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 (0.1.2.3) 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.

The Sunya Machina - Part I

The concept Sunya (zero) is a tantric machine. It sits at the root of arithmetic, a number that denotes paradox itself. Zero is the signature and symbol of something that does not exist and yet it could have or would have. Having zero mangoes means having no mangoes at all. And yet, despite the lack of mangoes, you do have something – a signifier of your lack.

In this way zero splits the continous flux of kala (time) into two channels – that which is, and that which isn't. It has now become customary to denote 'that which is' with the number '1' and that which isn't with '0'. Therefore, zero is a seed (and generator) – it necessitates the birth of 1. However, wherever there is difference (is/isn't), there must be a repetition. One is zero as seen from a Universe with one less dimension.

As soon as '1' takes birth, it usurps the not-being of '0' as a kind of 'being somewhere else' (in some other Universe) and sees it's own holographic reflection/repetition, giving rise to a duality instead of a union – the number 2.

Two is to be understood therefore as '1 more of that which we called 1 before' or 'both is and isn't are now here'. Even then, these three concepts above cannot by themselves describe all of existence – the superset of which they are merely subsets. That Superset, the Totality of Existence would consist of the following entities (and in brackets we have the symbols associated with each concept):

Totality of Existence (Kala) = That which isn't (0) + That which is (1) + That which is and isn't both(2) + The remaining cosmos (3).

The number 3 therefore denotes the final doubt of the Descriptor, the sign that even though he may have attempted to describe in exact terms universal phenomenon, there is something that will always be out of his grasp, because the Descriptor is a mere subset of that which is being described. Not only does the Sunya Machine obviate the next 3 terms, it generates all the numbers till infinity without the help of symbols like 4, 5, 6, 7, 8, 9. How does it do that?

The number line of integers from 0,1,2,3....and onwards to infinity is a kind of cellular automaton. Euclid proved long ago in the fundamental theorem of arithmetic that every natural number greater than 1 can be written as a unique product of prime numbers. The original section of the number line (0,1,2,3) is a conceptual machine that 'generates' the rest of the numbers as an output. Merely two unique symbols (0 and 1) are sufficient to represent all the numbers in binary, and number systems with a base greater than 2 are justified by their practical circumstance.

Abhijit Bhattacharjee's “polar place value number system” is a system based on his conjecture (the Bhattacharjee Conjecture) that any kind of number, even fractional number – can be expressed as additions or subtractions of powers of three. He has provided an algorithm in Pascal to calculate these factors for any given number. If proven rigorously by a mathematician who knows the jargon that is valid in academia, this finding could stand as a more profound observation than Euclid's fundamental theorem. Euclid showed that all numbers can be produced by prime numbers, but Bhattacharjee is trying to show that all numbers can be produced by the numbers upto 3. This is a significant reduction/compression in the algorithmic complexity of the number system. Marvin Minsky has written in an email to Abhijit that more mathematicians should take note of his work. Meagre doubts that one might have will be clarified only by a careful study of his Pascal algorithm.