The Root of the iQuad Coin
The Story and Logic of the Henriques Equivalency
The previous four blogs have delineated the conceptual architecture of the iQuad Coin, and explained how the iQuad Symbol works to frame the unique human subject via the Human Identity function and then links that idea to the complex unit circle. To fully understand the iQuad Coin, we need to go back to its roots, which are found in something called the “Henriques Equivalency.” First developed back in 2001, on its face the Henriques Equivalency looks like a mathematical equation, and is given as:
Although it looks like a basic mathematical equation, that could be a bit misleading and it should not be interpreted that way. I have put it in quotation marks here because it is better interpreted a kind of metaphysical gateway that connects many different domains of thought, including mathematics, modern physics, observer-observed behavioral relations, and subjective beauty.
As will be made clear in what follows, I called it the “Henriques Equivalency” not because I considered it to be objectively brilliant and true, but rather the reverse. That is, although I personally saw it as beautiful and true, I also saw that one can argue from a conventional mathematical or physics perspective that the Henriques Equivalency is misguided, misleading, and ugly. Indeed, I call this perspective the “Anti-Equivalency,” and it definitely is a valid opinion to hold. However, when I initially stumbled on it, I was convinced that the Equivalency held some deep meaning and utility, despite the fact that it did not immediately sync up with conventional knowledge systems in math and physics. I am glad I maintained that view.
Ultimately, I found my way through a labyrinth of justification systems from the Henriques Equivalency into the Euler Identity/Formula and, finally, the iQuad path, iQuad Symbol and the architecture of the iQuad Coin, which now serves as a central structure in the Unified Theory Of Knowledge (UTOK). This blog tells the story of where the Henriques Equivalency comes from. It gives an overview of why, according to UTOK, we need a metaphysics of the knower. From there, it transitions to an imaginal conversation with my mother to explain it the basic logic of the Equivalency and how it was derived and why it is significant. The next blog will summarize the derivation more formally and then show the relationship with the Euler Identity/Formula and how that gave rise to the Radical Mathematical Humanistic Equation, which set the stage for the iQuad path.
According to UTOK, physics is missing a theory of physicists. That is, physics does not have a clear framework for understanding human knowers and how humans generate epistemological and ontological claims about the physical world (or anything else, for that matter). There are, of course, some philosophically inclined physicists who have grappled with this problem. For example, John Wheeler was a famous physicist who developed a model of the “participatory universe.” He argued that the fundamental root concept in the universe was information, and that we needed to consider how human knowers perceive the known to see knowledge as a whole. Wheeler’s view shares some significant affinities with UTOK’s metaphysics. Below is a depiction of Wheeler’s conception of the participatory universe, which includes the eye looking back on the world. I have placed the ToK System, which maps both science and reality, and the iQuad Coin, which maps the specific conscious knower in relationship to Wheeler’s core depiction:
David Deutsch is another physicist who has grappled with these issues. His work, The Beginning of Infinity, emphasizes that “good explanations” are central to knowledge, and argued strongly that we need a framework for understanding the universe that includes explanations themselves. He criticized most physical theories of everything as being overly reductive and lacking in this way.
Those familiar with UTOK will know that one of its central considerations is the metaphysics of the human knower. In a way that is very much aligned with how Deutsch talks about the evolution of good explanations, the Tree of Knowledge System depicts science as emerging out of the Culture-Person plane of existence to develop accurate justification systems that map complexity and change. Like both Wheeler and Deutsch, UTOK argues that we must include both scientific knowers and systems of justification to have a complete picture of the world and our knowledge of it. This diagram shows how the ToK places the human knower and scientific knowledge in relationship to measurement.
As we have been describing in this blog series, we can add the iQuad Coin to this picture. As was described in Part I on the iQuad Symbol, the two central features of the Coin are that it connects (or entangles) the specific human subjective knower with mathematics, especially the complex unit circle and its representation of the real and imaginary axes to generate the complex number plane. The combination of the ToK System and the Coin bridges generalizable scientific knowledge with the subjective epistemological perspective of a particular knower. Given this, it perhaps will not come as a surprise to learn that that the Henriques Equivalency is an idea that bridges “real behavioral frequencies” in the world with “measured behavioral information” obtained by an observer to model the real.
So, what, exactly, is the Henriques Equivalency? It is a “quasi-mathematical equation” that is given as follows:
I put quotations around the Equivalency because it is not to be interpreted as a “standard” or “conventional” mathematical representation. Rather, it is better interpreted as giving a metaphysical statement. Specifically, it is a statement that represents the logical relationship between observers and observed that is obtained by looking at both knowledge of the physical world and how physicists justify that knowledge. As the circle and frequency wave suggest, we can think of the Henriques Equivalency as representing a kind of “geometric alignment” of scientifically observing behavioral frequencies, such that the observer is aligned or rendered equivalent with with the observed. The connection is perhaps clearer when we show that the Equivalency is the root of the Coin. In fact, we have basically been working backwards in this blog series relative to what I have been working on chronologically. The Equivalency comes along long before the Coin. The goal of this blog is to get to the root of the Coin and show where it comes from via the Henriques Equivalency.
Factoring the Human Knower In and Then Out of the Scientific Knowledge Equation
Before getting into the specifics of the Henriques Equivalency, let’s take a moment and elaborate more on what it means to say that the ToK System and larger UTOK includes the specific human knower, whereas most theories of everything in physics do not. One way to see this is to have a look at this picture of the universe from Stephen Hawking’s The Universe in a Nutshell.
On the left is the Big Bang singularity, and on the right is an observer looking back through spacetime via what is called a “light cone.” As suggested by this diagram, for Hawking, and many other physicists, the knower in physics is simply a measurement device that is “observing” empirical data.
From a UTOK perspective, however, this is conceptual framework is not enough. Rather, for a full account of the scientific knower, one must turn data into processed information via human observers and then that is placed in human knowledge systems about what that data means. More specifically, we need to be able to frame how data become etched into human scientific justifications (i.e., Deutsch’s explanations) for understanding the natural behavioral patterns in the universe. This requirement relates directly to my comment at the beginning of this essay that we need a theory of physicists (i.e., humans scientifically knowing about the physical world), as well as of physical processes (i.e., patterns of energy and matter behaving across scales). Put in philosophical terms, we need a descriptive metaphysical system that properly relates the epistemic knowing process of science with the ontic world.
The structure of the ToK System is such that it works to first factor in human knowing, and then factor it out. This is in direct contrast to Hawking and many others who interpret empirical data as simply being observed or measured, but fail to connect that to human knowers and the nature of scientific knowledge in cosmic complexified space. As described in the Tree of Knowledge Manifesto, in January and February of 2001, I developed seven different representations of the ToK System in an attempt to show the various ways it captured the knower-known relations. Figure 6 was titled: Science Generates Objective Knowledge by Factoring Out the Knower.
Interestingly, it closely resembles the Hawking diagram above. Indeed, my parents gave me that book for Christmas in 2001, and as I flipped through the pages, I landed on that diagram. When I did I jumped up off the couch and shouted, “That is my diagram! Only it is an inverted mirror image of it because Hawking is missing the human knower!” Here is the correspondence, and what I mean:
As suggested by this representation, rather than just reducing the human knower and human knowledge to a measurement device, the ToK System frames both the specific knower and scientific knowledge in context. The “knower” in the diagram refers to the specific individual human. The “anti-knower” refers to the epistemology of modern natural empirical science. That is, modern empirical science is structured to factor out the unique individual perspective and leave behind a generalizable nomothetic architecture of the essences in nature.
Importantly, as this series of blogs on the iQuad Coin makes clear, I did not have the iQuad Coin framed back in 2001. Indeed, the Coin is not minted until 2018. What I had was the Henriques Equivalency, but I did not really know how to frame what meant or how to include it in the overall system back then. However, with the addition of the Coin placed in relationship to the Tree, we can see that the Tree represents the general scientific knowledge achieved by factoring out the specific, unique, subjective human knower. This diagram shows the relationship between the Coin and the Tree, as it maps scientific knowledge.
The caption that I placed on this diagram in the 2001 ToK System manifesto reads: After snowballing through time, one small branch of complexity peers back on its first origins. By factoring itself out of the equation. This quote captures how my first focus with the ToK System was to develop a system of knowledge that was grounded in the natural science view that factored out the unique, particular subject. Now, with the iQuad Coin, we can return to this formulation and properly place the unique subject back into relation to the generalized objective knowledge produced by science.
The Henriques Equivalency
With this background, we can now turn to the Henriques Equivalency and share where it comes from. As suggested by the above overview, the Equivalency emerges from the way the ToK System frames the relationship between the ontic reality and the processes by which physicists, as human knowers, develop mathematical-empirical models of that ontic reality.
One way into the Henriques Equivalency is to explain why it is so named. One reason has already been mentioned. I gave it my own name because there is a “subjective” element to it. That subjective element was initially a major hurdle. After all, the last thing one thinks of when one looks at a mathematical equation relating to physics is that it is subjective. We will come back to this when we dive into the “Radical Mathematical Humanistic Equation.” However, knowing that the Equivalency connects to the iQuad Coin, and it links human subjectivity and mathematics, hopefully there are already some notions about how such a bridge might be constructed.
Why is it called the “Equivalency”? The Equivalency is so named because it represents the place where there is the greatest possible equivalency between the ontic-kinetic energy in the world and measured behavioral information obtained by an observer. We can think about it as the place where the ontic kinetic energy becomes the information processed by measurement device for our scientific knowledge. If that is what is meant by the term, what then is meant by the equation? The 2pi i f = 1 represents the mathematical/logical relations that are “leftover.” That is, by making equal the empirical knower with the empirically known, the Equivalency leaves behind mathematical conceptual operators.
This likely sounds pretty abstract, especially if you have no background in physics or math or the ToK System. To really understand the Equivalency, a reader needs to have a basic understanding of modern physics. In addition, one needs to have some elementary knowledge of: (a) the quantum of action, or “Planck’s constant”; (b) the Planck-Einstein relation equating the energy of a photon to its frequency times Planck’s constant (E = hf); and, finally, (c) the “matrix mechanics equation” developed by Max Born, Werner Heisenberg, and Jordan in 1926, which Born had etched into his tombstone, which is given as:
I understand that many folks will not have knowledge of these concepts, and thus the ultimate meaning Equivalency will be hard to follow. In addition, those who do have knowledge of these concepts in math and physics will initially find my logic problematic, given conventional frames. For all these reasons, the Henriques Equivalency will generally be seen either as a conglomeration of confusing ideas for those who are not informed or as blatantly erroneous for those who do have knowledge of modern physics. As such, we must acknowledge that, at first glance, the Henriques Equivalency has all the hallmarks of language game quackery. That is why (a) it is called the Henriques Equivalency and (b) I am explicitly acknowledging the “Anti-Equivalency” justifications upfront and (c) I sat on it for over 20 years prior to being clear enough about the maze of logic that surrounds it to share it publicly. As suggested by this blog series, the reason I am now finally in a place to publicly share the Henriques Equivalency is because it grew into the iQuad Coin. The iQuad Coin solves the “Anti-Equivalency” problem, and allows me to place the Anti-Equivalency justifications in proper relation to the claims made by the Henriques Equivalency.
The evolution through the maze of ideas has happened in several steps. First, there was the Equivalency itself, which I arrived at in 2001. That is the primary point of this blog, and it emerged from how I was using the Tree of Knowledge System as a lens to track how physicists justified knowledge in quantum mechanics and general relativity. Second, sometime between 2003 and 2004, I learned of the Euler Identity and Formula via the excellent book Where Mathematics Comes From. Indeed, that book gives a “mathematical idea analysis” of the Euler Identity and Formula that allowed me to see that it represented a key piece of the puzzle. Prior to that book, I had been completely unaware of the Euler Identity. This fact serves as a testament to my mathematical ignorance. One way to interpret what I saw in the Equivalency is that it pointed to the existence of the Euler formula in physics. The third step was formalizing union between the Euler Identity and the Henriques Equivalency in what I called the Radical Mathematical Humanistic equation. The next blog in this series will explain these steps.
The fourth step was the development of “the iQuad path to the Garden,” which was finalized on October 27, 2017. And finally, in 2018, I constructed the iQuad Coin via the iQuad Symbol, which connected the complex unit circle to the Human Identity function. I fully acknowledge that this a lot to both follow and swallow. For those who are skeptical but intrigued, let me offer this diagram as a teaser for where the metaphysical operations of this thing ultimately is going.
This depicts the basic logical architecture at work. In step one, we have the most elementary kind of logic, whereby negative one plus one equals zero. In step two, we have an “equivalent complexified representation” of that logical relation by replacing the negative one with an “Euler Coin” and replacing the positive one with the iQuad Coin. In step 3, we have a picture of Leonhard Euler on the iQuad Coin. He is the person who developed the Euler Identity, which is, objectively, the most beautiful equation in mathematics. Of course, then there is me on another iQuad Coin. In this context, it represents the bridge between my subjective belief in the beauty of the Henriques Equivalency, and my particular conscious self, captured by the Coin. Together these two Coins can be interpreted as the “Radical Mathematical Humanistic Equation,” which I will explain in the next blog in this series. However, prior to making that move, we need to obtain more clarity on the logic underlying the Henriques Equivalency.
Explaining It To My Mother
We are now in a place to deconstruct the logic of the Henriques Equivalency. I do so here in a dialogue format. A few years ago, I tried to explain the basic structure of the Equivalency to my mother. Unfortunately, I did not succeed. As is probably clear by now, the moves I am making are both unusual and complicated. And, to follow them without getting lost, one requires some basic background knowledge in mathematics and physical concepts. My mother (AKA Mimi to her grandchildren) is smart and has her doctorate in early childhood education/development. However, she does not have a strong background in either math or physics. Consequently, much of what I was trying to explain was Greek to her, and so when I tried to explain it, things went sideways quickly.
Later I went back and developed an “imaginal conversation” of what it might have looked like if I had been more effective in my framing. What follows walks her through the key steps that gave rise to the Equivalency. After she read this, she reply that she definitely had a better understanding of what I was trying to do, and why it might be important.
Me: Hi Mom, I have this cool thing I want to tell you about.
Mom/Mimi: That sounds good, you know I like hearing about your new ideas.
Me: Well, this one is pretty abstract. But let me try to explain it in a way that makes sense.
Mimi: I will give it a try.
Me: You saw the show Genius, right?
Mimi: You mean the one about Einstein? Yes. I saw that. It was interesting to see his story.
Me: And you know that Einstein played a key role in “overturning” Newtonian physics, right?
Mimi: Yes, but I am not sure I really understand that.
Me: The details require some study. But at the most basic level, Newton’s ideas were split into two.
Mimi: What do you mean?
Me: I mean that there were two revolutions in physics that happened around the turn of the 20th Century that changed Newton’s model of how the universe was structured and how matter behaved. Einstein was involved in both.
Mimi: Ok.
Me: One of those sets of ideas was called “relativity.” First, Einstein developed the idea of special relativity. That was in 1905. That was also the year he generated his most famous equation, E = mc2.
Mimi: What did that have to do with Newton?
Me: Well, for one thing it challenged the idea of an “absolute reference” for space and time. Einstein realized that if the speed of light had an absolute limit, that had lots of implications for space and time, and in particular how space and time were experienced or observed by different observers. In contrast, Newton thought that space and time were absolute.
Mimi: Can you give me an example?
Me: One helpful example is the twin paradox. Einstein realized that as things move relative to the speed of light, time slows down for the one moving in the relative frame of reference. So, imagine two twins. Both are aged 25. One twin gets into a rocket that goes almost the speed of light and he travels around the earth in this rocket moving almost at the speed of light for ten years. Then he lands. If the traveling brother was going close to the speed of light, he might only have aged 1 year, whereas his brother who stayed behind would have aged 10 years. Put in concrete terms, the travelers watch would have only registered 365 days, while his brother’s watch went through 3650 days. So you could then have identical twins who were 9 years apart in age.
Mimi: That is weird. You mean time slows down as your speed approaches light?
Me: That was one of the implications of Einstein’s special relativity. Einstein advanced this work further in 1915 with his formulation of “general relativity”. The formulation of general relativity incorporates gravity and explains effects that gravity and mass have on space and time. Namely, that mass “curves” space and time.
Mimi: I don’t think I understand that.
Me: It is another unusual aspect of modern physics that is outside our normal ways of thinking. The only point for us to understand here is that Einstein’s ideas about the positions of observers and space and time upended Newton. Einstein showed that calculating such things were dependent on the observer’s frame of reference. It is also helpful to note that Einstein’s general relativity is about very large things, such as the shape of galaxies. For example, it was general relativity that led to the idea of the Big Bang.
Mimi: That is the notion that the universe began as a point and then exploded into being?
Me: Basically, yes.
Mimi: Ok, so you said there were two ideas that emerged. What was the other one?
Me: That was “quantum mechanics.” It is about the behavior of the very, very small. As in atoms and even smaller, such as particles like electrons.
Mimi: Ok. So what happened there?
Me: Well, actually it started with Max Planck. He realized that energy had to come in packets or digits or quanta. The unit he generated was Planck’s constant, which is h. We can think of this as being something like the smallest possible unit of behavior, called the quantum of action. It is like the universe is made up of these indivisible digits of behavioral information.
Mimi: I am not sure I understand that.
Me: This is another issue that is complicated. A consequence of the universe coming in quantized digits of behavior means that at the very small level you cannot measure or observe things without disturbing them. One way to think about it is, at the very small level, things exist as a sea or wave of probability, and then you measure it and, boom, somehow the “wave collapses,” and it becomes a specific, fixed result. But before you measured it, it was just a bunch of possibilities.
Mimi: Why is that important?
Me: One reason is that it means the act of measuring or observing changes what one sees. For example, you can think of before one measures something it is a wave of probability and then it becomes a fixed point of information. It also means that at the quantum/sub atomic level there is no independent act of observing. One of the fundamental assumptions of Newton was that an independent observer could watch what was happening without altering it. There are other things that are weird to. Like the idea that nature is “quantized” or that there is an inherent randomness to how things behave at the fundamental level.
Mimi: Ok. That sort of makes sense, but I am not sure I fully understand.
Me: Well, it takes a long time to “fully understand,” and there are many things about this that I do not fully understand. But we can consider these as building blocks. Let me draw this out for you. Here is a picture that maps out some of the things I am talking about. The center cone represents the classical Newtonian world of matter in the form of things like rocks and planets. The bottom of the grey cone is quantum mechanics. The very large is general relativity:
Mimi: Ok. I can see that.
Me: So, Newton’s frame was split at the extremes, by quantum mechanics and general relativity. And the two are not commensurate, meaning they do not logically or mathematically “play well together.”
Mimi: What do you mean by that?
Me: I mean that the assumptions and equations that describe quantum mechanics do not mesh at a deep level with general relativity. Einstein spent the last half of his life, right up until his dying days, to try to build a “grand unified field theory” that merged quantum mechanics with general relativity. He never succeeded.
Mimi: Is that what you are working on?
Me: Not really. Or, rather, I am working on a part of the issue from a different perspective. That is, I am coming at these things in a new way that might be a part of the solution.
Mimi: So what are you doing?
Me: So, one of the things that both general relativity and quantum mechanics point to is the idea you need to include the “observer” at some level to have a full understanding. Newton assumed that there could be a completely independent observer that could go anywhere and observe anything without changing it. This makes sense at one level. For example, it seems silly to say that the moon only exists because we look at it. Indeed, our looking at the moon or sun or other galaxies or how apples fall to the earth does not seem to be relevant for how they behave. However, both general relativity and quantum mechanics show that for a more complete understanding of the universe as a whole, we need to include a frame for understanding the observer in relationship to the observed.
Mimi: Ok.
Me: With the Tree of Knowledge System, I had developed a new way to look at the problem. The reason is that the ToK includes a theory of the knower, in a particular the human knower. That is, most physical theories do not include a theory of physicists or how knowledge is generated in the physical sciences. However, the ToK System does that. Here is a diagram that builds from the first diagram that adds the human knower.
Notice, the first diagram is what we saw before. And now we have it matched up with a ToK diagram. The ToK diagram includes the place of the knower, the circle with the person riding the surfboard.
Mimi: Ok, yes, I can see that.
Me: I can use the ToK System to help decode how human knowers know things.
Mimi: I am not sure what you mean by that.
Me: Because I developed Justification Systems Theory, I can see mathematics and physics as a special kinds of justification systems, that have particular rules that legitimize those knowledge systems.
Mimi: You mean like the basic logic of math?
Me: Yes. One way to think about it is that logic “legitimizes” how mathematicians deduce proofs and make claims.
Mimi: And physics?
Me: They use data from things like light and sound to measure/observe empirical behavior and generate math equations that map the behaviors they measure in the natural world.
Mimi: Ok. I think I am following this at a basic level.
Me: Good. It is also the case that the ToK System puts justification systems in the universe. That is, it places science at the Culture-Person plane of existence. And it shows how scientific knowledge maps reality. Here is the classic ToK System diagram. The circle on the left is reality divided up into Matter, Life, Mind and Culture, and the oval on the right encapsulates scientific knowledge about that reality.
Mimi: Ok, I think I see that. You mean your ToK map shows where Life is and where Mind is and where Culture is. And it shows that science exists inside of Culture, but also it sort of comes out of Culture to map reality?
Me: Exactly. And this idea I had, this Equivalency, makes a jump that connects the human knower and measurement with observed behavior. Specifically, it makes them “equivalent” and then “factors them out” and “leaves behind” mathematical concepts that serve as bridging functions.
Mimi: I don’t think I follow.
Me. I know that is a mouthful. Basically, I found a point where the knower equals the known and there were interesting mathematical operations that were sort of leftover. The result is an “equation” that reads as 2pi i f = 1.
Mimi: Ok, yes, I remember seeing that. That is that circle and the wave, right?
Me: Exactly. The mathematics of that circle, actually a complex unit circle, and that wave frequency sign were leftover when I developed my special case where knower = known.
Mimi: I don’t really know what you mean by a special case.
Me: The special case refers to the idea of someone measuring a “single frequency photon.”
Mimi: What is a photon again?
Me: You can think of a photon as a digit of light behavior. Light is made up of electromagnetic energy. A photon is a packet of that energy. And a single frequency would involve a single electromagnetic rotation. If you go back to the diagram above with the knower riding the surfboard, you will see E = hf + mc2 where the Big Bang is. Do you see that?
Mimi: Yes.
Me: You can think of that as representing the kinetic and potential energy in the universe, especially if we were to add the strong and weak nuclear forces, but we don’t need to worry about that because we are talking about electromagnetism. A photon is a “particle” or “digit” or fundamental unit of electromagnetic radiation. A photon has no mass, so that part is zero.
Mimi: Ok, so with a photon you are left with just the E = hf, right?
Me: Yes, that is called the Planck-Einstein relation and it means that the kinetic energy of a photon equals its frequency, f, times Planck’s constant, which is the “h”.
Mimi: And that is the thing you told me is the smallest digit of behavioral information, right?
Me: Yes. Very good to see that. We are going to connect that to another equation. Do you see pq - qp = h/2pi i (I) at the bottom where quantum mechanics is?
Mimi: Yes. I have no idea what that means. Shouldn’t that just be zero? I mean p times q should equal q times p, right? So that would just be zero.
Me: Good point. However, and I know this is complicated, the p and q are not simple variables. They are what are called “matrices” and they do not “commute” like ordinary numbers. So, in this case pq and qp are not identical, and so, no, in this case it does not equal zero.
Mimi: What are they, then?
Me: For our purposes, they are best thought of as observable measurements of variables that are entangled or are “superimposed” in a wave of probability space. For example, we can think of q and p as the position and momentum of a particle. So, you have the observed position and momentum at one point and at then another. Because of the nature of quantum mechanics, you have to think of the whole thing as a kind of a system. Observing one variable influences the knowability of the other and all you can know is the probabilistic relations between the two. For our purposes, just think of this as representing the observational matrix of position and momentum at one point and then another.
Mimi: Ok. This is starting to be a stretch for me to follow.
Me: You can also think of that equation as the tool by which human knowers measure quantum behavioral information.
Mimi: Ok. Are you somehow going to connect the two equations? I know that the Equivalency relates the behavioral frequencies of the observer and observed.
Me: Excellent, and yes! I found myself fascinated by a special case where the observer and observed are essentially equivalent. The idealized case involves the behavior of a single frequency photon. Take a look at this diagram.
Me: The red and blue wave represent the frequency of a single photon going through one electromagnetic oscillation. This diagram shows the pq — qp = h/2pi i (I) as a kind of behavioral information measurement framework. The little device in the upper corner represents measurement, and then you have the pq on the left and then the qp on the right and the frequency in the middle. And that equals Planck’s constant divided by 2pi i times the “I”, which is called an Identity matrix. In this case that is one.
Me: The picture shows a single photon of electromagnetic wave behavior as kinetic energy in the world aligning with the behavioral information measurement framework. This is an idealized case in that I do not think physicists have observed a single frequency photon. My understanding is that they only recently have observed single photons, but this would still be very far away from observing a single oscillation. But the actual reality of this is equivalency is not what is important. What is important is the concept.
Mimi: I am not sure what you are getting at with the concept.
Me: Well, the basic logic is laid out on the other side of this slide. The pq — qp represents the observed or measured behavioral information. In the case of a single frequency photon, you have a case where there is the closest possible representation, overlap, or connection between the kinetic energy and the measured behavioral information. This means that, in this special case of a single frequency photon, you get E = pq — qp, as seen by a specific knower. So if those to sides are equal…see if you can make the next steps.
Mimi: I don’t really want to be tested.
Me: I don’t mean to test you. Just see if you can make the basic steps if we say that E = pq — qp.
Mimi: Well, if those are equal, then are you saying in your special case that the “hf” part is equal to the “h/2pi i” part?
Me: Exactly! And because the h’s cancel that reduces to 2pi i f =1. That is the Equivalency.
Mimi: That is it?
Me: Yep. The math is actually super simple.
Mimi: I don’t understand why you saw this as important.
Me: What I see in the Equivalency are fundamental conceptual or mathematical operators that represent the structure that allow for a bridge between measured behavioral information and kinetic energy in the world. Consider that the “h” is the fundamental unit of behavioral information. In the case of the Equivalency, it cancels out and what I saw remaining were the mathematical or conceptual operators that humans are using to map the physical world. One way to think of this is that I moved to “factor out both the physical world and the physicist measuring it” and “leave behind” the basic mathematical operators that allow for physicists to connect their core measurements with the kinetic and potential energy in the world.
Mimi: Ok, but I am not sure why this is important.
Me: It is important to me because it gave rise to a fascinating picture of these “operators.” For example, when I see this, I see the 2pi i as an equation for a circle that can reduce to a point. It can be considered an epistemic frame through which someone looks at the world. In addition, when I see the f, I see a wave of kinetic energy that exists in the world. And the special case is when the real behavioral frequency equals the information observed.
Mimi: Ok, so let me see if I am following you. You created this “ideal scenario” where the observer equals the observed, and got this 2pi i f =1 equation. And in it you saw that it held a picture of the ideas or basic mathematical concepts that physicists use to map behavior in the world?
Me: That is right. That is what I thought.
Mimi: And why did you think it was important?
Me: I thought it was important because, from the ToK point of view, it highlighted that for a complete understanding, we needed a theory of a human mathematical physical knower. And the ToK helped via this analysis point the way. It gave us a new angle to arrive at new insights on how humans develop knowledge systems in science. Here is the picture of what I was talking about.
This picture suggests that really there were THREE revolutions that needed to happen to correct Newton. That is, we needed General Relativity, Quantum Mechanics, AND a Metaphysical, Metatheory of the Mathematical Physical Knower. And it is this point that the ToK helps place on the field of understanding.
Mimi: But didn’t you bring it to a physicist, and didn’t he say it did not make sense?
Me: Yes. And he was right from a conventional standpoint. Indeed, I knew about those problems, and he highlighted them. That meant that I had some work to do to clarify exactly what I was saying and what I was not saying.
But Mimi: Ok, but why didn’t the physicists see it that way?
Me: Good question. There are a couple of obvious “wrong” things about what I did. For example, there is a time dimension on one side. That is, in the Equivalency, on one side you have the f, which stands for frequency of oscillations per second. But there is only a 1 on the other side. And you can’t have a time dimension on one side of an equation and not on the other. There is also the fact that I blended matrix algebra with conventional math, which also is not ok. And, when you start fooling around with the equation, it doesn’t really fit with standard math systems. So, there are a number of reasons it can be said to fail. But all of these were not relevant from the way I was looking at it. That is why it really is a “metaphysical gateway” rather than a traditional math equation. Of course, to see it as a metaphysical gateway, you need to be able to see the world via the lens of the ToK, and no physicist knows how to do that yet.
Mimi: Is that it? Were you just convinced that they did not understand your language and was that the end of it?
Me: No. What we have covered here is the story and logic of what I saw to derive the Henriques Equivalency. Although I thought it was a powerful idea in 2001, I also knew it was not justifiable at that point. That is, the conventional criticisms given by experts were just too strong, and I really did not know how to justify what it meant.
Mimi: So you had to add to this argument to make it meaningful?
Me: Exactly. What happened was that, in late 2003 into 2004, Andee (my wife) brought me home a book called Where Mathematics Comes From. It is great and it explains how humans generate mathematical operations. And at the end it analyzes something called Euler Identity, which is the most beautiful equation in mathematics.
Mimi: Yes, I remember you talking about that. You said that your Equivalency related to that.
Me: Well, first I should say that it is a testament to the magnitude of my mathematical ignorance that I did not even know that the Euler Identity existed when I developed my Equivalency. From an outside perspective, that should just prove that I am a quack. But from inside my justification system, that ignorance is actually evidence for the validity of the Equivalency. That is, the Equivalency points to the way the Euler Identity and its general application via the Euler formula serve as underlying conceptual operators in modern physics.
Mimi: I can’t follow that.
Me: I know, that is a whole separate story. Do you think you get the basics of the Equivalency now, though?
Mimi: I am not sure. I hear you saying that this 2 pi i f = 1 equation is something that emerges in a special case when the observer equals the observed. And this told you that these are the mathematics or something, what did you call them…?
Me: Mathematical conceptual operators.
Mimi. Right, you said those mathematical conceptual operators allowed physicists to bridge between the world of energy and the world of measurement. And you thought that was important. But others did not, and they pointed out that you had done some stuff that they did not think was allowed. But you understood why there was a problem and found a few insights and it eventually turned into the iQuad Coin, right?
Me: Yes, the Henriques Equivalency serves as the fundamental root of the iQuad Coin. And the construction of the Coin confirms for me at least that I was onto something that bridged mathematics and human subjectivity in a whole new way.
Conclusion
Via the iQuad Coin and Tree of Knowledge, the Unified Theory Of Knowledge provides a new descriptive metaphysical system that places the human subjective knower in relationship to both the ontic reality and scientific knowledge about that reality. The root of the Coin is found in the Henriques Equivalency, and this blog summarizes the story of how the Henriques Equivalency came about. In the next blog, I will share the basic steps that allowed the Henriques Equivalency to be derived and will then sync it up with the Euler Identity, which gives rise to the “Radical Mathematical Humanistic Equation.” That will then serve as the ground for the iQuad path to the iQuad symbol and the minting of the iQuad Coin. With that, the conceptual architecture of the iQuad Coin will be fully laid out.
