Clarifying Analytic and Holistic Interpretations of "2" (9)

Before moving on, I will briefly here explain what the holistic interpretation of dimensional numbers relates to!

All experience is conditioned by two fundamental sets of polarities which can be referred to as internal/external and whole/part respectively.

So in both physical and psychological terms (which in dynamic terms are complementary) a ceaseless interaction takes place as between these two opposite sets of polarities.

In geometrical terms we can represent these as 4 equidistant points on the unit circle (in the complex plane) with the sets horizontal and vertical with respect to each other.

These 4 points (i.e. + 1, – 1, + i and – i) then represent the indirect quantitative interpretation corresponding to the 4 dimensions.
The startling implication of this is that the qualitative notion of "4" (i.e. "fourness") implicitly implies the unconscious recognition of both real and imaginary opposites in their unified state!

Then we attempt to express this state of "fourness" in an indirect quantitative manner, (conforming to 1-dimensional interpretation) it entails the circular paradox of opposites in both real and imaginary terms. 

Thus once again our intrinsic experience of  dimensions - as relating to the qualitative aspect of number - relates directly to the unconscious (intuitive) aspect of experience.

Furthermore - though this goes beyond our present purposes - the intrinsic notion of space and time in physical terms, when appropriately understood, conforms directly with this mathematical interpretation of dimension (as the holistic qualitative nature of number).  

However, once again, in standard mathematical terms, based on merely (conscious) rational notions, the qualitative aspect is invariably reduced in a merely quantitative manner!


Just as with a compass, we can then view 4 dimensions as representing four equidistant points on the unit circle (represented by the four roots of 1).

Then all other dimensions can be viewed as representing unique configurations with respect to these 4 basic co-ordinates.

Now we can express the n roots of 1 by the equation,

1 = s^t , i.e. 1 – s^t  =  0.

Now as we have seen, in all cases one of these roots i.e. 1 –  s = 0, (where s = 1) represents the 1-dimensional case. Here ordinal (qualitative) meaning of number is directly reduced in a cardinal (quantitative) manner.   

Therefore to isolate the remaining non-trivial roots (where a distinct ordinal meaning is preserved) we divide 1 – s^t  =  0, by 1 –  s = 0 so as to obtain,


1 + s¹  + s²  + s³  +….. + s^{t – 1}    = 0

I refer to this as the (finite) Zeta 2 function which is designed to complement the well know Riemann zeta function (which I refer to as the Zeta 1 function).

Therefore using this terminology,

ζ₂(s) = 1 + s¹  + s²  + s³  +….. + s^{t – 1}    = 0.

Now initially I define this function for prime values of t (where solutions for s are unique).

Therefore in the simplest case where t = 2,

ζ₂(s) = 1 + s¹   = 0.

So s (i.e. s¹)  =  – 1.

This therefore represents the first of - what I refer to as - the Zeta 2 zeros. 

Now, the crucial role that this zero plays is that it enables one to "convert" the inherently qualitative notion of the 2nd (of two dimensions) indirectly in a quantitative manner.

And as we have seen (with the example of the crossroads) the very way we recognise a right turn when the other turn is posited as left is through negation of this polar direction. 

So right (in this context) is  – 1 (with respect to the left turn as + 1).

So what is happening here - though in this case it seems intuitively obvious - is that we are recognising that in simultaneous two-dimensional terms (when both N and S directions of approach are considered) that the two turns are paradoxically both left and right.

Therefore when we isolate a direction unambiguously as left (when approached in a N direction) then - relatively in this context - the 2nd turn must be right.

However we can switch the frames of reference through approaching the crossroads from the alternative S direction. So now - what was formerly identified as right (the 2nd turn) is now + 1 and what was formerly left is now – 1.

So it is important to realise that + 1 and  – 1 in this context are merely relative and interchangeable.

In other words though we we are using  quantitative means of expression to represent left and right turns, the two numbers + 1 and  – 1 and are now understood in a relative interdependent manner.

So we can perhaps now sum up the crucial difference as between the quantitative and qualitative notions of "2".

With the quantitative notion of "2", the two units are independent. (So + 1 is absolutely separated from – 1).

By contrast with the qualitative notion of "2" (as "twoness") the two units are interdependent with each other (and thereby interchangeable). (So + 1 can only be relatively separated from – 1 in an arbitrary context, which can be reversed where + 1 and  – 1 now switch positions with each other.

Pu more simply, the quantitative notion of "2" concurs with linear (either/or) logic; the qualitative notion of "2" concurs with circular (both/and) logic.      

Clarifying Analytic and Holistic Interpretations of "2" (8)

I have been clearly proposing that Mathematics is now in need of discovering a radical new manner of interpreting its symbols, which is directly based on "higher" dimensional understanding of a refined intuitive nature. Indirectly such understanding can be expressed in a circular rational fashion, that appears paradoxical in terms of conventional linear (i.e. 1-dimensional) interpretation.

And as we have seen the circular number system, based on the various roots of 1 (in the complex plane), when given an appropriate holistic interpretation, provides the vehicle for expressing such "higher" dimensional understanding in an indirect quantitative manner.

In this way, 2-dimensional interpretation - though the simplest to appreciate - serves as the prototype for the understanding of all other dimensions (> 2).

However, one may validly ask! If such understanding is completely unrecognised - at least in formal terms in present Mathematics - where can it be found?

Though I am dealing with this issue at length in some of my other blogs, I will briefly summarise my findings here.

The full spectrum of (potential) psychological development - as I see it - comprises three main areas.

The first relates to the specialisation of conscious type understanding (which I classify as 1-dimensional interpretation).
In our present intellectual culture - especially with respect to Mathematics and the Sciences - such interpretation (of an analytic nature) largely defines all recognised discourse.

The second relates to the specialisation of unconscious type appreciation of a highly refined spiritual intuitive nature (which relates to all dimensions (≠ 1).

Indirectly as we have seen such appreciation (of a true holistic nature) can be indirectly expressed in a precise manner, though circular rational interpretation (that appears paradoxical in terms of conventional understanding).

Historically such appreciation (of a deeply contemplative nature) has found expression in the great mystical traditions, East and West.

Unfortunately, its vast potential implications for the understanding of both Mathematics and Science have not yet been investigated to any significant degree.

So maybe we are only now reaching for the first time in history, the point where it can be generally recognised that - like the electromagnetic spectrum - that further bands of distinct intuitive energy exist, that have vast repercussions for our understanding of even the most commonplace mathematical relationships.

Indeed I am convinced that the greatest revolution in our intellectual history - much more profound than the physical revolutions of the 20th century - now awaits.

The third band of potential development - which I generally refer to as "radial" - is then concerned with the increasing specialised interaction of both conscious and unconscious aspects of understanding.

Now, clearly we are not remotely at this stage in Mathematics (which would require initially substantial progress with respect to the unrecognised holistic aspect of appreciation).

If however we were to look at some future golden age of Mathematics, I would outline 3 key areas.

1) Analytic - where specialisation of quantitative type relationships takes place. We are currently at that stage in our mathematical development.

2) Holistic - where corresponding specialisation of qualitative type relationships takes place. Consideration of this aspect has all but been completely removed from accepted understanding.

3) Radial - where growing specialisation with respect to the interpenetration of both the quantitative and qualitative aspects of mathematical relationships will take place. This offers therefore by far the greatest scope for the most creative and productive development of Mathematics.

In fact ultimately, both 1) and 2) can only be coherently interpreted in the context of the more comprehensive understanding of 3). 

This means therefore, for example, that the analytic aspect (of quantitative interpretation), which of course will continue to develop and flourish in its own right, will ultimately be interpreted in a merely relative - rather than absolute - fashion. 

However, though its relevance has not been directly appreciated with respect to Mathematics, 2-dimensional understanding has already played in different ways a significant role with respect to cultural understanding.

It appears for example in the paradoxes of the Greek philosopher Heraclitus in such statement as 

"the way up is the way down; the way down is the way up"

It also plays - as one would expect a major role in the teachings of the major spiritual traditions e.g. Taoism.

And this famous Buddhist sutra offers a striking example

"Form is not other than Void;

Void is not other than Form.

In philosophical terms it shows up in the approach of Nicholas of Cusa  with his "coincidence of opposites" (and who perhaps comes closest to seeing its distinctive mathematical relevance").

It plays a major role in the dialectic of Hegel.

It also plays a key role in Jungian Psychology (where conscious and unconscious are seen to play complementtary roles with respect to development).

It even plays an important role in quantum physics (though the full philosophical implications of this have not yet been recognised).

So for example it is now accepted that physical phenomena such as light can manifest themselves in two complementary ways (as particles and waves). 

In fact I am stating something quite similar in emphasising than number likewise can manifest itself in two complementary ways (which I refer to as Type 1 and Type 2 respectively).

So the Type 1 aspect - like particles - relates to the distinct independent features of number. However by contrast the Type 2 aspect - in a manner akin to waves - relates by contrast to the interdependent features of number (where it gets spread out as it were over all related numbers).

In fact I would go much further!

The quantum mechanical features of physical reality, such as the complementary particle/wave nature of matter, have their deepest roots in the very nature of number (when correctly understood in a dynamic interactive manner).