For some years now I have derived great benefit from Matthew Watkin's web-site which provides a wonderful set of mathematical resources on the Riemann Hypothesis and other topics. In a section on "number theory and Jungian archetypes" taken from "inexplicable secrets of creation" he has a very interesting quote from Alain Connes (whom we have mentioned before).

"I believe I have found a very nice framework [to prove the Riemann Hypothesis] but this framework is still awaiting the main actor. So there is the stage - it is perfectly well arranged and so on - but we are still expecting the heroine to come and complete it."

I would suggest rather that the heroine has been on the stage all along without however being noticed.

And this relates to a deep cultural blindness in the West with regards to the feminine with important implications for - among other things - religious and mathematical practice.

For example in the (Roman) Catholic Church, Mary as the Mother of God is given a unique position. However this has led to a somewhat disembodied notion of the feminine archetype.

So in theological terms with the virgin birth, it is believed that Mary conceived Jesus (without sexual intercourse taking place). This in turn has led subsequently within the Church to a somewhat unfortunate notion of the feminine where an unduly spiritualised archetype operates, that is greatly separated from its complementary physical side.

This in turn I believe has played a large part in the cultural exclusion of women from both clerical ministry within the Church (and by association all its power structures) and by extension many professions which have long misleadingly been viewed as male pursuits.

In other words because an exaggerated emphasis has been placed on the spiritual aspect (deemed especially appropriate for a celibate clergy) this unfortunately has led to strongly negative connotations with respect to its complementary physical aspect. So woman has frequently been viewed as the "temptress" and therefore likely only to ensnare any unfortunate man if granted true equality.

In a lesser form, this negative aspect of the feminine archetype is seen in terms of women being the "weaker" sex and somewhat "irrational" which of course is a liability where serious Mathematics is involved.

So in like manner as that within the Catholic Church, traditionally a somewhat disembodied notion of the feminine has characterised the mathematical profession.

Thus it is considered acceptable on occasion - especially as for example when confronted by the mystery of prime numbers - for mathematicians to project deeper unconscious feelings in a manner allowing for the use of feminine archetypes. However clearly this does not extend to acceptance of the feminine dimension at the abstract level of actual mathematical practice.

Here quite simply there is no place for the feminine in formal terms with Mathematics being viewed merely as a linear rational pursuit (which of course perfectly befits the masculine principle).

It is perhaps not surprising therefore that very few women have made a significant mark on the Mathematics profession.

There was indeed one woman Marie-Sophie Germain at the time of Gauss who made a lasting contribution (though hardly of the first rank). And it is instructive in this regard that when initially writing to Gauss she felt compelled to conceal her feminine identity by using a male pseudonym.

Then more recently there was Julia Robinson who did admittedly make a significant contribution to the solution of the 10th problem on Hilbert's famous list of 23 great unsolved problems.

And there is a truly excellent article on the Riemann Hypothesis "Prime Time" written for the general reader by Erica Klarreich but beyond that I cannot really think of anyone!

As regards Alain Connes' "nice framework" I am not in a position to comment in any specialised manner.

However as I already had formed several ideas of my own that paralleled in a holistic manner what he was attempting to achieve, I have maintained a keen philosophical interest in the nature of his pursuit!

Thus Connes set out to build a quantum energy system (based on the prime numbers) with energy levels corresponding to the non-trivial Riemann zeros. And apparently he has managed to achieve that goal.

However, coming from a dynamic interactive perspective, I had long considered that the Riemann zeros necessarily entailed such an energy system in the first place, which then served as the prototype for all derivative quantum energy systems (with similar sorts of spacings to the Riemann zeros).

Therefore I would see his work as simply restating the Riemann Hypothesis (in an admittedly more realistic physical context).

However the problem of providing a solution will - necessarily - remain as intractable as ever (as it is not in fact capable of solution within the axioms of the present mathematical approach).

So we come back to the heroine who is indeed on the stage (if only one had eyes to notice).

To appreciate this more clearly, we will contrast just two features typically associated with the masculine and feminine principles.

So rationality (or more specifically linear rationality) is associated with the former and intuition with the latter respectively.

Also independence is associated with the masculine and interdependence (in a capacity for relationship) with the feminine.

Now as we have already seen in the conventional approach to the prime numbers, independent isolated frames of reference are used, with an attempt to separately study both the individual nature of the primes and their general distribution from a quantitative perspective.

And again the masculine principle is to the fore in the manner in which - from a formal perspective - Mathematics is seen solely as a rational (linear) pursuit.

However we have already addressed the central issue with respect to the primes, where it is now apparent that they contain two aspects which mutually are encoded in each other. So from one perspective, the non-trivial zeros (from which their wave structure is derived) are encoded in the primes. And in reverse, the primes are encoded in the non-trivial zeros. So the mystery of the primes relates to the manner in which both their individual and collective aspects are mutually interdependent.

And such mutual interdependence properly relates to the feminine principle!

Likewise the very appreciation of such interdependence - by its very nature - is not amenable to (linear) rational but rather holistic intuitive appreciation (which then indirectly can be interpreted in a paradoxical circular rational manner).

And once again this points directly to the feminine principle!

So, as I say, when properly viewed, the heroine not alone can be seen to be clearly present but as occupying a central position on the stage.

However the considerable difficulty associated with this observation, is that the Riemann Hypothesis can no longer be solved through exclusive reliance on the masculine principle; in fact it is not even capable of proper appreciation in this manner.

In other words, when appropriately understood, Mathematics contains both quantitative (masculine) and qualitative (feminine) aspects, with - in the context of the primes - the Riemann Hypothesis the condition required for their mutual identity!

And through failing to recognise this vital qualitative aspect, conventional practitioners will remain continually blind to the already existing presence of the heroine on the mathematical stage!

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