A key problem that I faced for many years related to the provision of a satisfactory holistic mathematical interpretation of imaginary numbers.
Ultimately familiarity with Jungian psychology provided the key to the breakthrough.
It is amazing how so often the actual symbols and terms used in mathematics with respect to quantitative interpretation are deeply suggestive of their true holistic nature!
For example rational numbers (as quantities) bear a direct relationship with - what is termed - the rational paradigm (as qualitative).
We also saw that the holistic interpretation of irrational numbers leads to the need for a paradoxical type of circular logic which - quite literally - seems irrational in terms of accepted linear understanding.
We also saw that the very symbols used to represent 1 and 0 are (with small variations) the line and the circle. So the holistic interpretation of the binary digits incorporates both linear (1) and circular (0) interpretation.
Then we found that the holistic nature of prime numbers is deeply rooted in the nature of primitive behaviour (with respect to both physical and psychological reality).
Likewise in common scientific terms what is "real" is generally interpreted as that which conforms to conscious understanding (of a linear rational nature).
Imaginary i.e. what relates to the imagination, has the corresponding connotation of more intuitively inspired understanding that pertains directly to the unconscious.
So this would suggest from a qualitative perspective that the imaginary notion in mathematics in some way incorporates holistic meaning that properly pertains to the unconscious.
However we can do a little better than that in providing a precise interpretation of what an imaginary number entails (from a holistic qualitative perspective).
In quantitative terms the imaginary number i is defined as the square root of - 1 (which is not therefore reducible in a real manner).
Now we have already defined in holistic terms the 2nd dimension
- 1 as the (unconscious) negation of what has been formerly posited in real (conscious) terms.
Such unconscious negation is of a dynamic nature entailing interaction with the former posited direction of experience. So through such interaction we have the fusion of opposite polarities in the generation of spiritual intuitive energy. In this respect the process is remarkably similar to the manner in nature by which matter and anti-matter particles likewise interact to generate physical energy!
However this second intuitive dimension of understanding is strictly of an empty holistic nature (and therefore not directly amenable to rational interpretation).
However indirectly it can be given a reduced linear rational expression through obtaining the square root of this negated direction ( - 1).
In this manner - though again of a necessarily indirect paradoxical manner - holistic intuitive understanding properly pertaining to unconscious meaning can be formally incorporated in an acceptable scientific manner.
It took a long time for Mathematics to recognise the importance of imaginary numbers and then their comprehensive expression (together with the real aspect) of complex numbers.
I think it is only fair to say that the full acceptance of complex numbers in quantitative terms has truly revolutionised mathematical understanding. For example the Riemann Hypothesis which is our immediate concern here, would have been impossible to conceive in their absence!
However what is still completely missing from present mathematical understanding is any corresponding understanding that imaginary numbers need to be equally incorporated with real in a complex qualitative appreciation of mathematical symbols.
So mathematical interpretation has both real (analytic) and imaginary (holistic) aspects.
But remarkably only the real aspect - related to mere quantitative appreciation - is recognised.
So the equally important imaginary aspect - related to scientific qualitative appreciation - is totally ignored.
So Mathematics while recognising the importance of complex numbers (real and imaginary) as quantities, attempts such interpretation from within a qualitative approach (that is solely real).
And, as I will demonstrate in later contributions, this restricted approach breaks down with respect to appropriate interpretation of the Riemann Hypothesis.
Indeed I would now see a far deeper relevance to the Riemann Hypothesis.
Riemann made truly extraordinary strides with respect to the quantitative aspect of prime number behaviour through the sophisticated use of complex techniques of analysis.
However appropriate interpretation of what these results entail, actually points to a corresponding need to adopt a complex approach in qualitative terms (incorporating both conventional and holistic mathematical understanding).
Indeed ultimately the Riemann Hypothesis points to an essential condition that is required to ensure the consistent interaction of both aspects!
So in this context I will once again outline the basic programme for a comprehensive approach to Mathematics
1) Conventional Mathematics; this is the real aspect of mathematical understanding geared directly to the quantitative interpretation of mathematical symbols. Unfortunately Mathematics at present is almost completely limited to this real aspect.
2) Holistic Mathematics; this the imaginary aspect of mathematical understanding geared directly to the qualitative appreciation of mathematical symbols. This has been my own speciality now for more than 40 years!
3) Radial Mathematics; this is the most comprehensive approach entailing both real (conventional) and imaginary (holistic) aspects in a close interactive manner.
I would classify my own recent resolution of the Riemann Hypothesis as a very preliminary version of the radial approach (that still remains closer to holistic rather than conventional appreciation). However as the Riemann Hypothesis in this context relates more to a philosophical rather than strict quantitative truth, such an approach is still adequate to resolve the matter.
As Jung portrayed so well, the unconscious is always present in conscious understanding (in an unrecognised manner). What then happens is that the unrecognised aspect projects itself involuntarily on to conscious understanding.
Ultimately the desire for meaning and fulfilment is of an unconscious holistic nature. However this then becomes projected on to conscious phenomena (that are misleadingly seen to contain this meaning).
In this sense mathematical truth is never strictly of an absolute nature with unconscious blindness all the stronger for the fact that it is not formally recognised.
Indeed it is such blindness that has successfully blotted out recognition of the complementary holistic element of mathematical understanding which - quite literally - is still not seen as having any relevance.
From a quantitative perspective, the imaginary number i results from obtaining the 4th root of unity and is therefore lies on the circle of unit radius (in the complex plane).
In corresponding reciprocal fashion the imaginary number i (as qualitative) results from raising 1 to 4 (i.e. the 4th dimensional expression of 1). It likewise is of a circular nature (i.e. entailing a circular logical explanation).
The four roots of 1 in quantitative terms are 1, - 1, i and - i respectively. The four dimensions in qualitative terms therefore entail both the positing and negating of phenomena with respect to both real and imaginary polarities.
The two real polarities relate to conscious internal and external aspects which in dynamic interactive terms are always positive and negative with respect to each other.
It has to be remembered that even at its most abstract level, mathematics represents experiential activity which necessarily entails a dynamic relationship as between the knower and what is known. Though in linear terms these polarities are understood as corresponding with each other in absolute terms, strictly this is not so with an experiential dialectic (that continually changes) taking place.
The two imaginary polarities then essentially relate to the dynamic manner in which whole and part notions interact in experience.
Again in conventional mathematical understanding (which is qualitatively 1-dimensional) whole and part notions are reduced in terms of each other.
However what is properly whole - relatively - is of an infinite nature whereas the part is finite.
And as we have seen the whole pertains directly to intuitive and the part to rational understanding respectively.
When appreciated in this light the whole - while necessarily related - is not confused with the parts; likewise the parts - again while necessarily related - are not confused with the whole. Using spiritual language the whole maintains a distinct quality that transcends (in an infinite manner) the parts; equally the parts maintain a distinct identity through which the whole is made immanent (again in an infinite manner).
Thus switching from (transcendent) whole to (immanent) parts and in reverse manner (immanent) parts to (transcendent) whole entails a corresponding switch as between positive and negative with respect to imaginary polarities.
This distinction as between real and imaginary is of the first magnitude intimately affecting all understanding at its deepest levels. For example - properly understood - any concept and its related perceptions are real and imaginary with respect to each other. So for example if specific numbers (as parts of the number system) are real, then the corresponding (whole) number concept is imaginary. Once again whereas specific numbers possess an actual finite identity, the number concept to which they relate is potentially of an infinite nature. Also in reverse fashion if we now take the number concept as real, the number perceptions to which they relate are potentially imaginary (i.e. through which the infinite concept is made immanent).
When viewed in this manner, just as experience keeps switching as between conscious and unconscious aspects of understanding, in qualitative mathematical terms it likewise continually switches as between real and imaginary aspects.
Seen in this new light, scientific phenomena are necessarily of a complex nature (with real and imaginary aspects). This is even evident in normal speech. For example a person may speak of buying a "dream" house. So here the house is given a real identity as a finite specific conscious object (that can be identified locally); however equally it possesses an infinite aspect pertaining to unconscious desire as the embodiment of holistic meaning.
And strictly speaking this is true of all phenomena (including mathematical symbols). Whereas they have a real identity (as conscious) they equally possess an imaginary identity as in some way fulfilling a quest for meaning. Indeed without this imaginary aspect (pertaining to the unconscious) it would not even be possible to pursue mathematical truth!
Though expressed in a rational linear manner, the remarkable feature of imaginary numbers (as quantities) is that they are actually expressive of the alternative circular logical system (based on the complementarity of opposites).
So even at a quantitative level two logical systems are at work. Indeed it is this latter circular aspect of imaginary numbers that gives them amazing holistic properties in many quantitative contexts. Once again this is why Roger Penrose keeps referring to the magic of complex numbers.
However once again what is greatly missing is true appreciation of why complex numbers possess such holistic properties! Thus as we have seen, recognition is likewise needed in qualitative terms of the complex nature of mathematical activity.
As I have stated the true relationship as between a number quantity (and its corresponding dimension) is as real to imaginary. As real pertains directly to what is quantitative and imaginary to what is qualitative respectively, this implies the real to imaginary connection!
Now the holistic (i.e. qualitative) number system is defined as 1 raised to a dimensional number which varies.
Remarkably when we raise 1 to a real number e.g. a fraction, a transformation takes place so that the result is circular (lying on the unit circle in the complex plane).
(Though not recognised in conventional terms this equally applies in a complementary qualitative manner to all whole number dimensions!)
Then in reverse fashion, when we raise 1 to an imaginary number dimension, again a transformation takes place so that the result is now real (in a linear manner).
So once again the appropriate relationship as between a number quantity and its related dimension is as linear to circular (or alternatively in more scientific terms real to imaginary).
Again all of this is deeply relevant to understanding the true nature of the Riemann Hypothesis.
The Riemann Zeta Function is defined with respect to dimensional numbers that are complex. The implication therefore is that both quantitative and qualitative transformations pertain to the resulting numbers that materialise.
And - as we shall see - a wide range of these quantitative results cannot be given a conventional linear interpretation. However they can be given a coherent holistic mathematical interpretation!
The Riemann Zeta Function clearly demonstrates that mathematical operations can generate numerical results whose meaning is qualitative rather than quantitative.
The clear implication therefore is that we cannot properly understand such behaviour without incorporating holistic mathematical interpretation.
We can also perhaps now see why complex numbers have proven so valuable in uncovering some of the mysteries of the primes.
Prime numbers embody in their inherent nature patterns of behaviour that are linear and circular with respect to each other. Likewise when converted into more scientific amenable language, this implies that prime numbers embody behaviour that is both real and imaginary (in qualitative terms).
It is therefore by no means surprising that complex methods of analysis would be needed to uncover many of the quantitative secrets of prime numbers.
However, what is not recognised is that it then requires complex methods of qualitative interpretation to ultimately make sense of this quantitative behaviour.