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Monday, September 22, 2008
The Sunya Machina - Part II
One does not need more than the number 3 to represent and manipulate the entirety of mathematics. One possible way that the existence of the numbers ( can be justified is described in our introductory note. Another way is to hold up three fingers of any hand, and now ask yourself - "How many things do I see?" The correct answer is not 3, it is 4 - three fingers and the hand.

Even zero needs at least one symbol to be represented as zero, and the numbers (0,1,2,3) are four in number. When we examine the number 4, however, one realizes that from this point onwards, the river of numbers is a mere continuation of the idea that is self-contained in (0,1,2,3). From 4 begins a mechanical repetition unto infinity, and especially tiresome is the arbitrary selection of ten digits as the looping point of this decimal alphabet. As the Ulam Spiral shows, perhaps the distribution of prime numbers is not as mysterious as it seems. Lets us stop at 3 then, and observe the universe that is arranged before us. There is a tail that emerges from Om's behind. Om is the shape of Ganesha, lord of numbers - Om is therefore a heuristic diagram of the Sunya machina.

Instead of mundane increments, we now have patterns that emerge from the pure duality of difference, repetition and conceptual enclosure. To understand why primes emerge and and where they are prone to occur, it now becomes necessary to understand division and multiplication within the philosophical framework of the Sunya Machina. Since I am no more than a thumb-twiddling vedic crystalpunk, you might want to know what all this means from the mouth of a real mathematician like Dorian Goldfeld, who writes in an essay called Beyond the Last Theorem:

But there is another way of looking at a circle. Consider a clock, an antique twenty-four-hour model with a single hand that swings around the dial once a day, pointing first to “high midnight,” then to 1:00 A.M. and so on. The clock has no idea what day it is; as far as it is concerned,3:05 P.M. today is indistinguishable from 3:05 P.M. tomorrow, or next week or on any date you might imagine. In mathematical terms each point on the circular dial sets up an equivalence class comprising all the moments in the past, present and future at which the hand points precisely to that point. Schematically, the clock dial takes a time line marked with equally spaced integers (the midnight points),twists it into a shape like a Slinky, and then collapses the Slinky into a circle.

What the circle does for the one-dimensional flow of time, it can also do for the infinite one-dimensional space of the real number line. In that case the circle becomes a set of equivalence classes of pure numbers. Formally, for any number x, the equivalence class is defined to be the set of all numbers of the form x + nc, in which c is the circumference of the circle and n is any positive or negative whole number.

At first glance the two descriptions of a circle—one in terms of algebra, the other in terms of equivalence classes—could hardly be more different. But they are indeed equivalent, linked by the Pythagorean theorem and some elementary geometry.


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