The Mathematics of Symmetry Order
I. Zero as the Whole of All Numbers
Most of us expect there to be some direct relationship between mathematics and reality, but what single number in ordinary math symbolizes the everything of numbers? We all are accustomed to using words such as Universe, existence, or being, which can be meant to symbolize the whole of all that exists. Why then, if it is so easy to refer to the universe as a whole, why is it impossible for a number to represent the whole of all numbers? What is so different about the nature of mathematics which makes all numbers impossible to represent as a whole?
We might conclude that words such as Universe or being in fact have no meaningful application to nature, or perhaps our modern mathematics only partially represents reality. In the same way that the physical process of time could quite easily be secondary to a greater physical reality of timelessness, like a story in a book compared to the book itself, it is also possible that mathematics and even much of science describes a sub-system of the whole; a definitive reality within an infinite reality.
There is actually a way to combine together or sum all real numbers into a single whole number. If we sum all positive numbers with all negative numbers, then the total combination of all in question would sum to zero, as shown below.
(1 + (-1)) + (2 + (-2)) + (3 + (-3)) +... = 0 + 0 + 0 + ... = 0
Except mathematicians recognize there is a problem in the consistency of the result of such equations. In fact several different equations sum all real numbers yet each yields a different product:
0 + 1 + (2 + (-1)) + (3 + (-2)) + (4 + (-3)) + ... = 1 + 1 + 1 + ...
0 + (-1) + ((-2) + 1) + ((-3) + 2) + ((-4) + 3) + ... = (-1) + (-1) + (-1) + ...
So we conclude in math that the sum of all real numbers is undefined, which makes sense because if the sum of all numbers did indeed equal zero then we would be faced with the most complete of logical contradictions, since zero would simultaneously represent nothing and everything. Instead, zero represents nothing and there is no ultimate number that represents all numbers, and the logical consistency of math is preserved.
However, one rather significant issue has been overlooked in this matter. It is said that the sum of all real numbers is undefined, but logicians and mathematicians made a mistake when investigating the simple idea that all real numbers sum to zero. They failed to consider switching the value of zero away from nothing. They failed to consider the option of assigning zero the value of everything, a value equal to the combination and whole of all numbers.
In actuality we have in the past tested the hypothesis that all numbers might sum to zero, against a mathematical system where the value of zero is already pre-set to be nothing. In ordinary math, all values are relative to zero as nothing, so of course we would discover that all real numbers do not sum to zero. If it were not so, the logical consistency of mathematics would be destroyed. Yet we can as an alternative allow zero to transform into the sum and whole of all numbers, it just can't be done half way. As the saying goes, it's all or nothing. The proper test of zero as the sum of all numbers requires that we allow the value of zero to have a value greater than all other numbers. At first this seems nonsensical because we are in fact switching into an entirely different set of axioms.
If zero contains all other numbers, and becomes the largest value in a mathematical system, what then is the value of number one, or two? Which is greater, one or two, if zero is greater than both? If zero is the largest value, the only way there can be lesser values is if we remove some measure of value from the whole of zero. For example, suppose that we take away a (-1) from zero. What remains? Zero is suddenly no longer an absolute value containing all other numbers. Something has been removed from it. But what value does zero transform into to show that loss? The answer is simply that it becomes the value 1.
If we remove a negative one from zero the value of zero records that loss by
becoming a positive one. In other words, one contains all numbers, except (-1)
is removed. The number two is the sum of all numbers except (-2) is removed, and
so on, and so on.
Figure 1. The value of 2 can be drawn on a number line as shown below:
Just for the sake of clarity, switching to the negative, the number (-1) is a combination of all numbers except that a positive 1 is removed. In removing a positive two the whole shows that loss by becoming the number (-2), and so on, and so on. The larger the number, the more is removed, and thus larger numbers have ever smaller values. What makes this system most relevant and easily relatable to the SOAPS model presented, is that the content and subsequent value of numbers decreases as we count toward greater numerics.
All we are really doing is considering a system of values where positive and negative values combine rather than cancel. But because values on either side of zero in such a system decrease rather than increase, as we count into ever larger numerics, the numerical value diminishes. So the value of ever larger numbers move toward becoming infinitesimally small. In fact the values converge toward two points of infinitesimal value, an infinitely small positive value, and an infinitely small negative value.
In what I shall refer to as Symmetry Math, there are three extreme values, or three ultimate numbers. There is zero, the sum of all numbers, which I refer to as the number Omega. Then in the same way the number one in ordinary math bounds the infinite decimals that exist between zero and one, there is a value of positive infinity which I refer to as the number Proto, and a value of negative infinity referred to here as the number Eleat.
In symmetry math, zero and the entire spectrum of values are infinite and yet entirely definitive. Infinity in this system is not merely a series or a process. As we remove a part from the whole, we can only create other values which are themselves infinite and definitive as well. We still have a logically consistent system of values, but unlike our present finite system of values, in this system, all values are infinite, because every positive value contains all positive numbers, and every negative value contains all negative numbers. For example, a positive one contains all the positive numbers and also all the negative numbers except a (-1). The number one million contains all positive numbers, it merely contains fewer of the negative numbers than does the nearly whole positive one.
The absolute smallest positive value that is possible in this system is produced by removing all negative numbers from zero, by grouping separately the positive from the negative (the extreme of grouping order). In removing all the negative numbers all the positive numbers remain. This is the smallest possible value in symmetry math. Nothing more can be removed. One of the more elegant features of this system is that although the two smallest numbers are points of infinitesimal value, Proto and Eleat are themselves each half of the whole.
+∞ + (-∞) = ∞ or Ω or 0
Infinities in Cosmology
The idea of an infinite Universe was until recently often treated with skepticism. The nature of the infinite has long been an unresolved mystery in both physics and mathematics, although there are a few tolerated infinities, such as electrons and black holes. And there are infinite series equations which mathematicians consider to be definitive, because they express convergence.
However, any hope of ever discovering the Cosmos is finite vanished with the return of data from the Boomerang and Maxima balloon born telescopes , and more recently the Wilkinson microwave anisotropy probe  further verified that the geometry of deep space is flat, indicating profoundly that if we could observe galaxies at a common age the universe would extend infinitely in all directions without end. It follows logically, and most cosmologists agree, that if the geometry of the universe is flat, then the first moment of time, the alpha state, is necessarily also flat and infinitely extended.
We no longer need question whether the universe is infinite or not. Only now we have arrived at a question that seems less scientific or at least far more difficult to answer. How infinite is the Universe? Is existence bounded in any way? Evidence for an infinity of galaxies or space-time bubbles was not entirely unexpected, but what of the utter chaos of possibilities, all conceivable temporal universes and beyond, the majority completely unlike our own. Are there any identifiable boundaries to what exists?
I would suggest, based on this work, that the physical existence of all possible states may be the extent to which existence is radically infinite, satisfying an existential requirement of nature that all things exist. The SOAPS serving as a timeless foundation, limits the dimensions of temporality to a multiverse of space-time bubbles. The case for a mode of timelessness  is no less compelling than the case for a many-worlds universe, and without question only the profound nature of both positions have delayed their inclusion into science.
Identical to the SOAPS model, there are three identifiable extremes in this math, and most intriguing, the values of the smallest numbers, Proto and Eleat are themselves infinite, which I suggest represent in mathematical form the positive and the negative alpha states. The alpha state in our past is a positive singularity, a body that is spatially flat and infinite, yet it is the smallest value possible in nature. Internally alpha is smooth and uniform, due to being all positive, and from our perspective is in a constant state of inflation, yet because macrocosmically it contains merely half of the whole, it is the smallest value in our reference of values, a tiny point in our past, that expands only as negative values are added to it.
The process of transformation from the alpha state to zero is indeed similar to the creation and annihilation of two virtual particles, although we have no need of, and so eliminate the creation phase. The alpha state itself, the first moment of time, like all states, simply exists. As Richard Feynman remarked, �Time is a direction in space� and in fact the progression of time can be understood as a fourth dimension of spatial directions that due to backward causality (imbalance to balance) inevitably originate from the alpha state, and travel probabilistically through the body of all possible states, initially diverging into an expanding state space, but finally converging in state space toward the omega state, which is also in a constant state of inflation relative to our position in space-time.
To complete the virtual particle scenario, we place ourselves within the positive virtual particle, while the identical negative particle (anti-spacetime) integrates slowly and invisibly, expanding our positive volume and moving us ever nearer toward neutrality. This would explain why negatively charged electrons are point particles, since a negative density cannot exist spatially in our positive volume. The evolution and expansion of our universe is equally analogous to counting from the number Proto to the number Omega. The universe grows spatially because of an influx of negative density into a positive density, which creates a partially independent positive volume of space, which we call space-time, with density variations relative to the contrast gradient.
In ordinary math we count upward into an endless staircase of numbers, with no finality or boundary, and thus reality modeled by such a system has no ultimate or macrocosmic definition. In that math fundamentally counts things, there is naturally a number that represents nothing, yet no number represents everything. If we instead switch into a mathematical mode that is able to represent the greater infinite universe as a whole, then naturally we find that the system represents reality in an entirely different way. In symmetry math, infinity isn�t a never ending process, but rather the infinity of numbers is bounded by infinite extremes. Engulfing the finite, the entire symmetry mathematical plane is real, complete, and consequently quiescent and timeless. In symmetry math zero represents everything, and because the smallest values of this system still represent half of the whole, there is no such thing as nothing in this system. We haven�t merely reversed values, we have changed the very nature of our system of understanding.
In considering the new axioms of this system, we would not expect the values of the symmetry plane to be derived from an elementary first thing somehow emerging from nothing or an empty set. There is no axiom of nothing in this system from which we question the existence of the rest. Here, absolute zero is not a nothing but rather a whole form and not the absence of form. And all the finite parts come with the whole. A positive and negative infinity are the two fundamental numbers, the poles, from which all definite values and numbers assume form, like binary numbers that grow into virtual worlds. Without a nothing or a non-existence the values of symmetry math are assumed to exist as we would imagine possibilities to exist, permanently. In a philosophical study the two poles can be related to something and nothing, two singularities, the simplest of any two meanings, although this nothing is merely the negative of form or anti-form, which is itself form, just as anti-matter is matter.
In relation to order, in the same way that there are two distinct kinds of order in nature, we have found here two entirely different ways of seeing zero and all other numerical values. Symmetry math is as logically consistent and as valid as common math, of no use within the abstract world of individual things, yet immeasurably valuable in cosmology where a mathematical value for the universe as a whole is of critical importance in any attempt to understand for example, the implications of the many worlds theory, or how to conceptualize the realm of all possible states, or how to appreciate a geometrically flat universe.
Ordinary math is based upon a perspective derivative of grouping order, or thingness. This unique mathematical system is derived from symmetry order, its foundational axiom reflecting the innate singleness and wholeness of existence. As a perspective it doesn't see isolated or separate objects. Although its values are definitive, it represents physical form only as a single unified pattern.
This value system does not threaten the validity of common mathematics in any way. Each is built upon a perspective. Two apples will remain more apples than one. What this new mode does, is allow a radical shift of perspective, so that we can also see the universe as a symmetric undivided whole, where we know that for there to be a positive two apples (matter), there must be a negative two apples (anti-matter) set apart from the pattern that we observe.