More than 22 centuries ago, Aristosthenes gave the world his delivery on the prime numbers. Many have wondered about the nature of prime numbers, and many deliveries have been written about prime numbers since. In my book "In Search of a Cyclops" (published in 2000 as "The Proof of Nothing," with both versions available at pentapublishing.com), I take a different look at prime numbers. I use their sequencing as the explanatory basis for all numbers, and through the prime numbers I was able to discover a special matrix of all numbers. It is a delivery that does not give the number 3 the most important position. But the intriguing and controversial matrix does explain the numerous occurrences of threes we find all around us.
First of all, the matrix is indeed very much like that familiar six-number based matrix in which prime numbers are most often visualized ‹ containing all numbers and repeating the specific prime number positions in rows of six. Naturally, this context of six number groupings immediately show-cases why three would indeed be quite important. However, the six-number based matrix discussed in this article is quite peculiar, unlike the standard delivery ‹ even when it is only different in one particular position. Where any subsequent deliveries of this matrix deliver six numbers with each having contents, the very first delivery is 0, 1, 2, 3, 4, 5 (and not the Œusual¹ delivery of 1, 2, 3, 4, 5, 6). The first delivery of the matrix therefore has six numbers with only five of them having actual contents, and one number having no contents. As said, all subsequent deliveries have six numbers ‹ and those all come with contents.
Many of us like to avoid the subject matter of number zero ‹ especially in a scientific realm ‹ for it has often nothing but confusion to add. But in this case of a matrix of 0, 1, 2, 3, 4, 5, the 0 shows the very important empty position. The empty position occurs only once, and is subsequently filled up in each following round of this matrix, for instance, with 6 ‹ in 6, 7, 8, 9, 10, 11. Any number divisible by 6 in any 6-pack is taking up this specific position in the matrix.
To understand the significance of the first line, and following this matrix further in light of the prime numbers, it is important to ask ‹ once again ‹ if the number 1 is really not a prime number. Many may think we have gone down this road before, and that we should not re-address this issue. Yet knowing the nature of prime numbers is vital to understanding the matrix of six positions. The first aspect to remember is that prime numbers are numbers that cannot be divided by more than two numbers; they are divisible two times only as in just by themselves and 1. The fundamental context for prime numbers is therefore the lack of even a third option to multiply a number and then getting to a prime number. This simple set up reveals why number 1 is special.
Although it is true that 1 can be divided only by itself, it is also the square of itself, and it therefore doesn't follow the rule of a prime because a prime number can never be the square of any number. Yet this peculiar aspect helps direct our attention to the true nature of prime numbers. Where both the prime numbers and the number 1 have two parts available for multiplication (the number itself and 1), the number 1 differs from them in that not one, but both parts of multiplication can be used again and again without changing the result (1 x 1 leads to the same result as 1 x 1 x 1 x 1). The prime numbers, however, deliver only an identical result when one of the two numbers is multiplied, and not the other. When multiplying with 1 several times over, for instance, as in 1 x 1 x 1 x 7, this always results in an outcome of the prime number, an unchanged outcome no matter the number of times we multiply with 1. Yet when multiplying the other part, for instance in 1 x 7 x 7 x 7, any further multiplication with 7 already leads to the answer being no longer the same result. What we consider to be prime numbers are numbers that all have one part (and one part only) that can be used over and over again without changing the outcome. Yet having this infinitive aspect existing twice (or ad infinitum) in number 1 explains the duality that Euclid used in the Fundamental Theorem of Arithmetic to declare 1 a non-prime number. What 1 is really only guilty of, though, is containing the fundamental character of a prime number twice. It is a fascinating number ‹ with 1 being both the number itself, and the square of itself. A doubled prime phenomenon is captured within this one number.
Not just number 1, but the entire matrix has a dual character. That shouldn¹t come as a surprise to anyone, since Aristotle already declared our universe to be one of a dual nature. Yet Aristotle¹s delivery of an absence of a third way is incomplete. The first indications that reality is not just duality-based, of course, are the plentiful deliveries of threes on this threes.com website ‹ threes appear to be everywhere. But that could easily be explained by the six-number basis, with pairs of threes (2 X 3 = 6) still fitting the Aristotelian way. Aristotle showed us a context that was easy to grasp, and gave us a perspective that was easy to use in our daily life. But it made capturing reality from an overall perspective more difficult. Only when Rotterdam-born intuitionistic mathematician L.E.J. Brouwer (1881-1966) provided a way out, by showing that a third way can exist, only then did a full vision become available. Still, many important figures on our earth have shown that up to this day it is hard to give up on the pure Aristotelian view when stating that "if you are not with me, you are against me."
What Brouwer showed was that a third way is indeed possible under the condition that it has nothing to do with the proposed duality. An easy example is the gender neutral bacteria placed next to the male/female division of the world. Though many would agree with Aristotle that there is only duality ‹ if you are not male, you are female! ‹ a bacteria is not captured by the male/female division of our reality at all. A similar quick overall view is, for instance, the realm of answers with yes¹ in opposition to no.¹ But if we claim these two answers to be all there is, then a full delivery has been blocked from view. The overall delivery requires an important position for no answer¹ as well. A third position is possible if it is based on any kind of not-acknowledging the duality. The absence of a third way Aristotle proposed is incorrect, yet it only needs altering slightly to become correct: absence is the third way.
Going back to the matrix of 0, 1, 2, 3, 4, 5 and following Aristotle¹s creed on duality, we can split these six number right down the middle in two sets of three: [ 0, 1, 2 ] and [ 3, 4, 5 ]. The special condition of the empty position is captured only in the first set of three. And Aristotle¹s delivery exists then right there in this first set, for here we find the two numbers with contents of 1 in opposition to 2. As most of us will agree, the empty position is not really there, which holds especially true as long as we are engaged in believing that either just 1 or just 2 holds the truth; so we basically end up with just two positions with claims of being seen as paramount. Let¹s translate these two numbers in words. Though always a dangerous step in science (one can easily become the butt of jokes), most likely not many people will see any problem viewing 1 as unity and 2 as the symbol for duality. If there were ever two numbers totally in conflict with each other, then these are the two numbers. If we believe in unity then duality is the evil¹ number that needs to be put in a minor second position, but if we insist all is based on diversity, then unity is unreal and cannot exist except as a man-made condition ‹ an abstract. Either way, the empty position is declared to be unimportant, and that is ultimately done incorrectly. Only all three positions, 0, 1, and 2, deliver us a basic set of overall conflicting positions: empty, unity, and diversity. This full set of three numbers [ 0, 1, 2 ] supports both Aristotle¹s view fully, while also explaining Brouwer¹s intuitionistic delivery at the same time.
If three numbers can deliver the basics, why then should we see six numbers as the matrix instead? The matrix of six numbers captures the full integrity of both unity and duality in mutually enforcing ways, with using the empty spot to 'outplay' the other number. A duel automatically follows when both numbers 'try to overtake' that empty spot for their own gain. Where 1 can be seen as unity, the number 5 is subsequently used as the number representing the whole, as the concept that enforces the idea of unity. Confronted with duality and diversity, number 1 is by itself not enough to do the trick of capturing all. Next to that, number 1 can be seen as static, immobile, unrealistic, so the number 5 as the fully encompassing number comes to the rescue. Just like the synergistic whole being more than the parts, number 5 at the end of the six-pack of this matrix is more than the sum of the previous members. Synergy is that additional aspect of the whole set that now exists, though nothing in specific was added. As an example of synergy, a bicycle is a lot more than the identical number of separate bike parts all laid out on a floor. And like five fingers to a hand, the thumb may represent 1, but it is number 5 that represents the entire dexterity set. Singularity exists in specifics, but requires the also singular overall position that encompasses the separate parts.
Well, doesn¹t that prove duality¹s point? If number 1 cannot fully express the whole, and 5 must come to the rescue, we have ended up with 2 numbers expressing the whole! Should we then not see each number as a specific part, and as such duality and diversity are then truly the basic grounds of our universe. And if that is the case, singularity cannot exist in our universe, and only fools would claim in the name of science that singularity exists.
Too bad the number zero spoils the perfect end game here too. With an escape hatch for 1 created by zero, duality is not the full realm of reality either, simply because duality cannot claim the empty spot all the time. The singular overall picture of 5 is powerful enough to capture duality, but duality can claim nothing was added, and 4 is all there is. Truthfully though, the empty position cannot be captured by either 1 or 2; it can be taken in by either one, but only for as long as it is guarded and enforced.
Letting go of guarding any powerful position, that position of power becomes either empty (and unimportant) or it will simply become enforced by guards of the opposing camp. So, only the six numbers 0, 1, 2, 3, 4, 5 together deliver the entire scheme. Those who read my philosophical article on threes.com know these numbers point to a very peculiar and very familiar construct of positions (see the Pyramid and Three). For those interested in the full prime number delivery, I must refer to pentapublishing.com.
The basic numbers to capture our reality is then 5, while the basic number of positions is 6. The six positions contain 1/ the empty spot, 2/ the singularity that is found with both 1 and 5, and 3/ the duality that is captured by 2 and 4. Truly, the only odd number in this set is 3. An independent three really does not exist; it should be considered the freebie of the matrix. If three is found to exist, it exists only as part of a series, such as the earth being the third rock from the sun. If we find an apparent three, something special must be going on. An 'explanation' why three is so fascinating can be found looking at the first three numbers of the matrix again; adding up these numbers [ 0, 1, 2 ] results in an outcome of three as well, creating an illusory moment of perfection. In reality, no condition exists where only three equal parts belong to the same condition.
That is, except when the empty set is part of the group.
As a last interesting point, people who are familiar with the prime numbers know about the two versions in which they are most often portrayed. Prime numbers are placed often either in a six number pack or a 30-number grouping. It is peculiar that these two different sets of numbers capture the prime numbers so well. An explanation why these two sets are most appropriate has not been given until now. The matrix of 0, 1, 2, 3, 4, 5 can explain the importance of both versions. With a six-number based matrix that starts out with one of the six numbers not having any contents, the multiplication of numbers with contents is five times six. With trying to make prime numbers appear in almost neatly conditioned positions, we find that the sequence of six numbers and the sequence of 5 x 6 numbers are the best ways to do so.