We made the distinction yesterday as between implicit qualitative recognition of the 1st dimension as negative (where it remains completely ignored in formal mathematical interpretation), and full explicit recognition which inevitably leads to a redefinition of the nature of Mathematics (whereby both quantitative and qualitative aspects are recognised).
So once again, a mathematician may well recognise the important role of intuition with respect to important new discoveries. And this inherently requires to a degree - sometimes marked - the temporary negation of customary rational understanding. This then allows deeper holistic insight to incubate in the unconscious which is essential in enabling an important new breakthrough. But unfortunately such a mathematician will then formally interpret this new finding in a merely reduced rational manner (with the 1st dimension as positive solely recognised).
As I live in Dublin I can identify with the inscription on Brougham Bridge in honour of William Rowan Hamilton.
"Here as he walked by on the 16th of October, 1843, Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication
i^2 = j^2 = k^2 = ijk = - 1"
So this inscription indicates well how the "discovery" essentially relates to a sudden illumination (releasing holistic intuition into consciousness). Notice how this does not happen in the normal sequential manner of successive rational linkages spread out in linear time! Rather it represents the present moment thrust as it were into linear time (where the relationship of all aspects of the problem to each other are understood simultaneously). Indeed so fearful was Hamilton at losing such inspiration that he felt compelled to carve the equation immediately into the stone at the bridge (though alas no record of this now remains).
However in formal terms, Mathematics has nothing to say about the role of intuition in understanding, or its important dynamic interaction with rational type understanding.
So, in the most accurate sense, conventional mathematical interpretation thereby offers but a reduced and ultimately quite distorted account of the nature of mathematical truth.
In other words, in the qualitative mathematical manner that I now use these terms, Conventional Mathematics is entirely defined within a merely (positive) 1-dimensional framework of interpretation, where qualitative is reduced to quantitative meaning. However proper incorporation of quantitative with qualitative requires recognition that all other numbers (as dimensions) have an important potential role to play in mathematical interpretation.
So once the negative 1st dimension - which remains merely implicit in conventional mathematical interpretation - is then explicitly recognised, the very nature of Mathematics changes from an absolute fixed to a relative dynamic approach, which necessarily entails the interaction of both quantitative and qualitative aspects.
Now, we have already looked at the nature of 2 as a dimensional number, which immediately arises through explicit dynamic recognition of the negative aspect of linear understanding. So 2-dimensional interpretation contains both positive and negative aspects, in dynamic relationship with each other (as the complementarity of real opposites).
There is a remarkable evidence of this provided - when appropriately interpreted - by the Riemann Functional Equation. So s, representing a dimensional number (i.e. power) of the Function on the RHS, can be given a complementary expression on the LHS, now expressed with respect to the dimensional number 1 - s.
This suggest therefore that there are intimate connections as between 2 as dimensional number and - 1 (on opposite sides of the equation). What this means in effect is that we must keep switching as between quantitative and qualitative (and qualitative and quantitative) type understanding with respect to interpreting both sides of the equation.
Therefore when we explicitly recognise the holistic intuitive significance of the result for the Function, with - 1 as dimension on the LHS, this immediately leads to a corresponding recognition of the rational nature of the result for 2 (as dimension) on the RHS. In other words whereas the numerical result (π^2)/6 makes sense from a rational linear perspective on the RHS, this is not so with respect to the corresponding result (- 1/12) on the LHS! And the reason for this is that the LHS result does not conform directly to a linear quantitative, but rather a circular qualitative interpretation (of a holistic kind).
The deeper implication of this is that proper interpretation of the nature of the Riemann Zeta Function cannot be carried out from within the conventional mathematical perspective. As the real secret of the primes relates to this fundamental relationship as between its quantitative and qualitative aspects, clearly this is completely missed from a mere 1-dimensional perspective (where qualitative meaning is inevitably reduced in quantitative terms).
So just as the Riemann Zeta Function is uniquely undefined in quantitative terms where s = 1, equally it remains uniquely undefined in qualitative terms (in terms of overall interpretation) likewise where s = 1.
We can now suggest a further important connection with the Riemann Zeta Function.
We have already defined 2 (as dimensional number) as the rational interpretation of the complementarity of opposite real poles.
However the very nature of reason is to separate poles. So we are attempting therefore to express with the number 2 (as positive dimension) what properly relates to the true nature of interdependence in an indirect rational manner (which tends to give it a somewhat independent identity).
Therefore to move to the true intuitive meaning of what is implied by 2 (as dimension) we must negate such rational interpretation.
Then when we successfully negate any lingering independent element we are left with the intuitive recognition of true interdependence (which is nothing in phenomenal terms).
Now this is deeply illuminating as the value of the Riemann Zeta Function (the first trivial zero) for which s is - 2, = 0.
This strongly suggests that this numerical value corresponds directly to the holistic qualitative - rather than specific quantitative - meaning of 0. So once again, whereas we can interpret values on the RHS of the Functional Equation (> 1) in quantitative terms, corresponding values on the left are - relatively - of a qualitative nature.
In short whereas the positive sign with respect to any dimensional number, represents its rational interpretation, the corresponding negative sign represents its direct intuitive recognition (through negation of independent rational elements).
This explains therefore in qualitative terms, why the Riemann Zeta Function = 0 for the trivial zeros (i.e. negative even integer values of s). In all cases, these represent the complementarity of opposites where pure interdependence arises. And such interdependence is directly grasped through intuitive recognition (which implies negation of indirect rational understanding provided through the positive even number dimensions). And this recognition = 0 in phenomenal terms.
My own route to this understanding was based on a deep resonance with the work of St. John of the Cross, who deals very well with the negative dimensions (from a mystical contemplative perspective).
So the "dark nights" or purgations are directly concerned with the experience of negative dimensions (in qualitative mathematical terms).
The active purgations relate to the odd numbered dimensions (especially 1). The passive purgations relate to the even numbered dimensions. And the direct goal of such passive purgation in St. John's terms is "nada" i.e. nothing (= 0 in qualitative terms).
Finally he talks of "nights of sense" and "nights of spirit". The former would relate in scientific terms to empirical perceptions whereas the latter would relate to more deep rooted theories and concepts. And we will later demonstrate a further startling holistic mathematical result that arises through the dynamic interaction of perceptions (as parts) and concepts (as wholes) respectively!
Thus once again we can see in the process of discovery of the greatest scientists and mathematicians (e.g. recently with Andrew Wiles) long periods spent in the intellectual wilderness. These implicitly in a mathematical context, constituted active nights of sense and spirit i.e. where attachment to former customary perceptions and concepts required considerable erosion before essential new insights could successfully develop.
Just to complete this section we return to the fact that what is true in psychological terms has - by definition - a complementary meaning from a physical perspective.
Now just as interdependence in psychological terms leads to the generation of spiritual energy (in the form of holistic intuition) equally interdependence in physical terms leads to the generation of physical energy. However the mysterious feature of such energy as with light, is that it has no phenomenal existence in itself, but rather only indirectly through interaction with other phenomenal processes.
So if you look at the world through contemplative eyes, you will realise that because mass represents just another form of energy, that phenomena essentially do not exist! Rather what we term "physical reality" relates to arbitrary appearances of a merely relative nature that have no ultimate substance.
However the point that I am making is that such realisation is equally consistent with a more comprehensive mathematical interpretation of number, where quantitative and qualitative aspects are equally recognised (through the marriage of reason with the contemplative vision).