Thursday, May 25, 2017

Number and Development (10)

I will start this entry by contrasting the respective meanings in Type 1 and Type 2 terms of the fractions 1/3, 2/3 and 3/3 respectively.

In Type 1 terms these would be given as (1/3)1, (2/3)1 and (3/3)1 respectively.

So from the standard quantitative (analytic)  perspective (1/3)1 represents 1 of 3 (equal) parts i.e. one third;
(2/3)1 represents 2 of 3 (equal) parts i.e. two thirds.
(3/3)1 represents 3 of 3 (equal) parts i.e. 1 as a whole unit,  

However in Type 2 term, these fractions would be given as 11/3, 12/3 and 13/3 respectively.

Now in standard quantitative terms, these represent the 3 roots of 1 i.e. .5 + .866i, .5 .866i and 1 respectively.

However these also have an important qualitative (holistic) interpretation. And in dynamic interactive terms, when the Type 1 interpretation relates to the quantitative (analytic) aspect, then the corresponding Type 2 interpretation relates to the corresponding qualitative (holistic) aspect.

And the qualitative (holistic) interpretation of  1/3, 2/3 and 3/3 (i.e. 11/3, 12/3 and 13/3) would be expressed as 1st of 3 (related) dimensional units, 2nd of 3 (related) dimensional units and 3rd of 3 (related) dimensional units, respectively.

Of course, reference frames continually switch with respect to experience. So if we now start with the Type 2 aspect defined in the standard analytic manner (as the 3 quantitative roots of 1), then in Type 1 terms 1/3, 2/3 and 3/3 now take on a complementary holistic qualitative meaning as the 1st of 3 (related) base units, the 2nd of 3 (related) base units and the 3rd of 3 related base units respectively.

Now the key significance of a prime from the qualitative (holistic) perspective is that each of its ordinal members (with the exception of the last) is always uniquely defined.

In other words, when we omit consideration of the last unit (i.e. given in general terms as the nth of n units), the indirect numerical designation of all other ordinal positions for each prime, by definition, cannot be replicated with respect to any other prime.

So again from the quantitative (analytic) perspective, the primes collectively are unique as the fundamental factors or "building blocks" of the natural numbers.

However, from the qualitative (holistic) perspective, each individual prime group is unique in that all its natural number ordinal members (except the last) -indirectly represented in turn by the various roots of 1 (except the last) - are unique for that prime group.

So from this perspective, 5 (representing a prime group of individual members) is unique in that its 1st, 2nd, 3rd and 4th members - indirectly represented in turn in a qualitative manner by the 1st, 2nd 3rd and 4th roots cannot - by definition, be replicated with respect to the ordinal members of any other prime group.

It was at this time that I gave intense consideration as to the precise significance of the "trivial" last root i.e. 1 (which exists for every prime group).

Then it slowly dawned on me that it was the interpretation with respect to this root that naturally occurs in the standard ordinal interpretation of number (where ordinal notions are reduced in a cardinal manner).

So the number system starts with 1 (which or course is not prime). Then when we move on to consideration of 2, in standard analytic terms the 1st unit is unambiguously fixed as 1 with 2nd unit now in likewise manner fixed with the last remaining unit of 2.

Then when we move on to consideration of 3 the 1st and 2nd units have already been unambiguously fixed with the 1st two units, so that the 3rd unit is now likewise unambiguously fixed with the last unit (of 3).

We started with the quantitative definition of 3 (as cardinal number) = 1 + 1 + 1.

If we now define this in ordinal terms, 3 = 1st + 2nd + 3rd units.

However because in conventional mathematical terms, each ordinal position is unambiguously fixed in an absolute manner with the last unit of its corresponding cardinal number group (= 1), from this perspective,  

1st + 2nd + 3rd = 1 + 1 + 1 = 3.

So, therefore from the quantitative (analytic) perspective, a perfect correspondence appears to exist as between cardinal and ordinal notions!

However from the corresponding qualitative (holistic) perspective, all ordinal positions are merely relative and can be interchanged with each other.

And as these are indirectly given in quantitative as the corresponding roots of 1, from this perspective,

1st + 2nd + 3rd =   .5 + .866i .5 .866i + 1   = 0.

In other words, strictly speaking from the holistic qualitative perspective, the pure interdependence of all ordinal positions has no cardinal meaning!

So therefore the important point to grasp is that in standard quantitative (analytic) terms, each new ordinal unit is unambiguously fixed with the last unit of the number group in question. 

However the key point regarding the qualitative (holistic) interpretation of number is that ordinal positions can be undercharged with each other.

Thus in a group of 3 for example what is 1st from one relative perspective can equally be 2nd and 3rd from two other equally valid perspectives. Likewise what is 2nd from the first perspective, can be equally 1st and 3rd from the other perspectives, and finally what is 3rd from the first can equally be 1st and 2nd from the other perspectives.

So the holistic appreciation of ordinal positions implies the interdependence of each individual member with each other member of the respective group.

The analytic appreciation implies by contrast the independence of each individual member, whereby the ordinal position is unambiguously fixed with this member.

So as we have seen from this latter perspective, 1st is unambiguously identified with the last member (of a group of 1), 2nd with the last member (of a group of 2), 3rd with the last member (of a group of 3) and so on.

It was at this stage that I suddenly saw how a striking complementarity in fact existed as between all this work on the holistic nature of ordinal numbers and the famed Riemann zeros. 

In fact I could see now that were in fact two sides, as it were, of the same coin.

This insight arose from the attempt to isolate the truly unique holistic solutions indirectly implied in general terms by the t roots of 1 (except the default root of 1).

So the t roots of 1 are obtained from the equation, 

xt – 1 = 0, or alternatively as better suits our purposes 1 xt = 0.

Thus to eliminate the default root (where x = 1), we divide by 1 – x, to obtain

1 + x1 + x2 + x3 + ... + xt – 1  = 0.

So I started to refer to this as the Zeta 2 in contrast to the complementary type Riemann zeta function (i.e Zeta 1) where

1– s + 2– s + 3– s + 4– s + ....  = 0.

So the famed non-trivial zeta zeros represent the solutions for s to this equation.

But note the complementarity as between both functions! 

Whereas the former (Zeta 2) is of of a finite nature that be extended without limit, the latter (Zeta 1) is infinite in nature.

Likewise whereas the sequence of natural numbers appear as dimensional powers (with respect to the Zeta 2), they appear as a sequence of base numbers (with respect to the Zeta 1).

Also, whereas the unknown to be solved from the equation is a base number (with respect to the Zeta 2), it is a dimensional power (with respect to the Zeta 2).

The realisation of the complementary nature of these two functions was then to prove invaluable in "unearthing" the true significance of the Riemann (Zeta 1) zeros.      

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