I have continually referred on these blogs to the enormous central weakness underlying all conventional mathematical interpretation i.e. its complete absence of any genuine notion of interdependence.
It seems quite remarkabe that it is left to someone like myself - who would be considered a complete outsider by the mathematics profession - to address this all important issue that is steadfastly ignored by its own practitioners.
By its very nature Mathematics is built on linear rational notions of fixed relationships in a static independent manner. It therefore can only deal with interdependent notions – which essentially relate to the qualitative holistic aspect of mathematical activity – in a reduced and distorted quantitative fashion.
Now this problem cannot be addressed through the development of ever more complex specialised procedures within the present accepted framework of Mathematics. In fact these will only serve to blind us further to the fundamental problem which persistently is avoided i.e. that for all its great successes, such Mathematics is built on highly limited assumptions.
So what in truth represents but an extreme – though admittedly very important – special case has been misleadingly elevated to represent all valid Mathematics.
However – when properly appreciated – the true nature of Mathematics is dynamic in nature and incomparably greater that what has yet been imagined. In fact instead of the existence of just one valid system (in absolute terms), potentially an unlimited number of alternative dynamic mathematical systems exist (each possessing an important relative validity).
As we have seen Conventional Mathematics represents the special case where interpretation is - literally - 1-dimensional (linear) in nature. And the very nature of this system is that qualitative meaning such as holistic interdependence - in every context - is thereby reduced to quantitative interpretation.
The very way of properly dealing with interdependent relationships entails a circular – rather than linear – approach that is inherently based on the dynamic notion of complementarity.
I have made the important observation before that it is only when the dimension = 1 that an absolute – rather than relative – interpretation of mathematical symbols occurs. And here the qualitative aspect is thereby reduced to the quantitative in a static fixed manner.
For all other dimensions ≠ 1 a truly relative interpretation of mathematical symbols ensues where quantitative (analytic) and qualitative (holistic) aspects interact with each other in a dynamic complementary manner.
As is well known the only value for which the Riemann Zeta Function is undefined is where the dimensional value s = 1. For all other dimensional values of s, he function is indeed defined. What this implies again is that the Riemann Zeta Function – when correctly interpreted - entails the relationship as between the analytic and holistic aspects of number. Therefore the one point where it is undefined is where a pure analytic interpretation takes place (as in Conventional Mathematics).
This is why I repeatedly stress that the true nature of the Riemann Zeta Function (and its accompanying Riemann Hypothesis) cannot be properly understood within the framework of Conventional Mathematics (which uniquely does not formally allow for a holistic interpretation).
The simplest dynamic interpretation entails 2 dimensions i.e. a 1st and 2nd respectively).
The structure of these dimensions is inversely related to the quantitative nature of the two roots of i.e. + 1 and – 1 respectively.
The corresponding qualitative interpretation entails the complementary (interdependent) relationship of these two dimensions.
Now the 1st dimension (i.e. + 1) relates to the standard linear rational approach (which literally posits phenomena in experience in a rational conscious manner (which is of an analytic quantitative nature). The 2nd dimension – by contrast negates conscious interpretation in recognition of an alternative holistic qualitative type appreciation that is inherently of an unconscious (intuitive) nature.
What the latter truly entails is that conscious understanding is always – by definition – conditioned by polar opposites interpretations (that are equally valid in nature).
Therefore to affirm interpretation (in respect of one arbitrary pole) is always limited as interpretation according to the opposite pole is equally valid. Thus in order to switch reference frames as between poles – which is the very means through which experience becomes dynamically interactive – we must first negate identification with arbitrary interpretation (based on one pole as reference point).
I have illustrated this countless times before in relation to directions at a crossroads. If we approach the crossroads from one arbitrary polar direction, (e.g. travelling “up” a road) left and right have an unambiguous linear interpretation. Likewise if approached from the opposite direction (travelling “down” the road again left and right have an unambiguous linear interpretation.
However in terms of each other both of these interpretations are clearly paradoxical. So whereas linear understanding always implies interpretation in terms of one arbitrarily fixed reference frame, corresponding appreciation of interdependence entails the simultaneous recognition of both reference frames.
Now again this explains exactly why Conventional Mathematics is lacking any true notion of interdependence! By definition such recognition – at a minimum – entails two separate poles (as possible reference frames).
The opposite pole to quantitative is qualitative! Therefore in this regard before we can truly recognise notions of interdependence then we must recognise two separate poles (as reference frames) that are quantitative and qualitative with respect to each other!
Then within each frame (considered in relative separation) unambiguous type linear results will apply in rational terms. However when one then simultaneously combines both reference frames in an intuitive complementary manner, deep paradox results.
The important point therefore is that the unambiguous understanding of (single) independent reference frames conforms to linear interpretation in rational terms. By contrast the paradoxical appreciation of (multiple) interdependent reference frames conforms to a uniquely distinct type of appreciation which in direct is always of an (unconscious) intuitive nature. However indirectly it can be conveyed in a circular rational manner (based on the complementarity of opposite poles such as quantitative and qualitative).
Now my own approach is deliberately designed to conform to this model.
Thus I start by defining two aspects of the number system Type 1 and Type 2.
Initially quantitative results can be obtained under both types (in relative separation). However when we combine both in a truly complementary manner, their relative interdependence (which truly reveals their qualitative holistic nature) then becomes apparent.
So we can see in this two-dimensional approach that we continually alternate between the 1st dimension of interpretation that is linear and the 2nd dimension that is circular.
This enables us to properly preserve therefore (linear) analytic notions of quantitative independence with corresponding (circular) holistic notions of qualitative interdependence.
Now in an earlier blog I demonstrated what in fact is a striking example of this dynamic interactive approach with respect to both the Zeta 1 and Zeta 2 Functions.
Once again both of these Functions can be studied in relative separation from each other yielding important results (of a quantitative linear kind).
However when we attempt to combine both (in an interdependent manner) the truly complementary behaviour of both becomes readily apparent. One could validly say therefore that qualitative holistic understanding resides in the very ability to recognise this rich network of interconnecting complementary relationships with respect to the Functions.
I will just elaborate again on an especially striking example from previous blogs (Complementary Zeta Functions 5 & 6) .
I established that the terms in the harmonic series (i.e. for the Zeta 1 Function, where s = 1) is composed of the values corresponding to the Zeta 2 Functions (less 1) that range over the reciprocals of all natural number values of s ≠ 1.
So notice the complementarity here! Each term (as part of the infinite Zeta 1 expression) represents the whole of a corresponding infinite Zeta 2 expression (less 1).
Likewise whereas the Zeta 1 is defined for s = 1, the Zeta 2 is defined in terms of all other natural numbers ≠ 1. And finally we have complementarity where a whole number power (for Zeta 1) is paired with its reciprocal values for Zeta 2.
The requirement to subtract 1 from each Zeta 2 expression is explained by the difference between addition and multiplication with the additive identity (leaving an expression unchanged) = 0 and the multiplication identity (likewise leaving an expression unchanged = 1).
The deeper significance of this can be explained as follows.
In the cardinal (Type 1) approach each natural number is defined by a unique combination of prime number factors (in quantitative terms).
However when we understand more fully we realise that each of these prime numbers represents a hidden sub-atomic world as it were (in Type 2 terms) where each prime number is already defined in a unique ordinal manner by all its natural number members.
In this sense the corresponding quantitative uniqueness of every natural number except 1 (in terms of its prime factors) is only possible because each of these prime numbers is itself uniquely defined in a qualitative ordinal fashion by its prime number members!
However an equally fascinating reverse form of complementarity exists.
This time we start with the Zeta 2 expression where s = 1, which contains the infinite sequence of terms 1 + 1 + 1 + 1 ..
We now look at the corresponding Zeta 1 Functions which now in reverse complementary fashion range over all the natural number values for s (except 1).
Then when we sum up this infinite sequence of Zeta 1 expressions (again subtracting 1 in each case) the resulting sum = 1.
Therefore in this case the whole infinite sum of all the Zeta 1 Functions (with 1 subtracted in each case) = each individual term in the Zeta 2 expression.
Again the deeper significance of this is that just as the subatomic qualitative natural number ordinal structure is internally contained in each prime (in cardinal terms) thereby ensuring the subsequent unique quantitative structure of the natural numbers, this atomic quantitative prime number structure is already contained in each ordinal natural number member (thus ensuring the subsequent uniqueness of its internal qualitative structure). So each unique (qualitative) part is contained in the collective (quantitative) whole; also the collective (quantitative) whole is contained in each unique (qualitative) part!
In other words through this set of complementary relationships we grow in appreciation of the ultimate truly paradoxical nature of the relationship between the primes and natural numbers.
Thus the great question regarding the number system is ultimately resolved whereby the primes and natural numbers are seen as perfect reflections of each other in a pure ineffable manner.
So we are led through ever more refined rational reflection in attempting to grasp the two-way relationship of the primes and natural numbers with each other, to finally let go of all remaining remnants of thought through finding the answer revealed in total mystery.