Formally Deriving the Henriques Equivalency

The previous blog told the story how I arrived at the Henriques Equivalency. Given as “2pi i f = 1”, the Equivalency looks like a mathematical equation, but that is a bit misleading, as it does not work like that. Rather it is better framed as a metaphysical gateway into a multiplicity of different domains of thought. That gateway culminated in the construction of the iQuad Coin, such that the Equivalency functions as the root of the Coin. This blog gives the details regarding the formal steps I took to derive the Equivalency. Ultimately, the key idea that allows for the Equivalency to be derived is the notion that behavior is the fundamental concept that ties together the modern empirical natural science enterprise. That is, behavior, and its observation, ties together science metaphysically, epistemologically, and ontologically. It was by seeing how scientific knowledge is framed by the concept of behavior that I was able to generate a new linkage between “ontic kinetic energy” and “epistemic measured behavioral information.” This was the linkage that allowed me to equate the observed and observer, and then “factor them out” and leave behind the mathematical conceptual operators represented by the Equivalency.

When I showed my friend and mathematics professor Dave Pruett the geometric depiction of the Henriques Equivalency, with its circle and wave frequency shapes, he immediately proclaimed that struck him as capturing “wave-particle duality.” I was pleased that he made that association, as I too had seen that connection. Although this was a happy correspondence, I do not want to suggest here that Professor Pruett was a fan of the Equivalency. He found the moves I made justifying it to be confusing, and not to his liking as a professional mathematician. As such, I think it is safe to say he was in the “Anti-Equivalency” camp, which I defined in the previous blog as folks who do not see it as true, good, or beautiful. Indeed, in my opening remarks of the first Theory Of Knowledge Society Conference held at JMU in 2018, I had some fun with this fact, and teased Dave a bit by announcing to the entire conference that, for the next presentation, Professor Pruett was going to come up an extoll the mathematical virtues of the Equivalency…Not!

The reason I could be flippant about this issue was because I had already minted the iQuad Coin earlier that year. This meant I had developed a more complete system of justification around the nature of the Henriques Equivalency and its relationship to the Anti-Equivalency position. That is, via the Coin, I could now fully hold and justify both my affiliation for it and simultaneously empathize with those who found it inelegant or misleading. The reason is that the Coin completes the picture because it allows for a fully adequate frame for both human idiographic subjectivity and full mathematical coherence. The iQuad blog series gives the logic for this argument and the basic claim is visually made below, whereby the Equivalency evolves into the iQuad Coin.

Whereas the previous blog used an imaginal conversation with my mother to walk folks through the key features of the Equivalency, the goal of this blog is to more clearly show how I derived it. We can begin with a set of eight formal statements that delineate the logic of the Equivalency.

Observing Behavioral Frequencies as a Fundamental Frame for Natural Science

The central insight that allowed me to realize the Henriques Equivalency is located in my realization that behavior is the central concept in natural science. That is, behavior is a unifying concept that ties together the conceptual architecture of the modern empirical natural sciences. As suggested by the conclusion given in the slide above, the concept of behavior grounds the natural science in three ways: (1) metaphysically (i.e., its concepts and categories consist of objects/entities, fields, and change); (2) epistemologically (i.e., natural science analyzes behavior from a third person empirical perspective); and (3) ontologically (i.e., natural science has mapped the stratified layers of nature into behavioral patterns that operate at various levels and dimensions of complexification). It was finding a concept that tied science together in this way that set the stage for deriving the Equivalency. The Equivalency then results in bridging to a fourth key domain in natural science, that of mathematical operations.

As described in the previous blog, most “theories of everything” in physics are woefully inadequate when it comes to the metaphysics of knowledge and the knower. UTOK is a game changer in this regard. To give just one example, consider how this depiction of the ToK shows how it maps both the ontic reality (i.e., Matter, Life, Mind and Culture) and scientific onto-epistemology (i.e., the physical, biological, psychological, and social sciences) places them in proper relation.

The clarity afforded by the ToK System’s theory of science and reality is enhanced when we elaborate on how and why behavior is the central concept in natural science. This blog summarizes the argument and this academic paper offers it more completely, so will not repeat the whole argument here. Rather I will simply summarize how we can use behavior to frame science metaphysically, epistemologically, and ontologically.

The Descriptive Metaphysics of Behavior: Defining Behavior as Change in Entity-Field Relations

To understand how the UTOK frames behavior as the central metaphysical concept in science, we can return to a quotation from the first paper I produced on the Tree of Knowledge System back in 2003.

The most general definition of behavior is change in an object–field relationship, which can be algorithmically represented as (X)(Xo)t1 — (X)(Xo)t2, where X is the object, Xo is the field (not X) and t is time. This is important because it highlights that all sciences are sciences of behavior. Physics is the science of the behavior of objects in general. Particle physicists study the behavior of very small objects (e.g., fermions) using quantum theory, and cosmologists study the behavior of very large objects (e.g., galaxies) using the theory of relativity (Greene, 1999). If it is agreed that physicists study the behavior of objects in general, then it logically follows that other scientists study the behavior of certain objects in particular. Chemists study the behavior of molecular objects; biologists study the behavior of living objects. This analysis highlights that there are obviously significant problems with defining psychology as “the science of behavior.” It is not the fact that animals behave that makes them unique; it is that they behave so differently from other objects. The key then becomes defining the subset of behaviors that psychologists’ study.

This quotation has two key elements in it. First, there is the important conclusion that psychology cannot be the science of “behavior in general,” but instead must be focused on specific kinds of behaviors in nature. This is key both because it shows how behavioral psychology is misguided, and it shows why behavior has been overlooked in science more generally. The fact that the concept was birthed in psychology is one of the reasons it has not been attended to in much depth by folks in physics and biology, and only somewhat by folks in philosophy of science (e.g., Gilbert Ryle’s logical behaviorism).

The second point is more directly relevant to Equivalency. It describes the key metaphysical concepts and categories that go into behavior. As such, this describes the first point in the logic of the Equivalency, which pertains to the descriptive metaphysics of behavior. Note, there is a slight word difference, with the quotation framing behavior as “object-field change” and the formal statement using the word “entity.” Because the “X” in the behavior algorithm could be something like a wave or pattern, over the years I decided that “entity” was a better description, although thinking of “X” as the “object of analysis” still works. The key is to understand is that there is a “figure-ground” relationship between the object/entity and the field, and that science is about framing entities as objects of analysis in fields and analyzing patterns of change.

The point here is that ‘entity/field change’ provides the descriptive metaphysics for the concept of behavior. Back in 2003 I did not frame this in terms of metaphysics, but rather I described it using the somewhat unclear term “bottom up” to describe my approach. I did not have a good understanding of metaphysics at that time. As a psychological scientist, my training had largely equated philosophy with epistemology. I thought “metaphysics” referred to things that could not be empirically verified. [For a good example of this kind of attitude about metaphysics among science oriented thinkers, consider that the scientific realist Jim Rutt often says that when he hears the word “metaphysics” he reaches for his pistol]. I would later realize this was not a proper understanding of metaphysics. As this chapter on a “Metaphysical Empirical” approach to psychology makes clear, metaphysics is much better framed as the pre-empirical concepts and categories that one uses to carve up reality. Framed this way, I now see metaphysics as absolutely central to the scientific enterprise.

The key point here is that there is a basic metaphysical architecture that grounds science and frames the world in terms of entities and fields and change. Our confidence that this is a reasonable claim increases when we look at the basic definitions of science. Consider that the Lexico-Oxford dictionary defines science as “the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment.” If we agree that behavior can be defined as changes in entities and their relations across time, then, by implication, the concept of behavior can be considered to include the structure of those entities. However, the reverse is not true, as structure is static and does not include the dynamic aspects of behavior. Via the framework provided by the ToK System, we can also fold physical into the broader concept of natural, with the same logic. Natural includes physical, but physical does not include all that is natural (e.g., Life, Mind and Culture). This gives us a definition of natural science as the systematic observation, description, and explanation of behavioral patterns in the natural world.

Behavioral Patterns Frame the Ontology of Natural Science

In the context of scientific philosophy, we can say that ontology refers to the claims science makes about the nature of reality. This podcast between Sean Carroll and Daniel Dennett provides a good discussion of the essential insight that frames the second point in the Equivalency derivation. This is the fact that behavioral patterns in nature represent the key ontological concept in science. By this I mean the theories and descriptions of the things in nature are described by science in terms of entities changing in fields. The legitimacy of this claim is made clear when we think about the ToK System depicting the universe as an unfolding wave of behavioral complexification. It becomes even clearer when we use the Periodic Table of Behavior to organize the sciences into 12 different floors, framed by the four dimensions of complexity (Matter, Life, Mind and Culture), each of which are divided into three primary levels (part, whole, group across scales).

The Equivalency is about getting at the fundamentals, and the UTOK posits “Energy” as the most fundamental ontological concept, which is made clear in the above depiction of the Periodic Table of Behavior. (See here for a more detailed analysis of this argument, which describes the fundamental state at the beginning of the observable universe an “Energy Information Singularity”). This brings us to point 2 in the logic of the Equivalency, which states that the ontological concept of kinetic energy is the most ontic entity that there is. Put in plain language, we can say that, if anything is fundamentally real in the world, it is kinetic energy. Point 3 moves from that claim to the more specific Planck-Einstein relation, which is the finding from physics that the kinetic energy of a photon is equal to its frequency of oscillation times Planck’s constant. (Planck’s constant is the “quantum of action,” which quantizes the amount of kinetic energy change).

In sum, we can state that behavioral patterns in nature at various levels and dimensions represent the core ontology of modern empirical natural science. The most ontological-into-ontic concept is kinetic energy, and the kinetic energy of a photon is a function of its frequency times the quantum of action. From this “realist” assertion that frames how behavior maps onto natural scientific ontology, we can then move to the epistemological aspect of how behavior frames science and the role of empirical observation.

Observing Behavior Frames the Epistemology of Science

The epistemology of science has to do with observation, measurement, quantification, and experimentation to develop valid models of complexity and change in nature. As a psychological scientist, I learned all about the epistemology of science by understanding things like measurement theory, reliability, validity, analyses of variance, and randomized controlled clinical trials. These are all tools for observing how the world behaves and then making justified conclusions about it. As this blog on the empiricism makes clear, the epistemology of science is based on a third person, intersubjective/objective empiricism. This refers to the fact that the language game of science is framed by behaviors that can be observed by any general observer who can interpret data obtained via measurement. This is in contrast to first person empiricism, which refers to the subjective experience of being.

Consider, for example, that when I see the Equivalency, I experience it as beautiful. That is a subjective empirical observation and is not what is meant by scientific empiricism. Rather, scientific empiricism is grounded in data that can be replicated and expertise affords the capacity to objectively interpret what the data mean (e.g., experts in particle physics are trained to interpret what the patterned swirls mean when they look at computer images of particles smashing into each other. Here is a visual depiction of the key elements of behavioral observation.

This depiction is structured to help us bridge to points four and five in the logical derivation of the Equivalency. That is, it shows how the general process of observation is connected to the Born, Heisenberg, Jordan matrix mechanics equation. The equation was placed on Born’s tombstone.

We can then apply the “special case” given in point 6 and then show how that results in points 7 and 8. This was the graphic shared in the previous blog, which depicts the logic of these last three steps. The blue-red wave represents a single oscillation of an electromagnetic wave (i.e., a photon). The pq and qp represent the matrices of position and momentum. This then is assessed by the Identity Matrix (the I in the Born equation) times Planck’s constant divided by two pi times i.

The Equivalency is the special case where the ontic kinetic energy is essentially identical to the epistemically measured behavioral information. I interpreted the Equivalency as being the point at which the empirical observer equals the empirically observed. And I interpret the Equivalency as representing the “mathematical conceptual operations” that afforded the logical bridging function between the two.


The Henriques Equivalency emerged via the insight that behavior was the central concept that underlies and organizes natural science. It frames the metaphysics, ontology, and epistemology of science, such that observing behavioral frequencies to describe and explain patterns and structures in the natural world is a great definition of modern empirical natural science.

The Equivalency was meaningful to me because I arrived at it via a new perspective on this set of relations afforded by the ToK System. It was by understanding science via metaphysical, ontological, and metatheoretical perspective given by the ToK System that enabled me to frame how human knowers in general and, in this case physicists in particular, generated knowledge claims. My conclusion was that the Equivalency pointed to the mathematical conceptual operators that would round out the network of understanding. That is, I could then add mathematics to this picture.

When I initially generated the Equivalency, I did not really know how to interpret it. It was this weird “quasi-mathematical equation,” but was not really a math equation. I thought the derivation was sound, but was not certain what it meant. I used to call the derivation a “pseudo-proof” in an effort to communicate that I was looking at this via a different angle. Of course, that claim would not instill much confidence in folks like Dave Pruett who were trained in the standard way of thinking about mathematics. It is hard to argue with this, and thus I knew the anti-equivalency stance was clearly justified within this context. Indeed, that was why I did not advertise the Equivalency when I first found it. I just did not have enough knowledge to know how to work my way through the labyrinth of justification systems with coherence and clarity.

Nevertheless, I knew that the Equivalency was an important idea because it intersected with so many different ideas. Indeed, that is why I ultimately came to call it a “metaphysical gateway.” This is depicted below:

The next step in the journey came in 2003 when I learned about the Euler Identity and Formula via the book, Where Mathematics Comes From. That built the bridge from my “mathematical conceptual operators” to their “mathematical idea analysis” of the Euler Identity. In the Euler Identity/Formula, I saw the mathematical conceptual operators that I had pulled out via the Equivalency. This relation is show on the right side of the slide above. The next blog tells of that connection and how I ended up merging the Equivalency with the Euler Identity to generate the Radical Mathematical Humanistic Equation, which on sits on top of the seed in the UTOK Garden.



The Unified Theory of Knowledge (UTOK) a new consilient vision of natural science, psychology, psychotherapy, and philosophy that is oriented toward the cultivation of wisdom.

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Gregg Henriques

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