Wednesday, April 21, 2010

Monstrous Moonshine - mysterious connections

Sometimes a degree of ignorance can be a virtue.

I am by no means properly acquainted with the quantitative mathematical intricacies of modular functions (and the important expression relating to the j-function).

However in attempting to look at issues from a holistic mathematical perspective, I am not surprised that intimate connections have been demonstrated as between dimensions in the Monster Group and coefficients of the j-function.

After all the Monster Group relates to an amazingly symmetrical object as group of rotations in 196,883 (Euclidean) dimensional space with group order:

|M| = 808017424794512875886459904961710757005754368000000000

= 2^46 * 3^20 * 5^9 * 7^6 * 11^2 * 13^3 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71

Modular functions likewise relate to objects with supersymmetrical properties that can be transformed in an (infinite) variety of ways while remaining unaltered.

It is intriguing however from a holistic mathematical perspective that such functions are defined in complex terms with respect to hyperbolic space.

What immediately strikes me here is that - whereas the Monster is defined with respect to linear (i.e. Euclidean) dimensions - modular functions are defined by contrast with respect to circular notions (interpreted in a quantitative manner).

We have seen that in holistic mathematical terms that the unconscious direction of experience is negative (relative to the rational conscious).

Hyperbolic space can be understood as having negative - as opposed to positive - circular curvature. This is therefore extremely difficult to visualise from a rational perspective (suggesting that its real significance relates to intuitive holistic meaning of a qualitative kind).

Therefore though not directly accessible in quantitative circular terms, it would however be accessible - when appropriately understood - in a qualitative manner.

Also it is vital to recognise that even though i (representing the square root of - 1) can indeed be given an indirect quantitative interpretation in Conventional Mathematics, that it properly relates to holistic intuitive meaning that is indirectly expressed in a linear rational manner.

Roger Penrose for example keeps pointing to the magic of complex numbers in that they can so frequently be demonstrated to possess amazing holistic aspects. However the key to deeper appreciation of why this is the case requires corresponding qualitative understanding of a complex kind i.e. combining both real (quantitative) and imaginary (qualitative) interpretation.

The famous Taniyama-Shimura conjecture - the proof of which opened up the way for Andrew Wiles to solve Fermat's Last Theorem - establishes a key link as between modular forms and elliptical curves (that every elliptical equation with integer solutions has a corresponding modular form).

An elliptical function corresponds to the expression

y^2 = ax^3 + ax^2 + bx + c (with a, b, c and d integers).

Now in quantitative terms, solutions to such an equation will be two-valued (with positive and negative results). This suggests in corresponding qualitative terms the need for a two-valued logic using 1-dimensional (linear) and 2-dimensional (circular) interpretation.

So in this context the modular represents an appropriate circular quantitative expression (i.e. in complex space) of the elliptical equation.

Though such conversions are quite frequent in Mathematics from a quantitative perspective, the huge missing ingredient is the lack of any coherent qualitative (holistic) mathematical means of interpreting what is involved.

Therefore though Conventional Mathematics is truly ingenious in deriving and proving results (from a quantitative perspective), in many important cases it lacks the means to qualitatively explain why such results can in fact arise!

And Monstrous Moonshine is just one very important example of this problem!

However the deepest clue as to what is qualitatively involved with Monstrous Moonshine is provided by the Fourier expansion of the j-function (pardon the notation!).

j(r) = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + .....

r here is known as the half period ratio which itself expresses the ratio of two complex numbers.

q is especially interesting from a holistic mathematical perspective as it directly involves - what I have referred to as - the fundamental Euler Identity, e^(2pi*i) = 1

So q = e^(2pi*i*r)

Now the significance of this is of paramount importance (from the holistic mathematical perspective) as the fundamental Euler Identity provides the very basis for the derivation of all circular dimensions.

So in qualitative terms the dimension k (as qualitatively understood) is given by the identity

1^k = e^(2pi*i*k) which has the same structure as

1^(1/k) = e^[2pi*i*1/k)] as quantitatively interpreted.

So again the qualitative structure of the 2nd dimension (k = 2) is given

as 1^2 = e^(2pi*i*2) which has the same structure as the quantitative expression:

1^(1/2) = e^[2pi*i*(1/2)] = e^(pi*i) = - 1;

And we have already provided the qualitative explanation of this 2nd dimension as the dynamic negation of (unitary) rational form (of a positive conscious nature), which like the fusion of particle and anti-particle in physical terms, is the very means through which holistic energy of an unconscious nature is generated in experience.

With respect to the j-function therefore q = e^(2pi*i*r), implies that,

1^r = e^[2pi*i*r] in quantitative terms.

Therefore the actual qualitative dimension involved here is 1/r (the inverse of the half period ratio) with the same structure resulting.

Thus the important conclusion then follows that the j-function actually involves a mathematical expression involving the use of circular - as opposed to Euclidean - dimensions.

So from this perspective the j-function could be simply seen as yet another of these fascinating mathematical conversions whereby a linear quantitative is transformed into an equivalent circular expression.

As John McKay first noticed, there is a direct link as between coefficients in the j-function and corresponding dimensions in the Monster Group.

Thus the first relevant coefficient 196884 = 196883 + 1 (i.e. just one more than the smallest numbers of linear dimensions in which the Monster exists).

Then all other coefficients can be derived from simple linear expressions of corresponding dimensions in which the Monster can be constructed.

So for example the 2nd coefficient 21493760 = 21296876 + 196883 + 1. (In this context 21296876 represents the next lowest number of dimensions in which the Monster can be constructed).

The Monstrous Moonshine quest was therefore to prove that there was indeed a solid (quantitative) mathematical basis for such connections (rather than just mere coincidence) and Conway and Norton were subsequently able to achieve this while Richard Borcherds then established connections with string theory.

However once again, so even though Conventional Mathematics can indeed demonstrate such quantitative links, it cannot provide a satisfactory qualitative explanation as to why such links arise (due to the lack of formal development of any holistic mathematical aspect).
So what I have been briefly attempting to demonstrate is that the two areas, the Monster Group and the j-functions, actually employ complementary notions of dimensions (that are linear and circular with respect to each other).

Indeed just as there is strong link with 24 in the construction of the Monster, likewise this is also the case with the j-function.

The j-function is defined as:
j(r) = 1728 * J(r) where r again is the half period ratio and J(r) an expression involving this half period ratio and - what is referred to as - the elliptic lambda function. This in turn relates to an expression of ratios involving the ubiquitous Fundamental Euler Identity (which as we have seen is the very basis for defining circular dimensions).

Now 1728 = 12^3 and is thereby closely related to 24.

So 1728 = 24 * 24 * 3.

Other fascinating connections have equally been made. For example Andrew Ogg seems to have first noticed that the very numbers used in the construction of the Monster Group are the supersingular primes which form the basis for the construction of many j-functions. And there also appears to a connection as between the Monster Group and the strange and mysterious number 163!

However this would still be consistent with my qualitative notion of circular dimensions as the number 163 famously arises in the expression e^{[(163)^1/2]*pi} which is very close to a whole number.

From a holistic mathematical perspective, prime numbers represent an extreme both in linear and circular terms. They are the most independent in linear terms (as the building blocks of the number system) and likewise the most interdependent in circular terms (with respect to their general distribution). 163 especially embodies these extremes being famously involved in the solution to the quadratic expression x^2 - x + 41 (where x = 1 to 40, generates prime numbers).

Also I note that the fraction 4/27 appears in the elliptic lambda function, J(R).

Well 163 - 1 = 162 = (27/4) * 24. So could there even be in 163 some link with the j-function?

Friday, April 16, 2010

Monstrous Moonshine - 24 dimensional space

In the early 1980's I was very interested in Jungian psychology (and especially with respect to his theory of Personality Types). Part of the attraction arose from the fact that implicitly Jung formed many of his key notions in a manner amenable to holistic mathematical interpretation.

I mentioned in the last post the (true) qualitative circular notion of dimension and contrasted this with the merely reduced quantitative linear interpretation that predominates in conventional mathematical and scientific understanding.

As four dimensional space-time is so important (as conventionally understood) it is only reasonable to assume that an important circular interpretation of such dimensions can be equally given.

It was here that familiarity with Jungian concepts proved valuable.

Jung organised his understanding of Personality Types around 4 key functions that are often shown as equidistant points on the circle. So this bears obvious comparison with the four roots drawn on the circle (of unit radius) in the complex plane, which when qualitatively interpreted give the four circular dimensions of space-time. Indeed in his description of these functions, Jung used language, amenable to mathematical interpretation, in defining two (conscious) rational and two (unconscious) irrational functions respectively. A key breakthrough for me was the realisation that in more correct holistic mathematical terms, these functions constituted two real and two imaginary functions with complementary positive and negative aspects in each case.

So right away we can see that the (true) qualitative notion of dimension is mathematical in a holistic sense. Furthermore both physical and psychological behaviour dynamically conforms (in complementary fashion) to such dimensions.

Thus from the circular qualitative perspective, reality as we know it, in dynamic experiential terms, is based on four fundamental dimensions of space-time that are real and imaginary with respect to each other (with positive and negative polarities).

Once again, "real" in this context relates to distinctive phenomena that can be consciously understood. Positive and negative polarities imply that all these have both an external and internal direction (that are - relatively - positive and negative with respect to each other).

So for example when a mathematician attempts to understand any phenomenon an interaction inevitably arises as to between the (subjective) knower and what is (objectively) known. And when one reflects on it, such a two-way dynamic interaction is unavoidable even at the most abstract level of interpretation.

Thus - strictly - all truth is relative even in mathematics. The reason therefore that mathematical truths can appear absolute is because of the interpretation involved. Thus linear interpretation always entails reducing the internal mental contribution of understanding to what is objective.
So for example we view a mathematical proof as "objectively" true (in absolute terms). Alternatively it can involve reducing the objective aspect to its internal mental interpretation as "subjectively" true (again in absolute terms).

In mathematics general hypotheses relate to the latter aspect; the empirical testing of such hypotheses relates to the former. However because of the absolute nature of interpretation involved, in both cases a direct correspondence is assumed (which in dynamic interactive terms is not strictly valid).

In other words conventional understanding here is one-dimensional i.e. as consciously understood in absolute positive terms.

So allowance for the true dynamic interaction of both real polarities implies that mathematical truth is merely relative.

Thus at the two-dimensional level of understanding all mathematical truth has both positive and negative polarities that continually interact (and thereby change with respect to each other in experience).

Actually this is deeply relevant with respect to the present status of the proof that all finite simple groups have now been classified.

This proof is far from absolute and really represents - as correctly is true of all mathematical proof - an existing consensus among the mathematical community involved that all the major issues have been successfully dealt with. However this entails that we must remain open to the possibility that such a consensus could conceivably break down in the light of new discoveries.

So interestingly in this important respect the present "proof" with respect to finite simple groups is ultimately pointing to the need for a new qualitative interpretation - that goes beyond the merely linear - of dimensions!

The 3rd and 4th dimensions in this dynamic circular context relate to the crucial relationship as between whole and part that ultimately is imaginary (of an unconscious nature). Now again familiarity with Jungian concepts can help to see what is involved here.
Jung realised well that the unconscious when unrecognised always projects itself into consciousness in a hidden manner (as the shadow side of personality).

Now because in formal terms Mathematics is understood in a merely rational manner, this implies from a Jungian perspective that the unconscious aspect remains completely unrecognised.

And this imaginary (unconscious) aspect is - as I have consistently stated - the qualitative holistic aspect of mathematical understanding, that is vitally important, yet completely ignored in conventional terms.

Probably the most prevalent form of reductionism that pervades all conventional mathematical understanding, is that between whole and part.

When one attempts to understand the relationship between whole and part in merely reduced conscious terms, whole are thereby reduced to part notions (with the whole simply viewed as the sum of its constituent parts!).

So for example when we add two numbers say 2 + 3 the (whole) total = 5 is viewed in merely quantitative terms as the sum of the constituent part numbers.

Though of course such a reduced approach is extremely useful in a wide variety of contexts it cannot properly preserve true qualitative (i.e. holistic) meaning.

To preserve the true qualitative distinction as between whole and part, imaginary (qualitative) notions must be used; in this way the (material) part is seen in some measure as reflecting the universal whole (that properly is of an empty spiritual nature); in turn the universal whole reflects in turn the various parts. Therefore in this way the parts can be related to the whole and the whole to the parts without being directly confused with each other.

Now the ability to properly preserve such distinctions depends on the quality of the imaginary intuition generated (pertaining to the unconscious). So seen from this qualitative perspective, mathematical activity is no longer in formal terms just "real" (i.e. pertaining to the rational conscious) but also "imaginary" (i.e. pertaining to the intuitive unconscious).

Once again, the formalised understanding of the rational aspect, which must necessarily be indirectly fulled by intuition, comes from Conventional Mathematics.

However the formalised understanding of the intuitive aspect, which indirectly must be conveyed through appropriate rational forms, comes from Holistic Mathematics.

So seen from the qualitative holistic perspective, the four dimensions dynamically relate to the most fundamental polar distinctions that can be made with respect to all phenomena i.e. internal and external distinctions on the one hand and whole and part distinctions on the other.

Indeed we cannot even begin to understand in any phenomenal context without implicitly making these fundamental distinctions!

Coming back to Jung he went on to use his functions (and two attitudes) to define 8 personality types. Subsequently this work was to be significantly extended in the Myers Briggs classification to 16 Personality Types.

Though I found the Myers-Briggs Typology very useful, I gradually came to the conclusion that my own personality did not readily fit into the system and that indeed there were 8 missing Personality Types.

The key problem here is that in the Myers Briggs approach, each Type is defined with respect to either/or distinctions. For example one is defined as either E (extrovert) or I (introvert); as either S (sense oriented) or N (intuitive); as either T (thinking) or F (feeling) and finally as either J (judgement) or P (perception).

However I could see that there was another class of Personality Types that was inherently based on the complementarity of these opposite poles with 8 more resulting.

So in a more comprehensive system 24 distinct Personality Types would exist.

It then struck me that these 24 Types could be derived from the various configurations of the original 4 dimensions. In other words each Personality Type could be fruitfully interpreted to combine these four fundamental polar co-ordinates in a unique manner.

On further reflection I then realised that the holistic mathematical configuration for the 24 types could be given as 24 dimensions (corresponding structurally to the 24 quantitative roots of 1).

So viewed from this holistic mathematical perspective, each Personality Type represents a unique manner in which space and time can be experienced with 24 different dimensions resulting.

Thus the key point about this appreciation of "higher" dimensions is that dimensions are no longer understood as separate but combined in varying configurations with each other.

Later using this approach, I made a direct connection with String Theory. In holistic mathematical terms, psychological and physical aspects of reality are complementary. Therefore the 24 Personality Types (as the fundamental dimensions in which the psychological experience of space and time is organised) have a complementary explanation as 24 Impersonality Types (as the fundamental dimensions in which the physical nature of space and time is correspondingly organised).

And this latter interpretation I could see as relating directly to the vibration of a string in 24 dimensions!

When one reflects on the matter it makes little sense at the level of string reality to attempt to conceive of space and time as separate dimensions. Rather in varying ways they remain entangled with each other. So each dimension here actually relates to a distinctive configuration with respect to such entanglement!

However we cannot really grasp this while attempting to stick to a linear notion of dimension; rather in this context the true circular qualitative notion is more appropriate.

Thus conventional understanding of space and time in physics again reflects - from a qualitative perspective - merely 1-dimensional appreciation!

So without any knowledge at this stage of the Monster Group, I had already come to see, using a distinctive holistic mathematical approach, that 24 dimensional reality was of special significance and could see a clear connection as between this new interpretation of dimensions and string theory. In particular, it provided an explanation, through using the true qualitative interpretation, of how "higher" dimensions can be shown to conform to actual experience.

When interpreted in a linear (Euclidean) manner, 24 dimensional space has an important connection with the Monster Group.

We can perhaps appreciate its significance by initially considering the efficient packing of (circular) coins in 2-dimensional space. Now if we attempt to lay out the coins in touching rows (with the position of coins in each row matching those above) we will not get the most efficient packing arrangement. The most efficient solution - thereby minimising the space between the coins - can in fact be achieved by arranging the coins in alternative fashion with the RH axis - drawn from the centre of the coin - in one row matching the LH axis of the coin immediately above (or below).

This most efficient arrangement implies that each coin will touch 6 others (whereas in the previous example only 4 would be touched).

Now one could imagine a somewhat similar arrangement for packing spherical objects (approximated by oranges) in 3-dimensional space. Here in fact the most efficient arrangement would lead to each object being touched by 12 others!

One could pursue such packing arrangements into higher dimensional space. Though one cannot properly visualise these arrangements, the mathematical properties entailed can be clearly articulated.

The importance of 24 in this context is that it lends itself - unlike other dimensions - to an amazingly efficient packing arrangement i.e. with respect to 24-dimensional hyperspherical objects.

Indeed the most efficient arrangement possible, in work that also has strong connections with efficiency in sending communication signals, was provided by John Leech in what has become known as the Leech Lattice.

So Leech was able to demonstrate that with 24-dimensional space, the most efficient packing arrangement would entail that each hyperspherical circular object would be touched by 196,560 others.

If we consider for the moment the packing in two dimensions of (linear) square objects or in 3 dimensions cubes, this could be efficiently done without any space at all between the objects. And we could extend this thinking to hypercubes in 24 dimensions!

So seen from this perspective the efficient packing of hyperspheres represents the attempt to accommodate as closely as possible circular with linear quantitative notions (which is achieved when as little free dimensional space as possible is left over). In other words - though a most valuable exercise - it actually represents an attempt to reduce the quantitative dimensions as much as possible to the objects (thereby contained).

Now in fascinating reverse fashion, when one views the 24 Personality Types, representing the corresponding qualitative circular notion of dimensions, once again we have the attempt to accommodate linear and circular aspects. In other words the system of the 24 Personality Types, as outlined, is actually the attempt to harmonise as closely as possible the (linear) rational conscious with the (circular) intuitive unconscious.

So again with respect to the linear aspect we defined 8 "real" Personality Types (with an orientation primarily to conscious reality); then we also had 8 "imaginary" Personality Types (defined by a corresponding orientation to unconscious reality); finally we had eight complex types (defined primarily by the need to reconcile both conscious and unconscious).

However whereas in the quantitative case, we attempt to reduce any free dimensional space, in this qualitative treatment of dimensions, the goal is the opposite so as to free up as much space and time as possible (through minimising rigid attachment to quantitative phenomena).

The successful minimisation of object attachment in psycho spiritual terms, requires that all 24 Types be successfully harmonised in personality. In other words psychological integration requires identifying strongly not with just one Personality Type but in being able to recognise the equal contribution of all (with each providing a unique valid perspective on reality)!

Furthermore applying this to string reality we can equally say that 24-dimensional circular space provides a particularly suitable environment for successful physical vibrations with respect to both material and dimensional aspects of the string (in a manner that literally frees up space and time so as to facilitate such dynamic interaction).

Indeed, as I have mentioned in an earlier blog, a fascinating holistic mathematical explanation can be given for the importance of the 24th dimension (in qualitative mathematical terms).

Once again the nth dimension in this qualitative context is structurally related to its corresponding nth root (i.e. (1/n)th dimension in quantitative terms.

We can obtain any nth root of unity as:

Cos (2pi)/n + i Sin (2pi)/n where 2pi = 360 degrees (derived in turn from the fundamental Euler Identity, e^(2i*pi) = 1.

(Indeed this same Euler Identity plays an important role in the j-function to which unexpected numerical connections with the Monster Group have been demonstrated to exist resulting in the term "Monstrous Moonshine"!)

Now when we concentrate on the absolute magnitude of both Cos and Sin values the sum of such values when squared will always fall within a range between 1 and 2.

For example when n = 1, n = 2, n = 4 the extreme minimum value will result = 1.

In a psychological qualitative context this is due to the fact that these dimensions themselves represent respective extremes of pure rational linear understanding (n = 1), pure intuitive understanding (n = 2) and pure imaginary understanding (n = 4).

Then when n = 8, the squared total = 2. This in turn represents the fact that the 8th dimension represents another extreme expressing an attempted holistic balance as between both real and imaginary (conscious and unconscious) understanding.

This would suggest that the best radial position (allowing for the maximum in terms of balanced analytic and holistic understanding) would occur at 1.5 (as the mean of both extremes) and this indeed occurs when n = 24!

So with no reference whatsoever to the conventional mathematical treatment of dimensions, I had already discovered a compelling reason in holistic mathematical terms as to why 24-dimensional reality is important from a circular qualitative perspective! And once again such reality needs to be properly considered in a dynamic interactive manner where a very close relationship exists as between all 24 qualitative dimensions (that in turn bear a direct structural correspondence with the 24 quantitative roots of unity).

The Monster Group was first constructed by Bob Griess in 196,884 dimensions.

This in turn split into 3 subspaces (all with an intimate connection with 24)

So 98,304 + 300 + 98,280 = 196,884

The first of these numbers 98,304 = (2^12)*24

The second number 300 = the sum of the 1st 24 natural numbers (1 + 2 + 3 +....+ 24).

The final number 98,280 = 196,560/2 (i.e. half of the number that appeared in the Leech Lattice for touching hyperspheres).

Interestingly 98,280 = 98,304 - 24 (thus establishing a connection between 1st and 3rd numbers).

We have also already seen that 1^2 + 2^2 + 3^2 +....+ 24^2 = 70^2 (1)

Also 196,883 is the minimum no. of dimensions in which the Monster can be constructed = 196,883 = 196,884 - 1.

And 196,883 = 47 * 59 * 71

So this last prime is just one greater than the number 70 appearing in (1).

Finally these 3 primes 47, 59 and 71, have a range of 24 (71 - 47), and can be further shown to be related to 24 in an interesting manner.

47 = (2 * 24) - 1

59 = (2.5 * 24) - 1

71 = (3 * 24) - 1

Wednesday, April 14, 2010

Monstrous Moonshine - clarification of dimensions

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!

Monday, April 12, 2010

Monstrous Moonshine - role of the unconscious

As is well known the term "Monstrous Moonshine" was coined by John Conway with respect to an unsuspected mathematical connection demonstrated as between the Monster Group and the j-function (an important part of number theory related to modular functions).

The Monster Group itself appeared somewhat like the final piece in a massive jigsaw puzzle.

The whole area of Group Theory - that has now assumed great importance in Mathematics - arose out of the attempt to classify the different types of symmetrical objects that can arise.
A simple geometrical example of a symmetrical object is provided by the 3-dimensional cube which has six similar faces. Now there are many different ways of rotating the cube (48 in all) so that its symmetry remains unchanged. So the quest that arose in the 19th century was to classify the "atoms of symmetry" or basic building blocks as it were for all symmetrical objects, in what were termed finite simple groups.

In some respects this parallelled the similar quest to find the basic building blocks of the natural number system (i.e. the prime numbers).

Thus a great number of symmetrical objects are based on the different rotations possible with respect to various prime number permutations.

However other simple groups exist where the number of indivisible permutations do not correspond to prime numbers!

In the classification of the various simple groups that subsequently arose, most conveniently fell into four coherent families. However like separated islands, many exceptions remained (26 in all) that were lumped into a Sporadic Groups category, the largest of which is the Monster.

The total number of symmetries in this group - which cannot be decomposed into smaller sub-groups - is truly enormous and greater than all the quarks in the Universe. The lowest number of (non-trivial) dimensions in which they exist is 196,883 (47 * 59 * 71). It is not surprising therefore that this has been termed the Monster Group.

Then by surprise it was discovered that the number 196,883 and other larger dimensional numbers associated with the Monster bore a close relationship with coefficients of the j-function (a seemingly unrelated area of Mathematics).

It is in this connection that the term "Monstrous Moonshine" was coined by Conway.

"The stuff we were getting (i.e. the unexpected connections) was not supported by logical argument. It had the feeling of mysterious moonbeams lighting up dancing Irish leprechauns"

From my own holistic mathematical perspective as an "Irish leprechaun", I find this comment very interesting as it actually points to a deep unrecognised limitation of Conventional Mathematics.

The moon is commonly used to signify unconscious life. Indeed one remnant of this association is in the term lunacy (derived from the Latin word for the moon). So lunacy or madness would reflect a deep disorder with respect to the unconscious.

Then in the spiritual life "dim contemplation" is often used to refer to the faint passive light (like reflected moonlight) that greatly facilitates understanding of a universal holistic quality based on pure unconscious intuition.

As I have continually maintained there are in fact two equally important aspects in balanced (i.e. psychologically symmetrical) understanding of Mathematics.
One is the quantitative rational aspect which obtains its specialised expression in Conventional Mathematics. However the other - largely unrecognised - is the qualitative intuitive aspect which obtains its specialised expression in what I refer to as Holistic Mathematics.

As we will once again see the mathematical notion of Euclidean dimensions (on which the classification of the finite simple groups is based) is but a reduced rational notion based on merely linear logical concepts.

Indeed there is a strong paradox in this mathematical approach to symmetry based - as it is - on classification of irreducible building blocks (i.e. the finite simple groups).

Symmetry in any context entails the holistic relationship as between the various part aspects involved. However such holistic appreciation (literally of the whole context) should not be confused with successful analysis of the various parts. Though both aspects (whole and parts) are necessarily interrelated they should not be identified with each other. However, because of sole recognition of the quantitative aspect (suitable for partial analysis), in any relevant context Conventional Mathematics can only proceed by attempting to reduce the whole to the parts.

So the point I am making - which will be developed in further posts - is that true holistic understanding is needed to qualitatively appreciate what is entailed by "Monstrous Moonshine".

Another obvious paradox about the mathematical drive to understanding symmetry comes through recognition of the extremely important role played by many of its leading proponents who display classic symptoms of Asperger's Syndrome. For example John Conway, Simon Norton and Richard Borcherds would seem to fit readily into this category.

Indeed one could go further and argue that success in the highly specialised abstract task of finding the unique building blocks for all symmetrical objects has been greatly facilitated in this context through Asperger's Syndrome.

And I do not wish by this to question the merit of this achievement which has been truly outstanding; rather I am suggesting that from an overall comprehensive mathematical perspective it is quite unbalanced. Thus a much more rounded appreciation (requiring qualitative rather than quantitative mathematical notions) likewise needs to be provided.