Revisiting Whitehead’s Abstractive Hierarchy

Abstract

In Whitehead’s theory of “events”, the primary focus is on the critical assessment of abstraction. Modern science’s heavy reliance on abstraction has resulted in what Whitehead calls “the fallacy of misplaced concreteness”, where the abstract is mistaken for the actual. To address this issue, Whitehead replaces the traditional category of abstraction with eternal objects and defines the abstractive hierarchy. He aims to clarify the metaphysical status of abstraction and concreteness while dissolving their binary opposition. It is important to note that there are two possible approaches for interpreting an abstractive hierarchy. The differences between these approaches stem from the question of whether abstraction and complexity should be considered identical concepts. Based on their respective understandings, the abstractive hierarchy will exhibit different structural features, as well as varying positions for the concepts of abstraction and eternal objects within Whitehead’s metaphysical thought.

1. Introduction

Alfred North Whitehead’s early philosophical reflections were a response to the crisis of abstraction that modern science was facing. He succinctly labels this issue the fallacy of misplaced concreteness, “the accidental error of mistaking the abstract for the concrete” [1] (p. 52). This error is particularly evident in the ontological debates about the substance–attribute division. According to Rene Descartes, substance exists independently of any other entity. He also suggests that each substance has a singular attribute that forms its essence with all other attributes being contingent upon it. John Locke makes a distinction between the primary and secondary qualities: primary qualities are inherent properties like extension and mass, whereas secondary qualities are related to sensations or ideas generated through the body’s interaction with sensory faculties, such as smell and color. Descartes and Locke simplify the intricate fabric of actual existents into abstract entities or qualities and then promote these abstractions to the status of concrete realities. This misplacement leads to the dualistic tenor that came to characterize modern philosophy: the primary attribute of the mind is considered to be thought, whereas that of the body is extensional.

Whitehead’s substitution of “events” for the static, isolated, and passive category of “substance” aims to reconstruct the metaphysical foundations of science, which precipitates the central task of his event theory: “harmonising them by assigning to them their right relative status as abstractions” [1] (p. 88). In the chapters “Abstraction” and “God” of his seminal work, Science and the Modern World, Whitehead proposes the idea of abstractive hierarchy to give an account of an actual occasion in terms of its connection with the realm of eternal objects. This part of Whitehead’s philosophy is crucial to understanding his later cosmology, making it “the first chapter of metaphysics”. However, the abstruseness of it prompted R.M. Martin, Whitehead’s student and analytical philosopher, to comment that “this difficult chapter is usually neglected by commentators, and some of Whitehead’s ablest followers say that they have not understood it” [2] (p. 66).

Martin attempts to reconstruct Whitehead’s concept of abstractive hierarchy through type theory, aiming to rebuild the content of the “Abstraction” chapter. This strategy elucidates the hierarchical and analytical nature of the realm of eternal objects but fails to incorporate the concept of relational essence. A more fundamental issue is that Martin’s approach implicitly defines abstraction as an intellectual activity. This is perplexing, as Science and the Modern World largely critiques the “the fallacy of misplaced concreteness”. Why would Whitehead himself revert to a position that dichotomizes the knower and the known, phenomenon and essence, to envision a purely intellectual realm? Similar issues appear in more recent research, expressed even more explicitly. Van Haeften points out that “the chapter (referring to the ‘Abstraction’ chapter) is mainly about the intellectual activities of the mind” [3], and “Abstraction is a procedure of mind, eternal objects are the time-and-again recognized entities of awareness” [4]. The abstractive hierarchy is, then, “able to make progressive mathematical abstractions, by which we get (in thought) to higher and higher grades of complexity”. In this approach, eternal objects become passive objects of cognition. This is clearly incompatible with Whitehead’s metaphysical system, as it fails to address how it is possible for eternal objects to have ingression into actual occasions. The abstraction in the realm of possibility is characterized by togetherness; yet, when discussing the hierarchy of abstraction, this togetherness with concrete entities seems to be overlooked.

The reason why abstraction is regarded as a rational “operation” [4] largely stems from the neglect of the potential inconsistency between Whitehead’s two terms, “abstraction” and “complexity”. Whitehead’s formulation easily leads one to equate these two terms; if this is carried out, the structure presented by the abstractive hierarchy will place actual entities at the lowest level of the hierarchy. This results in the entire realm of possibility being seen as “knowledge” abstracted from actual entities, where “the endpoint of the hierarchy is an infinitely complex eternal object that constitutes the form of the occasion’s togetherness” [5]. The current approach is clearly unsatisfactory. There are some clues pointing in another direction, collectively forming strong evidence that the connotations of terms such as abstraction, eternal objects, and abstractive hierarchy are not as straightforward. The central task of this paper is to re-establish Whitehead’s fundamental position when he wrote Chapter 10 (and Chapter 11): an eternal object is not a dispensable epistemological concept but rather a crucial metaphysical concept; it does not passively become known but enters the actual occasion as an element of the flow of process. The key lies in the structural reconstruction of abstractive hierarchy, based on a further exploration of the concepts of abstraction and complexity.

Section 2 organizes Whitehead’s relevant discussions and provides a basic explanation of the involved terms and postulations. Section 3 addresses Martin’s treatment of abstractive hierarchy, which may be the only systematic reduction attempt made to date. Martin’s work clearly illustrates the structure of abstractive hierarchy, particularly if the distinctions between the concepts of abstraction and complexity are not made. Section 4 focuses on the connotations of abstraction and complexity, as well as Whitehead’s use of them, to show that the two concepts must be viewed as independent. Section 5 reconstructs the structure of abstractive hierarchy. Based on a re-understanding of abstraction and complexity, it discusses abstraction from possibility and abstraction from actuality, along with the hierarchical structures each presents. Section 6 summarizes and discusses why my interpretative approach is more advantageous than the previous one, as well as how this explanatory framework may alter our understanding of some of Whitehead’s statements.

2. Some Metaphysical Postulates of Abstractive Hierarchy

In the chapter “Abstraction”, Whitehead introduces several important concepts and related postulations. They are referred to as postulations because Whitehead employs a purely assertive manner in making these determinations. The most important concept in this chapter is “eternal object”. In Whitehead’s words, “Eternal objects are thus, in their nature, abstract. By ‘abstract’ I mean that what an eternal object is in itself—that is to say, its essence—is comprehensible without reference to some one particular occasion of experience” [1] (p. 159). The ideas of being abstract or universal can be traced back to the inquiries of ancient Greek philosophers into the word “essence”. Aristotle’s strategy, in the pursuit of unchanging essences regarding the classification into species and genera, initially advanced human intellect but ultimately distorted “the true vision of the metaphysical situation” [1] (p. 170). To avoid the essentialist implications of traditional terminologies, Whitehead introduces the concept of eternal objects as a substitute. “Abstract” is thus reformulated as a constitutive concept within his system, no longer denoting intellectual activities aimed at extracting, generalizing, or isolating so-called essences from concrete multiplicities.

Eternal objects implicate two metaphysical postulates. One may be called the principle of specificity; as Whitehead says, “Each eternal object is an individual which, in its own peculiar fashion, is what it is” [1] (p. 159). Whitehead points out that the immutable specificity of an eternal object constitutes its individual essence. Eternal objects represent the absolute specificity of the abstract while maintaining a relational connection to actual occasions,1 whose connection is described by the term “prehension”. An actual occasion α prehends every eternal object with the gradation of prehension inherent to α. This prehension of eternal objects by actual occasions is an aesthetic synthesis, a creative unification generating an evolving organism and novel values. The degree of prehension of an eternal object is contingent upon the actual occasion synthesized within it in various modes.2 The highest grade of prehension incorporates the individual essence of an external object as a factor in the aesthetic synthesis, whereas the lowest grade excludes it. Color mixing, where different proportions of primary colors produce various hues, could be a quotidian example of prehension. By proportionally adjusting primary colors, myriad hues can be created. For instance, cyan results from a 0:1:1 ratio of red, green, and blue, entirely excluding the individual essence of the eternal object red in the aesthetic synthesis; notably, however, the actual occasion cyan still prehends red.

The second metaphysical postulate concerning eternal objects is the principle of relatedness. As Whitehead states, “An eternal object, considered as an abstract entity, cannot be divorced from its reference to other eternal objects, and from its reference to actuality generally” [1] (p. 160). The principle of relatedness initially appears to conflict with the principle of specificity; how can the individual specificity of eternal objects be conceived if they are inherently bound to their relations to others? Whitehead addresses this apparent contradiction by introducing the concept of relational essence, a cornerstone of his philosophy of organism. The relational essence posits that the relationships between eternal object A and other eternal objects are intrinsic to A’s essence. The relational essence implies that A’s relationship with actual occasions is nothing more than the pattern or proportion in which A appears in conjunction with other eternal objects within that occasion; in other words, the principle of specificity does not entail the substantial independence of eternal objects. The principle of relatedness, serving as a general principle for the prehension of eternal object A into actual occasions, provides the foundation for all possible relational modes between A and all other eternal objects, thereby signifying a potentiality of actualization. Consequently, Whitehead designates the set of eternal objects as the realm of possibility, wherein each eternal object occupies a unique position within this universally interconnected systemic complex. Notably, although the realm of possibility encompasses every conceivable relational mode, it is not an extrajudicial domain. The general relationships among eternal objects are subject to at least one constraint in that they must include a selection of spatio-temporal relations and involve the status of actual occasions within the same spatio-temporal relations. The spatio-temporal continuum is the locus of relational possibilities wherein only eternal objects with determinate positions can be prehended and expressed in actual occasions.

The relations among eternal objects are intrinsic with each eternal object relying on the anchoring provided by all others. This seemingly precludes the epistemic possibility of knowledge unless one can simultaneously apprehend all eternal objects and their interrelations. Such omniscience, ostensibly the province of divinity alone, negates finite truths and renders knowledge unattainable for humanity. To reconcile the apparent incompatibility between finite truths and intrinsic relations, Whitehead posits the analytic character of the realm of eternal objects by stating the following: “The status of any eternal object A in this realm is capable of analysis into an indefinite number, of subordinate relationships of limited scope” [1] (p. 164). The analytic character of the realm of possibility makes finite truths possible, the rationale for which is likewise inherent in the concept of relational essence wherein the relational essence of each eternal object determines the complete unified system of relational essences because each object is an intrinsic part of the system in all its potential relations. Relational essence is neither unique nor exclusive to any eternal object; each eternal object’s relational essence determines or constrains a complete, unified system of relational essences. Whitehead posits it as the rationale for logical variables, with relational essence presupposing a set of eternal objects that can serve as the range of values for these variables, expressible without reference to their individual essences. In other words, relational essence is not merely an intrinsic attribute constituting the essence of eternal objects but also an inherent determination constituting the essence of the general relational system among eternal objects. Whitehead’s inspiration derives from Leibniz’s concept of perspective, in which each monad (fundamental individual) contains a perspectival view or holographic landscape of the entire universe, which Whitehead employs to elucidate the analytic character of the realm of eternal objects.

The principles of specificity and relatedness coupled with the analytic character of the realm of eternal objects collectively settle the theoretical foundation for the abstractive hierarchy. A finite set of eternal objects in a particular mode (finite relation) constitutes an eternal object, which is complex. The eternal objects that make up a complex eternal object are referred to as the “components” of that eternal object, and if any of these eternal objects are themselves complex, their constituent parts are referred to as the “derivative components” of the original complex object. As an essential term concerning eternal objects, “complexity” refers to its analyzability into a relationship of component eternal objects. When a complex can no longer be further analyzed, it is called a “simple eternal object”. A set of simple eternal objects with zero complexity constitutes the lowest complexity of the abstractive hierarchy.

Whitehead defines the abstractive hierarchy as follows [1] (p. 168).

There is “an abstractive hierarchy based upon g”, where g is a group of simple eternal objects, and this set of eternal objects satisfies the following conditions:

(i)

The members of g belong to it and are the only simple eternal objects in the hierarchy;

(ii)

The components of any complex eternal object in the hierarchy are also members of the hierarchy; and

(iii)

Any set of eternal objects belonging to the hierarchy, whether all of the same grade or, whether differing among themselves as to grade, are jointly among the components or derivative components of at least one eternal object which also belongs to the hierarchy.

Whitehead further stipulates that the complexity grade of an eternal object’s constituents must be lower than that of the eternal object itself. Accordingly, any member of such a hierarchy, which is of the first grade of complexity, can have as components only members of the group g, and any member of the second grade can have as components only members of the first grade and members of g, and so on for the higher grades. The abstractive hierarchy may be finite or infinite. In the former case, the system will possess a grade of the maximum complexity, which is characteristic of this grade where a member of it is a component of no other eternal object belonging to any grade of the hierarchy. The abstractive hierarchy is infinite if no grade of the maximum complexity meeting the aforementioned criteria can be identified.

3. Martin’s Approach

Whitehead’s overly concise explication of the abstractive hierarchy inevitably gives rise to significant interpretative difficulties. Martin considers that, whereas conditions (i) and (ii) in the definition of the abstractive hierarchy are lucid, condition (iii), serving as the condition of connexity, is articulated unsatisfactorily. This inadequacy stems from both the ambiguity of the language employed and the fact that (iii) precludes the possibility of a finite abstractive hierarchy. Roberts elucidates its contradictory nature by arguing that when one posits a finite abstractive hierarchy with a grade of maximum complexity, the condition of connexity necessitates that members of this highest grade must be components of other eternal objects in the hierarchy. This, however, would require a grade of even higher complexity, contradicting the initial assumption of a maximum grade, thus demonstrating the impossibility of a finite abstractive hierarchy [6]. The condition of connexity continuously demands an eternal object of a higher grade composed of or derivatively composed of members of the current set, thereby rendering a finite abstractive hierarchy impossible. Consequently, Martin reformulates the condition of connexity (iii) as a disjunction of propositions (iii)′ and (iii)″:

(iii)′ Given any set of eternal objects belonging to the hierarchy, whether all of the same grade or whether differing among themselves as to grade, every member of that set is a component or a derivative component of at least one member of the hierarchy.

(iii)″ There exists at least one eternal object belonging to the hierarchy containing as components all other eternal objects belonging to the hierarchy, whether all of the same grade or whether differing among themselves as to grade.

In light of the modified condition of connexity, Martin endeavored to reformulate Whitehead’s definition of the abstractive hierarchy employing type theory. Consider an abstractive hierarchy H predicated upon g of type 1; g is a class of simple eternal objects. An m-adic relation ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type n is a vertex of the abstractive hierarchy H if and only if it satisfies the following conditions:

(1)

${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ contains at least one relatum of type 2, one of type 3, …, and one of type n-I;

(2)

Every member of g is a relatum of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$;

(3)

Every relatum of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type 3 is a class of or relation between or among members of g;

(4)

Every relatum of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type 4 is a class of or a relation between or among relata of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type 3;

. . .

(n − 1) Every relatum of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type n-l is a class of or a relation between or among relata of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type n − 2.

The abstractive hierarchy H based on g with vertex ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ is, then, a class of type n + 1 whose members are the following:

(1)

${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ itself;

(2)

Given any component β of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type 3, the unit class $\stackrel{n-3\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\overbrace{\{\dots \{\{}}$β$\stackrel{n-3\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\overbrace{\left\}\right\}\dots \}}}$ is a member of H;

(3)

Given any component β of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type 4, the unit class $\stackrel{n-4\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\overbrace{\{\dots \{\{}}$β$\stackrel{n-4\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\overbrace{\left\}\right\}\dots \}}}$ is a member of H;

. . .

(n − 2) given any component β of ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ of type n-l, the unit class {β} is a member of H.

The definitions above pertain to finite hierarchies; Martin posited that extending these definitions to infinite hierarchical structures necessitates the employment of transfinite types.

Martin’s reformulation in type theory largely arises from his perception of an evident consonance between the abstractive hierarchy and type theory coupled with his assertion that type theory was “the only logic Whitehead knew” [2] (p. 75). It seems that Martin’s attempt to apply type theory appears somewhat tenuous and contradicts aspects of Whitehead, particularly his explicit stipulation that “The base of an abstractive hierarchy may contain any number of members, finite or infinite. Further, the infinity of the number of the members of the base has nothing to do with the question as to whether the hierarchy be finite or infinite” [1] (169). According to Martin’s definition in type theory, a set g with infinite members precludes the conceivability of a vertex ${\mathrm{V}}_{\mathrm{g}}^{\mathrm{m}}$ for a finite abstractive hierarchy, a conclusion clearly at odds with Whitehead’s articulation. Upon further examination, it becomes evident that Martin’s predicament extends beyond this contradiction.

4. Abstraction and Complexity

The key to understanding the idea of abstractive hierarchy lies in the terms “abstraction” and “complexity”. In Whitehead’s text, the categories of complexity and abstraction are frequently juxtaposed, seemingly indicating congruence in their extension and intention. This alignment finds support in some of Whitehead’s assertions, such as “we pass from the grade of simple eternal objects to higher and higher grades of complexity, we are indulging in higher grades of abstraction from the realm of possibility” [1] (p. 168). By substituting the eternal object for the traditional philosophical concepts of abstraction, Whitehead likely aims to reformulate abstraction through the lens of complexity, thereby allowing for a self-contained definition of abstraction within the abstractive hierarchy independent of external concepts. An example Whitehead employs to elucidate abstraction may corroborate this assertion. Suppose A and R(A, B, C) are eternal objects in the realm of possibility, with A being a constituent of R. At the level of R(A, B, C), all possible relations of A, except R, are evidently excluded, leaving only the determinate relation R. Thus, one might argue that A is more abstract in the more complex R(A, B, C). Whitehead’s example is highly suggestive, and he never explicitly distinguishes between complexity and abstraction in his text. The aforementioned signals have led many scholars, including Martin, to consider “abstraction” and “complexity” as different expressions of the same referent within the abstractive hierarchy.

Equating abstraction with complexity would situate actual occasions at the lowest grade of the associated hierarchy, a counterintuitive notion that also contradicts the definition of “eternal objects”. One definitive feature derivable from the definition of any abstractive hierarchy, whether situated in the realm of possibility or associated with actual occasions, is that simple eternal objects invariably occupy the lowest grade of complexity. Despite this difficulty, Martin posits a structure for the associated hierarchy based on abstraction–complexity equivalence:

Presumably the individuals, or entities of lowest or first type, are actual occasions and only such. The simple eternal objects (of zero grade but second type) then comprise the classes of and relations between or among actual occasions. Complex eternal objects of first grade (but third type) are classes of or relations between or among such classes and relations (and perhaps actual occasions also). And so on for all higher types and grades.

[2] (p. 69)

In Martin’s system, which structurally inverts the associated hierarchy, actual occasions reside outside any grade of complexity and can only be situated at the base of the abstractive pyramid, thus presenting how eternal objects are abstracted from actual occasions rather than how eternal objects are prehended and synthesized into actual occasions.

The structural difficulties in the associated hierarchy necessitate a reconsideration of the potential incongruence between abstraction and complexity. Relevant terminological definitions support this reassessment. “Complexity” is intentionally equivalent to “analyzability” because “the complexity of an eternal object means its analyzability into a relationship of component eternal objects” [1] (p. 167). This concept, dependent on the abstractive hierarchy, describes a grade’s position within the system. Conversely, “abstraction” is intrinsic to the definition of eternal objects because “to be abstract is to transcend particular concrete occasions of actual happening” [1] (p. 159). Whitehead conceives it as the conceptual counterpart to concrete, typically appearing as a predicate of eternal objects, and being complete in itself without a dependence on the abstractive hierarchy. One can posit that these concepts differ intentionally, and their extensions do not entirely coincide.

The following provides the most compelling defense for the distinction between complexity and abstraction: “abstraction from possibility runs in the opposite direction to an abstraction from actuality. Accordingly the simple eternal objects represent the extreme abstraction from an actual occasion; whereas simple eternal objects represent the minimum of abstraction from the realm of possibility” [1] (p. 171). Substituting complexity for abstraction in this proposition yields the contradictory statement that simple eternal objects represent the extreme complexity from an actual occasion, which is clearly inconsistent with previous assertions because simple eternal objects are not even complex. Therefore, the distinction between abstraction and complexity must be conceived as feasible; otherwise, Whitehead’s text becomes incomprehensible.

5. Abstraction from Possibility and Abstraction from Actuality

Whitehead distinguishes between “abstraction from possibility” and “abstraction from actuality”, asserting that the two run in opposite directions. Considering the distinction between abstraction and complexity, there is reason to believe that abstractive hierarchy is present in two structural forms. One form is the associated hierarchy arising from actual occasions wherein abstraction and complexity exhibit an inverse relationship. Actual occasion α creatively synthesizes new occasions toward the more complex end, manifesting the infinity of the associated hierarchy, whereas at the more abstract end, α ultimately traces back to the set of simple eternal objects. The other form is the abstractive hierarchy that solely involves eternal objects, in which abstraction and complexity exhibit a synchronous, positive correlation. Whitehead states, “we pass from the grade of simple eternal objects to higher and higher grades of complexity, we are indulging in higher grades of abstraction from the realm of possibility” [1] (p. 168). Only in the latter case does the abstractive hierarchy approximate the structural form defined by Martin. At this point, abstraction is, as Van Haeften noted, merely an intellectual activity.

Associated hierarchy is a form of abstractive hierarchy arising from actual occasions. Whitehead posits that the associated hierarchy is infinite, a characteristic determined by the nature of actual occasions. He considers an actual occasion as “a becomingness” [1] (p. 176), which he introduced in the “God” chapter. An actual occasion is a process or a creative synthesis prehending all eternal objects in a specific mode, and it is the associated hierarchy that determines the mode of prehension. Whitehead states, “Eternal objects inform actual occasions with hierarchic patterns, included and excluded in every variety of discrimination” [1] (p. 174). In other words, an actual occasion represents a constraint imposed upon possibility. Given a set of simple eternal objects g (A, B, C, etc.), the modes of prehending the eternal objects of g are innumerable. This means an actual occasion α can be expressed as R(A, B), which has already excluded all other possibilities, actualizing in a certain mode R.

An actual occasion, as a continuous, becoming process, is not isolated; rather, its position and potential connections with other occasions can be discerned within the associated hierarchy. The internal connections between an actual occasion α and others can be categorized in alternative ways with different definitions of past, present, and future. For instance, α may be further prehended into an actual occasion α′ in a higher grade, where α is an actual occasion in the past serving merely as a complex eternal object. α′ will, in turn, synthesize other occasions as “future” and be prehended into α″. The process of becoming and synthesis toward the future is infinite, consequently rendering the associated hierarchy infinite.

In associated hierarchy, the lowest grade of complexity comprises a set of simple eternal objects, representing the highest degree of abstraction. The second grade of complexity consists of complexes derived from the prehension of first-grade eternal objects having the full potentiality to actualize as an actual occasion or event with a slightly diminished degree of abstraction compared to simple eternal objects. The third grade of complexity is synthesized with second-grade actual occasions (possibly in conjunction with simple eternal objects) serving as the “future” of second-grade actual occasions, and so forth.

Whitehead argues that actual occasions, in their full concreteness, transcend the limitations of conceptual abstraction and thus cannot be fully expressed through concepts. Although concepts encompassing memory, anticipation, imagination, and thought constitute elements within an experient occasion, they represent a specific mode of inclusion characterized by abruptness. Whitehead describes abruptness as being exhausted by a finite complex concept [1] (p. 172), which serves as the vertex of the hierarchy. There is a limitation that breaks off the finite concept from the higher grades of illimitable complexity. Consequently, finite concepts prove inadequate in fully capturing the infinite process of becomingness that fundamentally constitutes the ontological nature of actual occasions. Whitehead addresses this abruptness as the distinguishing feature between the finite concepts in the mental and infinite complexity of actual occasions. The same reasoning applies to the abstractive hierarchy that solely involves eternal objects; it can only be finite.

This approach to abstractive hierarchy aligns more closely with Whitehead’s metaphysical intentions in all aspects. He aims to elucidate the prehension of eternal objects by actual occasions through associated hierarchy, thereby dissolving the binary opposition between abstraction and concreteness. The relationship between actual occasions and complex eternal objects is dialectical. In associated hierarchy, the distinction between the two lies solely in the abruptness of the latter, which detaches from the infinite complexity of becoming and yet becoming. The associated hierarchy represents the reconciliation of abstractness and concreteness, illustrating how actual occasions can be regulated from the perspective of possibility and how possibility can be regulated from actual occasions.

In addition, Whitehead’s concepts of becomingness and abruptness reaffirm the completeness of abstractive hierarchy. The infinite nature of the abstractive hierarchy associated with actual occasions, which is determined solely by the inherent becomingness, is independent of the cardinality of the base set g. The notions of becomingness and abruptness sufficiently articulate the finite or infinite nature of the abstractive hierarchy, aligning seamlessly with Whitehead’s original text. In this regard, it becomes apparent that modifying the condition of connexity (iii) in the definition of abstractive hierarchy or introducing type theory as a foundational basis may be dispensable.

6. Conclusions and Discussion

In Martin’s approach, Whitehead’s theory of the abstractive hierarchy is not regarded as an integral part of his metaphysical thought. Martin’s analysis and definitions suggest that he views Whitehead’s work as merely constructing a hierarchical system of abstract concepts. Subsequent research has still not exceeded the scope of Martin, consistently viewing abstraction and abstractive hierarchy as merely concerning “knowledge”. But there is no motivation for Whitehead to envision a binary oppositional abstractive hierarchy where eternal objects devolve into absolute abstractions in the realm of possibility, while actual occasions, relegated to the lowest grade of the associated hierarchy, revert to static, isolated entities. This leads to inconceivable fissures in Whitehead’s theory. Whereas he initially conceives abstractions as combined with concreteness, he summarily dismisses this combinatorial nature when discussing abstractive hierarchy. Existing research may potentially misapprehend Whitehead’s theoretical aspirations.

I assert that abstractive hierarchy is an integral component of Whitehead’s metaphysical system. Based on the distinction between abstraction and complexity, the divergent characteristics of the abstractive hierarchy that solely involves eternal objects and the associated hierarchy arising from actual occasions can be clarified. The structure of associated hierarchy indicates that eternal objects are not merely concerned with the mind or knowledge, but are involved in the becomingness of actual occasions. It is essential to remain vigilant that the ‘fallacy of misplaced concreteness’ is a central concern of Whitehead’s philosophy, and it runs through Whitehead’s Science and the Modern World and Process and Reality. In the context below, he explicitly states that two descriptions are required for an actual entity: one of its potentiality for objectification in other entities’ becoming and another of its own becoming process, both of which are analytical. Whitehead further elaborates “that the first analysis of an actual entity, into its most concrete elements, discloses it to be a concrescence of prehensions, which have originated in its process of becoming” [7] (p. 23). These discussions provide textual support for my argument. “Analytical” is not only a property of complex eternal objects but also a characteristic of actual entities, implying that actual entities cannot occupy the lowest grade of the abstractive hierarchy in any case.

A potential difficulty for my argument is that Whitehead never mentions abstractive hierarchy in Process and Reality; it appears he has abandoned this concept. How can the disappearance of this important concept be explained? One possible explanation is that the abstractive hierarchy has already been incorporated into the notion of extensive continuum. In the corresponding chapter, he states the following: “All actual entities are related according to the determinations of this continuum, and all possible actual entities in the future must exemplify these determinations in their relations with the already actual world. The reality of the future is bound up with the reality of this continuum” [7] (p. 66). Considering that Whitehead does not discuss the abstractive hierarchy solely concerning eternal objects in Process and Reality, the more refined “extensive continuum” replaces abstractive hierarchy and provides similar functionality. The extensive continuum should be conceived as a structure akin to the associated hierarchy, enabling it to elucidate how actual entities are intimately connected within the world.

The new interpretation of abstractive hierarchy may pose challenges to the study of Whitehead’s thought. An example concerns the realm of possibility. Van Haeften argues that the realm of possibility “is dependent on the reality of the temporal world” [3]. Meanwhile, Florian Vermeiren defends the position of “the identification of the extensive continuum with the realm of eternal objects in Science and the Modern World” [5]. From the idea of abstractive hierarchy, I argue that only eternal objects with defined positions in the spatio-temporal continuum can be prehended and synthesized in actual occasions. The extensive continuum concerns ‘actualization’, but it does not imply a limitation on the entire realm of possibility; otherwise, imagination would not be possible. The reason for this divergence lies in the view that the realm of possibility is merely an intellectual abstraction, while the extensive continuum is considered prior; thus, the latter is seen as a limitation on the realm of possibility. Another example concerns the discussion of eternal objects. Van Haeften argues that “eternal objects are abstract in the sense of being able to return in other ingressions” and “their being is their possible recognition in natural experience” [4]. I do not consider eternal objects to “return in other ingressions” as objects of recognition; rather, they are parts of ingression. Abstraction is not “full of peril” [3]; rather, “the approach to intellectuality consists in the gain of a power of abstraction” [7] (p. 254). What Whitehead cautions against is the fallacy of misplaced concreteness, he continually seeks to eliminate the binary opposition between abstraction and concreteness that arises from the fallacy.

Similar divergences can be found in discussions about the nature of relations, connectivity, and other concepts. These issues are intricate and far-reaching, beyond the scope of this article. What the latter aims to do is to start from the abstractive hierarchy, providing a coherent and reasonable explanatory framework for Whitehead’s philosophy—particularly the chapters “Abstraction” and “God” in Science and the Modern World.