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.

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