The Whole Part

Chapter 1
Introduction

A book about wholes, parts, and references. This book examines the world, both our experience of it and our thoughts about it, while focusing on the relations between wholes and parts. It examines our concepts, both how they are shaped by the world, and how they shape the world. These examinations use two analytic techniques: mereology (or analysis in terms of parts and wholes) and reference (or analysis in terms of references).

[Figure 1.1: A whole and a part that is a reference
to that whole.]

To illustrate these mereological and referential relationships diagrammatically, various material and mental things are represented with ellipses and arrows. For example, Figure 1.1 shows that:

• Wholes are composed of parts (via diamond arrows).

• Parts may be references (via triangle arrows).

• Parts may referentially contain their wholes (in virtue of both arrows).

More detail about the notational and diagrammatic conventions used in this book, and UML diagrams in particular, is presented in appendix Ideographic Conventions. The main content of this book is presented in four parts:

Part 1 introduces the book and provides a brief philosophical background of the main topics.

Part 2, entitled Theory of Mereology and Reference, is a short introduction to spaces and the dimensionality of spaces, particularly as they are structured by wholes, parts, and references. Wholes and parts form mereological spaces; wholes are formed by uniting parts, and parts are formed by dividing wholes. References and referents form referential spaces; references are formed by representing their referents, and referents are formed by being represented. Finally, there is a section introducing identity conditions for things within these different spaces and a section examining various edge cases.

Part 3, entitled The Basic Model, applies the theory of mereology and reference to human experience, thereby establishing the basic model of human cognition. Three spaces are constructed, which can be distinguished by their referential or epistemic level: physical space, subjective space, and conceptual space. Six relations between these spaces are also introduced: sensation, action, conceptualization, visualization, interpretation, and symbolization. In order to thoroughly understand these spaces, they are analyzed in terms of their parts and dimensionality.¹

Part 4, entitled Practical Implications, is an informal look at various popular topics and paradoxes, cast within a mereological and referential framework. It explores questions such as: how does thought interfere with the perception of one’s surroundings? What is the role of emotion with respect to concepts? Is consciousness a property of only one type of epistemic space, or are there different types of consciousness associated with different spaces? Although no attempt is made to answer these questions definitively, the formal framework introduced in the earlier chapters provides a clear way to analyze them.

Chapter 2
History

A brief history of the relation of wholes, parts,
and references to the study of mind. As this work presents a basic model of mind, its focus is epistemological rather than ontological. In practice, however, the determination of which facets of experience belong to mind as opposed to external reality is a matter of some debate. This chapter is a brief history of that debate, whose scope is the psychological and philosophical background within the Western and Indian traditions.

The science of wholes and parts (and to a lesser extent, references) in both traditions has a long history, going back as far as Plato in Greece (400 BCE) or the Vedas in India (3500 BCE). This science is closely related to the study of one and many, since the numeric relationship of a whole to its parts is one to many. Wholes, parts, and references are also related to the study of universals and particulars, and absolute and relative truths, although the precise relationship in these cases is more complex. The study of references in particular is relevant to understanding several important epistemic distinctions such as that between matter and mind, and between the intuitive and rational aspects of mind.

Chapter 3
Mereological Space

A mereological space is structured with wholes and parts. Mereological spaces are structured by the whole/part relationship. Both physical spaces and sensory spaces can be structured mereologically; for example, a tree is a part of a forest and the sensation of a tree is a part of the sensation of a forest. However, the relation between a referent and its reference is not mereological. In other words, symbols are not necessarily either parts or wholes of what they symbolize.

All parts of a mereological whole share the dimensionality of that whole. For example, there are no 2‑D surfaces in a 3‑D world, and there are no 3‑D objects in a 4‑D world; therefore, it is not correct to conceive of a 2‑D line existing in a 3‑D space from a mereological point of view (although it may be a useful approximation or an interesting epistemological limit). This means that abstract elements such as points, lines, and planes are incapable of constituting physical space. In other words, parts have a spatial extent along every dimension of the space that contains them, just as a piece of paper has more than two dimensions.

Further, for any two entities A and B in a mereological space, one of the following relations holds:²⁰

1. Neither is a part of the other (i.e., the entities are disjoint).

2. A is a part of B (i.e., B is a whole of A).

3. B is a part of A (i.e., A is a whole of B).

4. Both are parts of each other (i.e., the things are identical).

5. There is no definite parthood relationship between the entities (i.e., A is neither a part of B, nor is A not a part of B).

These relations are illustrated graphically in Figure 2.3a. The fifth case is somewhat problematic, because there is no simple answer to the question “Is A a part of B?”. To state that either “A is a part of B” or “A is not a part of B” would be incorrect: one part of A is a part of B and another part of A is not a part of B. It is tempting to say that “A is a part of B” AND “A is not a part of B”, but expressing that is beyond the capacity of ordinary (bivalent) logic.

[Figure 2.3a: The five possible mereological relations.]

[Figure 2.3b: Recategorizing overlap
as three non-overlapping parts.]

In order to ensure that the part operation imposes at least a partial order on a given space, overlap must be prevented by categorizing any overlapping space as a new entity, as in Figure 2.3b.²¹ Epistemologically, however, it may make more sense to extend logic so that it accurately describes uncertainty, rather than removing all overlap (which may be impossible in some cases).²² To illustrate this, a continuous version of logic is introduced in appendix Formal Summary.

Chapter 4
Referential Space

A referential space is structured with
references and referents. Referential spaces are structured by the reference/referent relationship. The things that are named are called referents, and the things that name them are called references. References are directional, in that references point to their referents.

[Figure 2.4: References stratify things
according to their referential level.]

References are not mereological because the relationship between a reference and its referent has nothing to do with whether they are parts of one another, although references and the referents on which they depend often form parts of a mereological space. References are discrete, although the referential relation may in general be continuous (in which case it is called reflection).

Referential space itself is typically divided into subspaces of referenced objects and referencing objects. Because references can in turn be referents, a stratification of referential level is established. As Figure 2.4 illustrates, references form referential chains (depicted vertically) that may pass through multiple referential levels (depicted horizontally), mapping onto referents in lower levels. Things on the bottom level of the diagram have a referential level of zero and are not references to anything else (they are “things in themselves”). The referential level of each successive level is one more than the level to which it refers. Therefore, referential level indicates the distance of a reference from ground, or its ultimate object of reference.

Chapter 5
Combining Spaces

The combination of mereological and referential spaces yields spaces unlike either one. Most of this book uses hierarchies to structure spaces because of their relative simplicity. Hierarchies are a structure whose nodes are described using the metaphor of a family tree: parents are depicted above children and siblings are depicted next to one another. Hierarchy can also be described using the tree metaphor, although the trees are typically depicted as branching downwards instead of upwards, such that parent and child nodes are called “root” and “branch” nodes.

Different types of hierarchies use different structuring elements as relations. This chapter examines the two different types of hierarchies that correspond most closely to mereological and referential spaces: meronomies and taxonomies.

Chapter 6
Epistemic Universes

A presentation of three epistemic universes. The next three chapters examine the physical, subjective, and conceptual spaces consecutively. Since each space is boundless from its own point of view, they are also referred to as universes. For example, the subjective universe is a complete whole or totality in so far as all things can be experienced, and a given individual can never have experience outside of it. Similarly, conceptual space is also a universe in that all things can be conceptualized.⁴³ That said, they are universes determined by the subjective perspective, so things that form a subjective universe from one point of view do not do so from another.

For clarity, it is important to explicitly specify the intended epistemic space of things that exist in multiple epistemic spaces. For example, the idea of an orange, the sensation of an orange, and the orange itself must be carefully distinguished from each other. Therefore, this book follows a convention for referring to the different types of entities as that is depicted in the taxonomy of Figure 3.6a, where the root node (a thing) is a generic type that can be used in any epistemological space, and its subtypes are scoped to each subsequent epistemic space (which mirrors the spaces as depicted in Figure 3e).

[Figure 3.6a: The types of things within each epistemic space.]

[Figure 3.6b: The parts of objective, sensory, conceptual, and symbolic spaces are respectively known as
objects, sensations, concepts, and symbols.]

Epistemic types corresponding to four of those spaces are illustrated in Figure 3.6b.⁴⁴ The subscripts (or lack thereof) designate their associated entities as either objects, sensations, concepts, or symbols. For example, orange_φ refers to the concept of an orange (for more details, see appendix Typographical Conventions).

Chapter 7
Epistemic Relations

Six epistemic relations are defined between the objective, sensory, and conceptual spaces. This chapter looks at the six relations between the three epistemic universes of the basic model. These six relations may be summarized as follows with respect to the objective (O), sensory (N), and conceptual (C) spaces, and where symbolic space (V, a subspace of sensory space) is denoted explicitly:

• Sensation (Θ): Sensation is a causal function. It changes N as a result of O.

• Action (Δ): Action is a causal function. It changes O as a result of C.

• Conceptualization (Φ): Conceptualization is the whole function. It creates increasingly conceptual wholes as one moves from N to C.

• Visualization (Ψ): Visualization is the part function. It creates increasingly sensory parts as one moves from C to N.

• Interpretation (Ω): Interpretation is the referent function. It activates referents in C in virtue of symbolic references in V.

• Symbolization (Ξ): Symbolization is the reference function. It activates symbolic references in V in virtue of referents in C.

Chapter 8
Recursion

The interaction of subjective spaces and their relations creates higher‑order cognition and symbolic spaces. Symbols are references to concepts that form symbolic space. Symbols are specialized types of sensation: for example, the word wind_ξ is a part of our sensory space that becomes a symbol for the concept wind_φ. The combination of concepts that are composed of sensation and symbols that refer to concepts establishes the basis for a referential recursion between sensory and conceptual spaces that is illustrated in Figure 3.8a.

[Figure 3.8a: Symbolic space is the part of sensory space
that references conceptual space.]

Figure 3.8b depicts an unrolled version of this sensory/conceptual recursion. Each node’s label is annotated with a superscript indicating the conceptual order, and to the right of each node is a number that indicates the epistemic level.

[Figure 3.8b: Epistemic level indicates distance from ground.]

The left side of Figure 3.8b shows the approximate locations of three popular epistemic divisions, each of which traditionally corresponds to a mereological distinction rather than a purely referential distinction:

• Self/Other: The self/other distinction epistemically corresponds most closely to the distinction between sensations and objects.

• Body/Mind: The distinction between body and mind is approximately the division between nonconceptual and conceptual.

• Subsymbolic/Symbolic: The distinction between subsymbolic and symbolic mental content is made at the level of symbols. This is also the division between abstract universals and concrete particulars.

The loop that is formed by conceptualization (Φ) and symbolization (Ξ) allows human minds to create symbolic hierarchies. This loop enables recursive composition, whereby conceptual wholes are represented by symbols, and those symbols in turn form the basis for subsequent conceptual wholes. These hierarchies can be understood by reversing this process, which entails interpretation (Ω) and visualization (Ψ).

In linguistics, the hierarchies thus constructed correspond to the deep structure of language (see [Chomsky, 1995]). In mathematical set theory, there is an analogous structure called the Zermelo or von Neumann Hierarchy. Cognitively, the construction of higher-order concepts out of previously existing concepts is leveraged both long-term to define words as higher-order concepts and short-term to understand sentences that are dynamically-constructed.

The depth of these hierarchies can be quantified using the notion of epistemic level. Epistemic level is a measure of the number of transitions (or arrows) that must be traversed to reach ground, which has an epistemic level of zero by definition. Epistemic level therefore increases every time things are sensed, conceptualized, or symbolized, and decreases with every act, visualization, or interpretation.

Chapter 9
Concrete and Abstract

Concrete or zeroth-order concepts are wholes of sensation. Because they have the dimensionality of those sensations, they are concrete, open-dimensional events. Similarly, entire sentences that are formulas for constructing zeroth-order concepts result in concepts of the same dimensionality. In contrast, individual parts of speech represent abstract concepts and have only a portion of the dimensionality of sensory events, which is exactly what makes them universally applicable. For example, the color blue would not be able to color all the blue things if it had a particular spatial location.

As another example, take the sentence “She dances beautifully”. Most people probably accept that without a dancer of some sort, there can be no dance. Similarly, there can be no dancer without a dance; a dancer does, will do, or has done a dance. The co-occurrence of the dancer and the dance is a necessity; they constitute a concrete whole that corresponds to a complete sentence, that cannot be validly referenced using either a noun (the dancer) or a verb (to dance) exclusively.⁹⁴ More poetically, neither a dancer nor a dance has ever existed independently of the other.

As abstract parts of speech are combined, they create concepts that are increasingly concrete. For example, the phrase “she dances” exists more concretely than its subject (“she”) or its predicate (“dancing”); the entire sentence is more real, meaningful, or concrete than its noun or verb phrases individually.⁹⁵ The physical things that are designated by nouns are abstract concepts that lack a temporal dimension, and thus they cannot form a concrete part of a high-dimensional space (unless those nouns are implicitly or explicitly modified by verbs). Although nouns are meaningful in an abstract sense, their dimensionality is lower than that of physical space because they lack a temporal extent (nouns are abstract precisely because they generalize over the temporal dimension). Therefore, they are dimensionally incomplete, and only concrete concepts are completely meaningful as parts of a high-dimensional space.⁹⁶

The distinction between concrete and abstract can be understood by analogy: although the adjective “quick” has some meaning, it is easy to recognize as incomplete. It begs the questions, “Quick what? What is it that is quick?”. However, nouns such as “me” are also incomplete and should raise similar questions: “I did what?”. Further, nouns are not merely incomplete in terms of sentence structure, but in terms of corresponding to anything concrete⁹⁷ (presumably because material solidity is regarded as sufficient for existence under the classical (Newtonian) view of space).

The implications of incorrectly reifying nouns or other parts of speech are both practical and significant. Since desire affects things as we understand them, misunderstanding the nature of things renders us less capable of achieving our desires. For example, although we will be unhappy if we like “tasting sugar” but mistakenly seek “tasting salt”, we will probably be able to learn from our mistakes in a straightforward way. On the other hand, desiring the wrong type of things is a more subtle and pernicious problem. For example, if we like the taste of sugar but end up seeking sugar as a substance, we might collect sugar far beyond the time or capacity that we have to taste and enjoy it, a mistake that is considerably more difficult to detect and correct.

Chapter 10
Identity

The relation of identity is defined under the assumption that events are unique, so by default, no two events are identical. Similarly, all sensations are different: there is always a dimension along which any two sensations differ. However, when sensation is collected into concepts, the symbols that refer to those concepts enable generalization by higher-order concepts. In virtue of that generalization, differences between individual concepts are forgotten, and they become conceptually identical. This nominal identity is not entirely subjective since it captures aspects of truth about the world, but in virtue of forgetting, it does not capture the entire truth about the world.

Because nominal identity requires the association of a symbol and an object, it is invariably approximate. This inexact relationship often manifests in non-obvious ways, such as the paradox of the heap (see [Hyde & Raffman, 2018]):

If there is a heap of sand, and grains are removed one at a time, at what point is the heap no longer a heap?

While it is clear that a heap exists at one point and does not at another, it seems odd that a single grain of sand could make the difference between a heap and a non-heap. If a single grain of sand cannot make such a big difference, then heaps_φ and other concepts apply to their objects to varying degrees, which is problematic for anyone who believes that propositions about objects in the world can be fully true or false.⁹⁸

In virtue of generalization, the same object can be described in many different and valid ways. For example, the following large-scale and small-scale descriptions of an apple given by people in different lines of work illustrate this point:

• Sociologist: “The apple is a food-stuff which the proletariat can turn into cider and feed to the masses to keep them from revolting.”

• Psychologist: “The apple often stops the hunger neurons from firing, thus contributing to the cessation of the apple-gathering response.”

• Biologist: “The apple is a fruit, whose sweetness has been selected by evolution to provide for the disbursement and fertilization of the tree’s dicotyledonous progeny.”

• Chemist: “The apple is a complex of medium-chain, starchy hydrocarbons. It contains approximately twenty grams of fructose.”

• Physicist: “The apple contains primarily carbon, hydrogen, and oxygen. It warps spacetime in virtue of its mass.”

Each statement is only a partial truth with respect to the apple object. The variety of these statements demonstrates that the characterization of an apple depends on an observer’s perspective, and that none of these statements provides an exhaustive account. Even a single observer characterizes the apple differently at different times: the observer-when-hungry categorizes the apple as something to eat, while the observer-when-fed categorizes the apple as merely a fruit. These different levels of description are not inherently exclusive of each other, but only one formulation exists at one time when they are formulated symbolically. In virtue of that, emotions and thoughts often isolate a single story from all of the possible narratives that could be told about an apple.

Chapter 11
Absolute and Relative

Imagine a town in which “… all the children are above average”.⁹⁹ Although it is a pleasing image, such a town cannot exist; in order for someone to be above average, someone else must be below average. Nonetheless, it serves as an excellent and very explicit example of the relativity of properties.

The properties of an object, such as being average, are either relative or absolute because they are extrinsic (and inessential) or intrinsic (and essential) to that object. In other words, the relative properties of blueberries such as being something that grows on bushes, depend on the relationship of those blueberries to other things. Their absolute properties, such as high fructose content, depend on aspects of blueberry-matter and are theoretically independent of the relationship of blueberries to other things. However, the distinction between these two types of properties is not always clear; for example, the relative property “eaten by people” is related to the absolute property “fructose content”. Ultimately, it may not be possible to completely separate the absolute nature of a thing from all of its possible (relative) interactions.¹⁰⁰

If all properties are relative, however, things become meaningless. Therefore, one might conclude that only certain properties are absolute, or have a definable essence that is independent of other things. But which properties are absolute? Is the mass of a tree absolute, since it does not depend on the mass of non-tree things? However, the mass of a tree is defined in kilograms, and kilograms are in turn defined relative to the mass of a certain volume of water at sea level.¹⁰¹ This chain of reasoning seems to indicate that even abstract properties of an object such as its mass are relative to external objects.

The implicit resolution to this issue is that things are relative to one another in virtue of their wholes, while they are absolute in virtue of their parts. Even the expression of absolute properties such as mass is relative, however, because expression is necessarily symbolic. On the other hand, our sensory experience of heaviness is inexpressibly unique and experienced absolutely (or in a bottom-up sense).

The philosophy of reductionism makes this identification between parts and the absolute explicit by claiming that relative descriptions are exclusively external (or whole-based) and that absolute descriptions are exclusively internal (or part-based). Reductionism also claims that a thing is fully known when one knows its parts. For example, to understand people and their behavior, one must study psychology. To understand how psychology works, one must study physiology and the mechanism of the brain. To understand how physiology works, one must study biology and the mechanism of the neurons within the brain. To understand how biology works, one must study chemistry or physics and the mechanism of the molecules and quarks that make up the neurons, et cetera.

While there is no question that reductionistic analysis results in a very detailed explanation, it does not necessarily entail an increase in explanatory power. In other words, although a description that uses small parts is more detailed, it may be unnecessarily complicated. Similarly, the operation of the whole is not caused by the operation of parts any more than the operation of the parts is caused by the operation of the whole; causation requires that causes temporally precede their effects, which is not the case for spatial wholes and their parts.

Therefore, appeal to microscopic objects is not always necessary (or even beneficial), since movement at a macroscopic level can often be fully described in terms of macroscopic objects. For example, the neural or neuroanatomical description of a person is substantially less useful to most people than a physiological description, since we can more easily make physiological changes like getting exercise than direct changes to our chemical composition. Further, although the microscopic level does offer a more detailed analysis, it substitutes rational and relative thought for intuitive and absolute knowing, which is not always beneficial.

Chapter 12
Negation

Sensation derives meaning by referring to objects; that meaning is absolute in so far as it is not relative to other sensations. Similarly, the bottom-up aspect of concepts is defined by the sensations of which those concepts are composed. The top-down aspect of concepts, on the other hand, is defined relative to other concepts; concepts without sensory input rise to the foreground only when others recede to the background. As a result, concepts formed by a bottom-up process do not have a negation, whereas concepts formed by a top-down process do have a negation.¹⁰²

In other words, what is absolute is regarded as a whole; it is not a part, and therefore it is not relative to some larger thing. Because the absolute is whole and does not have a complement, therefore it has no negation. Similarly, the reason that conceptual parts do have a negation is that they are defined top-down, relative to a larger conceptual space. Specifically, they have complements with respect to the concepts that are their wholes, and those complements form their negations (i.e., with respect to those wholes). Thus, concepts in and of themselves are neither strictly relative or absolute: they are absolute in virtue of the sensation of which they are a whole, and they are relative in virtue of the larger concepts of which they are a part.

At the endpoints of the sensory–conceptual continuum are purely analytical entities that have no absolute aspect (or negative entities), and unanalyzed things that have no relative aspect (or positive entities). These two types of entities offer an excellent example of the separability of the relative and the absolute.¹⁰³

Negative entities such as a hole_φ are defined in relation or relative to other concepts, but they have no (absolute) sensory content of their own.¹⁰⁴ Therefore, holes are known only in relation to a larger context; for example, a hole in the ground is known only in virtue of being surrounded by earth.¹⁰⁵ At the other end of the spectrum are positive entities, which are purely sensory or absolute, and therefore do not have a negation. For example, “the negation of a tree” is not meaningful at a sensory level: it does not entail any other sensation, and it is not clear what the not-tree looks like. In other words, there is no such thing as a sensory opposite, since all sensation is positive appearance.¹⁰⁶

Most entities have both relative and absolute aspects; therefore, the relative aspect of an entity has a negation while the absolute aspect does not. From a subjective point of view, anything is absolute if it is left unanalyzed and experienced purely bottom-up, and anything is relative if it is analyzed and experienced in virtue of top-down influence. More simply, every entity is absolute as a whole and relative as a part.

Chapter 13
Intuition

Intuition is not the same as rational thought, but it is not irrational either; it is multi-rational, capable of understanding multiple concepts at the same time. Therefore, intuition is able both to use concepts and to transcend some of the limitations of rational thought. Although intuition is sometimes characterized as nonconceptual since it is non-symbolic, it is more properly characterized as multi-conceptual since it utilizes the content of multiple concepts (just as a picture is worth a thousand words). Because conceptualization is often relatively course, intuition is often associated with sensation rather than conceptualization, although intuition could not represent any mereological relations if it had no part/whole structure.

As opposed to intuition, rational thought is constructed of statements, where each statement has a syntactic structure that corresponds to a single conceptual hierarchy. Although there are exceptions such as puns that may correspond to two or more conceptual hierarchies, most statements are structurally unambiguous. Intuition, by contrast, uses an extremely limited syntax (i.e., mereological parthood without symbols), but it can use all words at once. Hence, intuition is a structure which corresponds to a dense mass of mereological and referential relations between multiple, overlapping concepts. Therefore, for the intuitive mind, everything is related to everything else.

Historically, the division between intuition and rational thought has manifested in many ways. Evans and Stanovich, reporting on this division in the context of Dual Process Theory, sum up this situation as follows:

The distinction between two kinds of thinking, one fast and intuitive, the other slow and deliberative, is both ancient in origin and widespread in philosophical and psychological writing. Such a distinction has been made by many authors in many fields, often in ignorance of the related writing of others.

Jonathan Evans, [Evans & Stanovich, 2013].

In the language of Dual Process Theory, intuition is a product of System 1 and rational thought is a product of System 2. In terms of the basic model, System 1 consists of sensation and zeroth-order concepts, while System 2 consists of symbols and higher-order concepts. As System 1 is multi-conceptual and operates in parallel, it cannot be symbolic. Conversely, symbolic processing must happen serially, so System 2 is only capable of conceptualizing a single concept at a time.

The informal argument that System 2 necessarily operates serially is based on inhibition. In particular, the argument from inhibition that symbols must operate serially is that symbols are defined in terms of both the presence and absence of other concepts. In other words, symbols are defined in terms of what they are not, in addition to what they are.¹⁰⁷ Further, the strength of top-down activation must outweigh bottom-up activation, otherwise it would be impossible to conceptualize things that are not present. In other words, symbols must be capable of representing and reactivating what they represent in the context of unrelated bottom-up activation, and the way to do that is by strongly inhibiting the negation of that concept.

Type 2 thinking, by inhibiting all irrelevant activation and forgetting numerous other conceptual relations, emphasizes a particular meaning. Therefore, while Type 2 thought is capable of knowing the parts and wholes of a concept, it causes the isolation of that idea by inhibiting its complement.

In contrast, Type 1 knowing does not isolate things in this way; it does not take things out of context or understand things solely in terms of their parts, but rather knows things in relation to everything else; no relations are completely inhibited. Type 1 knowing is therefore not only capable of knowing many concepts about something, but it is capable of knowing those concepts simultaneously (i.e., since concepts do not inherently interfere with one another). Additionally, because conceptual operations are performed in parallel, intuition is extremely fast.¹⁰⁸ In fact, the intuitive mind knows so much so quickly that it cannot be adequately described in language. As a result, intuition is often expressed metaphorically, since relating intuitions to one another bypasses the need to define them explicitly. ¹⁰⁹

Chapter 14
Animal Cognition

The differences between animal and human cognition are remarkable given the relative lack of neuroanatomical and genealogical differences.¹¹⁰ Unfortunately, understanding these differences is inherently difficult for at least two reasons. First, animals are reticent to talk in public, so it is difficult to hear about their experience first-hand. Second, although humans talk voluminously, it is difficult to isolate the aspects of cognition that are uniquely human, other than vague indications of “consciousness” or “language”. Therefore, in order to contextualize the theory of animal cognition, it is helpful to have a more general theory about where all things fit; inanimate objects and plants should have a place in the theory as well. Not surprisingly, these other forms of life are categorized by the basic model according to their degree of referentiality, or their epistemic level.

At the level of epistemic ground are inanimate objects, which lack the ability to sense or to act. For example, rocks do not sense or react to a neighboring brook in virtue of any internal mechanism or references.¹¹¹

A tree, on the other hand, senses things; for example, it senses the sun, and reacts to that sensation by growing toward it. Thus, the tree may be said to be conscious of itself in the same way as a rock, but it also has awareness of its referents, such as the sun. In terms of the basic model of cognition, plants sense, but they do not conceptualize because they do note have a mechanism by which they can form unitizations. For the same reason, animals with nervous systems can conceptualize things.

To summarize, non-referential (or reflexive) consciousness is something that all things have, referential awareness is something that only plants and animals have, and conceptual awareness is something that only animals have. The further differentiation between animal and human cognition is primarily attributed to the human capacity for words and symbols. The smallest change to the basic model of cognition that is capable of producing this difference is the removal of interpretation (Ω).¹¹² Removing interpretation (and possibly also symbolization, as shown in Figure 4.14) from the basic model results in a model of animal cognition that allows sensation and concrete concepts, but not symbols or higher-order concepts. This model predicts that animals may sense, visualize, and to some extent conceptualize in the same manner as humans, but they cannot engage in symbolic thought, and as a result, animal cognition is necessarily concrete.¹¹³

[Figure 4.14: Animal cognition, which lacks
interpretation (Ω) and symbolization (Ξ).]

As symbols enable abstraction, animals can form concrete unitizations but not abstract generalizations, and therefore they cannot think abstract thoughts. Further, they do not experience objects as nominally identical, since identity requires higher-order concepts. This difference in cognitive structure therefore entails differences in what is a priori for animals as opposed to humans.¹¹⁴

Behaviorally, the lack of interpretation can be understood in relation to the distinction in linguistics between signs and symbols: animals understand words as signs, but not as symbols. Signs are concrete, and often indicate that something else is impending (e.g., sit_φ, stay_φ, “play dead”_φ, etc.); they are always embedded in a causal context because they form a single spatiotemporal unit. On the other hand, symbols represent something and can be used to directly activate the concept that they represent.

Symbols enable a dramatic evolutionary advance because they enable the easy formation of recursively constructed concepts, and therefore the creation of linguistic structures of arbitrary complexity (see [Chomsky, 1995]). Linguistically, therefore, while animals may generate and understand certain verbal behaviors, the underlying deep structure of those utterances is significantly different from human syntax. Since the concepts of animals are always zeroth-order, characterizing how animal communication is different from human language requires identifying what parts of speech correspond to zeroth-order concepts. The basic model of cognition indicates that animal cognition is formed of entire sentences. Thus, what animals lack is the ability to create and comprehend abstract parts of speech such as nouns and verbs. In other words, since animals cannot form concepts corresponding to abstract parts of speech, their sentences are limited to be one word long and to represent entire events.

Expressed slightly differently, the theory of animal cognition proposed here is that animals comprehend the world as composed of events, rather than spatial objects and temporal processes. On a syntactic level, animals comprehend entire sentences, but not parts of speech such as nouns and verbs. This is a direct result of not comprehending space or time as independent entities on a cognitive level (see [Rogers, 1995]). In other words, the concepts of time and space as independent entities in human cognition are only possible in virtue of the abstract generalizations that are enabled by symbolization. Studies of animal behavior reinforce this finding by demonstrating that behavior is unitized, or that learning to perform a given spatial task entails learning both a time and rate at which that task is performed [Gallistel, 1980]. ¹¹⁵

Chapter 15
Bottom‑Up and Top‑Down

Bottom-up and top-down processes in cognition imply a vertical dimension that can be measured using the notion of epistemic level. Sensory parts are responsible for bottom-up activation, and symbolic wholes are responsible for top-down inhibition.¹¹⁶

The role of bottom-up processes in cognition is fairly straightforward: concepts are recognized in virtue of sensory input, especially if those concepts are emotionally significant. The role of top-down processes in both what is perceived and how it is perceived is a bit more complicated, and often underestimated. As a result of that top-down influence, our perception operates with significant distortion and even creation, an interference that often goes unnoticed. Perhaps this is inevitable, since the cognitive detection of what is not perceived requires extraordinary investigation.

A simple example of bottom-up and top-down effects in the visual system is the phenomenon of blind spots, which are areas of the visual field for which there is no sense data. Figure 4.15 can be used to see your blind spots—or better yet, can be used to not see them. First, close one eye and look at the dot on the side opposite of your open eye, and move the figure slowly closer and farther from your face. At a certain distance, the dot that you are not directly looking at disappears from your visual field.

[Figure 4.15: Blind spots.]

The blind spot in each eye is created by a small patch of missing retina, medial to the point directly behind the pupil where the optic nerve connects to the eyeball. As a result, there is no sensation from the corresponding part of the visual field. Blind spots demonstrate both that there are areas of reality that we expect to sense that we do not, and that we are not aware of this lack of sensation. Further, blind spots are not only sensory holes that we do not see and of which we have no awareness, but are areas of illusion. To more clearly demonstrate this fact, if you pass a pen in front of the blind spot that you identified during the previous experiment, you will not see a hole in the pen; rather, the missing part of the pen is filled in by your mind. Therefore, blind spots are characterized by both bottom-up, sensory deficits and top-down illusions.¹¹⁷

These missing bits of sensation and areas of hallucination clearly demonstrate that the world as we perceive it is not the world as it is; our sensations are filtered to make them comprehensible. Although this is not necessarily a bad thing, the lack of awareness of this alteration is problematic. As another example of top-down influence, consider the following question:

What part of your mind are are you using?

Most people who read this sentence see it as identical to the following sentence: “What part of your mind are you using?”. As the first sentence is not syntactically well-formed, they are not identical unless the first sentence is mentally altered to conform to our expectations and understanding. This type of correction is often beneficial; for example, it automatically corrects the syntax of ill-formed sentences such as the one above and causes false positive percepts of dangerous creatures. However, it is harmful if it creates a world that always conforms to our understanding and in which we recognize only familiar things, since living in such a world would not enable our understanding to increase.¹¹⁸

Dreaming and hallucinations are even more extreme examples of playing a creative role in what we sense. If the dream of a tree can arise without a tree, then no external cause at all is necessary for perception. Although this does not invalidate the more common case in which the tree-sensation arises in conjunction with a tree-object, it does make the relationship between the two more tenuous. As a rather extreme example that a number of philosophers and novelists have used to illustrate this issue, there is no way to tell if we are dreaming or not when sensations do not depend on the presence of objects.¹¹⁹

As these examples illustrate, top-down and bottom-up processing offer both an opportunity to balance and a risk of imbalance. That balance is particularly necessary in contexts such as learning a new activity, since top-down cognition is essential for many activities, but it also hinders expert performance. In such cases, therefore, it is beneficial to be able both to think about things and to not think about things.¹²⁰

Chapter 16
Attention

Attention is an operation that restricts the domain of perception from an omnipresent awareness to a localized awareness. This selectivity causes an individual to be relatively blind to things outside of the aperture of attention, a phenomenon called inattentional blindness.¹²¹ This process is analogous to putting blinders on a horse: although it causes the horse to be less distracted, the horse also perceives less. On a positive note, however, the decreased aperture of awareness allows the horse to more fully analyze what is perceived.

Attention is governed by both bottom-up and top-down processing, which gives rise to two types of attention called stimulus-driven and goal-directed attention, respectively. In terms of the basic model, attention is determined bottom-up by conceptualization and top-down by visualization. Bottom-up sensory activation is prone to grab attention if it is particularly intense, unexpected, or emotionally valent. Top-down conceptual inhibition guides attention to what is relevant by inhibiting conceptual wholes outside the scope of attention. As a side effect, top-down attention is limited to concepts that already exist.

Attention has a similar relationship to both concepts and emotions. Just as concepts guide attention and the subsequent scope of attention limits subsequent concepts, attention is also guided by emotion and that scope limits subsequent emotions. In both cases, we are conceptually and emotionally reactive primarily to the content of awareness, and it is attention which in turn governs the aperture of that awareness.

Attention, in addition to being viewed in an impersonal causal chain, is a skill that we are practicing all the time, regardless of whether we are conscious of doing so or not. Training in attention explicitly often entails exercising both its mereological and referential aspects.

Mereological training in attention consists of exercises such as directing awareness to areas of space of of which one is infrequently aware as in body scan meditations, or sustained awareness of a particular location as in meditation on an object. This process happens routinely as a side effect of various activities such as physical exercise or piano practice.

Referential training in attention explicitly directs awareness to different epistemic levels. For example, one may direct awareness exclusively to the occurrence of thoughts or to the referential content of those thoughts, where the latter awareness is of a higher referential level than the former. This type of training figures prominently in various forms of contemplative practice and meditation.¹²² In terms of the basic model of cognition, meditation that enables one to remain at the epistemic level of concrete concepts develops one’s intuitive ability.

Chapter 17
Emotions

Despite the profound importance of emotions, much less is known about them than is known about thoughts.¹²³ Perhaps this is due to their relative complexity, the tendency of cultures to preserve teachings from higher epistemic levels better than teachings from lower epistemic levels, or to the fact that emotions are not well-suited to being described symbolically.

The cultural dissociation of identity from emotionality is so prevalent that it is encoded in numerous social and legal systems. For example, there is no legal responsibility for actions that are committed “in the height of passion”. On the other hand, crimes committed when “thinking too much” are considered premeditated, which increases the amount of responsibility. But if we are responsible for what we think, why are we not responsible for what we feel? One possible answer is that identification with thoughts rather than emotions is an attempt to define ourselves as what we have control over. However, the claim that we have control over what we think about is dubious, because most of us cannot even control whether we are thinking or not. In other words, if one does not have sufficient control over one’s mind to start and stop thinking, then how could it be possible to have the fine degree of control required to choose the content of thought?

Historically, the reluctance to scientifically examine emotions is probably related to their antagonism with rational thought. However, the depiction of cognition as consisting of exclusively bottom-up emotions and top-down symbols is an overly simplistic characterization, since emotions play a large role in reasoning, just as emotions have their reasons. Although strong emotions can certainly encourage illogical conclusions, we think certain thoughts as opposed to others precisely because of their emotional valence and their conceptual context.

The role of emotions is particularly complex because they can become attached to symbols, or to references themselves instead of their referents.¹²⁴ For example, consider the profound differences between liking the sensation of “peanut butter”_ψ, liking the concept of “peanut butter”_φ, and liking the symbols “peanut butter”_ξ.

• Liking the sensation of peanut butter means that you enjoy the taste of peanut butter or the mouth feel of dollops smeared on an apple or carrot.

• Liking the concept of peanut butter means that you like the concept rather than the experience. Perhaps peanut butter is good because it is buttery, vegan, and not as expensive as cashew butter.

• Liking the symbol of peanut butter means that you like the actual symbols (i.e., the words “peanut butter”). Although it may seem odd to like symbols, they may become significant triggers if they are conditioned by their association with desirable and undesirable objects.¹²⁵

The emotional valence of concepts becomes the emotional valence of objects in a process called cognitive fusion. Cognitive fusion is an operational hazard of the way cognition works because it causes emotional confusion between concepts and objects (i.e., as a result of the concept inheriting the emotional significance of its referential content). The occurrence of the concept may additionally cause physiological responses that are inherently rewarding beyond the actual experience, such as activation of the parasympathetic rest/digest mechanism. As a result, thinking has the potential to condition concepts even further by creating a feedback loop that does not involve the presence of the object.

At least in some instances, emotional attachment to symbols rather than their intended objects seems to be a mistaken form of cognition.¹²⁶ It can certainly be disadvantageous if it biases which thoughts do and do not occur in virtue of how much we like the thoughts themselves as opposed to the content of those thoughts. Evolutionarily, it may be simply a side effect of emotions conditioning sensory space, and symbols making use of that same sensory space to represent concepts. On a practical level, even if emotional attachment to symbols is not viewed as an inherently negative thing, disentangling the attachment to symbols from the attachment to the concrete phenomena that those symbols represent is a formidable task.¹²⁷

Regardless of one’s views on the proper role of emotions with respect to epistemic level, two things are clear and worthy of reflection. One is that emotional attachment to higher epistemic levels in general causes strong top-down influence, which can be disadvantageous. Another is that lack of equanimity with respect to symbols at any given epistemic level causes bias in that top-down influence. The combination of these two, therefore, is especially problematic.

Chapter 18
Beauty

Where do we experience beauty? Do we experience it in the space of our bodies, or in a space that is coextensive with the world that we perceive? Folk wisdom holds that beauty is in the eye of the beholder, and psychologists hold that the positive feelings associated with beauty originate in the dopaminergic neurons of the limbic system. Similarly, the world points to us and tells us that our sense of beauty is subjective, or in our heads. However, we experience beauty in the objects of the world, and tell the world that we experience beauty out there. If beauty can only be present in a single location, then which location do we choose?¹²⁸

This incongruity with respect to the location of beauty is typically taken as evidence for indirect realism, rather than support for the argument that consciousness is extended. In other words, there are objective aspects of sensation that are due to the world and subjective aspects that are due to us, and beauty is regarded as one of the latter.¹²⁹ However, these internal references to an external world provide two possible points of view with respect to the location of beauty.

From the objective or material perspective, references are not treated as referential, so the experience of those references is located within the body. This is the predominant view within science as a whole: it views human bodies as parts of larger wholes and minds as collections of references to that larger whole that are contained within the body. Awareness of the external world is enabled by referentially bringing the external world within the brain or body, which is often assumed to be the location of consciousness.

From the subjective or mental perspective, on the other hand, experience is constituted by the content of those references, and as a result, the experience of beauty is located throughout the world.¹³⁰ In other words, since we perceive the referential content instead of the references themselves, beauty is located in beautiful objects rather than in the head. Even if the objects are beautiful because of our previous experience with them, then beauty is a relation in the world which we perceive or not in virtue of having that experience.

If both truths are compatible and valid from different points of view, then it would be a mistake to let the concepts of others interfere with our experience of beauty or vice-versa. Therefore, to cease feeling that external objects are beautiful because we are told that nothing is objectively beautiful would be a terrible mistake.¹³¹ If we find roses beautiful and someone else does not, all that is entailed is that we have the concepts that allow us to see their beauty because of our experiential history, while others do not.

The question about where beauty resides can be seen as part of a larger question about whether objective space should be seen as the intersection of all subjective perspectives or as the union of all subjective perspectives. For example, it is common to see pine trees as green, and it is less common to see them as beautiful. This is typically taken as evidence that greenness is an objective aspect of pine trees, while their beautifulness is a subjective aspect. Although there is no question that beauty is harder to define, the further assumption that beauty is purely subjective while green is purely objective amounts to an intersectional thesis. According to this thesis, the physical universe is itself entirely devoid of beauty; beauty is merely the subjective judgment of an observer.

The unional thesis entails that objective reality actually contains all subjective truths. For example, a pine tree is both beautiful and ugly from different yet valid points of view. According to this thesis, each of us knows aspects of a larger, multifaceted truth in which things in the world are actually beautiful and ugly from different perspectives. In other words, beauty and ugliness are relations between our perceptions and our prior experience that also exist in the world.

In slightly different terminology, the fact that the objective point of view must be valid for all observers removes beauty and other properties from objective space, because the translation of what is beautiful from one perspective to another is considerably more difficult than moving from one spatial location to another. It is worth considering, however, that just as standing in a different location brings some of the subjective observations of different observers into accord with one another, knowing each other’s prior experience with the world may bring into accord subjective judgments such as what is beautiful.

Chapter 19
Kindness

To understand intersubjective kindness, consider the following analogy:

The right hand looks after the left hand without thinking in a discriminatory way, “I am the right hand. I am taking care of the left hand.”

Thich Nhat Hanh [Hanh, 1998].

This analogy illustrates that kindness to oneself does not require justification; it is completely natural. To better understand kindness, it is useful to categorize behavior along two dimensions: self/other and dual/nondual.

Kindness entails that the self/other distinction is commutative. For example, if you have a sense of joy when someone else is pulled over for speeding instead of you, your cognition is not commutative with respect to the division between self and other. This commutativity is captured by various formulations of the golden rule, a rule that holds we should be equally kind to ourself and others. If kindness is not commutative, then it is either selfish, in virtue of which you make yourself happy at the expense of others, or it is otherish, in virtue of which you make others happy at your own expense.

Kindness can also be understood with respect to dualistic and nondualistic cognition, where duality is seen as the result of using top-down symbolic processing in cognition. Selfishness and otherishness are both inherently dualistic, since it is not possible to consistently and intentionally benefit oneself or another without reifying the duality between oneself and another; therefore, nondualistic selfishness and nondualistic otherishness do not exist (i.e., if one acts without conceptual duality, one may be kind or unkind, but not exclusively with respect to self or other).

Acting with kindness presents different obstacles depending on the presence or absence of duality. The potential hazard of dualistic selflessness is that unconditional love is necessary to unite the self and other, since they exist at a conceptual distance from one another (i.e., dualistically). If that love remains conditional, conceptual dualism is prone to tip the emotional scales in one direction or the other. On the other hand, the potential hazard of nondualistic selflessness is that if the nondualistic stance is a result of ignorance or lack of conceptual differentiation, then attempts to benefit self or other will not be reliably successful.

Because nonduality is difficult to talk about without reinforcing dualism, nondual traditions often express either that everything is self or that there is no self whatsoever. In both cases, the subject/object duality disappears.¹³² For example, if we identify with our body, we might enjoy eating savory foods or feel sad when we hurt. However, if we identify with our family or some other larger whole, then it doesn’t matter if we personally win a race or have some good thing happen to us, as long as the person who does is a part of our larger group; this is just what it means to identify as a part (of a group), rather than as a whole (of ones bodily parts).

As this discussion indicates, kindness depends crucially on the location of the self/other conceptual boundary. A more nuanced solution might preserve intellectual duality while allowing emotional nonduality; in other words, one should care for both self and other, but not confuse self and other by failing to intellectually distinguish between them. This is difficult to achieve in practice, since a person may have various habitual tendencies of which they are not conscious. For example, one might take too much credit for their own thought, rather than acknowledging the causal role in one’s thought of the entities that are perceived or referenced. This situation is similar to a prideful mirror that imagines it is entirely responsible for moving its reflections about, when in fact its (mental) references are causally moved largely by the movement of the reflected entities themselves.

Chapter 20
Overlap

The parts of the world are overlapping and interdependent.¹³³ Therefore, although the use of mereological analysis to draw boundaries between things is useful, the associated tendency to consistently rely on a single partition of reality into concepts is problematic.

Unfortunately, the cognitive tendency to view the world as consisting of some number of non-overlapping parts is difficult to avoid. A particularly pernicious example of the lack of overlap is exemplified by what Daniel Dennett calls the Cartesian theatre, which exists in various explicit and implicit forms (see [Dennett, 1991]). It arises from the view that the brain contains mental representations of the world and that consciousness is the spatially-withdrawn witness of these representations, rather than being something that is unified with them. Unfortunately, theories of consciousness that posit an internal witness often involve an infinite regress, since any witness would have to form an internal witness of their own to understand the external representations that they witness. More recent philosophical theories hold that consciousness is a kind of field that exists within the brain and encompasses (and has direct awareness of) its representations. But if consciousness occurs in an extended region, why is it necessarily restricted to occurring within the brain? After all, it has not been observed there any more often than it has been observed in the world.

In general, the lack of overlap between concepts is an operational hazard of higher-order cognition; our rational minds necessarily traffic in mutual exclusivity as a result of using higher-order concepts. Although the creation of higher-order concepts out of existing concepts is more efficient than building new concepts out of sensation directly, higher-order concepts are hierarchical, and hierarchy does not represent multiple overlapping concepts very well. Therefore, it is important to balance the convenience of higher-order concepts with the precision of zeroth-order concepts. In light of this cognitive bias against the formation of overlapping concepts, it is necessary to be diligent to recognize such overlap in the world. As a particularly interesting case of overlap, consider the mereological and referential overlap of two people.

[Figure 4.20a: Mereological overlap of two people.]

Mereological overlap of two people is shown in Figure 4.20a. This kind of overlap is unusual, because it entails sharing body parts (although it happens in cases of conjoined twins and (arguably) during pregnancy).¹³⁴ Referential overlap of individuals occurs if the subjective space of one individual overlaps the subjective space of another. This is shown in Figure 4.20b, where the numeric subscript denotes a given individual.

[Figure 4.20b: Referential overlap of two people.]

Although overlap of bare sensation is trivial because it occurs whenever two individuals see the same thing, the overlap of subjective spaces in general requires that the two conceptual spaces are the same. Thus, it is less clear to what degree the subjective spaces of two individuals are identical. For example, when you perceive a pine cone, and I perceive the same pine cone (from the same physical perspective), do we perceive the same thing? In other words, assuming an identical object, are the appearances of that object identical to each of us?

Because subjective experience cannot be verified by anyone other than the subject, perhaps there is no way to be certain. We might communicate with one another to arrive at an answer, although using communication to verify identity creates further difficulties, since we may use the same words but mean two different things. If we believe that mentality and physicality are tightly coupled, we may use the principle of Occam’s razor to argue that two individuals with the same sensory organs and concepts have identical perceptions in response to the same object.¹³⁵ In that case, the subjective spaces of two individuals literally overlap because they perceive the same thing, whether in virtue of extended consciousness or internal representation. In more detail, two minds sensing the same object have similar perceptions up to the epistemic level at which a significant difference occurs, whether that difference occurs in virtue of different sense organs or subsequent conceptualization. In this way, referential overlap establishes a basis in virtue of which two minds are identical, even if they do not mereologically overlap.¹³⁶

Chapter 21
Self and Other

One of the most important conceptual dichotomies, and perhaps the first concept that we form as children, creates self and other. So how exactly is the concept of self defined? The anthropologist Margaret Mead wrote that “self-consciousness is constituted by adopting the perspective of the other toward oneself.” [Zahavi, 2008]. This entails that the self is a higher-order concept; it entails not just sensing oneself, but knowing oneself on a relative level as an abstract and enduring object.¹³⁷ One’s self-identity is often an emotionally laden subject. Therefore, from an emotional point of view, the question “who are you?” is often related to the question “what is the largest whole of yourself that you strongly care about?”. In modern times, self is typically defined as a whole consisting of bodily parts. In older times, the self more often existed only as a part of a tribe. In both cases, however, the differentiation between self and other relies on both mereological distinctions (such as body/world) and referential distinctions (such as subjective/objective).¹³⁸ Therefore, the rest of this section explores the notion of self in virtue of how it exists both mereologically and referentially.

A mereological theory of self is defined in terms of wholes and parts, and determines characteristics such as the spatial location and size of the self. Several options for the boundary of a mereological self are shown in Figure 4.21a. The location of the self is most often coextensive with the body or a specific part of the body, such as the heart or brain. Mereological theories of self tend to be materialistic, and therefore emphasize connections between objects in physical (or mereological) relation to one another. Mereological theories also tend to deny consciousness, although in modern culture, consciousness and the subjective perspective are more often ignored than explicitly denied.¹³⁹

Of the possible locations of material self-identification, identification with the body is particularly intuitive because it is the largest object that affords relatively consistent sensations while also being contiguous within the field of experience. Contiguity is especially important because it facilitates concept formation; although the body is composed of many parts and is in turn a part of many larger wholes, it is typically viewed as a single mereological whole. As linguistic evidence for this fact, a synonym for person is individual, which literally means “unable to be divided”. The importance of the bodily self in our daily lives and routines is cemented by its use as the legal definition of a person. For example, taking care of the self is a legal mandate in the modern world, while taking care of others is optional (except in the case of one’s adolescent children).¹⁴⁰

[Figure 4.21a: Analysis of the self in terms of parts and wholes.]

A referential theory of self, on the other hand, is defined in terms of references and referents, and is more closely linked with idealistic philosophies. For example, to identify only at the epistemic level of referents (as opposed to references) leads to identification with what is experienced, rather than with who does the experiencing. Although it might seem strange to identify with the experiences themselves rather than the experiencer, the opposite extreme of exclusive identification with the experiencer is perhaps even more paradoxical, since an experiencer remains unknown unless it is identical with the experience of that experiencer—and on most accounts, it is not. In other words, if you identify as the witness rather than the witnessed, then you do not know yourself from the non-referential point of view, but only at an epistemic distance (i.e., just as you would know someone else).¹⁴¹

Referential theories of self such as idealism remove this subject/object dilemma by positing that there is no epistemic ground, or no material level of being which consists of ultimate referents. As a result of encouraging identification with perceptual content itself rather than the perceiver, this view downplays the distinction between the perception of our bodies and the perception of the world, since the location of the references is less relevant.¹⁴²

[Figure 4.21b: Analysis of the self in terms of references.]

Referential identification is illustrated in Figure 4.21b, which depicts a brain as a collection of references to both itself and to a body and world outside of itself. To identify with the references themselves implies a materialistic view and identification with the brain. To identify with the referents of those references implies identification with the referential content of the mind: the brain, body, and world as experienced.

Another way to characterize self-identification utilizes both a mereological self and an epistemic level at which one relates to that self. For example, one might identify with the bodily self, yet relate to that body in various different ways. This is illustrated in Figure 4.21c, which shows three referential spaces from which one may relate to the bodily self: objective, sensory, and conceptual. The self is viewed within each of these spaces differently, as either an object (body), a sensation (body_ψ), or a concept (body_φ), respectively. In other words, even though one identifies with the physical body in all three cases, there is a profound difference in how one acts, feels, or thinks with respect to that body.

[Figure 4.21c: Depiction of bodily (physical) self,
sensation of that body, and the concept of that sensation. ]

Chapter 22
Stratified Self

To some extent, any attempt to define a self may be criticized as trying to identify a complex, concrete entity within a simple, abstract context. For example, the question “Who am I?” uses an abstract pronoun “I”, with which we are intended to identify in order to formulate an answer. However, this identification is problematic if it is incorrect to identify with anything abstract in the first place.¹⁴³ Refusing to engage with the question is not a good solution, however, since our actions depend on our understanding, even if that understanding is implicit. Therefore, this section explores a stratified self-concept, and attempts to disentangle where we are (mereologically) from what we are (referentially).

Overview

One of the main endeavors of this book is to explore the aspects of our experience that are due to the nature of our minds. To do that, it is helpful to determine what can be known independently of the world, or a priori. By identifying and then removing what we contribute to our experience, we can know the world better as it is.

The basic model of cognition relies on the theories of mereology and reference to describe both how our minds are situated in the world and how in turn our minds give rise to mereology and reference. The choice of mereology is motivated by the fact that the existing symbolic formalisms of mathematics and logic run counter to our intuition about the world. In particular, the most common forms of topology and logic work best only when applied to the abstract geometric and symbolic atoms they take as primitive. As Einstein noted:

Geometry […] is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.

Albert Einstein [Einstein, 1924]

On one hand, it is clear that abstract sciences cannot involve objects directly, since symbols are inherently abstract and cannot be made concrete without losing their linguistic usefulness. However, geometry and logic are used precisely for their ability to describe real-world scenarios, so it is essential that they fit the world as well as possible. The proposal here is to use mereology as a basis for both symbolic logic and topology, in order to take better account of physical objects (which are clearly not composed of zero-dimensional points). To do so, high-dimensional objects are introduced as mereological primitives, which form a subsymbolic foundation. That foundation is augmented with (nominal) references to create symbols that allow the creation of abstract low-dimensional entities, such as geometric primitives. Beginning with high-dimensional objects is motivated by the following observations:

• Point-free topologies such as mereotopology are necessary for a subsymbolic model of mind, since point-set topology is inherently atomistic due to its dependence on partless points. In other words, mathematical models of cognition that require everything to have a symbolic description do not leave room for the ineffable, and therefore they are poor models of intuition.

• Mereotopology as a mathematical model of mind provides an excellent fit for human cognitive development. Further, it is clear that geometric thought precedes algebraic thought during the course of development.

• Purely reductionistic or point-based approaches do not make good psychological models, as cognitive science clearly demonstrates that our minds consist of both bottom-up and top-down processes.¹⁵⁵

Some people might argue that logic and topology do not need to be cognitively or intuitively correct, since they work in practice. However, if they are used as models of reality, they will inevitably produce an error of the type this book has argued against: treating objects as if they were equivalent to our symbolic representations of them. Therefore, any reduction of this type of mismatch (i.e., by using a non-atomistic model) will be beneficial for both thinking about and interacting with reality.

Topology

The most common form of topology is called point-set topology, and was the branch of mathematics that first succeeded at enumerating the continuum. The desire behind this enumeration was the unification of concrete, continuous space and abstract, discrete points. To do so, continuous N-D spaces are defined as infinite numbers of points of zero-size. While there is nothing inherently invalid about this approach, it is problematic from a psychological point of view because neither space nor matter is symbolic.¹⁵⁶

A classic example of one of the problems associated with the use of points as a concrete rather than an abstract formalism arises because the points that divide two shapes must belong to either one or the other of those shapes. A shape that contains the points on its surface is called closed, while a shape that does not contain its surface points is called open. The problem is that two open shapes can never come into contact with one another because neither can contain the points on their common boundary. Similarly, two closed shapes cannot come into contact, because between any two points on their respective surfaces are an infinite number of other points. While this lack of contact may be regarded as a mere intellectual curiosity, objects in the world are, in fact, connected. Therefore, this problem cannot be ignored by any model of reality that claims to be accurate.

Mereology

Mereology is a nominalistic version of set theory that was developed by the Polish mathematician Stanislaw Lesniewski. The operation of reference in the current work is roughly equivalent to Lesniewski’s naming operator (ε). Historically, mereology was viewed as a mathematical contender to set theory, a mathematical formalism for collections of things due primarily to the German mathematician Ernst Zermelo. A mathematical set can be implemented in the basic model by combining two operations, reference and whole. In fact, the operation of whole is the primary reason that mereology is an excellent fit for subsymbolic cognition; while set is an intransitive and discrete operation, whole is both transitive and continuous.

In mereotopology, which is the science of mereology applied to topology, it is often assumed that parts have the same dimensionality as their respective wholes. In the version of mereotopology developed here, the universe in which topological analysis begins is continuous and open-dimensional. The combination of these two premises seems to suggest that mereotopology does not allow points and lines; however, they are permitted in a nominal sense, as abstract entities. For example, zero-width points on an interval are nominal boundaries: they do not constitute that interval, but merely divide it. The situation is analogous to using a sharp knife to cut a loaf of bread in half: there is no part of the loaf which is not displaced to one side or the other by the knife. Although the location of the cut may be given a name (e.g., “0.5”), there is no bread associated with it. In mereotopology, therefore, real numbers are understood as nominal boundaries, or a collection of names for the divisions of an interval, rather than entities which constitute that interval.¹⁵⁷

Points cannot be divided; therefore, point-set topology begins with points, which are the smallest objects it will ever know, and uses sets to construct a universe from those points. This approach is not valid from a psychological perspective because mental space does not have a smallest element out of which other things are built; it is always possible to imagine something smaller.¹⁵⁸ For the same reason, the complementary topological approach that begins with the largest possible object and divides it is also invalid. Therefore, the version of mereotopology presented here starts in the middle, taking arbitrary spatiotemporal volumes as primitive, and uses them to create unions and intersections as necessary.

As opposed to points, the objects that mereotopology takes as primitive are neither infinitely small nor singular. They are concrete by definition and are linguistically better represented by mass nouns than count nouns. As there is no requirement to have partless parts or wholeless wholes, space is assumed to be both continuous and unbounded (i.e., open above and below). Therefore, mereotopology is inherently neither holistic nor reductionistic, and it analyzes objects just as cognition does: by examining them in terms of both their smaller parts and their larger wholes. Using a set of medium-sized objects as a starting point for analysis underscores another facet of psychological compatibility: such a starting point is constructive, in the same way as mental development itself (i.e., medium-sized objects are epistemically prior to both atoms and universes). Therefore, objects are always unknown to some degree, in that their parts and wholes are never fully specified.¹⁵⁹

As the aim of the version of mereotopology developed here is to be a good fit for cognition, it does not follow a standard mereological development. For a more formally rigorous theory, see GEMTC as developed by Roberto Casati and Achille Varzi [Casati & Varzi, 1999]. For a thorough overview of mereological systems, see the book by Peter Simons [Simons, 2000].

Logic

The reason that a model of cognition requires subsymbolic logic can be demonstrated with the following proposition:

The soccer ball is white.

The problem is that this statement is neither true nor false, since some parts of a soccer ball are white and other parts are not.¹⁶⁰ Symbolic logic, however, requires the subject “soccer ball” to be singular; it is not allowed to have parts (or at least, the subject is necessarily treated as a singular and partless whole by any predicates). In other words, predicates require a symbolic subject in order to yield an all-or-none, true-or-false answer. Therefore, traditional symbolic logic does not work for objects that are continuous. On the other hand, mereological logic allows predicates to be applied to continuous shapes in addition to discrete symbols, so it provides a better fit for actual objects.¹⁶¹

[Figure Aa: The five mereological relations.]

In order to illustrate the continuous nature of mereotopology, consider again the five mereological relations presented in section Mereological Space, depicted in Figure Aa. The fifth case is problematic because it cannot be described using classical logic and the parthood relation, unless a new symbol is introduced to name the region of intersection. Intuitively, however, it is clear that A and B overlap; they are parts of each other to some degree, and to a complimentary degree, they are not parts of one another.

[Table Ab: Shape Logic]

Table Ab is a truth table that illustrates how mereological logic gives a complete description of the situation in Figure Aa, where true is denoted as 1 and false is denoted as 0. The first column of row (5) can be read as “20% of A is a part of B and 80% is not”. This solution is therefore an improvement over symbolic logic because it accounts for the overlap in case (5) without having to explicitly refer to the entity created by the intersection, and therefore it avoids unnecessary symbolic proliferation.

The formula that describes parthood as a continuous function of x and y that yields a value between zero and one is:

A.1 The Basic Model

The basic model of cognition is a foundation
for mereology and symbolic logic. The set of equations that corresponds to the basic model of cognition are developed according to the following guidelines:

• A developmental or constructive approach is followed as much as possible. The model is constructive in that it begins with a universe constituted by some number of objects that create sensations. Conceptual wholes are created by taking unions of these sensory features, and parts are created by taking intersections.¹⁶²

• Theoretic devices such as axiomatic reduction are not attempted without psychological motivation, since the primary purpose of the model is to describe cognition.

• The details of psychological implementation are omitted, since they would interfere with creating a model that is easily understood.

• The desired outcome of becoming familiar with the model is a clarification of thought.

[Figure A.1: The basic model of cognition,
with sensation (Θ) and action (Δ)
mapping to/from subjective subspaces.]

A.2 Zeroth-Order Logic

Zeroth-order logic is implemented using
mereology and reference.

[Figure A.2: a priori cognition.]

The a priori operations of the basic model correspond to part (Ψ), whole (Φ), reference (Ξ), and referent (Ω). All of these operations are used on the right-hand side of the definitions in zeroth-order logic to produce variables on the left. The use of references created with the referential operator (Ξ) as variables on the right-hand side of definitions, however, requires higher-order logic.

The transition from basic model operations to logical values is made by defining zeroth-order truth as being real: a thing is true to the degree that it is a part of the universe. That degree is represented by a continuous truth value between 0 and 1, as described in the section Logic at the beginning of this appendix. In terms of visualization, a continuous version of parthood can be written as:

x is a part of y (or Pxy) in proportion to the visualization formed by x and y as a fraction of the visualization of x.

For simplicity, however, the truth values that are used in the following section are binary (i.e., following Boole, they are true and false instead of continuous values; see [Boole, 2003]). Therefore, parthood is written as:

As a final note, all variables (e.g., x and y) in the following equations are assumed to be concepts, so the subscripts indicating conceptuality are omitted (i.e., it is implied that x=x_φ and y=y_φ unless otherwise indicated).

A.3 Higher-Order Logic

Higher-order logic is characterized by the use of references. Higher-order logic is presented in two sections: first-order logic and abstract logic.

First-order logic is equivalent to a mereological formulation of modern (first-order) symbolic logic. Therefore, the section on first-order logic uses familiar mereological primitives and existential quantification.

Abstract logic is the basis for creating abstract entities. All abstract entities are defined nominally (i.e., in terms of references). Abstract entities are created by the intersection of higher-order entities; that process causes them to be entities of lower dimensionality, therefore they are incapable of being concrete parts. This reduction of dimensionality is also the basis for introducing boundaries and geometric primitives:

• A point is the boundary between two intersecting lines.

• A line is the boundary between two intersecting planes.

• A plane is the boundary between two intersecting solids.

These geometric entities are not explicitly defined here, since their development follows from the fact that the intersection of two higher-order entities can produce an entity of lower dimensionality.¹⁶⁵

B.1 Symbolic Conventions

An assortment of profound squiggles.

[Table B.1a: Epistemic Spaces]

[Table B.1b: Epistemic Relations]

[Table B.1c: A priori Symbols]

[Table B.1d: Logical Symbols]

B.2 Typographical Conventions

Conventions relating to the way things are written. In a conversation about a rock, it is sometimes unclear if the thing to which is referred is a physical thing (the rock object), a sensory thing (the rock sensation), a conceptual thing (the rock concept), or a symbolic thing (the name for the rock concept). To disambiguate between these different uses, the following conventions are adopted:

• “Rock” without typographic augmentation generally refers to the rock object.

• “Rock_ψ” refers to the sensation of the rock.

• “Rock_φ” refers to the concept of the rock.

• “Rock_ξ” refers to the symbol of the rock.

These particular lower-case greek letters are used because they are the result of visualization (Ψ), conceptualization (Φ), and symbolization (Ξ) operations. In particular, Rock_ψ = Ψ(Rock_φ), Rock_φ = Φ(Rock_ψ), and Rock_ξ = Ξ(Rock_φ).

Further, the concept of a rock may exist as a zeroth-order concept or a higher-order concept. To indicate the order of a concept, a superscript may be used:

• “Rock⁰” indicates a zeroth-order concept.

• “Rockⁿ” indicates an n^th-order concept (where n is some positive integer).

• “Rock⁺” indicates a higher-order concept (i.e., order 1 or more).

Finally, there are several other typographic conventions which have somewhat specific connotations in this work:

• Forward slashes are used to indicate dichotomies, as in subjective/objective.

• Hyphens are used for making compound-words.

• Emphasis is denoted like this, as are titles, foreign words, and mathematical variables.

• Glossary entries and intra-document references are underlined, and are hyperlinks in electronic versions of the book: self-reference.

• Bibliographic references and URLs are enclosed in square brackets to avoid syntactic ambiguity, as in [http://theWholePart.com].

• The car having the wheel as a part is equivalent to the wheel having the car as a whole.

• Parentheses are allowed to nest (sometimes (if one is careful to close them)).

• This work uses logical punctuation. For example, punctuation is placed within quotes only if it belongs to the “quoted material”.

• The affirming negation of an object x is written as not-x, and the non-affirming negation of that object is written as non-x.

B.3 Ideographic Conventions

Conventions relating to diagrams. The diagrams used in this work follow a syntax which is loosely based on UML (the Unified Modeling Language). UML depicts things and relations as shapes and arrows. Things of various kinds are represented with ellipses, as in Figure B.3a.

[Figure B.3a: Things are represented as ellipses.]

[Figure B.3b: Relations of various kinds are depicted
using lines with various arrowheads.]

The relations between things are depicted using the different types of arrows shown in Figure B.3b. Relations with a solid line are structural, while those relations with a dashed line indicate transformation or movement.

• Association: Association depicts an arbitrary relation between two entities. Since association has no arrowhead, it is a bidirectional relation. The labels on the association line above indicate that it is many-to-one, or that many of the things on the left are associated with one of the things on the right.¹⁶⁶

• Composition: The composition arrow points from a part to its whole.

• Reference: The reference arrow points from a reference to its referent.

• Generalization: The generalization arrow points from a subtype to its supertype (i.e., the thing on the left is a type of the thing on the right).

• Causal Flow: Causal flow shows general movement. The relations of the basic model of cognition are all variations of this arrow.

[Figure B.3c: Example part and type hierarchies.]

Things and relations are used to create hierarchies, as shown in Figure B.3c. The relative left-right position of the children is not meaningful, since no structural element has been drawn between them to establish a relation. That said, it is common to draw hierarchies with the largest or most general node at the top and proceed downwards. The labels on the meronomy at left (0.4 and 0.6) indicate the volume of each part relative to the whole. Since the volume of the table top and table legs add up to 1.0, there are no other parts of the table besides the top and the legs (assuming that the top and the legs do not overlap).

The introduction of labels to indicate the degree of parthood can also be used to depict overlap. As overlap is not defined in UML, it can be visually depicted as bidirectional composition, where the diamond arrowheads corresponding to composition are annotated with the degree to which they are a part. For example, Figure B.3d shows overlap between two parts, where 30% of part₁ overlaps with 40% of part₂.

[Figure B.3d: Overlap of two parts.]

References or reflections can also be depicted as bidirectional, as in Figure B.3e, which illustrates that just as mirrors reflect people that look at them, people reflect mirrors at which they look (in virtue of their mental references).

[Figure B.3e: Reflection of a person in a mirror,
and a mirror in a person.]