Saturday, November 19, 2016


© Edward R. Close, November 19, 2016

Such a simple thought, and yet so profound! I want to begin this post by reiterating the most important point of my last post:
Pure mathematical thought is a true reflection of reality at its deepest level.
In my opinion, mainstream science has gone astray; confusing conceptual mathematical tools with pure, mathematical logic that accurately reflects the nature of reality. By missing this distinction, scientists have gone down the wrong path by applying infinitesimal calculus to quantum reality, where it doesn’t apply, resulting in paradoxes and illogical conclusions. In past posts we have seen how rectifying this mistake allows us to discover new science and answer questions and explain puzzles that are inexplicable in the current mainstream paradigm.
Accept, if you will, that the statement in bold italics above is literally true; that pure mathematical logic is not an invention of the human mind, but something very real that is there to be discovered by any intelligent life form. While the symbols expressing it might be different for different forms of intelligent life, the truths they reflect will always be the same. There is only one reality - and we are capable of knowing it.

If the facts of pure mathematics accurately reflect reality, then exploring the logical structure of mathematics will yield valid information about objective reality. Of course, we must be very careful to continually check our findings against hard data, because consciousness (mind) has more degrees of freedom than the dimensional domains of space and time. For the past forty or fifty years, theoretical physicists have constructed a number of elaborate theories (e.g. string theories) that are internally consistent, but cannot be checked against objective reality.

The reason mainstream science has gone astray is easily understood when we take a look at the underlying assumptions of Newton’s calculus and compare them with the more basic logic of the calculus of distinctions (CoD), and ask what calculus is appropriate for application to the real world. The physical universe as we experience it through the physical senses is quantized, meaning that mass, energy, space and time only occur in whole-number multiples of quantum units. But Newtonian calculus assumes that physical measurements are infinitely divisible. This assumption works very well on the everyday scale of measurement but fails at the quantum scale.

The Calculus of Distinctions is a more primitive form of calculation that applies to the whole spectrum of reality. The Calculus of Dimensional Distinctions (CoDD) developed for the new, more comprehensive paradigm (TDVP) and the Calculus of Infinitesimals (Newtonian calculus) are actually sub-sets of the CoD. The basic units of measurement in the CoDD are quantum units, defined as measurements of the smallest possible measures of our quantized reality, based on real data. Thus, by definition, they cannot be divided into smaller units, making the CoDD applicable at the quantum scale. We know that mass and energy are quantized, but what about space and time?
In his note to the 15th Edition of Relativity, Einstein said: 

Physical objects are not in space, but … are spatially extended. In this way the concept of “empty space” loses its meaning.

In our exploration of the mathematical structure of the CoDD, we find that space and time also prove to be quantized, and consistent with Einstein’s statement, the concept of ‘empty’ space-time has no meaning.


Albert Einstein also said:

I am convinced that purely mathematical construction enables us to find those concepts and those law-like connections between them that provide the key to the understanding of natural phenomena.

Einstein liked to use thought experiments, and I’d like you, the reader, to participate in a slightly different kind of thought experiment here. Consider the nature of human awareness: there is no question that our awareness extends beyond the physical brain located in the bony cage of the human skull. I know this is true, because I am simultaneously aware of my body from the hairs of my head, to the tips of my toes and fingers, and I am also aware of my existence in the room in which I sit and type these words. Visualize with me the sphere in which you exist, and of which you are aware. It doesn’t matter whether you think of it stopping at the boundaries of your skin, or at the walls of the room you are in, at the edge of some limited sphere like this planet and its atmosphere, or even the much greater sphere of awareness, all of the visible universe. Pick a physical object within your sphere of awareness. In this instant, that object and every other object within your sphere of awareness can be exactly located using only three numbers.

Most people are familiar with Cartesian coordinates. Several other types of mathematical coordinate systems for locating points in a finite field of awareness have been developed: Polar, cylindrical, spherical, curvilinear, skewed coordinates … etc. Mathematicians have devised these different coordinate systems for application to specific kinds of problems. The Cartesian coordinate system, with three orthogonal dimensions, and six directions of motion: up, down, right, left, front and behind, is the easiest to work with, but in any of these coordinate systems, the location of each and every physical object of which we are directly aware through our physical senses, can be exactly located with just three numbers defining its position relative to any reference frame. The geometry of this three-dimensional spatial field is defined by the axiomatic geometry of Euclid.

There are other types of geometries that have been developed for specific purposes, but they are all defined in reference to Euclidean geometry. Euclidean geometry is the most natural geometry of our physical perception, so we will visualize our thought experiment in Euclidean space. For Euclidean space of more than three dimensions, I prefer the term Euclidean Dimensionometry.

Mass, Energy and Consciousness in our Sphere of Awareness
Planck showed us that energy is quantized; Einstein showed us the equivalence of energy and mass in the simple but profound equation E = mc2. Through the general theory of relativity, Einstein also saw that space and time exist only as extensions of the substance of reality; and application of the CoDD to the conscious drawing of distinctions has shown us that all real phenomena are aspects of one thing, which manifests as combinations of mass, energy and consciousness, and that any conceptualized reality without all three is meaningless.

It is logical that in such a unified reality, there should be a basic unit of measurement to which all measures of reality can be related by mathematical equivalence, and to which all measures can be normalized. We call that unit the Triadic Rotational Unit of Equivalence, or ‘TRUE unit’ for brevity in written discussion. The TRUE unit is defined by the mass and volumetric equivalence of the free electron spinning away from an ionized Hydrogen atom. The volume of the spinning electron shrinks in size and its angular velocity increases rapidly until, before it shrinks to zero, it reaches the relativistic limit of light speed. Its volume at that point is the minimum quantum volume possible. The mass and volume of this truly minimum quantum unit, which we call the TRUE unit, is then the basis of all CoDD measurements.

The Existence of ‘Extra’ Dimensions
Like most people, you probably think that you are aware of only three dimensions. This idea is often heard expressed in statements like: “If there are more than three dimensions, why can’t I see them?” This misconception arises from imprecise, and for our purposes, improper uses of the word ‘dimension’. In common usage, the word dimension can mean a variety of things. For example, you may hear: “The exact dimensions of the room are ten feet by twelve feet, six inches, with an eight-foot ceiling” (meaning measurements); or: “The many social dimensions of the problem must also be considered” (meaning aspects); or, “Maybe UFOs come from another dimension” (meaning somewhere unknown).

We do experience dimensions in our sphere of awareness, but we cannot ‘see’ dimensions; we see objects that have measurable extent in our field of vision. In our thought experiment, we shall adhere to a very precise definition of the word dimension: A dimension is measurable in the units of a variable of extent only.

All finite forms existing in our sphere of awareness are made up of distinctions of three types that are naturally drawn in the fabric of reality by human beings: (1) distinctions of extent, (2) distinctions of content, and (3) distinctions of conscious intent or impact.

(1) The three dimensions of spatial extent are measured in multiples of units of like feet, meters, miles, and kilometers. But we are also aware of a fourth dimension with measurable extent, a dimension of duration, or extent in time.

(2) Distinctions of content are measurable in multiples of units like mass, energy and thought. Unlike distinctions of extent, distinctions of content are not dimensional. They do not have direct relationships to the measurable dimensions of spatial extent, because spatial extent alone does not determine the content of an object. For example, a cannon ball has a very different content than a balloon of the same size, and while the contents of thoughts are reflections of physical impressions, including images, sounds and other sensations, they take up no physical space. Electrical and chemical reactions in the brain may trigger thoughts containing mental images, sounds and sensations, but they are not identical with our awareness of thoughts and mental sensations.

(3) Distinctions of conscious intent or impact reflect a higher level of complexity than distinctions of extent and content. How is this higher level of complexity reflected in the logical structure of the CoDD? Are there, in fact, more than four dimensions existing in our field of awareness?  When we analyze reality in terms of the DoDD, we find that the answer is yes. There are dimensions of consciousness, i.e. extent, and within those dimensions, there are distinctions of content as thoughts.

Dimensional Domains
As we recognize the existence of an increasing number of dimensions, it is helpful to think in terms of dimensional domains. One of the most important invariant relationships of dimensional domains is: Each n-dimensional domain is embedded in an (n+1)-dimensional domain. A one-dimensional domain (a line), for example, is embedded in a two-dimensional domain, a plane, and a plane can be seen to be embedded in the volume of a three-dimensional domain. With increasing numbers of dimensions, each dimensional domain in our sphere of awareness becomes increasingly more subtle and complex. The inclusion of time as a fourth dimension complicates a timeless three-dimensional domain by introducing changing forms, motion and relative velocities. The inclusion of additional dimensions of time and consciousness introduces potentials and qualities unimaginable in the simple 4-D domain of space-time.

Now I want to put what we’ve learned from this sphere-of-awareness thought experiment into historical context and connect it with some of the concepts we have discussed in previous posts:

The CoD represents mathematical logic that is conceptually prior to its division into separate disciplines like geometry, number theory and algebraic representation. The thread of this primary and fundamental mathematical logic runs through the teachings of Plato, Diophantus and Pythagoras, the methods of Gauss, Fermat and Cantor, and has surfaced again more recently in Lie and Grassmann algebras where number theory, algebra and geometry have been at least partly reintegrated for application to quantum physics. The logical structure of the CoDD and the Newtonian calculus are subsets of the logical structure of the CoD.

When we analyze the sphere of awareness in terms of distinctions, we see that as long as we are drawing distinctions on the macro scale of our physical perceptions, Newton’s Calculus of Infinitesimals works perfectly well, but when we consider that physical objects are actually made up of stable combinations of much, much smaller objects: electrons, protons and neutrons, and that neutrons are made of combinations of quantized up-quarks and down-quarks, we must use the Calculus of Dimensional Distinctions with finite minimal quantum units to measure and describe them. Then, because the structure of pure mathematics is a reflection of the structure of reality, we can look at the mathematical structure of the Calculus of Dimensional Distinctions to help us understand the nature of quantized reality in our sphere of awareness.

The Basis of the TRUE Quantum Unit is Real Data
Using Large Hadron Collider (LHC) data, we defined the most basic quantum unit of measurement as the minimal equivalent normalized mass and energy of the electron, and by applying relativistic principles to the electron stripped from the Hydrogen atom in the process of ionization, we determined its minimal quantum volume and thus defined the true minimal quantum unit of calculation for the CoDD, and called it the TRUE quantum unit. Applying TRUE unit analysis to the stable elementary particles that make up our physical universe, we found that up-quarks contain 4 TRUE units of mass, and down-quarks contain 9 TRUE units of mass. We also found that protons consist of a total of 24 TRUE units, and neutrons contain 38 TRUE units.

Dimensional Extrapolation and Fermat’s Last Theorem
We can project our dimensional-domain awareness from one dimensional domain to the next with the mathematical procedure of Dimensional Extrapolation using a specific form of the Conveyance Equation: (X1)m + (X2)m + (X3)m +…+ (Xn)m = (Xn+1) (See THE BASIS OF TRUE UNIT ANALYSIS posted October 15, 2016) that happens to be the Diophantine form of the Pythagorean Theorem: (X1)2 + (X2)2 = (X3)2.

When Fermat’s Last Theorem was applied to the Conveyance Equations describing the combinations of the TRUE units of the elementary particles, and their integer solutions were found, we discovered that a third form of reality, which we called ‘gimmel’, not measurable as mass or energy, but contributing directly to the total angular momentum of any spinning structure, had to exist for electrons, protons, neutrons, and atoms to continue to exist as stable objects. Previous discussions, including one posted October 28, 2016, “MORE REVELATIONS ABOUT THE NATURE OF REALITY”, provide derivations of the TRUE unit and gimmel in more detail.

How are the CoD and CoDD Related to Conventional Mathematics?
In conventional geometry, a mathematical singularity is a point with zero extent, i.e., zero dimensions; a line is a one-dimensional concept consisting of an infinite number of points, potentially stretching to infinity in opposite directions; a plane is a two-dimensional domain; and a three-dimensional domain is a space. Physical objects on the macro scale on this planet at least, have weight and occupy space.

In a quantized reality, comprised of combinations of quanta that contain integer multiples of the quantum unit, a reality where space and time do not exist in the absence of matter and energy, the point, defined in the Newtonian Calculus as a mathematical singularity, with zero dimensions, simply does not exist. In a quantized reality, the nearest thing to a point would be a single quantum unit, which would be a three-dimensional domain the size of a free electron, and, conceptually, the line might be replaced by a string of electrons placed in a row. But electrons all carry the same electrical charge, and repel each other. So no such thing exists in a quantized reality. Similarly, two-dimensional planes are conceptual, not objectively real, because they have no volumetric extent capable of containing mass and/or energy. So the first level of our sphere of awareness is a three-dimensional domain containing objects composed of integer combinations of quantum units. Non-substantial geometric figures of less than three dimensions, like segments of straight lines and planes, as well as regular polygons and even circles, are only approximated in the real world of quanta and quantum combinations.

What else have we learned from TRUE analysis?
By applying TRUE analysis to the atomic elements of the Periodic Table, we found that electrons existing within the structure of stable atoms have at least 106 TRUE units of gimmel. This also led to the discovery that the elements most supportive to organic life have a volume of (108)3 TRUE units with many more units of gimmel than elements less supportive or necessary for life. We also found, as reported earlier in these posts, that the application of TRUE analysis in a nine-dimensional spin model yields answers to a number of questions that have perplexed physicists for decades, like why quarks combine in triples, the Cabibbo mixing angle, and other puzzles.

How do we know there are more than four dimensions, and how many are there?
Awareness of a three-dimensional domain is only possible from the vantage point of awareness of a larger dimensional domain, i.e., a domain of four dimensions or more. A four-dimensional domain exists in our sphere of awareness because, thanks to memory and imagination, we are aware of the passage of time. This is true because, while we experience only one quantum of time, at a time, we remember previous events and anticipate a logical succession of those events, thus becoming aware of a linear progression of time. Clearly, the fourth dimension, as time line, is quite different than the first three dimensions, with subtle meanings in our sphere of awareness not possessed by the first three, and it seems to have only one direction. The dimensions of the 4-D event domain are related to each other and mass and energy mathematically by the transformation equations of relativity.

The Conveyance Equation also yields quantum combination equations which play a central part, as demonstrated in previous posts and publications, in the determination of the mathematical properties of the different dimensional domains. By repeated applications of the process of Dimensional Extrapolation, we see that the Dimensionometric structure of the CoDD consists of sequentially embedded domains totaling nine finite dimensions embedded in a transfinite domain, finally embedded in an infinite substrate. It is also worth noting that the various string theories, and brane theories developed in an attempt to accommodate both relativity and quantum mechanics, require several dimensions beyond the three of space and one of time.

We have shown previously that, mathematically, after each triad of embedded dimensions, the next triad must include a new type of number representing the rotational projection into the next dimensional domain. In a quantized reality, the first three dimensions are characterized by integers. The new numbers derived by Dimensional Extrapolation for the 4th through 6th dimensional domains turn out to be the so-called “imaginary” numbers written ‘Ai’, where ‘A’ is an integer and ‘i’ equals the square root of minus one. After three Ai dimensions, the new numbers derived for the 7th through 9th dimensional domains turn out to be ‘complex’ numbers of the form A + Bi. Finally, after three A + Bi dimensions, any additional domains, of ten dimensions or more, are ‘hyper-complex’ and transfinite, if they exist, because all of the roots of unity after m = 9 are complex numbers. So, our sphere of awareness is potentially one of nine finite dimensions descending through a transfinite domain from an all-embracing infinitely continuous substrate. As human beings, however, with the limitations of a specific physical form equipped with very limited physical senses, we are only directly aware of three spatial dimensions, one quantum moment in time and three dimensions of consciousness. But the mathematical logic of the CoDD, strongly suggests that the three Ai dimensions are dimensions of time.

Someone asked: "Are the three dimensions of time ‘past, present and future’?" No. The present is the quantum moment of one’s immediate experience, and that moment along with the past and future define the timeline of one’s individual experience.


What is the nature of the three dimensions of time? Two or more non-congruent timelines, like those of multiple conscious entities, suggest two-dimensional time, and the potential awareness of two or more timelines, suggests three-dimensional time (see the invariant relationship of embedded domains mentioned above under the heading “Dimensional Domains”).

How do I envision the three dimensions of consciousness?
I see the three dimensions of consciousness as analogous to the three dimensions of space, providing a framework for thoughts and images in what we call mind. The three dimensions of space and the three dimensions of consciousness are most likely causally linked to the three dimensions of time. Just like the concept of empty space is meaningless without mass and energy, the concept of time without events is also meaningless. Thus I see reality as a unified structure of space, time and consciousness that would be completely meaningless if any one of the three were absent. This implies that the structure of our sphere of awareness actually consists of seven dimensions, not just three and the structure of the CoDD implies that, in addition to space-time, there are at least two more dimensions, and a postulated transfinite realm that are potentially knowable. Further exploration of the logical mathematical structure of the CoDD may give us some additional clues about what the other domains will be like, if we can find ways to expand our awareness to directly include them.

Why do elementary particles, planets, solar systems, and galaxies rotate and spin? After years studying the mathematical structure of the CoD as reflective of the structure of reality, and applying the CoDD and TRUE analysis to the compound structures of physical reality, I have become increasingly convinced that the full answer to this question is to be found in the nature of the other two dimensions of time, the transfinite domain, and the infinite substrate of space, time and consciousness. But the best answer that I can offer now, is that it may be that our limited view of reality from the reference frame of space-time, where we have heretofore accepted the illusion that we have a uniquely stationary position as conscious observers at rest in an otherwise dynamic, ever-changing reality, most likely distorts our view of the ultimate nature of reality significantly. The illusion that we are stationary, even though we are actually whizzing around with dizzy angular velocities, spinning with the surface of this planet at up to 1000 mph, rotating around the sun still much faster, and wheeling at a yet even faster rate though the cosmos with our galaxy, puts us at the center of a nexus of opposing forces of expansion and contraction.

How are mass, energy and gimmel related to consciousness? So far, I have found no conclusion other than that mass, energy and gimmel must be finite manifestations of the logical structure of the infinite substrate, which I speculate may be the ever-existing essence of consciousness, the source of all things, without which nothing would exist.

It is also my opinion that we exist in a reality that has no absolute beginning or end, only the illusion of beginnings and endings, arising from the apparent ever-changing panorama available from our limited frame of reference. And, if we are ever able to see the whole scope of reality from infinity to infinity, then, for us, I believe the illusion of separation will disappear.

Finally, it is my hope that the reader who has followed me this far can see the potential within TDVP for the scientific expansion of human knowledge to include provable principles of paranormal phenomena, currently considered by mainstream science to be beyond the realm of real science. I have some ideas about this that I hope to discuss in future posts.


  1. Good on you, Ed, and the reason, since my mystical-initiation of 1980, I call myself a 'Cosmic' - The Whole is part of me and I am part of It, life after life, ad infinitum!

  2. Hi Edward, you said, "A dimension is measurable in the units of a variable of extent only.", and by extent, you appear to mean spatial extent. You then go on to say that consciousness and time consist of 3 dimensions each. Yet these are not obvious dimensions of spatial extent.

    Let's consider our familiar time that scientists often call a dimension. They can justify this only because they have created units for it and draw graphs using those units, and so it is converted to a spatial extent. But unlike space, we cannot see time. We can experience it as a subjective duration, but even that varies with the nature of events during that duration. If anything, duration is a dimension of consciousness. I suggest that time is no more a dimension than is colour, yet you would likely say that colour does not have extent and so cannot be called a dimension. Time as a construct of our own making should not underpin a theory of existence.

    You also allow that consciousness has dimensions. Would one of these be the dimension of colour? Then why stop there? Surely consciousness would have many more dimension than three. To begin with, all qualia would be dimensions of consciousness, as well as the emotions, and any other subjective feeling that has a range of expression. You might consider the possibility that there are potentially infinite dimensions of consciousness, not all expressed in the same being.

    My point is that limiting the use of dimensions to concepts that have spatial extent seems arbitrary. Even you ignore that limitation when it suits you by allowing time and consciousness to be dimensions. Is this a case of a nice mathematical construct or theory driving a description of reality?


  3. Hi William,
    First, thank you for reading my posts. Your comments are a litle puzzling to me however, if you have read my last post carefully; but it is possible, of course, that I didn't make it clear. There are 3 kinds of distinctions and 3 kinds of extent in the CoDD, spatial extent is just one of the 3 kinds of extent. And I must say that I don't share your opinion that we justify treating time as a dimension only because we have created units for it and draw graphs using those units. The real units for time are the Ai units, where A is an integer and i is the square root of minus one, unfortunately called "imaginary" numbers because they cannot be located in "real" space (spatial extent). The name stuck, even though it is abundantly clear now that they are very real. When imaginary numbers crop up in the mathematics describing real phenomena initially defined over the feld of real numbers in one, two or three dimensions, another orthogonal dimension is indicated. The new dimension, however is not one of spatial extent. In TDVP, and in this post, perhaps too briefly, I have defined 'dimension' very carefully, in order to avoid the kind of broad usage of the word that is often seen in less technically logical writings. As I tried to explain in defining the three kinds of distinctions, distinctions of content have no direct relation to extent. See my cannonball/balloon analogy in the post. The description of spheres of awareness in TDVP are built on the rules of parsimony and economy. There are only 3 kinds of quantifiable extent. Qualia, e.g., have no direct relationship to extent; they are contents of consciousness. Calling anything that can be observed or measured in any way was part of the basis of David Hilbert's program for a complete axomaticization of mathematics. Gӧdel's incompleteness theorems burst Hilbert's dream of encompassing all mathematics in an axiomatic system defining all variables as dimensions of Hilbert space. Dimensions defined mathematically are measurable in variables of extent only.

  4. Oh, Ed, thank goodness for my only having been blessed with the layman's basic simplicity of Y= X Squared plus One - Roll on Homo Sapiens Cosmos, hopefully by my next life! Amun!

  5. Hi Edward, I'm not surprised if I misunderstood what you are saying, since your theory is quite complex. It may be impossible to really understand it without following the math. Your theory seems based on mathematics, and you then extrapolate to experience. This seems to be the way physics has progressed over the last 100 years or so.

    Your math tells you that there are these dimensions, and so you have to give them meaning. To quote you, "I see the three dimensions of consciousness as analogous to the three dimensions of space, providing a framework for thoughts and images in what we call mind. The three dimensions of space and the three dimensions of consciousness are most likely causally linked to the three dimensions of time. Just like the concept of empty space is meaningless without mass and energy, the concept of time without events is also meaningless. Thus I see reality as a unified structure of space, time and consciousness that would be completely meaningless if any one of the three were absent." The last sentence seems reasonable, but you seem unable to explain in detail how the theory relates to experience.

    The view of time as a subjective human construct is not just my opinion. You implied it in your Transcendental Physics book where you proposed a continuous process of creation by the perceptual process. I believe that you took the position that all that could possibly exist is represented in primary consciousness. So when we have an experience, it is the movement of our awareness through a sequence of existing states. So where is the rational time dimension in that scenario?

    But now you go so far as to say that there are three dimensions of time, and that is even harder to relate to our experience. The same can be said for your three dimensions of consciousness. Because your math calls for only three dimensions, you cannot say that qualia are dimensions of consciousness. You must define them as contents of consciousness not related to extent. Let's take the concept of a constant sound or light. These certainly have dimensions of intensity which can vary from low to high, i.e., they have extent as well as content.

    Maybe you could explain how your three dimensions of time and of consciousness relate to our experience. What could one measure to validate these theoretical constructs?

    1. Thanks again William. My commenmts here will not be in any spefic order related to your comments, but hopefully I'll be able to touch on all or most of your comments/questions. Measurements can be nominal or ordinal. The way we've defined dimensionality, extent has to be measurable nominally, measurement of content, on the other hand can be nomiminal or ordinal. With content, like light, low, medium, high, etc. intensity measurement is ordinal. So, by definition, light intensity is not dimensional. There are several levels and kinds of consciousness. Distinctions drawn in/by Primary C are different than distinctions drawn by/in individualized connsciousness. The awareness of a conscious individual consists of a series of distinctions, experienced from that individual's relatively limited framework that forms a line through time. Two non-coincident timelines form a plane. awareness of that time 'plane' requires awareness of the third dimension of time. I must go now, but maybe this is a start.

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