Learn How to Mathematically Not Count Non-stuff

 

From silence the music emerges until it finishes in silence.  Play the rests!  They are as much the music as any of the notes.  --Chicago musicians

 

It's all in the edit.  -- Kami

Both the initial and ultimate reality or meaning of numbers, including the values of various power levels, is discernable both in and from backgrounds of silences or spaces or not-ness-es from which "ones" substantiate or differentiate.  The concept may be considered a differentiation of negative and positive definitions as applicable to things.  However, in a certain way, a positive and, simultaneously, a negative description of the same thing may be construed as functionally the same definition.

[a] salutation--this one at the beginning of Bhᾱskara's book on mathematics, the Vija-Gaṇita:  'I revere,' he says, 'the unapparent primary matter ... for it is the sole element of all which is apparent ... the arithmetic of known quantity ... is founded on that of unknown quantity; and ... questions to be solved can hardly be understood by any, and not at all by such as have dull apprehensions, without the application of unknown quantity ...'  Robert Kaplan in The Nothing That Is, p. 60.

A Non-Basis of Numbers as in the Powers' Pattern

About One-ness from which patterns emerge:

For the power of 0, none of every one of all emptiness-es differentiate into substantive ones.  "Beats" of both stuff and non-stuff are theoretical and a premise.  The theory is that ones exist and are substantial compared to the backgrounds, also "beats", that mark and define the non-existence of something.  That a meaning of substance and not-substance exists is both an initial and the ultimate power given to numbers.  Assuming ones exist and may be differentiated from a background (even if not differentiated) is the foundation of numbers and of all of the powers of numbers.  (See also On the Existence of Numbers.)

[This puts a new twist on Albert Einstein's "Not everything that can be counted counts, and not everything that counts can be counted."]

Numbers Found

The numbers of any power of n are founded upon one-things at an initial rate of:  n "one-things" per n+1 "beats"--where beats are either empty backgrounds or substantial one-things.   Silent beats of not-ness define or "package" or differentiate sets of n substantiated ones by the not-ness of specific "packages of things". 

Remember there is a fundamental (initial or final) silent, not-any-thing-at-all, emptiness that adds the "beat" of space that defines "not everything-that-is" rather than specifically "not such and such".  The total number of "foundational ones" per x "beats" according to the power of n would be (x - 1 [or not -1])/(n+1) and, if n > 1, then multiply by (n-1)/n.  The total number of "beats" that are silent, empty, backgrounds for a power of n is (x - 1 [or not -1])/(n+1) which, if n > 1, is multiplied by 1/n and finally add [or not] 1.

For the power of 1, half of x total "beats" of definition--not including the over-all not-ness--are substantiated ones and countable.  Once we count ones, we establish a set of symbols that are integer numbers.

Notice that numbers are treated primarily as sets of sum-able/countable things and secondarily as symbols.  Existential knowledge grants symbols "accountability" and informs treatments of whole categories of number, including the powers.  

The information of One-ness / Not-ness is destined to inform Two-ness which will inform Three-ness, etc. 

For the power of n this means that x total "beats" of definitions may be treated as:  primarily sets of size-n of "one-things" with every (n+1)th silence defining the sets.  (The one-things can be named integers by counting them.)  The integers then can be parsed generally into thing-sets of size (n-1) "one-type-things".  Remove every nth integer to delineate these new sets-things.  (The not removed sub-sets-of-counted-things are accounted for by summations that name each summation in integer terms.)  Next, the newly accounted, named numbers are treated generally as set-things of (n-2) of "one-type-things".  Remove every (n-1)th value. (The not-removed sub-set-values can also be accounted for in integer terms by summations.)  Etc. until a proposed set of "one-things" reaches the limit that is n - n.  The last naming of "one-things" gives x^n values.

Numbers exist, IN SEQUENCES . . .

Notice that number sequences between powers sometimes differ invisibly judging by numerical values alone.  Considering non-number data or definitions of what number is not, sequences are not really the same sequences (with the exception of the "x totality").

For the power of 0

For the power of 1

For the power of 2:

The power of 3:

x = 0 or non-definition totality; y = 1's or the definition of something-s with "anti-definitions" also implied; z = integers or "the count" of 1's; so then a zth case or zth value of other sequences is:  z in equations or algorithms that have integer coefficients, integer power variables, and integer terms.  The powers' pattern is a handsome result. 

The reality of definition as both "something" and "not something" (or something's background) and thus the reality of an ineluctably sequential accounting-symbolism are always present in number.  So power levels will have meanings even as their integer series [z] all seem the same.

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Last modified: 12/16/05