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Symmetry Mathematics
New Axioms that define the Universe as a Whole

What is the largest number you can think of, no wait, what is the largest number of all? What is the total sum of all numbers? Of course the answer is that there isn't an answer to this question. But let's put it another way. What is the greatest universe of all ? What if we imagine all things that exist? Can we at least find a single concept, a simple word, that includes all things combined together into one single whole universe? Is there such a word? Sure, this is easy. The word everything does that. Also there are words such as Universe, or existence, or being, which can be meant to symbolize everything that exists.

What about math? How many numbers in mathematics symbolize an everything in the number world? Is there some place on the real number plane which symbolizes the sum or the whole of all numbers? Interestingly, the answer to this question is no. As you know, there is always a next greater number. There is just something different about the nature of the system of mathematics which makes it impossible to represent all numbers at some point within the number system itself.

We could use the term positive infinity to refer to all the positive numbers combined together, but such a term would not actually represent a completed sum or whole. Since there is always a next greater number in this group there cannot be a single definite value. Positive infinity is more like a never ending process; a series of numbers, rather than a number. The same is true of the infinity of negative numbers. Like the positive side, there isn't a completed unified whole. 

But what if we combine together all the positive numbers with all the negative numbers? Isn't there an answer to that equation. At first it seems like if we try to sum all numbers into a single ultimate number; all the rational and the irrational, the even and odd, the total combination of all would be zero, as shown below. Wouldn't that be strange if the sum of all numbers equaled zero. We could then say everything equals zero, couldn't we. And that really wouldn't make sense, because the meaning of zero is very related to the word nothing. 

(1 + (-1)) + (2 + (-2)) + (3 + (-3)) +... = 0 + 0 + 0 + ... = 0

This equation makes it seem like zero is the sum total of all real numbers. There is always a negative value for every positive value, as shown above with integers. However, there is a problem. It is possible to sum all numbers several different ways, and the sum does not always have the same answer. Several equations sum all real numbers yet each yields a different product. The two equations below add up all integers but as you can see, they have different sums:

0 + 1 + (2 + (-1)) + (3 + (-2)) + (4 + (-3)) + ...  =  1+ 1 + 1 + ...

next:

0 + (- 1) + ((-2) + 1) + ((-3) + 2) + ((-4) + 3) + ...  =  (-1) + (-1) + (-1) + ... 

These two equations and the first that equals zero each include all integers yet we find three different solutions to the same equation, and consequently it is said in mathematics that the sum of all real numbers is undefined. Which really kind of makes sense. Otherwise, zero would be a mathematical nothing and an everything simultaneously. Instead zero represents nothing and there is no ultimate number that represents all numbers. 

Zero cannot represent both nothing and everything in the same mathematical system of values, and as long as we remember that, we can discover a mathematical system, very similar to ordinary math, and yet very different, because in this different system, zero represents everything, and there is no number to represent nothing.

It is said that the sum of all real numbers is undefined but logicians and mathematicians made a mistake in formulating the rules concerning zero. We tested the hypothesis that all numbers might sum to zero, using a mathematical system where the value of zero is pre-sets to be nothing. In ordinary math, all values are relative to zero as nothing, so of course we discovered that all real numbers do not sum to zero. If it were not so, the logical consistency of math would be destroyed.

No wonder no one has followed through, to consider zero as the genuine sum of all numbers. If zero were considered appropriately, even for a moment, to be the summation of all numbers, its value would have to be acknowledged as a number greater than all other numbers. Do you see what I am saying? Its a bit radical. If we sum all numbers instead of cancel all numbers, we alter the entire value system, and suddenly you have what appears at first to be nonsensical. If zero is the greatest value; the sum of all numbers, what then is the value of number one, or two? Which is greater, one or two, if zero is greater than both?

How do we resolve this? Is it just nonsense? Or are we at the threshold of discovering something important? Naturally in order to find out we must explore new terrain. However, keep in mind, that ordinary math functions in relation to the definitive world of things that we observe each day. It is a valid system. And yet that system cannot describe the universe as a whole, because in counting things we count upward into an endless abyss of numbers. If we wish to understand and describe the universe with a mathematical system that respects the universe as a whole, then we have to see the world in an entirely different way. Presently we see the world as if everything is more than nothing. What follows is a way of seeing the world as if all that we know is less than everything. In one system nothing is a foundational axiom. In the other it has no place or meaning.

(1 + (-1)) + (2 + (-2)) + (3 + (-3)) +... = 0 + 0 + 0 + ... = 0

We begin by looking at the simplest most straightforward way of summing all numbers shown above. For a moment we will say the correct sum is zero. So we will now switch the value of zero away from nothing and make zero the largest value in the mathematical system. Zero no longer equals nothing. Zero now equals everything, and contains every number within it. Every positive and every negative number on the real number plane sums to zero, making it the largest of all numbers. We have left ordinary math but we have landed somewhere else.

If zero becomes the sum and the whole of all numbers, then 1 is the sum of all numbers, except (-1). In other words, like zero, the number one contains all numbers, except that a (-1) is missing. Switching now to the negative, the number (-2) is a combination of all numbers except that a positive 2 is missing, which would otherwise create the balance of zero. In removing 2 the whole shows that loss by becoming the number (-2). The number (-2) is otherwise whole.

It feels odd at first but simply keep in mind that we are no longer counting finite things, so we are not saying here that two things are less than one thing, but rather, one is less than the whole of all numbers contained in zero. In symmetry math we treat zero as a unified whole, a completed infinity, and only when we remove a part from the whole, do we create other numbers which are infinite and complete as well. 

2|equals the set of all real numbers except (-2) or (-1)+(-1). Thus the symmetry value of 2 can be drawn on a number line as shown below:

We are not merely reversing the general value system of mathematics, we have changed the very nature of the system. In essence, there isn't any number in this system that represents nothing. There is no basic duality of something/nothing like that which exists in ordinary mathematics. An empty set in this system is recognized as the ultimate combination of all sets. And naturally, the values on either side of zero are less than the whole set that is zero. 

In the same way that there are two distinct forms of order in nature, there also exists two entirely different ways of seeing numbers, or in other words, there is a whole other value system in mathematics every bit as valid as the one we presently use, which I will refer to as Symmetry Mathematics

Our present mathematical system is based upon a perspective derivative of density order, or thingness. Surprisingly, there is also a unique mathematical system based upon symmetry order, or wholeness, with a dramatic switch and modification to numerical values. Using symmetry math we can envision both the physical and mathematical universe as a singular complete whole. I hope that my overall work reveals how solid and far reaching the concept of symmetry order is in cosmology, physics and mathematics.

One shouldn't assume this unique value system affects our normal value system. Each is built upon a perspective. Two apples are still more apples than one. We can still divide up and see the world from a finite or definite perspective, in which case the infinite becomes indefinite, but now we can also see the world as an undivided whole, where all apples are part of a unified singular universe. In symmetry math we fully realize that for there to be a positive two apples (matter), there must be a negative two apples (anti-matter) removed from the whole. And so the two positive apples are less than the whole of the four apples combined.

Next page:  Extreme Values of Symmetry Math
 
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Copyright © 1999, 2000 by Devin Harris