Monday, March 4, 2013

The Number 2

We have repeatedly seen that there are two distinct aspects to number that are quantitative (analytic) and qualitative (holistic) with respect to each other.

Expressed another way, one aspect is properly geared towards interpretation of the cardinal aspect and the other towards the ordinal aspect of number respectively.

And as stated here this requires two distinct interpretations with respect to the number system which I refer to as Type 1 and Type 2 respectively.

Whereas the Type 1 relates directly to the (recognised) conscious aspect of linear rational interpretation, the Type 2 relates to the (unrecognised) unconscious aspect, which indirectly can be given a rational interpretation in a circular logical manner.

Properly understood therefore the actual experience of number entails the dynamic interaction of both conscious and unconscious in experience entailing in turn the dynamic interaction of the analytic and holistic aspects of number (entailing both linear and circular logical type interpretation).

As we have seen the inherent nature of addition and multiplication relate to both Type 1 and Type 2 aspects of number understanding respectively.

Therefore the key point once again is that we cannot possibly come to grips with the deeper issues underlying prime numbers and the Riemann Hypothesis from within the conventional mathematical perspective (which is based solely on a reduced absolute Type 1 interpretation).

Now I will seek here to illustrate these points with respect to the number 2, appreciation of which is sufficient to grasp the essential nature of the Riemann Hypothesis.
Now the standard Type 1 interpretation is based on the notion of 2 as representing a collective whole unit in a merely quantitative manner.

So if we try here to subdivide into its individual units we get 1 + 1, where the separate units are treated as fully homogenous (i.e. lacking any qualitative distinction).
So 2 in this context more comprehensively means 2 ^ 1 i.e. where 2 is defined in linear terms with respect to the default 1st dimension.

Therefore geometrically, numbers in Type 1 terms are represented as points plotted on the 1-dimensional line. So when we consider a number initially expressed with respect to a different dimensional value, its ultimate value is expressed in a merely reduced linear (1-dimensional) fashion.

For example from this perspective 2^2 = 4 i.e. (4 ^ 1).

In another words, though strictly a qualitative distinction is now involved with 2 ^ 2 representing 4 square (2-dimensional) units, this qualitative transformation is simply ignored with the result expressed in a reduced linear (i.e. merely quantitative) manner.

Now I have raised the fundamental problem with this approach in that in ignoring the qualitative distinction (expressed through the dimensional number) that no means in fact exist for making ordinal distinctions as between numbers.

In other words if we seek to interpret numbers in a merely quantitative manner, then we have no basis for making ordinal rankings (which assumes a qualitative relationship as between numbers).

Again though this as I say is absolutely fundamental and strictly undermines the very basis of all conventional mathematical interpretation, it remains completely ignored within Mathematics. This is why I have long been of the view that such Mathematics - despite its admitted great advances - is simply not fit for purpose (as it is based on highly reduced and ultimately untenable assumption regarding the very nature of its relationships).

So the alternative Type 2 interpretation of number starts from the opposite basis of viewing each number with respect to its individual members in distinct ordinal terms.
So from this perspective the number 2 comprises a 1st and 2nd member (each of which is qualitatively distinct).

However as this understanding is of a very distinct nature it cannot be reflected through Type 1 understanding.

So here we reverse the relationship as between number (as base quantity) and number (as dimension).

Thus in the Type 1 system where 2 = 2 ^ 1, 2 represents the base quantity and 1 the (default) dimension.

However in the Type 2 system, 2 = 1 ^ 2 so that now 1 represents the (default) base quantity and 2 the dimensional number (expressing qualitative distinction).

So again for example if we square the number 1, no quantitative change in the unit takes place; however a qualitative change to square (2-dimensional) units occurs.

And in the Type 2 we concentrate on the nature of such qualitative change!

To highlight the qualitative nature of this system we must switch to a circular type representation of number.
So the number 2 as dimensional number comprises two distinct ordinal members (i.e. 1st and 2nd).

Now an inverse relationship exists as between the qualitative nature of these units and the indirect (reduced) quantitative interpretation.

So the inverse of the 1st member as first root (where the number 1 now represents ordinal identity) remains unchanged as 1 ^ 1 = 1; however the inverse of the 2nd member as 2nd root (where 2 now represents ordinal identity) 1^ (1/2) = – 1.

So what we have achieved here - through the two roots of 1 - is an indirect quantitative means (on a circular number scale) of expressing the unique identity of the two ordinal members of 2.

And of course by extension we can express - again in an indirect quantitative manner - the unique identity of all the ordinal members of any number n by taking the corresponding n roots of 1!

So the various roots of 1 possess an as yet unrecognised significance in providing the indirect quantitative means of uniquely identifying the ordinal nature of number with respect to any group.

I have raised this key issue before as to how the meaning of 2nd for example is purely relative depending on the size of the group to which it belongs! So this problem of giving 2 a distinctive ordinal meaning is solved by considering the value of the 2nd of n roots (where n can be any natural number).

So each ordinal member of a group is given a - relatively - separate (indirect) quantitative identity; however the true holistic nature of the group requires the interdependence of these members in a qualitative manner.

And as we have seen the sum of all n roots of 1 (except where n = 1) = 0. So the qualitative interdependence of a group requires the (circular) combination of all its separate members.

Again I have illustrated before the nature of this interdependence with respect to road directions.

However as it fruitfully illustrates the true Type 2 nature of number, I will do so again.

Clearly left and right designations at a crossroads have a merely relative identity depending on the direction from which the crossroads is defined.

So if we approach a crossroads travelling towards it in an upward northerly direction, we can unambiguously identify for example, a left turn (which we can designate as + 1).

In this context the opposite right turn can be designated as – 1.

However this designation has a merely relative identity (depending on the arbitrary direction from which the crossroads is approached). So from the opposite direction (approaching the crossroads while moving in a downwards southerly direction) once again a left turn has an unambiguous meaning (as + 1) and a right (as – 1).

However this designation of left and right will be precisely the opposite as in the first case.

So the pure holistic appreciation of the interdependence of the directions (left and right) requires the ability to simultaneous combine both reference frames where what is left from one perspective is right from the other; and what is right from one is left from the other.

So in number terms, the simultaneous qualitative interdependence of both can be expressed as (+) 1 – 1 = 0 (or alternatively – 1 + 1 = 0).
So this apparently simple example of a crossroads implicitly entails the Type 2 appreciation of the nature of 2.

So each member of the group can be given a relatively separate ordinal identity in an indirect quantitative manner (i.e. within a given frame of reference) as + 1 and – 1 respectively.

However the overall holistic identity of the two members in qualitative terms requires the simultaneous recognition of both reference frames (creating paradox from a linear rational perspective).
In this way the quantitative recognition of the two members of 2 as separate, is cancelled out in pure holistic recognition of their qualitative interdependent nature.

Again I have often used the example of a crossroads to illustrate the nature of polar reference frames.

However such polarised reference frames are the very nature of experience (which is inevitably conditioned with respect to all phenomenal understanding by external and internal, and equally quantitative and qualitative polarities).

So when we look at the number system in an appropriate dynamic manner, we recognise that a ceaseless interaction as between these opposite polarities is necessarily involved.
Therefore any distinct knowledge of a quantitative nature is merely relative, and ultimately becomes completely cancelled out through a pure holistic qualitative recognition.

In psychological terms this entails the ceaseless interaction of both (conscious) rational and (unconscious) intuitive aspects of understanding which cannot be reduced in terms of each other.
However the unconscious qualitative aspect can be indirectly translated in a circular rational fashion that seems paradoxical in terms of conventional (linear) reason.

Even more remarkably, this circular understanding can be further indirectly translated in a linear rational fashion (that is imaginary).

So, just as we have real and imaginary aspects to number in quantitative terms; equally we have real and imaginary aspects to number (in qualitative terms).

As I have stated the key aspect with respect to number thereby points to the ultimate nature of the relationship of its quantitative (analytic) to its qualitative (holistic) aspect and this is what the Riemann Hypothesis is essentially all about!

The Riemann Zeta Function is defined, as we know in conventional mathematics terms, with respect to the complex plane (where real and imaginary possess a merely quantitative meaning).

However properly understood this complex plane should be equally defined so that real and imaginary are also given distinctive qualitative meanings.

Now from the quantitative perspective the one value where the Function remains underfined is for s (the dimensional value) = 1.
However this equally has an interpretation in qualitative terms which means that the Riemann Zeta Function remains uniquely undefined fo s = 1 (i.e. in terms of conventional linear rational understanding).

What this implies is that the Riemann Function entails the relationship - with respect to the Functional Equation - of - relatively - quantitative (analytic) on one side of the Zeta Function > .5 with qualitative (holistic) meaning on the other side < .5.

So the Riemann Hypothesis where the real part of s = .5 relates to the point where analytic (quantitative) and holistic (qualitative) meaning directly coincides.

So again the reason why the Zeta Function remains uniquely undefined in qualitative 1-dimensional terms is because this represents the one (and only) dimension where mathematical meaning is exclusively defined in a merely analytic (quantitative) manner.

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