We will be returning again to further clarification of this new approach to the Riemann Zeta Function.
I am conscious with each blog that I am repeating points already made but as this new formulation represents a radical departure from the conventional approach, I feel that it is valuable to keep reflecting it, as it were, from slightly different angles so as to facilitate a better appreciation of what is on offer.
So, in this slightly more refined formulation of the new Riemann Function, I am emphasising three strands (reflecting the three Mathematical Types (Type 1, Type 2 and Type 3 respectively).
The Type 1 strand reflects the traditional formulation of the Riemann Zeta Function based on a (mere) quantitative approach. So here the Function is defined in the complex plane for every value of s (representing the dimensional power of the Function) except for s = 1 (where it is undefined).
The Type 2 strand reflects the (unrecognised) formulation of the Riemann Zeta function based on the complementary qualitative approach. The significant difference here is that the values of s are now interpreted in qualitative terms (as distinctive multi-dimensional means of holistic interpretation), so that s and the resulting values from the Function are in each case interpreted in a unique manner!
As this holistic strand - despite its equal importance to the quantitative - remains totally undeveloped within Mathematics, my insights here are based very much on my own work carried out intensively over the past 45 years.
So here again the Function is defined in the complex plane (which remember again is now defined in an appropriate qualitative manner) for every value of s, except for s = 1.
The striking relevance of this last point once again is that this excludes the conventional (Type 1) approach entirely as an appropriate means of qualitative interpretation of the Zeta Function. And as the "music of the primes" fundamentally relates to this qualitative aspect, it simply entails that the Function does not properly lend itself to such interpretation in conventional terms. So the qualitative nature of the primes thereby completely eludes the mere quantitative approach!
The Type 3 strand is ultimately the most important by far offering a comprehensive interactive map of the Zeta Function. It combines both Type 1 and Type 2 strands, demonstrating how both quantitative and qualitative aspects (defined in relative isolation in the first two approaches) now interact in a vast multitude of dynamic complementary type relationships.
And at the heart of this understanding stands the Riemann Hypothesis as the central piece of the jigsaw, that seamlessly completes the whole complex tapestry in the ultimate identity of quantitative with qualitative (and qualitative with quantitative) interpretation.
Unfortunately there is further bad news for the Type 1 approach here.
Because of the truly dynamic interactive nature of the approach, both quantitative and qualitative aspects thereby undergo continual transformation.
Now originally in the Type 1 approach all quantitative type relationships are defined in a static (unchanging) manner.
However now, through continual interaction, the very nature of such quantitative type relationships subtly changes.
What this entails in effect is that a limitless number of possible qualitative interpretative models (corresponding to numbers as dimensions) can now be brought to bear on mathematical reality;, through such interpretative interaction, all quantitative relationships now likewise reflect the same limitless possibilities for change.
I illustrated this in a small way in yesterday's blog contribution, when I demonstrated how, through 2-dimensional interpretation (as qualitative) - which is the simplest of the alternative interpretations to demonstrate - the very nature of pi changes (in quantitative terms).
So we saw that pi is defined as a constant in quantitative terms (from a 1-dimensional perspective); however when then viewed in 2-dimensional terms, pi is redefined as having a merely relative approximate value (in quantitative terms) that is subject to uncertainty. So its constant nature is now understood as reflecting the combined interaction of both its quantitative and qualitative aspects (in an ultimately ineffable manner).
The upshot of all this means that when placed in its proper context, all quantitative relationships in Mathematics will ultimately need to be radically redefined. This clearly implies the that the Type 1 formulation - despite its apparent quantitative rigour - represents but an extremely reduced version of mathematical truth.
On a personal note the clear realisation of this last point has brought me now full circle in my own mathematical quest!
This journey had commenced in boyhood when I already sensed significant cracks in the mathematical edifice. This original disillusionment then reached its zenith at University where I realised clearly that the treatment of the all important infinite notion in Conventional Mathematics was totally unsatisfactory. So I was left with no option then but to attempt radical reconstruction of this mathematical world from outside the fold (trusting my own intuitions rather than conventional wisdom).
And finally now I am seeing clearly (at least for myself) the fruits of this long labour.
In other places I have further sub-divided the Type 3 (which I formerly termed radial) into three subsidiary classes.
Type 3 (A) represents the integration of both Type 1 and Type 2 aspects in a dynamic appreciation of the quantitative/qualitative interaction inherent in all mathematical processes.
However here the emphasis - relatively - is more on the qualitative side, with attention to the quantitative aspect primarily - though not exclusively - erepresenting the appropriate interpretation of already established relationships.
And this would accurately describe my own position. So, my mathematical focus started out with concern for such quantitative interpretation (at an embryonic level) and has culminated now in this much more developed position.
Type 3 (B) again requires a dynamic appreciation of the quantitative/qualitative interaction inherent in all mathematical processes. However here the emphasis - relatively - is more on using such appreciation to creatively generate new areas of mathematical enquiry, possibly culminating in radical new quantitative type hypotheses. In this it regard it would reflect a Type 1 approach (that has become considerably refined through deep contemplative vision)
Type 3 (C) would represent the most perfect expression of this approach, where one can display great mastery at a quantitative level in ways both highly productive and immensely creative, while also proving expert with respect to holistic type interpretation.
However this really represents a vision of the future which we have not yet remotely reached!
So with respect to the Riemann Zeta Function and the Type 3 strand, the Function is defined in the complex plane in both a quantitative and qualitative type manner (where now quantitative and qualitative are understood in dynamic interactive terms as inherent in all mathematical symbols, relationships etc.)
Thus, here the Function is understood in both a mutually complementary quantitative and qualitative manner for all values of s and corresponding results, except once again where s = 1 (in both a quantitative and qualitative fashion).
And finally once again the Riemann Hypothesis lies at the centre of it all (as the raison d'etre for the existence of everything).
Thus here the identity of both the quantitative and qualitative aspects of the primes both as numbers and as means of interpreting such numbers become identical in ineffable mystery.
From this enlarged perspective one may get some idea of how incredibly limited is our present conception of Mathematics.
It is as if we have confined ourselves for all our history to a tiny island when in truth vast continents lie completely unexplored close by.
Though it will be extremely difficult to emerge from such self imposed exile, marvellous vistas of completely new mathematical meaning await us in the future in what promises to be the most radical revolution in our intellectual history. And of course by extension because of the pre-eminence of mathematics this revolution will then quickly spread in a completely unprecedented fashion to all of the sciences.
And in this extraordinary new mathematical world, the long untenable quest to prove the Riemann Hypothesis will be remembered as a mere folly.