As I stated in the previous blog entry from the early 70's I underwent a period of intense (unconscious) darkness that lasted the best part of a decade.

So this was my "dark night of the soul" which ended for a considerable time any sustained intellectual work .

During the time I formed a strong resonance with the writing of St. John of the Cross who in a very intimate manner (because of the similar characteristics of experience) became my main spiritual support.

Ultimately even this kind of attachment proved a problem and I found myself - as in philosophical terms with Hegel previously - beginning to find limitations with his approach.

Now one may immediately question as to what possible relevance this might have for mathematical understanding! However in many ways it later proved decisive in terms of clarification of my holistic notions.

Now St. John distinguishes as between "active nights" and "passive nights". He further distinguishes as between an "active night of sense" which would relate directly to more superficial perceptions and an "active night of spirit" would relate directly to deeper conceptual constructs.

Customary experience is heavily based on the (analytic) conscious, which in a mathematical context would relate to linear rational interpretation. Unfortunately when we become unduly attached to such rational understanding, it blots out the corresponding holistic spiritual dimension (represented by intuition).

So in experiential mathematical terms, both "active nights" would be required to cleanse one of undue identification with the dualistic mode of mere analytic type interpretation of symbols (associated with what - I term - the Type 1 aspect).

This would then open a new more refined holistic type of mathematical experience heavily based on intuition (i.e. the Type 2 aspect).

However though rational understanding is inherently of a more paradoxical (circular) kind, indirectly problems can now arise due to undue attachment to the secondary conscious symbols used to mediate such understanding.

For example one may now see most relationships in terms of the dynamic complementarity of opposite polarities. However a certain rigidity can then set in (with consequent loss of the pure intuitive vision) due to undue attachment to such secondary symbols.

Indeed this was the big problem I found with Hegel's writings, where initial holistic insight of a strongly intuitive nature gradually gave way to an increasingly ponderous intellectual form of understanding (leading to a misplaced elevation of philosophy over authentic spiritual experience).

Therefore coming to terms with such secondary attachment to holistic type symbols requires a deeper form of cleansing or purgation in the "passive night of sense" and "passive night of spirit" respectively.

The goal of all this intense purgation in St. John's terms is "nada" i.e. nothing. Now what this really means is that through a deep detachment from any undue identification with primary phenomena of a (linear) analytic and secondary symbols of a (circular) holistic kind, one could then directly experience the spiritual light (of a purely intuitive nature).

Mathematical experience, as we have seen necessarily entails a dynamic interaction of reason and intuition.

Conventional interpretation reduces such experience in terms of the analytic extreme of mere reason.

However here in St. John we are presented with the holistic extreme in contemplative terms of pure intuition (representing psycho spiritual energy).

Thus when we apply this to number it implies that dynamic interpretation is bound by the two extremes of rigid analytic form (from an analytic perspective) and highly refined spiritual energy (from the holistic extreme).

And there is a further mathematical link in that nada = nothing (i.e. 0). However clearly this is now understood in a qualitative (holistic) rather than quantitative (analytic) sense.

So the deeper implication here is that all mathematical symbols ultimately can be given both an analytic (quantitative) and holistic (qualitative) interpretation.

In fact, this association - and deep reflection - on St. John's writings was later to prove decisive in leading me to a starting appreciation of the true nature of the Riemann Hypothesis through a radical holistic interpretation of the first of the trivial zeros!

However I gradually began to see problems - from my perspective - with his writings. In fact one of these had an intimate bearing on the fact that the values of the Riemann Zeta Function for negative even dimensional numbers (s) behave very differently than for those associated with the corresponding negative odd numbers!

St. John s places undue emphasis on passive purgation. For me, from a holistic mathematical perspective, passive purgation properly related to the even numbers. So for example the purification of 2-dimensional appreciation (entailing complementary opposite poles) would be of a passive nature.

However the odd dimensions (as revealed through their corresponding roots) were distinctively different in being more asymmetrical in structure. Thus I reckoned that purgation of attachment to symbols representing these dimensions would require a new more refined form of "active night".

Also healthy contemplative experience entails maintaining a balance as between development of the inner self (passively) and engagement with the world (actively).

And I felt that this balance was missing from St. John's writings perhaps explaining in part his difficulties in dealing more diplomatically with the problems surrounding his own Carmelite order at the time.

Also there is too much emphasis - certainly in his formal explanations - on the transcendent aspect of spiritual development. For example his most famous work (encompassing the "Dark Night") is entitled the "The Ascent of Mount Carmel". But just as the ascent of a difficult mountain is followed by an equally important descent, likewise it is similar in spiritual terms. So both transcendent (other-worldly) and immanent (this-worldly) directions of development need equal emphasis. Thus top-down integration of the personality (transcendent) should be balanced by bottom-up integration (immanent).

Indeed I came to realise to my own cost that undue emphasis merely on the stark demands of transcendent detachment ultimately is likely to give way to serious repression of the instinctive unconscious, culminating in depression.

Finally, at a later stage when studying the Enneagram, I began to realise more clearly that St. John's account is really representative, in an uncompromising extreme manner, of just one personality type (where both 4 and 5 characteristics are strongly in evidence).

However, despite these reservations, I have never encountered another writer who deals in such a profound spiritual manner with the existential dimension of human experience. So in this crucial sense I still regard him as truly unique.

Now this has very important implications for the full development of mathematical ability.

As stated so often in these blogs, current mathematical interpretation is exclusively associated (in formal terms) with mere analytic appreciation of a quantitative nature. Thus the emphasis is on the rational extreme of understanding (in an abstract 1-dimensional manner).

However, properly understood, there is an equally important aspect to Mathematics representing holistic appreciation of a qualitative nature.

However specialisation with respect to such holistic ability will require the corresponding extreme development of the unconscious requiring prolonged exposure to "dark night" cleansing of an intensive kind.

Only a small minority of highly motivated and spiritually gifted people have so far in our history been able to meet the demands of such refined contemplative development.

And even where such intuitive refinement has taken place to a marked degree, it has rarely been applied to mathematical interpretation.

So I have found in my own experience - even among those with a developed capacity for holistic understanding - that a major block still exists when applying this to Mathematics. In other words, the conventional reduced interpretation of mathematical symbols is so ingrained in our culture, that remarkably few ever seriously question its underlying rationale.

Thus we are still a long way from attaining a comprehensive vision of Mathematics. However the pace of technological change is now so rapid that this will quickly bring a corresponding need for unprecedented evolution in personal and social terms. And this could then pave the way for the greatest advance yet in our intellectual history where the nature of Mathematics and its associated Sciences will be utterly transformed.

So in my own programme, with respect to the evolution of Mathematics, I see 3 main stages.

1. Specialisation of the analytic aspect relating directly to linear (1-dimensional) reason. This certainly has now been attained but unfortunately at the the considerable price of completely blocking out recognition of its "shadow" holistic side.

2. Specialisation of the holistic aspect relating directly to intuition, indirectly expressed in circular ("higher" dimensional) rational terms. Once again though such specialised developed has indeed taken place at times with respect to the spiritual contemplative traditions, it has rarely been directly associated with mathematical and scientific interpretation.

3. Specialisation of both analytic and holistic aspects in dynamic relationship with each other, which will require the simultaneous balanced composition of both highly refined reason and intuition. This points to a future "golden age" which does not yet remotely exist. However, as I have stated, rapid transformation in our culture may now well speed up its eventual attainment.

## Friday, June 27, 2014

## Thursday, June 26, 2014

### Reflections on Experience (4)

I mentioned that in dynamic terms, that when we use a number to represent a dimensional power n, that strictly it then exists in a corresponding framework of n-dimensional time (and space).

Once again the standard Type 1 system i.e. 1

However the (unrecognised) Type 2 system i.e. 1

Thus the understanding of each of these numbers as a distinct dimensional power, is directly associated with a corresponding holistic configuration of time (and space).

So we will illustrate this point with respect to the simplest (non-trivial) dimensional number (as representing a power) i.e. 2.

Now to obtain this time configuration we obtain the two roots of 1, which are + 1 and – 1 respectively.

This implies that 2-dimensional time has two directions which are positive and negative with respect to each other.

I have explained before in several places in my various blogs the precise basis of this 2-dimensional notion of the experience of time.

Notions of linear time in fact require that we ignore the dynamic interaction of the opposite polarities (which necessarily underline all phenomenal experience).

So, for example, experience necessarily entails external (objective) and internal) subjective poles in relationship to each other.

Now with linear notions of time both poles are assumed to correspond directly with each other.

Therefore from one perspective we experience time as moving forward in a positive direction with respect to the external world (in relation to the self).

From another perspective, we likewise experience time as moving forward in a positive direction with respect to the internal self (in relation to the world).

Therefore because time is moving forward with respect to both poles (in dualistic isolation) we assume a direct correspondence as between them.

However, like the two turns at a crossroads, from an overall holistic perspective (where both frames are simultaneously acknowledged as interdependent) what is forward (positive) movement with respect to one pole is - relatively - backward (negative) movement with respect to the other and vice versa.

Thus in terms of the holistic appreciation of external and internal polarities, the movement of time is necessarily paradoxical with both a positive (+ 1) and a negative ( – 1) direction.

In like manner, movement in space from this holistic perspective is also paradoxical (with positive and negative directions).

So once again the 1-dimensional (linear) notion of time (and space) as moving in a forward (positive) direction is associated directly with analytic type interpretation (based on isolated i.e. independent reference frames).

This concurs directly with (conscious) rational type appreciation.

However the 2-dimensional (circular) notion of time and space, as the paradoxical relative movement in both forward (positive) and backward (negative) directions, is associated directly with holistic type appreciation (based on simultaneous i.e. interdependent reference frames).

This concurs directly with (unconscious) intuitive type appreciation (that is indirectly interpreted in a paradoxical rational manner).

So if we want to put it simply, the Type 1 aspect of the number system is directly analytic in nature geared to quantitative interpretation.

The Type 2 aspect of the number system is then directly holistic in nature geared to qualitative appreciation.

Then clearly from a comprehensive perspective, the number system must be understood in a dynamic interactive manner comprising both analytic (Type 1) and holistic (Type 2) appreciation.

Thus during my later years in college, I was already in the process of developing this vitally important - though formally totally neglected - aspect of mathematical understanding and had progressed sufficiently to formulate the nature of 2-dimensional understanding.

And just as all roots of 1 necessarily contain 1 (as a basic root), likewise all refined holistic understanding of "higher" dimensions starts with analytic notions (as appreciation of interdependence must necessarily be based on what - initially - is understood as independent).

One consequence that I then realised is that associated with each number (as dimension) is a unique means for holistic interpretation of mathematical reality.

So what presently is widely accepted as Mathematics, in fact represents just one possible interpretation (i.e. 1-dimensional).

However an unlimited set of other possible interpretations of Mathematics potentially exist (in accordance with other numbers representing dimensions).

Furthermore this standard interpretation (i.e. 1-dimensional) in some ways is the most limited of all as it is absolute in nature (and capable only of interpreting analytic type truth in a quantitative nature).

Actually this insight was to later lead me to recognising a remarkable feature of the Riemann Zeta Function (which is vital for appreciation of its true nature)!

All other interpretations (according to dimensional numbers > 1) necessarily are of a relative dynamic nature entailing both analytic (quantitative) and holistic (qualitative) aspects.

And the simplest - and in many ways most important - of these dynamic relative interpretations corresponds to 2 (as dimensional number).

Now essentially, all the "higher" dimensions entail increasingly refined holistic interpretations of the numbers as dimensions (and the corresponding configurations of time and space entailed).

However, it took some considerable time for me to obtain a certain mastery of those other dimensions that I realised as especially important in integral terms i.e. 4-dimensional and 8-dimensional.

Though an initial intense phase of illumination had opened the door as it were to this new holistic vision of Mathematics (which I could see would have immensely important consequences), the light quickly faded to be replaced by a prolonged period of intense darkness (lasting the best part of a decade).

What I did not properly realise at the beginning is that this attempt to coherently clarify the holistic nature of mathematical symbols was dependent on a deep specialisation with respect to unconscious type development. And the proper preparation for such development would therefore require an intense purgation of attachment to conscious dualistic notions of all kinds.

Thus I quickly found myself from the early 70's immersed in an underworld of unconscious darkness that, like I said, continued unabated for many years.

Once again the standard Type 1 system i.e. 1

^{1}, 2^{1}, 3^{1}, 4^{1},... is defined in terms the natural number quantities raised to the default dimensional power of 1. This automatically implies a corresponding 1-dimensional framework of time (based on linear movement in a forward direction).However the (unrecognised) Type 2 system i.e. 1

^{1}, 1^{2}, 1^{3}, 1^{4},... is defined in terms of the the default unit quantity, raised to the natural numbers as representing different dimensional powers.Thus the understanding of each of these numbers as a distinct dimensional power, is directly associated with a corresponding holistic configuration of time (and space).

So we will illustrate this point with respect to the simplest (non-trivial) dimensional number (as representing a power) i.e. 2.

Now to obtain this time configuration we obtain the two roots of 1, which are + 1 and – 1 respectively.

This implies that 2-dimensional time has two directions which are positive and negative with respect to each other.

I have explained before in several places in my various blogs the precise basis of this 2-dimensional notion of the experience of time.

Notions of linear time in fact require that we ignore the dynamic interaction of the opposite polarities (which necessarily underline all phenomenal experience).

So, for example, experience necessarily entails external (objective) and internal) subjective poles in relationship to each other.

Now with linear notions of time both poles are assumed to correspond directly with each other.

Therefore from one perspective we experience time as moving forward in a positive direction with respect to the external world (in relation to the self).

From another perspective, we likewise experience time as moving forward in a positive direction with respect to the internal self (in relation to the world).

Therefore because time is moving forward with respect to both poles (in dualistic isolation) we assume a direct correspondence as between them.

However, like the two turns at a crossroads, from an overall holistic perspective (where both frames are simultaneously acknowledged as interdependent) what is forward (positive) movement with respect to one pole is - relatively - backward (negative) movement with respect to the other and vice versa.

Thus in terms of the holistic appreciation of external and internal polarities, the movement of time is necessarily paradoxical with both a positive (+ 1) and a negative ( – 1) direction.

In like manner, movement in space from this holistic perspective is also paradoxical (with positive and negative directions).

So once again the 1-dimensional (linear) notion of time (and space) as moving in a forward (positive) direction is associated directly with analytic type interpretation (based on isolated i.e. independent reference frames).

This concurs directly with (conscious) rational type appreciation.

However the 2-dimensional (circular) notion of time and space, as the paradoxical relative movement in both forward (positive) and backward (negative) directions, is associated directly with holistic type appreciation (based on simultaneous i.e. interdependent reference frames).

This concurs directly with (unconscious) intuitive type appreciation (that is indirectly interpreted in a paradoxical rational manner).

So if we want to put it simply, the Type 1 aspect of the number system is directly analytic in nature geared to quantitative interpretation.

The Type 2 aspect of the number system is then directly holistic in nature geared to qualitative appreciation.

Then clearly from a comprehensive perspective, the number system must be understood in a dynamic interactive manner comprising both analytic (Type 1) and holistic (Type 2) appreciation.

Thus during my later years in college, I was already in the process of developing this vitally important - though formally totally neglected - aspect of mathematical understanding and had progressed sufficiently to formulate the nature of 2-dimensional understanding.

And just as all roots of 1 necessarily contain 1 (as a basic root), likewise all refined holistic understanding of "higher" dimensions starts with analytic notions (as appreciation of interdependence must necessarily be based on what - initially - is understood as independent).

One consequence that I then realised is that associated with each number (as dimension) is a unique means for holistic interpretation of mathematical reality.

So what presently is widely accepted as Mathematics, in fact represents just one possible interpretation (i.e. 1-dimensional).

However an unlimited set of other possible interpretations of Mathematics potentially exist (in accordance with other numbers representing dimensions).

Furthermore this standard interpretation (i.e. 1-dimensional) in some ways is the most limited of all as it is absolute in nature (and capable only of interpreting analytic type truth in a quantitative nature).

Actually this insight was to later lead me to recognising a remarkable feature of the Riemann Zeta Function (which is vital for appreciation of its true nature)!

All other interpretations (according to dimensional numbers > 1) necessarily are of a relative dynamic nature entailing both analytic (quantitative) and holistic (qualitative) aspects.

And the simplest - and in many ways most important - of these dynamic relative interpretations corresponds to 2 (as dimensional number).

Now essentially, all the "higher" dimensions entail increasingly refined holistic interpretations of the numbers as dimensions (and the corresponding configurations of time and space entailed).

However, it took some considerable time for me to obtain a certain mastery of those other dimensions that I realised as especially important in integral terms i.e. 4-dimensional and 8-dimensional.

Though an initial intense phase of illumination had opened the door as it were to this new holistic vision of Mathematics (which I could see would have immensely important consequences), the light quickly faded to be replaced by a prolonged period of intense darkness (lasting the best part of a decade).

What I did not properly realise at the beginning is that this attempt to coherently clarify the holistic nature of mathematical symbols was dependent on a deep specialisation with respect to unconscious type development. And the proper preparation for such development would therefore require an intense purgation of attachment to conscious dualistic notions of all kinds.

Thus I quickly found myself from the early 70's immersed in an underworld of unconscious darkness that, like I said, continued unabated for many years.

## Wednesday, June 25, 2014

### Reflections on Experience (3)

In yesterday's blog entry, I referred to the experiential basis of an inherently dynamic appreciation of the number system, whereby concepts of a (potential) infinite and perceptions of an (actual) finite nature continually interact in a relative manner.

Of course concepts can likewise be given a (reduced) actual and perceptions a (transformed) potential meaning. So the important point to grasp is that relative to each other, number concepts and perceptions are infinite as to finite (and finite as to infinite) in a dynamic complementary manner.

In psycho spiritual terms this dynamic appreciation is then associated with the two-way interaction of both intuition and reason in experience.

Unfortunately however, a much reduced interpretation of such experience is given in Conventional Mathematics, whereby the (potential) infinite is formally reduced in an (actual) finite manner with corresponding (holistic) intuition likewise reduced in (analytic) rational terms.

Thus instead of an inherently dynamic relative appreciation of number, a static absolute interpretation arises, which ultimately distorts the very nature of number (and by extension all mathematical relationships).

Another important aspect of this dynamic appreciation of number is that it is intimately related to our experience of space and time.

Though I had not yet progressed sufficiently to properly articulate the appropriate nature of time (and space) that would correspond to this dynamic appreciation of number, at least I was clearly aware that it would likewise be of a - relative - rather than absolute nature and would in effect require a new understanding of the true holistic nature of such dimensions.

After dropping Mathematics at the end of the 1st year, I continued my degree taking Politics and Statistics in its place. So in this sense I still kept one foot as it were in the mathematical camp.

However in an unexpected manner, the choice of Politics was to ultimately open the door to making the new holistic mathematical connections that I was seeking.

A major part of the course related to the political thinking of the great philosophical figures such as Plato, Aristotle, Hobbes, Locke, Rousseau and Hegel. As I had now become keenly interested in philosophy, I started to explore deeply their overall positions.

It seemed apparent to me that the various schools of philosophy were based too much on the dualistic emphasis (e.g. empiricism and idealism) of just one limited aspect of overall experience.

So I was searching for a much more holistic vision which could naturally synthesise all these various schools as constituent parts.

In this regard I found that the last mentioned Hegel in many ways to my taste with his sweeping evolutionary view of history. This interest owed much in fact to the brilliance of our lecturer - a Jesuit priest - who clearly was a big fan and well versed in his philosophy. In fact he devoted an entire year to the study of Hegel and these lectures quickly became the favourite hours in my week.

Now I may add that I was far from an uncritical admirer. I found him - unfortunately like many academic philosophers - a very poor communicator. Also - possibly in part due to his comfortable state position - he espoused an unacceptable form of nationalism which may well have contributed to the terrible rise of Nazism in the 20th century.

However, as I became attuned to his notion of the dialectic, I began slowly to see connections with my emerging holistic mathematical notions.

In particular I mentioned in the last blog entry how at an earlier stage, I had pondered deeply the two roots of 1, trying to make sense of what seemed to me a paradoxical situation.

Well now I believed I could give a coherent explanation to this problem.

I would put it like this! As the conventional treatment of number is 1-dimensional, this means that opposite poles of experience, which inherently interact in experience, must be reduced in terms of each other in a static absolute manner.

So in this context, our experience of number necessarily entails both external and internal aspects. The external aspect treats the number as an object (out there somewhere in abstract space).

The internal aspect relates to the mental construct i.e. perception of number which necessarily must co-exist with the external objective experience.

So in experiential terms, with respect to number, we have the interaction of both an (external) object and the corresponding (internal) perception.

Now conscious awareness of object and perception implies that they be posited (+) in experience. (So I am using + now in a holistic rather than analytic manner).

However to switch from object to perception (and perception to object) we must dynamically negate (in unconscious manner) what has been formerly posited.

So with respect to these two polarities, dynamic experience entails the continual positing (+ 1) and corresponding negation (– 1) of single individual reference frames.

I realised that this was the very essence of 2-dimensional - as opposed to 1-dimensional - interpretation. In other words, when we attempt to express the dynamic nature of experience, with respect to 2 polar reference frames, in a (1-dimensional) analytic manner, it appears as paradoxical, where what is + 1 (as a posited pole) can equally from the opposite perspective be represented as – 1 (a negated pole).

Indeed this is the very situation that arises at a crossroads. When using a single polar frame of reference (N or S) a turn at a crossroads has an unambiguous meaning. So if heading N up a road, one can clearly designate a left turn and right turn respectively. However when one switches the frame of reference by heading S, again one can designate left and right turns at the crossroads in an unambiguous manner. So both of these cases of unambiguous identification entail 1-dimensional interpretation (where just one pole of reference is used).

However when one now considers N and S simultaneously as interdependent (i.e. 2-dimensional interpretation), deep paradox arises, for what is left from one perspective is right from the other and what is right from one is left from the other.

So expressed in a reduced 1-dimensional manner, 2-dimensional interpretation implies a result that can be + 1 (a left turn) or – 1 (not a left turn i.e. a right turn) depending on the single pole of reference (N or S) used.

Now it must be remembered that our very experience of number, necessarily entails the continual dynamic switching of reference poles (such as external and internal)

So the deeper connection I was now able to make - which is mind-boggling in terms of its consequences - is that the proper interpretation of 2 (as a dimensional power or number) requires a unique form of interactive understanding based on the complementarity of opposite poles.

Furthermore, as Conventional Mathematics by its very nature is 1-dimensional (based on single reference poles) it - literally - cannot even recognise this reality.

Put another way, Conventional Mathematics can only incorporate analytic type interpretation (based on independent poles of reference).

However there is another aspect of mathematical interpretation that is inherently dynamic i.e. Holistic Mathematics, which is based on poles of reference that are understood as interdependent.

The simplest - and indeed most important - example of such interdependent holistic appreciation entails 2 complementary poles i.e. that are + 1 and – 1 with respect to each other.

However in principle we can extend this indefinitely. So for example in the important case where we simultaneously consider the interdependence of 4 poles, 4-dimensional interpretation is used.

And in more general terms where n poles are considered simultaneously, n-dimensional interpretation is used.

I mentioned before that the problem of multiplication relates directly to the fact that it entails interdependent, whereas addition entails independent notions.

So in additive terms,

1 + 1 = 2, i.e. 1

However in multiplicative terms,

1

The startling implication of all this is that we cannot therefore properly understand the multiplication of 1 by 1 in conventional mathematical terms. Conventional interpretation gives but a reduced perspective that distorts its very nature! In other words, the key notion of interdependence, which it entails, is necessarily reduced in an analytic (i.e. independent) manner.

So I had clearly shown (from my perspective) that a coherent approach to Mathematics necessarily entails both Type 1 and Type 2 aspects of the number system. Associated with Type 1 is the standard quantitative approach to number based on analytic notions of absolute independence. By contrast, associated with the Type 2 is an alternative qualitative approach based on holistic notions of interdependence. Both are clearly necessary in a coherent comprehensive approach.

Finally in this entry, I will answer my earlier childhood query, which implied that from the 2-dimensional perspective a theorem (such as the Pythagorean) could be true and false at the same time.

Now in 1-dimensional terms. external and internal poles are reduced in terms of each other (and treated as one). So therefore when a theorem proven as absolutely true, this thereby rules out the opposite that it is false. However from a 2-dimensional perspective, both external and internal poles are recognised.

So we could now interpret the proof of a theorem in two ways:

1) as externally true (in an objective sense):

2) as internally true (in terms of mental interpretation).

Now in 2-dimensional terms, both of these interactively arise in a dynamic relative manner. So from this perspective, the truth of any proposition is strictly of a relative approximate nature.

Thus if we now attempt to reduce this 2-dimensional relative truth in an absolute 1-dimensional manner, then two positions are possible.

1. We can absolutely affirm the truth of the theorem in an external objective manner. (+ 1). This implies therefore the opposite (i.e.absolute falsehood) with respect to the truth of the theorem as internal interpretation (– 1).

2. We can absolutely affirm the truth of the theorem in an internal subject manner (as mental interpretation). This implies therefore the opposite (i.e. absolute falsehood) with respect to the truth of the theorem in an external objective manner (– 1).

Again, what is truly remarkable is that a parallel form of interpretation applies to the two roots of 1 (in quantitative terms). These two roots represent therefore the reduced linear (1-dimensional) attempt to represent the "higher" 2-dimensional behaviour of number (that entails the dynamic interaction of opposite poles).

Of course concepts can likewise be given a (reduced) actual and perceptions a (transformed) potential meaning. So the important point to grasp is that relative to each other, number concepts and perceptions are infinite as to finite (and finite as to infinite) in a dynamic complementary manner.

In psycho spiritual terms this dynamic appreciation is then associated with the two-way interaction of both intuition and reason in experience.

Unfortunately however, a much reduced interpretation of such experience is given in Conventional Mathematics, whereby the (potential) infinite is formally reduced in an (actual) finite manner with corresponding (holistic) intuition likewise reduced in (analytic) rational terms.

Thus instead of an inherently dynamic relative appreciation of number, a static absolute interpretation arises, which ultimately distorts the very nature of number (and by extension all mathematical relationships).

Another important aspect of this dynamic appreciation of number is that it is intimately related to our experience of space and time.

Though I had not yet progressed sufficiently to properly articulate the appropriate nature of time (and space) that would correspond to this dynamic appreciation of number, at least I was clearly aware that it would likewise be of a - relative - rather than absolute nature and would in effect require a new understanding of the true holistic nature of such dimensions.

After dropping Mathematics at the end of the 1st year, I continued my degree taking Politics and Statistics in its place. So in this sense I still kept one foot as it were in the mathematical camp.

However in an unexpected manner, the choice of Politics was to ultimately open the door to making the new holistic mathematical connections that I was seeking.

A major part of the course related to the political thinking of the great philosophical figures such as Plato, Aristotle, Hobbes, Locke, Rousseau and Hegel. As I had now become keenly interested in philosophy, I started to explore deeply their overall positions.

It seemed apparent to me that the various schools of philosophy were based too much on the dualistic emphasis (e.g. empiricism and idealism) of just one limited aspect of overall experience.

So I was searching for a much more holistic vision which could naturally synthesise all these various schools as constituent parts.

In this regard I found that the last mentioned Hegel in many ways to my taste with his sweeping evolutionary view of history. This interest owed much in fact to the brilliance of our lecturer - a Jesuit priest - who clearly was a big fan and well versed in his philosophy. In fact he devoted an entire year to the study of Hegel and these lectures quickly became the favourite hours in my week.

Now I may add that I was far from an uncritical admirer. I found him - unfortunately like many academic philosophers - a very poor communicator. Also - possibly in part due to his comfortable state position - he espoused an unacceptable form of nationalism which may well have contributed to the terrible rise of Nazism in the 20th century.

However, as I became attuned to his notion of the dialectic, I began slowly to see connections with my emerging holistic mathematical notions.

In particular I mentioned in the last blog entry how at an earlier stage, I had pondered deeply the two roots of 1, trying to make sense of what seemed to me a paradoxical situation.

Well now I believed I could give a coherent explanation to this problem.

I would put it like this! As the conventional treatment of number is 1-dimensional, this means that opposite poles of experience, which inherently interact in experience, must be reduced in terms of each other in a static absolute manner.

So in this context, our experience of number necessarily entails both external and internal aspects. The external aspect treats the number as an object (out there somewhere in abstract space).

The internal aspect relates to the mental construct i.e. perception of number which necessarily must co-exist with the external objective experience.

So in experiential terms, with respect to number, we have the interaction of both an (external) object and the corresponding (internal) perception.

Now conscious awareness of object and perception implies that they be posited (+) in experience. (So I am using + now in a holistic rather than analytic manner).

However to switch from object to perception (and perception to object) we must dynamically negate (in unconscious manner) what has been formerly posited.

So with respect to these two polarities, dynamic experience entails the continual positing (+ 1) and corresponding negation (– 1) of single individual reference frames.

I realised that this was the very essence of 2-dimensional - as opposed to 1-dimensional - interpretation. In other words, when we attempt to express the dynamic nature of experience, with respect to 2 polar reference frames, in a (1-dimensional) analytic manner, it appears as paradoxical, where what is + 1 (as a posited pole) can equally from the opposite perspective be represented as – 1 (a negated pole).

Indeed this is the very situation that arises at a crossroads. When using a single polar frame of reference (N or S) a turn at a crossroads has an unambiguous meaning. So if heading N up a road, one can clearly designate a left turn and right turn respectively. However when one switches the frame of reference by heading S, again one can designate left and right turns at the crossroads in an unambiguous manner. So both of these cases of unambiguous identification entail 1-dimensional interpretation (where just one pole of reference is used).

However when one now considers N and S simultaneously as interdependent (i.e. 2-dimensional interpretation), deep paradox arises, for what is left from one perspective is right from the other and what is right from one is left from the other.

So expressed in a reduced 1-dimensional manner, 2-dimensional interpretation implies a result that can be + 1 (a left turn) or – 1 (not a left turn i.e. a right turn) depending on the single pole of reference (N or S) used.

Now it must be remembered that our very experience of number, necessarily entails the continual dynamic switching of reference poles (such as external and internal)

So the deeper connection I was now able to make - which is mind-boggling in terms of its consequences - is that the proper interpretation of 2 (as a dimensional power or number) requires a unique form of interactive understanding based on the complementarity of opposite poles.

Furthermore, as Conventional Mathematics by its very nature is 1-dimensional (based on single reference poles) it - literally - cannot even recognise this reality.

Put another way, Conventional Mathematics can only incorporate analytic type interpretation (based on independent poles of reference).

However there is another aspect of mathematical interpretation that is inherently dynamic i.e. Holistic Mathematics, which is based on poles of reference that are understood as interdependent.

The simplest - and indeed most important - example of such interdependent holistic appreciation entails 2 complementary poles i.e. that are + 1 and – 1 with respect to each other.

However in principle we can extend this indefinitely. So for example in the important case where we simultaneously consider the interdependence of 4 poles, 4-dimensional interpretation is used.

And in more general terms where n poles are considered simultaneously, n-dimensional interpretation is used.

I mentioned before that the problem of multiplication relates directly to the fact that it entails interdependent, whereas addition entails independent notions.

So in additive terms,

1 + 1 = 2, i.e. 1

^{1 }+ 1^{1}= 2^{1}. (This again represents the Type 1 definition of number based on a default 1-dimensional interpretation implying single independent frames of reference).However in multiplicative terms,

1

^{1 }* 1^{1}= 1^{2 }(By contrast, this represents the Type 2 definition of number based in this case on the simultaneous interdependence of two reference frames).The startling implication of all this is that we cannot therefore properly understand the multiplication of 1 by 1 in conventional mathematical terms. Conventional interpretation gives but a reduced perspective that distorts its very nature! In other words, the key notion of interdependence, which it entails, is necessarily reduced in an analytic (i.e. independent) manner.

So I had clearly shown (from my perspective) that a coherent approach to Mathematics necessarily entails both Type 1 and Type 2 aspects of the number system. Associated with Type 1 is the standard quantitative approach to number based on analytic notions of absolute independence. By contrast, associated with the Type 2 is an alternative qualitative approach based on holistic notions of interdependence. Both are clearly necessary in a coherent comprehensive approach.

Finally in this entry, I will answer my earlier childhood query, which implied that from the 2-dimensional perspective a theorem (such as the Pythagorean) could be true and false at the same time.

Now in 1-dimensional terms. external and internal poles are reduced in terms of each other (and treated as one). So therefore when a theorem proven as absolutely true, this thereby rules out the opposite that it is false. However from a 2-dimensional perspective, both external and internal poles are recognised.

So we could now interpret the proof of a theorem in two ways:

1) as externally true (in an objective sense):

2) as internally true (in terms of mental interpretation).

Now in 2-dimensional terms, both of these interactively arise in a dynamic relative manner. So from this perspective, the truth of any proposition is strictly of a relative approximate nature.

Thus if we now attempt to reduce this 2-dimensional relative truth in an absolute 1-dimensional manner, then two positions are possible.

1. We can absolutely affirm the truth of the theorem in an external objective manner. (+ 1). This implies therefore the opposite (i.e.absolute falsehood) with respect to the truth of the theorem as internal interpretation (– 1).

2. We can absolutely affirm the truth of the theorem in an internal subject manner (as mental interpretation). This implies therefore the opposite (i.e. absolute falsehood) with respect to the truth of the theorem in an external objective manner (– 1).

Again, what is truly remarkable is that a parallel form of interpretation applies to the two roots of 1 (in quantitative terms). These two roots represent therefore the reduced linear (1-dimensional) attempt to represent the "higher" 2-dimensional behaviour of number (that entails the dynamic interaction of opposite poles).

## Tuesday, June 24, 2014

### Reflections on Experience (2)

Though with some misgivings - as I was already at odds with its fundamental rationale - I opted to pursue a degree in Mathematics at University (together with Economics).

Very much in the context of the times (mid to late 60's) in Ireland it was a broadly based Arts degree, which also included as subjects English Literature and Latin. Looking back, I consider that this in fact proved very healthy for my overall intellectual development and would consider that present degree programmes have in the main become much too specialised in an increasingly narrow manner.

However at the time I did not enjoy the experience at all (especially the 1st year) as my growing conflict with standard approaches came to a head not only in Mathematics but equally with respect to Economics.

With the benefit of hindsight it is now much easier for me to understand what was really happening at this time.

I would put it like this! Just as electromagnetic energy can be viewed as a spectrum with many distinctive bands, properly understood intellectual understanding should be seen in the same light.

However standard intellectual discourse especially with respect to Mathematics and the Sciences is presently heavily concentrated on just one narrow band of the overall spectrum.

This is what I refer to as linear (1-dimensional) understanding. In any context for discussion, this approaches truth in terms of dualistic notions based on just one polar reference frame. So typically for example in Mathematics the external (objective) is abstracted from its internal (subjective) pole; likewise the independent (quantitative) is likewise abstracted from the interdependent (qualitative) pole.

However just as natural light forms just one small band on the overall electromagnetic spectrum, properly appreciated, such dualistic linear understanding likewise forms just one small band on the overall spectrum of possible mathematical and scientific understanding.

Now, in the various spiritual contemplative traditions, it has long been recognised that many further stages of potential development exist with respect to intuitive understanding of an increasingly refined nature. Indeed detailed accounts testifying to the universal features of such states (both East and West) have been made available by the spiritual pioneers of the various religions..

However what has not yet been properly realised is that these advanced intuitive states of an increasingly holistic nondual nature, have not only relevance in a contemplative spiritual context, but equally have enormous implications for the advanced appreciation of all mathematical and scientific relationships.

So a major breakdown with respect to the rigid dualistic notions that especially inform our existing understanding of Mathematics (and related Sciences) was taking place with respect to my personal development during this period.

Thus I was already moving towards this distinctive holistic approach to Mathematics (and also Economics) for which no formal recognition whatsoever existed in the culture (especially at University level).

I remember for me it proved an intensely lonely and disillusioning experience with a marked decline in ability and motivation for what was conventionally acceptable, together with a deeper emerging holistic appreciation (in which no one displayed the slightest interest).

In fact I found my time at University pursuing Mathematics such an alienating experience that it led me to drop the subject altogether at the end of the year.

However, a much deeper philosophical questioning regarding the standard assumptions of Mathematics had taken place which especially centred around the treatment of the infinite notion.

I could readily see that that this key notion was being treated in a reduced rational manner, which was strictly nonsensical.

For example it was customary to define an infinite number as greater than any finite number. But to me this lacked any coherence! If one designates any finite number in actual terms (regardless of how large) another finite number necessarily can be found that is greater than it. Thus we cannot meaningfully approach the infinite notion from an actual finite perspective. Therefore a key qualitative distinction separates the finite from the infinite notion with both necessarily occurring in a dynamic interactive framework.

So, for example, the infinite notion (with respect to number) strictly applies in a merely potential manner to all numbers, whereas the finite notion always necessarily applies in a specific actual - and thereby limited -context.

Crucially therefore, both notions strictly can only meaningfully apply to number in a dynamic relative (i.e. approximate) context.

Indeed there were strong parallels here with the existential crisis I was going through at the the time with all the comforting belief systems in my life seemingly breaking down. In such a situation one searches for a deeper resolution to the problem through developing an authentic belief system, intimately based on personal conviction. So I now was led to adopt the same approach with respect to Mathematics with a willingness to challenge all conventional notions that did not stand up to serious scrutiny.

In fact this experience, though chastening at the time, was to quickly lead on to a truly dynamic interactive appreciation of the nature of the number system.

Initially this understanding was based heavily on philosophical type understanding. However I was later enabled to interpret it all coherently in a new holistic mathematical manner.

So the dynamic interaction of finite and infinite notions is inseparable from our very understanding of number.

The number concept is infinite in the sense that it applies potentially to all numbers. However any specific number identified (as a corresponding perception) is then necessarily of a finite nature.

Thus in dynamic terms, the very ability to posit (or determine) specific numbers (as perceptions) always implies another set of finite numbers (that always - by definition - must remain undetermined). In this sense our knowledge of number necessarily takes place against a background of uncertainty.

From this dynamic perspective, it makes no sense to attempt to approach the infinite notion from a finite perspective. So once again they are qualitative and quantitative with respect to each other.

In terms of experiential understanding, the very appreciation of the infinite is directly associated with intuition, whereas finite notions are directly interpreted in a rational manner.

Thus the dynamic interaction of both finite and infinite in number terms is replicated in psycho spiritual terms through the dynamic interaction of both reason and intuition.

However because of the reduced nature of Conventional Mathematics, in formal terms this relationship is defined in a merely static rational manner. Thus the inevitable interaction of reason and intuition in experience is reduced in merely rational terms. This of course equally implies that the true - merely potential - nature of the infinite notion is likewise reduced in a merely finite actual manner. This then leads directly to the confusion of the (potential) infinite with (actual) finite notions.

Therefore to put it briefly, I quickly realised that Conventional Mathematics - despite its appearance of great rigour with its modern treatment of limiting notions - could only operate by effectively reducing the infinite notion (in every context) in a finite rational manner. This much sought for rigour was in fact at bottom just a sham, diverting attention away from the crucial limitations imposed by its reductionist approach.

This opened up for me the much greater issue of how finite and infinite (or alternatively quantitative and qualitative) notions can be consistently related with each other within a mathematical context..

Indeed properly understood, this is the very issue to which the Riemann Hypothesis points.

However, this crucial overriding need to establish dynamic consistency as between quantitative and qualitative, is not even recognised within Conventional Mathematics (as its axioms are already based on the reduction of qualitative to quantitative meaning).

Very much in the context of the times (mid to late 60's) in Ireland it was a broadly based Arts degree, which also included as subjects English Literature and Latin. Looking back, I consider that this in fact proved very healthy for my overall intellectual development and would consider that present degree programmes have in the main become much too specialised in an increasingly narrow manner.

However at the time I did not enjoy the experience at all (especially the 1st year) as my growing conflict with standard approaches came to a head not only in Mathematics but equally with respect to Economics.

With the benefit of hindsight it is now much easier for me to understand what was really happening at this time.

I would put it like this! Just as electromagnetic energy can be viewed as a spectrum with many distinctive bands, properly understood intellectual understanding should be seen in the same light.

However standard intellectual discourse especially with respect to Mathematics and the Sciences is presently heavily concentrated on just one narrow band of the overall spectrum.

This is what I refer to as linear (1-dimensional) understanding. In any context for discussion, this approaches truth in terms of dualistic notions based on just one polar reference frame. So typically for example in Mathematics the external (objective) is abstracted from its internal (subjective) pole; likewise the independent (quantitative) is likewise abstracted from the interdependent (qualitative) pole.

However just as natural light forms just one small band on the overall electromagnetic spectrum, properly appreciated, such dualistic linear understanding likewise forms just one small band on the overall spectrum of possible mathematical and scientific understanding.

Now, in the various spiritual contemplative traditions, it has long been recognised that many further stages of potential development exist with respect to intuitive understanding of an increasingly refined nature. Indeed detailed accounts testifying to the universal features of such states (both East and West) have been made available by the spiritual pioneers of the various religions..

However what has not yet been properly realised is that these advanced intuitive states of an increasingly holistic nondual nature, have not only relevance in a contemplative spiritual context, but equally have enormous implications for the advanced appreciation of all mathematical and scientific relationships.

So a major breakdown with respect to the rigid dualistic notions that especially inform our existing understanding of Mathematics (and related Sciences) was taking place with respect to my personal development during this period.

Thus I was already moving towards this distinctive holistic approach to Mathematics (and also Economics) for which no formal recognition whatsoever existed in the culture (especially at University level).

I remember for me it proved an intensely lonely and disillusioning experience with a marked decline in ability and motivation for what was conventionally acceptable, together with a deeper emerging holistic appreciation (in which no one displayed the slightest interest).

In fact I found my time at University pursuing Mathematics such an alienating experience that it led me to drop the subject altogether at the end of the year.

However, a much deeper philosophical questioning regarding the standard assumptions of Mathematics had taken place which especially centred around the treatment of the infinite notion.

I could readily see that that this key notion was being treated in a reduced rational manner, which was strictly nonsensical.

For example it was customary to define an infinite number as greater than any finite number. But to me this lacked any coherence! If one designates any finite number in actual terms (regardless of how large) another finite number necessarily can be found that is greater than it. Thus we cannot meaningfully approach the infinite notion from an actual finite perspective. Therefore a key qualitative distinction separates the finite from the infinite notion with both necessarily occurring in a dynamic interactive framework.

So, for example, the infinite notion (with respect to number) strictly applies in a merely potential manner to all numbers, whereas the finite notion always necessarily applies in a specific actual - and thereby limited -context.

Crucially therefore, both notions strictly can only meaningfully apply to number in a dynamic relative (i.e. approximate) context.

Indeed there were strong parallels here with the existential crisis I was going through at the the time with all the comforting belief systems in my life seemingly breaking down. In such a situation one searches for a deeper resolution to the problem through developing an authentic belief system, intimately based on personal conviction. So I now was led to adopt the same approach with respect to Mathematics with a willingness to challenge all conventional notions that did not stand up to serious scrutiny.

In fact this experience, though chastening at the time, was to quickly lead on to a truly dynamic interactive appreciation of the nature of the number system.

Initially this understanding was based heavily on philosophical type understanding. However I was later enabled to interpret it all coherently in a new holistic mathematical manner.

So the dynamic interaction of finite and infinite notions is inseparable from our very understanding of number.

The number concept is infinite in the sense that it applies potentially to all numbers. However any specific number identified (as a corresponding perception) is then necessarily of a finite nature.

Thus in dynamic terms, the very ability to posit (or determine) specific numbers (as perceptions) always implies another set of finite numbers (that always - by definition - must remain undetermined). In this sense our knowledge of number necessarily takes place against a background of uncertainty.

From this dynamic perspective, it makes no sense to attempt to approach the infinite notion from a finite perspective. So once again they are qualitative and quantitative with respect to each other.

In terms of experiential understanding, the very appreciation of the infinite is directly associated with intuition, whereas finite notions are directly interpreted in a rational manner.

Thus the dynamic interaction of both finite and infinite in number terms is replicated in psycho spiritual terms through the dynamic interaction of both reason and intuition.

However because of the reduced nature of Conventional Mathematics, in formal terms this relationship is defined in a merely static rational manner. Thus the inevitable interaction of reason and intuition in experience is reduced in merely rational terms. This of course equally implies that the true - merely potential - nature of the infinite notion is likewise reduced in a merely finite actual manner. This then leads directly to the confusion of the (potential) infinite with (actual) finite notions.

Therefore to put it briefly, I quickly realised that Conventional Mathematics - despite its appearance of great rigour with its modern treatment of limiting notions - could only operate by effectively reducing the infinite notion (in every context) in a finite rational manner. This much sought for rigour was in fact at bottom just a sham, diverting attention away from the crucial limitations imposed by its reductionist approach.

This opened up for me the much greater issue of how finite and infinite (or alternatively quantitative and qualitative) notions can be consistently related with each other within a mathematical context..

Indeed properly understood, this is the very issue to which the Riemann Hypothesis points.

However, this crucial overriding need to establish dynamic consistency as between quantitative and qualitative, is not even recognised within Conventional Mathematics (as its axioms are already based on the reduction of qualitative to quantitative meaning).

## Monday, June 23, 2014

### Reflections on Experience (1)

It might help perhaps to put much of what I have been discussing in my various blogs in better context by relating key ideas to my own personal development.

What I believe has distinguished my own approach from a comparatively early age is a strong belief that Mathematics is very different from what we customarily imagine and that in effect the accepted paradigm represents but a reduced and very limited version of a much greater mathematical reality.

A key personal preoccupation, evident from a very early age, was the relationship as between addition and multiplication.

Before I become aware of logs (at around the age of 6), I had been experimenting with my own system of converting multiplication to addition.

Basically I operated with what would be identified as a base 2 system.

So to give a simple example, if we wished to multiply 8 by 32, the first number (in a base 2 system) could be expressed as 2

Thus multiplication of the two numbers (on the LHS) would be represented as the sum of the two powers (with respect to base of 2) on the RHS.

So 8 * 32 = 2

Now in principle we could express any number as a power of 2. Therefore to multiply numbers, we would express each number as 2 raised to a certain power and then add the two powers before calculating the result (i.e. 2 raised to the combined dimensional power).

I was later to discover that what I had discovered formed the very basis for logarithms.

Here, however instead of a base 2, I was initially introduced in primary school to the base 10 system.

Therefore in multiplying any two numbers, we first obtain the logs (to the base 10). In other words this expresses the power the number 10 would be raised to, in order to obtain the number in question.

Then in multiplying the two numbers we would add the logs (representing these dimensional powers).

Then to obtain our final answer we would obtain the anti-log i.e. the number that corresponds to 10 (raised to the combined sum of powers involved).

So to take our example again of 8 * 32!

In the base 10 system this would be 10

Then adding the logs (i.e powers) we get 2.4082.

Then by obtaining the anti-log (10

So what fascinated me regarding this early mathematical adventure was how multiplication from one perspective (i.e. the numbers to be multiplied) represented addition (with respect to dimensional powers).

Thus I already formed an inkling of the fact that - rather than just one universal definition - numbers possessed two complementary meanings, with the first relating to a number (raised to another number) and the second relating to the corresponding power (to which the original number is raised).

Later while still attending primary school in Ireland, I began to seriously question the very manner in which multiplication is interpreted.

At the time we were working in class through simple arithmetical problems relating to the practical application of multiplication to areas and volumes.

This was before the metric era where inches, feet, yards etc provided the customary means for measuring lengths.

So if for example we had a rectangular field with length 80 yards and width 60 yards, the area of the field i.e. 4800 (i.e. just less than an acre) would be expressed in square yards.

Therefore though the starting measurements (i.e. length and width) are expressed in linear (1-dimensional) units, the corresponding area is expressed in square (2-dimensional) units.

So in multiplying the length by the width, both a quantitative and a qualitative transformation takes place with respect to the units involved.

So 80 * 60 = 4800 in quantitative terms; however we have now also switched from a linear (1-dimensional) to a square (2-dimensional) interpretation with respect to the nature of the units. In other words a qualitative transformation with respect to the (dimensional) nature of the units is likewise involved.

And here through reflection on an (apparently) simple arithmetical problem, I could see how a fundamental distinction separated the nature of addition and multiplication.

So if we seek to deal with the two lengths from an arithmetical perspective, no (qualitative) change in the nature of the units is involved.

Thus 80 + 60 (both measured in 1-dimensional units) = 140 (also measured in 1-dimensional terms). So here, the operation of addition can be interpreted in a merely quantitative manner.

However 80 * 60 (both measured in 1-dimensional units) = 4800 (now measured in 2-dimensional units).

So in contrast to addition, the operation of multiplication must be interpreted in both a quantitative and qualitative manner.

I also could see clearly at the time that a qualitative transformation would always be necessarily involved whenever multiplication is involved. With the multiplication of 2 numbers we would move to 2-dimensional units, with 3 numbers 3-dimensional units and so on!

Thus the repeated attempt, in conventional mathematical terms, to treat numbers as "abstract" quantities represents a gross form of reductionism, which thereby conceals the true nature of multiplication.

So quite simply in conventional mathematical terms, whenever numbers are multiplied, the qualitative nature of the transformation involved is reduced in a merely quantitative manner.

Thus from this perspective, 80 * 60 = 4800, i.e. 80

I strongly realised - though of course I would have found it difficult then to properly articulate my reservations - that something truly fundamental with respect to mathematical truth was involved here and so attempted to explore the issue further.

Therefore in order to highly the key distinction involved (with respect to addition and multiplication), I concentrated on the simplest possible case.

Thus in terms of addition, 1 + 1 = 2.

Now expressed more fully with respect to linear (1-dimensional) units,

1

However, when we now apply multiplication to these two units,

1

So in the first case (with respect to addition), a quantitative transformation with respect to the units takes place (with the dimensional power or exponents of the units remaining at their default value of 1).

However by contrast (with respect to multiplication), a qualitative transformation with respect to the dimensional nature of the units takes place. (with now the base numbers remaining at their default value of 1).

What is remarkable is how the number 2 in both operations is associated with distinctive meanings!

Thus in the first case, 2 has the standard quantitative interpretation; however in the latter case 2 now relates to a distinctive qualitative interpretation!

So we have here in this simple example, the genesis for two distinctive interpretations of the number system.

The first - which I refer to as - Type 1 is of the standard quantitative nature; however the latter - Type 2 - is of a distinctive qualitative nature.

I was still too young to appreciate that these two systems implied that the number system was necessarily of a dynamic interactive nature. However fundamental progress had already certainly been made.

Now because of long training in the reduced quantitative means of interpreting mathematical symbols, the eyes of professional mathematicians will glaze over at the very mention of the qualitative which they see as having no direct relevance to their discipline.

However I wish to repeat now that qualitative notions are necessarily involved in all mathematical relationships (though consistently confused with quantitative type interpretation).

Again this is of key relevance in distinguishing the very nature of multiplication.

Stated briefly, the quantitative aspect of mathematical interpretation pertains to the treatment of numbers as separate (i.e. independent of each other). This is embodied in the cardinal notion of number.

3 for example is interpreted as a whole unit which is defined in terms of homogeneous units (that lack qualitative distinction). So 3 = 1 + 1 + 1 (with these units interpreted in an absolutely independent manner).

However the qualitative aspect of interpretation pertains to the corresponding treatment of numbers as related (i.e. interdependent with each other). This, by contrast, is embodied in the ordinal notion of number.

So if I refer now to the ordinal notion of 3 (i.e. 3rd) by definition this has no meaning in the absence of its related context with other members of a number group.

Indeed, though it is moving much further ahead, this is the basis of how all numbers can be given two distinctive meanings. So 2 for example can be given both a cardinal and ordinal meaning respectively. The cardinal then relates directly to the quantitative notion of number as separate and independent from other numbers; the ordinal then relates by contrast directly to qualitative notion of number as related and interdependent with other numbers.

Now when one reflects on experience of number, it necessarily keeps switching as between cardinal and ordinal type meaning.

The cardinal recognition of number has always an implied ordinal meaning. For example if one counts out 2 numbers (in cardinal terms) this implies recognition of a 1st and 2nd member (in an ordinal manner).

Likewise if one ranks two numbers as 1st and 2nd, then this likewise implies the cardinal recognition of 2!

However because of the dominance of the quantitative approach within Mathematics, ordinal notions are simply reduced in a cardinal manner. Thus the understanding of the primes and the natural numbers is carried out exclusively with respect to mere cardinal notions.

In this way, the key nature of the number system is thereby lost in that it properly represents the continued interaction of aspects that are both cardinal (quantitative) and ordinal (qualitative) with respect to each other.

From a cardinal perspective, each (composite) natural number represents the unique product of 2 or more primes.

So for example 6 = 2 * 3.

I have already used one type of argument to suggest that both a quantitative and qualitative transformation is involved when we multiply these two numbers.

So we can imagine 3 as independent units separated in linear time (arranged for example in a row).

So we could represent this operation by two distinct rows of 3. Then we add up all the separate units we get 6 (which represents the quantitative transformation implied by the relationship).

However the very capacity to see the two rows of 3 as common, thereby enabling the multiplication by 2, implies the corresponding recognition of interdependence (i.e. as two similar rows) which is a directly qualitative distinction.

Therefore, 2 * 3 implies recognition of both (quantitative) independence in the recognition of the separate units of each row and (qualitative) interdependent elements with respect to the recognition of the common similar nature of both rows.

So once again Conventional Mathematics inevitably reduces this interactive understanding (entailing both quantitative and qualitative aspects) in a reduced - merely - quantitative manner.

Now a deeper issue raised by this multiplication process is that the qualitative recognition of shared interdependence does not belong to linear time but in fact entails moving to an entirely new appreciation of n-dimensional time (and space). So in the simplest case involving the multiplication of two numbers, 2-dimensional time (and 2-dimensional space) is involved. Now of course because of the reduced 1-dimensional nature of Conventional Mathematics (from a qualitative perspective), we fail completely to recognise this crucially important point and thereby consistently misrepresent - in formal terms - the true dynamic nature of our experience of number.

So even at a very early age (10 or 11) I was beginning to see - literally - completely new dimensions to Mathematics that I was intent on exploring further. Thus I was already beginning to realise that what is commonly represented as Mathematics (i.e. its absolute quantitative interpretation) represents but an - admittedly very important - special case of a much more comprehensive mathematical reality.

This was brought home to me in a startling way through continual reflection on the square root of 1.

It struck me as very strange how in Conventional Mathematics the square of 1 had just one unambiguous result, whereby the corresponding square root can be given two results which are direct opposites of each other!

So the square of 1 is given as 1, whereas the two square roots of 1 are given as + 1 and – 1 respectively.

Now I reckoned that it would not be considered possible for example in the matter of mathematical proof that a proposition could equally be given a positive truth value (+ 1) and a negative truth value (– 1).

This would thereby suggest for example that the Pythagorean Theorem could be both true and false!

The very nature of accepted mathematical proof is unambiguous and designed to rule out the possibility of such a situation. Therefore by definition if a proposition is proven true, it cannot also be false (and if proven false cannot be also true). Thus in the area of proof, the positive truth value rules out the negative, and the negative truth value rules out the positive!

However here with respect to one of the simplest mathematical operations, in quantitative terms, a positive truth value is deemed to include the negative, and the negative include the positive.

For me this represented blatant inconsistency with respect to the accepted linear (1-dimensional) logic on which conventional mathematical truth is predicated.

So I already began to suspect that associated with the square of a number was a "higher" 2-dimensional logic that operated in a very different manner from what was conventionally accepted.

Furthermore I suspected that there was a direct link as between this "higher" dimensional logic (in qualitative terms) and the corresponding roots of 1 (in quantitative terms).

This was indeed heady stuff! However I lacked the sufficient mental maturity (especially at a philosophical level) to make further progress at this stage.

So for the remaining primary and secondary years I maintained an uneasy balance as between conventional mathematical notions (which I already suspected were deeply flawed) and emerging personal notions (which however could not yet be properly articulated at this time).

What I believe has distinguished my own approach from a comparatively early age is a strong belief that Mathematics is very different from what we customarily imagine and that in effect the accepted paradigm represents but a reduced and very limited version of a much greater mathematical reality.

A key personal preoccupation, evident from a very early age, was the relationship as between addition and multiplication.

Before I become aware of logs (at around the age of 6), I had been experimenting with my own system of converting multiplication to addition.

Basically I operated with what would be identified as a base 2 system.

So to give a simple example, if we wished to multiply 8 by 32, the first number (in a base 2 system) could be expressed as 2

^{3 }and the second as 2^{5 }respectively. So 8 * 32 = 2^{3 }* 2^{5}. Thus with respect to the RHS, we would now add the dimensional powers (indices).Thus multiplication of the two numbers (on the LHS) would be represented as the sum of the two powers (with respect to base of 2) on the RHS.

So 8 * 32 = 2

^{3 + 5 }= 2^{8 }= 256.Now in principle we could express any number as a power of 2. Therefore to multiply numbers, we would express each number as 2 raised to a certain power and then add the two powers before calculating the result (i.e. 2 raised to the combined dimensional power).

I was later to discover that what I had discovered formed the very basis for logarithms.

Here, however instead of a base 2, I was initially introduced in primary school to the base 10 system.

Therefore in multiplying any two numbers, we first obtain the logs (to the base 10). In other words this expresses the power the number 10 would be raised to, in order to obtain the number in question.

Then in multiplying the two numbers we would add the logs (representing these dimensional powers).

Then to obtain our final answer we would obtain the anti-log i.e. the number that corresponds to 10 (raised to the combined sum of powers involved).

So to take our example again of 8 * 32!

In the base 10 system this would be 10

^{.9031 }* 10^{.1.5051}.Then adding the logs (i.e powers) we get 2.4082.

Then by obtaining the anti-log (10

^{2.4082}) we get 255.976 which approximates closely to 256.So what fascinated me regarding this early mathematical adventure was how multiplication from one perspective (i.e. the numbers to be multiplied) represented addition (with respect to dimensional powers).

Thus I already formed an inkling of the fact that - rather than just one universal definition - numbers possessed two complementary meanings, with the first relating to a number (raised to another number) and the second relating to the corresponding power (to which the original number is raised).

Later while still attending primary school in Ireland, I began to seriously question the very manner in which multiplication is interpreted.

At the time we were working in class through simple arithmetical problems relating to the practical application of multiplication to areas and volumes.

This was before the metric era where inches, feet, yards etc provided the customary means for measuring lengths.

So if for example we had a rectangular field with length 80 yards and width 60 yards, the area of the field i.e. 4800 (i.e. just less than an acre) would be expressed in square yards.

Therefore though the starting measurements (i.e. length and width) are expressed in linear (1-dimensional) units, the corresponding area is expressed in square (2-dimensional) units.

So in multiplying the length by the width, both a quantitative and a qualitative transformation takes place with respect to the units involved.

So 80 * 60 = 4800 in quantitative terms; however we have now also switched from a linear (1-dimensional) to a square (2-dimensional) interpretation with respect to the nature of the units. In other words a qualitative transformation with respect to the (dimensional) nature of the units is likewise involved.

And here through reflection on an (apparently) simple arithmetical problem, I could see how a fundamental distinction separated the nature of addition and multiplication.

So if we seek to deal with the two lengths from an arithmetical perspective, no (qualitative) change in the nature of the units is involved.

Thus 80 + 60 (both measured in 1-dimensional units) = 140 (also measured in 1-dimensional terms). So here, the operation of addition can be interpreted in a merely quantitative manner.

However 80 * 60 (both measured in 1-dimensional units) = 4800 (now measured in 2-dimensional units).

So in contrast to addition, the operation of multiplication must be interpreted in both a quantitative and qualitative manner.

I also could see clearly at the time that a qualitative transformation would always be necessarily involved whenever multiplication is involved. With the multiplication of 2 numbers we would move to 2-dimensional units, with 3 numbers 3-dimensional units and so on!

Thus the repeated attempt, in conventional mathematical terms, to treat numbers as "abstract" quantities represents a gross form of reductionism, which thereby conceals the true nature of multiplication.

So quite simply in conventional mathematical terms, whenever numbers are multiplied, the qualitative nature of the transformation involved is reduced in a merely quantitative manner.

Thus from this perspective, 80 * 60 = 4800, i.e. 80

^{1 }* 60^{1}^{ }= 4800^{1}. And strictly, this is in error. Stated in a more qualified manner, it represents but a reduced quantitative interpretation of a result that properly entails both quantitative and qualitative aspects.I strongly realised - though of course I would have found it difficult then to properly articulate my reservations - that something truly fundamental with respect to mathematical truth was involved here and so attempted to explore the issue further.

Therefore in order to highly the key distinction involved (with respect to addition and multiplication), I concentrated on the simplest possible case.

Thus in terms of addition, 1 + 1 = 2.

Now expressed more fully with respect to linear (1-dimensional) units,

1

^{1 }+ 1^{1}^{ }= 2^{1}However, when we now apply multiplication to these two units,

1

^{1 }* 1^{1}^{ }= 1^{2}.So in the first case (with respect to addition), a quantitative transformation with respect to the units takes place (with the dimensional power or exponents of the units remaining at their default value of 1).

However by contrast (with respect to multiplication), a qualitative transformation with respect to the dimensional nature of the units takes place. (with now the base numbers remaining at their default value of 1).

What is remarkable is how the number 2 in both operations is associated with distinctive meanings!

Thus in the first case, 2 has the standard quantitative interpretation; however in the latter case 2 now relates to a distinctive qualitative interpretation!

So we have here in this simple example, the genesis for two distinctive interpretations of the number system.

The first - which I refer to as - Type 1 is of the standard quantitative nature; however the latter - Type 2 - is of a distinctive qualitative nature.

I was still too young to appreciate that these two systems implied that the number system was necessarily of a dynamic interactive nature. However fundamental progress had already certainly been made.

Now because of long training in the reduced quantitative means of interpreting mathematical symbols, the eyes of professional mathematicians will glaze over at the very mention of the qualitative which they see as having no direct relevance to their discipline.

However I wish to repeat now that qualitative notions are necessarily involved in all mathematical relationships (though consistently confused with quantitative type interpretation).

Again this is of key relevance in distinguishing the very nature of multiplication.

Stated briefly, the quantitative aspect of mathematical interpretation pertains to the treatment of numbers as separate (i.e. independent of each other). This is embodied in the cardinal notion of number.

3 for example is interpreted as a whole unit which is defined in terms of homogeneous units (that lack qualitative distinction). So 3 = 1 + 1 + 1 (with these units interpreted in an absolutely independent manner).

However the qualitative aspect of interpretation pertains to the corresponding treatment of numbers as related (i.e. interdependent with each other). This, by contrast, is embodied in the ordinal notion of number.

So if I refer now to the ordinal notion of 3 (i.e. 3rd) by definition this has no meaning in the absence of its related context with other members of a number group.

Indeed, though it is moving much further ahead, this is the basis of how all numbers can be given two distinctive meanings. So 2 for example can be given both a cardinal and ordinal meaning respectively. The cardinal then relates directly to the quantitative notion of number as separate and independent from other numbers; the ordinal then relates by contrast directly to qualitative notion of number as related and interdependent with other numbers.

Now when one reflects on experience of number, it necessarily keeps switching as between cardinal and ordinal type meaning.

The cardinal recognition of number has always an implied ordinal meaning. For example if one counts out 2 numbers (in cardinal terms) this implies recognition of a 1st and 2nd member (in an ordinal manner).

Likewise if one ranks two numbers as 1st and 2nd, then this likewise implies the cardinal recognition of 2!

However because of the dominance of the quantitative approach within Mathematics, ordinal notions are simply reduced in a cardinal manner. Thus the understanding of the primes and the natural numbers is carried out exclusively with respect to mere cardinal notions.

In this way, the key nature of the number system is thereby lost in that it properly represents the continued interaction of aspects that are both cardinal (quantitative) and ordinal (qualitative) with respect to each other.

From a cardinal perspective, each (composite) natural number represents the unique product of 2 or more primes.

So for example 6 = 2 * 3.

I have already used one type of argument to suggest that both a quantitative and qualitative transformation is involved when we multiply these two numbers.

So we can imagine 3 as independent units separated in linear time (arranged for example in a row).

So we could represent this operation by two distinct rows of 3. Then we add up all the separate units we get 6 (which represents the quantitative transformation implied by the relationship).

However the very capacity to see the two rows of 3 as common, thereby enabling the multiplication by 2, implies the corresponding recognition of interdependence (i.e. as two similar rows) which is a directly qualitative distinction.

Therefore, 2 * 3 implies recognition of both (quantitative) independence in the recognition of the separate units of each row and (qualitative) interdependent elements with respect to the recognition of the common similar nature of both rows.

So once again Conventional Mathematics inevitably reduces this interactive understanding (entailing both quantitative and qualitative aspects) in a reduced - merely - quantitative manner.

Now a deeper issue raised by this multiplication process is that the qualitative recognition of shared interdependence does not belong to linear time but in fact entails moving to an entirely new appreciation of n-dimensional time (and space). So in the simplest case involving the multiplication of two numbers, 2-dimensional time (and 2-dimensional space) is involved. Now of course because of the reduced 1-dimensional nature of Conventional Mathematics (from a qualitative perspective), we fail completely to recognise this crucially important point and thereby consistently misrepresent - in formal terms - the true dynamic nature of our experience of number.

So even at a very early age (10 or 11) I was beginning to see - literally - completely new dimensions to Mathematics that I was intent on exploring further. Thus I was already beginning to realise that what is commonly represented as Mathematics (i.e. its absolute quantitative interpretation) represents but an - admittedly very important - special case of a much more comprehensive mathematical reality.

This was brought home to me in a startling way through continual reflection on the square root of 1.

It struck me as very strange how in Conventional Mathematics the square of 1 had just one unambiguous result, whereby the corresponding square root can be given two results which are direct opposites of each other!

So the square of 1 is given as 1, whereas the two square roots of 1 are given as + 1 and – 1 respectively.

Now I reckoned that it would not be considered possible for example in the matter of mathematical proof that a proposition could equally be given a positive truth value (+ 1) and a negative truth value (– 1).

This would thereby suggest for example that the Pythagorean Theorem could be both true and false!

The very nature of accepted mathematical proof is unambiguous and designed to rule out the possibility of such a situation. Therefore by definition if a proposition is proven true, it cannot also be false (and if proven false cannot be also true). Thus in the area of proof, the positive truth value rules out the negative, and the negative truth value rules out the positive!

However here with respect to one of the simplest mathematical operations, in quantitative terms, a positive truth value is deemed to include the negative, and the negative include the positive.

For me this represented blatant inconsistency with respect to the accepted linear (1-dimensional) logic on which conventional mathematical truth is predicated.

So I already began to suspect that associated with the square of a number was a "higher" 2-dimensional logic that operated in a very different manner from what was conventionally accepted.

Furthermore I suspected that there was a direct link as between this "higher" dimensional logic (in qualitative terms) and the corresponding roots of 1 (in quantitative terms).

This was indeed heady stuff! However I lacked the sufficient mental maturity (especially at a philosophical level) to make further progress at this stage.

So for the remaining primary and secondary years I maintained an uneasy balance as between conventional mathematical notions (which I already suspected were deeply flawed) and emerging personal notions (which however could not yet be properly articulated at this time).

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