Number as Dimension

We return here once more to the clarification of the notion of dimension as used in Mathematics. This is vital in turn for the clarification of what is meant by the quantitative and qualitative aspects of number.

Unfortunately because Conventional Mathematics is based inherently on a reductionist fallacy i.e. that number can be understood with respect merely to its quantitative aspect, it is perhaps not surprising that this key issue is effectively avoided.

 

If we define numbers as independent in an absolute quantitative sense, then this begs the question as to how numbers can be successfully related with each other (which requires the qualitative notion of interdependence).

The very fact that this is not readily appreciated as of the most fundamental importance only goes to show how ingrained this reductionist interpretation of number has become. In other words, we assume that this interdependent aspect of number behaviour (whereby numbers assume a qualitative relationship with each other) can be successfully understood in a merely quantitative manner!

 

The true inherent meaning of dimension is of a qualitative nature and relates to the manner in which the fundamental polarities of experience i.e. external/internal and whole/part are related.

1-dimensional interpretation in this context simply entails the attempt to understand such relationships in a uni-polar manner (i.e. using just one pole as an exclusive frame of reference).

So for example all mathematical experience necessarily entails the dynamic interaction of objective subjective (cognitive) aspects that are relatively external and internal with respect to each other.

1-dimensional thereby implies then that we fix interpretation with just one pole in an absolute manner.

So this leads to the standard view of numbers as absolute entities existing in an objective manner. Though one may recognise that strictly such numbers cannot have experiential meaning without the existence of mental constructs, somehow a belief persists that an absolute correspondence applies to both the external objects and the internal constructs.

In other words Conventional Mathematics essentially operates on the illusion that mathematical objects have an absolute existence (independent of our relationship with them).

1-dimensional interpretation equally leads to the standard view of number as existing in an independent quantitative manner. Though once again it may be recognised that a general dimensional context is required to enable an ordered relationship of these numbers, somehow the belief remains that this can be done in a merely quantitative manner.

In truth, the general context providing this capacity for ordered number relationships is of a qualitatively distinct nature. However a remarkable denial of this key fact pervades conventional mathematical interpretation.

So once again its 1-dimensional nature is demonstrated by the manner in which the qualitative dimensional aspect is reduced in a merely quantitative manner.

Now of course Conventional Mathematics can indeed give meaning to dimensions (≠ 1) in a quantitative manner.

So for example from this perspective 2³ = 8 (i.e. 8¹). Thus, though the qualitative context has here changed through use of 3 (as dimensional number) the numerical result is given in a reduced quantitative manner (in terms of 1 as dimensional number).

It is extraordinarily important therefore to grasp that Conventional Mathematics is defined in qualitative terms by its merely 1-dimensional nature.

This effectively means that variables are treated in an absolute - rather than relative - manner (where relative implies  dynamic interaction as between the opposite polarities that condition all phenomenal  experience).

When one grasps this point, one can then clearly recognise not alone why the Riemann Hypothesis can have no proof, but even more importantly why its true nature cannot be successfully understood in conventional mathematical terms!

 

As we know the only dimensional value (in quantitative terms) where the Riemann Zeta Function remains undefined is for s = 1.

In the more comprehensive understanding of this Function this also implies that only dimensional value (in qualitative terms) for which the Riemann Zeta Function remains undefined is also for s = 1.

This means that the Riemann Zeta Function (and associated Riemann Hypothesis) cannot be properly understood in the conventional mathematical manner.

Thus the Riemann Hypothesis essentially relates to:
(i) the condition with respect to number where both external (as objective results) and internal (as mental interpretation) are successively reconciled as ultimately identical and
(ii) the condition where both the quantitative (cardinal) and qualitative (ordinal) nature of number are likewise reconciled (as ultimately identical).

Therefore we cannot attempt to understand this relationship - which entails the dynamic interaction as between opposite polarities - in a reduced absolute sense (where objective results are divorced from cognitive interpretation and the quantitative aspect of number likewise divorced from its qualitative aspect). This is why the Riemann Zeta function cannot be successfully understood in a conventional (i.e. 1-dimensional) manner.

I can say this with considerable confidence having already been dimly aware of the problem from about the age of 10.

Even then I was seriously questioning conventional procedures. This started the long journey to get to the bottom of the problem (as I saw it) in the hope of offering a more authentic mathematical approach.

And after more than 50 years on this journey I believe that I have managed to come up with at least the general framework for a more comprehensive appreciation of Mathematics.

I have recounted before how I found as a child the conventional explanation of a square root deeply unsatisfactory.

For me there it seemed that an essential symmetry should be preserved as between the notion of a square on the one hand and a square root on the other.

So for example  we start with 1 and square we get - apparently - one unambiguous answer i.e. 1². However when we then get the square root we now have two possible answers + 1 and – 1.

The conventional explanation seemed to me even at this young age deeply illogical.

We would not accept in terms of the proof of a theorem for example that it could have equally have in qualitative terms a negative as well as positive truth value. This would be like saying that we could accept the proof of the Pythagorean Theorem for example as either true or false. However in the parallel quantitative context (in the context of a square root) it was indeed maintained that a number could have either positive or negative values! 

So I started to suspect - though I would not have been able to articulate my thoughts then in a coherent manner - that the qualitative nature of 2 (as a dimensional number) was quite distinct from 1.

And as Conventional Mathematics is defined qualitatively in terms of its merely 1-dimensional nature, this opened up the possibility of entirely distinctive logical approaches to Conventional Mathematics.  In other words the very inconsistency that in could see in the standard explanation of the two roots of 1 was due to the fact that 1-dimensional (either/or) logic was not adequate to explain this - apparently simple - problem.

Many years later (after long immersion in Hegelian philosophy and the wisdom of the great spiritual traditions) I was able to return to this problem with what I considered was a satisfactory answer.

Whereas 1-dimensional logic is characterised in an absolute (linear) either/or manner, 2-dimensional logic is characterised by a relative (circular) both/and approach. This then leads to paradox in terms of the 1-dimensional approach.

The Greek philosopher Heraclitus summed this 2-dimensional logic up well in his statement,

“The way up is the way down; the way down is the way up”

What is involved here has profound consequences for all mathematical interpretation.

 

If one fixes direction in terms of just one pole either “up” or “down” then movement along a road is unambiguous, whereby it can be consistently defined in terms of the given frame of reference.

If one now alters the frame of direction (in the opposite manner) then again unambiguous directions can be given in terms of this new reference frame.

 

So fixing polar reference frames (with just one pole as independent) i.e. as 1-dimensional, leads to unambiguous answers of an absolute nature.

 

However when we now simultaneously try to relate both reference frames as interdependent i.e. as 2-dimensional, this leads to paradoxical answers (in terms of 1-dimensional logic).  So what is “up” or “down” in this sense is purely dependent on context.

I have come to realise over the years that - quite remarkably - Conventional Mathematics, because of its 1-dimensional nature, is totally lacking any genuine notion of interdependence (and thus always reduces this notion, in any relevant context, to independence).

Alternatively we could say that Conventional Mathematics is totally lacking any genuine qualitative or holistic notion (thereby reducing it in a merely quantitative analytic fashion).  

So getting back to our example on directions, the directions “up” and “down” in 2-dimensional terms can be represented as + 1 and – 1  in relation to each other. However these are now understood in a merely relative fashion with positive and negative depending on context.

The deeper implication is that where the dynamic interdependence of two polarities is concerned 2-dimensional - rather than 1-dimensional - interpretation is required.

This intimately applies therefore to the interpretation of mathematical symbols which are inevitably conditioned by such dynamic interaction in experiential terms..So internal and external and quantitative (part) and qualitative (whole) polarities continually interact in experience and are related to each other in a dynamic complementary manner.

Thus coming back to the square of 1 and the corresponding square root of 1, one can perhaps appreciate now that this properly requires 2-dimensional - rather than 1-dimensional - interpretation.

 

Thus when we square 1 i.e. 1², we move - literally to 2 dimensions (which qualitatively are defined as both + 1 and – 1 in relation to each other (depending on context).

Now when we get the square root we are attempting to express these two polarities in a reduced absolute fashion. So what is both + 1 and – 1 (in 2-dimensional terms) becomes either + 1 or – 1 (in a 1-dimensional format) .

Strictly, whereas the cardinal (quantitative) notion of 2 represents - literally - a whole unit (without qualitative distinction), the corresponding qualitative notion of 2 entails its two ordinal members as individual units i.e. 1^st and 2^nd (without quantitative distinction).

Thus the two roots of 1 are obtained with respect to 1¹ and 1² respectively i.e. 1^1/2 and 1, = – 1 and + 1.   

Therefore, 1-dimensional interpretation is characterised by the use of single independent frames of reference (with respect to polar opposite interaction) in an isolated manner.

It thereby entails linear (either/or) logic in rational terms.

2-dimensional interpretation entails both 1^st and 2^nd dimensions. So initially it necessarily entails the 1st dimension in making unambiguous distinctions based on single independent reference frames. However there is now a clear recognition that these can now be made from two opposite directions!

Then the 2^nd dimension entails the simultaneous integration of both reference frames (as complementary opposites) where both are seen as interdependent. In a direct sense this implies holistic recognition of an intuitive kind (pertaining to the unconscious). However it is then indirectly translated in a circular logical (both/and) manner that appears paradoxical in terms of linear reason.

 

2-dimensional interpretation represents the minimum necessary to understand the number system in its inherent dynamic interactive nature, allowing for both analytic (quantitative) and holistic (qualitative) appreciation of mathematical variables or even more simply both the (relative) independence and interdependence of mathematical variables.

In an important sense, as I have explained in previous blog entries, all other natural number dimensions can ultimately be expressed in a 2-dimensional fashion.

From my early 20’s I spent several decades developing - what I referred to as - Holistic Mathematics in recognition of its completely neglected qualitative aspect.

Initially this quest was largely driven by the realisation that a qualitative mathematical interpretation could potentially be given for all stages of human and physical transformation (including rare contemplative states). With a highly developed contemplative state, the dynamic interaction as between the key polarities (underlying all phenomenal recognition) becomes increasingly more refined corresponding to ever higher number dimensional configurations.  In my own work I especially concentrated on the nature of interpretation corresponding to 2-dimensional, 3-dimensional, 4-dimensional and 8-dimensional appreciation respectively!

However it is only in the last decade that I have seriously sought to explore the implications of all this for appreciation of key mathematical problems such as the nature of the number system and the Riemann Hypothesis.

I am now of the firm opinion that despite a veneer of great rigour with its ultra-specialised understanding of so many topics, at a fundamental level, standard interpretation represents a greatly confused mess of highly reduced notions. (These unfortunately have become so reduced through the long unchallenged consensus regarding their use, that an almost total blindness regarding their shortcomings now exists).

I used to be of the opinion - while developing the importance of holistic mathematical notions - that standard interpretation would remain largely valid with respect to quantitative appreciation.

 

However I have come to clearly realise that all mathematical notions - including of course number - are properly of a dynamic relative nature. As quantitative and qualitative aspects are ultimately interdependent, it is therefore not possible to understand number properly in a merely reduced quantitative manner.

For example the correct appreciation of 1 and 2 in a qualitative ordinal manner (as 1^st and 2^nd respectively) requires 2-dimensional interpretation and therefore cannot be properly explained in conventional terms.

 

When one realises how quantitative and qualitative aspects are inevitably intertwined with respect to the appreciation of number, then the key issue  arising relates to the ultimate consistency of both aspects.  This indeed is the central message of the Riemann Hypothesis, which therefore can have no strict meaning in conventional (i.e. merely quantitative) terms.

Nothing less than a total revolution is now required in our mathematical understanding. This of course likewise entails a total revolution in what is meant by science.

I hope readers to this blog can get some sense of the importance of what is involved. Successful transformation with respect to our present civilisation urgently depends on the rapid realisation of its many implications.