Thursday, August 27, 2015

Zeta Zeros and the Changing Nature of Number (1)

In conventional terms, we think of number in an absolute manner as possessing an independent quantitative identity that remains unambiguous in every context.

This is especially the case in respect to cardinal numbers, where each number e.g. "2", has a definite fixed meaning in this sense..

This is also the case with respect to the ordinal notion of number, where each ordinal number is interpreted with respect to the last unit of a number group.

So 1st is - by definition - the last unit of a group of 1 member. 2nd is then defined as the last unit of a group of 2, 3rd as the last unit of a group of 3 and so on.

In this way, the ordinal notion of number is in effect reduced in a cardinal manner.

However there is a crucial difference as between the notion of number (defined as a specific number with respect to a given dimension) and the corresponding notion (where number refers directly to the dimension which now generally applies to all specific numbers).

In the first case number has an independent meaning. So when we define numbers in a 1-dimensional manner (as lying on the number line), each number e.g. 3 is given an absolute independent identity.

However when we probe into the dimensional notion of number, we are required to accept a corresponding interdependent notion of number (where each dimension is related to the others in an organised manner).

Thus to move from the notion of a 1-dimensional representation of an object (i.e. the line) to a 2-dimensional representation (e.g. a square) the 2nd dimension must be clearly related to the 1st.
So if the 1st dimension represents length, then the second (drawn at right angles will represent width).
So they clearly are not independent of each other but related in a definite manner as interdependent.
And then if we proceed further to 3-dimensional representation the 3rd dimension (the height) must again be related in definite manner (as interdependent with the other 2 dimensions).

Therefore though the base notion of number (as within a given dimension) is independent in a quantitative manner, the corresponding dimensional notion is interdependent - as the relationship of each of its dimensions - in a strictly qualitative manner.

Thus rather than being absolute, the true notion of number is strictly of a relative nature, with aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.

For in quantitative terms a number is defined as the sum of its unit parts.

So 3 (for example) = 1 + 1 + 1.

So here the units are all homogeneous (literally without qualitative distinction).

However the very notion of ordinal implies that 1st, 2nd and 3rd can be thereby distinguished in a qualitative manner.

However by defining 1st, 2nd and 3rd as the last units of a group of 1, 2 and 3 members respectively
1st + 2nd + 3rd thereby can be expressed as 1 + 1  + 1 (which reduces these ordinal notions to cardinal definition).

What this means in effect is that no choice is left with respect to the dimension chosen. For example with respect to 3 dimensions, when we identify 3rd with the last dimension of 3, then this means that the other two dimensions must have already been chosen. So for example if the 1st dimension is identified with length and the second dimension with width, then the last dimension thereby relates to the depth. If however we had identified 2nd - not with the last unit of 2 but - as the 2nd of 3 dimensions, then if the 1st dimension had been chosen as the length, the 2nd dimension would not have been fixed in meaning, but could have been chosen as either the width or height in this case.

Therefore the requirement of identifying the nth dimension in any context as the last dimension of n, in effect reduces ordinal notions in cardinal terms (where each dimension in effect is seen as clearly separate from the other dimensions).

Thus, when we remove this restriction of equating each ordinal number with the last member of the corresponding cardinal group, then a new relative notion of each ordinal number emerges.

For example instead of defining 1st as the first - which is also the last - of a group of 1, we could define it as the 1st of a group of 2, 3, 4, .....n members.

Therefore in this new context the meaning of 1st is now of a merely arbitrary nature, with an unlimited number of relative interpretations.

Indeed we can easily recognise this relative nature of ordinal numbers (without perhaps equal recognition of its mathematical significance!

So in a 1 horse race, coming in 1st would not be much significance. However if it comes in 1st in a race with -say - 40 horses, then - relatively - this is a much greater achievement. Therefore as the number of the cardinal group increases, the relative importance of any earlier ordinal member likewise increases. And this process is ultimately without limit.

 Now, remarkably there is a simple mathematical way of giving expression to all these relative interpretations of ordinal numbers, which in effect amounts to the dimensional notion of fractions.

We can for example easily imagine a circular cake that is divided into 5 equal pieces.
The fractions 1/5, 2/5, 3/5, 4/5 and 5/5 then represent the fraction of the whole cake represented by 1, 2, 3, 4 and 5 slices respectively. And in the final case where we have 5 part slices with respect to the total of 5 parts this represents 1 unit (now representing the whole cake).

However even here the true situation is extremely subtle as we move from part to whole notions which strictly entails the relationship of quantitative and qualitative aspects.

However we represent this in (reduced) Type 1 terms, where all fractions are expressed with respect to the dimensional power of 1 i.e. (1/5)1, (2/5)1, (3/5)1, (4/5)and (5/5)1.

However we can equally give a Type 2 meaning to these fractions, where now in an inverse manner, they represents dimensional powers with respect to a default base number of 1.

So in Type 2 terms we have 11/5, 12/5, 13/5, 14/5 and 15/5.  These in fact represent the 5 roots of 1 that give rise to a circular - rather than linear - number system.

They now acquire a fascinating qualitative type meaning where,

11/5 represents the 1st (in the context of 5), 12/5, the 2nd (in the context of 5), 13/5, the 3rd (in the context of 5), 14/5, the 4th (in the context of 5) and 15/5 the 5th (in the context of 5),

Now of course the last here i.e. the 5th (in the context of 5) is always 1. This corresponds to the fact that the default root of 1 = 0. We can for our purposes refer to this as the trivial root.

So to obtain the t roots of 1, we can set 1 = st , i.e. 1 –  st  = 0.

Now the default absolute root is represented as – s = 0. Therefore dividing – st  = 0 by  – s = 0, we thereby obtain  the remaining  – 1 roots.

The resulting equation for the non-trivial roots is given as:

1  +  s1  +  s2 + x3  + .... +  st – 1 = 0.

This is what I refer to as the Zeta 2 function, which complements the well-known Zeta 1 (i.e. Riemann) zeta function. 

Whereas the latter is related to the hidden holistic expression of the cardinal primes, the latter is related to the corresponding hidden holistic expression of the ordinal nature of each prime.

So if t is prime, then 1  +  s1  +  s+  s3 + .... +  st – 1 = 0 expresses the non trivial zeros for that prime.

For example if t = 5, then 1  +  s1    s2  +  s3  s4  = 0 expresses indirectly in quantitative terms, 1st, 2nd, 3rd and 4th (in the context of 5 members).

So whereas the Zeta 2 function starts from the premise that the cardinal natural numbers express unique combinations of the primes, the Zeta 1 function starts from the complementary premise that each prime consist of a unique combination of ordinal natural numbers!

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