The Henriques Equivalency, the Anti-Equivalency, and Their Potential Value in UTOK

Gregg Henriques
Unified Theory of Knowledge
12 min readApr 27, 2024

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The root of the iQuad Coin is found in something called the Henriques Equivalency, which is represented as 2πif = 1.

What exactly is the Equivalency? Well, the first thing that needs to be said about it is that it is not a “real” mathematical equation. I will return to this point when I explain the Anti-Equivalency.

However, just because it is not a real mathematical equation does not mean it is not valuable. After all, I developed it in 2001, and it ultimately leads to the iQuad Coin, which is one of the key philosophical pillars of UTOK.(Note, it is called the Henriques Equivalency because I, subjectively, find it valuable/beautiful…as you will see this is an important qualification).

This blog explains the Henriques Equivalency in simple terms and then explains why anyone with knowledge of math or physics will tell you it is not a real equation. However, this critique does not kill the value of the Equivalency. The blog concludes with the tantalizing suggesting that the relationship between the positive assertion of value and the negative assertion of value adds an important point of consideration that results in aligning the Equivalency with the formulation of human consciousness that I developed with John Vervaeke in Untangling the World Knot.

Deriving the Henriques Equivalency by Seeing a Simple Relation Between Two Fundamental Equations

I derived the Equivalency by using the lens of the Tree of Knowledge to see the place of scientific knowledge on the Culture-Person plane of justification, relative to physical processes located at the Energy/Information and Matter/Object planes.

This is important because it helps us to understand that, instead of being a real mathematical equation, the Equivalency is a formulation that represents the mathematical conceptual operators that frame the formal process of observing behavioral frequencies that emerged in quantum mechanics. Put differently, the Equivalency is more about physicists knowing about subatomic behaviors than it is about physical behaviors per se.

UTOK defines behavior as change in entity-field relations and characterizes science as a justification system was grounded in analyzing behavioral frequencies that can be measured. That is, scientific knowledge is socially constructed via an intersubjective, third person empirical lens to observe behavioral frequencies and develop accurate descriptions and explanations of such processes. This simple set of assumptions allowed me to see an interesting relationship between two important equations in physics.

The first equation is called the Planck-Einstein relation, and it is given as:

The E stands for energy, h is Plank’s constant, and f stands for the frequency of the electromagnetic wave. It means that the kinetic energy of a photon is a function of the frequency times Plank’s constant.

The second equation is the Born, Heisenberg, Jordan algebraic matrix equation for quantum mechanics. I came across it in the book, Q is for Quantum: An Encyclopedia of Particle Physics. It is given as follows:

In it, the I stands for the identity matrix, and p and q stand for momentum and position. And, crucially, they stand for matrices, not simple numbers. This is relevant because it makes what I do highly questionable from a formalized knowledge perspective.

But, formalized, conventional knowledge is not the only kind of knowledge there is. Indeed, it was my unique perspective of seeing physics through the lens of the Tree of Knowledge allowed me to see a connection between these two equations that manifests in the Equivalency.

When you know how to look through the lens of the ToK System, the connection is not hard to see, especially when we include the fact that I was thinking deeply about how physicists observe and map behavior. I had come to formally represent behavior as (X)(Xo)t1 — (X)(Xo)t2, where X is the entity, Xo is the field (not X), and t is time.

I had also realized that there was a deep connection between behavior and kinetic energy, such that behavior is, in essence, kinetic energy. (Note: Potential energy is potential behavior).

Because my perspective was embedded in the Tree of Knowledge System, I knew that we needed a theory of both physical behavior and a theory of physicists measuring behavior with instruments, observing behavior via conscious experience, and developing physical equations that accurately mapped behaviors in the world.

All of this comes together in the Equivalency as follows:

The Einstein-Plank relation describes kinetic energy.

The pq — qp frames behavioral information that can be measured in probabilistic space.

However, in a simple, idealized case of a single photon exhibiting a single oscillation, there is an identity between E and pq — qp (i.e., in the special case, E = pq — qp). In such an example, the Identity Matrix is 1. This means that there is also the following equivalency in the special case:

Even I can do the math from here. The two hs cancel and you get 2πif = 1. Here is a diagram putting the elements together:

What did this mean? I saw it as representing the mathematical conceptual operators that physicists had developed for observing behavioral frequencies. The 2πi portion connected to the measurement into observation process, whereby the f represented the frequency in the world.

Although I did not have the language then, I can now say that the 2πi represents the epistemic portion of knowing (i.e., measuring into observing behavioral frequencies), whereas the f represents the ontic (i.e., kinetic energy in the world that exists independent of observation). The reason it was important is that it was a special case where knower = known and thus left behind the core conceptual relations that afforded that equivalency.

I developed this set of insights in 2001. A year or so later, I would learn about the Euler Identity and Euler formula and was convinced there was a deep connection. That is, I think the Euler formula serves the function of the mathematical conceptual operations I uncovered in the Equivalency. The Euler Formula is as follows:

In it, I saw the set of mathematical conceptual operations that I had stumbled upon in the Equivalency. The following representation captures what I mean, as it frames the way I conceived of frequency (left side) and observation (right side) in the Equivalency:

I lacked the necessary skills to advance the formulation any further. But in the special case of the Equivalency, we could substitute it for the 1 in the Euler Identity and give UTOK’s Radical Mathematical Humanistic Equation:

Here is a depiction hangs in our UTOK Monastery that ties these elements together:

The Anti-Equivalency and Why It is a Valid Frame of Reference

In 2018, I hosted a conference called, Toward a Big Theory of Knowledge, and I played a bit of a joke on my friend, Dave Pruett. Dave is a professor of mathematics at JMU, and he and I shared many lunches discussing the nature of reality. He was a fan of my work in general, but he really did not like the Equivalency. The joke I played on him was that, in my talk, I was going through the speakers and I said that in his talk, Dave would be extolling the virtues of the Equivalency!

Of course, this was the farthest thing from the truth. Dave repeatedly warned me about making the Equivalency public. The reason is that, on the surface, it makes me look like a quack.

Let’s be very clear why this is the case. First, the f in the Equivalency refers to oscillations per second. Thus, there is a time unit on one side, but there is no time unit on the side of 1. This means there is a unit problem (i.e., a unit on one side of the equation but not on the other), which basically means that the technical mathematical operations were not done correctly.

Second, as was mentioned, the p and q in the Born, Heisenberg, Jordan equation are not numbers, but matrices, which means that you are mixing apples and oranges if you are combining them with a standard equation.

Finally, the Euler Identity is generally considered the most beautiful mathematical equation in the world. And Leonhard Euler is a great mathematician. I am not even a mathematician, yet alone a great one. Moreover, the relationship between the Equivalency and the Euler Identity/Formula is not clear, nor does it lead one to be able to do workable mathematical transformations.

The conclusion is clear. The Henriques Equivalency is bullshit. It is an example of quackery and arrogance of the worst kind. No serious person who knows anything about either math or physics would see this as valuable in their language systems. And the implication by yoking the Henriques Equivalency to the Euler Identity is that I (Henriques) has made some great insight, when clearly the reverse is true.

Ouch! How on earth can I see this as a viable justification AND believe in the value of the Equivalency?

Placing the Anti-Equivalency in Right Relation to the Equivalency

Mathematics and physics are language systems that play by very specific rules to arrive at refined knowledge claims.

On the surface, the Equivalency seems to mock those claims. It is crucial that refined knowledge systems are not infiltrated by bullshit, and so going negative on the Equivalency is totally justified from that vantage point.

Indeed, we can represent this as the (-)2πif position. Negative here represents the valence of the refined attitude of the knower who knows that the Equivalency is bullshit when it comes to traditionally formalized systems of mathematical and physical knowledge.

So where does this leave us? Well, we need to remember that I never publicly asserted the validity of the Equivalency until after I had developed the iQuad Path and the iQuad Coin. You see, prior to the Path and the Coin, I had no good way of justifying my position in a way that satisfied all perspectives.

However, once the iQuad Coin formulation emerged via the iQuad Path, a new way to frame the Equivalency versus Anti-Equivalency relation was possible. Both were justifications stemming from the human identity matrix, and that human identity matrix, represented as i^4 could be placed in the Euler Identity. Any mathematician will tell you this is a valid representation:

Of course, this does align with the iQuad Coin, as suggested by this diagram:

What is the lesson here? The first lesson is that we need to be very clear about the epistemic frames we are using to justify our claims.

The second is that, within traditional frames of reference of math and physics, the Anti-Equivalency is a totally valid position. Indeed, it is one that should be encouraged.

The third is that with the iQuad Coin framing the human identity matrix, the validity of being positive toward the Human Equivalency (i.e., (+)2πif) becomes a valid frame also. The reason is that the Equivalency is not a mathematical formula per se, but it is adjacent to math and physics…and the human psyche (see here for how the iQuad Coin aligns math, physics, philosophy and the psyche). And that makes it potentially a new kind of knowledge technology.

And the last part of this blog shares why this maybe be a very cool insight after all.

The Anti-Equivalency and the Three Kinds of Qualia

Why am I elaborating on this? After all, this is a technical set of claims for UTOK insiders, and I already delineated much of this in previous blogs on the root of the iQuad Coin and formally deriving the Equivalency. The reason is that I am planting seeds for framing human consciousness in a new light. Here I will lay out just the outline.

In our Cognitive Science Show Series, Untangling the World Knot of Consciousness, John Vervaeke and I outlined three different kinds of qualia. Qualia, of course, refer to the subjective conscious experiences that constitute the domain of Mind2. [In addition, it is worth keeping in mind that modern empirical natural science works in part by factoring out qualia.]

The first was adjectival qualia, and it is the most commonly referenced. It refers to the properties of perceptual entities, such as the greenness of a leaf. It is what most folks think of when they consider qualia.

The second, which stemmed from John’s work, was adverbial qualia. It refers to the indexing process that frames conscious attention around relevance and provides the hereness-nowness-togetherness that constitutes the core unity of conscious experience. John’s work demonstrates that it can be conceptually and phenomenologically separated from adjectival qualia.

Finally, during our conversation and alignment of John’s recursive relevance realization formulation with UTOK’s Behavioral Investment Theory, there was valence qualia. This aspect relates to how the agent values the experience, with the broadest, most basic being a positive/approach versus negative/avoid.

This brings us to a fascinating alignment with the analysis above. Consider the following representation of the Pro/Anti Equivalency:

Now align it with the three kinds of qualia and something very cool emerges. The pro/anti aligns with valence qualia, the 2πi aligns with adverbial qualia that frame the observation via relevance, and f aligns with adjectival qualia via the frequency of resonance that is realized.

Using the iQuad Coin as the frame for the human consciousness epistemic portal, we can depict the arrangement as follows:

In plain language, this suggests that there is a recursive relevance that frames potential realizations that are then evaluated positively or negatively depending on the investment orientation of the agent.

I will end it here for now, with this as a teaser. For those interested in where UTOK can take us, consider what has happened here. In in 2001, I used the Tree of Knowledge to see a special case where the knower/measurer/observer = the known/measured/observed. This relationship factored out Plank’s constant, and left us with mathematical conceptual operators that related the measurement frame to the frequency of behavior in time.

This Equivalency would ultimately function to root the iQuad Coin, which frames the unique, particular human identity in the world and bridges that frame to physics, philosophy, and math. The Coin emerges from the iQuad Path that connects nominal truths to deductive truths to scientific empirical truth claims about “objective beauty” to first person empirical claims about “subjective beauty” that frames the Henriques Equivalency as subjectively beautiful in a way that also legitimizes the claim that it is “objectively ugly” from the vantage point of the Anti-Equivalency.

By adding the pro/anti angle on the Equivalency we can now also see the alignment of three valence qualia that give us an understanding of how humans observe things. This means we can now coherently interrelate the real things in the world, with our constructed social ideas, with our mathematical ideal formulas.

Contrast that with the Enlightenment Gap and the current chaotic fragmented pluralism we are swimming in and you can see why I am excited about where UTOK might take us.

PS From the Blind Spot to the Seen Spot

The recently published work, The Blind Spot: Why Science Cannot Ignore Human Experience, makes an excellent case that the human knower is essentially factored out in science, but this is a profound error, as it also needs to be framed and factored in. The book explains why this leaves a gaping hole in our scientific and humanistic understanding, but it does not offer a clear frame for how this hole should be filled. Of course, filling this hole and replacing it with the “seen spot” is at the core of UTOK’s project.

PPS Extending the Anti-Equivalency to Anti-UTOK Claims

I hope it did not escape folks that the above analysis thus includes the justification systems of folks who are disinclined toward UTOK. The Anti-UTOK position is all part of the justification systems that make the world go round. We need to keep this angle and capacity to include various perspectives when considering the value and scope of UTOK.

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Gregg Henriques
Unified Theory of Knowledge

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