The Whole Part

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The Whole Part

http://theWholePart.com

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ArborRhythms Publishing

Eugene, OR


This work is in the public domain.

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All profits from the sale of this book are
donated to non-profit organizations.



Author:  Alec M Rogers


ISBN:  978‑0‑9830376‑3‑7

Library of Congress Control Number:  2020904097

Date:  April 26, 2020


Revision History:

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Preface

The Whole Part describes a basic model of cognition that uses parts, wholes, and references to analyze both our material and mental experience. 

The framework of continuous epistemological space, rather than discrete symbolic logic, is used to provide a formal foundation for thinking about reality. Mereological analysis of that space examines things in terms of their whole/part relationships, and referential analysis of that space examines things in terms of their reference/referent relationships. These analyses are used to illustrate the structure of our minds. Since our mental structures determine how reality is sensed and conceptualized, understanding these structures clarifies which aspects of our experience are due to the world and which are due to various facets of our cognition. 

You will particularly enjoy this book if you are interested in how our minds work, since it explores the structure and operation of our cognition in great detail. To do so, a basic model is constructed that provides an understanding of the relationship of wholes to parts, references to referents, and how those relationships influence and are influenced by cognition. This model is simple enough to be independent of various complexities in neuroscience and physics, although it is both motivated by those sciences and entirely compatible with them.

Finally, although everyone experiences both sensation and conceptualization, it is difficult to fully experientially discriminate between them. This ability is fairly critical; to love the idea of something in place of the thing itself is a recipe for trouble. Therefore, may the model developed in this book serve as a tool to better understand and positively transform our lives.


Sincerely,

∼alec


Background

Introduction

History

Reality

Parts and Wholes

Particulars and Universals

Mind and Matter

Sensation, Concepts, and Symbols

Cognitive Science

Summary

Theory of Mereology and Reference

Mereological Space

Wholes

Parts

Mereological Identity

Mereological Edge Cases

Referential Space

References

Referents

Referential Identity

Referential Edge Cases

Combining Spaces

Meronomies

Taxonomies

Dimensions

Orthogonality

Wholes of References

Beyond Hierarchy

The Basic Model

Epistemic Universes

The Physical Universe

Objective Parts

Objective Dimensions

The Subjective Universe

The Subjective/Objective Dichotomy

Sensory Parts

Sensory Dimensions

The Conceptual Universe

The Sensory/Conceptual Dichotomy

Conceptual Parts

Conceptual Dimensions

Epistemic Relations

Sensation

Action

Conceptualization

Visualization

Interpretation

Symbolization

Recursion

Symbolic Space

The Conceptual/Symbolic Dichotomy

Symbolic Parts

Symbolic Dimensions

Higher‑Order Concepts

Cognitive Meronomies

Cognitive Taxonomies

Language

Semantics

Syntax

Sentences

Practical Implications

Concrete and Abstract

Identity

Absolute and Relative

Negation

Intuition

Animal Cognition

Bottom-Up and Top-Down

Attention

Emotions

Beauty

Kindness

Overlap

Self and Other

Stratified Self

Conclusion

Acknowledgments

Formal Summary

Formal Summary

The Basic Model

Zeroth-Order Logic

Higher-Order Logic

Reference Material

Symbolic Conventions

Typographical Conventions

Ideographic Conventions

Glossary



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). 

To illustrate these mereological and referential relationships diagrammatically, various material and mental things are represented with ellipses and arrows in diagrams such as the following:

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

This diagram shows that:



To understand the notational and diagrammatic conventions used in this book, it is recommended to scan the appendices (more detail about UML diagrams in particular is presented in appendix Ideographic Conventions). The main content of book is composed of four parts: 

Part 1 introduces the book and provides philosophical background for 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 from Part 2 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. 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]


As this work presents a basic model of mind, its focus is epistemological rather than ontological. In practice, the determination of which facets of experience belong to mind as opposed to external reality is a matter of some debate. The scope of this chapter is 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). It 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 is particularly relevant to understanding several important epistemic distinctions, such as the distinction between matter and mind, and between intuitive mind and rational mind. 

Reality

Some of the earliest arguments about wholes and parts derive from Indian philosophical debates about what is ultimately real (paramarthasat) and what is relatively real (samvritisat). These debates are continued in Western philosophy, where the reality of wholes and parts is discussed in the context of natural kinds. The belief in natural kinds entails that the world is naturally or ontologically partitioned into various categories or types of real objects. Belief in natural kinds often leads to one of two general positions: one which asserts the reality of universals (or the abstract attributes of objects) and one which asserts the reality of particulars (or the concrete objects themselves). Therefore, these examples illustrate two different criteria for what is real: one that pertains to material divisibility and one that pertains to the distinction between abstract and concrete entities. 

Parts and Wholes

Throughout history, there have been numerous debates about material composition. In Indian philosophy, these arguments often begin by asking if the whole is the same as or different from its parts. If one assumes (either explicitly or implicitly) that only one thing can exist in a particular place at a particular time, then either the whole or the parts can be real, but not both.[2] In other words, wholes and parts are seen to be ontologically incompatible because they compete for real status at a given location. 

The tension between parts and wholes led many schools of philosophy to posit that parts are real and that wholes exist only in dependence on those parts.[3] These schools are called reductionistic because they reduce large-scale phenomena to small-scale phenomena, so that explanations of wholes are supplanted by explanations of parts. As a result, reductionists tend to believe that things have an intrinsic nature that is defined in terms of the parts of those things. All larger wholes or composites are only real as nominal collections of these smaller atoms. This type of definition is intuitively appealing because large things often visibly undergo more change than small things, since they can be broken into more parts. Similarly, the smaller something is, the less likely it is to be visibly broken down, which makes it relatively permanent (or more real). To explain the variety of experiences that arise from small or partless particles, several types of atoms are typically posited; historically, these often included air, earth, fire, water, and sometimes space. 

Despite the utility of reductionism, it is a bit one-sided as a theory of reality. For example, why would one believe that small internal things are always a better basis for causal explanation than large external things? Similarly, why would one believe that the small is the cause of the large, when causation requires a temporal separation between a cause and its effects that is not present between parts and their whole? 

To remedy the reductionistic bias, holistic philosophies emphasize the opposite point of view: the fundamental reality of the larger whole. Historically, the emphasis on whole-based (or holistic) as opposed to part-based (or reductionistic) explanations of reality often coincides with the schism between the spiritual and the secular. More precisely, spiritual and selfless traditions tend to gravitate toward whole-based explanations of reality, while materialistic and selfish traditions tend to gravitate toward part-based explanations. Perhaps this tendency is not surprising given that what is divine is most often considered as a whole, and one’s self is most often considered as a part.[4]

Numerous non-monotheistic traditions are also holistic traditions. For example, polytheistic Brahmanical traditions extol the universal soul (Brahman) as opposed to the individual soul (paramatman), and non-theistic Buddhist traditions often emphasize dependent co-origination, or the mutual dependence of all parts (pratityasamutpada). In modern Western philosophy, a number of explicitly holistic arguments were made by Hegel, who stated that parts become more complete in the context of their wholes.[5] Interestingly, Hegel’s philosophy is also soteriological. It states that a thesis (an object) entails an antithesis (which relates to that object), both of which necessarily lead to a synthesis (which is the whole consisting of both the thesis and the antithesis). As syntheses become new theses, thought increasingly encompasses the Universe: the absolute whole that is beyond all relativity.

The perceived incompatibility of wholes and parts has also led philosophers to adopt nominalistic stances.[6] The Indian philosopher Chandrakirti (ca. 650 CE), after thoroughly analyzing the ways in which a whole may relate to its parts, finds that wholes are neither the same nor different from their parts. Further, since a given object is not essentially either a whole or a part, he concludes that objects exist only in a relative or nominal sense.[7]

Particulars and Universals

An important distinction between two types of mental content is characterized by various Western philosophers in terms of the requirement to experience an external world. Knowledge about mind and various mental activities does not require such experience, so it is called a priori experience: such knowledge is “prior to” experience. Knowledge about the nature of matter does require experience of the world, so it is called a posteriori: such knowledge is gained only “after the fact” of experience. Typically, a priori experience is characterized by generic universals, and a posteriori experience is characterized by specific particulars. 

Not all philosophies regard universals as mental and particulars as physical. Platonism, one of the earliest holistic philosophies in the Western philosophical tradition, holds that universal properties are real and particular objects are not.[8] For example, unchanging universals such as horseness and brownness intersect to form ephemeral, particular horses. Similarly, the universal of horseness can be regarded as an abstract union of many particular horses. 

Holistic theories such as Platonism, which treat wholes as more important than parts, also tend to prioritize mental and immaterial entities over physical and material entities. This may be a result of the more obvious impermanence of large particulars; in other words, permanent material particulars must be small enough that they are not demonstrably subject to change. Therefore, large particulars are less compatible with holism since they tend to be ephemeral, and being real is commonly associated with permanence. Similarly, immaterial objects such as thoughts are sometimes regarded as permanent because immaterial properties are not visible, and therefore, not subject to visible change. 

As a result, particulars are often seen as small (material) parts, while universals are often seen as larger (immaterial) wholes. This conjunction is expressed in the following table, which should be understood as illustrating a common historical occurrence, rather than a logical necessity:

Table 1.2a: Universals and particulars with respect to the whole/part and mind/matter distinctions.

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This table expresses a tension, since wholes and parts are regarded as having the same material or immaterial substance, while universals and particulars are not. In other words, since wholes are composed of parts, both wholes and parts are seen as material if one is a materialist, or as immaterial if one is an idealist. Almost always, however, universals are regarded as abstract and particulars are regarded as concrete. The view presented in this book is that all sensations are unique; therefore, conceptual wholes composed of those parts are also particulars. However, conceptual wholes can become abstract when they are created top-down out of symbolic references, and it is that abstractness which is characterized as immaterial.[9] 

Mind and Matter

The relation between mind and matter is a primary concern for any model of cognition, since a model that describes how cognition works is bound to be problematic without some notion of what cognition is. In this work, although wholes and parts are necessary to specify the location of those references and their referents, the relation between mind and matter is characterized primarily by references. Words are probably the most well-known form of references, so the study of references is essential to linguistics. References are also essential to thoughts, and therefore, they are essential to psychology. In the context of this book, references are used to define mind. Mind is that which refers to something, in addition to whatever else it may be. In more philosophical language, mind has intentionality, while matter does not.

Of course, the discriminability of mind and matter does not entail that they are entirely different kinds of things. For example, the position called neutral monism maintains that there is only one kind of stuff, which exists prior to being differentiated into mind and matter. This view was described by William James, who said that mind and matter are both “pure experience”, and that the duality of knower and known is a derivative notion. In his view, knowing (or awareness) is not a thing; rather, it is a relation between two parts of experience (i.e., between mental references and their physical referents). 

Neutral monism can also be expressed in physical terms by substituting James’ singular mental event (or experience) with Baruch Spinoza’s singular physical thing (or “the one substance”). In the latter case, because substance takes on attributes of body and mind, even objects such as rocks have some form of mind (a doctrine commonly known as panpsychism). Although this is a very liberal interpretation of mind, it is accurate if one regards the various features of rocks such as its layers and chips as references to (or even as memories of) things that happened to the rock over its existence. This poetic interpretation, of course, does not entail that rocks are smart in the same way as are animals.[10] In particular, although matter may be conscious at some non-referential level, this reflexive or intransitive consciousness is different from intentional mental awareness, which requires some degree of reference.[11] 

The main point of this discussion is that objects belonging to different categories of things may also be seen as belonging to the same, more general category. By recasting objects in this way, it is possible to reinterpret dualistic theories as monistic theories. For nominalistic theories, this reinterpretation is particularly simple, since things are not inherently categorized as wholes/parts or referents/references, just as arbitrary objects are neither small nor big except in relation to other objects.[12] However, if one attempts to avoid characterizing reality by refusing to adopt any analysis or categorization whatsoever, then it becomes impossible to express the nature of its one constitutive type. 

Sensation, Concepts, and Symbols

The distinctions between the sensory, conceptual, and symbolic aspects of mind are essential to models of cognition. These distinctions are known by many names and run parallel to many well-known dichotomies. In Indian philosophy, for example, subjective (symbolic) truths are known as relative truths, and objective truths are known as absolute or ultimate.[13] In terms of Dual Process Theory, sensation and concepts correspond to System 1, and symbols correspond to System 2.[14] Several of these distinctions are shown in the following table:

Table 1.2b: The symbolic, conceptual, sensory, and objective distinctions as they most closely align with several other dichotomies.

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As this table suggests, an essential role of a basic model of cognition is to clarify various aspects of our experience by providing a more precise categorical scheme, such as the one shown in the first column. According to multiple philosophical traditions, this effort is especially difficult for nonconceptual categories because dividing things into wholes and parts, one and many, references and referents, or even mind and matter, are all conceptual (or relative) categorizations.[15] Therefore, to study nonconceptual experience from a conceptual point of view is often regarded as a bit problematic, if not outright paradoxical. However, by analyzing the symbolic/subsymbolic dichotomy from the symbolic point of view, it may at least be possible to better understand the subsymbolic aspects of experience by extricating them from the symbolic aspects.

One difficulty with this table is that it oversimplifies several of its constituent distinctions. In particular, the distinction between absolute/relative is sometimes understood as two omnipresent aspects of reality, while this table indicates that the conceptual category is more relative than the sensory category. Similarly, bottom-up and top-down are better understood as directions instead of absolute locations, so placing them within the context of this table conflates direction with location. That said, this table does approximate the alignment of several relevant distinctions across a wide range of literature, and it raises questions about the relation of these categories to one another which are pertinent to subsequent chapters. 

The distinctions presented in the table are extremely generic categories, so it clarifies things to explore universals that we encounter on a daily basis. Our own self is an important and commonly cited example, to which we return throughout the book. The study of the self has a long history; for example, the Nyaya school of Indian philosophy claimed that the self (or atman) exists within the world as an eternal, singular thing. Later Indian and Buddhist philosophy took exception to this, arguing that neither a personal self nor anything else in the world could be either permanent or singular, and categorizing the self as relative (samvritisat) rather than absolute (paramarthasat). Immanuel Kant made similar claims, enumerating several a priori categories such as quantity (including notions of both singularity and plurality) that can be known independently of the world, thereby ensuring their status as universals and further providing a cognitive foundation for mathematics. The basic model of cognition continues these traditions, distinguishing sensory and conceptual aspects of mind, and discussing how these aspects relate to the objective world.[16] 

Cognitive Science

Modern cognitive science characterizes the relation between subjective and objective (or mind and matter) as indirect realism. According to this theory, everything outside the body is represented inside the body, both symbolically and subsymbolically. However, this view exists within a context of numerous other possibilities:



Since the predominant view is that subjective universes are constituted by references to the larger, physical universe, the basic model developed in this book follows suit.[17] However, all of these philosophical points of view present some true aspect of reality. 

Summary


Leonardo da Vinci [Varzi, 2015]. 


This book uses two types of analyses: mereological and referential. Mereological and referential analyses of a given space are nominalistic analyses that create individual things within that space.[18] They are nominal in the sense that the structure of space is determined by the operation that is used to structure it. Mereological spaces are structured by the whole/part relationship, and referential spaces are structured by the reference/referent relationship. As a result of these analyses, things can be divided into four general types: wholes, parts, references, and referents. However, most things may be any or all of these four types of things, depending on one’s point of view. For example, a word written on a piece of paper is a whole of many cellulose fibers, a part of a book, a reference to a concept, and the referent of this sentence.

As an example, the following diagram shows the way that references play a role in mereological and referential spaces at the same time: 

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Figure 2: One whole and two parts, with a referential relationship between the two parts. 

To make this diagram more concrete, suppose that the diagram describes a room, a vase, and a piece of paper that has the word “vase” written on it (objects that are a whole, a part, and a reference, respectively). Both the vase and the paper are parts of the room, at least temporarily. The word on the paper (“vase”) is a reference to the vase object. Therefore, the word on the paper is both a thing in‑and‑of‑itself and a reference to something else, as indicated in the diagram by the two attached arrows. In other words, it has a dual role: it has a mereological role (in relation to the paper, the ink, etc.) and a referential role (in relation to the referent).[19] Even the vase plays a dual role, since it is both a part and a referent.

The multiple roles of the objects in this example create two different kinds of space: referential space (which is structured by references) and physical or mereological space (which is structured by wholes and parts). As the former space requires subjective interpretation, it may also be characterized as mental space. This illustrates that the concept of space should be understood very generally, as it is used as a metaphor to describe both physical and mental reality in the context of this book. Space itself should also be understood generally in that it is high-dimensional or open-dimensional (a view consistent with modern physics). Understanding physical space in this way is particularly important because a central thesis in this book is that the dimensionality of space is shared with all things within that space. For example, if the physical universe is four‑dimensional, then our world contains four‑dimensional objects as parts, and cannot contain three‑dimensional objects except as abstractions.[20] 


Mereological spaces are structured by the whole/part relationship. Therefore, for physical space to be structured mereologically means that there is a whole/part relationship in the world. Sensory spaces can also be structured mereologically; for example, the sensation of a tree can be a part of the sensation of a forest. However, the relation between a referent and its reference is not mereological (i.e., symbols are not necessarily either parts or wholes of what they symbolize). 

All parts of a mereological space share the dimensionality of that space. 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:[21]


  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 as follows:

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Figure 2.3a: The five possible mereological relations.

The fifth case in this figure is somewhat problematic, because there is no single, definite 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. 

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 follows:[22]

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Figure 2.3b: Reclassifying overlap as three non‑overlapping parts.

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).[23] Therefore, a continuous version of logic is introduced in appendix Formal Summary.


2.3.1 Wholes

Wholes are composed of their parts.

The following picture shows a whole that is composed of two parts:

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Figure 2.3.1: A whole in relation to two parts.

It is common to read figures from the top down, which implies that the previous diagram shows a whole being made of two parts. However, the orientation of the figure makes no functional difference, since the structural relationship is indicated entirely by the arrows. Therefore, from a procedural point of view, a branch like the one depicted can be understood as either a division or a collection. As a division, it creates parts out of a larger whole. As a collection, it creates a whole out of smaller parts. 

Because there is no mereological relation depicted between Part1 and Part2, one might assume that they do not overlap. However, this may only indicate that overlap is difficult to express using the part/whole relation in conjunction with symbolic logic. This inability to easily express overlap is especially problematic given the abundance of overlap in the world. For example, although a tree does not overlap itself, arbitrary parts of a tree do; the bark of a tree is partially coextensive with the trunk of a tree, so neither is entirely a part of the other. 

As a result, any overlap which is not analytically removed is most often represented in separate hierarchies (i.e., mereological space within a single hierarchy most often has non-overlapping parts). Further, it is common for the parts within a single hierarchy to form a partition of that whole. Parts form a partition of their respective whole if the parts do not overlap and the space of that whole is completely covered by those parts. For example, apples and oranges do not form a partition of fruits, because there are some fruits that are neither apples nor oranges.[24]


2.3.2 Parts

Parts compose their wholes.

The following diagram depicts exactly the same information as the diagram in the previous section: a whole that is composed of two parts.

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Figure 2.3.2: Two parts in relation to a whole.

Assuming that the whole depicted in the figure has exactly two non-overlapping parts illustrates that the same object can be nominally constituted by one whole or two parts.[25] In other words, because the boundary is nominal, the presence or absence of a boundary does not materially affect the composition of the whole. In other words, although the universe may be mentally divided into things, the dividing lines do not have any concrete existence.[26] Although this may seem trivial, it implies that the boundary that does the dividing does not occupy the same space as the things it divides. For example, when points divide lines, 0‑D things partition a 1‑D space. The same principle applies to lines dividing planes; the extent of the partitioning element is always zero along the dimension of the space that it divides. [27]

Although this may not sound significant, it is an important departure from point‑set topology, which is the field of mathematics that is typically used to describe space. Point‑set topology assumes the existence of partless parts (or points) that constitute spaces of higher dimensionality. For example, a three‑dimensional space is made up of an uncountable infinity of zero‑dimensional parts. While this analysis works mathematically, it is problematic from a psychological point of view (for further details, see appendix Formal Summary). 

Ignoring various intricacies of mathematical philosophy, several common principles of point-free-topologies are:





2.3.3 Mereological Identity

Mereological identity entails both internal identity and external identity.

Two things are mereologically identical if they have both the same wholes and parts.

If two objects have the same parts, they are internally identical. For example, the internal or material identity of a car may entail that it has four wheels, two axles, a body, a steering wheel, and an engine as parts. Internal identity is closely bound up with essentialism, since the essence of a thing is generally considered to be internal to that thing.[28] 

If two objects have the same wholes, they are externally identical. External identity is often stated in terms of functional properties. For example:

A thing is a chair if you can sit on it. 

However, all definitions using properties can be stated using the language of wholes and parts, so the previous sentence can also be expressed as:[29] 

A thing is a chair if it is a part of the class of all things on which you can sit.

In order to illustrate mereological identity, the following diagram depicts several of the wholes and parts of ravens, writing desks, and worms:

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Figure 2.3.3: The mereological identity of ravens and writing desks.

In this mereological context, ravens, writing desks, and worms all are parts of larger wholes. Ravens and writing desks are also wholes of smaller parts, while worms are atoms because they have no points.[30] Because ravens and writing desks have the same wholes and parts in this limited context, they are externally and internally identical, and therefore mereologically identical. In mathematical terms, there is an isomorphism between ravens and writing desks.


2.3.4 Mereological Edge Cases

There are two mereological edge cases: wholeless wholes and partless parts.

The edge cases of the whole/part continuum are whole‑less wholes and part‑less parts, or ultimate wholes and ultimate parts. Ultimate objects are particularly interesting because things with no wholes lack external properties, and things with no parts lack internal properties. This makes them paradoxical to talk about or even think about, although they figure into our mental lives frequently.

In mathematical mereology, a space with upper or lower limits on the operation of parthood is called bounded above or bounded below, respectively. If there are no such bounds, that space is called open (i.e., open above or open below).

2.3.4.1 Ultimate Wholes

A mereological thing that has no wholes is an ultimate whole, or a universe.

The ultimate whole is the biggest thing. Only one ultimate whole can exist within a single mereological space, since if two ultimate wholes existed in the same space, then there would be a larger whole that consists of both.[31] In an epistemological context, the ultimate whole is similar to the mathematical notion of the set of all sets. In an ontological context, the ultimate whole is known as the universe. 

The term “universe” derives from the Latin universum, meaning “[every object] combined into one”. Unfortunately, this term also designates an object that is not the largest thing (i.e., if one believes in things that extend beyond the universe, such as multiverses). In this context, the word “universe” is used as originally intended; the universe is that thing which is a whole of everything else. As with space, the universe is open-dimensional. Therefore, although the universe is typically described with spatial (i.e., atemporal) metaphors, it is better understood as a long‑lasting event because it has both a spatial and a temporal extent. 

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Figure 2.3.4a: Universes are wholeless wholes.

Because there is nothing to which the universe can be compared other than itself or parts of itself, it is difficult to define. You could call it large in relation to its parts, but that would entail that all things are large things. The universe cannot have external properties, so it cannot be big or small, heavy or light. Since definitions are always given in terms of other things (i.e., they are relative), it is not possible to define “the one without a second”.[32] In other words, to call the universe a unity implies that there is some plurality with which it can be contrasted, which is not the case. But even though it may be mistaken to make a relative statement about the universe, it may help to counter prior misconceptions. For example, the assertion that the universe is one entity might be correct in so far as it counters the assertion that the universe is a multiplicity of independent things. 

Conceptually, the universe is a space without limit. In order for a universe to be finite, which is implied by a whole that has no further wholes, space must be closed.[33] Closed space is paradoxical, though, since one could imagine going to the end of the universe, and then moving a bit beyond that point. If that is possible, then the universe is not closed. If that is not possible, it suggests that the obstruction is a thing that lies beyond the boundary, which also entails that the world is not closed. In other words, if boundaries exist only as the division between parts, then wholes which are not parts in any larger whole do not have boundaries. On that account, universes cannot have a boundary because there is nothing for them to be divided from, so they are unbounded by definition.[34] 

2.3.4.2 Ultimate Parts

A mereological thing that has no parts is an ultimate part, or an atom.

The ultimate part, or the atom, is defined in this context as the smallest thing. The term “atom” derives from the Greek atomos, which means uncuttable or indivisible.[35] For space to be atomic, or to have parts which are atoms, means that the process of creating parts cannot occur indefinitely. In other words, they are things that cannot be subdivided.

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Figure 2.3.4b: Atoms are partless parts.

Atoms have no parts in virtue of which they can be distinguished from one another. Therefore, all atoms are internally identical, and can only be distinguished from one another in virtue of their larger wholes. As a result, the atom represents the limit of reductionism as an explanatory strategy; it is not possible for reductionism to explain what an atom is in terms of its parts, because it has no parts.[36] 


Referential spaces are structured by the reference/referent relationship. The things that are referenced are called referents, and the things that refer to them are called references. References may also be known as symbols, signifiers, or names. References are directional, in that references point to their referents. 

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. However, the references and the referents on which they depend also form parts of a mereological space. References are discrete, although reference may in general be continuous (in which case it may be 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 depicted in the following diagram:

Image

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

As the preceding figure illustrates, references form referential chains (depicted vertically) that pass through multiple referential levels (depicted horizontally). References from higher levels map onto their referents in lower levels. Things on the bottom level of the diagram have a referential level of zero and are things in themselves, which entails that they are not references to anything else. The referential level of each higher level is one more than the previous level. Therefore, referential level indicates the distance of a reference from ground, or its ultimate object of reference. 


2.4.1 References

References refer to referents.

The word reference derives from the Latin referent, which means “bringing back”. References are things that are capable of referencing or denoting a referent. The existence of a reference implies that its reference can be found in the world. There may be multiple references for a single referent, but a single reference can only refer to a single referent. The following figure depicts the simplest relationship between a reference and its referent.

Image

Figure 2.4.1: A reference in relation to its referent.

References are both things themselves as well as things that refer to other things. References do not need to resemble their referents in any way, a principle known in linguistics as the arbitrariness of signs. Therefore, the importance of references qua references does not derive from what they are, but from what they represent or signify.[37] They play a crucial role in cognitive psychology, where the referenced objects of the external world become referential parts of the space of subjective awareness.


2.4.2 Referents

Referents are denoted by references. 

Referents are the things to which references point: they are referenced, denoted, or named by these references. 

Image

Figure 2.4.2: A referent in relation to its reference.

Referents may be denoted by multiple references, and they may in turn be references to other referents, but they do not have a mereological structure. Thus, although referents may have dimensionality in virtue of being parts of mereological spaces, their only structure within referential space is derived from their referential relationships. In other words, references and referents are atoms within discrete referential spaces. Consequently, there is no such thing as a half of a reference. 


2.4.3 Referential Identity

Referential identity entails that two references refer to the same referent.

The notion of identity can be applied to references in at least two ways. Referential identity means that two things are identical if they have the same referent. For example, the references “the first man to walk on the moon” and “Neil Armstrong” have the same referent, so these two statements are referentially identical. References also establish an identity between a reference and its referent. Since references can represent something unlike themselves, the validity of the reference/referent mapping is established via isomorphism. For example, isomorphism between a reference and its referent, such as the concept riverφ and the physical river, entails that the concept relates to other concepts just as the river relates to other physical objects. In this way,  isomorphism allows the reference to be independent of its referent, such that the concept of the river may be contextually similar to the physical river, without needing to have identical physical characteristics, such as being physically wet.


2.4.4 Referential Edge Cases

There are several referential edge cases: ultimate referents, ultimate references, self‑referential references, empty references, and full references.

There are two primary edge cases in referential spaces: ultimate references and ultimate referents. Ultimate references cannot be referenced, and ultimate referents are non‑referential. Unlike parts, which require a relation between exactly two objects, references have two additional edge cases that involve only one object: references that refer to themselves (or self-references) and references that refer to nothing (or empty references). This section also introduces full references, which are the complement of empty references.

2.4.4.1 Ultimate Referents

An ultimate referent does not reference any other referent.

Image

Figure 2.4.4a: An ultimate referent is not a reference to any other referent.

In some philosophical theory such as Kantian metaphysics, ultimate referents are things-in-themselves (ding an sich), which form a referential ground of being; without them, references become vacuous. In other words, ultimate referents are particularly significant because they create a grounding that gives all other references meaning. If this grounding is not present, then all references become meaningless. However, what are ultimate referents from one point of view may not be ultimate references from another, so while it may be necessary to begin with a referential ground, that referential ground is not necessarily unique. 

A profound example of references with no ultimate referents is a dictionary. All the words in a dictionary are defined by other words. Therefore, their definitions consist of references that are eventually circular. For example, if an alien who didn’t know any words tried to find the definition of a word in a dictionary, the definition of that word would necessarily contain other words that the alien does not know, which would entail looking up further words. This situation involves an infinite regress; if the alien does not know the meaning of any words to begin with, how is it possible to learn even one of them? The only way to break this endless cycle is to know the meaning of one or more of the words: in that context, those words become ultimate referents.[38] 

2.4.4.2 Ultimate References

Ultimate references cannot be referenced.

Image

Figure 2.4.4b: An ultimate reference is not a referent of any other reference.

NB: It is not possible to talk about ultimate references, because to do so would be to refer to them.

2.4.4.3 Self‑Referential References

Self‑Referential references refer to themselves.

Image

Figure 2.4.4c: A self‑reference refers to itself.

Self‑referential references seem possible, unlike the obviously impossible example of a part that contains its whole. However, self-reference is problematic even if it is possible, since self-reference is the basis for innumerable paradoxes. A common example of self-reference is the Liar’s Paradox, attributed to Parmenides (ca. 485 BCE):

I am lying.

It seems rather innocuous at first, but it presents a question that is difficult to assess. If it is assumed to be true, then it becomes false. At the point of being negated, it becomes true once again. Establishing the ultimate validity of this sentence is therefore impossible, as it involves an infinite regress.

Probably the most significant thing about Parmenides’ statement is that it is self‑referential: it describes itself. Hence, a reasonable first step in the elimination of paradoxes eliminates self‑reference. Unfortunately, the recognition of self‑reference is confounded because reference does not have to be immediate; it can be a multi‑step, circular phenomena, as illustrated by the two statements below:


  1. The statement below is false.
  2. The statement above is true.


The paradox in this case is more difficult to spot, but it is still an example of self-reference if both statements are considered as a single whole that refers to itself.[39] The first statement is neither true nor false until it is evaluated. Under the assumption that it is true, the truth of the second statement is negated. This negated statement makes the assertion that the first statement is false, which contradicts the original assumption, and so on, such that the truth conditions never converge. 

2.4.4.4 Empty References

Empty references refer to nothing.

Image

Figure 2.4.4d: An empty reference does not refer to anything.

Although almost all referential relations involve a referent, there is one important exception: the empty reference, which is a reference whose referent does not exist or which refers to nothing. Since it has no referent it is not clear that it is even a reference, although it is widely used in practice. 

2.4.4.5 Full References

Full references refer to everything.

Image

Figure 2.4.4e: A full reference refers to everything.

Another unusual reference that bears mention is the full reference, which refers to everything, or the mereological sum of all things at the referential level to which it refers. It is particularly significant because it forms a trivial isomorphism between two referential levels, since it is necessarily well-defined (i.e., it refers to all referents at a given referential level). In other words, it does not require any discrimination between various objects of reference.


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 next to one another. Hierarchy can also be described using the tree metaphor, although the trees are depicted as branching downwards instead of upwards. When using the tree metaphor, 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 closely to mereological and referential spaces: meronomies and taxonomies. 


2.5.1 Meronomies

Meronomies are hierarchies that are structured with the composition relation.

Meronomies are mereological hierarchies in which the children are parts of the parent. Meronomies simultaneously represent divisions of a larger whole and collections of parts, so if the trunk is‑a‑part‑of the tree then the tree is‑a‑whole‑of the trunk, and vice‑versa. The following figure depicts the meronomy of a tree, whose parts are a trunk, roots, branches, and leaves. It depicts those parts using the composition relation, which is often called the has‑a relation, as in “a tree has‑a trunk”.

Image

Figure 2.5.1a: A meronomy is a hierarchy that uses the composition relation.

It is not possible to create meronomies with abstract child nodes; the rationale behind this is based on their cognitive structure, and is discussed in section Cognitive Taxonomies. Therefore, even though it is true that a pine is a tree, a maple is a tree, and an elm is a tree, meronomies like the following are invalid (where the superscript “+” denotes that the entity is abstract):

Image

Figure 2.5.1b: A meronomy is invalid if its child nodes are abstract.

On the other hand, it is possible to create meronomies that have a discontiguous but concrete root node. A meronomy with a discontiguous whole is shown in the following diagram, where the whole (“All trees”) is composed of parts that are concrete collections of different trees (“All pines”, “All maples”, and “All elms”):

Image

Figure 2.5.1c: A discontiguous meronomy.

Although possible, concrete discontiguous meronomies are somewhat unusual, perhaps because it is easier to form concrete concepts of contiguous objects than of discontiguous objects (for more detail, see section Symbolic Space). 


2.5.2 Taxonomies

Taxonomies are hierarchies that are structured with the generalization relation.

Taxonomies are type hierarchies, in which the child things are kinds of the parent things. The primary difference between meronomies and taxonomies is that taxonomies are composed of abstract things: they are composed of types or classes of things rather than concrete things, and they entail the use of references (taxonomic construction is described in section Cognitive Taxonomies).

The following taxonomy illustrates the abstract type tree, which is composed of three other types of things: pines, maples, and elms (where the subscript φ denotes that the entity is a concept). The empty triangle denotes the generalization relation, which is often called the is‑a relation, as in “a pine is‑a tree”.

Image

Figure 2.5.2: A taxonomy is a hierarchy that uses the generalization relation.

In common English usage, the difference between meronomies and taxonomies is that a pine is‑a tree, but a trunk is‑a‑part‑of‑a tree (or a tree has-a trunk). However, the collection of all pines is a part of the collection of all trees, and the collection of all trunks is also a part of the collection of all trees, which illustrates the transitivity of meronomies (a feature not shared by taxonomies). 


2.5.3 Dimensions

Dimensions are axes along which a thing can be differentiated.

Dimensions are the measurable extents of space or things in space. The number of dimensions of a given space is the number that are necessary to specify a unique location within that space. The general notion of dimensions used here encompasses three of the four types of scales: ordinal, nominal, and interval (ratio scales are omitted). These types of scales correspond to sorted, unsorted, and measurable dimensions, as illustrated in the following examples:




2.5.3.1 Ordinal Dimensions

Ordinal dimensions are sorted by a structural element such as the composition relation.

In an ordinal dimension, things are sorted. More precisely, an ordinal dimension imposes an order or partial order on the elements that it structures. As an example, finishing first, second, or third in a marathon constitutes an ordinal dimension. Knowing the position does not tell you exactly what the finishing time was, but it does convey that one time is greater or lesser than another.

A diagram depicting an ordinal relationship is shown below. Because parts must be smaller than wholes, an order is established between the nodes labeled “tree”, “leaf”, and “branch”. Ordinal relations are transitive: if a leaf is a part of a branch, and a branch is a part of a tree, then a leaf is a part of a tree. Similarly, if a dog is an animal, and a Labrador is a dog, then a Labrador is an animal.

Image

Figure 2.5.3a: Ordinal dimensions sort their elements.

As can be seen from the diagram, an ordinal dimension imposes an order which is not present among siblings (e.g., there is no order between different types of animals).

2.5.3.2 Nominal Dimensions

Nominal dimensions are constituted by unordered entities, which are distinguished only by name.

A nominal dimension is unordered, since it has no basis to assign relative positions to things. The following figure depicts types of animals and parts of trees, both of which are nominal dimensions because the position of the child nodes is not significant (and therefore they must be distinguished by name).

Image

Figure 2.5.3b: Nominal dimensions name their elements.

For taxonomies, nominal dimensions create nominal identity. Nominal identity can be understood as the creation of a parent type which contains all of the children as subtypes or tokens, which thereby creates an equivalence class over the children. In the example above, cats and dogs are the same in that they are both animals.[40] Nominal identity is explored further in section Wholes of References.

2.5.3.3 Interval Dimensions

Interval dimensions are formed by combining nominal and ordinal dimensions.

An interval dimension introduces an additional relation between its parts that results in a measurable distance (or metric) between those parts. As a numerical example, 1 is the same distance from 2, as 2 is from 3, as 3 is from 4, precisely because the distance from one to the next is the same number (1). The following figure depicts structures corresponding to this numerical example:

Image

Figure 2.5.3c: Interval dimensions combine ordinal and nominal dimensions.

The same node cannot appear more than once in a meronomy, so the meronomy on the left depicts the repeated unit element (1n) as different things. These different things form an equivalence class in the taxonomy on the right, which is what makes them the same (i.e., all “1”).


2.5.4 Orthogonality

Two dimensions are orthogonal if a change in one dimension does not necessitate a change in the other.

When working with multiple dimensions, it is useful to characterize them with respect to one another as being either dependent or independent. If dimensions are independent of one another, they are called orthogonal. For example:



Diagrammatically, orthogonality determines how hierarchies are combined with one another. For example, the following figure shows two taxonomies dividing the same whole (x) into types a, b, c, and d:

Image

Figure 2.5.4a: Two hierarchies representing different divisions of the same whole.

These hierarchies can be combined in two ways: either by grafting parent nodes together or by appending the branches of one hierarchy to each of the terminal nodes of the other. The criteria for choosing one method as opposed to the other is to both prevent overlap of the nodes and to ensure that the space is fully partitioned.

The first possibility for combining these two taxonomies is to create a single tree with a parent (x) and four child nodes (a, b, c, and d). For example, if the a/b and c/d distinctions both correspond to color, and no object can have multiple colors, then the two dependent hierarchies can be combined as follows: 

Image

Figure 2.5.4b: A combined (1‑D) hierarchy.

This type of combination is not appropriate when the a/b and c/d dimensions are orthogonal.[41] For example, if the a/b dimension represents “young animals”/“old animals”, and c/d represents “stupid animals”/“smart animals”, then combining these dimensions in a flat hierarchy would not allow categorizing animals that are both smart and young. 

In the case where the hierarchies are orthogonal, combining them requires appending the branches of one tree to each of the terminal nodes of the other tree, as follows: 

Image

Figure 2.5.4c: A combined (2‑D) hierarchy.

This kind of combination increases the depth of the hierarchy, and the additional structure allows additional information to be encoded. Each of the four different choices corresponds to a path from the root of the tree to a terminal node, or a selection from each of the two constituent categorizations, young/old and smart/stupid. As a result, the previous dilemma of how to categorize smart, young animals is resolved (and corresponds to the rightmost path in the tree). 


2.5.5 Wholes of References

Wholes of references are wholes of parts that are also references in a referential space.

In mereological space, things can be both wholes and parts. In referential space, things can be both references and referents. The combination of mereological and referential spaces results in further subcategories:



The arbitrary combination of wholes, parts, and references in the context of cognition is not always useful, since many of these combinations do not exist. For example, it is not possible to form parts of references, so that combination is not explored further. It is possible, however, to form wholes-of-parts-that‑are‑references (or wholes of references), and that construct is widely used. 

A whole of references exists within mereological space; however, each of the parts of that whole (i.e., the references) also exist within referential space. Wholes of references are unlike wholes of parts for the following reasons:


  1. Wholes combine their parts in a single space.
  1. The dimensionality of continuous parts is equivalent to the dimensionality of their wholes.
  1. References are discrete (i.e., they are atoms).
  2. Combining discrete entities into a single space creates a discrete space whose dimensionality is one higher than its constituent referential atoms (i.e., since combining discrete entities in a single space is not possible without doing so). 


In a cognitive context, wholes of references are called symbols, which are explored further in section Symbolic Space. For now, it is sufficient to note that wholes of references are the basis of generalization (the operation used to construct taxonomies). Therefore, wholes of references are the basis of nominal identity (i.e., two things are nominally identical if they are tokens of the same type). 


2.5.6 Beyond Hierarchy

Hierarchies are a pragmatic but limiting structure for comprehending reality.

Although hierarchies are useful and easy to grasp conceptually, there are several problems associated with the reification of hierarchies:


  1. Hierarchies have only one root, which reinforces the notion that all parts have a single whole.
  2. A single hierarchy categorizes things in only one way, so an object is often seen as exclusively a part, whole, reference, or referent (or some combination thereof) within that hierarchical context. 
  3. Hierarchies do not have overlapping nodes; therefore, they implicitly reinforce the notion that there is a single correct way to partition reality.


The following figure illustrates two alternatives to hierarchies: 

Image

Figure 2.5.6: Three different ways to organize nodes.



The basic model of cognition begins with the universe, or physical space, and defines two further subspaces from the perspective of an individual. Everything within physical space that is experienced by an individual constitutes subjective space, and everything within subjective space that is conceptualized by that individual constitutes conceptual space. In summary: 



These spaces are defined in terms of reference: subjective space contains references to physical space, and conceptual space contains references to subjective space. They are epistemic spaces, since the notion of reference in a psychological context corresponds to knowing. In other words, for one thing to reference another thing is for it to know about (or represent) that other thing. Therefore, minds are collections of references, in addition to whatever else they may be. 

These epistemic spaces can be depicted in virtue of their referential/reflective relationship to one another as follows:

Image

Figure 3a: Three referential spaces.

In addition to existing in a referential relation to one another, these three epistemic spaces are parts of one another: subjective space is a part of physical space and conceptual space is a part of subjective space. This is graphically depicted in the following two diagrams, both of which express the same thing: 

Image

Image

Figure 3b: Different ways of depicting spaces as parts of one another.

Using the second type of notation (UML), these three spaces can also be depicted as parts that are references to each other: 

Image

Figure 3c: Three spaces as referential parts.

When these spaces are viewed as parts of one another, the complements of those parts with respect to their wholes creates two further spaces, objective space and sensory space.[42] Objective space is not sensed (O=U‑S), and sensory space is not conceptualized (N=S‑C).



It is also useful to distinguish a symbolic space that is a specialized part of sensory space, and whose symbols reference concepts:



Diagrammatically, these six spaces can be depicted as follows:

Image

Figure 3d: The five spaces (U, O, S, N, C, V), seen as referential parts of one another. 

The objective (O), sensory (N), and conceptual (C) spaces, and the relations between them, form the basis for a basic model of cognition that is used throughout the book. Due to the way in which they are defined, they form a partition of physical space. They are created by two dichotomies: the subjective/objective dichotomy and the sensory/conceptual dichotomy. 

The subjective/objective dichotomy is the basis for two important relations, sensation and action, which are depicted in the context of objective and subjective spaces as follows: 

Image

Figure 3e: The relations between the subjective and objective spaces.



As a model of cognition, Figure 3e corresponds closely to behaviorism, although stimulus and response are replaced with the more humanistic terms sensation and action. Further, the model developed here presents subjective space as a mind with a point of view, rather than as an object. 

The sensory/conceptual dichotomy divides subjective space into conceptual and nonconceptual content, which can be represented as follows: 

Image

Figure 3f: Dividing subjective space into sensory and conceptual spaces.

As mentioned, subjective space consists of references to physical space. Subjective space represents mereological and referential relations between objects in the world by structuring the references that it contains. That structure entails relations between the conceptual and sensory spaces, as illustrated in the previous diagram.[43] 

Sensory and conceptual spaces, although they are modeled as discrete entities, form a continuum whose content is sensation, and which is collected into conceptual wholes as one moves from sensory space to conceptual space.[44] Movement in the opposite direction is also possible, which entails creating increasingly sensory perceptual parts from conceptual wholes. These two operations are called conceptualization and visualization, respectively. In terms of the last chapter, conceptualization and visualization correspond to the mereological operations of whole and part:



In addition to the mereological relations corresponding to whole and part, there are two referential relations which capture symbolic relations between conceptual and sensory space (or more specifically, between conceptual and symbolic space): 



These two relations associate concepts and symbols, and thus map between conceptual space and symbolic space. Symbols are modeled as sensations that have symbolic meaning, so the symbolic space that they form is a subspace of sensory space. Hence, symbols are significantly different from concepts; while concepts provide understanding in virtue of being wholes of sensory content, symbols are representations that enable language and Type 2 thought (symbols are explored in section Symbolic Space).

Combining Figure 3d with Figure 3f and adding interpretation and symbolization yields the following model: 

Image

Figure 3g: The basic model in detail.

In the interest of simplicity, this diagram is simplified into the following basic model of cognition, whose three nodes are mutually‑exclusive spaces, and whose edges (or dashed arrows) depict the flow of information between those nodes:[45]

Image


The next three chapters examine the physical, subjective, and conceptual spaces in detail. Since each of them is boundless from its own point of view, they are referred to as universes. The subjective universe is a complete whole or totality, since a given individual can never have experience outside of it. Similarly, conceptual space is also a universe, in that nothing can be conceptualized which is not a concept.[46] 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.

It is important to indicate the intended epistemological space when referring to something, as it can be complicated to keep track of things that exist in multiple epistemic universes. 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 they exist in each epistemological space. These types are represented in the following taxonomy, where the root node (thing) is a generic type that can be used in any epistemological space, and its subtypes refer to things within each subsequent epistemic space (which mirrors Figure 3g):

Image

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

The following four epistemic types are of primary importance in this work:[47]

Image

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

The subscripts (or lack thereof) designate their associated entities as either objects, sensations, concepts, or symbols. For example, orangeφ refers to the concept of an orange (for more details, see appendix Typographical Conventions).


3.6.1 The Physical Universe

All events are parts of the physical universe.

The physical universe includes everything from the physical point of view.[48] It occupies the full range of every dimension which is attributed to it, including the temporal. It contains all things as events, which form parts of it. Even minds are a part of the universe, whether minds are equated with brains or something else. Thus, from the objective point of view, sensations and concepts are kinds of objects (i.e., they have a physical realization). 

Image

Figure 3.6.1: Sensations and concepts <pdf>
</pdf>depicted as types of objects.

3.6.1.1 Objective Parts

The parts of objective space are called objects.

What are the primitives of reality? Are there things out of which reality is composed, such that there is a unique decomposition into certain parts and not others? Or if the world may be both conceptually and physically partitioned in numerous different ways, is it at least possible to characterize which types of things constitute valid parts? 

One way of answering the question about which types of things exist relies on the dimensionality of those things. For example, the theory called presentism maintains that physical things are 3‑D entities that arise and pass away each instant. The theory called eternalism posits that physical things are 4‑D entities, which have an inherent temporal aspect of which is perceived only a temporal slice.[49] 

If one subscribes to the theory that objects share the dimensionality of the objective universe, then knowing the dimensionality of space decides between these two alternatives.[50] For example, if the universe only exists at one time, then it could contain only objects that exist entirely within the present, and not objects that have a temporal extent. Similarly, if the universe is at least 4‑D, then the objects within it are also at least 4‑D. For example, if a thing such as a bike ride is 4‑D, then the parts of that bike ride (such as the start, middle, and finish of the bike ride) are also 4‑D events; however, the bicycle itself is not a 4‑D part, and therefore does not exist as a concrete entity. 

For most people, treating objects as 4‑D things entails a shift in both mental perspective and in talking about them. For example, 4‑D things are not alterable (or mutable): only objects without a temporal extent can undergo change or vary as a function of time. In other words, if objects do change, they must change in a dimension other than the four which serve to define them as objects. Further, if 4‑D things are concrete, then 3‑D things are abstract, a notion that runs counter to our typical worldview.[51]

3.6.1.2 Objective Dimensions

The objective dimensions form a conditional space.

To examine objective space, it is necessary to take a perspective which is outside of any particular subjectivity, or which is valid from within every subjective perspective. In order to do that, objective truths form a set of relative laws. Therefore, descriptions of the universe take the form of conditional propositions, or experiments that one can perform in order to test their validity. If the statements are valid for everyone, then they are objectively true.

This underscores an important fact about the dimensionality of objective space: the number of dimensions of physical space is the number of dimensions that are used to formulate its physical laws. 

The Number of Objective Dimensions

There are at least four physical dimensions.

The universe is often defined as “everything that exists”, which is slightly misleading in that it denotes only the present moment. In particular, the notion of what is current turns out to be subjective; the sequence of events in 3‑D space depends on the observer’s frame of reference, and is not the same for all observers. Therefore, the Euclidean concept of an extended 3‑D space that exists at a single time is untenable: there is no single time for all positions. A spacetime consisting of at least four dimensions is necessary in order to paint a consistent picture of an objective space that is valid for all observers and which does not depend on subjective reference point.[52] 

However, the necessity of four dimensions does not entail an upper limit; the dimensionality that is ascribed to space grows as necessary to accommodate the physical equations that are used. On a practical level, the number of dimensions may be limited by the number required to change from one subjective reference frame to another, since that is all that it takes to make different observations agree with one another. That said, although the equations that express physical laws such as relativity are easily expressed in high-dimensional spaces, they can also be expressed in spaces of arbitrary dimensionality or other coordinate systems.[53] For example, some theories of physics such as string theory use ten or more dimensions, while people from Flatland get along with only two. 

On a theoretical level, there is no reason to posit any particular upper bound; space has as many dimensions as are used to describe it (for a related thought experiment, see section Symbolic Dimensions). Therefore, the physical universe is characterized as open-dimensional.

The Nature of Objective Dimensions

The physical dimensions are usually described as Euclidean and continuous.

Do the physical dimensions extend infinitely in all directions, or are they finite? Are they continuous and infinitely divisible, or are they discrete? Most people are either implicitly or explicitly committed to Euclidean dimensions, or to dimensions that extend in orthogonal directions from an arbitrarily assigned origin. This understanding of dimensionality is probably the simplest possibility, so it is adopted here.[54]

To ask if space continues infinitely is equivalent to asking if it is bounded, or if it is an ultimate whole. As discussed previously, boundaries are paradoxical if they are used as endpoints to finite spaces, because the nature of boundaries is to divide things. One-sided endpoints, however, do not provide two sides to divide. Therefore, space is most often understood as boundaryless and extending infinitely. 

Just as the spatial extent of the universe may extend infinitely, so may its temporal extent. Finite temporal boundaries are generally expressed as moments of creation or destruction. These two temporal endpoints are often regarded in the same way, but not always; some people believe the temporal dimension extends infinitely in only one direction. Most physicists, for example, believe in a beginning of time called the big bang, but debate about whether time will have an ultimate end, which they call the big collapse. In non-European cultures, non-linear views of time are relatively more prevalent. For example, members of the North American Hopi tribe see time as circular and the Indian Vedic tradition envisions epochs of time as recurring perpetually.

Space itself is generally regarded as infinitely divisible and thus continuous, although there is some controversy about the continuity of matter that manifests in modern theories of quantum physics (for more details about topological continuity, see appendix Formal Summary).


3.6.2 The Subjective Universe

The subjective universe is that part of the physical universe that is experienced by an individual.

Although subjective space is only a part of the physical universe, it is everything from the subjective point of view. There is no way to experience anything else, because all objective reality is necessarily contained within sensation. Only through subjective experience can one learn about the objective world: a world independent of experience, in which objects persist independently of any observation of them. This book follows an inverted development in this respect, in that it begins with the physical universe.[55] 

The subjective universe consists of both sensation and conceptualization; this chapter focuses on sensation, since it is epistemically prior to conceptualization (which is addressed in the next chapter). The term “sensation” as used in this context covers all nonconceptual experience; for example, even emotions are sensed, even though the sensation of emotions is quite different from the sensation of external phenomena. 

From the sensory point of view, objectsψ and conceptsψ are kinds of sensations (i.e., they have a sensory realization).[56]

Image

Figure 3.6.2a: Sensations of concepts and objects, depicted as types of sensations.

3.6.2.1 The Subjective/Objective Dichotomy

The referential division between subjective/objective is similar to the mereological division between body/world.

A subjective space is defined as that part of the physical universe which is experienced by some individual. In virtue of this, the subjective perspective is localized in both space and time. One learns about objective space in virtue of both direct experience with the world and cultural transmission. For example, during the stage of development known as object permanence, children learn that there are things that are not experienced, but which still exist (i.e., in objective space). These objects collectively form objective space, and are the complement of subjective space with respect to the physical universe. In virtue of these two spaces, the subjective/objective dichotomy is formed.

The subjective/objective dichotomy is very different from the body/world dichotomy (where “world” in this context means the world without that particular body). By definition, the subjective/objective boundary is determined referentially, while the body/world boundary is determined mereologically. As a consequence, if someone does not sense the hair on the back of their head, then it is not a part of their subjective space, even though it is a part of their body. Conversely, the hair that someone sees on the back of someone else’s head is a part of their subjective space.

Image

Figure 3.6.2b: Physical space can be divided mereologically (as body/world) and referentially (as subjective/objective).

These two dichotomies within physical space greatly influence how people define the self/other dichotomy. The self/other dichotomy is most often characterized as creating non‑overlapping parts of a larger whole, which is to some extent the result of a mereological definition rather than a referential definition. Perhaps the self, understood as a collection of parts that are references, is more easily seen as parts than references. This bias may occur because parts are a necessary prerequisite of references to parts, because of cultural transmission, or due to a legacy of language and thought that establishes the border between self and other as the border of the nervous system. But do we experience our references as things within us, or do we experience what our references reference, and therefore as the objects outside of us? Further, if we experience our mind as outside of us, then to identify with our mind is to identify with objects beyond our skin. As an intuitive argument in support of this claim, it is somewhat odd to identify with something that is not experienced (such as bodily parts that are not sensed), and perhaps just as odd not to identify with what is sensed (such as sensation of things external to our body).

One reason to identify specifically with bodily sensation, as opposed to sensation in general, is that bodily sensation is relatively stable compared to the rapidly changing subjective universe. For example, turning one’s head may change one’s view in dramatic ways: it may rapidly move mountains into and out of sensory space. This relative impermanence is probably a principal reason why mountains are excluded from the self‑concept; sensation corresponding to visual imagery such as mountains is not consistent, as compared to bodily sensation.[57] In other words, the physical body offers more consistent sensation in virtue of stable proprioception; although the sensation of the body changes, it generally does so at a slower pace. So even though subjective and objective space are defined as exclusive, their material overlap across time is relatively much greater than the overlap between body and world across time (e.g., which occurs when eating or breathing). 

3.6.2.2 Sensory Parts

The parts of sensory space are called sensations.

Sensation provides the content of subjective experience in response to various internal and external objects. To sense an object entails awareness of the many sensations due to that object; the more sensation, the better the detail or clarity of the perception (in general). However, the parts of sensation are not necessarily small. For example, a small object may be experienced as an intersection of relatively large sensory features such as color. In other words, sensations are partless because they are unanalyzed, not because they are points.

Just as the objective world can be viewed as a collection of many things, the corresponding subjective sensory space is composed of many sensory parts. However, sensations form an unorganized feature space, and are only fully individuated and organized when they are collected into wholes by concepts.[58] 

Taken together, sensation and conceptualization constitute perception. There are two basic theories about the directness of perception: direct realism maintains that objects are perceived directly and indirect realism maintains that only a reference to those objects is perceived. Since the parts of the subjective universe are understood as references to an external reality, the basic model of cognition at least partially endorses the latter view. This position is motivated by the observation that if perception is not experienced in virtue of subjective references, then there is no way to account for intersubjective perceptual differences. In other words, intersubjective perceptual differences such as hallucinations or the perception of beauty suggest that beings experience reality as subjectively filtered, as opposed to having direct experience of objective (or non-referential) reality.[59] 

That said, the distinction about the directness of perception may be a false dichotomy if it is possible to experience reality in both referential and non-referential ways (a thesis explored in section Stratified Self). One reason for understanding this distinction as a forced choice may be that the subjective/objective division is often defined as a single mereological or referential boundary, a definition which forces consciousness to be distal to the object of perception. However, it may be that at least some areas of consciousness such as the brain are known both directly and indirectly, or both referentially and non-referentially.[60] 

3.6.2.3 Sensory Dimensions

Sensation is open-dimensional and is typically partitioned into several internal and external modalities.

Traditionally, sensory space is divided according to modality, or in virtue of our various specialized sense faculties. This typically results in five external senses (smell, taste, touch, hearing, and sight) and a number of internal senses (mental, emotional, and several others). 

Image

Figure 3.6.2c: Categorization of internal and external sensation.

As compared to the five external senses, the internal senses are not well understood.[61] Presumably, this is because knowledge of internal senses relies more heavily on subjective experience that is unique to each individual, which is not available to intersubjective verification. Therefore, although internal sensations may be well known at an intuitive level, they are difficult to communicate about, and often not well-organized conceptually.[62]

The Number of Sensory Dimensions

Sensory space is open-dimensional.

Concepts organize sensation into discrete units of perception (or percepts). Sensation itself is open-dimensional, and can be collected in arbitrary ways; in fact, dimensionality itself can be viewed as the result of the conceptual overlay on top of sensation. However, sensation can accommodate that conceptual overlay precisely because it supports discrimination. 

The notion of an unstructured sensory space that is structured by concepts goes back at least to Kant, and probably earlier. In virtue of that conceptual structure, sensation is guided by (top-down) attention. As Carolyn Dicey-Jennings mentions, “… attention transforms conscious experience from a pre-objective space to an objective space by invoking a common spatiotemporal framework” [Jennings, 2005].[63] The creation of a spatiotemporal framework is also to some degree hard-wired; for example, the dimensionality of visual sensation increases when moving posteriorly along the optic nerve, from sensation toward conceptuality. In particular, although sensation is 2‑D when measured close to the retina of either eye, it becomes 3‑D farther along the neural pathways, where the input from each eye is combined to determine the distance of a given object.[64] 

The Nature of Sensory Dimensions

Sensory dimensions allow ordinal discrimination.

As sensation is conceptualized, it is individuated into percepts that are neither inherently small nor orthogonal. Even if the atoms of referential awareness correspond to individual neurons from the objective point of view, those neurons subjectively reference an aspect or region of the world that is often large and distributed, and which overlaps the aspects and regions of the world that are represented by other neurons. In other words, although minds can be physically divided into small material constituents such as neurons, neurons are not small constituents from a referential or subjective point of view. In terms of what they refer to, which is precisely what makes neurons meaningful from the subjective perspective, neurons often detect large-scale features such as color, edges, and linear orientation. Therefore, sensation should not be identified with a set of points from the subjective perspective; rather, it should be considered as a number of arbitrarily large, conceptually unanalyzed atoms. 

An important aspect of sensory space is that it supports discrimination. In particular, it must be possible to discriminate sensations for which there are no concepts, since that forms the basis of conceptualization. As numerous psychophysical experiments demonstrate, the number of possible sensory discriminations is so large that sensory space can be regarded as approximately continuous. Using the sensation of taste as an example, scientists claim that there are five dimensions of taste (sweet, bitter, sour, salty, and umami are reported in [Huang et al., 2006]). Assuming that one can discriminate twenty different intervals along each of the five dimensions of taste, that would create a space of 205 possible sensations, or more than three million different tastes.[65] 

Another characteristic of sensory dimensions is that they are not linear with respect to the physical quantities that they represent. For example:



Further, not only is cortical area disproportional to the bodily area that it represents, but that proportion changes over time. For example, if you take piano lessons, the size of the cortical area that is dedicated to your fingers will increase.[67] This can be taken as evidence (at least for those who believe in this sort of mind/brain correlation) that even the perception of a constant stimulus changes drastically throughout the course of our lives. 


3.6.3 The Conceptual Universe

The conceptual universe is that part of the subjective universe that is conceptualized by an individual.

Although conceptual space is only a part of the subjective universe, it is everything from the conceptual point of view. Thus, from the conceptual point of view, objectsφ and sensationsφ are kinds of concepts (i.e., they have a conceptual realization). 

Image

Figure 3.6.3a: Concepts of objects and sensations depicted as types of concepts.

Concepts are known in relation to one another. Therefore, knowing a concept entails knowing its mereological context within conceptual space, or knowing both its conceptual wholes and parts. For example, a tree stump is known as both a brown wooden thing in virtue of its brown parts, and as a tree of some particular species in virtue of its larger wholes. Similarly, the concept waterφ may be learned both via sensation of waterψ and as a kind of liquid (i.e., waterφ is an abstract concept that is a part of the abstract concept liquidsφ). 

3.6.3.1 The Sensory/Conceptual Dichotomy

The referential division between conceptual/perceptual is similar to the mereological division between brain/body.

The mereological body/brain dichotomy is analogous to the referential sensory/conceptual dichotomy. More precisely, the brain contains references which form conceptual space when they are subjectively experienced. Similarly, the body is approximately the location of the references which form sensory space when they are subjectively experienced.[68] The difference between these two is that the brain/body dichotomy treats references as non-referential physical entities, and the sensory/conceptual dichotomy treats references in virtue of their referential content. In slightly different terms, minds are located in brains from the third-person perspective, while brains are located in minds from the first-person perspective. 

As with the subjective/objective dichotomy, sensory and conceptual spaces are defined not to overlap. However, the boundary between what is sensed and what is conceptualized changes extremely quickly, and therefore the overlap of content across time is much greater for the sensory/conceptual distinction than for the body/brain distinction.

Image

Figure 3.6.3b: The self can be divided mereologically (as brain/body) and referentially (as sensory/conceptual).

The sensory/conceptual dichotomy is a simplification of a continuum of perception, ranging from purely sensory content at one end to highly categorical and conceptual at the other (although the content itself, since it is specified referentially, does not change). 

Conceptual and sensory spaces overlap partially in virtue of visualization, a top‑down process that strongly influences what is and what is not sensed. That doesn’t mean that modalities such as thinking and seeing are the same, but that they both contribute to the same extended perceptual space; as a result, you can see what you are thinking. More precisely, if you are thinking, then you cannot see what you are not thinking (more detail about this difference is presented in section Attention). 

3.6.3.2 Conceptual Parts

The parts of conceptual space are called concepts.

Concepts can be categorized into two main types according to their composite structure: concepts composed of sensation and concepts composed of symbols. Concepts of sensation are called zeroth-order, and concepts of symbols are called higher-order. The distinction between zeroth‑order concepts and higher‑order concepts is similar to the distinction between Type 1 and Type 2 thought in the context of Dual Process Theory. 

Because zeroth-order concepts are formed by making wholes of sensation, the meaning of zeroth-order concepts derives directly from their sensory content.[69] They are also guaranteed to have a spatiotemporal context, which makes them concrete. 

Higher-order concepts, on the other hand, are higher in that they are composed of symbols which in turn reference lower‑order concepts. This use of existing concepts to construct new concepts is efficient, because it does not require learning directly from experience. Unfortunately, the requirement that new concepts are built using existing concepts entails a granularity that is not always a good fit for the objective experience which must be represented. Further, in order to understand higher-order concepts, the sensory content of those constituent symbols must be visualized, which causes higher-order concepts to be abstract. 

As a result of the composite nature of higher order concepts, and in virtue of the top-down focus they require, such concepts are notoriously polarizing. For example, people often assert that a given conceptual proposition is entirely true or false, without allowing for any middle ground. However, this polarity is not an inherent characteristic of concepts; conceptual space can support bottom-up intuition and top-down visualization, both of which operate in parallel, and which often act in concert. 

The interpretation of sensation in terms of symbolic content, on the other hand, forces concepts to be discrete and to occur serially.[70] As a result, people who persistently structure their experience in virtue of symbolic space are particularly prone to black‑and‑white thinking, since the effect of symbolic thought is to exclude experience which does not fit the conceptual model.[71] It is therefore useful to differentiate these two modes of conceptual activation, one which is bottom-up, subsymbolic, and commonly associated with zeroth-order concepts, and one which is top-down, symbolic, and more often associated with higher-order concepts.[72] 

3.6.3.3 Conceptual Dimensions

Conceptual dimensions form a space which is extended by creating concepts of concepts. 

Concepts are wholes of sensation. They are similar to transparent containers, for which sensation provides the content. As a result, the nature of the container depends on the content. In particular, if continuous parts are collected, the whole forms a continuous space, while if discrete parts are collected, the whole forms a discrete space. In both cases, the collection of parts results in a whole of larger size, and therefore the effect of conceptualizing is to increase conceptual granularity. 

In addition to creating wholes, concepts composed of symbols create nominal dimensions, and therefore nominal identity. Implications of this fact are discussed further in section Symbolic Space.

The Number of Conceptual Dimensions

The number of conceptual dimensions is unlimited.

The dimensionality of a single concept depends on the order of that concept. The dimensionality of a zeroth-order concept composed of sensations is generally equal to the dimensionality of its composite sensation, as it does not increase or decrease dimensionality.[73] 

The dimensionality of a higher-order concept, however, is always one more than the dimensionality of its constituent symbols. This is because symbols are discrete atoms; therefore, concepts of symbols are discrete and are interpreted as 1‑D unordered collections. Those concepts can also be interpreted as having a dimensionality that is one greater than the dimensionality of what their constituent symbols represent. In this way, the dimensionality of referential space is extended by creating a nominal dimension that ranges over its references.[74] This process is illustrated in section Symbolic Dimensions.

The Nature of Conceptual Dimensions

Conceptual dimensions are continuous or discrete in virtue of their contents.

As with the number of dimensions, the nature of conceptual dimensions also depends on the order of the concepts in question. Concrete zeroth‑order concepts are wholes of sensation that exist in the same mereological space as their parts. Therefore, they are effectively (if not entirely) continuous, since the granularity of sensation is extremely fine. They are also concrete, just as sensation is concrete. For example, if one learns the zeroth-order concept “small dog”φ in virtue of experience with several small dogs (dog1, dog2, dog3, …), that concept exists as a concrete conceptual union of dog experiences, with numerous contextual associations. By forming the concept in this way, one may learn that such dogs bark a lot at a high pitch. 


This chapter looks at the six relations between the three epistemic universes: sensation, action, conceptualization, visualization, interpretation, and symbolization. 

Image

Figure 3.7: The basic model of cognition.

The six relations between the objective (O), sensory (N), and conceptual (C) spaces of the basic model may be summarized as follows:





3.7.1 Sensation

Sensation affects sensory space as a result of objective space.

Image

Figure 3.7.1: Sensation (Θ) is the dual of action.

Sensation creates sensations in sensory space from objects in objective space. There are two important points to note about sensation. 

The first is that it creates a mass of sensation, rather than individuated sensations. Sensation is individuated, combined, and divided by later conceptualization and visualization, though no new sensation is produced in this process. In other words, sensation provides the entire content of our experience (i.e., sensation includes things like feelings and emotions).

The second is that sensation is the dual of action. Just as action causes effects in the world, sensation causes effects in us. However, the English grammar of both words implies that the person is the agent, since the person both senses and acts. This asymmetry is unfortunate, since sensation is caused by the action of the external object (i.e., it is similar to saying that a baseball senses the bat that hits it, when in fact it doesn’t have much of a choice in the matter). 


3.7.2 Action

Action affects objective space as a result of conceptual space.

Image

Figure 3.7.2: Action (Δ) is the dual of sensation.

Action changes the physical world according to the intent of an individual, through acts such as walking or the creation of sound waves.[75] When the sounds are spoken words, action also has symbolic content.[76] 


3.7.3 Conceptualization

Conceptualization creates conceptual wholes from sensory parts.

Image

Figure 3.7.3: Conceptualization (Φ) is the dual of visualization.

The creation of concepts entails the bottom-up collection of sensation and concepts into a single whole. Concepts can also be composed of symbols, in which case they are called higher-order concepts. Conceptualization can operate in parallel, although emotional attachment or top-down symbolic activation may enforce serial operation. 


3.7.4 Visualization

Visualization creates sensory parts from conceptual wholes.

Image

Figure 3.7.4: Visualization (Ψ) is the dual of conceptualization.

Visualization entails the top-down projection of concepts to sensory space. It may be seen as a transformation of concepts to sense data, or at least a transformation of percepts from a higher to a lower epistemic level. It therefore operates in a direction opposite to conceptualization. 

Visualization is not necessarily visual, although visual imagery is often used to describe this process (as in the word imagination). Since partial visualization makes conceptual parts or percepts out of larger conceptual wholes, but does not necessarily create “raw” sensation, it may also be called understanding.[77] Visualization is capable of turning multiple concepts into a single concrete unit, which may be the mechanism behind the psychological process of chunking.[78]


3.7.5 Interpretation

Interpretation activates concepts in conceptual space from symbols in symbolic space.

Image

Figure 3.7.5: Interpretation (Ω) is the dual of symbolization.

Interpretation is an act of dereferencing that creates meaning from a sensory reference. These perceptual references are known as symbols. In the context of cognition, they are known specifically as cognitive symbols, to differentiate them from verbal or written symbols. 

Interpretation entails the activation of a concept based on its name. In other words, symbols are symbolizations of their corresponding concepts, and concepts are interpretations of their corresponding symbols. Interpretation is therefore similar to conceptualization, but unlike conceptualization, the interpretation of a sensation is not a mereological whole of that sensation. 


3.7.6 Symbolization

Symbolization creates symbols in symbolic space from concepts in conceptual space.

Image

Figure 3.7.6: Symbolization (Ξ) is the dual of interpretation.

Symbolization is an act of naming that takes place between a sensation (or the reference) and a concept (or the referent). The sensation becomes the symbol for that concept, and forms the basis for symbolic thought. For example, the vocalized word earthξ is the name for the concept earthφ.[79] 


As mentioned previously in section The Basic Model, symbols are references to concepts that form symbolic space. For example, the word windξ is a part of our sensory space that becomes a symbol for the concept windφ. A symbol is therefore a specialized type of sensation. The combination of concepts that are composed of sensation, and symbols that refer to concepts, sets up the basis for a referential recursion between sensory and conceptual spaces, which is illustrated in the following figure:

Image

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

The loop that is formed by conceptualization (Φ) and symbolization (Ξ) allows human minds to create symbolic hierarchies. This loop enables recursive composition, such that sensory wholes are represented by a symbol, which forms the basis for subsequent wholes of symbols. These hierarchies are similarly understood by reversing the 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, this compositional hierarchy allows the construction of higher-order concepts out of previously existing concepts, which is leveraged both long-term to define words as higher‑order concepts, and short-term to understand dynamically constructed sentences. 

The depth of these hierarchies can be quantified in virtue of their epistemic level. Epistemic level is a measure of the number of transitions or arrows that must be traversed to reach ground, where ground is defined to have an epistemic level of zero. Epistemic level therefore increases every time things are sensed, conceptualized, or symbolized, and decreases with every act, visualization, or interpretation. The following figure depicts unrolling the sensory/conceptual recursion: 

Image

Figure 3.8b: Epistemic level indicates distance from ground.

The right column of the preceding figure displays the epistemic level, and each node of the previous figure is annotated with a superscript indicating its conceptual order. The left side of the figure shows the locations of three popular epistemic divisions:





3.8.1 Symbolic Space

Symbolic space is the space of symbols, or sensations that reference concepts.

Symbolic space is the subspace of sensation whose parts are interpreted symbolically. Therefore, symbolic space is literally a mental sense, which is distinct from the conceptual understanding to which it corresponds. Thus, although the term “symbol” often corresponds to both internal thoughts and external written or spoken words, symbolic space consists exclusively of cognitive symbols. 

Cognitive symbols (or simply symbols) are meaningful in terms of both their conceptual content and their relation to other symbols. The relation of symbols to one another is determined by their common wholes, and is explored further in section Language.

3.8.1.1 The Conceptual/Symbolic Dichotomy

Although symbols are parts in mereological space, they are references in referential space.

Symbols may be experienced either non-referentially as sensation, or in virtue of their referential content as symbols. The results are dramatically different, since the sensation of a symbol is different from understanding the concept which that symbol represents. Cognitively, this decision is a bit of a forced choice, because the sensation of a symbol and the conceptualization of that symbol tend to destructively interfere with one another.[80] 

As an example of this interference, consider perceiving symbols without additionally interpreting them, as happens when listening to speakers of an unfamiliar language. Although one is aware of various auditory characteristics of the words such as the pitch, volume, and timbre, the meaning of the words is not understood. In contrast, interpretation of the words when listening to a familiar language often causes the sensory detail of the sound to be lost.[81]

3.8.1.2 Symbolic Parts

The parts of symbolic space are called symbols.

An essential characteristic of symbols is that they must activate or re-present the same concepts as would perceiving the physical object which they designate. However, unlike the sensations that trigger that concept, they are not required to resemble that object or its parts at a sensory level. 

In fact, because symbolic understanding competes with bottom-up perception of environmental stimuli, symbols must inhibit all sensory content that destructively interferes with their associated concepts in order to activate their intended meaning. Although this prevents irrelevant bottom-up activation from interfering, it also prevents any other symbols from being active simultaneously. Therefore, symbols necessarily operate serially, rather than in parallel.

3.8.1.3 Symbolic Dimensions

Because symbols are discrete references, their collection into wholes forms a discrete, symbolic space.

A symbolic space is composed of discrete referential atoms. Atoms are similar to geometric points, although they have unit extent rather than no extent.[82] This may be depicted as follows:

Image

Figure 3.8.1a: An atom (zero discrete dimensions).

Symbolic dimensions, or the dimensions formed by collecting symbols within a referential space, are synthesized by combining references. For example, a reference can be combined with references to its left and right, thereby forming a line of references. In the following figure, an atom is iterated in an arbitrary direction. The result is a line:[83]

Image

Figure 3.8.1b: A line (one discrete dimension).

This process can be repeated: a line, iterated in a direction orthogonal to its length, creates a plane. 

Image

Figure 3.8.1c: A plane (two discrete dimensions).

Similarly, many planes collected along an orthogonal dimension form a 3‑D cube.

Image

Figure 3.8.1d: A cube (three discrete dimensions).

In the physical universe, the fourth dimension is called time (at least by physicists).[84] As with the previous dimensions, a novel dimension is created by iterating a lower‑dimensional object along a new axis that is orthogonal to the existing ones.

Image

Figure 3.8.1e: A timeline (four discrete dimensions).

The depiction of a 5‑D object is a particularly interesting example of how each successive dimension is produced, because few people have explicitly extended the conceptualization (or visualization) of dimensionality that far.[85] Implicitly, however, the fifth dimension is used all the time; it is the dimension of possibility. Understood in this way, the fifth dimension is not a difficult concept to grasp, although it is not explicitly treated as a spatial dimension.

To conceptualize the five‑dimensional world, it is helpful to imagine another earth that is similar to ours, which exists at the same place and time (or at the same spatial and temporal coordinates), but which occupies a different fifth dimensional coordinate. In philosophical terms, it is a possible world, and not the actual one. 

Visually, since four dimensions are represented by a world‑line, the fifth dimension is represented by multiple world‑lines:

Image

Figure 3.8.1f: Multiple timelines (five discrete dimensions).

Because the introduction of the fifth dimension enables the discussion of possible worlds, the fifth dimension is referred to as the dimension of modality.


3.8.2 Higher‑Order Concepts

Higher‑order concepts are concepts that are composed of symbols.

Concepts may be either concrete or abstract. Concrete concepts are also called zeroth-order concepts, and are conceptual wholes that are constituted by sensation or possibly other concrete concepts. Abstract or higher‑order concepts are wholes of symbolic references, and as such, their order is one more than the dimensionality of their constituent concepts. For example, if treeφ is defined as the whole composed of pineφ and mapleφ, and both pines and maples are abstract first-order concepts, then the nominal dimension used to contain these two concepts makes treeφ a second-order concept. 

The ubiquity of concepts has led to numerous different descriptions in many different contexts. For example, the following terms correspond (at least approximately) to concepts: wholes, generalizations, unitizations, categories, abstractions, generalities, sets, classes, and equivalence classes. Clearly, the meaning of these terms can become imprecise given their somewhat differing use in multiple contexts. The following table clarifies how some of those terms are used here:

Table 3.8.2a: Two Kinds of Concepts.

Image

Several of these distinctions require further comment, especially the critical distinction between concrete and abstract concepts. 

Contiguity: Higher-order concepts of symbols are formed by the enumeration of category members; therefore, they are conceptually granular or chunky. They also tend to represent objects which are discontiguous, since contiguous objects are easy to represent with zeroth-order concepts. It may be that zeroth-order concepts are necessarily contiguous. In either case, they are almost certainly more difficult to form due to cognitive constraints, since they go against the principles of Gestalt concept formation.

Unitization: Unitization refers to the process of making successive wholes, or creating successive unions of parts, without increasing the order of the concept. It is thus the conceptual operation that corresponds to creating meronomies from concrete parts. For example, the concepts headφ, torsoφ, armsφ, and legsφ may constitute the concept of a bodyφ without necessarily involving any  intermediary symbols. Although it is possible to form a meronomy of abstract parts, is generally not intended (this is explored further in the next section, Cognitive Taxonomies).

To understand the difference between unitizations and generalizations, imagine an apple and an orange. The unitization of these two (i.e., their zeroth-order concept) is the union of all sensations of the apple and the orange; it is a concrete, mereological fusion. On the other hand, the abstract generalization of an apple and an orange is the intersection of all parts and wholes of apples and oranges 

Abstraction or generalization: Abstraction entails the formation of a categorical prototype, whose properties are common to the sensations of the constituent concepts of that category.[86] As mentioned previously, when zeroth-order concepts are formed, they are wholes of their composite sensation. Therefore, they are concrete, just as that sensation is concrete. However, when higher-order concepts are understood, the visualization of their constituent symbols makes them abstract. 

The reason that symbols become abstract is that they are combined with one another in order to isolate specific properties. For example, the property greenφ may be isolated from green trees and green moss by forming the intersection of the properties of those objects. For example, the meaning of the higher-order concept “pet dogs”φ involves the intersection of petφ and dogφ concepts, which makes sense even to a person who knows what pets and dogs are, but who has never met a pet dog.[87] 

Symbols are visualized by negating the opposite of their referent concepts (where the opposite means all concepts which are not parts or wholes of the given concept). When two symbols are combined in this top-down manner, the resulting concept exists as the intersection of the two concepts. This process will make a concept abstract if the intersection lacks any location (which happens if the constituent concepts have disjoint spatial locations). 

However, it is not true that the visualization of concepts is always abstract. Take as examples the zeroth-order concepts chairφ1 and chairφ2, which are composed of the union or unitization of the sensations derived from two chairs, chair1 and chair2. There may be a further zeroth-order concept, chairφ3, that is the conceptual unitization of these two chair concepts. Understanding that concept entails only visualization, and does not require forming the intersection of two symbols, so the resulting concept remains concrete. 

The combination of these examples yields two further observations. First, a given concept can be defined both in an absolute sense (via zeroth-order, concrete sensation) and in a relative sense (via higher-order, abstract concepts). This entails merely that things in the world are known both by experience and definition.[88] However, it blurs the distinction between the order of concepts as they exist in practice, since few if any concepts exist purely in a zeroth-order or higher-order way.

Second, the nature of a higher-order concept that is formed out of the two symbols chairξ1 and chairξ2 seems a bit problematic because the physical intersection of two distinct concrete entities is necessarily empty. The intersection of the chairφ1 and chairφ2 ideas is not necessarily empty, however, since they share a number of abstract wholes (or properties), such as being places to sit.[89]

3.8.2.1 Cognitive Meronomies

Knowing entails structural relations between concepts.

To know the relations between objects in the world entails structuring the references to those objects in a particular manner. In order to examine the different ways in which this might be achieved, imagine a world in which there is a cat named “Felix”, who has a paw (or “Felix’s paw”). The knowledge structure according to that state of affairs is shown in the following diagram, where the relations shown by dashed arrows indicate that:


  1. catφ = Φ(Felixψ): Felixψ is a catφ.
  2. pawφ = Φ(“Felix’s paw”ψ): “Felix’s paw”ψ is a pawφ.
  3. catξ = Ξ(catφ): the name of the concept catφ is catξ.
  4. pawξ = Ξ(pawφ): the name of the concept pawφ is pawξ.


Image

Figure 3.8.2b: Arrows A–D illustrate several ways of knowing that “cats have paws”.

How can we use a diagram to represent that cats have paws? In other words, what is the structure of conceptual space such that the referential structure is isomorphic to the structure in the world between cats and paws?

There are at least several options, which correspond to the solid edges in the diagram:


  1. catφ is a whole of pawφ. This option seems like the most obvious answer. Together with relation (1), it creates concepts which are composed of both sensory and conceptual content. It is a concrete relation, however, so it does not capture the universality of “cats have paws”.
  2. catφ is a whole of pawψ. This option creates a concept catφ that is composed only of sensations (e.g., such as Felixψ and “Felix’s paw”ψ). 
  3. catφ is a whole of pawξ. A potential drawback of this option is that the concept catφ becomes constituted by both sensation (e.g., Felixψ) and symbols (e.g., pawξ), which makes it an uncomfortable amalgamation of different epistemic levels.[90] 
  4. “things with paws”φ is a whole of catξ. This option is problematic because it involves a usage of paws which is not connected to the concept of pawφ, which potentially leads to conceptual proliferation. 
  5. (not depicted) In order to know if cats have paws, the existing concepts can be visualized in sensory space, where the parthood relationship can be determined by inspection. This option is necessary for visualization, but it is not really a solution to the issue of knowledge because it does not store knowledge between concepts (or in conceptual form). 


The cleanest solution allows this relation to be modeled in two different ways, which express slightly different things. Option (A) expresses the relation “concrete cats have concrete paws” as a concrete meronomy, and option (D) expresses the relation “abstract cats have abstract paws” as an abstract taxonomy. One notable aspect of this solution is that there is no necessary connection between the concrete concept “paws” and the abstract property “things with paws” (this is not a very elegant finding, although it may well be in accord with cognition). Another notable aspect is that “things with paws” is typically modeled in prototype theory as a property, although in the basic model properties are modeled as wholes.[91] 

3.8.2.2 Cognitive Taxonomies

Higher‑order concepts are composed of symbols that reference concepts, rather than composing concepts directly.

In order to explore how higher-order concepts are constructed, the diagram below shows several relations and entities in the objective, sensory, and conceptual spaces. Objective space  is depicted at the top level, which causes the sensations aψ, bψ1, and bψ2 in sensory space. These sensations are conceptualized as aφ and bφ in conceptual space. Finally, these concepts are united in a higher‑order concept, cφ.

Image

Figure 3.8.2c: The structure of a higher‑order concept.

This diagram can be made more concrete by visualizing it as a description of a world that consists of one dog and one cat.[92] Concepts (aφ and bφ) are formed by perceiving the dog once (aψ1) and the cat twice (bψ1, bψ2). The dog and the cat are both kinds of the abstract type animal (cφ).

Although this diagram accurately represents the structure of knowing corresponding to the basic model, the generalization relation that is indicated by empty‑triangle arrowheads that point away from cφ is a shorthand that has not been defined in terms of the relations present in the basic model of cognition. Therefore, the rest of this section explores how generalization can be implemented using wholes (conceptualization), parts (visualization), referents (interpretation), and references (symbolization). The following abstract taxonomy is used as the example model that requires transformation:

Image

Figure 3.8.2d: A taxonomy using the undefined generalization relation:
Tree
φ = generalization( Pineφ, Mapleφ, Elmφ ).

The most obvious choice for implementing this structure uses the operation of whole to create a meronomy, although that does not capture the intended meaning because of the abstract leaf nodes. In other words, an abstract tree does not have abstract pines and maples as parts; rather, pines and maples are kinds of trees. Therefore, meronomies such as the following are generally inaccurate:[93] 

Image

Figure 3.8.2e: An invalid conceptual meronomy with abstract conceptual elements
Tree
φ = whole( Pineφ, Mapleφ, Elmφ )

Valid methods for implementing the taxonomy are illustrated in the following two diagrams, each of which has different cognitive consequences. The first method entails turning the conceptual wholes back into sense data, and then collecting that sense data into a new conceptual whole:

Image

Figure 3.8.2f: A sensory meronomy
Tree
φ = whole( Pineψ, Mapleψ, Elmψ )

The second method entails creating symbolic references to the wholes and then collecting those symbols into a new, higher-order conceptual whole:

Image

Figure 3.8.2g: A symbolic meronomy
Tree
φ = whole( Pineξ, Mapleξ, Elmξ )

The difference between these two structures is that the first one does not increase the conceptual order of the concept treeφ. In other words, the treeφ of the sensory meronomy is of the same order as its constituents, while the treeφ of the symbolic meronomy is one order higher than its constituents as a result of creating symbolic references to them. 


3.8.3 Language

Languages are dynamic systems for expressing and understanding conceptual space.

Symbolic space is a discrete space that is composed of symbols and which represents conceptual space. In order to express that symbolic space intersubjectively, humans use words and language. Language involving syntactically complex sentences is possible only for humans, as abstract parts of speech require higher-order concepts.

Language can be understood as a way to create meaning from conceptual structures, which are in turn derived from a series of abstract parts of speech.[94] The next three sections briefly explore three aspects of language: semantics, syntax, and sentences. 

3.8.3.1 Semantics

Semantics is the study of symbolic meaning.

Concepts have two kinds of semantics, one with respect to the sensory and conceptual parts that they compose, and one with respect to the conceptual wholes of which they form a part. These two kinds of semantics form the absolute and relative meaning of those concepts. Mathematically, they are analogous to the intension and extension of a set.

To learn an object as a zeroth‑order concept requires emotional motivation and direct experience of that object. When such learning opportunities happen a sufficient number of times, a conceptual whole is formed that consists of concepts corresponding to the parts of that object, and which in turn forms a part of the larger conceptual contexts in which that object appeared. Thus, the semantics of zeroth‑order concepts depends on their concrete parts and wholes, or their mereological context. 

To learn what an object is as a higher‑order concept requires knowing its abstract (symbolic) parts and wholes. For example, the meaning of the abstract concept treeφ derives from pines and maples being kinds of trees, and trees in turn being kinds of plants. Further, the meaning of the concept treeφ also derives from things it is not.[95] As the following diagram illustrates, trees are plants that are not house plants.

Image

Figure 3.8.3a: Trees, presented in an abstract taxonomic context.

This diagram may not be a typical model of the cognition of most people, in that it depicts treeφ as a concept that is a composite of other higher-order concepts. More likely, treeφ is a basic category for people not living in nature, that is learned without conceptually differentiating the different types of trees.[96] 

3.8.3.2 Syntax

Syntax is the study of symbolic combination.

Syntax allows a collection of preexisting symbols to produce a novel semantic result. For example, while the concept iceφ is known on a concrete basis to residents of Canada, it may be known only on an abstract basis to residents of a hot country with no refrigeration (i.e., where there is no ice). In other words, residents of that hot country know what iceφ is only by its definition: solid, cold water.[97] This process of using syntax to recombine known concepts is used both dynamically to understand language, and as the underlying structure to create higher-order concepts.

In order to minimize the complexity of syntactic analysis, this section focuses on the following sentence (and ignores the definite article, “the”):[98]

The green frog croaked loudly.

Under the assumption of a binary-branching syntax where the VP modifies the NP, the syntactic production rules for this sentence are:[99]

S VP (NP)
NP ADJ (N)
VP ADV (V)

These rules are sufficient to create the deep structure or parse tree of the sentence, but they are not sufficiently detailed to show how various parts of speech relate to concepts. Therefore, it is necessary to address how adjectives (ADJ), adverbs (ADV), nouns (N), and verbs (V) are understood in relation to cognition. To begin, the example sentence is analyzed into the following parts of speech:

greenADJ, frogN, croakedV, loudlyADV

Applying the syntactic rules given above to these parts of speech results in the following formula for the meaning of the sentence:

Image

Figure 3.8.3b: The deep structure of the phrase “green frog croaked loudly”.

Since each of these parts of speech correspond to cognitive symbols, the parsed sentence may also be represented as:

loudlyξ( croakedξ ) ( greenξ( frogξ ) )

As the sentence is constructed, the symbols of the sentence are visualized one after the other. Since the noun “frog” is the deepest part of the structure, it is the first concept to be visualized or understood. 

This syntactic process can be modeled with more precision by augmenting the intuitive understanding of nouns and verbs with a more formal notion that includes their dimensionality. In particular, the recombination of abstract parts of speech such as nouns (which lack any temporal aspect) and verbs (which lack any spatial aspect) results in sentences that can be visualized in a concrete space. In this example, if the dimensionality of the entire sentence is n, then the dimensionality of frog is n-x, and the dimensionality of croaking is x.[100] Therefore, combining these concepts restores the original dimensionality of the event (n). Other parts of speech play similar syntactic roles, such as when definite articles reduce the generality of their associated count nouns. 

The result of this syntactic process (i.e., the restoration of concrete dimensionality) is necessary in order for abstract concepts to be visualized. For example, the visualization of the abstract subject frogξ entails seeing its wide mouth and big eyes as it squats on a lily pad. The subsequent visualization of the successive symbols of the sentence further limits the scope of that visualized space by filtering out what they are not. For example, greenφ removes the possibility of the frog being brownφ, and the visualization of the verb croakξ prevents the frog from just sitting there, doing nothing. 

Applying this process to the original sentence results in four acts of successive visualization, which combine in concrete space:

Ψ(frogφ), Ψ(greenφ), Ψ(croakedφ), Ψ(loudlyφ) 

To generalize this process, the original grammar can be extended so that the terminal nodes consist exclusively of sensation, which creates a transformation from sentences to concrete sensory space, where the meaning of the sentence can be visualized:

S VP (NP)
NP ADJ (N)
VP ADV (V)
ADJ
Ψ (Φ (adjξ)
N
Ψ (Φ (nξ)
ADV
Ψ (Φ (advξ)
V
Ψ (Φ (vξ)

3.8.3.3 Sentences

There are fundamentally two types of sentences: sentences that express events and sentences that express identity.

The previous section explored the structure of a sentence about a physical event. There are also sentences about language itself, which express relations on a higher epistemic level and are used to define words. The distinction between sentences about the world and sentences about language mirrors a fundamental dichotomy known in many contexts with different terminology: synchronic/diachronic, knowledge/news, a priori/a posteriori, synthetic/analytic, de re/de dicto, necessary/contingent, etc. Distinguishing between these two types of sentences is vital; to mistake one type of sentence for the other leads to subtle but serious confusion. 

Sentences about events can be understood as formulas to dynamically construct zeroth-order concepts, while sentences about language describe higher-order relations. Sentences about language typically take the following form: word is a definition (or part is‑a whole).[101] Therefore, examples of this kind of sentence should be interpreted as definitions that express logical relations, as opposed to contingent statements about the world:



The underlying structure of these sentences is significantly different from the structure of sentences about events. Consider the example made famous by Helen Keller, who at the age of twenty-one learned her first word: waterξ. Her understanding entailed knowing that the word waterξ meant waterφ. It is tempting to model this as an association between the concept of water as an experienced word, and the concept of water as physically experienced:

water‑wordφ = water‑objectφ

The equivalence relation, however, does not exist in the basic model as an association. It can be expressed, however, as two symbols that belong to a common whole (which establishes an equivalence class): 

whole( ref( water‑wordφ ) , ref( water‑objectφ ) )

Therefore, the equality between these two concepts is achieved by the introduction of symbols that designate each of them, and a metaconcept which creates an equivalence class that contains the abstract water-object and water-word concepts. This can be graphically depicted as follows:

Image

Figure 3.8.3c: The proposed structure behind knowing “water is water”.



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 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 without hesitation that without a dancer, there can be no dance. With a bit more hesitation they might conclude that without a dance, there can be no dancer; a dancer does, will do, or has done a dance. The dancer and the dance are parts that always occur together in reality; they constitute a mereological whole which cannot be validly referenced using exclusively a noun (the dancer) or a verb (to dance), and which therefore must be referenced using a complete sentence.[102] More poetically, neither a dancer nor a dance has ever existed without each 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.[103] A person or any other abstract concept corresponding to a noun that lacks a temporal dimension cannot exist, and cannot be a concrete part of a high-dimensional space without being modified by a verb. Although nouns have abstract meaning, their dimensionality is lower than that of physical space because they lack a temporal extent (nouns are abstract precisely because they are generalizations over the temporal dimension). Therefore, they are dimensionally incomplete.[104] This is just what it means to be abstract: only concrete concepts are completely meaningful as parts of a high-dimensional space. 

The distinction between concrete and abstract can be understood by an 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?”. For nouns such as “me”, incompleteness should raise similar questions: “I did what?”. In contrast to this, however, most people do not recognize the referents of nouns as incomplete; material solidity is regarded as sufficient for existence if one holds a classical view of space (i.e., and not spacetime). This is a mistaken view, since nouns are not just incomplete in terms of sentence structure, but in terms of corresponding to any concrete thing.[105] 


The relation of identity is defined under the assumption that events are unique, and that by default, no event is identical with anything else. Similarly, all sensations are different: there is always a dimension along which any two sensations differ. However, when sensation is collected into concepts, it becomes possible to generalize over the symbols that refer to those concepts. In virtue of that generalization, differences between individual sensations are forgotten. As a result of this forgetting, things become conceptually identical. This nominal identity is not entirely subjective since it captures aspects of truth about the world, but in virtue of forgetting, neither is it entirely objective.

Because nominal identity requires the association of a symbol and an object, it is invariably approximate. This fact 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. Since most people acknowledge that a single grain of sand could not make such a difference, then heapsφ and other concepts apply to their objects only to varying degrees, which poses problems for anyone who believes that propositions about the world can be fully true or false.

[106]In virtue of this 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:




Imagine a town in which “... all the children are above average”.[107] 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 example of how some things are relative to others, since being average is a 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 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”. In fact, it may not be theoretically possible to separate the absolute nature of a thing from all of its potential future (relative) interactions.[108] 

Although it is clear that some properties are relative, things become meaningless if everything is relative. Therefore, many people maintain that some 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. Kilograms, in turn, are defined relative to the mass of a certain volume of water at sea level.[109] This process indicates that even abstract properties of an object such as its mass are relative to external objects. 

The implicit resolution to this issue is that concepts and sensations are relative to one another in virtue of their wholes, while they are absolute in virtue of their parts. Therefore, the expression of properties such as mass is necessarily relative, because expression is necessarily symbolic. Our sensory experience of things like heaviness, however, is undeniable in that it is experienced absolutely (or in a bottom-up sense). 

The philosophy of reductionism makes this alignment explicit by stating that relative descriptions are exclusively external (or whole-based) and absolute descriptions are exclusively internal (or part-based). Reductionism further claims that a thing is fully known when one knows its parts. For example, reductionism implies that 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. 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. Although this may initially seem counterintuitive, causation requires that a cause temporally precedes its effect, which is not the case for a spatial whole and its parts. 


Sensation derives meaning by referring to objects; that meaning is absolute as 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; certain concepts are present only when others are absent. As a result, (bottom-up) sensation does not have a negation, whereas (top-down) concepts always have a negation.[110] 

In other words, what is absolute is regarded as a whole, in that it is not a part relative to some larger thing. As a whole, it does not have a complement, and complements are the basis of 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 a complement with respect to the concepts that are their wholes, and that complement is their negation with respect to that whole. 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. 

This may be stated informally in terms of two kinds of negative entities:[111] negative sensations (which are purely absolute) and negative concepts (which are purely relative). Negative sensations do not exist, in that the opposite of sensation does not entail anything. Negative concepts do exist, in that the opposite of a concept entails the not-concept. In other words, the sensory features that constitute zeroth‑order concepts are always present in the world, while the sensory features of higher‑order concepts are not always present in the world. As an illustration, a holeφ is defined relative to other concepts, but it has no (absolute) sensory content of its own.[112] Therefore, holes are known only in virtue of their relation to a larger context; for example, a hole in the ground is known only in virtue of being surrounded by earth.[113] 

The claim that negative sensations do not exist means that “the negation of a tree” is not meaningful at a sensory level: it does not entail any positive sensation, and it is not clear what the not-tree looks like. In other words, there is no such thing as a sensory opposite except for the lack of sensation, since all sensation is positive appearance.[114] 

The claim that negative concepts exist, on the other hand, entails the conceptual presence of the conceptual complement. For example, if something is an animal then it is a not-vegetable and a not-mineral; conceptual negation is therefore called an affirming negative, since the relative aspect of a concept is defined in terms of what it is not. 


Intuition is able both to use concepts and to transcend some of the limitations of rational thought. Intuition is not rational thought, but it is not irrational either; it is multi-rational, capable of understanding multiple concepts at the same time. Intuition is sometimes characterized as nonconceptual, in that it is non-symbolic, but it is a form of understanding which is multi-conceptual, in that it utilizes the content of multiple concepts, just as a picture is worth a thousand words. Although it is natural to associate intuition with sensation as opposed to conceptualization, since intuition is subtle and conceptualization is often relatively course, intuition could not know 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 conceptual hierarchy. Although there are exceptions such as puns that may be interpreted using two or more conceptual hierarchies, most statements are structurally unambiguous. Intuition, by contrast, uses an extremely limited syntax (i.e., mereological parthood), but it uses all words at once. Hence, intuition is a structure which forms 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 the 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. Similarly, sensation and zeroth-order concepts are used by System 1, while System 2 also uses 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, which makes System 2 able to conceptualize only 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 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.[116] 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. 

Because Type 2 thinking requires the inhibition of all irrelevant activation, it emphasizes a certain meaning by forgetting numerous other conceptual relations. Therefore, Type 2 thought entails knowing the parts and wholes of a thing, but causes the isolation of that idea in virtue of inhibiting its conceptual complement. 


The differences between animal and human cognition are remarkable given the relative lack of neuroanatomical and genealogical differences.[119] 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, they are categorized by the basic model according to their degree of referentiality, or their epistemic level.

Inanimate objects such as rocks do not sense or react to a neighboring brook in virtue of any internal mechanism or references.[120] A tree, on the other hand, senses things; if nothing else, it senses the sun, and reacts to that sensation by growing toward it. Thus, the tree may be said to be conscious of what it is, just like the 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, since they have no mechanism by which they can form unitizations. Animals with nervous systems, however, can additionally conceptualize things. In summary, non-referential (or reflexive) consciousness is something that all objects 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 characterized as 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).[121] Removing interpretation (and possibly symbolization) from the basic model results in a model of animal cognition that allows sensation and concrete concepts, but not symbols or higher-order concepts. Therefore, animals may sense, visualize, and to some extent conceptualize in the same manner as humans, but they cannot engage in symbolic thought. As a result, animal cognition is concrete.[122] 

Image

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.[123] 

Behaviorally, the lack of interpretation (Ω) 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., sitφ, stayφ, “play dead”φ, etc); they are always embedded in a causal context. On the other hand, symbols are something, and they 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 is significantly different. To characterize how animal communication is different from human language, since the concepts of animals are always zeroth-order, it is necessary to identify what parts of speech correspond to zeroth-order concepts. The basic model of cognition indicates that animal cognition is formed of sentences, just as human language, but that what animals lack is the ability to create and comprehend abstract parts of speech such as nouns and verbs. Therefore, the concepts of animals always correspond to entire sentences. In other words, since animals cannot form concepts corresponding to abstract parts of speech, their sentences are limited to be one word long. 


Bottom-up and top-down processes in cognition imply a vertical dimension that is generalized by the notion of epistemic level. Sensory parts are responsible for bottom‑up activation, and symbolic wholes are responsible for top‑down inhibition.[125] 

The role of bottom-up processes is fairly straightforward: concepts are recognized by us 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 top-down influence, our perception does not operate without distortion, and that distortion often goes unnoticed. Perhaps this is inevitable, since the cognitive detection of what you do not perceive takes 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. The following figure 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.


Image


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, a blind spot is not only a sensory hole that you do not see and of which you have no awareness, but an area of illusion. To more clearly demonstrate this fact, if you pass a pen in front of the blind spot that you identified in 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 both as bottom-up, sensory deficits and as top‑down illusions.[126]

These missing bits of sensation and areas of hallucination clearly demonstrate that the world as we sense 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 unfortunate. 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, although most of us alter the first sentence to conform to our expectations and understanding. This type of correction is generally beneficial; for example, it causes false positive sensations of dangerous creatures and automatically corrects the syntax of ill-formed sentences. However, it is harmful if it creates a world that always conforms to our understanding and in which we recognize only familiar things, since that would not enable our understanding to increase.[127]

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 number of philosophers have noted, there is no way to tell if we are dreaming or not if sensations do not depend on the presence of objects.[128] 


Attention is an operation that restricts the domain of perception from the awareness of everything to the awareness of a specific thing. This selectivity of attention causes a phenomenon called inattentional blindness, which means that an individual is relatively blind to things outside of attention.[130] 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 mind to more fully analyze what is perceived.

In general, 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. In terms of the basic model, attention is determined bottom-up by conceptualization and top-down by visualization. Bottom-up sensory activation grabs attention if it is particularly intense, unexpected, or emotionally valent. Top-down conceptual inhibition guides attention to what is relevant by inhibiting what is irrelevant at higher conceptual orders; conceptual wholes outside the scope of attention are inhibited. 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 the scope of attention also limits subsequent emotions. In both cases, we are conceptually and emotionally reactive primarily to the content of awareness, and it is attention which governs the aperture of that awareness.

Attention is a skill that we are practicing all the time, regardless of whether we are conscious of doing so or not. Training in attention has both mereological and referential aspects. 

Mereological training in attention often takes the form of directing awareness to areas of space of which we are infrequently aware, as in body scan meditations or meditation on an object. This process also happens routinely as a side effect of activities like piano practice, which expands the degree of cortical representation of the fingers. 


Despite the profound importance of emotions, much less is known about them than is known about thought.[132] Perhaps this is due to their relative complexity, the tendency of culture to preserve teachings from higher epistemic levels better than teachings from lower epistemic levels, or to the fact that emotions are inherently not suited to being described symbolically. For whatever reason, the cultural dissociation of identity from emotionality is so prevalent that it is encoded in various social and legal systems. 

For example, there is no legal responsibility for actions if they are committed “in the height of passion”, unlike crimes committed when “thinking too much”. But since we are responsible for what we think, why are we not responsible for what we feel? One possible response 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 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 bottom-up emotions and top-down conceptuality as antagonistic processes is an overly simplistic characterization, since emotions play a large role in conceptualization (and intuition has its 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 instead of referents.[133] 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.



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 such as the parasympathetic rest/digest mechanism that are inherently rewarding (i.e., independent of the actual experience). In that case, the concept has the potential to condition the associated concepts even further by creating a feedback loop that involves maladaptive thought patterns. 

At least in some instances, emotional attachment to symbols rather than their intended objects seems to be a mistaken form of cognition.[135] 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 a negative thing, disentangling the attachment to symbols from attachment to the concrete phenomena that those symbols represent is a formidable task.[136] 


Where do we experience beauty? Do we experience it in the space of our bodies, or in the space that is coextensive with the perceived world? 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, each of experiences beauty in the objects of the world, and tells the world that we experience beauty out there. If beauty can only be present in a single location, then which location do we choose?[137]

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 there are subjective aspects that are due to us.[138] Beauty is regarded as one of the latter, made possible by our qualitative evaluation of references of an external reality. 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, and so 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 scope of consciousness, the location of which is restricted to the body. 

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.[139] 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. 

If both truths are non-conflicting and valid from different points of view, however, 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.[140] 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, and others do not. 

The question about where beauty resides can be seen as 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. Many scientists would argue from this 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, and the only beauty is in the interpretation as beautiful by an observer. 

The unional thesis entails that objective reality 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.  



To understand intersubjective kindness, it is useful to look at a simple 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].

In order to understand what is meant by the term kindness, behavior can be categorized 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, which generally connotes that you should be equally kind to yourself and others. If this relationship 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 duality and nonduality. Selfishness and otherishness are both inherently dualistic, since it is not possible to consistently and intentionally benefit oneself or another without reifying the distinction between oneself and another. If kindness is nondualistic, then it strives to achieve maximal happiness regardless of whose happiness that is. 

The four alternatives created by these two axes may be summarized as follows:



Different nondual traditions often express either that everything is self, or that there is no self whatsoever. In both cases, the subject/object duality disappears.[141] For example, if we identify with our body, we might enjoy eating savory foods or feel sad when we hurt. However, if we consider our family or some other larger whole as our self, 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 with a group. 


The parts of the world are overlapping and interdependent.[142] Therefore, although the use of mereological analysis to draw boundaries between things is useful, the associated tendency to consistently rely on a single partitioning of reality into concepts is problematic. 

The characterization of 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 continues to exist in various modified 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 positing something that is mingled with them. However, theories positing 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 its representations. But if consciousness is extended, 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. In other words, 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 easier than building new concepts out of sensation directly, higher‑order concepts are hierarchical, and hierarchy does not represent multiple overlapping concepts very well. Therefore, we must balance the precision of zeroth-order concepts with the convenience of higher-order concepts. 

In light of the cognitive bias against forming 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 individual people. 

Mereological overlap of individuals is unusual, because it entails sharing body parts (although it happens in cases of conjoined twins and arguably during pregnancy).[143]

Image

Figure 4.20a: Mereological overlap of two people.

Referential overlap of individuals occurs if the subjective space of one individual overlaps the subjective space of another. This is shown in the following diagram, where the numeric subscript denotes a given individual:

Image

Figure 4.20b: Referential overlap.

Although overlap of sensation is trivial because it occurs whenever two individuals see the same thing, the overlap of entire subjective spaces requires that the two subjective spaces are the same. However, 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 to each of us identical? 


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.[145] One’s self-identity is often emotionally related to this sense of self. Therefore, from an emotional point of view, the question “who are you?” May also be asked as “what is the largest whole of you that you strongly care about?”. In modern times, self is often defined as a collection of bodily parts, and the brain is included as one of these. In older times, the self only existed as a part of a tribe. In both cases, however, the differentiation between self and other relies on both mereological distinctions such as between body/world and referential distinctions such as between subjective/objective.[146] 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, which determines things such as its spatial location and size. 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, since the idealistic perspective is better described referentially.[147]

The diagram below depicts several options for the boundary of a mereological self:

Image

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

Of the possible sources 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 seems especially important; although the body is composed of many parts and is a part of many larger wholes, it is typically viewed as a single mereological whole. Even linguistically, a synonym for person is the word individual, which literally means “unable to be divided”. The importance of the bodily self is cemented in our daily lives and routines in that it is used 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 children).[148] 

A referential theory of self, on the other hand, is defined in terms of references and referents. For example, to identify only at the epistemic level of referents (as opposed to references) leads to identification with what is experienced, rather than what does the experiencing. Although it might seem strange to identify with experiences themselves rather than the experiencer (since those experiences are not inherently limited to experiences of our bodies), the opposite extreme of exclusive identification with the experiencer is 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 as witnessed (or from the non-referential point of view).[149]

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 in which ultimate referents exist. As a result, this view encourages identification with perceptual content itself and therefore downplays the distinction between the perception of our bodies and the perception of the world. The idealist doctrine was famously summarized by George Berkeley in his statement “Esse est percipi”: to be is to be perceived. 

Referential identification is illustrated in the following figure, 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 mind or awareness, which is identical to their referential content: the brain, body, and world. 

Image

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

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 a bodily self, yet relate to it in various different ways. This is illustrated in the following figure, which shows three referential spaces within which one may characterize the bodily self: objective, sensory, and conceptual. The bodily (material) self is viewed within each of these spaces differently, as either an object (body), a sensation (bodyψ), or a concept (bodyφ), respectively. 

Image

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


To some extent, any attempt to define one’s 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.[150] Refusing to engage with the question is not an ideal solution either, since our actions depend on our understanding, even if that understanding is implicit. Therefore, this section explores a stratified self-concept, and attempts to detangle where we are mereologically from what we are referentially. 

Boundaries

Although there are many different forms of conceptual dualism, the primary form of dualism is that between self and other. The location of the self/other dichotomy is contentious because the location of that boundary is not well-defined from exclusively a mereological or referential point of view. 

From the mereological point of view, the body/mind boundary does not distinguish subject and object as entirely different kinds of things. We cannot experience the objective world except as it presents to us through subjective experience.[151] Similarly, there are no boundaries from the perspective of an outside observer looking inwards; psychology has been exploring the brain for a long time, expecting to find the location of consciousness, but it has not found any boundary where one crosses from the objective world into the subjective world. 

Similarly, from the referential point of view, the body/world boundary does not distinguish subject and object as entirely kinds of things. In other words, mind knows the body at a referential distance, in the same way that it knows the world. Although this may seem a bit disconcerting, there is a lot of psychological evidence behind it. For example, the stories that we tell about the reasons for our own actions are very often wrong, which indicates that we learn and theorize about ourselves in much the same way that we learn and theorize about the rest of world.[152] 

Since the self/other boundary cannot be completely and precisely defined either mereologically or referentially, we must find an alternate way to disentangle these points of view.

An Interaction

The mereological and referential perspectives are illustrated in the following table, where the mereological body/world distinction creates the columns, and the referential body/mind distinction creates the rows.[153] 

Table 4.22a: The self/other distinction viewed from mereological (body/world) and referential (body/mind) perspectives.

Image

From the objective or third-person perspective, there is no access to referential content (i.e., selfb and selfc are known only from the subjective perspective). Therefore, as everything is known non-referentially and materially, the distinction between the body and mind of other individuals collapses. As a result, an individual’s self is identical to the mereological body from the objective perspective, or selfa and selfb taken as an undifferentiated first column. 

From the subjective or first-person perspective, referents as material things are initially unknown, and the world is contained within the mind. In other words, there is no other from the subjective point of view since there is no a priori distinction between body and world. Therefore, self is initially identified with what is perceived from the subjective perspective, or selfb and selfc taken as an undifferentiated second row. 

To summarize these two points of view, the mind is a part of the body from the objective or materialistic perspective, and the body exists within the mind from a subjective or idealistic perspective. In terms of references, the objective perspective locates us where the references are located, and the subjective perspective locates us where the referents of those references are located.[154] 

Other as Negative Entity

In Table 4.22a, other is a negative entity. It exists in the same way as a hole in the ground; both are things that exist for a given subject on an exclusively conceptual level, neither of which has any sensory content. In other words, there is never consciousness of other within any given epistemic level, and since there is no experience of other, experience of self is not relatively meaningful. In more poetic terms, the self/other dichotomy exists at the level of the head, but it does not exist at the level of the heart. As a result, although neither objective nor subjective perspectives are abandoned, subject/object dualism collapses because the axes cease to be dependent (i.e., they are not orthogonal). It is an open question how the divisions of different epistemic levels should relate to one another, but any conflation should not occur in virtue of our ignorance.

Therefore, while it may make sense to conceptually discriminate between self and other, to feel inherently differently about them may indicate a confusion with respect to epistemic levels. That is, it may indicate an unjustifiable projection of a difference from one epistemic level onto another (i.e., where that difference does not exist). In terms of Table 4.22a, this entails identification with selfb but not selfc, even though both are known only at an epistemic distance. Even if such a projection is not a result of any confusion, a more detailed understanding of how we exist simultaneously on multiple different epistemic levels is sure to be of benefit in eliminating any confusion.

Self as Intersection vs Union

One method to integrate the objective and subjective points of view allows the objective perspective to say where the body is and the subjective perspective to dictate where the mental experience is, and does not force these perspectives into agreement. In other words, it allows these perspectives to be independent. According to this method, selfb in Table 4.22a constitutes an interaction that does not exist except as a mixture of subjective and objective views, a localized and referential self which is valid from neither point of view on its own, and which exists only because the mereological and referential points of view are (erroneously) forced to terminologically share a common self.

On the other hand, if the self is regarded as a combination of the body from a mereological perspective and the mind from a referential perspective, then there are two main options to define the self: self as an intersection or self as a union. The view that the self is an intersection results in the identification of the self with selfb in Table 4.22a. Informally, it may be expressed as a result of taking the mind to be inside the mereological body and the body inside the referential mind. The view of the self as a union results in the identification of the self with selfa…c in Table 4.22a. Informally, it may be expressed by saying that the mind exists outside of the mereological body and that the body extends beyond the referential mind.[155] 

Increasing Continuity

Although dualism is often criticized for dividing things, it may also be criticized for being insufficiently differentiating (or not dividing enough). To remedy this criticism as applied to Table 4.22a, the mereological body/world and referential body/mind divisions are expanded. As before, the columns represent mereological size, and the rows represent epistemological level: 

Table 4.22b: A stratified view of the self.

Image

In this table, the notion of self is spread across several epistemic levels and mereological boundaries. It illustrates that the more highly referential the self is (which corresponds to the lower rows of the table), the less localized it becomes.[156] The table may therefore be viewed as a discrete approximation of a continuum from a proximal, dense, and non-referential self to a distal, sparse, and referentially-distant self.[157] 

In this table, the self is constituted by the lower-left triangle and is composed of references from various epistemic levels. It does not contain any others, which exist as the unknown negative entities forming the upper-right triangle.[158] According to this model, the various selves should be understood as somewhat independent of one another. In virtue of these multiple aspects of self, the location of each aspect of self varies. For example, self1 is located in the body, and self2 is located in a neighborhood. In other words, the self of each level is colocated with its referential content.[159]

The Nondual Perspective

This book has analyzed subjective experience from a very dualistic perspective, so it is fitting to end with a note about nonduality.[160]

From a psychological point of view, nonduality corresponds to a lack of top-down symbolic inhibition, which would otherwise create dualism in virtue of creating heavy-handed visualization. By preventing too much emotional attachment to symbols, nonduality leaves space for bottom-up influence. 

Of course, this understanding of nonduality does not express the actual subjective experience of nonduality. The nondual perspective can be embraced only to the extent that it is experienced ineffably or non-symbolically: even structuring our experience with relatively true notions obscures our lived experience. In other words, we need to feel it. No amount of knowing in a relative way can replace knowing in an absolute way. That said, both the dualistic and nondualistic perspectives are necessary, so hopefully the theory explored in this book will help to make the two less polarizing. As the Zen master Dogen wrote:


To study the self is to forget the self.

To forget the self is to be actualized by myriad things. 

When actualized by myriad things, your body and mind as well as the bodies and minds of others drop away.

Dogen Zenji [Dogen et al., 2013]


Conclusion

It seems that in a conclusion, people generally summarize the book and talk about the future. If you’ve read the book, you don’t need a summary, so let’s talk about the future.

Personally, I can’t see the future; perhaps some of you can. Maybe it doesn’t matter, since you can’t change the future that you’ve seen; if you could, that would imply that what you saw was not the future. Further, if you try to change an unavoidable future, you will likely play a causal role in the outcome you were trying to avoid.[161] So I guess the best that I can do is to express my hope.

I hope that after reading this book, some of the numerous ways in which our intellectual understanding affects our lived experience are now clearer. I hope also that the basic model of cognition will be beneficial to you, particularly with respect to understanding ourselves. I have tried not to be heavy‑handed about it, but I strongly believe that people are more interconnected than many of us realize, and I believe that undoing any linguistically or culturally transmitted sense of isolation goes a long way in allowing our hearts and minds to open, and enabling us to more easily recognize the wholes of which we are a part.



May it benefit all beings,

alec


PS: I hope that reading this book has been both enjoyable and intellectually stimulating, and I apologize for any mistakes I have made or any ignorance I have unwittingly transmitted. 


Acknowledgments

I would like to thank all of the people and things that made this book possible:

To all of my professors at the many colleges I have attended, and especially to those supporting my thesis and other academic works. 

To Angelica Kaufmann, Cathy McMahen, Ken Parker, and others for their astute observations and excellent editing.

To my family, for the immeasurable kindness they have shown over the years.

To my lovers, to whom I apologize even though they said that is part of the problem.

To my friends, who have shown encouragement to open my heart and lift my gaze.

To my coworkers, for both the opportunity to work with them and the walks after lunch.

To my computer, even though we have a bit of a stormy relationship, and all the wonderful open-source software, such as Inkscape, Graphviz, and Gimp (and also to Scrivener).

Last but not least, to nature, hot yoga, bike rides, pints of nondairy ice cream, and the many other things that kept me going during the difficult times.





This appendix is divided into three main parts: 


  1. A summary of the basic model of cognition.
  2. Zeroth-order (mereological) logic.
  3. Higher-order (symbolic and abstract) logic.


Each stage is developed out of its predecessor; the basic model gives rise to mereology, and mereology gives rise to symbolic logic. This introduction summarizes the motivation behind this formulation in terms of topology, mereology, and logic.

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 “refeology” to describe both how our minds are situated in the world and how our minds give rise to mereology and reference in turn. The use 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. 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 for nominal or abstract low-dimensional entities, such as geometric primitives. Beginning with high-dimensional objects is motivated by the following observations:



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 will be beneficial for both thinking about and interacting with reality. 

Topology

The most common form of topology is called point-set topology. Historically, point-set topology 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 space is not symbolic.[165] 

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 there are an infinite number of other points. While some people might regard this lack of contact as a mere intellectual curiosity, objects in the world are connected. Therefore, this problem cannot be ignored by any accurate model of reality.

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). Mathematically, mereology was viewed as a 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 current work by combining two operations, reference and whole. In fact, this is one of the reasons 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 might suggest that mereotopology does not recognize 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 serve only to 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.[166] 

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.[167] For the same reason, the complementary topological approach that begins with the largest possible object and divides it is also invalid. As a result of these considerations, the version of mereotopology presented here starts in the middle with spatiotemporal volumes or simply objects as primitive, and combines or divides them 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 neither holistic nor reductionistic, and analyzes objects by examining them in terms of both their smaller parts and their larger wholes, just as cognition. Using a set of continuous 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 either atoms or universes). Therefore, objects are always unknown to some degree, in that their parts and wholes are indeterminate.[168] 

As an aside, the version of mereotopology developed here is a good fit for cognition, but it does not follow a standard mereological development: its primary purpose is to demonstrate the bridge between subsymbolic cognition and logic. GEMTC is a more formally rigorous theory that was 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 is demonstrated by 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.[169] Symbolic logic, however, requires the subject “soccer ball”  to be singular; it is not allowed to have parts (or at least, the subject is treated as a singular and partless whole by any applied predicates). Because symbolic predicates treat the subject symbolically, they 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 truth to apply to continuous shapes in addition to discrete symbols, so it provides a better fit for actual objects.[170]

In order to illustrate the continuous nature of mereotopology, consider again the five mereological relations presented in section Mereological Space:

Image

Figure Aa: The five mereological relations.

The fifth case is problematic because it cannot be described using classical logic and the parthood relation without introducing a new symbol 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.

The following truth table 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:

Table Ab: Shape Logic

Image

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; therefore, it avoids unnecessary symbolic proliferation. The formula that describes parthood as a continuous function between zero and one of x and y is:

Image


The set of equations that correspond to the basic model of cognition are developed according to the following guidelines:



Image




Equation 1.1: Physical Space

Image

The basic model begins with the physical universe (U). 

Physical space is equivalent to the combination of the subjective (Si) and the objective (Oi) spaces of a given individual (i).

Equation 1.2: Subjective Space

Image

Subjective space (S) is defined to be all references to physical space from a given individual’s perspective (i). 

Subjective space is equivalent to the union of an individual’s sensory (N) and conceptual (C) spaces.

Equation 1.3: Objective Space

Image

Objective space (O) is defined to be that part of physical space (U) which is not subjective space (S).

Objective space is the result of action (Δ).

Equation 1.4: Sensory Space

Image

Sensory space (N) is defined to be that part of subjective space (S) which is not conceptual space (C).

Sensory space is the result of bottom‑up sensation (Θ) and top‑down visualization (Ψ).

Equation 1.5: Conceptual Space

Image

Conceptual space (C) is defined to be all references to sensory space (N). 

Conceptual space is the result of conceptualization (Φ) and interpretation (Ω).

Equation 1.6: Symbolic Space

Image

Symbolic space (V) is composed of references to conceptual space (C). 




Equation 1.7: Sensation

Image

Sensation (Θ) is a causal function that maps from O to N. 

Sensation may be written as the sum of smaller acts of sensation (Θi).

An act of sensing (Θi) produces a sensation (xψ) from an object (x).

Equation 1.8: Action

Image

Action (Δ) is a causal function that maps from N to O. 

Action may be written as the sum of smaller acts (Δi).

An act (Δi) produces an object (x) from a concept (xφ).

Equation 1.9: Conceptualization

Image

Conceptualization (Φ) is the whole function, that maps from N to C. 

Conceptualization may be written as the union of smaller acts of conceptualization (Φi).

An act of conceptualization (Φi, Φj) produces a concept (xφ) from some number of sensations (yψ) or concepts (zφ). 

Equation 1.10: Visualization

Image

Visualization (Ψ) is the part function, that maps from C to N. 

Visualization may be written as the intersection of smaller acts of visualization (Ψi).

An act of visualization (Ψi, Ψj) produces a sensation (yψ) or concept (zφ) from larger conceptual wholes (xφ).

Equation 1.11: Interpretation

Image

Interpretation (Ω) is the referent function, that maps from V to C.

An act of interpretation (Ωi) produces a concept (xφ) from a symbol (xξ).

Equation 1.12: Symbolization

Image

Symbolization (Ξ) is the reference function, that maps from C to V.

An act of symbolization (Ξi) produces a symbol (xξ) from a concept (xφ).

Equation 1.13: Epistemic Level

Image

The epistemic level of a thing is the total number of epistemic operations that must be taken in order to reach ground.

Equation 1.14: Conceptual Order

Image


Image

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 equations in zeroth-order logic and produce truth values on the left. The use of references as variables on the right-hand side, 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. As all operations of the basic model are continuous, it would more closely approximate cognition to produce a continuous logical value between 0 and 1 as in the section Logic at the beginning of this appendix. In terms of visualization, a continuous approximation of parthood can be stated as:

Image

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 expressed as:

Image


Visualization (Ψ): Given some number of entities, their intersection creates a part.



Equation 2.1: Part

Image

x is a part of y if the visualization of x and y (which is their intersection) is equal to the visualization of x (i.e., the intersection of x and y removes nothing from x).

Equation 2.2: Whole

Image

x is a whole of y if and only if y is a part of x.

Similarly, x is a whole of y if the visualization of y and x is equal to the visualization of y (i.e., the intersection of x and y removes nothing from y).

Equation 2.3: Everything

Image

Everythingφ is defined as the combination of all possible sensations. 

It can be written with the circle symbol.

Equation 2.4: Nothing

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Nothing is defined as the absence of everything.

Equation 2.5: Truth

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Equation 2.6: Reference

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x is a reference to y if x is produced by symbolizing y.

Equation 2.7: Referent

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Equation 2.8: Negation

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The negation of a variable is its sensory opposite.[172]

The falsity of a variable is the truth of that variable’s negation.

Equation 2.9: Conjunction

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The conjunction of two entities is the truth of the concept that combines both.

Equation 2.10: Disjunction

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The disjunction of two entities is the truth of the concept that combines the visualization of each.

Equation 2.11: Proper Part

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x is a proper part of y if x is a part of y and y is not a part of x.

Equation 2.12: Referential Level

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The referential level of a reference is one higher than the referential level of its referent.

Equation 2.13: Dimensionality

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The dimensionality of a part that is not a reference is equal to the dimensionality of the whole. 

The dimensionality of a whole of references is one more than the dimensionality of the referent of that reference.[173] 


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 and therefore incapable of being concrete parts. This reduction of dimensionality is also the basis for introducing boundaries and geometric primitives:




Equation 3.1: Existence

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A reference exists if its referent is true, in the sense described in the previous section.

Equation 3.2: Universality

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A reference is true for all entities if there is no referent of that reference for which it is false.

Equation 3.3: Atoms

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A thing is an atom if it has no parts.

Equation 3.4: Universes

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A thing is a universe if it has no wholes.

Equation 3.5: Empty Reference

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A reference is empty if its referent does not exist.

Empty references are denoted using the null-set operator.

Equation 3.6: Full Reference

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A reference is full if it is a reference to a universe. 

Equation 3.7: Overlap

Image

Two objects overlap one another if there is some third object that is a part of both.

Equation 3.8: Underlap

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Two objects underlap one another if there is some third object that is a whole of both.

Equation 3.9: Complement

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The complement of a part (x1) with respect to a whole (y) is a second part (x2) such that:

  1. x2 is part of y.
  2. All parts of x1 or x2 are parts of the whole y.
  3. x1 and x2 have no parts in common.
  4. There is no part of that whole which does not overlap with either x1 or x2. 

Equation 3.10: Connected

Image

Two entities are connected if they overlap or their boundary exists.

Equation 3.11: Internal Identity

Image

Two things are internally identical if they have the same parts.

Equation 3.12: External Identity

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Two things are externally identical if they have the same wholes.

Equation 3.13: Mereological Identity

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Two things are mereologically identical if they are both internally and externally identical.

Equation 3.14: Referential Identity

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Two references are referentially identical if they are equal, or their referents are referentially identical.

Equation 3.15: Nominal Identity

Image


Equation 3.16: Boundary

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Table B.1a: Epistemic Spaces

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Table B.1b: Epistemic Relations

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Table B.1c: A priori Symbols

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Table B.1d: Logical Symbols


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:



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

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 is used:



There are several other typographic conventions which have somewhat specific connotations in this work:




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:

Image

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

Different things are related to each other using the following different types of arrows. 

Image

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

The relations with a solid line are structural: only those relations with a dashed line indicate transformation or movement. 



Things and relations are used to create hierarchies as in the following diagram. 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 root node at the top and proceed downwards.

Image

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

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.

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 may be annotated with the proportion to which they are a part. For example, the following figure shows overlap between two parts, where 30% of part1 overlaps with 40% of part2:

Image

Figure B.3d: Overlap of two parts.

References or reflections can be depicted as bidirectional: 

Image

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

This diagram illustrates that just as mirrors reflect people that look at them, people reflect mirrors at which they look (in virtue of their mental references). 


Glossary

Image

absolute: A thing is absolute if it does not depend on other things and relative if it does depend on other things. Conceptually, the parts of a thing form its absolute nature, and the wholes of a thing form its relative nature. 

abstract: Concepts are abstract if their dimensionality is lower than a concrete concept. The degree of abstraction is related to its conceptual order: zeroth-order concepts are concrete and higher-order concepts are abstract. 

atom: An atom is an indivisible thing.

awareness: In the context of this book, awareness is intentional; awareness is always awareness of something else. The term consciousness is used when other meanings are intended.

basic categories: Basic categories are “… ‘in the middle’ of a general-to-specific hierarchy. Generalization proceeds ‘upward’ from the basic level and specialization proceeds ‘downward’.” [Lakoff, 2008]. Experimentally, basic categories are basic in the sense that they are known better, as measured by cue validity.

boundary: Boundaries are nominal objects that have an extent of zero along the dimension which they divide. Therefore, boundaries create parts that are connected. For a more substantial discussion, see [Varzi, 2015].

chunking: Chunking is the cognitive process by which multiple concepts in short-term memory are turned into a single concept. 

complement: The complement of a part is defined as the whole of that part with that part removed.

concept: A concept is a part of the conceptual universe. It serves as the basis of thought, it is a whole of sensations, and it is in turn referenced by symbols.

conceptual order: Conceptual order indicates the number of symbols that must be traversed in order to reach ontological ground. Higher‑order concepts consist of symbols, or references to constituent concepts. It is also possible for concepts to be composed of other concepts directly (as unitizations), in which case they have a higher epistemic level but not a higher order. 

concrete: See abstract.

consciousness: In the context of this book, consciousness is understood as phenomenal consciousness, rather than access consciousness (see [Block, 1995]). Consciousness may be reflexive or intransitive, in which case it is not mediated by references. See also awareness. 

contiguous: A thing is (spatially) contiguous if its parts are connected. A thing is (spatially) discontiguous if its parts are not connected. 

deep structure: The deep structure of a sentence, as opposed to its surface structure, is the underlying syntactical structure that determines how its semantic value can be computed from its various syntactic parts (i.e., the words).

dichotomy: A dichotomy is an operation that divides a whole into two parts. It is the simplest way to make parts out of a given whole. 

dimension: A dimension is an axis that allows discrimination along its length. It may be nominal, ordinal, interval, or ratio. 

direct realism: Direct realism entails that the mind directly engages with external reality. Indirect realism entails that the mind consists of representations of an external reality, and that there is consciousness only of these representations (i.e., and not of external reality itself). Indirect realism is also known as representationalism, or the Representational Theory of Mind (RTM). It is often contrasted with views such as the Computational Theory of Mind (CTM), which posit that mind is a result of mental operations. For a modern treatment, see [Fodor, 2008], [Lycan, 2019], or [Pitt, 2018]. 

discontiguous: See contiguous.

Dual Process Theory: The theory that the mind can be roughly divided into two systems or processes, one intuitive (System 1) and one rational (System 2). In the Dual Process context, intuitive thinking is called Type 1 and symbolic thinking is called Type 2.

epistemic level: The epistemic level (or referential level) of an entity is its distance from some ontological ground. See also conceptual order and section Recursion. 

epistemological priority: One entity is epistemologically prior to another if it has a lower epistemic level.

epistemology: Epistemology is the study of knowledge. Ontology is the study of being or existence. 

equivalence class: An equivalence class is a set of things whose members are equal (or equivalent). Equivalence classes are created by wholes of references. See nominal identity.

existence: Existence is a property exclusively of references. If a reference exists, then it can be validly dereferenced (i.e., its reference has a location).

extension: See intension.

external identity: See internal identity.

feature space: A feature space is a space that consist of features. For example, visual space is a 3‑D space augmented with the feature of color.

finitism: Mathematically, finitism entails that infinity is a process and infinitism entails that infinity can be completed (as required for infinitesimal points or infinite sets). For more details, see the essays section of the companion web site, [http://theWholePart.com/essays].

Flatland: Flatland is a two‑dimensional world that was originally described by Edwin Abbott [Abbott & Stewart, 2008]. Three-dimensional people do enter that land, but they have unexplainable and magical properties from the point of view of the people of Flatland.

generalization: Generalization refers to the way that a higher-order concept becomes abstract. For example, a conceptual whole can generalize across its parts by isolating their common features. See also unitization.

hierarchy: A hierarchy is a tree‑like structure that consists of one or more dimensions. Two prominent types of taxonomies are meronomies, which are hierarchies composed of concrete parts and wholes, and taxonomies, which are hierarchies composed of abstract parts and wholes.

holism: Holism explains a system in terms of its wholes. Reductionism explains a system in terms of its parts. Both are biased views in that they presuppose that such explanations are more valid or fundamental than other alternatives.

idea: The idea corresponding to a given concept is the activation of that concept along with its mereological context, or all of its parts and wholes. 

indirect realism: See direct realism.

infinity: Infinity is often understood as a number that is bigger than any other number, although it is better understood as a limit process with no upper bound (see also finitism).

intension: The intension of a set is a characteristic property of all members of that set. The extension of a set is the enumeration of the elements of that set. 

intentionality: Intentionality is a philosophical term that indicates a sense of aboutness. For example, awareness is intentional because it is always awareness of something (and therefore inherently dualistic). 

internal identity: Internal identity establishes the identity of an object in terms of its parts. External identity establishes the identity of an object in terms of its wholes. 

isomorphism: Isomorphism, which literally means “the same shape”, is a relation between two things that expresses equality or congruence. Isomorphism can be evaluated in mereological space or referential space. A reference is isomorphic to its referent if they are internally and externally identical within their respective domains.

mereology: Mereology is the study of parts and wholes. As a mathematical science, it was originally developed by the Polish philosopher Stanislaw Lesniewski.

metaconcept: A metaconcept is a higher-order concept whose purpose is to equate a word and its object by composing both of their symbols. It is a key component of the proposed mechanism for language: the symbol of the word activates the metaconcept, which causes the subsequent visualization and interpretation of the symbol for the object. 

negative entity: A negative entity is a concept that cannot possibly be sensed, such as a hole in the ground, which is known only in virtue of the lack of sensation of dirt.

nominal identity: Nominal identity entails that two entities are identical in virtue of being designated by the same name. For example, me from eight years ago is the same me that woke up today because I have the same name now that I did then.

nominalism: Nominalism is a philosophical doctrine that claims that objects in the world are only nominally identical from moment to moment, and not intrinsically identical (see nominal identity). Nominalism is opposed to philosophical realism, which maintains that reality is composed of natural kinds. Believers in natural kinds typically believe that objects have an essence in virtue of which they remain the same from moment to moment. For example, I am the same person that I was eight years ago because of certain enduring and essential properties.

object: In the context of this work, an object is a physical thing, as opposed to sensations and concepts, which are different forms of mental things. 

object permanence: Object permanence means knowing that unobserved objects continue to exist in an external world. This entails that a concept is developed for an object that exists independently of the sensation due to that object. Object permanence as a developmental stage was first studied by the psychologist Jean Piaget. 

ontology: See epistemology.

open: If a space is open, then there are no ultimate mereological limits such as largest universes or smallest atoms. Dimensionality may also be open; an open‑dimensional space has an unlimited number of dimensions.

orthogonal: Two dimensions are orthogonal if a change in one dimension does not entail a change in the other. In two dimensions, orthogonal lines form a right angle to one another.

overlap: Two objects overlap if they share a common part. Two objects underlap if there is some larger whole that contains both.

panpsychism: Panpsychism is the belief that everything has a psyche, or that mind and matter are everywhere coextensive. See [Goff & Seager, 2017].

part: A part is a thing that is contained in another thing, which is called its whole. If the part is smaller than that whole, it is called a proper part. Similarly, a thing is a whole if it has parts.

partition: A partition of a thing is a complete and exact decomposition of that thing into some number of parts. Every bit of the whole is contained in some part, and no bit is contained in more than one part.

percept: A percept is a combination of a concept and its associated sense data. A percept is a part of perceptual space, which exists in a continuum between sensory and conceptual spaces. 

realism: See nominalism.

reductionism: See holism.

reference: A reference is a representation of a thing, as opposed to the thing itself. Linguistically, references may be either lightweight signs or heavyweight symbols. A reference refers to a referent.

referent: See reference.

reflection: Reflection is a continuous version of (discrete) reference.

refeology: Refeology is a term coined to denote the study of references. It is related to semiotics, the study of signs and symbols, although it includes the study of cognitive signs and symbols. Refeology derives from the Latin refero (“to bring back”) and the suffix logy (“study, discussion, science”).

relative: See absolute.

self-reference: See self-reference.

sensation: A sensation is a part of sensory space. It constitutes nonconceptual mental content, and it is a subjective referent to either an object or a concept (in the latter case, it is called a symbol).

set of all sets: The set of all sets (or the Universal Set) is that set which is composed of all sets. If not created using constructivist principles, it creates paradoxes such as the Russell-Zermelo Paradox (see [Irvine & Deutsch, 2016]).

space: Space is used in this book as a metaphor for both reality and mind. It is a subsymbolic replacement for discrete entities.

subsymbolic: In the context of cognitive science, theories such as connectionism offer a subsymbolic alternative to traditional symbolic approaches. In mathematics, a similar distinction exists between point-free topologies such as mereology and point-set topology. See also symbol.

symbol: A symbol is a sensation that references a concept. The term “symbol” as used in this book entails a cognitive symbol, as opposed to verbal or written symbols that exist independently of mind. See also subsymbolic.

thing: A thing is a generic term that is further differentiated into objects, sensations, concepts, and symbols; for further details, see section Epistemic Universes.

token: A token is an instance of a class. For example, there are two tokens of the word “a” in the previous sentence, but those tokens consist of a single type. A type is a class of things.

type: See token.

underlap: See overlap.

unitization: Unitization refers to the process of making wholes as unions of parts. It does not entail generalization.

universe: A universe is a space which is an ultimate whole from a given point of reference. 

whole: See part.


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[1] Dimensionality is particularly well-suited to non-symbolic analysis, since it is an aspect of space, rather than any particular mereological entity.

[2] The implicit assumption that only one thing can exist in one place and time is possibly due to the structure of symbolic cognition. Although it reduces redundancy, it also tends to prevent overlap, which is generally undesirable.

[3] The belief that only atoms are ultimately real is a view shared by both Therevadan Buddhism and Greek atomistic philosophy.

[4] In fact, God is often considered to be the ultimate whole, a belief that forms the basis of both pantheism and panentheism. Similarly, consistently regarding one’s self as a whole is the basis of egotism.

[5] The Buddhist assertion of pratityasamutpada is stronger than Hegel’s thesis about the completeness of parts, because it expresses that things do not exist as separate entities.

[6] Chandrakirti frequently talks about a whole and its parts as they relate to a person, in which context they are referred to as a “self” and its “aggregates”.

[7] There are several arguments in Indian philosophy that a whole is neither the same as its parts nor different from them. For example, a whole is not the same as its parts because a thing cannot be both one and many at the same time. Another argument is that even though various parts of a whole could be missing, the whole continues to exist, therefore none of the particular parts are essential. However, a whole cannot be other than its parts, since removing all the parts will also remove the whole. For more information, see the section on sevenfold reasoning in [Thakchoe, 2011] or similar arguments in [Wasserman, 2018].

[8] Platonism is a denial of materialism because it posits that material things are actually a bundle of ideal forms.⁠. Platonism (or Platonic Realism) and Materialism are both construed in this context as types of realism: one accepts the reality of universals and the other accepts the reality of particulars. In this book, universals are interpreted as valid mental phenomena, regardless of their ontological status.

[9] The relationship between abstraction and materiality is explored further in subsequent chapters. Briefly, abstraction entails that things lose some of their characteristic (particular) properties, such as their specific location within space or time. For example, an abstraction based on two green things may have no properties associated with place or time if the green things are not collocated, but it has the property of color since greenness belongs to both.

[10] Although as Steven Pinker said, “Rocks are smarter than cats, because rocks have the good sense to go away when you kick them.” (see [Pinker, 2007]).

[11] In this book, “consciousness” can be reflexive or intransitive, while “awareness” is always intentional.  In other words, consciousness can be direct or pre-reflective, while awareness is necessarily reflective or awareness of something else. Therefore, arbitrary matter may have some form of consciousness, but only a mind can have awareness. Reflexive consciousness is known in Indian philosophy as svasamvedana, where it is likened to a lamp that illuminates itself (see [Kellner, 2011]).

[12] This point of view does not imply that “everything is relative”. For example, although a rock may be called soft, it does not make a decent pillow.

[13] Discerning the relative and the absolute in the Indian philosophical context is of great importance, since doing so is a precursor to correctly understanding and operating in the world.

[14] This categorization scheme is revised slightly in subsequent chapters, where concrete concepts are distinguished from abstract concepts.

[15] For example, the Kantian tradition defines several categories of things that are inherently a priori, such as number. Similarly, the Buddhist tradition defines any verbal expression as relative (or imputed) truth.

[16] Even if the world cannot be expressed directly, it is possible to make conditional statements about it.

[17] The less common view is that the physical universe contains references to the subjective universe, although this is certainly true if references are understood in a very general sense. This book tries not to take sides about any ultimate ground of reality, although it necessarily adopts a point of view for the purposes of presentation.

[18] The use of the term “things” is deliberately generic; this term is delineated further in section Epistemic Universes

[19] According to an Indian metaphor called the Net of Indra, the universe is like a net of jewels, each of which reflects all the others. This suggests that all objects are references and have this dual role. Shantideva (ca. 8th century CE) similarly characterizes all objects as having a relative nature and an absolute nature.

[20] It may clarify things to think in terms of “events” and “spacetime” instead of in terms of “things” and “space”, since the former terms connote having more than three dimensions.

[21] These relations are stated using the improper part operator (≤) instead of the proper part operator (<). If proper parts are used as the basis for our meronomy, then case (4) is equivalent to case (1).

[22] A partial order is one which has equivalence classes (e.g., classes of objects for which neither of those objects is a part of the other). A space is totally ordered if all objects in that space are either parts or wholes of all other objects in that space.

[23] It also makes sense practically if there is a cognitive dis‑ease with overlapping parts when overlap is not accommodated in one way or another (i.e., if our choice of logic affects our tolerance of ambiguous situations in the world).

[24] Partitions can be easily formed by recursively creating dichotomies of a whole, since a part and its complement always create a partition.

[25] The number of things in a mereological space is therefore a function of our point of view. For example, whether a forest is considered to be one forest or one thousand trees does not change the reality of the situation, since in both cases the forest has the same spatiotemporal extent, or shape.

[26] However, there is usually some basis in the world in virtue of which the division is made. In more technical language, fiat (or nominal) boundaries often correspond to bona fide (or actual) boundaries.

[27] If this were not the case, the partitioning elements themselves would be parts. In point-set topology, this has side effects such as preventing objects from making contact with each other.

[28] For reasons discussed later, human minds seem particularly prone to the essentialist view. However, the argument against essentialism that no thing is entirely independent of its context has been made by numerous movements including Platonism, Gestalt psychology, and Indian philosophy.

[29] Mathematically, this corresponds to the extensional definition of a superset, rather than the intensional definition of the set itself.

[30] Writing desks have feathers in virtue of their quill pens.

[31] At least if one assumes the mereological thesis of unrestricted composition.

[32] “One without a second” (ekam evadvitiyam) is a euphemism for Brahman, the ultimate reality underlying all phenomena (that is mentioned in Chandogya Upanishad, chapter 6 section 2).

[33] Space could also be non‑Euclidean, such as a toroidal space that wraps around like a donut. However, such a space (if open-dimensional) would need to be finite in some dimension, otherwise it would intersect with itself.

[34] This discussion of spatial boundaries also applies to temporal boundaries, a topic that is addressed in section The Physical Universe.

[35] The name “atom” stuck to the particle it was used to name, even though that particle ceased to be regarded as indivisible due to the discovery of sub-atomic particles. The quest to find increasingly small particles has continued, and today it remains an open question whether the physical universe has such smallest units or not.

[36] In modern physics, particle definition is made by appeal to various properties such as handedness and spin. If these particles cannot be distinguished in virtue of their parts, then it becomes a considerable problem to explain why different properties are associated with different particles (since in absence of parts, these particles have no internal differences).

[37] Although references can be understood in terms of what they are (i.e., in some other space), they are references because they can represent the thing to which they refer.

[38] Note that this does not entail that ultimate referents are independently meaningful: they may contribute only a portion of meaning, and/or derive their meaning from part/whole relations with other referents. In either case, the meaning of an ultimate referent cannot be entirely referential.

[39] These two statements, when seen as a whole from a larger perspective, entail self‑reference. In other words, they prevent the referential space in which they occur from being well-ordered.

[40] Although it sounds odd to say that a cat is a dog, it is also odd to say that one cat is another, or even that a middle‑aged cat is the same cat as that cat when it was a kitten.

[41] Orthogonality for Euclidean spaces can be understood as axes that exist at right angles to one another.

[42] Note that objective space is only objective from the point of view of a particular individual; it may overlap with the subjective space of a different individual.

[43] These cognitive mereological and referential relations are depicted with Greek letters rather than with UML notation, since they only constitute parts and wholes from the subjective perspective. In other words, neither conceptual nor sensory spaces are physical parts of one another; rather, they are constituted by references which are subjectively experienced as parts of one another.

[44] The basic model of cognition allows wholes of wholes and parts of parts, although they are not depicted as a transition from sensory space to conceptual space. This omission is done for the sake of simplicity.

[45] The decision to establish conceptual space as the exclusive source of action is not entirely accurate, but it is the best fit at the current granularity of the model.

[46] To use a more concrete example, our house may be a part of the world which we enter and leave, but if we never leave it, it is our universe. People may come and visit, and tell us about the world outside, and we may form an idea of the universe outside, but we still form that idea from within our house.

[47] The combination of sensation and conceptualization is referred to as perception. Therefore, concepts in conjunction with their sensory content are sometimes called percepts.

[48] Implied in this statement is that even mental phenomena are physical, in addition to whatever else they may be. In other words, what is subjective from one point of view is objective from another point of view.

[49] Four‑dimensional objects are also called occurrents (see [Simmons, 2000]).

[50] To support the theoretical position that all parts of the physical universe share its dimensionality, the relationship of N‑D objects to several experiments in modern physics is explored in the essays section of [http://thewholepart.com].

[51] Typically, the distinction between abstract and concrete is taught as the distinction between “child” vs “my child”, where “child” is seen as a general type and “my child” is seen as a specific instance of that type. However, if 3‑D objects are abstract, then “my child” is actually an abstraction of which “my child on Tuesday” is a concrete instance.

For further discussion, see [Ingram and Tallant, 2018] or [Markosian, 2016].

[52] The notion that space is 4‑D, or that time is an integral part of space, was proposed by Hermann Minkowski in 1908. Therefore, spacetime is more formally known as Minkowski space.

[53] For example, a two‑dimensional coordinate could be expressed using two real numbers and a Euclidean coordinate system, such as the point at [y=1 inch, x=1 inch]. However, the same point in space could be located in a number of different ways; using polar coordinates, it could be specified as [angle=45 degrees, radius=1.414 inches].

[54] One alternative to Euclidean dimensions are spherical dimensions (or in the N‑dimensional case, hyperspherical dimensions). To visualize this, imagine that you are an ant traveling on the surface of a sphere. If you go far enough in a given direction, even though you are traveling in a straight line with respect to the surface, you will end up where you started.

[55] This choice is a partial endorsement of the claim that “if a tree were to fall in the woods and no one was there to hear it, it would make a sound”. If you believe that there is no objective domain above and beyond the many subjective domains of individual experience, feel free to substitute the term “multi‑subjective universe” for the term “physical universe”.

[56] It may be that concepts cannot be sensed, because there is no feeling within the brain (i.e., the brain lacks sensory neurons such as nociceptors). In that case, the left branch of this diagram would be incorrect, although the symbols that reference concepts can be perceived in either case.

[57] However, as Ralph Waldo Emerson said, “A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines.”

[58] Hence, to talk of sensations as count nouns may be slightly misleading, since the lack of individuation of sensory parts renders them more like mass nouns (i.e., prior to being quantified). If desired, the word “percept” can also be used, although that presumes some degree of conceptualization.

[59] That said, one might argue that all aspects of reality can only exist in virtue of a subjective/objective interaction. For example, consider the way a single coin appears elliptical from one point of view, and circular from another. These two views indicate that the coin in itself is not exclusively circular or elliptical.

[60] This amounts to the claim that individuals have experiences of multiple overlapping epistemic levels. This can also be viewed as a psychological variant of the ontological claim that all of reality has both a relative and an absolute nature (a claim made by the Buddhist philosopher Shantideva).

[61] Although the external senses have a fairly common categorization, the internal senses do not. For example, there is no commonly agreed-upon categorization of human emotions.

[62] The science of subjectivity is difficult because of the inability to conceptualize and independently verify subjective experience. In other words, it is difficult to arrive at a consensus opinion about what is being referred to when talking about subjective experience because a given subjective domain does not present itself to multiple observers, although it is possible under the assumption that one can behaviorally respond to sensations for which there are no concepts.

[63] In the context of this book, since “objectivity” is used in the sense which is independent of subjectivity, the term “objective space” as used here should be understood as a subjective space that becomes increasingly intersubjective in virtue of its conceptual organization.

[64] The input from each eye is combined neurally to form a disparity map to determine the distance of objects, since monocular vision does not produce the phenomenon of depth.

[65] Twenty intervals grossly underestimates our discriminative capacity. In psychophysics, the lower limit of differentiation is called the just noticeable difference, which has been shown in studies of vision to be affected by the firing of a single neuron (see [Baylor & Lamb, 1979]).

[66] Since fingers occupy a larger amount of sensory space than do our forearms based on the amount of cortex devoted to them, it seems reasonable to conclude that humans literally sense more of their fingers than of their forearms even though fingers occupy a smaller amount of physical space.

[67] This effect is known as neuroplasticity.

[68] There is an obvious disconnect in the analogy between “body” and “sensory space” in that sensory space extends referentially further than the body does mereologically. However, the body is the largest part of sensory space that is mereologically consistent, and thus easily defined.

[69] The sensation that is available to collect into concepts is determined by both sensory (or bottom‑up) activation and conceptual (or top‑down) inhibition.

[70] Both zeroth‑order and higher-order concepts can operate in parallel with one another after they have been learned experientially, since both can be activated bottom-up. When activated by symbols, however, they are limited to serial operation.

[71] Because symbolic thought causes an inhibition of what is unattended, the top‑down influence of a single concept can make us perceive (and thereby know) less. For example, a hungry mind categorizes experiences only as food or not‑food, while other concepts become invisible. This is contrary to how concepts are implicitly believed to work, since the occurrence of a concept is typically seen as an indication of knowing more (see section Attention).

[72] These systems overlap in practice, which somewhat blurs the distinction.

[73] A counterexample to this generality happens in the visual pathway, where the dimensionality of the representation increases (see section Sensory Dimensions).

[74] The reason that collecting even open-dimensional atoms increases the dimensionality is that atoms are discrete and have unit extent. Therefore, by collecting them, the direction in which they are collected becomes divisible, which was not previously possible.

[75] As mentioned previously, action is a mapping from subjective space to objective space, although it is depicted here as originating from conceptual space for simplicity.

[76] Although the basic model treats the physical universe as the source of everything, it is also a destination; just as our sensation is caused by the world, the world is caused by our actions. Given the coexistence of these universes, perhaps the determination of which universe existed first is merely the result of one’s point of view. For example, perhaps the scientific tradition that maintains that the physical universe existed before there was anything to perceive it, and the spiritual and idealist traditions that maintain that the creation of the universe required an act of perception, are compatible and complimentary points of view.

[77] However, understanding entails knowing both the parts and wholes of an object, so visualization seems like the best English word to describe this process.

[78] This is probably why various chunking strategies often integrate concrete cues, such as walking along a path.

[79] Children often learn words by going through a subvocalization process of internalizing words. Symbolic space becomes a separate “mental sense” to the extent that symbols are no longer associated with auditory sensation; the degree to which symbols remain vocalized varies from individual to individual.

[80] This interference is described in a number of sources such as [Baddeley, 2001]. The necessity of this destructive interference is an important question, since it would be advantageous to have access to both non-degraded semantic and episodic content.

[81] The trade-off between sensation and conceptualization applies not just to other people’s speech, but to one’s own thought. In other words, it is possible to either sense or conceptualize one’s own thought process, depending on the epistemic level to which one pays attention.

[82] The dimensionality of atoms in this section refers to their differentiable dimensionality. Thus an atom, even if it exists as a part of a high-dimensional space, does not have any discrete dimensionality because it is not differentiable along any dimension.

[83] A line constructed in this way is not entirely equivalent to a mathematical line, since it begins with a discrete atom instead of a point: therefore, this process increases the dimensionality of a discrete space, not a continuous space.

[84] Time is said to move in only one direction, so it seems different from the other spatial dimensions, however its apparent unidirectionality is a result of using the direction of entropy as a measure of time.

[85] The most obvious options for introducing a fifth dimension are either adding a dimension after the temporal or adding a new dimension in the penultimate position and preserving time as the last coordinate. Although it seems quite counterintuitive, the second option allows objects such as the Necker Cube, which has four spatial dimensions and no temporal dimensions.

[86] Generalizations may be seen as propositions or propositional functions. Propositions are functions that produce a truth value as a result. As a simple example, isaTree(x) indicates the presence of a tree.

[87] Further, the dimensionality of an abstract concept is less than the dimensionality of its constituent concepts. In more detail, the dimensionality of the reconstituted meaning is determined by the degree to which the constituent concepts of a higher-order concept overlap. Therefore, concreteness can be viewed as a continuum which increases with dimensionality.

[88] In psychological terms, categories are defined using both exemplar theory and prototype theory.

[89] In other words, since the ideas of the two chair symbols entail the activation of all parts and wholes of those chair concepts, the intersection of those two symbols is not empty in virtue of the common parts and wholes.

[90] This is obviously a bit weird, but that doesn’t rule it out, because so are humans. Intuitively, however, a conceptual mixture of different referential levels does not seem correct. For more astute observations on this issue, see [Lewis, 1991].

[91] Prototype theory is proposal within cognitive psychology that pertains to the understanding of cognitive categories, and which is often contrasted with exemplar theory.

[92] In this diagram, the objective universe is not depicted as divided into a dog and a cat, which would imply that they exist as natural kinds. That said, objects are labelled as individual entities elsewhere in this book as a matter of convenience.

[93] They are not necessarily invalid, since an abstract tree does have abstract bark as a part. An equivalent concrete meronomy is presented in section Meronomies.

[94] Since the syntax of language is extremely complicated, the presentation in this section necessarily omits numerous details. However, it is sufficient to show how the basic model of cognition interfaces with the study of syntax and semantics.

[95] This epistemological context is important for all concepts of which one knows the definition, although it is essential to understand objects with which one does not have direct experience.

[96] The categories at the epistemic level where concepts change from concrete to abstract are known in cognitive science as basic level categories.

[97] Equivalently, ice is understood as everything after removing all things that are not solid, not cold, and not watery.

[98] Definite articles are modifiers which reduce the abstraction of the object to which they are applied. For more information, see [http://theWholePart.com/essays].

[99] The Binary Branching Hypothesis is a theory in generative linguistics that syntax trees cannot have more than two branches.

[100] It might make sense in this case to let x equal 1, which would correspond to the temporal dimension, although the VP can carry modality, which suggests that x may in general be higher.

[101] Although the copula is‑a is very common in this context, several forms of the verb “to be”, the verb “means”, and other words can also be used. At least in English, the symbol that is being defined typically precedes its definition.

[102] The sentence is the smallest linguistic referent which can be validly dereferenced. For example, even proper nouns (understood as 3‑D objects) must combine with a verb to validly refer to an object in the world.

[103] Alternatively, noun phrases and verb phrases are less independent than entire sentences; things which have both spatial and temporal parts are more independent, or less abstract.

[104] The argument that nouns and verbs cannot be validly dereferenced independently from one another can also be made on the basis of physics: spatial things (nouns) and temporal things (verbs) never exist as separate entities in physical reality, since spacetime requires both spatial and temporal ordinates. Further, because they are dimensionally incomplete, they cannot be visualized. Although it is possible to imagine an object not moving, one might argue that there is an implicit verb in that case, “not moving”, as opposed to a low-dimensional visualization.

[105] Although it would be possible to understand nouns as high-dimensional or permanent (4‑D) entities, in that case they would not be able to be modified by a verb since verbs require temporal change.

[106] This limitation is removed in a variant of classical logic called fuzzy logic

[107] Garrison Keillor maintains this is true of the town called Lake Woebegone, in the introduction to his radio show, A Prairie Home Companion.

[108] The latter is more or less what the scientific method seeks to express in a set of conditionals called experiments.

[109] The original definition of the kilogram was the mass of a liter of water at sea level. Since that mass can vary, the kilogram was subsequently defined in terms of International Prototype Kilogram, a particular object located in France. Since the mass of that object has also been found to vary, the kilogram is currently being redefined.

[110] This is prefigured by differing use of negation of Buddhist and Western logic; Buddhist logic often uses non-affirming negation, while Western logic emphasizes affirming negation.

[111] See [Casati, 2009] or [Reicher, 2019] for a discussion of non-existent objects in general.

[112] Negative concepts are therefore quite similar to the Buddhist conception of self (understood as the “mere I”), in that they exist on a relative level but not on an absolute level.

[113] Therefore, holes are ultimate examples of the relative; we can say where a whole is, but we cannot say what a hole is, since a hole has wholes but no parts.

[114] Sensations do have a spatial complement (i.e., sensation in other spatial locations), although that space is known only in virtue of subsequent conceptual wholes.

[115] The relation of emotions to these endpoints is particularly interesting, since emotional attachment to negative entities is entirely conceptual and emotional attachment to unknown entities is entirely sensory or nonconceptual.

[116] For example, in order to think of a dog, a symbol must conceptually inhibit all non-dog animals; otherwise, the visualized concept will not be sufficiently specific.

[117] The bandwidth of System 2, because it processes linguistic symbols serially, drastically limits the capability of concepts compared to intuition. However, System 1 tends to be biased in ways which may be undesirable, which leads to its characterization as irrational. See the excellent books by [Kahneman, 2013] or [Gladwell, 2007] for more discussion.

[118] Although the content of intuition cannot be described in language, it is possible to express how intuition works: the experience is ineffable, but the mechanism is effable.

[119] Animal cognition in this context refers to the cognition of most animals, with the exception of several species that have demonstrated an ability to form higher-order concepts, such as elephants, dolphins, and some higher primates (see [Koerth-Baker 2010]).

[120] However, they may have some form of limited consciousness of what it is like to be a rock. Under the assumption that all material things have some form of consciousness, references are conscious in virtue of that materiality, and are further aware of external things in virtue of being referential. References to references create awareness of awareness for the same reason, and a recursive process that involves references allows recursive awareness. All of these types of knowing are experienced in virtue of matter being conscious, but they have referential awareness only in virtue of the representational capacity of a nervous system.

[121] Presumably removing only symbolization is not sufficient, since it could be replaced with sensation of actions (e.g., self-talk).

[122] The cognitive difference between animals and humans entails that animals are more closely tied to their sensory lives than humans, since their minds are of a lower epistemic level. For example, animals are not in danger of becoming emotionally attached to abstractions, because they don’t form any abstractions.

[123] In particular, referential truths are a priori for humans, but not for other animals. However, mereological truths are a priori for both humans and animals.

[124] Studies of animal behavior reinforce this finding: behavior is unitized, such that learning to perform a given spatial task entails learning a time and rate at which that task is performed [Gallistel, 1980].

[125] Top‑down inhibition is similar to an understanding of how concepts appear to the mind called apoha theory, which was developed by the Indian scholar Vasubandhu. It is also similar to the spotlight model of attention.

[126] It seems probable that this top-down illusion is in fact present throughout perceptual space but noticed only in selective contexts (such as in this example).

[127] In other words, if one does not perceive anything that one does not already understand, then learning is prevented. Therefore, although it seems paradoxical, there is a sense in which greater understanding can be achieved by temporarily not understanding.

[128] For example, Zhuangzi dreamed that he was a butterfly, and upon awakening, could not be sure that he was not a butterfly dreaming he was a man (see [Hansen, 2017]).

[129] In more technical terms, the interference between symbolic and subsymbolic processing sometimes makes it desirable to temporarily stop System 2 so that one may rely on System 1.

[130] Although some sensation is still processed cognitively, information that is outside of attention is often inaccessible (in the sense of access vs phenomenal consciousness: see [Block, 1995]).

[131] In the type of meditation known as noting practice, meditators are often instructed to label their thoughts as thinking. That advice is significant in terms of epistemic level, because it implies that one should be aware of thoughts as thoughts, rather than engaging with the conceptual content of those thoughts.

[132] For example, there is no agreement about how many emotions there are. Similarly, the experience of emotions is often difficult to localize; fear may occur in the belly, or stress may occur in the shoulders, although mood is often expressed as an omnipresent phenomenon.

[133] As the variety of emotions is huge, this section focuses only on two emotions as they relate to the basic model: like and dislike (or attraction and aversion).

[134] Advertisers make use of this fact to create brand-names and logos that literally become delicious.

[135] The thesis that our symbols should not be allowed to become emotionally valent is similar to the spiritual injunction not to become attached to our thoughts. In that context, undesirable emotional attachment can be diminished by developing equanimity with respect to the occurrence of concepts, or temporarily prevented by preventing the concepts from occurring. There are a number of different theories about how that can be done most effectively, such as adopting a stoic view, getting more exercise, or practicing various forms of meditation.

[136] This is especially true because the consequence of removing the emotional valence of the referential content itself would be equivalent to removing any intuition. For further discussion of the role of emotion in intuitive wisdom, see the essay on omniscience and universal empathy at [http://theWholePart.com].

[137] Admittedly, this is a rhetorical question. As Thomas Nagel stated, “The subjectivity of consciousness is an irreducible feature of reality, and it must occupy as fundamental a place in any credible world view as matter, energy, space, time and numbers.” [Nagel, 1989].

[138] The philosopher John Locke called these the primary and secondary qualities of perception.

[139] While from an objective point of view representations may be seen as non-referential parts, the importance of their location as subjectively experienced should not be discounted. To ignore referential content in an account of the world is equivalent to entirely devaluing mind in favor of matter.

[140] In other words, the conflation of the subjective and objective perspectives may encourage identification with the references (i.e., the body and particularly the brain) as opposed to the referential content of those references (i.e., the subjective universe).

[141] Treating your neighbor with the same level of kindness as you treat yourself is not self-sacrificing if you consider your neighbor to actually be yourself, but neither is it selfish. The lack of a subject/object distinction is therefore creates a significantly different philosophy from the typical characterization of solipsism.

[142] Objects are often seen as independent rather than interdependent, perhaps because every time we pay attention to them, they become wholes.

[143] It also happens to some degree when kissing, or in virtue of blood and organ donation.

[144] In other words, because minds are defined referentially, identity between minds is expressed in terms of referential identity, not mereological identity.

[145] Practical assessment of this concept can be measured using something called the mirror self-recognition test, in which the subject is tested for their ability to recognize their self in a mirror.

[146] Another way of expressing the mereological/referential distinction is that consciousness is supported by one physical area, but represents another area, or that its base (or its physical location) is other than its place (or the location in which it is meaningful).See [Ganeri, 2012] for an extensive analysis of this distinction between the base of consciousness (asraya) and the place of consciousness (adhara) in different philosophical traditions.

[147] Materialism primarily explores connections between objects in physical (or mereological) relation to one another. It also tends to deny consciousness, although in modern culture, consciousness and the subjective perspective are more often ignored than explicitly denied. For a counter example, see [Dennett, 1991].

[148] Interestingly, our material possessions are also included in the self-concept from the legal point of view.

[149] The belief that one can know oneself directly, without (dualistic) experience of experience or experiencer, is called self‑reflexive or intransitive consciousness. This type of consciousness operates at the physical level of being, which contains only referents and no references.

[150] For this reason, many spiritual traditions either entirely reject a relative understanding of the absolute, or they caution that a relative understanding of the absolute is not equivalent to the ineffable experience of the absolute.

[151] Although we know that our subjective world is limited, it is not possible to experience the objective world without it becoming subjective. For example, we are taught that we are coextensive with our body, but subjectively our senses extend right through our skin; we sense as far as we can in all directions, but never sense objectivity.

[152] This does not mean that we do not have direct access to sensation, but rather that our subsequent attempts to explain ourselves conceptually are not necessarily correct (for example, see [Wilson et al, 1989]).

[153] “Mind” in this context entails all subjective referential content, or everything that is experienced.

[154] According to idealism, we subjectively experience ourselves as everywhere. In some sense, idealism is more open to the idea that the “owner” of a thought is as much the world as it is the individual. As a result, idealism tends to be kinder (because one’s mind belongs to others as much as oneself), and materialism tends to be more selfish (since it does not include selfc).

[155] This may also be viewed as a failure of correct translation. Speaking loosely, perhaps what is world from the objective point of view is mind from the subjective point of view.

[156] Referents are less localized than their references under the assumption that at least some references point outwards in addition to inwards.

[157] There is a limit to the referential level of a mind, since it becomes increasingly difficult.

[158] In other words, other only exists when multiple epistemic levels are combined into a single self; no epistemic level is independently aware of an other. For example, the “sensory other” is known only as self2 without the content of self1. The lack of an other at any given epistemic level puts the self/other dichotomy on a curious footing, since it always straddles epistemic levels.

[159] Analogously, my body is mine, but my mind is not (i.e., the “owner” of mind is as much the referential content as it is the physical references themselves). In more poetic terms, the thought of the tree belongs to the tree, not to me.

[160] The term nondualism is sometimes associated with mysticism, but there is nothing mystical about its use in this context, where it refers to the nonduality of subsymbolic mental content. Abstract experience is dualistic or symbolic, and concrete experience is nondualistic or subsymbolic. This does not entail that concrete experience is devoid of concepts, since intuition makes use of conceptual discernment. It does mean, however, that concrete experience is devoid of any single dualistic concept at the expense of the others, as happens when it is invoked by its symbol. At an even deeper level, nondualism may refer to reflexive or intransitive consciousness, although that is not discussed here.

[161] This becomes particularly poignant if you act unethically in the course of trying to prevent an outcome you believed would happen.

[162] For what its worth, I’m a pantheist, so scientists and earth-worshippers should feel included, too.

[163] Further, they are literally counter-intuitive in that the use of such logic prevents intuition from operating correctly.

[164] In other words, cognitive psychology has no need for points, although it relies rather crucially on  basic categories and medium-sized objects.

[165] In psychological terms, what point-set topology does is to completely replace the absolute with the relative, or approximate the concrete with the abstract. To do so, point-set topology requires  infinity, which involves a number of difficulties when applied to the real world.

[166] In other words, numbers are negative entities, and not parts of the space that they divide. As a result, partless particles and completed sets with infinite cardinality are not necessary, although they may be used if desired.

[167] NB: this is not an ontological claim about neurons, but rather an epistemological claim about experience.

[168] The fact that the primitives are to some degree unknown may make mathematicians uncomfortable, but it is clearly compatible with psychological development. Since objects are not fully known geometrical primitives or logically singular subjects, it is only possible to describe them in virtue of their interaction with other objects, which describes both what and where the object is relative to other objects.

[169] Equivalently, it is both true and false.

[170] This logic shares some features with a type of logic called intuitionistic logic, but it is sufficiently different in both motivation and practice that that foundation is not used.  It is also similar to fuzzy logic, although fuzzy logic expresses possibility. While combining the mereological approach with fuzzy logic would approximate human reasoning better by expressing degree of certainty, that approach is not followed here for brevity.

[171] As a result, the creation of proper wholes requires at least two parts, and the creation of proper parts requires at least two wholes.

[172] This refers specifically to affirming negation; non-affirming negation, which removes all certainty from a truth value, is not depicted as it requires making the truth value more complex.

[173] In fact, the reference is responsible for increasing the dimensionality, but the whole is required to create differentiability along that dimension.

[174] Although there may be a mapping between discrete referential entities and continuous parts of small size, that mapping can only be exact if one subscribes to the philosophy of infinitism.