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?
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?
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