Note 4 - Carl Jung and Holistic Mathematical approach
Carl Jung deserves special mention as someone who - at least implicitly - had a deep appreciation of the holistic nature of mathematical symbols.
One of his greatest followers Marie-Louise von Franz developed these notions in her book "Number and Time". Indeed the subtitle "Reflections Leading towards a Unification of Depth Psychology and Physics" itself is highly suggestive of the complementary application - in dynamic terms - of mathematical notions to both of these realms.
Von Franz says
"Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four"
Now it is certainly true that so many important structures come in fours. For example, we conventionally accept the four dimensions of space and time; we also accept four fundamental forces in nature.
Also in more psychological terms, Jung used four basic functions for structuring personality. Likewise Ken Wilber's use of four quadrants (applicable to all holons) is very important for his integral approach.
It is possible in fact to give a very interesting holistic mathematical explanation for the importance of four.
Phenomenal reality is based on duality (due to separation of opposite poles).
Now the important mathematical operations correspond in turn to horizontal, vertical and diagonal notions.
Addition represents a horizontal transformation (within a given dimension i.e. 1)
Multiplication involves a vertical transformation (i.e. the dimension thereby changes)
Finally, obtaining the power of a number leads to a diagonal transformation entailing both horizontal and vertical aspects. (It is even expressed in a diagonal manner!)
Now the remarkable fact about 2 (i.e. duality) that when it is combined with itself (in horizontal, vertical and diagonal terms) it leads to a total of 4.
This can be demonstrated quite easily
2 + 2 = 4 (horizontal)
2 * 2 = 4 (vertical)
22 = 4 (diagonal)
So in the sense we have defined it, 4 is invariant with respect to horizontal, vertical and diagonal transformations of 2 (duality).
Jung also had a keen appreciation of the importance of mandalas as symbols of integration.
Now such mandalas use the symbol of the circle which are divided - often very ornately - in a symmetrical fashion. (According to Jung the most popular symmetrical divisions are 4 and 8).
What is fascinating is that the basic geometrical structure of these mandalas corresponds directly with the holistic circular interpretation of number (esp. 4 and 8).
However the deeper mathematical significance of these mandalas has - I believe - not been properly appreciated (at least in formal terms). The indirect linear expressions of circular numbers (i.e. as roots of unity) yielding the geometrical patterns that comprise the mandalas, are - literally - reduced expression of oneness. Quite remarkably these provide a precise holistical mathematical means of explaining all the fundamental dynamic interactions
1) as between horizontal exterior and interior poles (within a given level)
2) as between vertical wholes and parts (between levels) and
3) and finally diagonal form and emptiness poles (both simultaneously within and between levels).
It is hardly surprising therefore that mandalas can serve as powerful archetypes of integration, as the full (holistic) mathematical understanding of these symbols leads indeed to an "Integral Theory of Everything" .