It has become crystal clear to me that there is non-trivial underlying order in all systems that have sufficient complexity. Sufficient complicity means the system must have dimension greater than or equal to the real number system in 4 dimensions. This realization came to me while studying the works of Frank P. Ramsey (a Mathematician, Philosopher and Economist) . In particular, his use of the concept of the pigeonhole principle (relating to the real numbers) in reverse. One can read the details of the pigeonhole principle for further clarity. In my own words I will state what it says and the underlying application of this fact. If one has n cubbyholes and n+1 objects to fit into those cubbyhole's then at least one cubbyhole has more than one object contained in it. That is at least two objects will be contained in one of the cubbyholes. If the pigeonhole principle is viewed in reverse we get the following concept. If one starts with an undivided infinitely long continuous structure then partitions said structure into a finite number of partitions at least one of the newly created partitions has one object. This statement is the crux of the argument. This statement belies the fact that the universe has an underlying order. I will note here that Ramsey did not state what that order is. I will, through example, glimpse some of the underlying order of the natural numbers as proof that the above statement is true.

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Ex. 1 Somewhere around 1964 a Mathematician S. Ulam was attending a convention and engaged himself in doodling on paper. His doodle was a construct of the real numbers arranged in a square that spiraled outward.He then went on to note the prime numbers located within the construct. What he noted, to his surprise was that the prime numbers in his construct were lining up along the diagonals of the square of natural numbers. This intrigued Ulam and he went on to construct a much larger square noting that there was a great degree of order in a construct that he felt should just contain random dots for all the real numbers. So, he discovered order in the natural numbers that no one had previously thought existed. This picture of prime number order appeared on the cover of Scientific American Magazine in 1964.

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Ex .2 About 30 years later a follower of Ulam noted his work and must have mused that maybe he could clarify and make the picture that Ulam discovered more clear. He was able to do this by using an Archimedian spiral. Mr. Sack's picture was indeed much more clear than that of Ulam. I will note here that another unique spiral exists (a logarithmic spiral) it is possible that this spiral will make the picture even more clear. I leave this project for study by another person. As an aside, it may be that the inverse of the logarithmic spiral may be a good candidate for study. Note:There are other spirals than the ones I have just mentioned.

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Ex. 3 A much older example from Mathematical history comes to mind. They are called the triangular numbers. These interesting numbers actually form triangles when placed on paper as dots. Note: The first triangular number is called T sub one and it can be found by substituting one in the equation (n)(n+1)/2 this gives a single dot. If one substitutes two into the equation we obtain the real number 3 and its easy to see that triad forms a triangle also. We can go on with the substitutions but suffice to say these substitutions all form larger and larger triangles for as large an n as one wishes to use. The interesting fact here is that this act of dividing the reals into two classes one of "triangles" and the other of non triangles. If one thinks of the pigeonhole principle here (in reverse) we get one filled set with objects (triangles) and another set that is empty of triangles. This is Ramsey's idea in a nutshell. I will note here that the square numbers are also another example of this concept. Square numbers are the sum of triangular numbers.

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Ex. 4 Modern day Fractals (discovered in the late 1970's) are still another example of an order arising from an iterated equation. They have very interesting properties one of which is scale similarity.

To sum things up, the pigeonhole principle in reverse is a very effective way to view a very complex system by using an important subset of the entire set and looking for order in the subset. That then indicates something about the order in the larger set. Thus, the universe is a sufficiently complex object and it must have underlying order also. Finally, this conclusion that the idea that (for sufficiently complex systems), randomness can not exist.