Tuesday, July 30, 2019
MATHEMATICAL LOGIC REFLECTS THE STRUCTURE OF REALITY
The theme of this presentation is that real mathematics is not a human invention. The language, and notation, the ways in which mathematics is described, may be human inventions, but the mathematical patterns of logical order observed as cause and effect, are not. They exist in nature. The universe is obviously dynamic, and the changes we observe and experience in the universe proceed in a well-ordered manner, and have apparently done so for billions of years; if that were not the case, there would be no such thing as science, nor even a universe. Our very existence depends on the continued orderly cycles of our natural environment, and that continued progression of orderly cycles depends upon stability at the very most basic level. That basic stability starts with the structure of the proton, the most stable compound structure in the universe. The breaking down, or radioactive decay observed in every other compound sub-atomic or atomic structure, has never been observed to happen to a proton. No one has seen a proton decay. If protons do decay, particle physicists estimate the half-life of that decay would be longer than the estimated big-bang age of the universe.
With the addition of an electron, balancing the positive charge of the proton, the remarkable stability of the proton is passed on to the hydrogen atom, and this simplest of the atoms becomes the most abundant element in the universe. Gottfried Wilhelm Leibniz declared that the question: “Why is there something rather than nothing?” is the first question science should seek to answer. To rephrase that seminal question, seeing that the proton is the basis of atomic stability, we must ask: Why are protons so stable? The answer turns out to be quite simple: Quarks, the quantum constituents that comprise protons, are not particles as physicists assume; they are energy vortices, spinning in three, six and nine dimensions, and protons are remarkably stability because two up-quarks and one down-quark are not just stuck together like tinker toys to form a proton, they merge like drops of water when they combine. The truth of this will be demonstrated in this presentation.
The main purpose of this presentation is to provide the rational foundation for, and the basic development of a primary quantum calculus capable of comprehensively representing the logical structure of reality. I will approach this ambitious task in three steps: 1) by recounting a brief overview of the history of the development of rational thought during the last 2,500 years or so, for the purpose of identifying the mathematical logic of the innate structure of reality; 2) by focusing on the most important ideas, concepts and theorems that point the way to a mathematics of reality that will re-unify mathematics, symbolic logic and the natural sciences; and 3) by describing some of the insights gained from the application of the integrative quantum calculus. The key ideas of the focus in steps #1 and #2 will include Gӧdel’s incompleteness theorem, G. Spencer Brown’s Laws of Form, and the principles of relativity and quantum physics.
The historical separation of the study of the innate logical structure of reality into separate intellectual disciplines like philosophy, number theory, algebra, geometry, quantitative analysis, the infinitesimal calculus, physics, chemistry, psychology, medicine, linguistics, anthropology, etc., etc. …,was necessary. It took a number of very dedicated, serious thinkers a few hundred years to construct the conceptual bases and specialized languages for each of these disciplines. Not even polymaths like Leibniz and Newton lived long enough to re-integrate the various disciplines into a consistent epistemology, and turn to the task of developing a reasonably complete ontology, i.e., an understanding of the true nature of reality.
Specialization, while necessary at first, quickly became a barrier to any comprehensive understanding of the holistic nature of reality. Today, specialists in one field can barely communicate with specialists in another field. Each discipline has its own language. For example, when I switched from the study of pure mathematics and theoretical physics to engineering in the middle of my academic career, to escape the ivory tower of academia, I was amazed to find that engineers had re-invented the wheel, so to speak, by developing their own languages, with special terminology and notations for basic parameters and mathematical procedures that had already been more efficiently defined in the natural sciences and pure mathematics. Specialization is modern science’s Tower of Babel.
As it turns out, the separation of disciplines, especially the separation of physics, mathematics and logic, into functionally independent disciplines, is a mistake that has led to the conflation of non-existent abstractions with reality, and logical consistency with truth. G. Spencer Brown partially rectified some of the confusion caused by this artificial separation of disciplines, and re-connected logic and mathematics, by introducing complex values analogous to complex number variables (expressed algebraically as a + bi) into the algebra of logic in his calculus of indications. But he fell short of re-connecting primary calculus with physical reality, even though he remarked on the similarity of the forms produced by the introduction of imaginary numbers into logic to the laminated onion-like structure of atoms. (Laws of Form p. 105) He also alluded to the role of consciousness in the development of the calculus of indications, but did not attempt to represent consciousness as a variable in the equations of the calculus, as we have done.
Brown re-connected mathematics and logic with a wonderfully functional interpretation of the calculus of indications for logic, in Laws of Form, but then observed that determinations of the truth or falsity of statements made in the calculus could not be related to existence or non-existence without risking the creation of logical paradox. He concluded that appealing to existence was, after all, not necessary in order to solve problems within the calculus. In addition, Brown concluded that while consistency and completeness could be demonstrated within the calculus, proof could only be established by appealing to a more comprehensive logic existing outside the calculus. This conclusion is consistent with Tarski’s theorem, and suggestive of Gӧdel’s incompleteness theorem. The connections between these theorems in pure theoretical mathematics and the natural laws of physical, mental and spiritual reality will be made clearer as we develop the calculus of distinctions, the quantum mathematics of reality.
To start with, we need to spell out some definitions in detail, so the reader does not have to speculate about any of the terms used, or supply definitions of his or her own, that may or may not coincide with the meaning intended.
Calculus: A calculus is a system of logic incorporating calculation. There is not just one calculus, several can be identified, associated with varying types of a priori assumptions.
Calculation: A calculation is a process that transforms symbolic representations of patterns of conceptual and/or existential distinctions from one form to another, different, but equivalent form. For example, 1 + 2 = 3 is an application of the fundamental mathematical operation of addition that transforms one symbolic representation, 1 + 2, to a different, but equivalent form symbolized by the integer 3.
Distinction: In the primary calculus of distinctions, a distinction is defined as anything that can be distinguished from its surroundings in any definable or describable way.
In the Calculus of Dimensional Distinctions (CoDD), A is equivalent to B, if and only if, they simplify to the same basic value of the calculus.
A complex CoDD expression can be simplified by substitution of an equivalent, simpler expression. Examples will be provided later in this discussion.
The replacement of one expression by another equivalent expression.
Existing versus Conceptual Distinctions
In this discussion, because the calculus is anchored in reality, I am able to equate the term ‘existing’ with true and ‘real’, as opposed to purely conceptual, signifying a mental image or construction that has no actual counterpart in the real world. By ‘real’, I mean either something existing physically, or something having measurable impact on physical reality, or both.
Consciousness: Consciousness is primary. That means that it cannot be defined in terms of anything else, and the correct answer to the oft-asked question “What is consciousness?” is: Consciousness is the a priori existential reality through which all things are perceived. It is that which gives rise to awareness, without which nothing could be said to exist.
Dimension: It is necessary to define the term ‘dimension’ very precisely for use in the calculus of distinctions, because currently, the term is used in a variety of ways in various fields of human thought, and thus can mean different things to different people. In the calculus of distinctions, it is limited to mean extension, nothing more, and nothing les: A dimension is measurable in units of a variable of extent in space, time or consciousness. Other types of variables are not dimensions, and should not be confused as such.
TRUE: In the physical universe, CoDD variables of space and time are quantized in multiples of the Triadic Rotational Unit of Equivalence (TRUE), the quantum equivalence unit based on the physical characteristics of the electron. Dimensional variables of extent in consciousness related to mental images, are reflections and contractions or expansions of the dimensional variables of extent in the physical universe.
Domain: Like dimension, the term ‘domain’ means a variety of things in common parlance, so we must also have a precise definition for the term domain in the calculus of distinctions. A domain is a region defined in terms of variables of extent. An n-dimensional domain is a region defined in terms of n dimensions. Space-time, e.g., comprises a 4-dimensional domain symbolized by 3S-1t, a volume is a 3-D domain, an area is a 2-D domain, a line is a 1-D domain, and point is a zero-D domain, known as a mathematical singularity in conventional mathematics.
Variables of Content, Extent and Intent
Content, i.e., energy and mass as energy equivalence, is quantized, and because of this, space and time, or space-time extent, which, as Einstein pointed out, is not fundamental because it has no existence of its own, must be considered to be quantized as well. Intent implies consciousness, and must be included because it impacts reality. The necessity of the quantization of space-time becomes very clear with the application of the calculus to phenomena measured in multiples of the quantum equivalence unit. Referring to a region of space-time smaller than the volume of the quantum equivalence unit has no meaning in quantized reality, because it leads to the confusion of particles and waves and a variety of problems related to the Pauli exclusion principle and what is known in contemporary quantum physics as ‘the measurement problem.’
This brings up some important points regarding the calculus of distinctions. When dealing with distinctions of extent, the calculus becomes the Calculus of Dimensional Distinctions (CoDD). In this calculus, a mathematical singularity, zero and nothing should not to be confused with each other. For the calculus to be properly related to the real world, the difference between these three concepts must be clearly defined.
The Concepts of a mathematical singularity, Zero and Nothingness
With the application of the calculus of distinctions to the real world, we must be clear that the state of non-distinction is not to be confused with the concepts of singularity, zero and/or nothingness. When counting objects, like apples or oranges, zero denotes the absence of a specific kind of object, not a state of nothingness. Similarly, the absence of a distinction does not indicate a state of nothingness, it just means that there are no distinctions in the field of awareness. The numerical value of zero is also sometimes confused with the concept of nothing. Zero can be assigned to a point on a line between other points representing positive and negative values. But, in quantized reality, dimensionless points (mathematical singularities), one-dimensional lines and 2-imensional planes do not exist; they are conceptual mental constructs only, giving rise to the confusion that something can be nothing, and that something might arise from nothing.
Distinguishing Conceptual Distinctions from Existential Distinctions
In today’s mainstream science, applications of macro-scale mathematical tools to quantum phenomena have led to serious confusions of abstract conceptualizations with existential distinctions, i.e., real phenomena. The inappropriate application of Newtonian calculus is perhaps the most obvious mistake of this sort, but there are even more basic instances of this kind of confusion. For example, the concepts of mathematical singularity (the point), line and area are often imagined to be structures that exist in the real world, but, in fact, they are purely conceptual and cannot exist in the quantized reality of the physical universe. Distinctions of less than three dimensions have no volume, and thus no capacity for substantial content. Replacing the concept of a point with one volumetric quantum equivalence unit (the TRUE) in the calculus of distinctions, and thereby accurately reflecting physical reality, allows us to clarify the difference between conceptual mental images and distinctions that exist in the real world.
Hilbert Space and CoDD n-Dimensional Domains
In previous publications, I have used the term Hilbert space, named after David Hilbert, the mathematician who extended Descartes’ 3-D space in which objects are located by three numerical coordinates, to n-dimensional space to accommodate non-Euclidean geometries. There are, however, significant differences between the classical concept of Hilbert space and the n-dimensional domains of the calculus of distinctions. In the CoDD, dimensions are limited to aspects of reality that can be measured in integral variables of extent. In the CoDD, Mass, energy, density, and other quantifiable features of objects existing in an n-dimensional domain, are measured using variables of content and/or combinations of variables of content, extent and intent, and should not be confused with dimensions, which are measured in variables of extent. To avoid this kind of confusion in the CoDD, dimensions are strictly limited to realities measured in variables of extent, and those variables are defined in terms of quantum equivalence units. In addition, in the CoDD domain, roughly analogous to Hilbert space, the dimensionless point is replaced by the multi-dimensional volume of the Triadic Rotational Unit of Equivalence.
A Triad of Variables are needed to Describe Existential Distinctions
Mass, energy, density, and other features of content in the quantized world, are measured using variables of content and/or combinations of variables of content, extent and intent. An existential object is only fully described by including variables of extent, content and intent or informational meaning associated with measurable impact. The three values possible in the CoDD are: 1) True, or real; 2) False, or non-existent; and 3) imaginary (analogous to a + bi), or extra-dimensional. With these concepts and definitions, as outlined above in mind, we can develop a useful functional notation for the calculus of dimensional distinctions.
(To be Continued)