Illustration of a Crossroads

However before moving on further, I wish to clarify further the true distinction as between the analytic (quantitative) and holistic (qualitative) interpretation of number with reference to the first prime “2”.

In this regard, it may be fruitful to consider the deceptively simple example of a crossroads.

If one - say, travelling N - encounters a crossroads, left and right turns can then be unambiguously identified. Likewise when one approaches the crossroads, travelling S, left and right turns can again be unambiguously identified.

So, with just one absolute frame of reference (i.e. travelling either N or S separately), left and right turns can be defined in an independent manner.

This conforms to 1-dimensional appreciation (which is the basis of standard mathematical interpretation).


However, if one now considers the situation where the crossroads can be approached simultaneously from both N and S directions, identification of left and right turns is rendered paradoxical.

For what is left (when travelling N) is right (when travelling S); and what is right (when travelling N) is left (when travelling S).

And this situation, where appreciation of two reference frames interchanges in complementary fashion, constitutes 2-dimensional interpretation, from a qualitative perspective.

So, we can generalise that where one has appreciation of n reference frames interchanging with each other in mutual fashion, that this constitutes n-dimensional interpretation.

And the key point here is that all integer dimensions (> 1) constitute relative - rather than absolute - holistic frameworks of understanding.


So analytic interpretation, where the dimension is 1, defines conventional mathematics and represents the special absolute case (where relative interaction is ignored).

However all other dimensional interpretations > 1 (where the relativity of 2 or more reference frames is recognised), properly require true holistic appreciation.

Thus, the paradoxical understanding that the turns at a crossroads can be both left and right (depending on the direction from which the crossroads is approached) implies 2-dimensional appreciation of a holistic nature.

So, if we initially identify (when travelling N) a left turn as the 1^st and a right turn as the 2^nd turn respectively, we can see, when now allowing for the simultaneous approach from both N and S directions, that in this context, what is 1^st can also be 2^nd and what is 2^nd can also be 1^st.

And this constitutes the holistic understanding of the number “2” (where both unit members are potentially interchangeable).

So when interpreting turns at a crossroads, the recognition that 1^st can be 2^nd and 2^nd can be 1^st is already necessarily present in an implicit manner.

One thereby potentially realises - before an actual direction of approach is given - that a left (1^st) can equally be a right turn (2^nd) and that a right (2^nd) can equally be a left turn (1^st).

When one is then given one clear unambiguous direction - say approaching the crossroads while travelling N - left and right turns attain a fixed independent meaning.

Thus if the left is identified as the 1^st, this remains fixed in actual terms, so that the right turn can then be unambiguously identified as the 2^nd turn.

And this actual appreciation of left and right turns (from either N or S separately) constitutes the analytic understanding of “2” where both units are considered as independent.


In all mathematical experience, properly interpreted, a continual interaction takes place as between analytic (actual) and holistic (potential) type understanding.

However, quite amazingly, in the specialised formal development of mathematics, the holistic aspect is entirely eliminated from interpretation.

And this has greatly distorted our fundamental appreciation of number!

So properly understood, number represents dynamic interacting patterns, entailing complementary analytic (quantitative) and holistic (qualitative) frames of reference.

However, through several millennia now of reduced interpretation, we have misleadingly convinced ourselves that the nature of number - especially in relation to the primes - is absolute.

And again - except in a special limited sense - this could not be further from the truth!


We can illustrate an additional important aspect of number from the crossroads example.

Using standard mathematical language, if we identify a left turn as + 1, then a right turn is thereby – 1 (not a left turn). So left and right in this context constitute opposite unit directions.

However in the 2-dimensional holistic appreciation of direction, we have seen potentially - before a N or S direction of actual approach is assigned - that what is left is right and what is right is left.

So this implies from the holistic perspective, that in this context, + 1 is equally – 1 and – 1 is + 1. 


We can get some appreciation of what this means from physics, where a matter particle when combined with a corresponding anti-matter particle results in a fusion of energy.

In fact it is very similar here in mathematical terms!

Therefore, when one understands a number - in this case the number “2” - in true holistic fashion, it leads to a psycho-spiritual fusion in energy, which we commonly identify as intuitive insight.

And what is very revealing here is that the understanding, to which such holistic intuition relates, is diametrically opposite to its analytic counterpart.


This in turn throws light on another significant shortcoming with respect to conventional mathematical interpretation.

It may well be recognised by most practicing mathematicians that important creative work is commonly fuelled by intuitive insight.

For example this applied especially to the great Indian mathematician, Ramanujan.

However in the formal presentation of mathematical results, (holistic) intuition is quickly reduced to (analytic) rational type explanation.

However intuition, which indirectly is expressed in a paradoxical (i.e. circular) rational fashion, represents the special contribution of the unconscious mind and is utterly distinct from rational type understanding in conscious terms.

So the truly fundamental role for mathematics, is to properly explain the relationship as between analytic (conscious) and holistic (unconscious) appreciation of its symbols, with a view to establishing their mutual consistency with each other.

And as we shall see, this is the real issue posed by the Riemann Hypothesis. However this crucial observation is destined to entirely elude the profession while the holistic (unconscious) aspect of mathematical understanding is steadfastly ignored.


So from the dynamic perspective that I am proposing here, we really have two extreme limiting positions in the understanding of symbols such as number.

The first extreme is the standard analytic mathematical perspective, where for example, the primes are represented as absolute unchanging forms.

However the opposite holistic extreme leads to the appreciation of numbers in pure relative terms as representing energy states (in physical and psychological terms).

Thus when we appreciate a prime in an actual cardinal manner (comprised of homogeneous independent units) as a “basic building block” of the natural numbers, we are at the analytic (quantitative) extreme of understanding.

However, when we appreciate a prime in true ordinal terms (as potentially comprised of unique interchangeable units) we are then at the opposite holistic (qualitative) extreme of pure intuitive understanding. 

In dynamic interactive terms, these two extremes are complementary in relative fashion, representing the twin aspects of number independence and number interdependence respectively. 

So, if we return again to the crossroads example, the holistic appreciation of “2” corresponds to the intuitive realisation of + 1 and – 1 rapidly interchanging in experience resulting in a pure psycho-spiritual energy state. 

Then when we attempt to reduce such 2-dimensional appreciation in a linear (i.e. 1-dimensional) fashion, through obtaining the two roots of 1, we again get + 1 and – 1 (however this time with positive and negative signs clearly separated).

So we commonly refer to 2, in analytic terms, as 2¹ (i.e. as a number on the 1-dimensional number line). Here 2, represents the independence of its homogeneous individual units.

However, in relative holistic terms, 2 is thereby properly expressed as 1². So 2 here represents the potential interdependence of its unique individual dimensional units.

And when we attempt to express this relationship i.e. x² = 1, in 1-dimensional terms, x = + 1 and – 1 (now as actual separated units).


This points to a hidden unrecognised aspect of the number system, where again to use a very close analogy from physics, number keeps switching as between its particle and wave identities.

So when in conventional terms, we square 1 (i.e. 1¹) we get 1², which could geometrically be represented, in analytic terms, as a square (of side 1 unit).

However when we attempt to return to this former 1-dimensional expression, through obtaining the square root, we now get two answers (+ 1 and – 1) that lie on the unit circle (in the complex plane).

Thus in moving from 1-dimensional to 2-dimensional format, the very nature of number changes from its analytic (particle) to (holistic) wave aspect. Thus number possesses both linear (independent) and circular (interdependent) aspects that keep switching in the dynamics of understanding.

Therefore the resultant roots relate to a different notion of number, which is relatively of a holistic (wave) nature, than the analytic (particle) notion with which we started.

The crucial conclusion again is that all numbers possess both analytic (particle) and holistic (wave) aspects, which in dynamic interactive terms, keep switching in two-way fashion as between their base and dimensional (and dimensional and base) aspects. (For this purpose, with respect to a number such as 1², 1 is the base and 2 the dimensional aspect respectively with both aspects continually switching in experience as between their quantitative and qualitative meanings! )