The iQuad Coin

Part I: Linking the Human Identity Function and the Complex Unit Circle

This blog begins a series on the iQuad Coin, which is a key aspect of UTOK, but it has not been elaborated upon much and needs more specification. In UTOK, the Coin works with the Tree of Knowledge System and the Garden to give a holistic account of the human subjective knower relative to natural science via the Tree and wisdom via the Garden. This blog introduces the Coin and shows how its two central features, the complex unit circle and the Human Identity Function, are linked. This linkage grounds the Coin and networks it into the UTOK architecture.

In his book, Incomplete Nature: How Mind Emerged from Matter, Terrance Deacon makes the case that the traditional systems of scientific thought that emerged out of the Enlightenment have tended to be physicalist and mechanistic in ways that fail to provide a clear, causal role for things such as goals, motives, wishes, and fears. His book explains both why these concepts fell out of favor as natural science developed, and why we need to bring them back to have a complete understanding of the natural world. Deacon framed these issues in terms of what he calls “abstentia” and “ententionality.” As he writes: “Longing, desire, passion, appetite, mourning, loss, aspiration — all are based on an analogous intrinsic completeness, an integral without-ness.” He defines this integral without-ness as abstentia and claims this is “a defining property of life and mind.” He further makes the case that abstentia gives rise to ententional entities (e.g., organisms, animals, and people). He further argued that “conscious experience is the quintessence of an ententional phenomena. It is intrinsically perspectival, representational, consequence-oriented, and normative.” By explicating abstentia and ententionality, Deacon uses the concept of incompleteness to develop a more complete naturalistic picture of the world.

The Unified Theory Of Knowledge (UTOK) affords us a naturalistic, scientific worldview that aligns well with Deacon’s assessment and analysis. According to UTOK, to have a truly effective metaphysical architecture for understanding the patterns of behavior in nature, one must scientifically differentiate the behavior of organisms from inanimate objects (i.e., Life from Matter), animals with brains from other organisms (i.e., Mind from Life), and self-conscious persons from animals (i.e., Culture from Mind). The vision logic of this argument is explicitly represented in the Tree of Knowledge System, which depicts how Life, Mind, and Culture can be effectively framed as both continuous with and distinct from the inanimate material dimension of complexity. UTOK posits that the fundamental reason for ontologically separating Life, Mind, and Culture from Matter is that the behavior patterns of cells, animals, and persons emerge from the dimensions beneath them via information processing systems and communication networks that have fundamentally novel causal properties.

The Tree of Knowledge System is the First Branch on the UTOK Tree of Life

This ontological emergence argument is extended via the Periodic Table of Behavior, which shows that to understand emergence in nature, we need to divide it into both the different planes of existence on the ToK System (i.e., Matter, Life, Mind and Culture) and into primary levels of analysis within each plane (i.e., parts, wholes, and groups across scales). As a function of this novel analysis of emergence, UTOK affords a natural scientific worldview that readily aligns with and accommodates Deacon’s concepts of abstentia and entensionality.

The UTOK also makes the crucial epistemological distinction between different kinds of knowing. In particular, UTOK claims that science is a fundamentally different kind of knowing than subjective personal knowing. The former is based on an exterior, third-person empiricism, quantification, and systematic logical analyses, whereas the latter is based on a unique, particular, interior, first-person qualitative empiricism. UTOK frames this difference via the relationship between the ToK System and the iQuad Coin.

Although the iQuad Coin has long been a part of UTOK, I have not elaborated on it much in writing. However, I am now in a place to explicitly articulate the role it plays in the UTOK system. In three recent episodes of UTOKing with Gregg (see here, here, and here), I provided the theorists John Vervaeke and Bruce Alderman a guided tour of key aspects of the iQuad Coin. This blog series seeks to delineate the structure of the Coin in more systematic detail.

The Basics of the iQuad Coin: The Human Identity Function and the Complex Unit Circle

As is typical of coins, there are two sides to the iQuad Coin. One side has the iQuad symbol on it, and the other has the UTOK Tree of Life. The UTOK Tree of Life should not be confused with the Tree of Knowledge System. Rather, as depicted above, the ToK System is the first branch on the UTOK Tree of Life.

In examining the iQuad symbol, there are two major aspects that need to be understood in order have a basic grip on why it is structured the way that it is. The first is the fundamental meaning of iQuad, which is that i raised to the 4th power = 1. This fact is represented by the corners, which list these numbers. In mathematics, ‘i’ is the imaginary number, which is the square root of negative one. It is called an imaginary number because there is no real number that fits this description. This is because when you square any negative number, you get a positive number.

Paul Nahin’s book, An Imaginary Tale: The Story of √-1, tells the story of how imaginary numbers emerged in mathematics over hundreds of years. Although there is much rich history to the story, the central move that solidified the place of imaginary numbers in mathematics was the realization by Caspar Wessel that imaginary numbers could be framed as orthogonal or perpendicular to real numbers. That is, the imaginary number line could be framed such that it was rotated 90 degrees from the real number line. This move sets the stage for what are called “complex numbers”. A complex number consists of a real and an imaginary number, which is often placed on a complex plane. To make this concrete, consider a grid such that the x-axis consists of real numbers (1, 2, 3, and so on) and the y-axis consists of imaginary numbers (i, 2i, 3i and so on). One could then plot a number on the grid. For example, the real number could be 2 and the imaginary could be 3i, and this would be located at point (2, 3i) on the complex number grid. (For more on complex numbers, see here or here).

If we draw a circle around the grid that intersects the real number x-axis at 1 and -1, and the y-axis at i and -i, then we have what is called a complex unit circle. With the complex unit circle, we now have set the stage for framing the most basic meaning of iQuad. By definition, if you square i, you get -1. Then if you square -1 you get 1. This means that i raised to the fourth power equals one. This is the first and most basic meaning of iQuad. We can take this one step further, which allows us to make a connection to the shape of the Coin. Following Wessel, we can think about each multiplication of i generating a 90-degree rotation around the complex unit circle. Each unit adds 90 degrees, with the fourth completing the circle at 360 degrees. This is shown in the diagram below.

The next thing to note about the iQuad symbol is that it is made in the shape of an “H.” And if you then rotate the coin 90 degrees, it then can be seen as being in the shape of an “I”. This, of course, spells “HI,” and this rotation can be initially thought of as a welcoming into the UTOK philosophy. More deeply, though, the “H” stands for Human and the “I” stands for Identity, and together they represent what is called the “Human Identity Function.” The Human Identity Function is about framing each person’s unique, subjective portal into the world and then using UTOK to build bridges between them.

The first move in grokking the Human Identity Function is to frame it in terms of the Human Identification Matrix. This refers to your capacity to form a matrix of identities between yourself and the world around you. Thus, you identify both as a particular person (e.g., your named self) and you identify with particular roles (e.g., husband, father, sports fan, whatever). You also readily identify things in the environment, such as chairs for sitting, dinner on the stove, and other people, and their selves and roles, etc. This cognitive mapping of the self-world relation is the identification matrix. It is the sense and meaning making schema you use to understand yourself and your relationship to the world. Although the language used here might be new to you, the basic idea should nonetheless be familiar. That is, it is just rather obvious that you open your eyes and automatically perceive the world around you. That is the work of your identification matrix.

Given that the complex unit circle and the Human Identity function are the two most basic aspects of the iQuad symbol, it makes sense wonder about the connection between the two. The full answer to this question is complex and multi-layered, and needs a series of blogs to make clear. The reason is that the iQuad Coin has its roots in something called the “Henriques Equivalency” that I developed back in 2001, and to understand how that evolved into the iQuad Coin takes a long time to unpack. Indeed, I am just now ready to lay out all the pieces of the argument that moves from the Henriques Equivalency to the Human Identity Function to the complex unit circle, and, ultimately, the Euler Identity and Formula.

Despite this, we can start to make some linkages by considering the 90 degree rotation, and the relationships between the concepts of real, imaginary, and complex. As discussed above, these words have clear and precise meanings in the world of mathematics and the complex unit circle. If we put them in commonsense terms from the standpoint of the Human Identification Matrix, we might say that “the real” corresponds to the things in the world, whereas the “imaginary” corresponds to ideas in our head. This is the first step, but we need to deepen the analysis.

To make this a bit more concrete, consider yourself in a furniture store and you see a man looking at the table. He has been approached by a salesperson and he offers the comment, “I see the table”. The first thing to do is to notice how easy it is for you to make sense of the situation. If they have been socialized in our culture, it is not hard for most people to identify tables, furniture stores, customers, and salespeople. You likely would even be able to automatically make useful inferences about why the man said, “I see the table.” For example, you might infer he was letting the salesperson know he is aware of the table and considering it, but does not want to commit to purchasing it. We can frame this capacity as arising from the Human Identification Matrix.

Now let’s apply the real, imaginary, and complex nomenclature to this scene. The simplest way to frame this is by saying that table is real and that the customer has an image of the table in his head. We can tentatively call this virtual image an phenomenological representation of the table “imaginary.” This is a useful move, but it raises a problem. There is a difference between something that is purely imaginary, such as fairies dancing on the head of a pin, versus an image that corresponds to something in the world, as is the case in this example with the table.

The exact nature of the relationship between the images we perceive, the imaginary ideas we completely make up, and the so-called “real” world is one of the most longstanding problems in philosophy. Because of this, let’s use the remaining word and call this relationship “complex.” Complex, then, refers to the dynamic relationship between images, imaginary concepts, and the real world. We can extend this further by saying we operate on or in a “complex” plane of existence, such that our perceptions are representing, mapping, or modeling the external world, and that there must be some kind of correspondence, else we could not interact with it the way we do.

This description linking the complex unit circle with the Human Identity function allows us to ground the basic frame for the iQuad Coin. It is a place holder for the unique human subject and, at the same time, creates a kind of association matrix with the complex unit circle. The rotation of H to I, and the fact that in the complex unit circle, imaginary units represent a rotation on the real is a loose association. In addition, we have made a stronger association between imaginary concepts inside of someone, relative to real things in the world, and the complex relationship between them. The two basic features of the iQuad Coin function to create some “associations and entanglements” between human subjectivity and mathematics. These threads of entanglements, adjacencies, and identities will grow through the blog series.

In Part II, I will expand on the Human Identity function and how it is structured to bridge one’s experience of being in the world to the world as it is mapped by UTOK, especially the Tree of Knowledge System.



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Gregg Henriques

Gregg Henriques


Professor Henriques is a scholar, clinician and theorist at James Madison University.