A SIMPLE ONLINE COURSE IN THE BASICS OF TDVP

A SIMPLE COURSE INTEGRATING THE SEARCH FOR TRUTH

One of the founders and first President of the Academy for the Advancement of Postmaterialist Science (AAPS), Dr. Gary Schwartz, has suggested that I prepare an online course designed to help people understand Transcendental physics and the Triadic Dimensional Distinction Vortical Paradigm (TDVP), the shift to a consciousness-based science. The previous post, “TDVP MATHMATICS AND THE STRUCTURE OF REALITY”, together with this post, articulate the basis for such a course.

In the previous post, I discussed the difference between the Aristotelean and the Platonic approach. The Aristotelean approach (Aa) emphasizes the use of mathematics as a tool to solve problems, while the Platonic approach (Pa) looks deeper and sees mathematics as a reflection of the logical structure of reality. The Aa is primarily descriptive, while the Pa is primarily explanatory. So, contrary to the belief of scientists like professor Rowan-Robinson, quoted in the previous post, neither is wrong, both are needed, and they must be integrated in the search for truth.

Publishers tell authors writing for the general public to avoid equations. They say something like: “With every equation, you lose a thousand readers.” Most mathematicians, however, think in equations, and most physicists use equations to describe physical processes. Equations are the sentences of the language of mathematics and science; and every branch of science and engineering has developed its own language. To understand Transcendental Physics and TDVP, you must set aside such specialized languages and learn a new language, a language integrating physics and geometry, engineering and theory; thinking and doing. That language is a primary calculus that I call the Calculus of Distinctions (CoD). It includes procedures like Dimensional Extrapolation {moving from an n-dimensional domain to an (n+1)-dimensional domain}, quantum equivalence units, called Triadic Rotational Units of Equivalence (TRUE) and Diophantine equations that I call Conveyance Equations, because they convey the logic of consciousness into physical structure. Are these intimidating terms for you? They won’t be if I’m able to develop a course comprised of clear, simple steps from the basics to the integrated language of TDVP.

Understanding the CoD is the first crucial step toward  understanding TDVP. Inspired by George Spencer Brown’s “Laws of Form”, I developed the calculus of distinctions in 1986, and published the basics in my second book, “Infinite Continuity” in 1990, and expanded on it in “Transcendental Physics” in 2000. Even now, more than 30 years after the first inspiration, only a few people, probably less than half a dozen, understand the importance of the calculus of distinctions, a system that integrates mathematics, science and consciousness.

I’m called a mathematician, physicist, logician and cosmologist because I’ve gone through the training and hazing of formal education: from high school math and science to college and university courses in applied and theoretical science and engineering, attaining degrees, and going on to post-graduate research. But in this integrated course, I must rise beyond, and go deeper than those specialized labels. I am no longer a physicist or an engineer, no longer an Aristotelean or Platonic thinker, a scientist or a lay-person, neither a materialist nor a spiritualist, I am an integrated human being. And the task I am setting for myself is nothing less than to explain in simple terms, the integration of the human search for truth, i.e., the integration of mathematics, science, philosophy, and spirituality.

THE BASICS

·      Physical reality is quantized. This simply means that the basic measures of physical reality, i.e., mass and energy, only occur in multiples of one very small unit, and that is the smallest quantum unit.

·      An exact equivalence exists between mass and energy. That means that mass and energy are simply two different forms of the same thing. This is expressed mathematically by the simple equation E = mc², where c, the speed of light, is defined as the distance travelled in a given unit of time, like miles per hour, or meters per second.

·      In observing, describing and measuring physical reality, there are two types of variables: variables of extent: width, length and depth, and variables of content: mass and energy. All other measurables are combinations of these.

·      To describe quantum reality in its simplest form, all units of measurement of the variable of observation and measurement must be normalized directly to the quantum equivalence unit of mass and energy. This is done very simply by naturalizing the speed of light to unity; i.e., by defining the speed of light as one, so that c = one unit of distance divided by one unit of time. In other words, c = 1/1 = 1. In this system of units, with c = 1, from which c² = 1, the equation E = mc² becomes E = m, signifying a one-to-one relationship between mass and energy.

·      Along with the two types of physical variables of content and extent, there is a third type of variables, the variables of consciousness. They are the variables of intent, resulting in impetus and impact on physical reality, leading to meaningful structure and purpose.

·      There are no “separate realities”. Quantum reality and macro-reality are one and the same, and that reality includes both physical objects and conscious beings. Therefore, a comprehensive science must include both consciousness and physical objects, and the way in which they relate to each other.

·      Consciousness, i.e., conscious reality, is a reality that is as important as, if not more important than physical reality. Without it, there is no you or me.

·      The reality we experience, in all of its complexity, begins with the simple drawing of a distinction. This is why we (Neppe and Close) called our first book together “Reality Begins with Consciousness”. All the various forms of reality arise from the drawing of distinctions.

·      The first and most important distinction is the distinction of self from other.

·      Secondary distinctions are formed in the “other”. They are formed, or are recognized as pre-existing by conscious entities.

·      All of natural atomic structure (called hadronic matter by contemporary science) is made up of electrons, protons and neutrons. But protons and neutrons are made up of up-quarks and down quarks. So the three basic building blocks of stable atomic structure, capable of forming molecules, hadronic matter and sustaining life, are electrons, up-quarks and down-quarks.

·      The basic quantum equivalence unit is derived from the physical characteristics of the electron, the elementary entity with the smallest mass among the three basic building blocks.

·      All of the basic units of measurement of mass, energy, space, and time can be naturalized to the quantum equivalence unit, which, because of the spinning of elementary particles, and the inclusion of consciousness, is called the Triadic Rotational Unit of Equivalence (TRUE), the true quantum equivalence unit. See derivations in Parts 8 - 13 of the IQNexus article referenced in the previous post. (Putting Consciousness into the Equations of Science: The Third Form of Reality (Gimmel) and “TRUE” Units (Triadic Rotational Units of Equivalence) of Quantum Measurement, the IQNexus Journal, 2015 Vol. 7, No. 4.

·      This basic quantum equivalence unit (TRUE) is the basic distinction of the calculus of dimensional distinctions (CoDD), a subset of the primary logic of the CoD.

·      All measurements are integer multiples of this unit, and all equations describing the combination of elementary entities including electrons, quarks, protons and neutrons will be Diophantine equations, i.e., equations with integer (whole numbers: 1, 2, 3, …) solutions. This allows us to use mathematical theorems (independently proved relationships) like the Pythagorean Theorem and Fermat’s Last Theorem, to answer questions like why quarks combine in threes and why fermions have one-half integral spin.

An understanding of these basic concepts should enable anyone to follow the logical development of TDVP, including putting consciousness into the equations of science, and the solutions and explanations of many problems that cannot be adequately addressed in the paradigm of current mainstream science.

THE ULTIMATE QUESTION AND ITS ANSWER

THE ULTIMATE QUESTION AND ANSWER

©Edward R. Close January 11. 2018

The year was 1685. Traveling across Germany in a Pferdekutsche (stagecoach), Gottfried Wilhelm Leibniz and a traveling companion stopped at an inn for a short rest and a drink. Leibniz was already famous. About as famous in Germany as Einstein was in the US in the 1950s. He was a polymath: a medical doctor, mathematician, linguist, philosopher, scientist; he had invented something called “the calculus”, and he had produced the world’s first mechanical calculator that could perform multiplications and divisions. When he identified himself, the people in the inn laughed. They could not believe that this mud-spattered, ordinary-looking man could be the most famous man in Europe!

“So, then, Mr. Philosopher, what is the most important question a human being can ask?” One of the scoffers queried.

Leibniz frowned, cleared his throat, and said: “The first question a man of science should ask, is ‘Why is there something rather than nothing?’”.

Everyone laughed, and returned to their bier steins, shaking their heads, not realizing that what they had just dismissed as nonsense was, in fact, the ultimate question.

Why is there something rather than nothing? Modern science’s answer is “the Big Bang”. But that is actually no answer at all. The idea of something exploding into existence from nothing, is nonsense, and you don’t have to be a scientist to know that it is nonsense.

To find the answer, we must go back another 1,435 years, to the time of Diophantus of Alexandria. Diophantus was a Greek mathematician who lived in the third century, and died at the age of 84, sometime between 285 and 299 AD. How do we know how old he was when he died? He left a mathematical puzzle on his grave stone, which, when solved, reveals his age.

Inscribed on his tombstone was the following poem (translated from the Greek):

Here lies Diophantus. Now this wonder behold,

Through art algebraic, the stone tells how old:

God gave him his boyhood for one-sixth of his life,

One twelfth more as a youth, his whiskers grew rife;

After another one-seventh, had marriage begun;

And five years later, came a bouncing new son.

Cruel Fate took the child of this master and sage

When he was only one-half his father's final age.

Consoling four years with the science of numbers,

The sage ended his life, and in this grave slumbers.

A high school algebra student today can solve this puzzle by writing and solving two equations in two unknowns. Hint: Let X = Diophantus’ age at the time of his death, Y = the son’s age at the time of his death, and use the clues given in the poem to construct simultaneous equations.

Diophantus specialized in finding whole number (integer) solutions for algebraic equations. As a result, equations for which there are integer solutions are now called Diophantine equations.

What does this have to do with Leibniz’s “ultimate” question? To answer this question within the question, we have to refer to the mathematical work of a man who was nearly 40 years old when Leibniz was born, and died when Leibniz was only 19.

In 1637, an obscure French jurist, an amateur mathematician named Pierre de Fermat, had already answered Leibniz’s ultimate question, but he probably didn’t know it! In the margin of a page in a book on Diophantine equations he wrote: “I have found a marvelous proof, but the margin of this book is to small to contain it.”

Fermat’s proof of what became known as “Fermat’s Last Theorem”, was never found, and the renowned French thinker of the day, Rene Descartes, called Fermat “an uneducated trouble-maker” and tried to discredit him as an amateur. As it turned out, Fermat was at least as good a mathematician as Descartes. You may recall Descartes’ statement: “I think; therefore, I am!” But, of course, this is no more an answer to the question of existence than the Big Bang answer, because it leads to the deeper questions of what is ‘thinking’ and what is ‘being’.

Fermat’s proof ultimately depends upon the logical method of infinite descent. If something is assumed to exist on a large scale, and can be reduced by logical steps to a version of the same thing, but on a smaller scale, then it can be reduced again and again, until you arrive at a version of the same thing at the smallest possible scale. Then it can be easily proved to be either true or false, existing, or not existing.

Notice that the Big Bang theory is a form of infinite descent. The idea that the universe is began as the result of an explosion from a dimensionless point 13.8 billion years ago is obtained by an infinite descent in space and time, starting with something, i.e., what we have now, descending to nothing. But, that makes no sense. So, maybe it was a very dense mass that existed before the big bang? But, if so, where did that come from? What existed before it exploded? Something or nothing? If it was something, then the big bang was not the beginning of everything, only the beginning of the expanding universe we have today. If it was nothing, then we have a paradox with the production of something from nothing. If your answer is “God created it”. Then God existed before the big bang, and you can’t say there was nothing before the big bang.

Is there an answer? Yes, there is an answer, and Pierre de Fermat discovered the key to it while studying Diophantus’ equations. But, as I said, Fermat himself, probably did not realize that he had the answer to the ultimate question. That realization had to wait for more than 300 years.

In 1900, Max Planck discovered that the matter and energy of the physical universe exist only in multiples of a very, very small unit. This means that in the physical universe, infinite descent to zero is impossible. Descent stops at the smallest quantum. This is, after all, what Fermat’s infinite descent is about. It stops with an integral form, not zero. The infinite part of the name infinite descent is used because it can start with any assumed object, however large, and then descends to its smallest possible integral form.

In 1986, I developed the calculus of distinctions, derived from George Spencer Brown’s calculus of indications, adapted it for use in a quantum reality, and applied it to the big-bang expanding universe theory. I published the results in a book titled Infinite Continuity in 1990.

Here are two excerpts from Infinite Continuity:

From the first paragraph of the Preface:

“The digital clock beside my bed read ‘1:11’. It was one-eleven am, January 16^th, 1986, and I had just awakened from a very vivid dream. A stone had dropped into a clear, dark pool, filling it with golden ripples of spreading light, and at the center of my awareness the pieces of an intricate puzzle had suddenly fallen together. Grabbing a pencil and paper, I began writing as rapidly as I could. So vivid were the details, so clear the understanding, that the heart of the vision flowed out onto the paper that night.”

From the last paragraph of Part V, Summary and Conclusions:

“We stand again on the threshold of a new scientific frontier. As science recognizes the connectedness of all things, the need to integrate the knowledge we have gained becomes more urgent. The formalization of a new science is required. This new science will encompass the existing paradigm, and yet transcend it by including the relationship of [Primary] Consciousness to matter and energy, and the relationship of individual consciousness to existential reality.”

The calculus of distinctions applied to quantum reality became known as the calculus of dimensional distinctions and one application developed in collaboration with Vladimir Brandin, was published in Elements of Mathematical Theory of Intellect, Moscow Interphysics Laboratory, Moscow, Russia, 2003.

Finally, in 2011, in collaboration with Vernon M. Neppe, MD, PhD, internationally renowned neuroscientist, I completed the proof of the existence of a third form of reality, which Vernon named gimmel, not measurable as mass or energy, and therefore non-physical. With that, the proof was complete, and the answer to Leibniz’s question was finally clear: There is something rather than nothing because there is no such thing as nothing.

This also validates Einstein’s statement in the final appendix of his book on relativity, written only a few years before his death, where he suggested that there is no such thing as empty space.

This answer bears emphasizing:

Fermat’s method of infinite descent applied to quantum cosmology proves that there never was nothing, there is no absolute beginning or end, only changes in form. Nothingness is a fiction, a meaningless concept conjured up by confusing changes of form with beginnings and ends.

A NOTE ON MY 1965 PROOF OF FERMT'S LAST THEOREM

FERMAT'S LAST THEOREM PROVED 30 YEARS BEFORE SIR ANDREW WILES' PROOF

Over the years I have submitted my proof to more than 50 professional mathematicians. The mathematicians who have  rejected my 1965 proof have done so primarily because of the belief that there cannot be a proof of FLT using simple mathematical concepts.The few who have actually tried to refute FLT65, have attempted to support this belief with the fact that the division algorithm may or may not apply to integer constants obtained by substituting specific integer values into integer polynomials and reducing them to single integers. But there is no proof that it is true for the integer polynomials of the Fermat equation, and so three of them have resorted to demonstrations that have no relevance to actual solutions of the Fermat equation, to try to make the point that FLT65 may not be valid.

Their idea that the division algorithm might not apply to the integer polynomials of the Fermat equation factor f(Z) = Z^p-1 + Z^p-2X + Z^p-3X² + ••• + X^p-1 arises from the fact that, for given integer values of X and Z, f(Z) can be reduced to a single integer (a constant), and if that single integer is not prime, in general, one of its integer factors may or may not contain the integer equal to the integer value of Z₁ – a.

FLT65 provides a way to determine whether or not any specific single integer value of f(Z) (a polynomial factor of the Fermat equation) can contain the specific single integer value of Z –a (a polynomial factor of Y in the Fermat equation) as a factor, using the division algorithm and its three corollaries. For an integer solution of the Fermat equation, f(Z₁) must not only contain Z₁ – a, it must be equal to (Z₁ – a)^p.

The division algorithm and its corollaries, by definition, apply to all polynomials with real number variables, so they apply to polynomials of integer variables in the same way they apply to all polynomials of real numbers because integers are real numbers which, along with non-integer rational and irrational numbers, comprise the field of real numbers. Finally, an integer solution of the Fermat equation, if there is one, is simply one of the infinite number of solutions to one of the Fermat equations, and the three numbers of any solution are a set of three numbers existing in the field of real numbers. FLT65 demonstrates the fact that for the Fermat integer polynomials f(Z) and Z – a, where both polynomials must be factors of Y^p, there are no integer values of a, X and Z for which Z – a divides f(Z), because the remainder will always be non-zero.

After more than 40 years, I still have hope that more mainstream mathematicians will join the small, but growing number of mathematicians who agree that there are no fatal flaws in the logic of FLT65.

Edward R. Close, June 26, 2017

For those who are not familiar with Fermat's Last Theorem, I've pasted in a previous discussion and some relevant links below.

The Basic FLT65 Proof

The following steps summarize the logic and mathematics of FLT65. For brevity, I will not present proofs of the steps here because they are so easily proved that they can be proved by a bright high school algebra student. If these steps aren’t obvious go to http://www.erclosetphysics.com/search?q=Fermat%27s+Last+Theorem+Part+1.

STEP #1: The first step in FLT65 was to provide a rigorous proof of the division algorithm and its three corollaries. The reason I provided this proof first, even though it was well known to mathematicians, was to show that it applies to all polynomials across the field of real numbers, including integers, and to highlight the fact that the uniqueness of the dividend and remainder allows the all-inclusive “if and only if” of Corollary III. These points were pointed out in FLT65.

STEP #2: If there is an integer solution for Fermat’s equation: xⁿ + yⁿ = zⁿ, to prove or disprove it, we need only consider n as prime numbers, p >2, and x, y, and z as relatively prime positive integers. Proof of this is included in FLT65 allowing us to proceed to Step 3.

STEP #3: Fermat’s equation can be rewritten as z^p – x^p = y^p, and since all prime numbers >2 are odd, factored as follows: z^p – x^p = (z-x)( z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = y^p

Similarly, z^p – y^p = (z-y)( z^p-1 + z^p-2y + z^p-3y² +•••+ y^p-1) = x^p.

For the next step, and throughout this discussion, keep in mind that we have assumed that there are integer solutions to Fermat’s equation, so the approach is to determine whether this assumption leads to a contradiction. If it leads to a contradiction, FLT is proved.

STEP #4: It is easy to show by simple algebraic division that the only common factor that may be shared between the factors of the Fermat equation is the integer p, and since x, y, and z are relatively prime integers, if either x or y contains p as a factor, the other cannot. See the proofs of this in the original proof in the link above. So we can let y represent the one that does not contain p. It then follows that the two factors of the left hand side of the first equation of step 3 are relatively prime and thus are perfect p-powers of integers. Thus, by inspection of

(z-x)(z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = y^p, we see that we can write

(z-x)= B^p, and (z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = A^p, where A and B are positive integers.

STEP #5: If there is an integer solution, then x and y are specific integers X₁ and Y₁, and the p-1 polynomial in z, f(z)=(z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁ ^p-1) = A^p, and B^pA^p = Y₁^p. That is, the two factors must be perfect p-powers, integers raised to the pth power.

STEP #6: In a positive integer solution, z >Y₁ >A, and by closure of integers, there is a positive integer a, such that A= (z – a), and by corollary II of the division algorithm, when f(z)=(z^p-1+ z^p-2X₁ + z^p-3X₁² +•••+ X₁ ^p-1) is divided by Z – a, the remainder is equal to f(a) = a^p-1+ a^p-2X₁ + a^p-3X₁² +•••+ X₁ ^p-1.

STEP #7: Corollary III of the division algorithm says that f(z) is divisible by z –a if, and only if, f(a) = 0. But, since f(a) is the sum of p positive integers, it can never equal zero. Thus by assuming there is an integer solution of z^p – x^p = y^p, we have produced a contradiction proving Fermat’s Last Theorem.

Discussion

Note that the case n = 4 is not addressed in this proof. It was overlooked in FLT65, but this was not a problem because there were several known proofs for n = 4, including one by Fermat himself.

So FLT65 is effectively a complete and valid proof of FLT; but approximately 90% of the mathematicians to whom the proof was submitted over the years did not respond at all. This is because, before Sir Andrew Wiles’ proof was accepted, professional mathematicians received hundreds of supposed proofs of Fermat’s last theorem per year.

If you’ve ever taught mathematics and had to evaluate proofs developed by students, you know it can often be very challenging and time consuming, and attempts at proofs by amateur mathematicians are usually filled with all kinds of errors. In addition, because in more than 300 years, so many first-rate mathematicians had tried to prove or disprove FLT and failed, most mathematicians consider reviewing such ‘proofs’ a waste of time.

I’m sure that this was the reason the first mathematician to whom I sent it rejected it. The reason he gave, however, was that, if FLT65 were true, it would also apply to the case n = 2. [When n = 2, we have z² – x² = y², which does have integer solutions known as the Pythagorean triples, e.g. 3,4,5]. Of course by giving this reason for rejecting FLT65, he revealed the fact that he hadn’t read it, because it is clear to anyone with basic math skills reading the first page that the method of proof of FLT65 doesn’t apply to the case n = 2.

Of the 10% who did respond, most gave the opinion that there had to be a mistake somewhere, but failed to point one out, or provide any mathematical argument supporting their opinion. Of the remaining recipients of FLT65, only a few provided any sort of mathematical demonstration supporting their opinions. Those responses are presented in the article accessed by the links provided above. Those arguments were all easily refuted. However, one of those demonstrations, actually offered with different numerical values by three reviewers, is worth mentioning here because it is a classic example of inadvertent misdirection, and it also shows how tricky a proof of FLT can be.

The argument they put forth was that the division algorithm and corollaries certainly apply to algebraic polynomials, but they may not necessarily apply to the integers obtained when, for specific integer values of z, a and X₁, the algebraic polynomials z-a, f(z) and f(a) are reduced to single integer values. This is an interesting conjecture, but none of the reviewers attempted to prove or disprove it, instead they offered what they thought were counterexamples to FLT65 for n = 3. They selected integer values of z, a, and X₁ that, when substituted into f(z) and f(a), produced an integer value for f(z) that contained the integer z-a as a factor, even though f(a) did not equal zero, appearing to violate corollary III of the division algorithm.

It is worth contemplating this argument a little more deeply for a moment, because by doing so, we expose the fact that such a demonstration is not actually a counterexample, but is in fact, an inadvertent misdirection, shifting attention away from the fact that z must be part of an integer solution to the Fermat equation. It is not hard to find positive integer values for z, a, and X₁ such that f(z) is divisible by z-a, and of course f(a)= a^p-1+ a^p-2X₁ + a^p-3X₁² +•••+ X₁ ^p-1 will still be non-zero because all the terms are positive integers. But because the values of z and a selected have no relation to the Fermat equation, these demonstrations have no bearing on the logic of FLT65.

FLT65 started with the assumption that there is an integer solution for the Fermat equation. This means that for a numerical example to be relevant, z must be part of an integer solution of Fermat’s equation. The issue is not whether you can find integer values for z, a, and X₁ that will make f(z) divisible by z-a; the relevant point here is that, if there is an integer solution, the value of z must satisfy Fermat’s equation. Then, because f(z) and z-a are both polynomials in z, the algorithm and corollaries apply, and the remainder must equal zero for f(z) to equal a perfect p-power, A^p, if the assumption of an integer solution is true. But, of course for Fermat’s equation, f(a) cannot equal zero. -- End of story!

I think these reviewers were so intent on trying to find a way to disprove FLT65, which they were convinced from the beginning could not be valid, that they were blinded to the fact that, if their ‘counterexamples’ were valid, they would actually have provided integer solutions for z^p – x^p = y^p, directly disproving FLT, and thereby also disproving Andrew Wiles’ proof.

So after fifty years, FLT65 still has not been refuted. Those who tried have failed, but only a few besides myself have accepted it as valid, and two of them have since passed away. Many of the mathematicians who have reviewed it believe it cannot be valid, and two even claimed to have refuted it, but their arguments were easily disproved. See the details in the links provided above. 

After fifty years, I would like to have closure; so anyone out there who believes the proof is faulty or incomplete is challenged to provide irrefutable mathematical proof that I can understand supporting that belief. If you can prove to me that FLT65 is wrong, I will acknowledge you proof and send you a check for $100.

Unlike Sir Andrew Wiles’ proof of FLT, which is hundreds of pages long, drawing on a very sophisticated knowledge and understanding of elliptic functions and modular algebra, FLT65 is a relatively simple proof relying only on basic mathematical principles. I believe that Pierre de Fermat will rest easier when FLT65 is recognized as valid, because it proves that he could have proved his famous theorem with mathematics available in 1637. If FLT65 is is at last recognized as correct, I, and poor Fermat will have closure.

       

UNDERSTANDING THE NATURE OF REALITY, WITH TWO CONTINUATIONS AND CONCLUSION

A NEW KEY TO UNDERSTANDING THE NATURE OF REALITY

©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’.

CONTINUATION #1

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 ﬧ_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 ﬧ_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, ﬧ₀ = ﬧ₁ = ﬧ₂ =  ⦰, 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: ﬧ₀ has no extent, no content, and no relevance to the structure of the universe, and both ﬧ₁ and ﬧ₂ have extent and relevance, but no content. Finally, geometry (dimensionality), pure mathematics, language, and logic are coherently integrated by equating ﬧ_n when n ≥ 3 with the logical condition called True, and  F 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.

CONTINUATION #2: QUANTUM CALCULUS

In the preceding Continuation, we defined the concept of a distinctions very broadly: the symbol ﬧ_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 ﬧ, 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. ﬧ ≡ﬧ    2.  ﬧﬧ ⇔ ﬧand 3.  ﬧ ﬧ ⇔    , 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.

THE QUANTUM CALCULUS

CONNECTING LANGUAGE, LOGIC & DIMENSIONAL MATHEMATICS

Conscious verbalization (English)	Symbols and Connectives of Logic	Algebraic Representation	Calculus of Distinctions Quantitative/Geometric Representation
Real, Existing, True	∃	X=A	ﬧ
Identical	A≡A	X≡A	ﬧ ≡ﬧ
Memory, Equivalence, Number	∀	f(t)=1,2,3….	ﬧﬧ ⇔ ﬧ
Appearance, Cancellation, Order	↔	F(t)=0	ﬧ ﬧ⇔   , or ⦰        *
Integer Variable	A	x = A = any whole number	A = ﬧ
Not A	∼A	x ≠ A	⦰
A or B	A ⋁ B	A≥X≥B	AB=ﬧ
A and B	A . B	A + B	ﬧ ﬧ ﬧ *
A implies B	A ⊃ B	A → B	AﬧB *

* 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 ﬧ 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:

S_nXⁿ+ S_nYⁿ = S_nZⁿ,

where S_n is the volumetric shape factor. Finally, dividing through by S_n, 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 ﬧ.  So, if n ≥ 3, the FLT equation translates to the CoD equation:

Xⁿ + Yⁿ = Zⁿ ⇒ ﬧ ﬧ ﬧ= ﬧ

Applying Basic Calculation Operation #2, we have: ﬧﬧ= ﬧ

And applying Basic CoD Operation #3, we have:  ⦰ =ﬧ, 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 ⇒⦰ + ﬧ = ﬧ⇒ ﬧ = ﬧ, 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.

CONCLUSION

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 ⇒ ﬧ ﬧ ﬧ= ﬧ, but by Basic CoD Operation #2,

2.)  ﬧﬧ ﬧ= ﬧﬧ ⇒ ﬧ, and by Basic CoD Operation #3,

3.) ﬧﬧ= ﬧ ⇒ ⦰ =ﬧ, 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!