Perhaps the single greatest problem with respect to conventional mathematical understanding relates to the standard treatment of dimensions, which is both highly reduced and unsatisfactory.
From my current perspective I see this problem as so fundamental that it is truly remarkable how it is persistently overlooked.
I generally describe the standard method of mathematics as the linear rational approach.
Now the line is literally 1-dimensional; so alternatively from a qualitative philosophical perspective, Conventional Mathematics is based on 1-dimensional rational understanding.
To see what this actually entails we can consider for a moment the natural number system.
Now, when we list the natural numbers i.e. 1, 2, 3, 4,...they are commonly defined - merely - with respect to their quantitative characteristics.
However implicit within such understanding is a (default) qualitative aspect (as dimension 1).
In other words implicitly each number as quantity is defined qualitatively with respect to the first dimension.
Thus in conventional terms whenever a number is raised to a dimension (other than 1) its ultimate value is defined with respect to the (default) 1st dimension.
For example we raise 2 to the power of 2 i.e. 2^2, the resultant value is given in reduced quantitative terms as 4 (i.e. 4^1).
Even as a child I could see that there was something wrong here. For example in geometrical terms we could represent 2^2 as a square (perhaps representing the area of a table with sides measured in metres). However quite clearly the 2-dimensional expression (4 square metres) is qualitatively different from the reduced 1-dimensional expression (i.e. 4 linear metres).
Thus whenever one raises a number to a dimension (or power) other than 1 - or alternatively multiply or divide two numbers - a qualitative as well as quantitative transformation takes place in the units involved.
However quite remarkably this qualitative transformation is conventionally ignored with numerical results expressed merely with respect to their reduced (i.e. 1-dimensional) quantitative characteristics.
So the first thing to clearly recognise here is this: even though - relative to the quantitative aspect of number - its corresponding dimension is correctly of a qualitative nature, a merely reduced (i.e. quantitative) notion of dimension is employed in Conventional Mathematics.
Though I did therefore recognise that there was a major problem here, it took me a long time to articulate - at least to my own satisfaction - the true mathematical nature of this alternative qualitative (i.e. dimensional) aspect of Mathematics.
Eventually I was to realise that this alternative system was of a circular - rather than linear - nature bearing a close structural relationship with the quantitative notion of number roots.
Of course, I do not for a moment question the great value of the conventional (quantitative) approach to Mathematics. My point is simply that there is - literally - an equally important qualitative dimension that is almost completely unrecognised.
So with respect to natural numbers we start by defining two separate systems.
In the first i.e. conventional approach, the quantitative aspect ranges over the natural numbers (with the qualitative dimensional aspect fixed as 1).
i.e. 1^1, 2^1, 3^1, 4^1,....
In the second i.e. holistic mathematical approach, the quantitative aspect now remains fixed as 1 (with the qualitative dimension ranging over the natural numbers).
i.e. 1^1, 1^2, 1^3, 1^4,....
Clearly from the conventional linear perspective, this second number system appears quite uninteresting (with the reduced quantitative value in each case = 1).
However from the correct qualitative perspective it is quite different.
Now the initial clue as to the nature of this new system is the realisation that in structural terms a unique inverse relationship exists as between the qualitative dimension and its corresponding root. So if D is the (qualitative) dimension is question, 1/D represents the corresponding (quantitative) root.
Therefore for example to find the form of the 2nd dimension we thereby raise in quantitative terms 1 to the power of 1/2.
In other words as regards the structural form of the relationship,1^2 (qualitatively) = 1^1/2 (quantitatively).
So in general 1^k (qualitatively) = 1^1/k(quantitatively)
We then use the expression derived from - what I term - the fundamental Euler Identity i.e. e^2ipi = 1,
So 1^(1/k) = cos(2pi/k) + i sin (2pi/k) where 2pi = 360 degrees.
So the structural form of dimension 2 is thereby given as
1^(1/2)= cos 180 + i sin 180 = - 1
So whereas 1^(1/2)= - 1 can be given a quantitative interpretation, 1^2 = - 1 can be given a corresponding qualitative (i.e. holistic mathematical) interpretation.
Now this number, - 1 lies on the unit circle in the complex plane.
Therefore the qualitative aspect of Mathematics is based ultimately on a circular - rather than linear - logical appreciation.
(Of course as I have continually stated, the most comprehensive form of Mathematics - which I term Radial Mathematics - combines the full interaction of both quantitative (linear) and qualitative (circular) understanding.
Put another way this entails recognition of an alternative binary system (linear = 1 and circular = 0), that potentially can encode in mathematical terms all transformation processes!)
Now perhaps we can appreciate clearly how Conventional Mathematics represents just a special case, where no distinction exists as between the (whole) qualitative dimension and its (part) quantitative root.
In other words when D = 1, both D and 1/D are thereby identical. So once again Conventional Mathematics represents the special case where no distinction can be made as between the quantitative and qualitative aspects of a relationship,
However with respect to all other dimensions a unique distinction does indeed exist as between (whole) dimensional numbers (as qualitative) and their reciprocal part expressions (as quantitative).
Put another way, to avoid the continual reduction of whole to part notions in Mathematics we must incorporate fully this (unrecognised) holistic aspect.
So we have seen that Conventional Mathematics is defined in linear terms by the 1st dimension.
However we have now uncovered the 2nd dimension (existing on the unit circle as - 1) which thereby has a circular logical meaning.
So in this context, we need now to holistically define in qualitative what is - 1.
Quite clearly we cannot just maintain the standard linear interpretation (which is merely quantitative!)
+ 1 in a holistic context relates to (unitary) form that is consciously posited in experience (in linear rational terms)
In corresponding terms - 1 in a holistic context relates to negation of such unitary form (which is the very means through which experience switches from conscious to unconscious appreciation).
So just as in physics when a particle particle fuses with its negative counterpart it leads to creation of material energy, likewise in psychological terms the dynamic negation (in unconscious terms) of what has been consciously posited, leads to the generation of spiritual intuitive energy.
In rational terms this leads to a new circular logical system that is inherently dynamic in nature and based on the complementarity of opposites.
Thus rather than the either/or approach of linear logic (where opposite polarities are unambiguously separated) we now employ a both/and circular logic (where opposite polarities are always paradoxical with precise meaning in any context depending on the arbitrary relative context employed.
Therefore the key implication of now incorporating this 2nd (with the 1st dimension) is the formal recognition that mathematical activity itself comprises both rational (conscious) and intuitive (unconscious) aspects.
So Mathematics is no longer formally defined in merely rational terms (which befits the 1st dimension) but now entails distinctive rational (conscious) and intuitive (unconscious) aspects which cannot be reduced in terms of each other. And once again this latter unconscious aspect, that is primarily of a holistic intuitive nature, is translated indirectly in rational terms through a circular logic of the complementarity of opposites (both + and -).
Holistic Mathematics represents the specialised development of the intuitive aspect of mathematical understanding just as Conventional Mathematics represents the specialised aspect of rational understanding.
However just as good mathematical work at the conventional level must be inspired by appropriate intuition, likewise in reverse manner, good mathematical work at the holistic level must preserve an appropriate manner of rational interpretation (in the form of mathematical symbols that are qualitatively understood).
Thus ultimately the two aspects are complementary (with enhanced appreciation of both arising from their mutual interpenetration).
Now a unique holistic mathematical interpretation is associated with every dimension (the structural nature of which is identical with its corresponding root).
Of particular relevance here is the holistic mathematical interpretation of the 4th dimension. Structurally this is identical with the 4th root of unity which is i.
The qualitative mathematical interpretation of i is of paramount significance and basically provides an indirect means of translating holistic appreciation (of an unconscious nature) through linear type symbols.
Put another way, it provides the rational means for mathematically incorporating qualitative understanding in an acceptable manner.
Indeed in this context whereas Conventional Mathematics represents the real aspect, Holistic Mathematics represents the corresponding imaginary aspect of a more comprehensive mathematical understanding.
So this comprehensive understanding i.e. Radial Mathematics incorporates in qualitative terms a complex rational approach (combining both real and imaginary aspects).
Once again, though it is of course true that Conventional Mathematics employs complex numbers (with real and imaginary parts) it does so within a merely reduced quantitative context.
Such an approach attempts to deal with complex quantities from a merely real qualitative perspective!
To sum up therefore the qualitative approach leads to the clear recognition that all mathematical understanding represents varying configurations with respect to real (analytic) and imaginary (holistic) understanding. And a key task of Holistic Mathematics is to provide the appropriate interpretation of each dimension (representing such complex configurations).
Now when we return to the Monster Group we can see that it is based on a merely reduced quantitative notion of dimension (as befits the Euclidean exploration of space).
However for every dimension that is quantitatively understood in this manner, there is a corresponding circular notion (existing on the unit circle in the complex plane) which can be given a coherent holistic qualitative interpretation.
Thus once again The Monster is at present defined analytically with respect to its component building blocks (as parts). However such understanding in itself does not facilitate the understanding of the qualitative relationship as between these parts in holistic terms. So ultimately an entirely different alternative mathematical approach has to be incorporated with present understanding.
In the next contribution I will suggest in the context of the Monster how this alternative understanding of dimensions can be incorporated!
From my current perspective I see this problem as so fundamental that it is truly remarkable how it is persistently overlooked.
I generally describe the standard method of mathematics as the linear rational approach.
Now the line is literally 1-dimensional; so alternatively from a qualitative philosophical perspective, Conventional Mathematics is based on 1-dimensional rational understanding.
To see what this actually entails we can consider for a moment the natural number system.
Now, when we list the natural numbers i.e. 1, 2, 3, 4,...they are commonly defined - merely - with respect to their quantitative characteristics.
However implicit within such understanding is a (default) qualitative aspect (as dimension 1).
In other words implicitly each number as quantity is defined qualitatively with respect to the first dimension.
Thus in conventional terms whenever a number is raised to a dimension (other than 1) its ultimate value is defined with respect to the (default) 1st dimension.
For example we raise 2 to the power of 2 i.e. 2^2, the resultant value is given in reduced quantitative terms as 4 (i.e. 4^1).
Even as a child I could see that there was something wrong here. For example in geometrical terms we could represent 2^2 as a square (perhaps representing the area of a table with sides measured in metres). However quite clearly the 2-dimensional expression (4 square metres) is qualitatively different from the reduced 1-dimensional expression (i.e. 4 linear metres).
Thus whenever one raises a number to a dimension (or power) other than 1 - or alternatively multiply or divide two numbers - a qualitative as well as quantitative transformation takes place in the units involved.
However quite remarkably this qualitative transformation is conventionally ignored with numerical results expressed merely with respect to their reduced (i.e. 1-dimensional) quantitative characteristics.
So the first thing to clearly recognise here is this: even though - relative to the quantitative aspect of number - its corresponding dimension is correctly of a qualitative nature, a merely reduced (i.e. quantitative) notion of dimension is employed in Conventional Mathematics.
Though I did therefore recognise that there was a major problem here, it took me a long time to articulate - at least to my own satisfaction - the true mathematical nature of this alternative qualitative (i.e. dimensional) aspect of Mathematics.
Eventually I was to realise that this alternative system was of a circular - rather than linear - nature bearing a close structural relationship with the quantitative notion of number roots.
Of course, I do not for a moment question the great value of the conventional (quantitative) approach to Mathematics. My point is simply that there is - literally - an equally important qualitative dimension that is almost completely unrecognised.
So with respect to natural numbers we start by defining two separate systems.
In the first i.e. conventional approach, the quantitative aspect ranges over the natural numbers (with the qualitative dimensional aspect fixed as 1).
i.e. 1^1, 2^1, 3^1, 4^1,....
In the second i.e. holistic mathematical approach, the quantitative aspect now remains fixed as 1 (with the qualitative dimension ranging over the natural numbers).
i.e. 1^1, 1^2, 1^3, 1^4,....
Clearly from the conventional linear perspective, this second number system appears quite uninteresting (with the reduced quantitative value in each case = 1).
However from the correct qualitative perspective it is quite different.
Now the initial clue as to the nature of this new system is the realisation that in structural terms a unique inverse relationship exists as between the qualitative dimension and its corresponding root. So if D is the (qualitative) dimension is question, 1/D represents the corresponding (quantitative) root.
Therefore for example to find the form of the 2nd dimension we thereby raise in quantitative terms 1 to the power of 1/2.
In other words as regards the structural form of the relationship,1^2 (qualitatively) = 1^1/2 (quantitatively).
So in general 1^k (qualitatively) = 1^1/k(quantitatively)
We then use the expression derived from - what I term - the fundamental Euler Identity i.e. e^2ipi = 1,
So 1^(1/k) = cos(2pi/k) + i sin (2pi/k) where 2pi = 360 degrees.
So the structural form of dimension 2 is thereby given as
1^(1/2)= cos 180 + i sin 180 = - 1
So whereas 1^(1/2)= - 1 can be given a quantitative interpretation, 1^2 = - 1 can be given a corresponding qualitative (i.e. holistic mathematical) interpretation.
Now this number, - 1 lies on the unit circle in the complex plane.
Therefore the qualitative aspect of Mathematics is based ultimately on a circular - rather than linear - logical appreciation.
(Of course as I have continually stated, the most comprehensive form of Mathematics - which I term Radial Mathematics - combines the full interaction of both quantitative (linear) and qualitative (circular) understanding.
Put another way this entails recognition of an alternative binary system (linear = 1 and circular = 0), that potentially can encode in mathematical terms all transformation processes!)
Now perhaps we can appreciate clearly how Conventional Mathematics represents just a special case, where no distinction exists as between the (whole) qualitative dimension and its (part) quantitative root.
In other words when D = 1, both D and 1/D are thereby identical. So once again Conventional Mathematics represents the special case where no distinction can be made as between the quantitative and qualitative aspects of a relationship,
However with respect to all other dimensions a unique distinction does indeed exist as between (whole) dimensional numbers (as qualitative) and their reciprocal part expressions (as quantitative).
Put another way, to avoid the continual reduction of whole to part notions in Mathematics we must incorporate fully this (unrecognised) holistic aspect.
So we have seen that Conventional Mathematics is defined in linear terms by the 1st dimension.
However we have now uncovered the 2nd dimension (existing on the unit circle as - 1) which thereby has a circular logical meaning.
So in this context, we need now to holistically define in qualitative what is - 1.
Quite clearly we cannot just maintain the standard linear interpretation (which is merely quantitative!)
+ 1 in a holistic context relates to (unitary) form that is consciously posited in experience (in linear rational terms)
In corresponding terms - 1 in a holistic context relates to negation of such unitary form (which is the very means through which experience switches from conscious to unconscious appreciation).
So just as in physics when a particle particle fuses with its negative counterpart it leads to creation of material energy, likewise in psychological terms the dynamic negation (in unconscious terms) of what has been consciously posited, leads to the generation of spiritual intuitive energy.
In rational terms this leads to a new circular logical system that is inherently dynamic in nature and based on the complementarity of opposites.
Thus rather than the either/or approach of linear logic (where opposite polarities are unambiguously separated) we now employ a both/and circular logic (where opposite polarities are always paradoxical with precise meaning in any context depending on the arbitrary relative context employed.
Therefore the key implication of now incorporating this 2nd (with the 1st dimension) is the formal recognition that mathematical activity itself comprises both rational (conscious) and intuitive (unconscious) aspects.
So Mathematics is no longer formally defined in merely rational terms (which befits the 1st dimension) but now entails distinctive rational (conscious) and intuitive (unconscious) aspects which cannot be reduced in terms of each other. And once again this latter unconscious aspect, that is primarily of a holistic intuitive nature, is translated indirectly in rational terms through a circular logic of the complementarity of opposites (both + and -).
Holistic Mathematics represents the specialised development of the intuitive aspect of mathematical understanding just as Conventional Mathematics represents the specialised aspect of rational understanding.
However just as good mathematical work at the conventional level must be inspired by appropriate intuition, likewise in reverse manner, good mathematical work at the holistic level must preserve an appropriate manner of rational interpretation (in the form of mathematical symbols that are qualitatively understood).
Thus ultimately the two aspects are complementary (with enhanced appreciation of both arising from their mutual interpenetration).
Now a unique holistic mathematical interpretation is associated with every dimension (the structural nature of which is identical with its corresponding root).
Of particular relevance here is the holistic mathematical interpretation of the 4th dimension. Structurally this is identical with the 4th root of unity which is i.
The qualitative mathematical interpretation of i is of paramount significance and basically provides an indirect means of translating holistic appreciation (of an unconscious nature) through linear type symbols.
Put another way, it provides the rational means for mathematically incorporating qualitative understanding in an acceptable manner.
Indeed in this context whereas Conventional Mathematics represents the real aspect, Holistic Mathematics represents the corresponding imaginary aspect of a more comprehensive mathematical understanding.
So this comprehensive understanding i.e. Radial Mathematics incorporates in qualitative terms a complex rational approach (combining both real and imaginary aspects).
Once again, though it is of course true that Conventional Mathematics employs complex numbers (with real and imaginary parts) it does so within a merely reduced quantitative context.
Such an approach attempts to deal with complex quantities from a merely real qualitative perspective!
To sum up therefore the qualitative approach leads to the clear recognition that all mathematical understanding represents varying configurations with respect to real (analytic) and imaginary (holistic) understanding. And a key task of Holistic Mathematics is to provide the appropriate interpretation of each dimension (representing such complex configurations).
Now when we return to the Monster Group we can see that it is based on a merely reduced quantitative notion of dimension (as befits the Euclidean exploration of space).
However for every dimension that is quantitatively understood in this manner, there is a corresponding circular notion (existing on the unit circle in the complex plane) which can be given a coherent holistic qualitative interpretation.
Thus once again The Monster is at present defined analytically with respect to its component building blocks (as parts). However such understanding in itself does not facilitate the understanding of the qualitative relationship as between these parts in holistic terms. So ultimately an entirely different alternative mathematical approach has to be incorporated with present understanding.
In the next contribution I will suggest in the context of the Monster how this alternative understanding of dimensions can be incorporated!
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