As we saw in yesterdays blog entry, the standard treatment of number is fatally flawed in one fundamental respect. This entails formal recognition solely of the quantitative aspect of number from an analytic perspective (where numbers are viewed in an absolute manner as independent entities).
However, in truth, an equally important qualitative aspect characterises numbers in a holistic manner, where they are now viewed in relative terms i.e. with respect to their interdependent relationship with each other.
Therefore, to properly understand number in a coherent fashion with respect to its quantitative and qualitative aspects, we must combine both in dynamic interactive terms, where they are now understood as both relatively independent and relatively interdependent with respect to each other.
And this requires the integrated combination of mathematical understanding in terms of both its analytic and holistic aspects.
Using yesterday's blog entry, I concentrated at length on interpretation of the prime number "3" to illustrate how both the analytic aspect (of quantitative independence) and the holistic aspect (of qualitative interdependence) arise.
And as we have seen, these two aspects are closely related to the cardinal and ordinal aspects of number respectively. I also was at pains to explain how the ordinal aspect - which properly is of a qualitative nature - is invariably reduced through standard mathematical interpretation in a merely quantitative manner.
In psychological terms, the analytic aspect of mathematical understanding is directly associated with rational - or to be more precise linear rational - interpretation; the holistic aspect is then directly associated with intuitive recognition, which can indirectly be conveyed in a circular rational manner.
However once again, though indeed the importance of intuition may well be admitted, especially for creative mathematical developments, its distinctive holistic nature is not properly recognised in conventional terms, where it is invariably reduced in an analytic fashion!
So when one maintains in cardinal terms that 3 = 1 + 1 + 1, this properly relates merely to the quantitative aspect of the relationship (entailing relative independence). Thus the homogeneous units here - literally - lack any qualitative identity!
Then when one alternatively maintains that 3 = 1st + 2nd + 3rd (units), this now properly relates to the qualitative aspect of the relationship (entailing relative interdependence). Here in complementary fashion, the collective sum of the units - literally - lacks any quantitative identity.
And as I explained yesterday, this can be demonstrated through representing the ordinal notions of 1st, 2nd and 3rd indirectly in quantitative manner through the 3 roots of unity.
Therefore the understanding of number necessarily entails combining both quantitative (analytic) and qualitative (holistic) aspects in a coherent fashion.
In the actual cognitive experience of number, one ceaselessly keeps switching as between quantitative notions (of a conscious nature) and qualitative notions (of an unconscious origin), as one moves between the cardinal and ordinal aspects of appreciation respectively.
However though our experience of number necessarily combines both aspects, formal mathematical interpretation then greatly distorts the inherent nature of the process, through reducing the qualitative aspect - in every context - in a merely quantitative manner.
So accepted mathematical interpretation (i.e. in a merely analytic rational manner), has led to the blotting out of its complementary holistic unconscious aspect to an extraordinary degree in our culture. Indeed so great has this blindness become, that it is no longer even recognised as an issue!
However if one is to achieve a clear understanding of the significance of the Zeta 2 zeros and then in close parallel the corresponding significance of the Zeta 1 (Riemann) zeros, it is vital to appreciate this key point.
To make this perhaps a little more accessible, I will deal now with the simplest case where a Zeta 2 zero arises i.e. in the case where t (representing a prime) = 2.
Therefore In this case the function is given simply as,
1 + s21 = 0. So s2 = – 1.
Thus this zero is the "non-trivial" root (of the 2 roots of 1) = – 1.
As we have seen, the other root + 1, represents the default case where the ordinal notion of 2nd (in the context of 2 reduces to 1. And as the trivial case of the one root of 1 leads automatically to the default notion case of the ordinal notion of 1st (in the context of 1) reducing to 1. So from the standard analytic perspective both 1st and 2nd can be identified in a cardinal manner as representing 1 unit!
Well! What this all really mean?
Let us return now to the simple example of the crossroads, that I have repeatedly used in this context!
From an analytic perspective, designation of left and right turns (at the crossroads) carry an unambiguous meaning.
So if one approaches the crossroads from a specific direction (say heading N) a left turn, for example, can be unambiguously identified.
Thus if one identifies this left turn as + 1, the corresponding right turn can be designated, in this context, as – 1 (i.e. not a left turn).
So + 1 and – 1 carry here the standard analytic meaning!
Now, if one then approaches the same crossroads from the opposite direction (heading S) again one can unambiguously identify left and right turns, designated in the same manner as + 1 and – 1.
Therefore in linear (i.e. 1-dimensional) rational terms, unambiguous identification of turns concurs with an interpretation taken within just one polar reference frame.
So, in the first case, the interpretation relates to the N direction of approach (to the crossroads). In the second case it now relates to the S direction of approach.
And within these single isolated frames of reference, unambiguous identification in standard analytic terms can take place.
However when we now wishes to switch to a 2-dimensional frame of reference, both N and S directions must be viewed simultaneously, which then creates immediate (circular) paradox in terms of the former unambiguous linear identification.
For what is left at the crossroads (when heading N) is now right (when heading S); and what is right heading N is now left heading S.
So from the 2-dimensional holistic perspective (which requires the simultaneous recognition of approaches from both directions), what is left is also right, and what is right is also left!
Put another way, in terms of our number designation, + 1 = – 1 and – 1 = + 1.
We can likewise explain this in ordinal terms! If we initially identify a left turn as 1st, then the right turn will now be 2nd.
So 1st and 2nd can be unambiguously identified in standard linear terms (which is the accepted rationale for the standard analytic interpretation of number).
However when one now interprets 1st and 2nd in a 2-dimensional manner, 1st and 2nd become completely interchangeable with each other. So what is 1st from one perspective is 2nd from the
other; and likewise what is 2nd from one perspective is 1st from the other!
And such mutual interdependence can only be grasped in a holistic - rather than analytic - manner.
In psychological terms, this holistic understanding implies direct intuitive recognition, whereas the analytic understanding implies direct rational interpretation.
Expressed in yet another fundamental manner, the analytic (rational) aspect relates to the quantitative aspect of number appreciation, with the holistic (intuitive) aspect relating to the corresponding qualitative aspect. And both aspects continually interact with each other in a dynamic two-way fashion.
We are now in a position to understand the extraordinary importance of the Zeta 2 zeros, as the solutions other than 1 for the various roots of unity. And as understanding necessarily commences in a linear manner, these must always be combined with the default case where the remaining root = + 1.
Essentially, they act as a bridge connecting both the analytic (quantitative) and holistic (qualitative) aspects of mathematical understanding.
Thus with two reference frames (concurring with the case where t = 2), + 1 and – 1 can be used in the standard analytic manner - as in our example - to designate in a relatively independent manner, left and right turns. Thus again if heading N, when + 1 designates a left turn, – 1 unambiguously identifies a corresponding right turn (i.e. as not left).
Likewise when heading S with + 1 designating a left turn, – 1 designates a right turn (in a relatively independent manner).
So + 1 and – 1 are here used in a relatively separate manner (with unambiguous interpretation resulting).
However to recognise the 2-dimensional situation, where one can approach the crossroads equally from N and S directions, one switches to a true holistic interpretation of the same number symbols.
So + 1 and – 1 are now used in a (circular) interchangeable manner as interdependent with each other (again in a relative manner).
And this latter use is highlighted by the fact that the resulting sum (in numerical terms) = 0. In other words, we are here recognising the merely relative notions of left and right, which have no strict meaning independent of each other.
In principle, though much harder to fully grasp in a true holistic manner, all the other Zeta 2 zeros (in combination with the standard linear default case) play the same role.
Thus, they enable within the context of a given number, the consistent conversion of both quantitative and qualitative aspects of interpretation i.e. where ordinal notions can be understood in both a relatively independent (as quantitative) and relatively interdependent manner (as qualitative) respectively.
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