Clarifying Analytic and Holistic Interpretations of "2" (7)

We saw yesterday how ordinal notions represent the reduced expression, in a 1-dimensional quantitative manner, of  "higher" dimensional qualitative notions of number.

Therefore the notions of 1st and 2nd (with respect to 2 numbers) represent the reduced interpretation of the two related "higher" dimensions (corresponding to 2).

In an indirect quantitative manner, these are expressed as the two roots of 1 (i.e. of 1¹ and 1² ) i.e. 1^1/2  and 1^2/2 which are  – 1 and + 1 respectively.

Though these two roots, – 1 and + 1, are clearly separated (at the reduced 1-dimensional level), they are understood intuitively as interdependent with each other at the corresponding "higher" 2-dimensional level of understanding, where the true qualitative nature of "2" (as "twoness") is directly apprehended.

However, indirectly their qualitative nature can be indicated at the 1-dimensional level through obtaining the sum of the two roots i.e. – 1  + 1 = 0.

In other words, the qualitative identity of "twoness" where "both" units are directly understood as interdependent, is strictly without any quantitative identity i.e. = 0!

Therefore, when we adopt this holistic interpretation of the nature of number, the various roots of 1 assume an extraordinary new importance.

So in general terms, the n roots of 1 provide the indirect quantitative means of expressing the notions of 1st, 2nd, 3rd,...., nth (with respect to a group of n numbers).

The "higher" dimensional interpretation here directly refers to the intuitive qualitative realisation of the group interdependence of the n units (which would be understood as clearly independent in an individual manner at the conventional 1-dimensional level of interpretation).

Put another way, at the standard 1-dimensional level of rational quantitative interpretation, each unit has an absolute fixed identity. 

However, by contrast, at the n-dimensional level of direct holistic intuitive appreciation, each unit is directly understood in intuitive terms as interdependent with every other unit. This now creates circular paradox in terms of standard analytic interpretation at  a rational level, as it implies for example that the 1st unit can equally be the 2nd, 3rd,...., nth unit. Equally each unit can switch identity with each other unit depending on the initial relative perspective that is adopted.

Once again we saw this clearly at work in the case of 2 dimensions with respect to the understanding of a crossroads.

So when we identify the left turn (heading N) as the 1st direction, the right turn (heading S) is the 2nd direction. However when we switch reference frames so as to approach the crossroads (heading S) the right turn is now the 1st and the left turn the 2nd direction respectively.

And in (indirect) quantitative terms, when the left turn (heading N) is + 1, the right turn (heading N) is – 1. However when the right turn (heading S) is + 1, then the left turn (heading S) is – 1.

So 1-dimensional rational interpretation (within isolated N and S polar reference frames) leads to unambiguous notions of fixed identity i.e. as 1st and 2nd in any context.

However 2-dimensional intuitive appreciation (within simultaneous N and S polar reference frames) leads to paradoxical notions of a merely relative identity (where 1st and 2nd switch locations)

In corresponding quantitative terms, 1-dimensional rational interpretation leads to unambiguous notions of separate identity in quantitative terms with respect to + 1 and  – 1.

However 2-dimensional intuitive appreciation, leads to paradoxical notions of  relative identity in qualitative terms, where + 1 and – 1 are interchangeable.   

Now it should be apparent from the crossroads example, that the consideration of 2-dimensional appreciation, necessarily includes 1-dimensional. So we cannot attempt to understand the nature of interdependence at the 2-dimensional level, without first considering independent identity at the 1-dimensional. 

And just as 2-dimensional includes 1-dimensional interpretation, likewise n-dimesnional includes 1, 2, 3,...., n – 1 dimesnional interpretation.

Therefore once again, the n roots of 1 holistically relate to the quantitative interpretation of n dimensions in reduced 1-dimensional terms, where the notions of 1st, 2nd, 3rd, ...., nth now can attain a relatively separate independent identity (depending on context).            

So one of the n roots of 1 (strictly n roots 1¹, 1² , 1³ ,......, 1ⁿ) is always necessarily 1, reflecting the 1st dimension, which defines the nature of conventional mathematical interpretation.

In other words ordinal notions of 1st, 2nd, 3rd etc which strictly have a qualitative - rather than quantitative identity - are simply reduced in conventional mathematical terms to their corresponding cardinal notions.

Therefore, though properly understood in qualitative terms, the notions of 1st, 2nd, 3rd are merely arbitrary (depending on context) they are given an absolutely fixed identity in conventional mathematical terms. So the the notion of 1st is identified unambiguously with the 1st unit (where it = 1). The notion of 2nd is likewise unambiguously identified with the 2nd unit (= 1), the 3rd unambiguously with the 3rd unit (= 1) and so on.

So to return to the simple example of "2"!

In conventional cardinal terms this is expressed as 

2 = 1 + 1.

Now if one likewise insists on an ordinal expression,

2 = 1st unit + 2nd unit.

However, as we have seen, by the very nature in which ordinal notions are defined in conventional mathematical terms,

this reduces to 2 = 1 + 1 (i.e. the cardinal definition).

Thus the massive reduction (in every context) of qualitative to quantitative type notions, is directly replicated in conventional mathematical terms by the corresponding reduction of ordinal to cardinal type notions.  This reflects the attempt to give number notions a merely fixed independent identity in quantitative terms. However the very ability to relate these numbers with each other requires a subtler interdependent  identity (where numbers can continually interchange with each other).

Once again this latter aspect appreciation, requires an entirely new holistic mathematical approach.

Now only does this have an immense unrecognised potential in its own right, but it is also vitally necessary to provide a coherent framework for all quantitative interpretation.       

Clarifying Analytic and Holistic Interpretations of "2" (6)

I mentioned before that the key issue with respect to Mathematics - especially in relation to number - is to ensure the consistency with respect to interaction of both its quantitative and qualitative aspects of interpretation.

And remember both of these aspects are necessarily involved with respect to a coherent dynamic appreciation of number!

So in effect this requires from one perspective, that a satisfactory means be found to convert the qualitative aspect of number interpretation (indirectly) in a quantitative manner; equally from the other perspective, a satisfactory means is required to convert the quantitative aspect of interpretation (indirectly) in a qualitative manner.

I then began to realise that the ordinal notion of a given number could be (indirectly) converted in a quantitative manner through obtaining the corresponding roots of 1.

Let is illustrate carefully what is unloved here with respect to the simplest case relating to the number "2".


Now the qualitative notion of "2" is expressed through the Type 2 aspect of the number system as 1².

 "2" here relates to the interdependent identity of two dimensions (i.e. directions), that strictly in this qualitative state have no (independent) quantitative identity. In fact - just as with the fusion of a matter and anti-matter particle in physical terms - "2" here refers directly to an energy state (from both physical and psychological perspectives).

Therefore when one appreciates the inherently dynamic nature of number, it is no longer feasible to attempt to abstract it in an absolute manner from the natural phenomena of experience. Rather number now represents the encoded nature of such phenomena (with these phenomena in turn representing the decoded nature of number). So in this sense a two way interaction characterises the relationship of number and natural phenomena (in physical and psychological terms).

So just as the Type 1 (quantitative notion of number) has strictly no direct meaning in qualitative terms, likewise in a complementary dynamic manner, the Type 2 (qualitative) notion of number has likewise no direct meaning in a quantitative  manner.

So once again in this context, 2¹ represents the Type 1 (quantitative) expression of "2", whereas 1²  represents the corresponding Type 2 (qualitative) expression.

Once again from a Type 2 perspective, neither of the two dimensions (directions of 2) can be separated! In psychological terms this entails an intuitive recognition of "2" as representing the interdependence of opposite external and internal polarities (in psychological terms). In a corresponding physical manner, it represents an energy state (again as the interaction of opposite poles).

However by now attempting to express this (qualitative) 2-dimensional state in the standard 1-dimensional manner, we can, in a sense, now indirectly express these two dimensions in a quantitative type manner.

In effect ,this entails obtaining the two roots of 1 (or more precisely the roots of  1² and 1¹ respectively).

Remarkably when now expressed in this manner, the two roots give quantitative expression to the qualitative notions of 1st and 2nd in the context of 2 members.

Therefore the intuitive notion of "twoness" (where dimensions cannot be separated), gives way, at the 1-dimensional level (based on the separation of such polarities), to the ordinal notions of 1st and 2nd These then can be indirectly expressed in a quantitative manner as + 1 and  – 1 respectively.


Therefore the notion of "twoness" - though again directly of an intuitive nature - is represented at the common rational level of discourse as relating to 1st and 2nd units (which can interchange locations).

We have already had a good example of this in the representation of the directions of a crossroads, where in relation to each both left and right are understood to be paradoxical.

So once again, though the understanding that a turn can be both left and right defies normal rational logic (in a 1-dimensional manner) it can be intuitively grasped (in 2-dimensional terms). Then when we express this in a reduced manner (as 1-dimensional) if the 1st turn is "left", the 2nd turn is thereby - relatively -"right". Likewise if the 1st turn is now "right", the 2nd turn is thereby - relatively - right.

And as we have seen in a quantitative manner, when the 1st turn as "left" is posited as + 1, the 2nd turn as "right" is thereby negated as – 1. However when the 1st turn as "right" is then posited as + 1, the 2nd turn as "left" is now thereby negated as – 1.

So the clear intuitive realisation of "twoness" (at the 2-dimensional) level can only be approached in rational (1-dimensional) terms in a circular manner, through paradox.  

Thus once again the intuitive notion of "twoness" is expressed at the rational level in a relative manner as both 1st and 2nd members (where 1st is equally 2nd and 2nd equally 1st).


This leads to the fascinating conclusion that ordinal notions, which are deeply implicit in our handling of all quantitative mathematical relationships at the accepted 1-dimensional level, in fact reflect the reduced conscious understanding of "higher" dimensional relationships that are directly apprehended in an intuitive (unconscious) manner!

Thus, the notions of 1st and 2nd (in the context of 2 units) requires 2-dimensional understanding in qualitative terms.

Then the corresponding notions of 1st, 2nd and 3rd (in the context of 3 units) requires 3-dimensional understanding in qualitative terms. The very meaning of 1st and 2nd now changes in the context of this "higher" dimensional appreciation!

In more general terms, the notions of 1st, 2nd, 3rd,......., nth (in the context of n members) requires n-dimensional understanding, in qualitative terms.

Therefore, our standard ordinal notions cannot be properly understood within the conventional mathematical framework, as it is geared to merely quantitative type appreciation of an analytic nature.

In fact a radically distinct form of Mathematics, which I customarily refer to as "Holistic Mathematics", is required. In other words - rather than the customary quantitative (analytic) interpretation of mathematical symbols - a distinctive qualitative (holistic) appreciation is required.

Thus every mathematical symbol has given both quantitative (analytic) and qualitative (holistic) aspects, both of which necessarily interact in the dynamics of experience.


The quantitative (analytic) aspect directly relates to rational interpretation (corresponding to 1-dimensional understanding). However indirectly this must be implicitly fuelled by intuitive type appreciation (of a qualitative nature).

The qualitative (holistic) aspect, by contrast, relates directly to intuitive appreciations (corresponding to "higher" dimensional insight). However indirectly this is then expressed in a quantitative type manner through circular (paradoxical) rational type relationships.


In conventional mathematical terms, the ordinal type appreciation of number is simply reduced in cardinal type fashion. In other words the qualitative aspect of number understanding is reduced in a quantitative type manner. We will see precisely how this happens in the next blog entry.

Clarifying Analytic and Holistic Interpretations of "2" (5)

When one considers carefully in conventional terms the simple mathematical relationship,

1 + 1 = 2,

a strange paradox can be shown to arise.

Once again because these two cardinal units are treated as homogeneous and independent (in a merely quantitative manner) this thereby creates an unavoidable uncertainty with respect to their corresponding ordinal identity (as qualitative).

Therefore, each unit can in turn be ranked as 1st or 2nd in relationship to each other!

Thus the very ability to rank the numbers in a fixed ordinal manner unambiguously as 1st and 2nd respectively, requires some arbitrary qualitative criterion, that is not inherent in the cardinal identity of the numbers.

For example if I look at two cars in a driveway, to unambiguously order them as 1st and 2nd, some qualitative criterion must implicitly be used.

If I base my decision - say - on the age of the cars (ranking the most recently registered as 1st) then - provided they do not have the same registration date - this problem of ranking can be solved.

However according to an alternative qualitative criterion - say the sizes of the respective cars - this ranking could be reversed with what was 1st (on the previous ranking) now 2nd and what was 2nd now 1st.

The implication of this is that in the absence of any qualitative criterion, each unit can potentially be ordered (i.e. related in an interdependent manner) as both 1st and 2nd.

Then in the more general case of the number n, each unit can potentially be ordered in terms of n ordinal positions.

So for example in the case of 1 + 1 + 1 + 1 + 1 = 5, each of the units here can potentially assume 1st, 2nd, 3rd, 4th and 5th positions in relation to each other.

Thus though from a cardinal (quantitative) perspective, each unit is given one fixed identity in actual terms, from the corresponding ordinal (qualitative) perspective, each unit can freely range over all numbers in a potential manner.

And once again from a psychological perspective, in mathematical terms, rational understanding - of an analytic nature - is directly identified with the fixed unambiguous nature of actual relationships; intuitive appreciation by contrast is directly identified with the corresponding extended nature of potential relationships (which appears paradoxical from the rational perspective).


So clearly once again both the cardinal (quantitative) and ordinal (qualitative) nature of number relationships can only be properly understood in a dynamic interactive context that entails both analytic (rational) and holistic (intuitive) type appreciation.

Put another way we can only hope to coherently understand mathematical relationships when both the conscious and unconscious aspects of understanding are properly integrated with each other in a balanced manner.


Because, as we have seen quantitative and qualitative aspects of number relate to distinct domains of understanding, the fundamental question then arises as to how they can be consistently related to each other.

Let's be clear here! This is by far the most fundamental issue that can be raised with respect to mathematical activity, yet it cannot even be recognised in conventional mathematical terms!

Once again in conventional terms the qualitative aspect is simply reduced in quantitative terms, so that this key issue of mathematical consistency cannot be approached from this restricted perspective!

And though we are running slightly ahead of ourselves at this stage, when properly understand this is of direct relevance to appreciation of the key significance of the famed zeta zeros!

For it is the very nature of the zeta zeros to provide this key interface as between the quantitative and qualitative aspects of mathematical understanding.


So looked at in an equivalent manner, if mathematical relationships can indeed be assumed to be consistent, we are faced with the key issue of somehow converting the qualitative type nature of number (indirectly) in a quantitative perspective.

Equally from the alternative perspective, we have the complementary issue of converting the quantitative type nature of number (indirectly) in a qualitative manner!

For example ordinal notions of 1st, 2nd and 3rd etc. do not possess a direct quantitative meaning. Rather they strictly relate to the qualitative relationship as between an interdependent group of numbers!

However, remarkably, we can indeed give these ordinal notions an indirect quantitative meaning in a fascinating manner.

Now once again, this issue does not arise in conventional mathematical terms, where ordinal meaning is quickly reduced in a merely quantitative manner.

I will explain precisely how this happens in future blog entries.

Clarifying Analytic and Holistic Interpretations of "2" (4)

Though the qualitative notion of "2" can - as we have seen - be represented through the Type 2 aspect of the number system (i.e. 1²) as "twoness", the problem is that this does not directly relate to rational type understanding.

Indeed if we go back to our example of the crossroads, the very manner by which one recognises that a turn can be simultaneously "left" and "right" illustrates the qualitative notion of "twoness".

Once again, when we use just one isolated polar reference frames (i.e. 1-dimensional interpretation) a turn at a crossroads is unambiguously either "left"or "right". So if one is heading N for example towards the crossroads,  a left turn can be unambiguously identified (or posited). So one can refer then to this left turn as + 1. This in turn means that the right turn is thereby excluded (or negated) in this context as – 1.

Now if one now changes the polar reference frame and heads S from the opposite direction towards the crossroads, once again the left turn can be unambiguously identified. So again this can be posited as + 1, with the excluded (or negated) right turn as – 1.

However what was unambiguously a left turn, when heading N is likewise unambiguously a right turn heading S. And what was unambiguously a right turn heading N is now unambiguously a left turn heading S.

Therefore when when we simultaneously combine both polar reference frames (N and S) in what represents 2- dimensional interpretation, each turn can be represented as both + 1 and  – 1.

Now this situation is completely paradoxical from a rational dualistic perspective. Such rational understanding  by its very nature is linear i.e. 1-dimensional, where opposite polar reference frames are clearly separated.

However the very essence of 2-dimensional interpretation is that it is directly intuitive in nature. Thus through one cannot rationally understand how in this context + 1 can also be – 1 (where only isolated polar reference frameworks are considered) one can however intuitively grasp how this can be so, through intuitively combining two opposite polar reference frames simultaneously).

The great importance of this illustration is that the very notion of "twoness" which again relates to the qualitative notion of "2" occurs directly in experience through the intuitive realisation that occurs from simultaneously combining opposite polar reference frames.

And once again this is fundamental with respect to the dynamic nature in which experience takes place.

In our crossroads illustration we used the opposite polar directions of North and South and Left and Right respectively.

It is then very similar with respect to actual experience where interaction continually takes place with respect to external and internal and also whole and part polarities. The latter set in turn directly relates to the corresponding distinction as between the quantitative and qualitative aspects of number!

So therefore put succinctly, the quantitative aspect of number appreciation relates directly to rational interpretation (that is 1-dimensional in nature).

However the qualitative aspect, by contrast, relates directly to intuitive appreciation (representing the simultaneous recognition of multiple "higher" dimensions i.e. reference frames).

The simplest - and in many ways the most important - example of qualitative understanding relates to the 2-dimensional notion of  "twoness".

What happens here is that two polar reference frames - that in rational terms are considered as independent - are now intuitively seen as dynamically interdependent with each other.

This is very similar to the physical level where both a matter and anti-matter particle are brought together. Though + 1 and – 1 with respect to each other as separate particles in their independent states, when brought together as interdependent they fuse together as physical energy.

Likewise when the intuitive realisation takes place in experience of the complementary nature of opposite poles (e.g. external and internal) a corresponding fusion takes place in an intuitive manner (as psychological energy).

Thus in experience, the qualitative nature of number relates directly to intuitive appreciation (of "higher" dimensions).

Once again the simplest case represents the qualitative appreciation of "2" which implies intuitive appreciation of 2 dimensions (i.e. the simultaneous recognition with respect to two opposite reference frames that are equidistant in a circular manner).

In like manner the qualitative appreciation of "3" implies direct intuitive realisation of 3 dimensions (as the simultaneous interdependent recognition with respect to three circular equidistant reference frames).

In general the qualitative appreciation of "n" implies direct intuitive realisation with respect to n dimensions (as the simultaneous interdependent recognition with respect to n circular equidistant reference frames).

In psychological terms, this qualitative recognition - which in some measure must be implicitly present to enable meaningful interpretation even at a quantitative level - is directly of an unconscious nature.

However because Conventional Mathematics is formally geared to the merely quantitative aspect of understanding, it remains completely blind to the unconscious dynamics by which understanding of number takes place.

Put another way, a coherent interpretation of even the simplest number operations requires that both conscious and unconscious type understanding be dynamically integrated in a balanced coherent manner.

We will see some of the basic implications of this in future blog entries!

Clarifying Analytic and Holistic Interpretations of "2" (3)

As we have seen, properly understood, number should be defined in a dynamic interactive manner entailing both base and dimensional aspects, which are - relatively - quantitative and qualitative with respect to each other.

Therefore the number "2" is dynamically defined as representing the interaction of its two aspects i.e.  2¹ and 1². Therefore from the perspective, where the base number "2" is considered in an quantitative  (independent) manner, then the corresponding dimensional number "2" must be then - relatively - considered as of a qualitative (interdependent) nature.

So we have the interaction here therefore of the quantitative notion of "2" with the corresponding qualitative notion of "2" (i.e. as twoness).

One must remember that all phenomenal experience - including of course mathematical - is universally governed by fundamental opposite polarities. Chief among these are external and internal poles, so that no number can be be objectively experienced (as external) in the absence of a corresponding mental perception (that is - relatively - internal). So ultimately what is objectively known cannot be divorced from corresponding subjective mental interpretation!
Thus any absolute notion of truth within Mathematics is untenable.

Likewise we have whole and part polarities, which directly entail complementary quantitative and qualitative aspects with respect to phenomena.

So once again we cannot experience a number in a quantitative manner without a corresponding qualitative counterpart recognition.

So once again number recognition is strictly relative with respect to recognition of both its quantitative and qualitative aspects.

As I have repeatedly demonstrated, the crossroads analogy can be very helpful here!

If standing at a crossroads one denotes a direction as left, then in relative fashion, the opposite direction is right. If however approaching the crossroads from the opposite direction, one now denotes this first turn as right, then the opposite is thereby left.

So depending on the context from which direction (N or S) the crossroads is approached, each turn can be both left and right (and right and left) respectively.

So it is quite similar in mathematical terms, which is governed likewise by twin opposite polarities that keep switching in direction.

Therefore in this context, if we denote the base  as quantitative, then the dimensional number is - relatively - qualitative in nature.

However, when we now switch the frame of reference - which repeatedly occurs in experience - the base number is qualitative, with the corresponding dimension assuming a quantitative meaning.

Now because of the reduced nature of standard mathematical interpretation, these critical distinctions are completely missed with number - whether representing the base or dimension - interpreted in a merely quantitative manner.

This in fact is tantamount to one who insists that both turns at a crossroads represent  the same directiont. And this arises from the solely linear (1-dimensional) interpretation that is employed, which necessarily operates within single isolated polar reference frames.

Thus operating in this manner with the single direction of movement as heading North towards the crossroads, one can indeed unambiguously identify, for example, a left turn at the crossroads.

Then when we again fix a single direction of movement, this time heading South towards the crossroads, one can unambiguously identify a left turn.

However when we now try to combine both of these isolated frames of reference, both turns at the crossroads are identified as left!

Thus to understand that the turns must be relatively left and right with respect to each other, one must implicitly move from a 1-dimensional (where N and S are considered separately) to a 2-dimensional manner of interpretation, where they are simultaneously considered.

Intuitively we can do this automatically in relation to understanding the turns at a crossroads.

However, when it comes to mathematical understanding, we are rigidly locked within a merely linear (1-dimensional) framework because of the massive reductionism involved. So the internal pole of recognition is not distinguished from the external; equally the qualitative pole of recognition is not distinguished from the quantitative.

So effectively this is like attempting to understand the directions at a crossroads where only the N direction with respect to (vertical) movement is recognised and solely the left direction with respect to (horizontal) movement!


Therefore with respect to our understanding of "2", we can switch frames of reference so that the base number is now qualitative, and the dimension - relatively - of a quantitative nature.

In fact when one reflects on it carefully this qualitative aspect of recognition is necessarily involved in all multiplication processes.

Let us take for example the simple example of 2 * 3.

Now imagine we have 2 rows with 3 coins in each row, this could then be represented as 2 * 3.

In the standard reduced manner of conventional mathematical interpretation, such multiplication could be represented merely in terms of addition where 2 * 3 = 3 + 3.

So from this perspective, the multiplication operation results in 6, interpreted in a merely quantitative (1-dimensional)  manner as representing independent units.

However, clearly there is something missing from this reduced interpretation for as we know in geometrical terms when we multiply 2 * 3, the units change from a linear (1-dimensional) to a square (2-dimensional) format. So a qualitative change in the nature of units occurs (as well as the recognised quantitative transformation).

And even to represent the two rows of coins (with 3 coins in each row) requires adopting a 2-dimensional format!

The crucial "missing" element is that the very ability to match off the the two rows, requires that we recognise their common identity. And this common identity relates to the qualitative notion of number interdependence (i.e. the common recognition of "threeness").

Therefore we initially must indeed recognise the independent identity of each of the three coins with respect to each row. This relates to the quantitative nature of the multiplication operation.

However the very ability to then match the two rows requires the common recognition of similarity with respect to each row (as now interdependent with each other). This then relates to the qualitative nature of the multiplication process.

Now in the pure case of simple multiplication, where we multiply individual units (recognised as independent), a mere qualitative transformation is involved in dimensional terms.  

However in the more general case of compound multiplication where we multiply numbers (≠ 1), both a quantitative and qualitative transformation is involved.


Thus to conclude this blog entry!

Both the base and dimensional aspects of number can be given a quantitative (as independent)  and qualitative interpretation (as interdependent) respectively.

However just like the turns at a crossroads, they always bear a complementary relationship with each other.

Therefore if the base is defined in a quantitative manner, then the dimensional number is - relatively - qualitative - in nature.

If however the base number is defined in a qualitative manner, then the dimension is now - relatively - quantitative - in nature.

Thus with respect to the number "2", it keeps switching in experience as between both its quantitative and qualitative aspects - with respect to the independent notion of "2" and interdependent notion of  "twoness" - in both base and dimensional terms.

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

In yesterday's blog, entry, I distinguished how the notions of independence and interdependence respectively are associated with two complementary aspects of number, dynamically interactive in nature, that are quantitative and qualitative with respect to each other.

So for example whereas we can denote the quantitative (independent) notion of "2" as 2¹, by contrast the corresponding qualitative (interdependent) notion in an inverse complementary manner as 1².

Thus whereas the first aspect (Type 1) relates to two number units of a linear (1-dimensional) nature the latter (Type 2) relates to one number unit that is square (2-dimensional) in nature.

Therefore, once again the key distinction as between both aspects is that the the former (Type 1) relates to units that are independent of each other, whereas the latter (Type 2) relates to units that have and ordered relationship that are thereby interdependent with each other.

Thus properly incorporating the key notions of number independence (as quantitative) and number interdependence (as qualitative) respectively entails again that both must be considered together in a dynamic interactive manner, where independence and interdependence have now a strictly relative - rather than absolute - meaning.


These dynamic relative distinctions are then vital in illustrating the key difference as between the operations of addition and multiplication respectively.

We can now illustrate this simple case of addition i.e.

1 + 1 = 2.

In conventional mathematical terms, both of these units are considered as homogeneous in a merely reduced quantitative manner where they are considered absolutely independent of each other (thereby lacking any qualitative distinction)..

However we can now more accurately relate addition directly with the Type 1 aspect of number.

So now 1¹ + 1¹ = 2¹.

Both units here are still considered explicitly as independent in a quantitative manner. However now the interpretation is strictly of a relative nature (that implicitly allows for the numbers to be related to each other).

There is a basic issue with conventional interpretation that is completely overlooked and in effect treated in a grossly reduced manner. For if the two units are in fact to be treated as independent in an absolute manner, then this begs the question as to to how they can be combined with each other to form the new number identity of "2"!

In other words, we can see clearly here that in conventional mathematical terms, the qualitative notion of number relationship (i.e. number interdependence) is simply reduced in a a merely quantitative (independent) manner.

Because of this basic reductionism, there is no way in conventional terms to properly distinguish the nature of multiplication from addition.

So again in conventional terms 1 * 1 = 1 (with the quantitative value remaining unchanged).

However when we now properly distinguish as between the Type 1 and Type 2 aspects of number,
this operation of multiplication is represented as

1¹ * 1¹ = 1².

In general terms in the number expression a^b, I refer to a as the base and b the dimensional number respectively.

Therefore in this simple case  we can see that with addition, the base number changes while its dimensional counterpart remains unchanged. This is associated explicitly with a quantitative change in the units involved.

Then, by contrast with multiplication, the dimensional number now changes while its base counterpart remains unchanged. This  in turn is explicitly associated with a qualitative change in the units involved. In other words in the multiplication of  1 by 1, we must explicitly recognise that the units are related to each other (i.e. as interdependent) in a qualitative manner!

Therefore by confining ourselves initially to the simplest possible operations representing addition and multiplication respectively, we can clearly distinguish how multiplication differs from addition.

So quite simply, expressing this finding appropriately in a dynamic interactive manner, the operations of addition and multiplication are quantitative and qualitative with respect to each other.

Thus once again whereas addition in our illustration was associated with a quantitative transformation in units, (with the qualitative dimension unchanged), by contrast multiplication was associated with a qualitative transformation in the units (with the quantitative base unchanged).    


The implications here are truly enormous, though apparently completely overlooked in conventional mathematical terms.

Put simply, we cannot coherently understand the simplest use of the operations of additions and multiplication respectively from within the present universally accepted mathematical perspective.

In fact put bluntly, we cannot coherently understand any mathematical relationship from this accepted conventional perspective.

Thus an intellectual revolution much more fundamental than that which has already partly occurred in physics with quantum mechanics is urgently required for the coherent understanding of all mathematical relationships.

For so long we have tried to reduce in every mathematical context qualitative to quantitative type meaning.

This has now created the considerable illusion that Mathematics is all about abstract quantitative type relationships (where qualitative considerations have no relevance).

In truth this is utterly unfounded. Just as we cannot properly reduce the understanding of water solely to its hydrogen atoms. in a much more fundamental manner we cannot properly reduce mathematical understanding to its mere quantitative aspect!

Clarifying Analytic and Holistic Interpretations of "2" (1)

I return once more to illustrate the crucial distinction as between the analytic (quantitative) and holistic (qualitative) aspect of number.

Once again from the additive approach each number can be expressed in quantitative terms as the combination of individual independent units.

So for example in the simplest case 2 = 1 + 1.

Because these units are defined in a merely homogeneous absolute quantitative manner, they are thereby lacking any qualitative distinction.

However if the units were indeed absolutely independent of each other, then there would be no means by which such numbers could subsequently be related to each other.

So this latter capacity by which (individual) numbers can be related to each other assumes the complementary notion of number interdependence (which is inherently qualitative in nature).

So complementing the quantitative notion of "2" is the corresponding qualitative notion of "twonness"  where the two units, that are assumed independent of each other in quantitative terms, are now understood as interdependent with each other,

Put another way, the quantitative notion of "2" could not exist without the corresponding qualitative notion of "twoness". Equally the qualitative notion of "twoness" implying the interdependence of both units could not exist without the quantitative notion of "2" (where we must start with the independent units).

Therefore the clear implication is that the true notion of number - far from being abstract in an absolute manner - is inherently of a dynamic interactive nature (comprising both quantitative and qualitative aspects).

And the notions of independence (by which numbers assume a separate individual individual identity) and interdependence (by which they are collectively related to each other) are thereby of a merely relative  nature.

Thus the conventional quantitative approach - which is accepted without question - is but a reduced interpretation of number that ultimately greatly distorts the very nature of mathematical activity!

I have expressed before how both aspects of number i.e. quantitative and qualitative can be represented.

With respect to the quantitative aspect, each natural number is defined with respect to the default dimension (i.e. power or exponent of 1).

So 2 from this quantitative perspective is defined as 2¹.

When a number is then raised to a dimensional power (≠ 1), its ultimate value is then given in a reduced quantitative manner.

So with respect for example to 2², we can perhaps readily appreciate that is geometrical terms, this would be represented by a square (with area of 4 square units).
Therefore as well as the quantitative transformation implied by the expression, we also have a qualitative transformation in the nature of units involved.

However with conventional number interpretation, this qualitative aspect is simply ignored with the value of 2²  given in a merely linear quantitative manner as 4 (i.e. 4¹).

With respect to its qualitative aspect, 2 is expressed in a complementary fashion as 1².

Therefore in this case, we are concentrating on the inherent qualitative nature of the number.

Now one could indeed attempt to express 1² as 1^{(1 + 1)}, just as we could equally attempt to express 2¹ 
as (1 + 1)¹,

However the addition of these units refers to two distinctive meanings (which are completely ignored in conventional interpretation)!

In the quantitative case - which I will refer to as the Type 1 aspect - the units involved are assumed to be independent of each other.

So again, 1 + 1,  in Type 1 terms refers to units that are understood independent of each other.

However in the corresponding qualitative case - which I will refer to as the Type 2 aspect - the units are now considered, by contrast, as interdependent with each other.

So now 1 + 1, in Type 2 terms, refers to units that are understood as related i.e. interdependent with each other.

Indeed if one reflects briefly on the nature of a 2-dimensional object, one can quickly appreciate that the 2 dimensions involved  i.e. length and width, cannot be meaningfully independent of each other.

Though the length and width, separately viewed, are both of a 1-dimensional nature, clearly the very appreciation of the two 2-dimensional object, requires that we can view these two dimensions as related to each other in an ordered manner.

Therefore if we tried to maintain the separate independent identity of the two dimensions (as in Type 1 terms), we have the consideration of 2 units objects (that are 1-dimensional in nature) represented as 2^1.

However when we now recognise the relationship of the two dimensions (length and width) as interdependent then we have the consideration of 1 unit object (that is 2-dimensional in nature) represented as  1².

The deeper realisation here is that all mathematical symbols can be viewed in both Type 1 (analytic) and Type 2 (holistic) terms. In the first case, we concentrate on the independent (quantitative) aspect of relationships. Then in the second case, we concentrate on the interdependent  (qualitative) aspect.

And crucially both of these aspects can only be properly understood in a dynamic interactive manner as complements of each other, where both quantitative independence and qualitative interdependence are understood in a relative - rather than absolute - manner!