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