Sunday, October 23, 2011

Wholes and Parts

An intimate mathematical relationship exists as between wholes and parts in both linear and circular logical terms respectively.

Now it is the very essence of the linear approach to give a merely reduced rational interpretation of wholes and parts (where the whole is viewed solely in quantitative terms as the sum of its constituent parts).
From a conventional (linear) perspective a cake for example might comprise four (equal) slices. Thus the whole cake in this context is interpreted as the sum of the four (4) part slices.

Then in reverse manner each individual slice is interpreted - again in quantitative terms - as a quarter (1/4) of the whole cake.


Now in (Type 1) mathematical terms the whole cake (containing the four slices) = 4^1.

Meanwhile the reverse relationship of each part to this whole = 4^(- 1) = 1/4.

Thus, fascinatingly, the very ability to switch in experience from whole to part recognition entails the corresponding ability to implicitly negate the 1st dimension. What this means is that understanding implicitly must switch from rational to intuitive mode before this important transition can be made. However when qualitative interpretation is absolute, this switch from positive to negative sign is ignored so that it formally remains at a merly rational level!

Thus it requires Type 2 mathematical understanding to provide this important insight of how intuition insight is necessarily involved in the understanding of the true relationship as between whole and part..

Therefore once again Type 1 interpretation - which is conducted in merely reduced rational terms - cannot explain the decisive qualitative means of switching from whole to part (and part to whole). In order words in confining itself to a merely (reduced) quantitative (Type 1) interpretation of its symbols, it ignores completely the equally important qualitative (Type 2) interpretation of the same symbols.

Thus the minus sign (-) has a well-defined meaning in quantitative (Type 1) terms. However the same sign has a largely unrecognised qualitative (Type 2) meaning as the negation of what is (consciously) posited in rational terms. In dynamic interactive terms, this implies the (unconscious) intuitive aspect of recognition. Thus from a Type 2 perspective, in transferring from whole to part recognition, linear (1-dimensional) interpretation of a rational kind must be (temporarily) negated in a qualitative form of holistic recognition. Thus though the part can then be likewise interpreted in a quantitative manner, the decisive switch from whole to part recognition requires Type 2 qualitative recognition.


However equally we have a circular number system that allows for a whole range of differing qualitative interpretations in each case entailing a unique configuration of both (conscious) rational and (unconscious) intuitive type appreciation. From this perspective the (whole) dimension given by the number 4, is qualitatively defined in Type 2 terms (as 1^4).

Then 1/4 i.e. 1 ^(1/4) represents the reverse quantitative relationship in part terms for this system.

In circular quantitative terms this is given as i (which lies on the unit circle in the complex plane).

Thus in reverse terms i refers to the qualitative meaning of 4 (as a whole dimension).

This clearly indicates again therefore that the very switch from whole to part recognition always entails a corresponding switch from qualitative to quantitative. However understanding here is much more refined allowing for imaginary (rather than real) appreciation.

So 4-dimensional interpretation from a qualitative mathematical perspective is of an imaginary (rather than real) rational nature, as the indirect expression of unconscious meaning.

One important physical application of this principle relates to quantum mechanics where the "higher" psychological interpretation of a qualitative nature is always necessary to properly interpret "lower" sub-atomic phenomena (of a physical nature). In other words if the manner of the "higher" intuitive recognition in qualitative terms is n-dimensional, the corresponding "lower" physical recognition with respect to physical "parts" in nature is 1/n. So once again their is a reciprocal qualitative to quantitative (and quantitative to qualitative) relationship as between whole and part.

Wednesday, October 19, 2011

Number as Dynamic Interaction

When properly appreciated all numbers represent a dynamic interactive process that necessarily entails both quantitative and qualitative aspects (which provide the very basis for such interaction).

A number perception representing a quantity is strictly speaking meaningless in the absence of its corresponding number concept (which relatively is of a qualitative nature).

So for example the number "2" representing a specific number perception has no meaning in the absence of the universal concept of number.

And this concept of number is strictly qualitative relating to what is potential and infinite. So in this qualitative sense the general concept of number applies potentially to all (as yet unspecified) numbers in an infinite manner.

However any specific number is necessarily of an actual finite nature. And this relationship of actual number perceptions (as quantities) and the potential number concept (as qualitative) provides the very dynamic for subsequent number interaction in experience.

Of course just as a left turn off a road becomes a right (and a right turn becomes a left) when one switches the direction of movement along the road, likewise through dynamic switching in experience number perceptions equally can attain a qualitative and the number concept - relatively - a quantitative aspect.

So from this latter perspective the number perception is seen to embody, as it were, the general property of number (that is qualitative in nature) while the concept is understood to apply to all actual numbers (in quantitative terms).

This thereby represents the reverse direction of the relationship as between quantitative and qualitative in number experience.

From a psychological perspective the dynamic interaction by which both the quantitative and qualitative aspects of number appreciation occur, requires in turn both rational (conscious) and intuitive (unconscious) understanding. In this interaction the intuitive aspect points to the holistic potential aspect of qualitative recognition (that switches as between concept and perception). The rational then points to corresponding actual quantitative recognition (that again switches - in relative fashion - as between perception and concept).

In the conventional (Type 1) approach, Mathematics is formally interpreted in a rational manner. This entails in turn that a merely reduced quantitative notion of number is given.

The starting point for a true interactive (Type 3) approach is the recognition that every specific number quantity implicitly implies a corresponding dimensional number concept that is - relatively - of a qualitative nature (and vice versa).

Thus for example "2", which conventionally in Type 1 terms is interpreted (solely) as a number quantity, equally has an implicit qualitative aspect (as representing a number dimension).

So in more complete terms, the natural number quantity "2" is defined with respect to a (default) number dimension 1 which, in experiential terms, is - relatively - of a qualitative nature.

So in this context 2 is properly 2^1.

Now, the very nature of 1, when used as a qualitative dimension is that qualitative is necessarily reduced to quantitative type meaning. Therefore though the number concept properly refers here to a potential - as opposed to actual - meaning, from a linear (1-dimensional) rational perspective, this is then interpreted in a merely reduced manner (as applying to all actual numbers).


However just as we can define the extreme quantitative (Type 1) approach in terms of a default dimensional number (i.e. 1), likewise we can define an extreme qualitative (Type 2) approach in terms of a default base quantity that is also "1".

So "2" in this approach representing a dimensional number is written as 1^2.
The significance of this latter approach is that each number represents a unique manner of qualitative interpretation of mathematical symbols.
Thus from this perspective we have a - potentially - infinite set of ways for the logical interpretation of mathematical relationships. And the qualitative structure of each logical system of interpretation exactly matches the corresponding root structure (in quantitative terms)! So just as the two roots of 1 - for example - in quantitative terms are + 1 and - 1, equally the qualitative logical corresponding to 2 (as dimensional number) is based on the complementarity of opposite poles (i.e. + 1 and - 1, taken as interdependent. So in quantitative terms we interpret the roots in a linear either/or logic (based on the independence of opposites) in corresponding qualitative terms we interpret the dimensional number in both/and terms (based on their interdependence).

Extending this realisation, every mathematical symbol that is given a specific quantitative interpretation in Type 1 Mathematics, can equally be given a holistic qualitative interpretation in Type 2 terms.

In Type 3 Mathematics we then attempt to interactively combine both Type 1 and Type 2 understanding.

Thus the simple mathematical expression 2^2, in Type 3 terms represents a dynamic interaction as between 2 as base number and 2 as dimensional number (with each number having both quantitative and qualitative aspects always in opposite relationship to each other).

Wednesday, October 12, 2011

Three Levels of Appreciation of Transcendental Numbers

I have mentioned on many occasions that three type of mathematics exist i.e Type 1, Type 2 and Type 3 respectively.

I will now interpret the meaning of a transcendental number such as e according to the three types.

In conventional Type 1 appreciation, e represents e^1. So when one concentrates merely on the quantitative interpretation of a number, it is always defined with respect to the (default) 1st dimension. So here we have - literally - a linear rational interpretation of number.

Properly speaking as we will see, a number such as e transcends mere rational interpretation. In qualitative terms, rational (linear) is suited to the interpretation of rational discrete numbers that are finite in nature.

However the very essence of e - and indeed all transcendental numbers - is that they combine both finite and infinite aspects in their very nature. Thus though the quantitative value of e can be approximated to any required degree of accuracy, its true value always remains unknown (leading potentially to an unlimited number of terms in its decimal sequence).

Though there are many ingenious ways of representing e, as with all transcendental numbers it cannot be the solution to a polynomial equation.


In Type 2 appreciation, in a somewhat inverse fashion e represents the dimension to which the (default) no. 1 is raised.

So e in Type 2 terms is represented as 1^e.Therefore as a dimension, it now takes on a holistic qualitative significance (with respect to its default base quantity).


This is a crucial point that is not addressed in Type 1 interpretation. From Type 1 perspective, when e is used as a power it still represents a number quantity. However properly speaking the relationship between the base quantity and its dimensional power is always quantitative as to qualitative (and qualitative as to quantitative).

So e in Type 2 terms takes on an appropriate qualitative holistic meaning. Now a transcendental number in qualitative always expresses a relationship between rational (discrete) and intuitive (continuous) notions!
In this context e has a special significance as the number which uniquely reconciles both aspects within its own nature.
So in psychological (and corresponding physical terms) with e, both the processes of (discrete) differentiation and (continuous) integration are reconciled.
It serves therefore as an especially advanced qualitative symbol of such differentiation and integration (where both aspects are indistinguishable).


In Type 3 terms, understanding keeps switching as between both quantitative and qualitative interpretation with respect to e as both base and dimensional number respectively.

So we understand e as a (base) quantity. Then attention switches to understanding e in holistic terms as a dimensional quality; then attention switches again to now understanding e in holistic terms also as a base quality; and finally in this cycle attention switches to understanding e also as a dimensional quality.

So in this dynamic interactive manner, e whether representing a base or dimensional number, possesses both quantitative and qualitative aspects which keep alternating in experience.

This means that in Type 3 terms a proof always entails both quantitative and qualitative aspects and is subject to the Uncertainty Principle.

For example the Type 1 proof that any algebraic number raised to an algebraic power can in truth be given both a quantitative (Type 1) and qualitative (Type 2) interpretation. (In qualitative terms this amounts to a holistic mathematical interpretation of how development proceeds from the psychic/subtle to the causal level).

Then in Type 3 terms the Uncertainty Principle necessarily applies to the dynamic interpretation that combines both. In this manner, in comprehensive Type 3 terms, all mathematical proof is subject to the Uncertainty Principle.

Mathematical Dimensions and Psychological Development (2)

We have see in the last contribution that when the quantitative to qualitative relationship is maintained that raising a rational number to a rational fraction results in an irrational quantity.

So for example in the best known case, the square root of 2 i.e. 2^(1/2) = 1 .4142... is an irrational number.

In corresponding fashion when rational perceptions are dynamically related to rational concepts that are fuelled by appropriate intuitive appreciation of a qualitative nature (algebraic) irrational perception (and later conceptualisation) results. Expressed in more common language this entails the more refined appreciation of phenomena that are inherently paradoxical in nature.

The very nature of an irrational number is that it necessarily combines both finite and infinite aspects. Thus an irrational number can be approximated in rational terms to any required degree of accuracy. However equally it possesses an elusive infite aspect in that its exact value can never be known.

Likewise in corresponding qualitative terms, irrational understanding with respect to perceptions and concepts (in what is sometimes referred to as the psychic/subtle realm) combines both finite and infinite aspects. Thus phenomena in experience still possess a distinct identity (of a dynamic relative nature). However equally they possess a numinous spiritual quality that is infinite in nature.

As always we can identify - though in truth considerable overlap may be involved - three stages at this level.

Firstly we have the unfolding mainly of the more superficial refined phenomena that are qualitatively irrational in nature. Now once again the very basis of rational understanding is that interpretation appears - especially in a mathematical context - unambiguous in nature. Thus for example the positing therefore of what is true, implies the corresponding negation of its opposite as false.

However in the very dynamics of understanding at this level, phenomena that are consciously posited in experience are quickly negated (in an unconscious manner), Thus propositions take on a merely relative i.e. paradoxical truth value.

The second stage then entails deeper conceptual structures of understanding that are also irrational (in qualitative terms).

Finally at the most advanced stage we have the growing interaction of both irrational perceptions and irrational concepts paving the way for a remarkable transformation of experience to a new level.

When the mathematician Hilbert detailed 23 unproven propositions at his famous address in 1900, one related to finding a proof that any rational (or irrational) number when raised to an irrational dimension (power) would result in a transcendental number quantity!

A transcendental number resembles an even more subtle form of irrational number.

For example the the square root of 2 and the well known constant pi are both irrational numbers. However whereas the former - and indeed any number of this type - can be expressed as the solution to a polynomial equation, the latter cannot be expressed in this manner.

There is a fascinating corollary in qualitative terms. Paradoxical (2-dimensional) understanding based on the complementarity of opposites is clearly paradoxical in terms of conventional linear (1-dimensional) reason that is unambiguous in nature.

However if we define reason in terms of the former (2-dimensional) variety then from this perspective it is now rational. Indeed Hegel did precisely this in his writings defining reason in terms of his dialectic while treating conventional logic as a "lower" form that he termed "understanding".
The trouble is that Hegel then effectively reduced the nature of such dialectical reason through his failure to emphasise the corresponding need for the necessary supporting intuition provided through authentic contemplation.

This in turn is a regular failing at the irrational (psychic/subtle) level where secondary rigid attachments to the paradoxical symbols in experience emerge.

The resolution of this problem requires the profound negation of such attachment. In this way one gradually develops the ability to preserve an increasingly harmonious balance as between (conscious) reason and (unconscious) intuition.

Put another way this implies maintaining an appropriate relationship (that is quantitative as to qualitative) as between both the paradoxical perceptions and concepts that typify the level.

It was eventually proven in 1934 that when a rational (or irrational) number is raised to an irrational power that a transcendental number quantity results. And remember this was one of Hilbert's 23 propositions!

Remarkably we can provide the qualitative corollary to this proposition by saying that when rational (or irrational) perceptions are appropriately related to irrational concepts that a transformation in understanding takes place whereby experience of a transcendental nature emerges. And this is the important transformation that enables successful transition from the psychic/subtle to the causal level.

Now, we can understand the true nature of a transcendental number with respect to the nature of pi, which represents the pure relationship of the circular circumference to its line diameter. In like manner transcendental understanding (which typifies the causal realm) represents the pure relationship between circular appreciation (that is paradoxical) and rational understanding (of a linear nature). In other words it points directly to the common relationship as between both.
Now the center of a circle equally represents the midpoint of its line diameter. In like manner it is through the still point of being (representing the naked will through pure volitional intent) that both circular and linear type appreciation are reconciled. In this way the transcendental structures properly evolve. Because this entails approximating ever closer to this still point of being (in both physical and psychological terms) I have always referred to the causal level in holistic mathematical terms as the point level!

In terms of development of such structures the most refined possible are of an imaginary - rather than real - nature.

A real transcendental perception (using holistic mathematical language in a precise manner) relates to a consciousness of a specific phenomenon as representing the refined interaction of both conscious and unconscious aspects of experience (with both operating in close harmony).
A real transcendental concept then represents corresponding conscious experience of general universal categories as again representing the refined harmonious interaction of both conscious and unconscious aspects of experience.

However an imaginary transcendental perception is even more elusive as representing the indirect recognition of a projection emanating from the unconscious where again both conscious and unconscious aspects of recognition with respect to its temporary phenomenal identity are maintained in close harmony. And then finally an imaginary transcendental concept would entail the corresponding recognition with respect to indirectly projected universal categories of experience. In other words when conscious and unconscious aspects of recognition become so closely related in experience so as to approximate simultaneous identity, then - by definition - remaining involuntary attachment to phenomena largely ceases.

Some 20 years ago when I wrote the "Number Paradigms" I recognised in holistic mathematical terms that the most refined conceptual structures possible in experience are - in holistic mathematical terms - of an imaginary transcendental nature and that these typify the most advanced stage of the causal level (approaching pure spiritual union).

It was only later that I was able to properly make the connection as between such understanding and the Euler Identity and realise its deeper significance.

Wednesday, October 5, 2011

Mathematical Dimensions and Psychological Development (1)

I have already pointed to the fact that - properly understood - every number expression represents a dynamic interaction as between a a base quantity and a dimensional number (that is relatively of a qualitative nature).

So what we might refer to in conventional (Type 1) terms as the number quantity 2, more accurately is expressed as 2^1 (where 2 is quantitative and 1 - relatively - of a qualitative holistic nature).

However because the very nature of linear (1-dimensional) understanding is to reduce qualitative to quantitative type meaning, from a Type 1 perspective, interpretation of numbers is invariably reduced in a mere quantitative manner.

However when correctly appreciated in holistic Type 2 terms, number expressions have an intimate bearing on the interpretation of all the main stages on the spectrum of psychological development.


As we know number is the best means we have for quantitative ordering in experience. In fact such ordering is synonymous with number appreciation. And seen from this Type 1 perspective we have various number types such as prime, natural, rational, irrational etc.

What is not recognised is that the same number types have a profound relevance for the qualitative interpretation of development.
In fact the holistic structure of each of the main levels of psychological development can be precisely matched to expressions entailing the main number types. Furthermore we can subdivide each "number" level into three coherent stages showing how the transformation into the next level on the spectrum takes place.

So, three main stages are involved with respect to unfolding of each level.

Firstly we have the unfolding of specific perceptions (characteristic of the level in question). In the second stage this gives way to unfolding of the more generalised conceptual understanding (again characterising the level). Now the relationship of such perceptions and concepts is as quantitative and qualitative with respect to each other!
Then in the third stage increasing dynamic interaction takes place as between perceptions and concepts causing a transformation to a new level (whose structure in turn corresponds with a new distinctive number type).


Development commences from a completely undifferentiated state where form is indistinguishable from emptiness. This relates in qualitative terms to - what I refer to as - the original numbers. So unity (1) is initially indistinguishable from nothingness (0). Then when development commences the first initial differentiation takes place leading to the birth of duality (2) in experience.

Though we still have great confusion, in a qualified sense we can identify three stages (1) where duality takes place with respect to incipient perception (2) when duality now takes place with respect to the incipient formation of concepts and finally where some level of interaction of both takes place. This can be identified with the archaic level.


All going well, the dynamic interaction of understanding that is quantitative and qualitative with respect to each other, leads to a transformation to the next level.

This level is then identified in terms of the holistic interpretation of the prime numbers that entails an intimate relationship of both conscious and unconscious aspects of experience. Indeed we can accurately use the word primitive to refer to such experience where holistic collective notions (pertaining to the unconscious) are continually confused with distinct specific notions (pertaining to the conscious). We can identify this with the magical level.

Again in a qualified manner we have the unfolding of three stages firstly with respect to specific perceptions of a primitive kind, then more general concepts (of a primitive nature) and finally the growing dynamic interaction of both perceptions and concepts.

When we raise a prime number to a prime number dimension in quantitative terms, a transformation takes place in that we derive a new (composite) natural number.

Likewise in holistic psychological terms when we relate specific perceptions of a primitive quantitative nature to their general concepts (that relatively are qualitative in nature) a transformation to a new level of understanding takes place which correlates with the holistic nature of natural numbers.

One of the key characteristics of natural numbers is that numbers are whole (and not yet divided into fractional components).
Likewise at this level (which relates to mythic development) wholes - especially with respect to concepts - cannot be properly broken into parts. Put another way, abstract ability is not yet sufficiently developed so that objects are still given a (whole) personality. Only later when the conscious aspect is further differentiated from the unconscious does true abstract ability of an impersonal nature properly unfold.

One fascinating feature of this level is that one learns to - temporarily - negate both perceptions and concepts (while holding them in memory). This is made possible through the greater level of phenomenal constancy characteristic of this level.
This literally means therefore that such dynamic negation of concepts in experience relates to negative rather than positive dimensions (in number terms).


Once again we can identify three stages 1) where natural perceptions (and their corresponding temporary negation) takes place 2) natural concepts (and their temporary negation occurs and finally (3) the growing interaction of both perceptions and concepts in both a positive and negative sense.
So during this level the natural numbers give way to the integers (where numbers can be both positive and negative) in holistic terms. And the scientific structure of the level is based on such understanding.

When we look at the simple number expression 4^ (- 1) we have an integer that is raised to the negative of 1 (as dimensional power). And this number expression in Type 1 terms leads to the generation of a fraction i.e. 1/4.

Likewise in development, when whole perceptions are related to concepts (that can be temporarily negated in experience) we generate analytic understanding of a fractional kind.


This leads to an important transformation in development whereby the mythic gives rise to the rational stages (that are so relevant in conventional scientific understanding).

What is remarkable here is that appropriate holistic mathematical interpretation implies that even the simplest rational task implies the ability to temporarily negate the linear (1-dimensional) mode of understanding.

Let us illustrate with respect to the task of cutting a cake into four slices.

Now, the cake obviously represents a whole and each slice another unique whole object. So the ability to recognise each slice also as a part requires the temporary negation of its whole status. This then enables the link of the slice to the whole status of the cake whereby it is now recognised as a part of that whole. So, initially each slice is posited as a whole perception and then negated (with respect to this status) enabling it to be thereby understood as a part of the greater whole of the cake.
So here we establish the link between whole and part perceptions which implicitly requires the ability to temporarily negate their respective identities while still holding them in memory. And the same ability also implicitly develops with respect to concepts. For example one can initially recognises a specific number such as 2 as a perception. However the recognition that this number belongs to the universal number class i.e concept of number requires the temporary ability to negate its specific status in recognition of its general identity. Then the reverse procedure of once again recognising 2 as a specific perception in turn requires the temporary negation of its number concept. Therefore - though still operating at a merely implicit level the continual dynamic positing and negating of both perceptions (as quantities) and concepts (in qualitative terms) takes place. Once again the (linear) rational level - which dominates conventional understanding of Mathematics and Science - has 3 stages.

The first corresponds to the rational appreciation of perceptions in the ability to break object perceptions into smaller parts and rearranging them again in composite wholes. This is generally referred to in Piagetian terms as conop.

The second stage corresponds to the ability to break general universal concepts into sub-categories and then synthesise them with respect to the original concepts. In Piagetian terms this is formop.

Finally the third stage entails the growing interaction of rational perceptions and concepts. When suitably refined - though this is not generally the case - this leads to a dynamic intuitive based form of rational understanding i.e. vision-logic.


Another remarkable transformation occurs in Type 1 quantitative terms when a rational number is raised to a fractional dimension (as power) in that an irrational number results.

For example in the well known case when 2 is raised to 1/2 we obtain the square root of 2 = 1.4142... which is an irrational number.


However, in Western culture there is very little evidence of the sustained growth of further more advanced stages of understanding (beyond the rational level).

One important reason for this is quite revealing. The source of the dynamic transformation that takes place in the emergence of each new psychological level results from the interaction of perceptions and concepts, that are implicitly experienced as quantitative and qualitative with respect to each other. This enables therefore a fruitful interaction of conscious and unconscious to emerge.

In formal terms the interpretations of Mathematics and Science directly reduce the qualitative in quantitative terms. As this leads to emphasis solely on the conscious aspect of experience, a considerable diminution in true holistic type qualitative understanding can result. For example Holistic (Type 2) Mathematics, which I am outlining in these contributions, is totally unrecognised at present by the mathematical profession!


In experiential terms therefore this can reduce the unconscious intuitive aspect of understanding (within which true qualitative appreciation is embodied) to such a significant degree that sustained progress beyond the rational level is not possible.
In other words the specialisation of rational understanding, which is so characteristic of Western culture can greatly reduce the role of the unconscious in experience with the consequence that dynamic transformation beyond the rational level is significantly impeded.

Sunday, October 2, 2011

Multiplication and Addition

The key problem in reconciling addition with multiplication is that they represent mathematical processes that are quantitative and qualitative with respect to each other. And as Conventional (type 1) Mathematics is based on a merely reduced quantitative approach this creates enormous difficulties in properly appreciating the nature of the problem.


As we have seen properly we have two number systems that are quantitative and qualitative with respect to each other.

1) In the conventional (Type 1) system, the natural numbers 1, 2, 3, 4, 5,.... for example respect quantities are defined with respect to a (default) dimensional value of 1.

So written in full, this system is represented as:

1^1, 2^1, 3^1, 4^1, 5^1,........


2) In the unrecognised (Type 2) system, the same natural numbers 1, 2, 3, 4, 5,.... represent qualitative dimension that are defined with respect to a (default) base quantity of 1.

So written in full, this alternative system is represented as:

1^1, 1^2, 1^3, 1^4, 1^5,........


Now with respect to the first system when we add two numbers, say, 2 + 3,
this is fully represented as

(2^1) + (3^1) = 5^1


However when we add the same two numbers, 2 + 3, with respect to the second system, this is fully represented as

(1^2) * (1^3) = 1^5.


So whereas addition of these two numbers is involved with respect to the first (quantitative) system, multiplication of the same two numbers (now representing dimensions) is entailed with respect to the second (qualitative) system.

This clearly entails that whereas pure addition (i.e. with respect to numbers that are all defined with respect to 1 as default dimension) is of a direct quantitative nature. Pure multiplication (i.e. with respect to numbers that are all defined with respect to 1 as default base) is by contrast of a direct qualitative nature.

Therefore we cannot ultimately hope to reconcile addition and multiplication without equal recognition of both Type 1 (quantitative) and Type 2 (qualitative) numerical systems.

In practice therefore where non-unitary values are given to both (quantitative) base and (qualitative) dimensional numbers, Type 3 Mathematics (representing the coherent interaction of both Type 1 and Type 2 systems) must be used for comprehensive understanding.


A further problem relates to the reconciliation of multiplication and exponentiation.

If multiplication is now treated in a (Type 1) quantitative manner, then
2 * 3 for example is represented as (2^1) * (3^1) = 6^1.


However with respect to the second (Type 2) dimensional system 2 * 3 is represented as

(1^2)^3

So multiplication with respect to the two numbers in the Type 1 (quantitative) system represents exponentiation with respect to the same two numbers in the second.

And just as (2^1) * (3^1) = (3^1) * (2^1) (with respect to the first)

(1^2)^3 = (1^3)^2 (with respect to the second).

Saturday, October 1, 2011

Nature of Number as Dimension

I have been thinking again in a deeper manner regarding the nature of number representing a dimension (or power).

Let us start with the convenient (default) case of 1.

Now clearly 1 can represent a unit quantity. So implicit therefore in the recognition of any specific object is the number 1 (as an actual finite quantity).

However when used to represent a dimension the number 1 takes on a distinctive holistic meaning (in a potential infinite manner).

So for example if we attempt to represent the number system on a straight line this automatically presumes a linear (1-dimensional) background that is - potentially - infinite.

Therefore though we can use the same symbol 1 to represent a base unit quantity or alternatively the linear dimension (within which such a number is expressed) clearly the meaning is very different in each case.

In the former case 1 represents a specific finite notion that is inherently quantitative in nature; in the latter case it represents a holistic - potentially - infinite notion that is of an inherently qualitative nature.


And as all numbers representing quantitative values must implicitly be expressed with respect to a corresponding number dimension (with the default value = 1), then every number expression - when properly appreciated - necessarily entails a relationship between two aspects which are quantitative and qualitative with respect to each other.


As we have seen, the default dimensional state of a number is 1. And as it is the very nature of linear (1-dimensional) understanding to reduce the qualitative aspect to the quantitative, this means in effect that the qualitative notion of number is effectively always ignored in Conventional (Type 1) Mathematics.


This also causes an important difficulty when dealing with dimensional values (other than 1) which are inevitably treated in a reduced linear manner.


For example 2-dimensional reality would relate to a potentially infinite plane (within which a 2-dimensional object can be placed). However because the qualitative nature of logical understanding remains 1-dimensional, in Type 1 Maths this entails considering the plane as (linearly) extended in two directions that are horizontal and vertical with respect to each other.


However once we depart from 1-dimensional qualitative interpretation, the true nature of dimension is revealed to be of a circular nature.

In fact - when again appropriately understood - this is actually demonstrated in Type 1 Mathematics through the notion of roots.

If we obtain the two roots of unity, they will lie as equidistant points on the circle of unit radius (in the complex plane). Now we can of course in quantitative terms recognise these as + 1 and - 1 respectively. However if we are to give an appropriate 2-dimensional interpretation in qualitative terms (as is appropriate) then we require a logical means of combining + 1 and - 1 as being both true. Now this is done through the paradoxical (both/and) logic of the complementary opposites where each pole like the left and right turns on a road has a merely relative validity.

So we can see here an important inverse relationship as between the 2 quantitative roots of the number 1 and the corresponding 2-dimensional qualitative interpretation (with which they are consistent).

Strictly speaking we do not have 2 roots of 1 i.e 1^1.

- 1 is indeed the (unique) square root of 1. + 1 is however the (unique) square root of 1^2. And 1^1 and 1^2 relate to distinct qualitative numbers (representing dimensions).



Now in a comprehensive appreciation, an even more subtle dynamic interactive understanding is required. Thus when we start with the base - say of 1^2 - here 1 is quantitative and 2 as dimension - relatively - qualitative. However equally the base number 1 can be given a quantitative meaning with the dimensional number 2, thereby in relative terms quantitative. For example all natural logs (representing numbers as powers) clearly have a quantitative meaning as do for example the values of s in the Riemann Zeta Function!

So in relative terms, where the base number has a qualitative meaning this implies that the conceptual nature of the number is highlighted in understanding.

Thus 1 for example can be seen in quantitative terms as a number perception (in relation to the qualitative concept of number). However in reverse terms it can be seen as the number concept (to which 1 relates). So in experiential terms both aspects necessarily interact with all numbers thereby possessing both quantitative (specific) and qualitative (holistic) aspects.

However this poses severe limitations on a mathematical approach that solely recognises the quantitative aspect. And this reduced interpretation is what we misleadingly refer to as Mathematics.

More properly it refers to Type 1 Mathematics. So enormous scope remains for the proper development of Type 2 Mathematics (focussing on the neglected qualitative aspect) and Type 3 Mathematics (where both aspects - quantitative and qualitative - are coherently related).