Future Vision of Mathematics

Before briefly making a few observations on L-functions let me outline some reflections on a future “golden age” of Mathematics.

Here I would see three distinct areas (where only one currently exists).

1) The first - which for convenience - can be referred to as Type 1 Mathematics, relates to the traditional analytic approach based on the reduced quantitative interpretation of mathematical relationships.

At present, mathematics i.e. certainly with respect to formal recognition, is exclusively identified with this type.

As its methods have become increasingly specialised in an abstract rational fashion, admittedly, enormous progress has been made. And this will continue into the long distance future with many new significant findings for example with respect to our current topic of the Riemann zeta function (and associated L-functions).

However an important present limitation, as we have seen, is the manner in which exclusive identification with the analytic, blots out the holistic aspect (with which creative intuition is more directly associated).

So while not wishing in any way to prevent the further progress of analytic type developments, eventually I believe it will be accepted that the Type 1 represents just one highly important aspect, which should not be exclusively identified as mathematics.

So a strictly relative - rather than absolute - interpretation will thereby eventually emerge for all its relationships.


2) The second - which I customarily refer to as Holistic Mathematics - represents the Type 2 approach.

My personal development has been somewhat unusual in this regard. Though very interested in mathematics as a child, my first serious reservations with the standard treatment of multiplication arose at that time. Therefore, from that moment I was already reaching out for a new holistic dimension, not catered for in formal terms.

So mathematics for me has very much represented a solo voyage of uniquely personal discovery.

And in adult life my abilities - such as they exist - have been largely centred on elaborating the hidden holistic dimension of mathematics. Then in the last 15 years or so, using these holistic insights, I have turned my attention to topics such as the Euler Identity and the Riemann Hypothesis with a view to providing a radical new perspective as to their true inherent nature.

Though retired from work for nearly five years, it feels only now that I have come full circle in being able to finally resolve (at least to my own satisfaction) those childhood reservations regarding multiplication.

However it is very difficult attempting to convey holistic mathematical insights to a professional audience (that does not formally recognise their existence).

So my most successful communication has been with talented generalists (with some mathematical background) seeking to integrate various intellectual disciplines in a more holistic manner.

However even here there has been considerable resistance to the belief - reflecting the deep-rooted nature of conventional assumptions - that mathematics itself is in need of radical re-interpretation.


Bearing the above comments in mind, I will now try to convey some of the flavour of my holistic mathematical pursuits.

As it is directly concerned with the qualitative nature of mathematical symbols, much of this work has related to the holistic clarification of the various stages of psychological development through the use of these symbols.


In terms of the physical energy spectrum, natural light forms just one small band on the overall spectrum of physical energy.

In like manner, the mental structures (based on accepted common sense intuitions underpinning linear logic) represent just one small band on the overall potential spectrum of psychological development.

Conventional mathematics represents specialised understanding with respect to this one small band.

However just as there are “higher” forms of physical energy (besides natural light) equally there are “higher” forms of psychological energy (besides the accepted intuitions of conventional mathematics).

These “higher” bands on the psychological spectrum have been extensively investigated by the major esoteric religious traditions East and West, where they are identified with advancing levels of spiritual contemplation of an increasingly formless nature.

However what has not yet been realised - except in a perfunctory manner - is the important fact that these bands are likewise associated with new forms of holistic mathematical interpretation (utterly distinct from conventional type notions).

In analytic terms, the study of higher mathematical dimensions entails understanding of an increasingly abstract rational nature (where the object aspect is increasingly separated from its subjective counterpart).

However in holistic terms, the study of higher dimensions by contrast entails appreciation of an increasingly intimate intuitive nature, indirectly transmitted in paradoxical rational terms, where both object and subjective aspects are seen as ever more interdependent with each other.

Now one of the extraordinary findings arising from these investigations is that all psychological (and indeed physical structures) can be given a distinctive holistic mathematical rationale.

And the holistic notion of number is intimately tied to these structures.

So from the holistic perspective, accepted formal mathematical understanding is 1-dimensional (in qualitative terms). This simply means that the qualitative aspect is formally reduced in quantitative terms.

However associated with every other number (≠ 1) is a distinctive means of interpreting mathematical symbols with a partial relative validity. So the absolute type understanding that we accept as synonymous with valid mathematical interpretation represents just one special limiting case of a potentially infinite set.

And this insight was later to prove of inestimable value in relation to a true dynamic appreciation of the Riemann zeta function.

Also, the holistic relative notion of number is intimately connected to a corresponding new holistic appreciation of the nature of space and time with a direct relevance in physical and psychological terms.

For example, the holistic notion of “4” relates to a dynamic appreciation of the corresponding 4-dimensional nature of space and time (with positive and negative directions in real and imaginary terms).

From a psychological perspective, this entails that all conscious (real) experience has - relatively - both positive (external) and negative (internal) aspects and also that unconscious experience, indirectly projected in a conscious (imaginary) manner, has likewise both positive (external) and negative (internal) aspects.


At a deeper level, the imaginary directions explain - in an indirect conscious manner - how in the dynamics of experience, whole switch to part (and in turn part switch to whole) notions.

In brief, akin to directions on a compass, all other holistic dimensions can then be looked on as providing unique configurations with respect to the dynamic relationship as between wholes and parts (in objective and subjective terms).

The famous Swiss psychologist C. J. Jung offered some broadly similar insights, while implicitly formulating his notions in a manner amenable to holistic mathematical interpretation.

So when one of his disciples Marie Louise Franz was later to say that “Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four”, it was this holistic (circular) - rather than analytic (linear) - notion of “4” that she had in mind. 

Just one final example I will offer, though all this represents but the tiniest glimpse into a potentially vast new field of investigation, is the holistic counterpart to the accepted binary system!

So the two binary digits 1 and 0 - given an independent interpretation in the standard analytic manner - can be potentially used to encode all information processes.

However, equally the two binary digits 1 and 0 - now given a holistic interdependent meaning as linear (1) and circular (0) type understanding respectively - can be likewise used potentially to encode all transformation processes

So for example, in this contribution, I have argued that the number system - and indeed all mathematics - should be interpreted as representing a dynamic interactive transformation process, entailing both quantitative and qualitative aspects.

And these two aspects relate directly to 1 and 0 respectively (in a holistic manner).

In fact, at the most general level, ultimately all differentiated and integrated processes both in physical nature and psychologically in human terms, are encoded in a holistic binary digital manner.


3) This, which I commonly refer to as Radial Mathematics or more simply the Type 3 approach, represents potentially the most comprehensive form of mathematical understanding, entailing the coherent integration of both analytic (Type 1) and holistic (Type 2) aspects.

When appropriately understood, all mathematics is intrinsically of a Type 3 nature (though not yet recognised because of the deep-rooted acceptance of reduced assumptions).

Indeed it is only in the context of Radial Mathematics (Type 3) that the other two aspects Conventional Mathematics (Type 1) and Holistic Mathematics (Type 2) can reach their fullest expression. So perhaps some day in the distant future, Type 3 will become synonymous with all mathematics.

However even within this category, I would distinguish three important sub-types, (a), (b) and (c) respectively.


Though rooted to a certain degree in the holistic aspect of Mathematics, sub-type (a) is mainly geared to the derivation of exciting new analytic type discoveries (with creative insight playing a key role).

There is no doubt that at least implicitly, Ramanujan belongs to this category. At a deeper level, I believe Riemann also belongs leading to highly original discoveries relying on a strong holistic dimension. However in neither case was the holistic dimension of mathematics explicitly recognised.

So in the future, even greater creative analytic discoveries in various fields will be possible, when mathematical talent is backed up with a truly mature holistic appreciation of symbols.


The second sub-type - while requiring appropriate analytic appreciation (the degree of which depends on the precise context of investigation) - is mainly geared towards the holistic interpretation of mathematical objects.

I would classify my own recent efforts as a most preliminary version of sub-type (b), operating necessarily at a very rudimentary level.

However this is still adequate for example to provide the bones of a distinctive dynamic appreciation of the number system with which to radically re-interpret the nature of the Riemann zeta function (and Riemann Hypothesis).

Thus, subtype (b) is not geared directly to analytic discovery, but rather a coherent dynamic interpretation of mathematical relationships. However because it is most creative at a deep level of enquiry, indirectly it can facilitate exciting new directions for analytic discoveries.


The final subtype (c) entails the most balanced version of both (a) and (b), opening up possibilities for the finest form of mathematical understanding, that is at once immensely productive and highly creative and readily capable for example of synthesising various fields of mathematical study.

One of its great benefits is that it can also provide the ready capacity to appreciate the potential practical applications of new mathematical discoveries.

The reason now why so much abstract mathematical analysis seems irrelevant in practical terms is precisely because it is understood in a manner that completely lacks a holistic dimension.

However with both aspects (analytic and holistic) properly incorporated, the practical applicability of abstract mathematical findings would be more readily intuited.

And from an enhanced dynamic perspective, all mathematical findings (which, when properly understood, are experientially based) have practical applications!

In this regard as a general principle, I would imagine that what is considered most important in abstract mathematical terms, is likewise potentially of greatest significance from an applied perspective.


Another great advantage of subtype (c) is that by its very nature, mathematics can now become readily integrated with the rest of human experience, allowing for the fullest expression of personality development.

Thus if we want a vision of what mathematics might look like at its very best, we would choose subtype (c) with respect to the Type 3 aspect.

However, we are still a very long way indeed from realising this wonderful reality, with the great lack yet of an established holistic dimension to mathematics, serving as the chief impediment.