May 7, 2022
The Radical Mathematical Humanistic Equation (Part I)
The Euler Identity Plus the Henriques Equivalency Equals Zero
How should we think about mathematics in relationship to human thought? What is the relationship between objective and subjective beauty? Is there a way to combine mathematics, subjective experience, philosophy, and the natural sciences into a single “equation”? The Radical Mathematical Humanistic Equation affords us a new way to approach these questions.
The Radical Mathematical Humanistic (RMH) Equation is one of the key ideas in the UTOK Garden. It resides on the top of the UTOK Seed, above the Educational Infinity Loop and the Wisdom Energy Icon.
In the original Garden diagram, the RMH equation is characterized as a “way to link mathematics to humanistic thought.” This blog walks the reader through the RMH equation step by step, starting with the word “link”. This is because the RMH equation is about making a set of linkages that connect major domains of thought together. These linkages are made via deductive and associative adjacent identities. A deductive identity is something like 1+1=2. This is true by logic and definition. The imaginary number raised to the fourth power equaling one is also a deductive identity.
To form an adjacent associative identity, we attach meaning to this relation. So, for example, in the UTOK language game, we call the i˄4 = 1 relation “iQuad.” We then proceed to create an adjacent associative identity linkage via the iQuad Symbol on the Coin, as described in the four part iQuad blog series (here, here, here, and here). The most basic adjacent associative identity is the connection between the human psyche and the complex unit circle. Thus, via the iQuad symbol, we tie the i˄4 = 1 relation with the Human Identity Function. In so doing, we create a matrix of associations between real, imaginary, and complex numbers and the processes by which humans operate based on knower — known relations. The iQuad blog series shows how the space between those concepts yields a fruitful knowledge architecture.
The RMH equation makes an adjacent associative identity between the Henriques Equivalency and the Euler Identity. Indeed, it is this space that ultimately gave rise to the iQuad Coin and its set of identity relations. To understand the RMH we need to have a basic grasp of both the Henriques Equivalency and the Euler Identity. I explained the Henriques Equivalency in more depth here and here. As such, I will only briefly summarize it.
The Henriques Equivalency is given as “2πif = 1”. It is in quotation marks because although it looks like a mathematical equation, it is better thought of as a kind of “metaphysical gateway” that fruitfully affords many adjacent associative identities. The Equivalency was derived by seeing a new relation between the Planck-Einstein Relation and the fundamental equation in quantum mechanics derived by Born, Heisenberg and Jordan. Specifically, it represents an idealized case whereby one measures the behavioral information of a single frequency photon. Conceptually, this is a unique instance when the “ontic kinetic energy” in the world is identical to the “measured epistemic behavioral information” obtained by a knower. As such, the Henriques Equivalency represents a special case where “observer/knower = observed/known.” It is in that special case that we obtain the “2πif = 1” relation.
What does this mean? As explained in the prior blogs, I interpreted this as the geometric or mathematical logical operations that are “leftover” when we see an equivalency between the ontic kinetic energy and the epistemic measured behavioral information. By leftover, I mean that we can interpret these concepts to be the mathematical conceptual operators that afforded the relationship to be seen.
I fully acknowledge that the moves I am making to derive this equivalency are, well, fuzzy. That is, not everyone would agree with the justification I am giving for why this is a reasonable and helpful set of moves. Indeed, many people who deeply understand quantum field theory and the algebraic matrix equation would not find my logic sound or useful. Whereas deductive logic carries with it a particular kind of force, fuzzy logic is, well, more subjective. I can decide that the Equivalency is both beautiful and practically useful because I am emphasizing particular aspects regarding what it means. For example, I can consider the Henriques Equivalency a useful mnemonic device that allows me to see the relationship between the Planck Einstein Relation and the fundamental equation in matrix mechanics and an observer making an empirical observation.
I can also add that I always found the representation of 2πif = 1 to be “beautiful.” That is, there is something about the relationship between the 2, π, i, and f equaling 1 that I find to be elegant and deeply meaningful. It speaks to my heart and has many associative meanings that I find resonate with me deeply. Of course, we can argue that beauty, like the fuzziness of fuzzy logic, resides in the eyes of the beholder. The point I want to make here is that I am linking the Henriques Equivalency with both fuzzy logic and subjective beauty. I am basically saying, if you see the world from my perspective, the Henriques Equivalency is true and beautiful.
Of course, you do not have to see it as true and beautiful. If you know math and quantum field theory, you might find it clumsy, inelegant, arrogant, wrong, and wrong-headed. This is what I call the Anti-Equivalency position. It is also the case that you may be completely ignorant. Maybe these terms seem new and weird and the whole argument makes your head spin. Or, maybe you are knowledgeable about them, but are neutral about the value of the Equivalency. That is, you see my point and logic, you see the Anti-Equivalency point, and you are neutral in your opinion. These four attitudes regarding the Henriques Equivalency (i.e., positive, negative, ignorant, and neutral) sum up the vast majority of positions human knowers could take. With the Equivalency framed as such, it forms adjacent associative identities with subjective opinion, fuzzy logic, and beauty. Now I want to link that set of associations with deductive logic and objective beauty. In so doing, I will have made a set of linkages between humanistic thought and mathematics.
The Euler Identity is Objectively True
As mentioned above, the RMH equation links the Henriques Equivalency to the Euler Identity. As such, we need to spend a bit of time on the Euler Identity. The Euler Identity is a very famous mathematical equation. I know just saying that will make some folks a little trepid. But as I hope is clear by now, I do not do advanced mathematics, and so I will not be taking you on that kind of trip. Rather all we will need to understand is how to describe the Euler Identity. As implied by the RMH, if we take out the Henriques Equivalency, we can see that the Euler Identity is given as:
You are likely familiar with π, i,1, 0, +, and =. The only other symbol is e, which is the natural log constant or Euler’s number. It is technically defined as: “the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.” What does this mean? For many outside math, logarithms feel strange and don’t come naturally. The easiest way to think about them is to connect them with exponents. An exponent is when you raise something to the power of it. So, 10 exponentially raised to the power of 3 is 1000. The logarithm can be thought of as the inverse of this. The log of 1000 with a base of ten asks what raises 10 so that we get 1000? The answer is 3.
The natural log constant is a special kind of base. As noted in this blog, “e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.” Although this is important if you get into the depths of the RMH equation, for now this is all I am going to say about e, exponents, and logarithms.
The most basic point is this: There is a thing called the Euler Identity. It states that the natural log constant raised to π times i equals negative one. The only thing you need to know at this point is that it is true in the objective, deductive sense. That is, it is not debatable or “maybe true,” but rather it was proven deductively true in 1748 by Leonhard Euler. There are many good books (here and here) and videos (here and here) on why it is true. Here is one done with the frame that is supposedly simple enough for Homer Simpson to understand.
The Euler Identity is Objectively Beautiful
The second thing we need to know is that the Euler Identity is objectively beautiful. How can something be objectively beautiful? Well, you can define objective beauty via an empirical criterion. That is, we can define objective beauty operationally by the criteria that when people see it, they define it as beautiful. Thus, if People Magazine lists the 10 most beautiful people in the world, we can say that the people on the list are objectively beautiful. By this definition, the Euler Identity is objectively beautiful. Many surveys have been done that have found that mathematicians rate the Euler Identity as the most beautiful equation in mathematics. In 2014, neuroscientists looked at the brains of mathematicians in scanners and found that, compared to the other major mathematical equations, the Euler Identity carried the neurological signatures of beauty more than the other equations, in addition to being rated the most beautiful based on self-report.
Why do mathematicians consistently rate the Euler Identity as being the most beautiful equation in the world? Here is one explanation:
The number 1, that most concrete of numbers, is the beginning of counting, the basis of all commerce, engineering, science, and music. As 1 is to counting, pi is to geometry, the measure of that most perfectly symmetrical of shapes, the circle — though like an eager young debutante, pi has a habit of showing up in the most unexpected of places. As for e, to lift her veil you need to plunge into the depths of calculus — humankind’s most successful attempt to grapple with the infinite. And i, that most mysterious square root of -1, surely nothing in mathematics could seem further removed from the familiar world around us. Four different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition.
As this narrative suggests, the Euler Identity is seen by many as being at the very center of mathematical knowledge. Indeed, we can that represented in this Map of Mathematics, where the Euler Identity is circled in yellow.
My hope is that some adjacent associative identities are forming in your mind. Specifically, I hope you can see that the Euler Identity holds as a deductively true, objectively beautiful identity and is networked into mathematics, whereas the Henriques Equivalency represents a fuzzy logic, subjectively beautiful identity.
I first developed the RMH in 2004. My initial goal was to try to merge them together. I was only partially successful in this way. The ultimate pathway to success was finding the iQuad Coin. Specifically, I needed a way to specify the placeholder of the human knower, such that we could have a frame for bridging the objective and subjective fields into a coherent correspondence. Of course, this is what the iQuad Coin does with its Human Identity Function.
In the above depiction, we can see the basic structure of what I am doing in terms of forming adjacent associative identities that uses the Coin to bridge the Henriques Equivalency to the Euler Identity. The logic here starts at a very basic level of negative one plus one equals zero. Then it makes an adjacent associative identity by replacing negative one with the Euler Identity and the positive one with the iQuad Coin. The only added move made here is a nominal one of the coins. The logic is deductive and pure.
We then shift to employ the Coin as a place holder of our subjective identities. That is, we can represent Leonhard Euler on an iQuad Coin for his human identity function, and we can represent me on one as well. Finally, we can replace my identity with the Henriques Equivalency and Euler’s personal identity for the math formula. The consequence is an adjacent associative identity matrix that is linking mathematics and its deductive logic to a metaphysical gateway that connects subjective experience and human identification matrix lenses to philosophy and mathematics and quantum mechanics and general relativity.
This summarizes the basic logical structure of the relationship between the Euler Identity, the Henriques Equivalency and the iQuad Coin. The next blog makes the bridge into more specifiable relationships. Specifically, we will connect the Euler Identity to the Euler Formula, which plays an important role in physics. Indeed, in the Euler Formula we will see a much deeper connection with the Henriques Equivalency, especially when we place it in relationship to the Tree of Knowledge System and how it maps the relationship between time and the growth of complexification.
