Tuesday, September 20, 2016


©Edward R. Close September 20, 2016
In my posts about TDVP, the scientific paradigm that puts consciousness into the equations of science, I have often mentioned the Calculus of Distinctions (CoD), a primary form of mathematical logic from which all known mathematic systems can be derived. I’ve also stated that I derived and/or validated most of the basic mathematical concepts supporting TDVP using the CoD. Even though I’ve published details of the CoD elsewhere, they are neither easily available, nor easily understood. This is a bit unfair to readers of my posts who have no idea what the CoD is. Until now, I justified leaving details about the CoD out of my posts for the general FB follower for the following reasons:

The concepts involved in the CoD are not trivial. It took me many years to understand their true importance. I developed the basic concepts of the CoD by expanding concepts from George Spencer Brown’s Laws of Form to include dimensionality and the consciousness of the observer between 1984 and 1986, and I first published the basic concepts of the CoD in my second book, “Infinite Continuity” in 1990. But for most people, learning the CoD, a new system of mathematical logic, seemed too onerous. On the other hand, I believed that the results obtained by applying the CoD, including explaining things the current materialistic scientific paradigm cannot explain, should be enough to get people interested.

Previously inexplicable things explained by application of the CoD included the Cabibbo angle in particle physics, why quarks combine in threes and why some elementary particles have an intrinsic ½ spin, just to name a few.  Explaining things that have puzzled mathematicians and physicists for years, in some cases, centuries, definitely go a long way toward proving the validity of the CoD and TDVP. But, however justified I may have been in the course of presentation I have followed, I must admit that, without at least a basic understanding of the CoD, anyone trying to understand my posts is missing an Important piece of the puzzle. So I decided to endeavor to rectify this state of affairs by explaining the CoD in terms that I believed anyone interested in TDVP could understand.

As I started to work on a simplified step-by-step explanation of the basics of the CoD, because CoD concepts integrate the logic of number theory, geometry and symbolic logic, I began to get deeper insights into the logical connections between the major branches of academic study investigating the nature of reality; specifically: philosophy, science, and mathematics, branches of thought that historically have been developed as if they were independent of each other, and that led me to an inspiration concerning the best way to present this introduction to the Calculus of Distinctions.

In the educational system we have developed over the past few hundred years, various aspects of philosophy, science, and mathematics are taught as separate subjects, and psychological improvement, spirituality, and religion are pursued via various practices as separate goals. While this may seem natural and even necessary in the development of human thought, it has led to a fragmentation of effort and departmental specialization to the point that people in one field cannot easily communicate with those in other fields. Researchers in theoretical physics, e.g., use terminology largely unknown to theologians, philosophers, psychologists, biologists and engineers. Mathematicians who call themselves number theorists and those working in applied mathematicians, might as well be speaking completely different foreign languages. But, I submit to you that reality is only one, not the disparate unrelated realities suggested in some disciplines. The illusion that different parts of reality might be governed by different, completely incommensurable laws is much more a product of the limitations of human observation, measurement and thought, than an actual multiplicity of realities.

Since the time of Pythagoras, the study of mathematics has become enormously sub-divided into a number of more and more abstract disciplines. Because of this, it is understandable that the casual reader of my posts might well think that the CoD is just another abstract side road in the multiplicity of super complex fields of inquiry. In fact, the truth is just the opposite.  The calculus of distinctions is the re-integration of conscious thought, the mathematics of arithmetic, algebra, geometry, and symbolic logic into one logical system.

Today, when one chooses, or, as is more often the case, is forced by public education to study mathematics, he/she will find the curriculum fragmented. The students’ first introduction to math may be in a course teaching them to do ‘applied math’ by learning how to punch keys on a calculator or computer. The only thing duller and less interesting than that are the details of addition, subtraction, multiplication and division that lurk behind the operation of a calculator. If, for some strange reason, a student decides to go further in academic mathematics, he or she will likely be indoctrinated into a series of progressively complex and boring courses including algebra, trigonometry, set theory, geometry, statistics, probability, and integral and differential calculus. If that’s not enough to convince a student to switch to some other major, we suspect there may be something mentally, physically, psychologically, or socially wrong with this individual.

Seriously, today’s math education systems are very poorly designed for anyone wanting to actually learn mathematics. They are generally not designed to teach students about mathematics all. Rather, they are designed to teach students how to use a few specific tools and techniques to solve numerical problems that someone thinks are important. In my opinion, many, perhaps most, people teaching math today know very little about mathematics. If you want to understand the deep nature mathematics, and how it fits into the larger picture of reality, you’re pretty much on your own. In today’s universities the focus is on learning more and more about less and less. The CoD reverses this trend.

This why I am eager to teach someone, anyone, about the Calculus of Distinctions. As I’ve said in previous posts and publications, the CoD starts at the beginning of the story, not in the middle as most formal math courses do. It integrates the basic ideas of conscious distinction, equivalence, number, dimension, substance and logic, into one consistent set of operations which allows us to re-integrate the disciplines of number theory, geometry, algebra, and symbolic logic, which never should have been separated in the first place. And, it allows us to put consciousness into the equations of science.

All knowledge and understanding begins with the conscious drawing of a distinction, the conscious awareness of self as different from the rest of the universe. This is where an understanding of the logic of mathematical reasoning starts, with your personal experience of knowing the difference between self and other, not with abstract concepts describing processes of calculation. This first step is described by G. Spencer Brown in Laws of Form as the basis of the Calculus of Indications. A distinction is further expanded and defined in the CoD as real, substantial and dimensional. In describing the reality we experience, secondary distinctions, i.e., distinctions in self and/or other, must have measurable extent and content and definable meaning. Measurable extent means dimensionality, measurable content means substance, and definable meaning means impact on experience or purpose.

In posts to come, I hope to make clear to you how developing the concepts of quantitative and qualitative thinking from the beginning of the conscious drawing of distinctions allows us to see the interconnectedness of all things and solve problems and answer questions not possible otherwise. I plan to post some important CoD proofs never seen before.

Because the ideas I am presenting in this series are sequential, each new post building on those that have gone before, I will add new posts to this post as ‘continuations’.

Readers familiar with G. Spencer Brown’s Laws of Form will notice some similarity in what follows here with his Appendix 2, which is the interpretation of his calculus of indications (CoI) for logic. In his original work, Brown established the fact that while distinctions may be drawn in any way we please, the Laws of Form are the same for any universe, and so the similarity in form should not be surprising. But it is a similarity in form only. Development of the CoD departs markedly from Brown’s adaption of the CoI for logic: Brown makes no distinction between real, perceived or conceptual distinctions, but because we are applying the logic of the CoD to the quantized reality which is our universe, we need to make it clear from the beginning that a real distinction drawn in our quantized reality is identified with an existent quantum unit or combination of existent quantum units.

The definition of a real distinction, then, is very simple; it must have three things: extent, content and relevance to the structure of reality. A hydrogen atom, for example, fulfills the requirements of a real distinction: it has extent because it occupies a finite volume of space, it has content because it has mass and energy, and it has relevance to the structure of reality because H atoms are important components of many organic and inorganic compounds forming much of the structure of the physical universe. But in TDVP we could not choose the Hydrogen atom to define the most basic quantum unit, because, as small as it is, it is made up of yet smaller real distinctions.

We found that the free electron was the best elementary particle to use to define the ultimate basic quantum unit for three reasons: 1. The electron has the smallest mass of any of the stable subatomic entities making up the elements of the Periodic Table, 2. by applying the principles of relativity and quantum mechanics to the spin dynamics of the electron as it is stripped from the Hydrogen atom, we are able to define the smallest possible quantum volumetric equivalence unit. 3. The mass, spin and energy of ionization are well established to several decimal places giving us all we need to define its volumetric equivalence as a unitary distinction.  Because it reflects three kinds of extent, three kinds of content and three kinds of meaningful impact to convey the logical structure of consciousness to the structure of the universe, we call it the Triadic Rotational Unit of Equivalence (TRUE).  When we used this unit derived from the free electron as unitary, we found that all other elementary particles exist as volumetrically combined multiples of the TRUE unit. Thus the TRUE unit is a real distinction and the real building block of the universe.

‘Volumetrically combined’ means that the elementary particles that make up the nuclei of atoms, are not just stuck together like tinker toys, their mass/energy equivalence volumes merge to form a larger volumetrically symmetric entity. And the fact that all larger and larger stable particles, e.g., protons, neutrons, atoms, molecules, etc. are multiples of the unit, means all stable particles represent whole numbers of TRUE units, and the simple equations describing the combining of particles are composed of integers (whole numbers). This allows us to use the CoD with the unitary distinction defined as the TRUE unit, greatly simplifying calculations. You will see what I mean in the examples to follow.
To set up the CoD to handle logical calculations involving real distinctions that are whole number multiples of the TRUE quantum unit we let n represent an n-dimensional distinction. Note that this is significantly different than Brown’s symbol of indication. The subscript n allows us to represent real, versus conceptual distinctions, because when n = 0, this symbol represents a point, a mathematical singularity; when n = 1, it represents a line; when n = 2, it represents a plane; and when n = 3, it represents a volume.

Because of the simple fact that points, lines and planes have no capacity to contain any real substance, for n to represent a real distinction as defined above, n must be equal to or greater than 3. (n≥3).  In addition, we will let  represent a real state of no distinction. 

Now, in terms of observations of the outside world we call the universe, 0 = 1 = 2 , because, recalling the requirements for a real distinction (A real distinction must have three things: extent, content and relevance to the structure of reality), distinctions of 0, 1, or 2 dimensions do not meet the requirements: 0 has no extent, no content, and no relevance to the structure of the universe, and both 1 and 2 have extent and relevance, but no content. Finally, geometry (dimensionality), pure mathematics, language, and logic are coherently integrated by equating n when n ≥ 3 with the logical condition called True, and  with False.

With these simple definitions and interpretations of the CoD, we have the basis for a surprisingly powerful method for testing the logical validity of a wide range of statements, including verbal statements, mathematical conjectures and scientific hypotheses. We can articulate the connections between language, symbolic logic and the CoD with one-to-one relationships. For the next Continuation, I will prepare a table displaying those relationships.

In the preceding Continuation, we defined the concept of a distinctions very broadly: the symbol n represented a distinction of any kind, and the dimensionality of the distinction, represented by the subscript n, could be any non-negative whole number from zero to infinity. In developing a calculus of distinctions for quantum reality, however, we will be dealing with real sub-atomic entities which have real content. Therefore, the distinction between real and conceptual must be maintained, and because there is no content when n = 0, 1 or 2, n is restricted to positive integers equal to or greater than three (n≥3). For simplicity of presentation, from here on in this presentation, we will drop the subscript and all distinctions represented by the symbol , will be understood to be at least three dimensional.
A calculus has to have clearly defined units and primitive initial operations that provide the basic processes of calculation applied to those units. We will take the TRUE quantum unit as the basic unit of the quantum calculus, and the CoD is the calculus of consciousness, so the primitive equations of the CoD must provide calculation processes that reflect the primary functions of consciousness. The primary functions of consciousness are: 1. the drawing of unique distinctions, 2. Memory, and 3. organizing distinctions.
The Primitive Expressions of these functions as basic CoD calculation operations are:
1.     2.  ﬧﬧ and 3.       , or
Because these expressions represent interactions of mind and matter in quantum calculation processes, they are rich with meaning and implications that will be important to keep in mind as we proceed. In addition to expressing the difference between empty geometric concepts and substantially real objects, notice that Expression #1, implies space, because two identical objects require twice as much space as one; #2 implies time, since without memory, there could be no awareness of the passage of time; both #2 and #3 and express logical operations that can be used to transform CoD expressions; and #3 reveals the difference between non-existence and zero. The blank space after the ⇒ and the empty set symbol, ⦰, represent the absence or non-existence of distinction, not an absolute state of nothingness.
With these concepts and notations, we can now articulate the connections between language, symbolic logic, algebra, and the CoD with the one-to-one relationships displayed in the table below. This table of relationships is not exhaustive, other relationships can be derived from them, but they are sufficient for the purposes of this post.

Symbols and Connectives of Logic

Calculus of Distinctions
Real, Existing, True

Appearance, Cancellation,

 ⇔   , or ⦰        *
Integer Variable
x = A = any whole number
A =
Not A
x ≠ A
A or B

A and B

A . B

A + B
A implies B

A  B

                                                       AB *

* In these CoD expressions, the smaller symbol or symbols to the left are considered to be nested in the larger distinction symbol to their right: The upper arm of the larger should extend over the smaller symbol or symbols, but Word would not allow this, and I haven’t been able to import a PDF file into the blog.

To illustrate the power of the application of the Calculus of Distinctions, let’s apply it to the most famous theorem in mathematics known as Fermat’s Last Theorem, a conjecture that went without a formal proof for more than 300 years, even though all of the world’s most brilliant mathematicians tried to prove it and failed.
Fermat’s Last Theorem (FLT) states that there are no integer solutions for the equation
Xn + Yn = Zn, where n is a whole number greater than 2.
This means that, if Fermat was correct, there are no three whole numbers that can be substituted for X, Y, and Z that will satisfy the equation when the integer exponent n is greater than 2.
The CoD approach to proving Fermat’s Last Theorem is simple: with the incorporation of Euclidean geometry, integer (whole-number) mathematics and symbolic logic, we can consider FLT as a real world problem, not an abstract number theory problem independent of the real world. By doing this, every term the equation is transformed into a distinction composed of a finite number of TRUE units of three or more dimensions, and the operational rules and logic of the CoD govern their combination.
(It is interesting to note, as an aside, that Fermat approached some of his number theory proofs by defining whole numbers in terms of geometric figures.)
So let’s recast the FLT equation Xn + Yn = Zn, in the form of a CoD combination of two distinctions to form a third distinction: Consider the combination of two real n-dimensional symmetric distinctions to form a third real n-dimensional symmetric distinction, all three composed of Triadic Rotational Units of Equivalence (TRUE units), where the TRUE unit is the smallest possible unit in quantized reality, determined by application of the principles of relativity and quantum mechanics to the mass and energy of the free electron stripped from the Hydrogen atom. It is important to know that its size in conventional units of measurement is irrelevant to this or any other application of the CoD because, as the basic unit of reality, its value is set equal to 1 and everything is normalized to it.
The mass, energy and volume of the TRUE unit are functions of the dynamic nature of reality, and because of spin, the shape of elementary n-dimensional symmetric distinctions is spherical. The volume of an n-dimensional sphere is always a fractional multiple of a power of π times the radius of the sphere raised to the nth power.  For example, for n=3: V = 4/3πR3, and for n=4: V = 1/2π2R4, etc. If the radii of two 3-D spherical distinctions are X and Y, then the sum of their volumes is 4/3πX3+ 4/3πY3, and setting the sum equal to the volume of another sphere, 4/3πZ3, we have:
4/3πX3+ 4/3πY3 = 4/3πZ3,
describing the combination of two spherical distinctions. Similarly, for n=4, we have:      1/2π2X4+ 1/2π2Y4 = 1/2π2Z4
And, for any n-dimensional symmetric distinctions, we have:
SnXn+ SnYn = SnZn,
where Sn is the volumetric shape factor. Finally, dividing through by Sn, we have:  
Xn+ Yn = Zn
We have mathematically reduced the equation describing the combination of two n-dimensional symmetric distinctions forming a third n-dimensional symmetric distinction (all three composed of the minimum possible units) to the sum of two integers raised to the nth power equaling another integer raised to the nth power, so we now have an equation identical to the FLT equation. The addition of integers to the n power in the FLT equation is completely analogous to combining particles composed of TRUE units, and this allows us to apply the CoD logic to the FLT equation. The FLT question now becomes: If the radii of X and Y are integers, can Z also be an integer? If we can prove that it cannot, we have proved FLT.
Applying the identities, equations and relationships presented in the table above, we can proceed as follows: We know that if n = 1 or 2, the CoD truth values of Xn, Yn and Zn are all equal to . But, if n 3, the CoD truth values of Xn, Yn and Zn are .  So, if n 3, the FLT equation translates to the CoD equation:
Xn + Yn = Zn ⇒   =
Applying Basic Calculation Operation #2, we have: =
And applying Basic CoD Operation #3, we have:   =, which is an absolute contradiction, implying that the equation is false, proving that if X, Y and Z are non-zero integers and n 3, then Xn + Yn cannot equal Zn, proving Fermat’s Last Theorem.

On the other hand, if n = 1 or 2, the Xn = Yn = Zn = , and the FLT equation translates as:
Xn + Yn = Zn + =
This, of course, simplifies to = , which is not a contradiction, and is what we would expect, since when n = 1 or 2, there are many integer solutions to the FLT equation: When n = 1, any two integers will add up to a third integer: For example, 1 + 2 = 3 and 3 + 4 = 7, etc. And, when n = 2, we have solutions like 32 + 42 = 52 and 52 + 122 = 132, known as Pythagorean triples.
Finally, we see that for n 3, the equation Xn + Yn + Zn = Qn ⦰ + = = , which is no contradiction, implying that three real symmetrical distinctions can combine to form a fourth, explaining why quarks combine in threes, and proving that there are integer solutions to this equation, which means that, while two integers raised to any power greater than 2 cannot be added to yield a third integer raised to the same power, the sum of three integers raised to powers greater than 2 can equal a fourth integer raised to the same power. To see how quick and easy this is compared to conventional methods, go on line and look for solutions to Diophantine equations. The first two solutions of this equation are: 33 + 43 + 53 = 63 and 13 + 63 + 83 = 93, which, translated into CoD equations in TRUE units, have proved very useful in TRUE analysis of the elements of the Periodic Table, as reported elsewhere in these posts.

The striking thing about this proof of Fermat’s Last Theorem is its amazing brevity and simplicity, especially compared to the 1994 proof by Sir Andrew Wiles, which is a very challenging proof of the modularity theorem for semi-stable elliptic curves, combined with Ribet’s theorem, requiring more than 150 pages of complex mathematics only accessible to professional mathematicians. This proof, once the logic of the CoD is understood, is even shorter and simpler than the proof I produced in 1965. I believe this is so remarkable that it bears repeating:
Applying the CoD to FLT in three simple lines:
1.) For XYZ 0, and n 3, Xn + Yn = Zn   = , but by Basic CoD Operation #2,
2.)    , and by Basic CoD Operation #3,
3.) = =, which is an absolute contradiction proving Xn + Yn  Zn.
Not only have we proved Fermat’s Last Theorem in three lines, in two more lines, we show why the FLT equation does have integer solutions for n =1 and 2. And in two more lines, we prove that, while two symmetrical particles cannot combine to form a third symmetrical particle, three symmetrical particles can, explaining why quarks combine in threes to form the basic building blocks of reality.

If this is not remarkable, I don’t know what is!


  1. Good luck, Ed! However, for this lifetime, I feel mystically obligated to follow and abide by the broad-brush tenets of Y= X Squared plus One, which have already been well covered on both our sites. May UF be with you! Amun!

  2. Every 'Item', constituting the Cosmos, and our evolving understanding of It on planet Earth in the system of the Sun, must have, de facto, a duality, which makes It Triadic - Otherwise, It would not exist! Ed has chosen and mathematically shown that 'gimmel' represents Its fundamental constituent - Good for Ed! This can only endorse my more broad-brush and mystically-inspired conception as to what the Cosmos and our place in It is all about - Represented by the formula 'Y= X Squared plus One', where 'gimmel' gives primary substance to the the intrinsic, cohesive and everlasting nature of the formula's 'One'! I can say no more in support of Ed's fine work and lifetime of dedication to his mathematical treatise, and hopefully this will eventually add greater significance to and understanding of 'Y= X Squared plus One', and what this formula has in store for humanity's future enhancement, ad infinitum! Amun!