When asked once what was the most important problem in Mathematics - as claimed in Constance Reid's book - the great mathematician Hilbert replied!
"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"
Now Hilbert would have been referring here to the non-trivial zeros.
Funny, it was the trivial zeros that proved more crucial in my own case in realising the true nature of the Riemann Hypothesis, which is ultimately concerned with the condition for the identity of quantitative and qualitative meaning.
It seems that Hilbert might even have had some intuition that this was the case which would make his quote especially apt.
For as all created phenomena represent the relationship of both quantitative and qualitative aspects with their roots in the primes, what could be more important than discovery of the ultimate relationship between both aspects?
However in formal terms Hilbert considered Mathematics very emphatically as being based solely on the masculine principle (of linear logical interpretation).
Indeed he held the belief that all problems in Mathematics could in principle be solved within its axiomatic system.
In this regard he was quickly proven to be mistaken, when Godel showed that there would always be important problems which would remain undecidable (which could neither be proved nor disproved). This was subsequently demonstrated in 1963 with respect to the Continuum Hypothesis (which was the very first problem on Hilbert's famous list of 23 unsolved problems!)
Incidentally Godel apparently held the belief that the Riemann Hypothesis would also likely fall into this category of undecidable propositions!
As for Riemann, it became subsequently clear that he did not think of Mathematics as a specialised abstract pursuit but rather combined his mathematical interests with problems from Physics.
So it would be very easy to believe if Riemann had been aware of subsequent developments with respect to Quantum Mechanics, that he would have quickly appreciated the relevance of the non-trivial zeros for physical quantum systems. Indeed it is likely that he would have been the first to make such a suggestion!
So we will return over the next few blogs to the key issue of what in fact the non-trivial zeros of the Riemann Zeta Function actually mean, where we will finally conclude that they are every bit as significant as Hilbert postulated (though for reasons that would strongly conflict with his definition of Mathematics).
So we start with the prime numbers 2, 3, 5, 7,.. which are the most independent of all numbers (with no factors other than each prime and 1). In this regard the individual primes represent the extreme example of the masculine principle serving as the building blocks for the natural number system.
However there is an equally fascinating aspect to prime numbers in that they are likewise the most interdependent of all numbers.
What this precisely relates to is the Fundamental Theorem of Arithmetic, where each natural number can be uniquely expressed as the product of one or more prime numbers.
Now, because of the merely quantitative bias of Conventional Mathematics, I believe that the true significance of this latter aspect of the primes has been continually overlooked.
As I have stated many times before, when we multiply numbers together a qualitative (dimensional) - as well as quantitative - transformation is involved.
In Conventional Mathematics the qualitative aspect is then ignored with the result expressed in a merely reduced quantitative manner.
So what is entirely missed therefore with respect to multiplication, is that inherently it entails a qualitative process (which - by definition - cannot be encapsulated in conventional mathematical terms).
Therefore it is no wonder that Brian Conrey and Alain Connes have been quoted as frankly admitting that a central issue exists with relation to the link between addition and multiplication that is not understood!
Well the simple answer is that whereas addition can indeed be considered within a merely quantitative perspective, multiplication necessarily entails both quantitative and qualitative aspects of transformation.
In my journey towards appreciation of the true nature of the Riemann Hypothesis, I had already formed a strong realisation of this fact from childhood.
I knew there was something wrong with the conventional interpretation of multiplication (and that both quantitative and qualitative aspects were involved). However obviously at the time I had not sufficient intellectual capacity to develop the implications of that insight further!
So when I later returned to the issue my first instinct was to develop an alternative mathematical approach (where the missing qualitative aspect could be properly explored).
So the wonderful thing about prime numbers is that while representing the extreme example of independence in isolation, yet when combined, they represent the extreme collective example of interdependence (and can generate every natural number in a totally unique manner). And this unique relational capacity represents the embodiment of the (unrecognised) feminine principle.
Therefore prime numbers combine both extreme autonomous (quantitative) and relational (qualitative) capacities in their very identity.
The overall pattern of the individual primes is now well recognised, where they become progressively less densely populated as we travel up the natural number scale.
However it is the reverse with respect to the non-trivial zeros - or more precisely the imaginary part of the dimensional numbers that give rise to such zeros - which become more densely populated as we progressively travel up (or down) the imaginary number scale.
The clue to what these zeros actually represent comes from their relational capacity.
Now for very low numbers on the scale say up to 30, relatively few primes are required to generate the natural numbers involved; likewise few options exist in terms of possible unique combinations of such primes.
However as we progress to larger natural numbers, the number of primes required for their generation steadily increases (and likewise the possible combinations entailing such primes).
So the non-trivial zeros can be appropriately understood as measuring this relational capacity of the primes (in their unique collective configurations with the natural number system).
As I have stated before, this relational capacity, by comparison with the individual nature of the primes, is properly of a qualitative (rather than quantitative nature).
Two key indications of this are already apparent with respect to their quantitative representation!
Some weeks ago I marvelled at how easy it was to move from the prediction of the (average) spread as between individual primes to the corresponding (average) spread as between non-trivial zeros. The key distinction is the manner in which 2pi plays a role in the formula for the latter!
Now 2pi precisely measures the circumference of the unit circle (on which the qualitative aspect of the number system is based).
Therefore from a qualitative perspective, this clearly implies that we move from a linear interpretation with respect to the spread as between individual primes to a circular interpretation with respect to the spread as between non-trivial zeros.
And as circular interpretation is the indirect rational means of portraying intuitive appreciation of a holistic kind, this clearly establishes that we require two distinctive forms of understanding for the interpretation of the individual primes on the one hand and - relatively - the non-trivial zeros on the other.
Put another way the non-trivial zeros relate to the qualitative (holistic) nature of the primes (which can be given an indirect quantitative measurement).
I have shown in other blog entries how the imaginary (from a qualitative perspective) then serves in turn as the indirect means of expressing circular type notions in a rational linear manner.
An lo and behold once again the non-trivial zeros line up obligingly as quantitative points on an imaginary scale of measurement.
We must however keep returning to the crossroads analogy to remind ourselves of what is happening.
Once again, when using isolated poles of reference, when I travel "up" a road I can label a turn at a crossroads unambiguously left. Then when I later travel down the same road, I can unambiguously label the other turn also as "left". So both designations as "left" are valid within isolated poles of reference.
However when we consider both turns as interdependent (in two-way relationship to each other), they are necessarily left and right (and right and left) with respect to each other.
Now, it is exactly analogous in mathematical terms. When we approach the primes from an individual perspective, they do indeed display quantitative aspects. Then when we approach the primes from the opposite general perspective (with respect to their overall frequency) again they display quantitative aspects. However when we consider both individual and general characteristics (in a two-way interdependence), they are necessarily quantitative and qualitative with respect to each other.
So when we look at either the individual or collective nature of the primes - or indeed equally the individual or collective nature of the non-trivial zeros - within isolated frames of reference, they will indeed all display quantitative characteristics. However when we consider these in two-way interdependence with each other, in each case, they are necessarily quantitative and qualitative (and qualitative and quantitative) with respect to each other.
So once again the primes comprise two extreme aspects with respect to their very identity; with respect to their individual nature, we have an extreme autonomous capacity (as independent); then with respect to their collective identity we have
an opposite extreme relational capacity (as interdependent).
However properly understood in relation to each other, these capacities are as quantitative to qualitative (and qualitative to quantitative) respectively.
However because of the mere quantitative bias of Conventional Mathematics based on linear logical notions, it has no way of properly interpreting the equally important qualitative aspect. So it necessarily reduces the qualitative aspect (in the holistic communication capacity of the primes) to mere quantitative interpretation.
And in doing this it misses out completely on the true nature of the primes!
I referred to Alain Connes' metaphor of the missing female heroine (relating to his quest to solve the Riemann Hypothesis) in an earlier blog entry.
Thus, though the heroine in truth is, when viewed properly, centre stage in ardent embrace of our hero, from a conventional mathematical perspective, she appears entirely absent.