Proof Day Five Years Later: The Logic of the iQuad Path

Five years ago, on October 27, 2017, I finalized what I call the iQuad Path into the Garden. The day is called “Proof Day” in the Unified Theory of Knowledge (UTOK) because the iQuad Path provides a “proof” that links together the various kinds of justification systems into a coherent network of understanding. This blog attempts to outline the basic logic of this claim.

Here is the iQuad Path:

As shown, the iQuad Path consists of four steps. The first three are as follows:

1. 1 = i⁴ = iQuad

2. The Euler Identity, given as e^πi + 1 = 0, is mathematically true.

3. The Euler Identity currently is the most objectively beautiful equation in mathematics.

These are the “exterior” truths of the iQuad Path. By exterior, I am alluding to Wilber’s Quadrants, and they are exterior in that sense of the word. More specifically, I am claiming that they can be considered generalizable truths that exist across human knowledge systems in that they transcend both the subjective and social cultural context of justification and can be framed as analytic truths.

Starting with the first step, i is the “imaginary number,” which is the square root of negative one. It is called the imaginary number because there is no real number that, when squared, gives a negative answer (i.e., recall from your basic math classes that -1 x -1 = 1). Nonetheless, mathematicians have found a way to incorporate imaginary numbers into the logic of mathematics. And if you raise i to the fourth power you get 1. This means that the first aspect of the step, 1 = i⁴, is a basic deductive identity. It is true based on the definitions and logical deductive relations found in mathematics. There is one additional element to this step, which is a naming function. I am naming this relationship “iQuad,” so in the current definitional system, i⁴ = 1 = iQuad.

Step 2 is similar to Step 1 in that it has both a nominal and deductive logic aspect. The Euler Identity is a fundamental mathematical truth that anyone with a mathematics background can confirm. And just as I named the mathematical relationship i⁴ = 1 iQuad, the mathematics community has named this relation the Euler Identity. Interestingly, some historians of mathematics do not like this name because the relation was not first discovered by Leonhard Euler, but was initially formulated by a man named Roger Cotes. But this fact just makes the point that there is a logical mathematical relation that we are naming, and these two processes can be separated. Whether we call it iQuad or not it is true that i⁴ = 1, and whether we call it the Euler or Cotes Identity, e^πi + 1 = 0 is also a valid mathematical truth claim.

Step 3 is a different kind of truth. It is not a deductive truth that has to be true, but It is important to note that we are making a jump here in our epistemology. That is, we have shifted the kind of truth claim we are making. In steps 1 and 2 we are making truth claims grounded in naming and mathematical logic; here we are making a truth claim about the state of the world. “Objective beauty” is defined here based on the intersubjective agreement of human beings in a particular time period that something is beautiful.

We can justify step three by doing a comprehensive survey of mathematicians. This has been done on several occasions, and the results are consistent, especially if one only includes mathematicians and “purely” mathematical equations (i.e., there are some equations in physics that have, on occasion, been found by physicists to be more beautiful, but that does not challenge the claim as operationalized here).

Thus, the argument of Step 3 in the iQuad Path is that, as of October 27, 2017, the Euler Identity is the most beautiful equation in mathematics. Of course, this has not always been the case and we can imagine that a different equation might come along and replace it as the most objectively beautiful in the future. But a strong empirical argument can be made that the Euler Identity is currently the most objectively beautiful mathematical equation in the world (see, e.g., here, and here, with some calling it the “God Equation”).

The first three truths provide an emerging network of justifications. These justifications are exterior in the sense that they transcend any particular subjective or cultural context. The fourth step on the iQuad Path is an “interior truth.” In contrast to exterior justifications, it refers to being “inside” of a particular subjective position and worldview. The specific interior position is inside my mind, situated inside the worldview given by UTOK represented by the Garden.

The fourth step states that:

4. The Radical Mathematical Humanistic Equation, given as e^πi + 2πif = 0, is subjectively true and beautiful to me, although it might not be that for you, but it might become objectively true and beautiful in the future, even if it is never that for you.

Note that this step introduces a term, “2πif” and substitutes it for the one in the Euler Identity. It then renames the whole thing the . This, of course, means that 2πif should equal one.

The relation 2πif = 1 is called the Henriques Equivalency. . Here is a blog on the Henriques Equivalency and here is a slide that depicts its core logic.

It is not a traditional mathematical relation, but rather better understood as a descriptive metaphysical gateway that connects many different ideas together. Here is a depiction of this claim:

I am going to pause here and take a step back and offer a couple of paragraphs on UTOK’s descriptive metaphysics to help anyone not familiar with UTOK place what is being set up in a larger context of justification. If you are familiar with UTOK basics, you can skip to the next section.

UTOK, the Enlightenment Gap, and the Coin-Tree Relation that Repairs It

Modern empirical natural science emerged as a new way of knowing during the European Enlightenment. At its core is something called “third person empiricism,” which refers to the fact that science is based on empirical data that can be observed by any trained observer. Crucially, this is radically different from first person empiricism, which refers to what an individual sees from their unique vantage point. The more common terms for these basic claims are that science is “objective and quantitative,” whereas our unique perspective on the world is “subjective and qualitative.” Although modern science has, of course, resulted in much new knowledge, a problem emerged that has never been solved. That is, there is no good way to put the objective systems of knowledge generated by science with the subjective knowledge we have of our everyday lives. This failure to generate a coherent synthetic philosophy is called “Enlightenment Gap.”

UTOK is a new descriptive metaphysical system that places our subjective experience of the world and our scientific knowledge of the world together into a coherent whole oriented toward collective wisdom. It works by first giving us the Tree of Knowledge System, a novel and better map of Big History. The ToK System helps us see how the world can be effectively divided into the domains of: (1) an Energy Information Implicate Order of quantum physics; (2) the Matter-Object plane of classical physics and chemistry; (3) the Life-Organism plane of biology; (4) the Mind-Animal plane of psychology; and (5) the Culture-Person plane of the social sciences. When we add technology as a category (i.e., things like cars and buildings), and we have a basic taxonomy for the things in the world.

Whereas the Tree of Knowledge System maps the objective behavioral patterns of various things in the world, the iQuad Coin is different. It frames the subjective, first-person qualitative experience of being in the world. But what exactly is the relationship between the Coin and the Tree? Actually, the technical and precise answer to this question is very difficult to say in a blog. Why? Because the iQuad Coin represents your specific human identity function (also called your psyche) and it emphasizes the way that we are all unique, particular subjects. Thus, the actual answer depends on the context of each person and how their unique experience of being in the world can be connected to the ToK System formulation.

The broader and more straightforward route to constructing the bridge between science and the psyche is via UTOK’s metatheoretical formulation for human mental behavior, which is given by Justification Systems Theory, Behavioral Investment Theory, and the Influence Matrix. These constructs link the science of psychology with your particular psyche. To see this, you could start with some interior reflection and likely identify the narrating portion of your subjective mind. This is your ego in the language of UTOK, and it links you to the Culture Person plane of existence. You could then introspect further and shine the light of conscious awareness on your primate structure via the lens of the Influence Matrix and your mammal-into-animal-into-organism structure via Behavioral Investment Theory. We could then move to Character Adaptation Systems Theory and the Wheel of Development and the Nested Model of Well-Being to develop a more comprehensive map of your functioning grounded in a unified theory of psychology. (See here for an example of a “well-being interview” that does this). We can think of this linkage between psychological theory and the particular human subject as the standard bridge between scientific theory and unique subjectivity given by UTOK.

The iQuad Path is the “Formal Path” into the Garden

Whereas the standard bridge links the individual subject with the generalizable formulations grounded in psychological theory, the iQuad Path generates a formal logical network of understanding that bridges the particular subject with the UTOK worldview. Rather than being framed by psychological theory interpreting human experience via an iterative process, the iQuad Path is framed by “associative adjacent identities.” An associative adjacent identity makes connections between entities and then derives meaning from their relation.

Although this might be a new way of describing it, it is, in fact, a very a common way of forming knowledge. As an example, look at this diagram that is take from gestalt psychology:

Here we can say that A = B = C = D in the sense of being adjacent associative identities. Of course, A, B, C, and D are not completely identical, but they are equal in that they are the same shape and color. And, when placed together, they end up forming a square. (Or something we meaningfully perceive as a square. The gestalt perspective highlights how much of perception emerges from our minds filling in gaps and abstracting patterns to form wholes).

With this set up, we can now ask: The four steps are a set of adjacent associative identities that create a gestalt that ties together several different kinds of justification systems into a whole. There are many different kinds of “justification systems” that we use to legitimize claims. This blog describes seven of them, including: formal/foundational, coherentist, correspondent, phenomenological-subjective, social constructionist, pragmatic, and axiological (i.e., ethical or value-based). The argument is that the iQuad Path connects all of these into a network of associative adjacent identities to give a wholistic picture of the justification system matrix. The “iQuad Proof” then is formed from the gestalt of the whole.

Recall that the first two steps were deductive logical truth claims. The third step is an empirical truth statement, which is a correspondent truth claim. And, importantly, it is a correspondent truth claim about a foundational truth being “objectively beautiful.” As such, it forms an associative adjacent identity that links axiological and analytic truth claims. We noted that these were “exterior” truths in that they can be claimed across cultural contexts. All one needs is to understand logic, naming, operational definitions, and data collected on the topic to see these truths.

The fourth truth is different. It is a subjective phenomenological truth. I believe the Henriques Equivalency is true and beautiful. Framed this way, the focus shifts from a general epistemic position to a specific subjective epistemic position, held as such. This is what the qualifiers in the fourth truth mean. There are some reasons folks might argue that my logic and the way I put variables together in the Equivalency is not valid or useful. I am the first to acknowledge that it operates more as a “meta-mathematical/meta-physical/meta-subjective” claim. Thus, in some ways, I am generating my own system of justification. If you disagree with the value or utility of the Henriques Equivalency, then we can call that the “anti-Equivalency” position, and it is a valid specific subjective epistemic position as well.

We can then ask if the Equivalency combined with the Euler Identity to form the Radical Mathematical Humanistic Equation is pragmatically true. My argument is that, yes, it is pragmatically true in the sense that if you learn it, it will help you see how justification systems are connected to each other. More specifically, as depicted in the earlier graphic on the Equivalency, the Equivalency places the human knower in the universe and identifies that human knower as a scientific knower that generates mathematical conceptual operators that link behaviors in the world to observations in the world. And it draws an association between two important equations in physics, and shows a potential relation. To fully understand it, you need to understand the Planck-Einstein relation, given as E = hf and the Born, Heisenberg, Jordan algebraic matrix equation for quantum mechanics. Then you generate the special case where an observer observes the single oscillation of a single photon, and out pops the Equivalency. The Equivalency identifies the mathematical conceptual operators. Specifically, it seems to identify the mathematical conceptual operators that show up in the Euler Formula, of which the Euler Identity is a special case.

In essence, what I am saying is “For me, this is a useful set of associative adjacent identities.” And what emerges from this claim is an architecture that is connecting many different justification systems into a network. If we return to the gestalt, we can then make the following connections:

Now we have developed a set of associative adjacent identities that links mathematical logic to empirical science claims about beauty to subjective claims about beauty to pragmatic truth claims about how to think about the relationship between physical theories and physicists (i.e., human knowers who built theories linking observations to behaviors via mathematical operations).

The final step completes the circle. It generates the equivalency between the Henriques Equivalency and the iQuad Coin. So now we can say that 2πif = i⁴. The iQuad Coin is the frame for the Human Identity Function. It is the placeholder for all human knowers. To achieve this logically, we replace the Henriques Equivalency with iQuad, which of course, is mathematically equivalent to one. This allows us to take the subjective aspect of the Henriques Equivalency claim “out” of the equation, and return it back to the undeniably true form.

And yet, in completing the path, we have linked together these different kinds of justification systems and the possible positions that specific human knowers might adopt. That is, we can all agree on the first three truths as our foundational ground and then we can debate on the value of the fourth truth. We can be pro, con, neutral or ignorant on this last point. And regardless of one’s position, they can all be placed as positions on the iQuad Coin as a function of the Human Identity Function.

In so doing, a gestalt network of identities is created that has a new meta-logic for science, subjectivity, and wisdom.



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Gregg Henriques

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