More on Dynamic Nature of Number (4)

We have seen how number - when properly understood - entails a dynamic interaction between its quantitative and qualitative aspects (which are complementary with each other).

This raises the key issue of consistency with regard to the two aspects, when used in relation to one another.  

So again in Type 3 (radial) terms, it is now clearly understood that both the quantitative aspect of number (as independent) and the corresponding qualitative aspect (as interdependence) necessarily are of a relative nature.
 

Type 3 understanding represents both Type 1 (cardinal) and Type 2 (ordinal) aspects of number (where both are understood in dynamic terms as complementary).

So  strictly the cardinal notion of number (as explicitly understood) requires implicit understanding of the corresponding ordinal notion.

For example In Type 1 (base) terms, the cardinal notion of number is understood in a quantitative manner.

Therefore, using the number "3" to illustrate, this is explicitly understood as the sum of independent homogeneous units (that lack qualitative distinction).  Therefore 3 = 1 + 1 + 1.

However implicitly, a qualitative relationship must be recognised as existing between these individual units. Otherwise there would be no means of arriving at their composite total = 3.

Therefore explicitly the adding of 1 + 1 + 1 (in an independent quantitative manner) implicitly requires the qualitative ordinal recognition of these units (as 1st, 2nd and 3rd respectively).

Then in a reverse manner in explicit Type 2 (dimensional) terms, the ordinal notion of number as comprised of related ordinal components, implicitly requires the cardinal notion of number (as comprised of independent units).

Therefore we can only explicitly recognise the 1st, 2nd and 3rd ordinal members of a group of 3 members, if we already implicitly recognise 3 in cardinal terms (as composed of single independent units).

So in dynamic interactive terms cardinal and ordinal notions of number, ultimately, mutually imply each other in a holistic synchronous manner.


The huge question then arises as to the consistency of quantitative (cardinal) and qualitative (ordinal) aspects with respect to each other.

This implies for example that an indirect quantitative means is required to convert - as it were - qualitative to quantitative notions (that then can be seen to be related in a consistent manner).

Now again, in conventional mathematical terms, this key issue is avoided through an (unrecognised) form of reductionism whereby qualitative (ordinal) notions are viewed in one limited manner.

If we start with just one item which in cardinal terms is 1, we can unambiguously view this as the 1st (of a group of 1).

If we then move on to two items, which again in cardinal terms is 2, we can once again view this additional member as the 2nd (of a group of 2).

Then if we move on to three items, which now in cardinal terms is 3, we can now view the additional member as the 3rd (of a group of 3).

In this way we can continue to unambiguously identify additional ordinal members, by always considering its as the nth member of a group of n.

We can in fact mathematically show how each additional ordinal member, thereby becomes indistinguishable from corresponding cardinal members


Now quantitative meaning relates directly to linear (i.e. 1-dimensional) rational interpretation (which accurately characterises the conventional mathematical approach).

Dimensional meaning - which relative to base cardinal interpretation is of an ordinal nature - is represented as 1ⁿ.(where n is the dimesnion defined with respect to the default fixed base of 1).

Therefore to reduce from n-dimensional to 1-dimensional format we obtain the solution of
 xⁿ = 1ⁿ

Therefore x¹ = 1^n/n  = 1¹ = 1 (in cardinal terms).

Thus for the 1st unit x¹ = 1; then for the 2nd unit x¹ = 1; likewise for the 3rd unit x¹ = 1.

So the qualitative (ordinal) notion of 1st + 2nd + 3rd  has now been successfully reduced in mere quantitative (cardinal) terms as 1 + 1 + 1.

Therefore, the linear interpretation of ordinal notions is only possible through treating each additional ordinal member as the nth member of the nth group.

In this way ordinal notions seem compatible with absolute cardinal notions of number (in a Type 1 manner).

However when we treat ordinal notions in a truly relative manner, within each group, this comforting position breaks down irretrievably (in effect requiring a greatly enlarged mathematical framework).

More on Dynamic Nature of Number (3)

Yesterday, we looked at the distinction as between the analytic and holistic approaches to number interpretation.

Once again with the analytic, an absolute type interpretation results. At a deeper level this represents a linear (i.e. 1-dimensional) approach, implying in any context single unambiguous polar reference frames. So the external is clearly separated from the internal pole; likewise the quantitative is clearly separated from the qualitative aspect.
Therefore the customary analytic interpretation of number is with respect to its independent (external) objective identity in a merely quantitative sense!

However with the holistic, a relative type interpretation by contrast results, which always entails the dynamic interaction of opposite polarities.
So number from this perspective, necessarily entails a dynamic interaction as between internal (mental) and external (objective) aspects; equally it entails a dynamic interaction as between quantitative (independent) and qualitative (interdependent i.e. relational) aspects.

In extremes, the holistic aspect in itself represents the complementary opposite of the analytic aspect.
Whereas the analytic extreme leads to the absolute interpretation of number as unchanging phenomena of form, the holistic extreme leads to the purely relative interpretation of numbers as energy states (that ultimately are ineffable).

Insofar as the natural numbers are concerned, the analytic interpretation corresponds directly with standard cardinal notions.

However a major (unrecognised) problem relates to the corresponding conventional treatment of ordinal numbers, which properly requires holistic - rather than analytic - appreciation.

When one reflects carefully on it, the ordinal notion of number implies qualitative as well as quantitative considerations.

Say we are trying to rank the exam results of pupils in a class of 20!
Now this requires the quantitative notion of 20 and the recognition of each pupil as independent.
So standard cardinal notions here apply with 2 = 1 + 1, and 3 = 1 + 1 + 1 and so on.

However the attempt to rank each student as 1st, 2nd, 3rd, etc strictly implies the qualitative notion of interdependence, whereby we can place the student positions in relationship to each other.

Therefore in the standard analytic interpretation of number, ordinal notions are therefore necessarily reduced in a merely quantitative type manner.
Indeed, in this context it is interesting how the qualitative notion is never even mentioned in connection with the ordinal treatment of numbers, where the seemingly safer more neutral term of "rankings" is employed.
However, once again, rankings necessarily entail the qualitative notion of the relationship between numbers.
And if we view natural numbers as absolutely independent (in a cardinal manner), well then this begs the significant question of how these numbers can be related with each other!

So clearly there is an enormous question, regarding the very nature of ordinal numbers, which is completely overlooked in conventional mathematical terms.

Now if we confine ourselves for the moment to a finite group of numbers, we can begin to appreciate how ordinal rankings are merely of a relative nature (that keep changing depending on context).

So if for example we envisage a class of 2, obtaining 2nd place might not appear an achievement.
as in this context it represents last place!

However as the size of the class increases, the relative significance of 2nd changes. So 2nd in the context of 20, has very different connotations from 2nd in the context of 2. Again 2nd in the context of 200 would appear even more impressive!

So the meaning of 2nd - as indeed the meaning of every ordinal number - continually changes as the corresponding cardinal magnitude of the finite group to which it relates, itself is increased.

It is only when we view the cardinal number set as infinite, that the ordinal nature of number appears unambiguous.

So therefore if we consider 1st, 2nd, 3rd, 4th etc as ordinal members of an infinite class, then their meaning appears as invariant, whereby they can be represented in identical terms with the corresponding cardinal point of 1, 2, 3,  4 etc. on the number line.

In this (1-dimensional) linear manner - which again underlines all standard analytic interpretation -
ordinal notions can seemingly be successfully reduced to their cardinal counterparts in similar fashion.


However there is n an important unappreciated paradox about what is involved here.

When for example, we refer to the cardinal numbers 1, 2, 3 and 4 for example, these - by definition - are given a limited finite identity.

However, when we refer to the corresponding ordinal numbers 1st, 2nd, 3rd and 4th, these, by contrast necessarily pertain - with respect to their unambiguous identity - to number relationships which entail an infinite class (in cardinal terms).

So ultimately, the seeming correspondence as between cardinal and ordinal numbers in conventional mathematical terms, is based on the fundamental reduction of infinite to finite notions. And this strictly, is the same problem that underlines the attempted reduction of all qualitative notions in merely quantitative terms.

Thus the remarkable conclusion that we have reached - which has profound implications for conventional mathematical understanding - is that the ordinal notion of number is inherently associated with the Type 3 mathematical worldview (where both quantitative and qualitative notions are related).

Ordinal notions thereby represent both the quantitative notion of the independence (of each individual natural number) with the qualitative notion of the interdependence (with respect to the relationship between these numbers).

Thus in Type 3 terms, we at last realise clearly that the very notions of independence and interdependence (with respect to number) are themselves necessarily of a relative nature.

What this means is that both cardinal and ordinal notions mutually imply each other in a relative -rather than absolute - manner.

Using Jungian notions, thus there is a hidden shadow side to the interpretation of the qualitative aspect of number (in a quantitative fashion).

Also, there is also a hidden shadow side to the quantitative aspect of number (in a qualitative manner).

More on Dynamic Nature of Number (2)

In yesterday's blog entry, I indicated how our experience of a number keeps switching as between both quantitative and qualitative aspects with respect to both base and dimensional expressions respectively in a two-way dynamic complementary manner.

In fact the situation in truth is even more intricate, with experience also switching as between both internal and external perceptions in each case.

So for example the quantitative (base) notion of an number such as "2" in an independent cardinal sense alternates as between both the external aspect (in the acknowledgement of the number "object") and also internal aspect (in the acknowledgement of the corresponding perception of the number "2").


The next key area is then to properly distinguish analytic from holistic type appreciation.

In brief the analytic approach - as I define it - attempts to separate the opposite key polarities of experience in an absolute type manner leading to a fixed unambiguous form of understanding.

Thus for example with respect to number, the external aspect  (as the number "object") is separated from the internal aspect (as number "perception) and with both in effect thereby reduced in terms of each other. More customarily the internal aspect is reduced in terms of the external so that we thereby attempt to understand the behaviour of number (i.e. as number "objects") in an unambiguous absolute type manner.

Likewise - and perhaps even more significantly - both quantitative and qualitative aspects are likewise separated in an unambiguous manner with the qualitative aspect then in effect reduced in terms of the quantitative. Thus for example - using again "2" to illustrate in conventional terms no clear distinction is made as between the quantitative  notion of "2" as an independent number and the corresponding qualitative notion of "2" (i.e. as twoness) whereby it is seen as interdependent with all other instances of "2".

And as I indicated in the last blog entry, this represents the precise reason why the crucial distinction as between the operations of addition and multiplication is not properly understood in Conventional Mathematics.

By contrast the basis of the holistic approach is that the opposite polarities (that govern all mathematical experience) are now considered in a dynamic relative interactive manner as complementary with each other.

So from a holistic perspective the notion of number necessarily represents a dynamic interaction as between its external and internal polarities (which are positive and negative with respect to each other).

Likewise in holistic terms, the notion of number necessarily represents a dynamic interaction as between its quantitative and qualitative aspects in  both quantitative and qualitative aspects (which are now real and imaginary with respect to each other),

And of course in this holistic context the very meaning of mathematical notions (such as positive and negative; real and imaginary, etc) themselves switch from their customarily understood analytic to a new distinctive holistic meaning.

So every mathematical notion, with a clearly defined meaning in analytic terms, can be given a coherent alternative meaning in a holistic manner.  

Ultimately the most comprehensive form of mathematical understanding entails the harmonious interaction of both analytic and holistic type meaning.


Therefore for a comprehensive mathematical worldview, I define 3 distinct types of Mathematics, that I term Type 1, Type and Type 3 respectively.

The first worldview relates to customary analytic type understanding of an absolute type nature. This is Type 1 Mathematics.
In formal terms, effectively all accepted Conventional Mathematics belongs to this one category.

The second worldview relates to the totally unrecognised (in formal terms) holistic type appreciation of mathematical symbols, which is of an approximate relative nature. This is Type 2 Mathematics.

Though completely unrecognised by the Mathematics profession, I have spent more than 50 years of my life in developing fundamental key notions of a holistic kind.

For example I have now long realised that all developmental processes (such as human transformation) are  of a holistic mathematical nature.
Thus in this context, Holistic Mathematics entails the elaborate mapping of all possible stages of development (physical and psychological) with their corresponding encoding in mathematical terms.

The third worldview, which is by far the most comprehensive entails the harmonious combination of both the analytic and holistic approaches. I generally refer to this as Radial Mathematics, which equally corresponds to Type 3 Mathematics.
In truth, as the analytic and holistic aspects are themselves complementary in nature, one cannot properly understand mathematical reality (with respect to any issue) without adopting this approach.

However it is important to appreciate that even though the analytic aspect (so heavily dominant in Type 1 Mathematics) is once again restored, it is done so in a relative - rather than absolute - manner.

So for example, if we assert the truth of the Pythagorean Theorem, for example, In Type 1 Mathematics, this will be understood in an absolute type manner. However in Type 3, though the proof still maintains an important validity, it is understood in a merely relative manner that necessarily is still strictly subject to uncertainty.

I will finish this entry by giving a simple illustration of the distinction between the three approaches.

In Type 1 terms the left and right turns at a crossroads are understood in an absolute type manner as unambiguously either left or right . This implies a linear (1-dimensional) approach where only one unambiguous direction (either N or S) in terms of approaching the crossroads is considered.

In Type 2 terms, the left and right turns are now understood in relative terms as paradoxically both left and right. This implies a circular (in this case 2-dimensional) approach where both possible directions (N and S) are simultaneously considered with respect to approaching the crossroads.

Thus what is left (approaching from a N direction) is likewise right (when approaching from the opposite S direction); and what is right (approaching from a N direction) is left (when approaching from the opposite S direction).

Thus Type 2 understanding is circular - rather than linear - in nature (though it must necessarily start with linear type appreciation). It is multi-dimensional in nature (with a minimum of 2 dimensions involved). 2-dimensional appreciation entails the simultaneous recognition of 2 opposite directions, serving as reference frames. Multi-dimensional in more general terms entails the simultaneous recognition of n distinct reference frames (that geometrically can be represented in holistic mathematical terms as the n roots of 1).

In Type 3 terms, left and right turns at a crossroads have a partial unambiguous linear interpretation as either left or right  (depending on relative context when N and S directions of approach are separated) while also having a holistic paradoxical circular interpretation as both left and right when the two reference frames (N and S) are simultaneously considered.

This type of understanding represents the changing frames in a movie.

At any given moment, just one frame will be in evidence; however because of the paradox created by opposite reference frames, these keep switching in complementary fashion. Thus a limited partial validity applies to the analytic interpretation associated with each frame, while the overall holistic appreciation of the complementary nature of these frames ensures that these partial interpretations continually change.

More on Dynamic Nature of Number (1)

I am continuing here my most recent refinements relating to the truly dynamic interactive nature of number.

To keep this at its  very simplest, we will illustrate here with respect to the number 2.

Now as will become quickly apparent, every number in this context is defined with respect to both a base and dimensional number aspect that are complementary in quantitative and qualitative terms.
Thus once again in the expression a^b, a is the base and b the dimensional number accordingly!

So it is important to appreciate how the base number (representing a quantity) is complementary to its (default) dimensional number, which relatively - has a qualitative meaning.

1) Thus when I refer to "2" as a number quantity, this is necessarily defined with respect to a (default) dimensional number of 1.

Therefore it is more properly written as 2¹.

Thus 2 is thereby a specific actual number that is defined with respect to an overall linear dimension that potentially relates to all real numbers.

Thus when we use 1 to represent this dimension it strictly carries a qualitative - as opposed to quantitative - meaning.

The very notion of quantitative implies independence (from all other numbers).
However to enable such a number to be then related with other numbers, we require the corresponding notion of general number interdependence which is - relatively - of a qualitative nature
And this number interdependence is provided by the dimensional notion of number (which in this default case represents the 1st dimension i.e. the number line).

So I represent 2 as a specific number quantity, by highlighting this number (which is emphasised here in an explicit manner) in black, while the default dimensional number of 1 is shown in light grey (to show that it remains merely implicit in this instance).


2) We next look at the reverse notion of "2", now written as 2¹.
The number "1" which now represents the dimensional aspect of 1, carries an actual quantitative meaning i.e. as applying to all actual numbers (in this context natural numbers) on the number line.

The number now "2" represents the qualitative aspect of 2, where this number is now understood as in common with all other classes of 2 objects. For example if we have 3 columns with 2 items in each row, then we can strike a one-to-one correspondence as between the three columns (where each contains "2" items).

What is vital however to appreciate here is "2" is not now being used in a quantitative sense (where it is viewed as independent of other numbers) but rather qualitatively, whereby "2" is now seen to be interdependent with the members of each column (i.e. each column is similar in containing 2 members)

This in fact represents the fundamental difference as between addition and multiplication.

With addition two number quantities are combined (without change of qualitative dimension).

So 2¹ + 3¹ = 5¹.

However when we multiply thee numbers we strictly combine both quantitative and qualitative meaning.

So  2¹ * 3¹ = 6¹ * 1².

Thus both a quantitative change in units, as well as a qualitative change in the dimensional nature of the units takes place. Thus is we have a small table with length 3 ft. and width 2 ft. respectively we can immediately recognise that its are is 6 square feet. However in the conventional treatment of number multiplication the qualitative dimensional aspect of transformation is simply ignored.

Therefore referring again to the rows and columns, we must recognise initially the independence of the rows and columns (as separate).
However equally in then multiplying 3 by 2, we recognise the common quality of twoness with respect to each of the 3 columns.

So once again 2¹ represents the situation where the base number "2" is now of a qualitative nature and "1" as dimensional number assumes a quantitative identity as applying to any actual (natural) number on the (1-dimensional) line. In this way we can multiply 2 by 1, 2, 3, 4,.......

However with multiplication a change takes place to the qualitative aspect.
So in this context "2" refers to the two-dimensional plane which now potentially stretches in two directions in an infinite manner

Thus in moving from 1) to 2), the meaning of both base and dimensional numbers switch.

In 1), the base number "2" is quantitative and the dimensional number "1" is qualitative; however in 2), the base number "2" is now qualitative, and the dimensional number "1" is quantitative.   

3) We now look at the interpretation of 1².  

"2" as dimensional number is now used in a qualitative sense, representing the simple multiplication operation 1 * 1. Once again the length and width of the unit square are not independent of each other but related to each other in a specific manner. Therefore "2" is here qualitative. 

However  "1" as base number has a quantitative meaning representing the one (2-dimensional) object that results. So just as in 1) we defined an actual number i.e. "2" with respect to the number line (as potentially infinite), here we are defining an actual object i.e. 1 object with respect to the 2-dimensional plane that is potentially infinite. 

Thus in this context, the base number is quantitative and the dimensional number qualitative respectively.   

4) Finally we look at the interpretation of  1².  

Here "2" representing dimension carries an actual meaning, whereby it can be applied to classes of 1 object. So for example if we were comparing areas of different fields, these would all be of a 2-dimensional nature, that would apply in the case of each field. In this sense each field (as a unit) would be in common with each other field so that "1" would now have a qualitative meaning. 

And "2" now representing the dimensional number (with an actual finite significance) would be thereby quantitative in nature.

Thus again in 3) the base number "1" is quantitative and the dimensional number "2" is qualitative; however in 4) the base number "1" is now qualitative and the dimensional number "2" is quantitative.


Thus in the dynamics of experience, both base and dimensional numbers (which psychologically are represented through corresponding perceptions and concepts) keep switching as between both their quantitative and qualitative meanings respectively in a dynamic complementary manner.  

Therefore in the simplest possible case the number "1" can have a base meaning (corresponding to the actual rational perception of "1" that is of an independent quantitative nature.
However equally "1" can have a base meaning (corresponding to the potential intuitive perception of "1") that is of a common qualitative nature as applying to all classes of 1 object).

Then "1" can have a dimensional meaning (corresponding to the potential intuitive concept of "1" that is of an common qualitative nature (i.e. the number line as potentially applying to all numbers).

Finally "1" can have a dimensional meaning (corresponding to the actual rational concept of "1" that is of a common quantitative nature (i.e. the number line as actually applying to specific numbers).