It is important to keep realising that all mathematical symbols can be used in two distinctive ways corresponding to analytic (quantitative) and holistic (qualitative) type appreciation respectively.
In psychological terms this in turn relates to both rational and intuitive understanding.
Though holistic notions, corresponding directly with intuitive type understanding - which indirectly can be expressed in a circular ration fashion - necessarily arise at an implicit level, in formal conventional mathematical terms they are treated in a grossly reduced manner (ultimately distorting their very nature).
So let’s be fully clear about this! The approach I am adopting in these blogs cannot be successfully appreciated or indeed criticised from within the accepted framework of Mathematics for the very attempt to do so distils it of its essential value.
So once again, we start by placing mathematical activity in its proper dynamic perspective where both analytic and holistic aspects with respect to all mathematical symbols necessarily interact.
In this context both aspects can be initially isolated in a relatively independent manner before fully appreciating the nature of their two-way interdependence.
And we have been addressing this issue (through recent blog entries) in the important context of number.
So we have seen that when we adopt the Type 1 approach to number (in an analytic quantitative manner) that prime numbers appear as the (cardinal) building blocks of the natural number system.
So each natural number (other than 1) can be uniquely expressed as a product of prime numbers.
However when we then adopt the Type 2 approach (in a holistic qualitative manner) the natural numbers appear as the (ordinal) building blocks of the prime number system.
So each prime number can be uniquely expressed as a sum of numbers i.e. roots that represent the ordinal identity of each natural number. And strictly speaking the one common root in all these cases i.e. 1 is not unique.
So for example we could express the 3 roots of 1 (correct to 3 decimal places) as
1, -.5 +.866i and -.5 -.866i
Now as the first root 1, is common to all cases where the n roots of 1 are obtained, strictly it is the remaining 2 that are unique. In other words these 2 roots can never appear as the solution for the any other roots of 1 (where n is prime).
Therefore once again there is a remarkable complementarity as between the Type 1 and Type 2 systems.
As we have seen even though the Type 1 approach is designed to deal with the quantitative aspect of number, it is necessarily defined with respect to the number 1 (representing the qualitative notion of dimension).
Then in reverse, though the Type 2 approach is designed to deal with the qualitative aspect of number (geometrically represented by equidistant numbers on the circumference of the unit circle in the complex plane) it necessarily includes the linear quantity 1 as one of these roots.
This simply illustrates that the Type 1 and Type 2 enjoy a relative rather than absolute separation from each other. So the cardinal aspect cannot be fully interpreted in a merely quantitative manner; likewise the ordinal aspect cannot be understood in a merely qualitative manner. So 1 (as qualitative dimensional number) necessarily underlies the Type 1 approach; 1 (as quantitative number) necessarily underlies the Type 2 approach.
Now from my perspective, a reduced quantitative version of what I am presenting here has been developed through an enlarged notion of number as adeles and the use of p-adic number systems based on the primes. Also Alain Connes has made extensive use of these number systems in his attempts to solve the Riemann Hypothesis.
However though once again one can indeed greatly admire his sheer genius and incredible abstract ability in attempting to approach the Riemann Hypothesis from this angle, for me it misses the essential point!
In other words for all his sophistication and mathematical brilliance, Connes is still using the same reduced – merely quantitative - approach in attempting to unlock the mystery of the primes.
However as this mystery in fact relates to the central relationship as between quantitative and qualitative type understanding, the Riemann Hypothesis not alone will never be proved this way; in fact it cannot be properly understood in this manner.
Indeed appropriate appreciation of the true nature of the primes would quickly show that the Riemann Hypothesis cannot be proved (or disproved) using conventional mathematical techniques.
By its very nature, it transcends the current interpretation of Mathematics.
In this way the Riemann Hypothesis, when placed in its appropriate context, serves as an invitation to embrace a much enlarged mathematical paradigm that gives equal recognition to both its quantitative (analytic) and qualitative (holistic) aspects.
Once again we can give a simple – indeed beautiful – explanation as to why the Riemann Hypothesis cannot be solved from such an enlarged perspective.
As we know in Type 1 terms (using quantitative analysis), the only point where the Riemann Zeta Function remains undefined is where s = 1.
In (complementary) Type 2 terms (using qualitative holistic interpretation), the only point where the Riemann Zeta Function remains undefined is again where s = 1.
However in this latter holistic context, 1 refers to the 1-dimensional means of qualitatively interpreting symbols (which defines the Type 1 approach).
So quite simply the Riemann Zeta Function cannot be appropriately understood using the conventional mathematical approach (which is 1-dimensional in nature).
What this directly implies is that the Riemann Zeta function establishes key complementary relationships (through the Functional Equation) as between both quantitative (cardinal) and qualitative (ordinal) type meanings.
As we have seen in Type 2 terms, for any value of s (as dimensional number other than 1) a dynamic relationship as between quantitative and qualitative type aspects is involved.
So putting it quite bluntly, Conventional Mathematics which is qualitatively 1-dimensional and thereby absolute in nature, is uniquely unsuited to interpretation of the true significance of the Riemann Zeta Function (and of course its associated Riemann Hypothesis).