then xRz. Thus, if x is an ancestor of y and y is an ancestor of z, then x is an ancestor of z.
The relation feels is transitive. Each actual entity feels all of the actual entities in its causal past, i.e., its actual world (Whitehead, Process 239). In the two simple nexus, D feels C, and C feels A. Thus C is in D’s causal past, and A is in C’s causal past. Consequently, A is in D’s causal past. And so D feels A. In general, for any three actual entities x, y. and z, if z is in the causal past of y and y is in the causal past of x, then z is in the causal past of x. And so, if x feels y and y feels z, then x feels z.
In these introductory remarks about the logic of relations, I have described three formal properties of relations. In summary. the relation of simple physical feeling is irreflexive, asymmetric, and transitive.
III. Nonseriality
‘There is a prevalent misconception," Whitehead asserted, "that ‘becoming’ involves the notion of a unique seriality for its advance into novelty" (Process 35). In terms of these three formal properties of relations, I want now to show how actual entities are ordered nonserially.
The integers do have a unique seriality; they are ordered serially by the relation is less than. This relation is irreflexive, asymmetric, and transitive. Also, any two integers are such that one is less than the other. In general, a relation R is connected when it has the following formal property: For any two distinct entities x and y. either xRy or yRx. The integers have a unique seriality because the relation is less than is irreflexive, asymmetric, transitive, and connected. Any relation that has these four formal properties is a serial relation (Carnap, Symbolic 123).
Equivalently. the integers are ordered by the disjunctive relation "either is less than or is equal to." This relation is reflexive, antisymmetric, transitive, and connected. A relation with these four formal properties is a simple order (Carnap, Symbolic 123). Note that a relation R is reflexive when it has the following formal property: For any x, xRx. And it is antisymmetric when it has this formal property: For any x and y, if (a) xRy and (b) x is not identical to y, then it is not the case that yRx.
Obviously, the relation being a parent of is not connected. Similarly, the relation feels is not connected. And so actual entities do not have a unique seriality. The relation feels is not connected because actual entities do not feel their contemporaries. In the B-C unison, B does not feel C, and C does not feel B. In general, for any two actual entities x and y. there are three mutually exclusive alternatives: Either (1) x feels y, or (2) y feels x, or (3) x does not feel y and y does not feel x (i.e., x and y are contemporaries).
Even though the creative advance of actual entities does not have a unique seriality, actual entities are still ordered nonserially. This claim can be understood as follows: Even though the relation feels is not connected, it is irreflexive, asymmetric, and transitive. Any relation that has those three formal properties is a strict partial order. The concept of "strict partial order" is a generalization
of the concept of "serial relation." For example, the relation being an ancestor of has those three formal properties, and so human beings have the nonserial order recorded in genealogies.
There is an equivalent generalization of the concept of "simple order": A relation R is a partial order when it is reflexive, antisymmetric, and transitive.15 A set whose members are interrelated by such a relation is a partially ordered set. The disjunctive relation "either is felt by or is identical to" is a partial order. And the set of actual entities as ordered by that relation is a partially ordered set. Although the creative advance of actual entities does not have a unique seriality, their creative advance is partially ordered.
IV. Durations
Influenced by Einstein’s theory of relativity, Whitehead rejected the "‘classical’ theory of time." In particular, because of the relativity of simultaneity, there is no absolute present. Included in the idea that the creative advance is not uniquely serial is the idea that there is not a unique present state of the universe. Nevertheless, there are durations, and a "duration is a cross-section of the universe." But each actual entity "lies in many durations"; it does not lie in a cross-section of the universe that is unique (Whitehead, Process 125). The nonserial creative advance involves a nonserial succession of durations.
A duration is a maximal set of mutually contemporaneous actual entities. Hence durations are not nexus. Although this paper is about nexus, the topic of durations is (as we shall see) quite relevant. More explicitly. a duration D is a set of actual entities that satisfies the following two conditions (see White-head, Process 320): The first condition is that each actual entity in D is contemporaneous with every actual entity in D.16 The second condition makes D a maximal set of contemporaries: Any actual entity that is not in D either feels or is felt by some actual entity in D. In other words, any actual entity that is contemporaneous with every actual entity in D is itself in D.17
This definition of durations can be understood in terms of the logic of relations. The relation is identical to is reflexive, symmetric, and transitive, and thus an equivalence relation. A relation R is symmetric when it has the following formal property: For any entities x and y. if xRy. then yRx. The relation is identical to partitions the set of unreduced fractions into mutually exclusive equivalence classes (e.g., {1/3, 2/6, 3/9, etc.}). But simultaneity is relative. There are actual entities X, Y, and Z such that X is contemporaneous with Y, and Y is contemporaneous with Z, but X is not contemporaneous with Z. Consequently. the relation of contemporaneity -- i.e., neither feels nor is felt by -- although reflexive and symmetric, is not transitive (Whitehead, Process 320). A relation that is reflexive and symmetric is a similarity relation (Carnap, Structure 21-22, 112-13). Although such a relation need not partition a set in which it is defined into mutually exclusive classes, it does divide such a set into (possibly) overlapping similarity circles. In short, durations are the similarity circles of the relation of contemporaneity (Lango 89-92).
V. The Partial Ordering of Durations
Whitehead’s conception of "duration" illuminates how the universe in its entirety is in a process of becoming. For the creative advance of actual entities involves a creative advance of durations. Contemporaneous actual entities come into being in a "unison of becoming" (Process 124). The process of becoming of a maximal set of contemporaneous actual entities is the process of becoming of a duration.
This creative advance of durations is not uniquely serial. Nonetheless, because actual entities are ordered nonserially, my view is that durations also are ordered nonserially. The key point is this: The relation feels induces the following relation between durations. Duration E is later than duration D when the following two conditions hold: (1) D and B are not identical. (2) Each actual entity that is in E but not in D feels an actual entity in D.
Obviously. the relation is later than is irreflexive. Moreover, it is transitive: Let F be later than E, and E later than D. Each actual entity in F (but not in E) feels an actual entity in E. And the latter either (a) is in D or (b) feels an actual entity in D. Therefore (by the transitivity of feels), each actual entity in F (but not in D) feels an actual entity in D. Hence condition (2) is satisfied. Also, by a similar transitivity argument, condition (I) is satisfied: If F were identical to D, each actual entity in F would feel an actual entity in F, contradicting the definition of "duration."
And, since any irreflexive and transitive relation is asymmetric, the relation is later than is asymmetric. However, it is not connected. For simultaneity is relative. And so each actual entity "lies in many durations" (Whitehead, Process 125). Consequently, there can be two durations that have the following property: actual entities in each duration feel actual entities in the other. Let D and E be two such durations. And let X be an actual entity in D that feels an actual entity Y in E. If B were later than D, Y would feel an actual entity Z in D (or Y would be in D). And so (by the transitivity of feels) X (in D) would feel an actual entity in D, contradicting the definition of "duration." Hence B cannot be later than D. By the same type of transitivity argument, O cannot be later than B.
Therefore, the relation is later than is a strict partial order. Hence, the disjunctive relation "either is later than or is identical to" is a partial order. And the set of durations as partially ordered by it is a partially ordered set. The partial ordering of actual entities induces a partial ordering of durations.
This temporal ordering of durations presupposes only the relation of simple physical feeling. It does not presuppose Whitehead’s theory of extension in Part IV of Process and Reality, a theory that evolved from his earlier method of extensive abstraction (287)18 In contrast, the concept of a "time-system" in his Principles of Natural Knowledge presupposes the method of extensive abstraction; note that, in a time-system, moments are "arranged in serial order" (114). The theory of extension in Process and Reality is problematic: What are regions? What is their ontological status? More generally. why is the theory of extension coherent with the categoreal scheme of Part I? I cannot try to answer such difficult questions in this paper. My main point is that implicit in the categoreal scheme is a conception of the temporal ordering of durations that is independent of the complexities of the theory of extension.
In general, all of the mathematical patterns discussed in this part of the paper are independent of the theory of extension. The logic of relations can be used to define very general mathematical patterns involving actual entities that are not specifically geometrical (or topological).
VI. A Definition of Nexus
According to the fourteenth category of explanation, a nexus is a multiplicity of actual entities in the unity of the relatedness constituted by their simple physical feelings of one another. Let us term any set of actual entities that constitute a nexus a nexal set. For the realist, each nexal set corresponds to a nexus. For the eliminativist, the nexal sets are the nexus. The subjectivist can relativize nexal sets to a subject actual entity. The transcendentist can understand the nexal sets as corresponding to nexus as they exist in the consequent nature of God. And the inclusivist and the abstractivist can understand the nexal sets as corresponding to (or as being) cohesive nexus.
Accordingly. in order to define the category of nexus, I shall provide a definition of the class of nexal sets.
Let me begin with an example, the B-A-C nexus. B and C are contemporaries, but they have a common past. Both feel A. This illustrates the following claim: Two actual entities in a nexus may be interrelated indirectly by means of simple physical feelings. They may be interrelated through the mediation of other actual entities. Now consider the nexal set that contains just B, A, and C. Any two actual entities x and y in that nexal set are interrelated in one of the following two ways: (1) x feels y or x is felt by y. (2) x feels or is felt by y through the mediation of a third actual entity in the nexal set. More explicitly. (2) there is an actual entity z in the nexal set such that (a) x feels z or x is felt by z, and (b) z feels y or z is felt by y.
This example can be expanded as follows: There is a fourth actual entity B. C feels E. But B is contemporaneous both with B and with A. (Although B feels A, E lies in many durations, including one that contains B, and one that contains A.) Hence there is a B-A-C-B nexus. Let us consider the nexal set containing just B, A, C, and E. Any two actual entities x and y in that nexal set are interrelated in one of the following three ways: (1) x feels y or x is felt by y; (2) There is an actual entity z in the nexal set such that (a) x feels z or x is felt by z, and (b) z feels y or z is felt by y. (3) x feels or is felt by y through the mediation of two actual entities in the nexal set. More explicitly. (3) there are actual entities z and w in the nexal set such that (a) x feels z or x is felt by z, (b) z feels w or a is felt by w, and (c) w feels y or w is felt by y.19
And so forth. But how much forth? Let us assume that there are two durations that satisfy the following three conditions: (1) One is later than the other; (2) they do not have members in common; and (3) the set containing all of the actual entities from both durations is a nexal set.20 In that nexal set, there are chains of actual entities, linked together by simple physical feelings, that span the entire universe. Let one such chain contain B, A, C, and B. Continuing the chain in one direction, there is an actual entity F that feels E, there is an actual entity G that is felt by F, there is an actual entity H that feels G, and so forth across the entire universe.
In light of these examples, I shall use the concept of "the ancestral of a relation" to define the category of nexus. An ancestor is a parent of a parent of a parent, and so forth. You are linked to your ancestors by chains of parents. The relation being an ancestor of is the ancestral of the relation being a parent of. The concept of ancestral can be understood roughly as follows: For any entities x and y, x has the relation "the ancestral of R" to y when the following condition is satisfied:21 Either (1) xRy, or (2) there is an entity z such that xRz and zRy. or (3) there are entities z and w such that xRz and zRw and wRy, or (4) there are entities a, w, and v such that xRz and zRw and wRy and vRy. or (5) -- and so forth. The meaning of "and so forth" is explicated by means of a more exact definition of "ancestral" (Carnap, Symbolic 146-48).
Consider the following relation: feels or is felt by. Very roughly speaking, any two actual entities in a nexus are linked by the ancestral of the relation feels or is felt by. More precisely, the class of nexal sets is defined as follows: A set N of two or more actual entities is a nexal set when it satisfies the following condition: For any two actual entities x and y in N, either (1) x feels or is felt by y, or (2) there is an actual entity a in N such that (a) x feels or is felt by z and (b) a feels or is felt by y. or (3) there are actual entities a and w in N such that (a) x feels or is felt by z and (b) z feels or is felt by w and (c) w feels or is felt by y. or (4) there are actual entities z, w, and v in N such that (a) x feels or is felt by a and (b) a feels or is felt by w and (c) w feels or is felt by v and (d) v feels or is felt by y. or (5) (and so forth). The meaning of "and so forth" is to be understood in terms of the concept of ancestral.22
The actual entities in each nexal set constitute a nexus through their simple physical feelings of one another.23
VII. Nonseriality and Nexus
The creative advance of actual entities does not have a unique seriality. Consequently, the actual entities in a particular nexus could be ordered nonserially. but they also could be ordered serially. For example, an "enduring object" is a society of "serially" ordered actual entities. Note that Whitehead’s definition of "serial ordering" also holds of the vastly many serially ordered nexus that are not societies (Process, 34). Let me define a general concept of "serially ordered nexus." A nexal set N of actual entities is serially ordered when it satisfies the following condition (cf. Davies and Priestley 3); For any two actual entities x and y in N, either x feels y or x is felt by y Hence there are no contemporaries in N. A serially ordered nexal set is a nexal set in which the relation feels is a serial relation (i.e., a relation that is irreflexive, asymmetric, transitive, and connected). A nexus is serially ordered when the nexal set of its members is serially ordered.
Whitehead held that physical objects (other than elementary particles) are societies of actual entities that are not serially ordered (Process 35). Nevertheless, they still are nonserially ordered. Of course, vastly many nexus of actual entities that are not serially ordered are also not societies. Let me define a general concept of "nonserially ordered nexus." Although the actual entities in a nexus are interrelated through their feelings of one another, some might be contemporaneous, some might not feel each other. Consequentl3; a nexal set is nonserially ordered when it contains at least two actual entities that are contemporaneous. A nexus is nonserially ordered when the nexal set of its members is nonserially ordered.
Note that any nexal set that contains only two contemporaneous actual entities is nonserially ordered. Hence this definition is very general. At the opposite extreme, let us say that a nexal set is entirely nonserially ordered when each of its members is contemporaneous with at least one of its members. A nexus is entirely nonserially ordered when the nexal set of its members is entirely nonserially ordered. Ordinary physical objects are societies of actual entities that are entirely nonserially ordered.
VIII. Temporally Fragmented Nexus
The importance of the category of nexus lies partly in its great generality I want now to illustrate further the enormous variety of sets of actual entities that are nexal sets. In particular, I shall provide an illustration that is especially relevant to the problems of identity through time and the nature of the human mind.
To common sense, a human mind retains its personal identity through the passage of time. Similarly, to Whitehead, a "human mind" is an "enduring object" (Process 109). That is, a human mind is a society of serially ordered actual entities that inherit a defining characteristic (Process 34). Thus a human mind has identity of character through time because it is a society. And it is a single entity because it is a nexus.
A standard view is that identity through time involves spatiotemporal continuity But it is questionable whether the human mind is thus continuous. An adequate theory of the human mind has to account for such phenomena as sleep, unconsciousness, comas, and amnesia. It is conceivable, then, that our minds are temporally fragmented, that our mentality sometimes ceases while we are asleep or unconscious. Now a nexus is a unity, not because it has spatiotemporal continuity, but because the actual entities in it have feelings of one another. Hence, because of the great generality of the conception of nexus, a nexus that constitutes a human mind can be temporally fragmented. Its identity of character can be transmitted across a temporal hiatus.
What, then, are temporally fragmented nexus? How are nexus of this type to be defined? To answer these questions, we need some definitions. Each duration is a cross-section of the universe. A set S of actual entities is in the past of the duration D just in case each actual entity in S is felt by some actual entity in D. A set T of actual entities is in the future of the duration D just in case each actual entity in T feels some actual entity in D (cf. Whitehead, Process 320). Consequently, part of a nexus can be in the past of a duration, and part of it can be in the future of that duration.
A nexal set N is temporally fragmented by a duration D when the following three conditions hold: (1) Some members of N are in the past of D; (2) some members of N are in the future of D; and (3) but no members of N are in D. Thus D is a temporal hiatus in the becoming of N (see Whitehead, Process 322). Since any two temporal fragments of a nexus can be separated by many durations, the temporal hiatus between two fragments of a nexus could be measured by the time spanned by intermediate durations. Thus a nexus is temporally fragmented when its nexal set is temporally fragmented by at least one duration.24 And a nexus is temporally continuous when its nexal set is not temporally fragmented by any duration.
IX. The Phases of a Nexus
My view is that using the logic of relations to understand nexus is compatible with understanding them as processes. In support of this view, I shall use that logic to define the phases in the process of becoming of a nexus.
An actual entity comes into being through an internal process of concrescence. "It repeats in microcosm what the universe is in macrocosm" (Whitehead, Process 215). Thus the creative advance of the universe involves a succession of durations. Each duration consists of mutually contemporaneous actual entities. Analogously. a nexus comes into being through a succession of phases. Each phase consists of actual entities in that nexus which are mutually contemporaneous. Just as a duration is a cross-section of the universe, so a phase of a nexus is a cross-section of that nexus.
The definition of nexal phases is built upon the following idea: Nexus are intersected by durations, and each such intersection is a phase of a nexus. Each duration is a maximal set of mutually contemporaneous actual entities. And so each phase of a nexus is a maximal set of mutually contemporaneous actual entities in that nexus. Accordingly, the phases of a nexus are defined (roughly speaking) by restricting the above definition of durations to the actual entities in that nexus.
More exactly. the definition is this: Let N be a nexal set. A phase P in N is a subset of N that satisfies the following two conditions: The first condition is that (I) each actual entity in P is contemporaneous with even’ actual entity in P. The second condition makes P a maximal set of contemporaries in N; (2) Any actual entity in N that is not in P either feels or is felt by some actual entity in P. In other words, (2) any actual entity in N that is contemporaneous with every actual entity in P is itself in P. The phases of a nexus are the phases in its nexal set.
In light of this definition, let me suggest an answer to a question raised above:25 What did Whitehead mean by "the contemporary nexus perceived in the mode of presentational immediacy"? (Process 126). This passage could he read as referring to a contemporary phase of a nexus -- i.e., the phase that is perceived in the mode of presentational immediacy. Let us call it the "presented phase." Perception in the mode of presentational immediacy objectifies the actual entities "within one particular duration": "the ‘presented duration"’ (Process 321). The presented phase is the intersection of the nexus with the presented duration.
X. The Ordering of Phases
The universe consists of actual entities interrelated by their feelings of one another. Hence, the universe is a nexus, and durations are its phases. We have seen how durations are ordered by the relation is later than. The intersections of durations with a nexus -- i e its phases -- are ordered analogously. In brief, the process of becoming of a nexus involves a strict partial ordering of phases.
The key point is this: The relation feels induces the following relation between the phases of a nexus: Let P and Q be phases in a nexal set N. Q is later than P when the following two conditions hold: (1) p and Q are not identical. (2) Each actual entity that is in Q but not in P feels an actual entity in P
Just as the relation is later than between durations is a strict partial order, so this relation between phases of a nexus is a strict partial order. This claim can be established by transitivity arguments similar to the ones above.
An actual entity has a first phase and a final phase of concrescence. Analogously. a nexus can have a starting phase or a terminal phase. B is a starting phase in a nexal set N when B is not later than any phase in N. And F is a terminal phase in a nexal set N when no phase in N is later than F. Just as there is not a unique present state of the universe, so a nexus might have more than one starting phase or more than one terminal phase (cf. Davey and Priestley 15). On the other hand, these definitions do not require a nexus to have either a starting phase or a terminal phase. The universe is a nexus, and there is controversy about whether it has starting phases or terminal phases.
XI. Examples
The causal past of an actual entity is a nexus, and so is its causal future. An actual entity’s causal past involves a nonserial succession of phases that (in a sense) converge on it, and its causal future involves a nonserial succession of phases that (in a sense) radiate from it.
When actual entity B feels actual entity A, there (usually) are actual entities "between" them, namely, those actual entities that the causal future of A has in common with the causal past of B. More exactly. the medium between A and B is the set that contains any actual entity that both feels A and is felt by B.26 When the medium between two actual entities is (as it usually is) a nexal set, the succession of its phases (in a sense) radiate from the earlier actual entity to the later one.
There are nexus with phases that are ordered nonserially. but there also are nexus with phases that are ordered serially. Enduring objects (e.g., human minds) are societies of serially ordered actual entities (Whitehead, Process 34). The phases in the process of becoming of a serially ordered nexus are quite simple: each is a set containing a single actual entity. Obviously, those phases are serially ordered.
In contrast, physical objects (other than elementary particles) are societies of actual entities that are nonserially ordered. Such a physical object is intersected by a multitude of durations, and each such intersection is one of its phases. A maximal set of mutually contemporaneous actual entities In such a physical object is a phase of that physical object. The process of becoming of such a physical object involves a succession of phases of mutually contemporaneous actual entities. If simultaneity is not relative over sufficiently small distances, some such physical objects have phases that are ordered serially. But physical entities of sufficient extent (e.g., galaxies) have phases that are ordered nonserially.
Arguably, physics is committed to the real existence of entities other than particles -- e.g., fields and waves. Thus Whitehead suggested that there is an "electromagnetic society" (Process 98). And he remarked also that a "train of waves" is a society (36, 98). The actual entities in such a society have "physical purposes of the "second species": "It is due to this second species that vibration and rhythm have dominating importance in the physical world" (Process 277). In particular, an "enduring object" can involve "physical vibration" (Process 279). Let me augment his discussion of the metaphysics of waves as follows: In addition, there are nonserially ordered societies involving physical vibrations -- e.g., light waves emanating radially from the sun. Such a society has a defining characteristic that is inherited from phase (of mutually contemporaneous actual entities) to phase (of mutually contemporaneous actual entities). And these phases are ordered nonserially. The actual entities in each such phase have physical purposes of the second species. Hence the phases thus ordered exhibit vibration and rhythm.
The idea of "creative advance" includes the creative advance of nexus of actual entities, a creative advance that unfolds in phases.
Notes
1. To situate my project in a broader historical context, see McHenry. He interprets Whitehead’s metaphysics as including the fundamental thesis that "the universe evolves by an asymmetric process of causality in which former actualities [i.e., actual entities] are prehended by latter ones, but not vice versa (90). In this paper, my aim involves exploring how that thesis pertains collectively to nexus of actual entities.
2. For example, Kraus has a chapter on the theory of concrescence and a section on eternal objects. But, although her index has many entries for the term "nexus," there is no part of her book that focuses on the topic of nexus. Instead, the term "nexus" is briefly introduced in a discussion that focuses on the topic of actual entities (51). And, shortly later, there is a section devoted to the topic of societies. A theme of my paper is that, to understand the derivative notion of societies, we first need to understand the category of nexus.
3. In Whitehead’s Ontology, I interpreted his categories of existence in terms of the logic of relations. Since that book considered the category of nexus incidentally, the present paper also supplements it.
4. More fully, an event is "a nexus of actual occasions, inter-related in some determinate fashion in one extensive quantum" (Process73). My view is that the actual entities in a nexus need not be interrelated in one extensive quantum. Hence the concept of "nexus" is broader than this concept of "event." Some nexus are not events.
5. The logic of relations is integral to the project about the foundations of mathematics in Principia Mathematica. However, this paper is not a study in the development of Whitehead’s thought, and so I shall not speculate about the relationship between that work and Process and Reality.
6. One of Whitehead’s examples of such a nexus with two members is this: "D in its nexus with C" (Process 226).
7. Lawrence asserts that the term "nexus" refers to "a group of actual occasions" (81); does his term "group" mean "set" (or "class")? Similarly, Nobo calls a nexus "an interrelated group" of actual entities (2); later, he uses the term "set" (21).
8. Concerning the subject of relation instances, see Mertz.
9. It would seem that Kraus holds the realist thesis about nexus. In briefly introducing the term "nexus," she states that a nexus is "as real, individual, and particular as the actual entities comprising it" (51). And it would seem that Christian holds the thesis (233); see also his discussion of Whitehead’s two uses of the term "class" (260n).
10. See Whitehead’s Process and Reality 61-65, 117-26, 311-18.
11. I am indebted to Lewis Ford for some illuminating comments on an earlier draft of this paper. In light of his comments, I have added this section.
12. Actual entities and nexus are "concrete" (Whitehead, Process 18), whereas "universals" are "abstract" (Whitehead, Process 20). Accordingly, the thesis that a nexus can be formed thus through the mediation of an abstract entity is termed "abstractivism."
13. In commenting on an earlier draft of my paper, Ford proposed that the category of nexus can be understood solely in terms of something like this thesis. His interpretation of that category involves his analysis of the compositional history of Process and Reality (and Science and the Modern World). My view is that, whatever its compositional layers, Process and Reality contains a coherent system of metaphysics, one that is summarized effectively in its second chapter ("The Categoreal Scheme"). Accordingly, I have focused on a key category, the fourteenth category of explanation.
14. In Whitehead’s eighth categoreal obligation, there are the words "anticipatory feeling" (Process 27). But how can there be a prehension of an actual entity that has not yet come into being? However this problematic question is answered, there cannot be a simple physical feeling of a future actual entity. Simple physical feelings are only of the settled past.
15. Davey and Priestley abbreviate the term "partial order" as "order," and thus they use the term "strict order" instead of "strict partial order" (2, 21). For the sake of emphasis, I retain the word "partial."
16. Utilizing the mathematical concept of "antichain" (Davies and Priestley 3), let us say that any set of actual entities that satisfies this first condition is an antichain. Hence a duration is an antichain of actual entities that satisfies the second condition.
17. In his Principles of Natural Knowledge, Whitehead defines "durations" in terms of percipient events: a duration relative to a percipient event is "that complete whole of nature simultaneous with the percipient event" (68). Hence what that work calls a "duration" is termed in Process and Reality a "presented duration" (321). The point is that, in contrast to Principles, the definition of durations in Process and Reality does not presuppose a conception of "perception." Instead, it presupposes the conception of "(simple physical) feeling." Note that it is of far greater generality than the definition in Principles When a presented duration corresponds to a strain-locus, it is (so to speak) flat (Process 322-23). Other durations can have many undulations in snaking across the universe -- as long as no undulation is so bent as to allow one actual entity to feel another.
18. His most definitive discussion of durations is in Part IV, presumably to show how the theory of time in Process and Reality is related to the theory in Principles. What this placement obscures is that there is a temporal ordering of durations simply in terms of feels.
19. Chiaraviglio construes nexus as sets of actual entities (88). His definition of nexus is as follows (88): The relation prehends is transitive and irreflexive. A set S of actual entities is a nexus when it satisfies at least one of these conditions: (1) For any two actual entities x and v in 5, there is a third actual entity that prehends x and prehends y. (2) For any two actual entities x and y in S. there is a third actual entity that is prehended by x and that is prehended by v. According to his definition, what I call "the A-B nexus" is not a nexus. Also, what I call "the B-A-C-B nexus" is not a nexus. Because his definition requires the mediation of just one actual entity (no more and no less), it is not sufficiently general.
20. This claim does not follow from the definition of "duration." However, it is reasonable to think that the claim holds of pairs of durations in our "cosmic epoch" (Whitehead, Process 91).
21 There are two kinds of ancestral (Carnap, Symbolic 147). For simplicity, I use the second kind of ancestral. If the first kind were used instead, the next sentence in the main text should be enlarged as follows: Either x is identical to y, or (I) xRy or (etc.).
22. Let me be more explicit about the role of the concept of ancestral in this definition. Let S be a set of actual entities. We define the relation F as follows: xF5y if and only if (1) x feels or is felt by y and (2) x and y are in S. If each actual entity in S is related by the ancestral of F5 to every actual entity in S,S is a nexal set.
23. Let me comment on my earlier and quite different definition of nexus in Whitehead’s Ontology 94. There the goal was to distinguish the category of nexus from the other categories of existence (Ontology 14-15). In the present paper, the goal is to have a definition that reflects the internal structure of nexus.
24. Human history is rich with putative examples: Political entities could be temporally fragmented (e.g., the Persian Empire). The society consisting in the users of a language (e.g., Hebrew) could be temporally fragmented (cf. Whitehead, Process 90). Also, could an "entirely living nexus" (Whitehead, Process 103-17) be temporally fragmented?
25. See the section above on inclusivism.
26. Whitehead’s definition of the "medium" between two actual entities is problematic: Suppose that actual entity A feels actual entity D. "The medium between D and A consists in all those actual entities which lie in the actual world of A and not in the actual world of D" (Process 226). The problem is that an actual entity Z may be both in the actual world of A and contemporaneous with D. But then Z does not feel D, and so A cannot feel D through the medium of Z.
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