We have briefly defined the 3 Types of Mathematics - Type 1, Type 2 and Type 3 respectively.

In terms of my own treatment of the full Spectrum of Development, Type 1 is associated with the first two Bands (i.e. Lower and Middle) with specialised understanding associated with the Middle Band.

Not surprisingly as development in Western culture largely plateaus at the 2nd level, what is understood as Mathematics has become synonymous with Middle Band understanding. However correctly understood this in fact is Type 1 Mathematics (geared to quantitative appreciation of symbols).

Type 2 Mathematics is then correctly associated with both the 3rd and 4th Bands (i.e. Higher and Upper Middle) with specialised appreciation related to the 4th Band.

Though the development of these Bands is generally associated with the contemplative mystical traditions and has been well articulated from a spiritual religious perspective. However the great potential importance of the refined cognitive structures associated with such development for both Mathematics and Science has been greatly overlooked. So a key motivation for my own writing has been to outline the nature of these increasingly circular type structures and then sketch out their qualitative integral scientific significance.

Finally Type 3 Mathematics is associated with the 5th, 6th and 7th Bands (i.e. Radial).

For both Type 3 (a) and Type 3 (b) such understanding would properly unfold with the 5th and 6th Radial Bands (with specialised appreciation in each case related to the 6th Band).

Full specialised appreciation of Type 3 (c) Mathematics - which offers the greatest scope for the mature balanced interpenetration of both its quantitative and qualitative aspects - would then unfold with the 7th Band.

Mathematics of course provides an indispensable tool for scientific understanding.

So matching the 3 Types of mathematical we equally have 3 corresponding Types of scientific understanding.

Once again Type I relates to conventional quantitative notions of science. Once again in our culture Science is misleadingly identified with mere Type 1 appreciation.

Type 2 relates to qualitative notions in - what I refer to as - Integral Science.

Type 3 then relates to a more comprehensive form of scientific understanding where the mature interpenetration of both quantitative and qualitative aspects takes place.

Type 3 (a) entails a mainly quantitative analytic emphasis that is however appropriately informed with corresponding qualitative notions.

Type 3 (b) entails by contrast a mainly qualitative holistic emphasis that is appropriately informed with corresponding quantitative notions.

Type 3 (c) entails the most balanced mix of both quantitative and qualitative appreciation enabling greatest scope for scientific work that is both highly creative in vision and immensely productive of results.

My own mathematical and scientific contributions are mainly designed to be of the Type 2 variety (as these in practice are largely unrecognised and greatly misunderstood).

## Thursday, June 30, 2011

## Monday, June 20, 2011

### Three Types of Mathematics

Though I have been running a number of personal blogs now for some time, up to this I have not dedicated a general blog to Mathematics, especially of the qualitative variety that I have been developing.

Admittedly I have running a specialised blog on the Riemann Hypothesis (with respect to the qualitative approach) and also many entries in my "Integral Science" blog directly relate directly to qualitative mathematical material.

One of the reasons for this delay is that I did not wish to confine such a blog merely to - what I term - Holistic Mathematics. As Holistic Mathematics necessarily has close connections with the other varieties of Mathematics I wanted scope to deal - where appropriate - with issues relating to all types.

In fact it was only when replying to Elliot Benjamin recently on Frank Visser's "Integral World" site that it struck me that the overall types of Mathematics that I see possible relate to distinctive bands with respect to the psychological spectrum of consciousness. So this immediately suggested the title for this long delayed blog of "Spectrum of Mathematics".

Basically I would see that from a comprehensive perspective, three great areas of mathematical activity should exist (of which only one at present is formally recognised).

In the past I have referred to these branches as Conventional Mathematics, Holistic Mathematics and Radial Mathematics respectively. However recently I have come to the opinion that it would perhaps be better to refer to these more simply as Type 1 Mathematics, Type 2 Mathematics and Type 3 Mathematics respectively.

Now I hope in the course of future contributions to this blog to go into the nature of these three Types in considerable detail.

However to place some initial perspective on this approach I will here attempt to simply map the three Types to the relevant Bands on the psychological Spectrum.

In my present approach I define 7 major Bands. For a brief summary of these Bands see Update on Classification of Stages.

The first two Bands i.e. Lower and Middle are primarily in this mathematical context related to the gradual differentiation (in the Lower) and then specialisation (in the Middle) of rational type consciousness conforming to unambiguous either/or type logic. This is what I refer to as linear reason.

Type 1 Mathematics which is heavily based on a (reduced) quantitative approach to meaning relates heavily to the rational understanding corresponding to this 2nd Band on the Spectrum.

The next two Bands by contrast are geared to the continual unflolding and then specialisation of contemplative intuitive awareness. Now, when indirectly expressed in rational terms, this leads to the increasing use of circular (paradoxical) type appreciation based on a both/and type logic (of the complementarity of opposites).

In mathematical terms this leads to the use of mathematical symbols in a qualitative - rather than quantitative - type context.

So Type 2 Mathematics in its specialised form would relate heavily to the 4th Band of the Spectrum (where intuitive type awareness is consolidated).

The final three Bands on the Spectrum relate to - what I refer to as - radial development. This entails the increasing interpenetration in ways both highly creative and immensely productive of both the specialised quantitative and qualitative aspects of understanding.

So these Bands are directly related to the development of Type 3 Mathematics.

However even here I would distinguish three sub-categories.

With Type 3 (a) though a degree of specialisation has been attained with respect to both aspects (qualitative and quantitative) the mathematical emphasis is still mainly on the quantitative (using a developed intuition to fuel quantitative type appreciation).

With Type 3 (b) though again a degree of specialisation has been attained with respect to both aspects the mathematical emphasis is mainly on the qualitative side (where one uses holistic mathematical symbols to express qualitative type appreciation).

In my own case insofar as I would see any Type 3 mathematical ability it would decidedly be of this second variety!

With Type 3 (c) the most fruitful balance as between quantitative and qualitative type appreciation would be attained leading to possibilities for the highest expression of mathematical understanding (with respect to both its quantitative and qualitative aspects).

This would be most likely to be obtained for the rare number of individuals who achieve substantial development with respect to all Bands on the Spectrum.

It must be stated from the outset that in our present Western culture that development largely plateaus at the second Band of development which explains why Type 1 Mathematics is so dominant.

Indeed even where development takes place to a significant extent with respect to the further Bands, so far its important mathematical implications have not been properly articulated.

However I do believe that we have reached a stage in history where dramatic changes will take place in a relatively short period. And for some this will lead to exciting new interpretations of mathematical symbols that will herald the starting development of the two latter types of Mathematics.

Admittedly I have running a specialised blog on the Riemann Hypothesis (with respect to the qualitative approach) and also many entries in my "Integral Science" blog directly relate directly to qualitative mathematical material.

One of the reasons for this delay is that I did not wish to confine such a blog merely to - what I term - Holistic Mathematics. As Holistic Mathematics necessarily has close connections with the other varieties of Mathematics I wanted scope to deal - where appropriate - with issues relating to all types.

In fact it was only when replying to Elliot Benjamin recently on Frank Visser's "Integral World" site that it struck me that the overall types of Mathematics that I see possible relate to distinctive bands with respect to the psychological spectrum of consciousness. So this immediately suggested the title for this long delayed blog of "Spectrum of Mathematics".

Basically I would see that from a comprehensive perspective, three great areas of mathematical activity should exist (of which only one at present is formally recognised).

In the past I have referred to these branches as Conventional Mathematics, Holistic Mathematics and Radial Mathematics respectively. However recently I have come to the opinion that it would perhaps be better to refer to these more simply as Type 1 Mathematics, Type 2 Mathematics and Type 3 Mathematics respectively.

Now I hope in the course of future contributions to this blog to go into the nature of these three Types in considerable detail.

However to place some initial perspective on this approach I will here attempt to simply map the three Types to the relevant Bands on the psychological Spectrum.

In my present approach I define 7 major Bands. For a brief summary of these Bands see Update on Classification of Stages.

The first two Bands i.e. Lower and Middle are primarily in this mathematical context related to the gradual differentiation (in the Lower) and then specialisation (in the Middle) of rational type consciousness conforming to unambiguous either/or type logic. This is what I refer to as linear reason.

Type 1 Mathematics which is heavily based on a (reduced) quantitative approach to meaning relates heavily to the rational understanding corresponding to this 2nd Band on the Spectrum.

The next two Bands by contrast are geared to the continual unflolding and then specialisation of contemplative intuitive awareness. Now, when indirectly expressed in rational terms, this leads to the increasing use of circular (paradoxical) type appreciation based on a both/and type logic (of the complementarity of opposites).

In mathematical terms this leads to the use of mathematical symbols in a qualitative - rather than quantitative - type context.

So Type 2 Mathematics in its specialised form would relate heavily to the 4th Band of the Spectrum (where intuitive type awareness is consolidated).

The final three Bands on the Spectrum relate to - what I refer to as - radial development. This entails the increasing interpenetration in ways both highly creative and immensely productive of both the specialised quantitative and qualitative aspects of understanding.

So these Bands are directly related to the development of Type 3 Mathematics.

However even here I would distinguish three sub-categories.

With Type 3 (a) though a degree of specialisation has been attained with respect to both aspects (qualitative and quantitative) the mathematical emphasis is still mainly on the quantitative (using a developed intuition to fuel quantitative type appreciation).

With Type 3 (b) though again a degree of specialisation has been attained with respect to both aspects the mathematical emphasis is mainly on the qualitative side (where one uses holistic mathematical symbols to express qualitative type appreciation).

In my own case insofar as I would see any Type 3 mathematical ability it would decidedly be of this second variety!

With Type 3 (c) the most fruitful balance as between quantitative and qualitative type appreciation would be attained leading to possibilities for the highest expression of mathematical understanding (with respect to both its quantitative and qualitative aspects).

This would be most likely to be obtained for the rare number of individuals who achieve substantial development with respect to all Bands on the Spectrum.

It must be stated from the outset that in our present Western culture that development largely plateaus at the second Band of development which explains why Type 1 Mathematics is so dominant.

Indeed even where development takes place to a significant extent with respect to the further Bands, so far its important mathematical implications have not been properly articulated.

However I do believe that we have reached a stage in history where dramatic changes will take place in a relatively short period. And for some this will lead to exciting new interpretations of mathematical symbols that will herald the starting development of the two latter types of Mathematics.

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