Let me begin by reiterating that I really enjoyed your essay! Of course, one is not expected to be able to cover all aspects of a given theme. So, please understand my response as just a set of thoughts that emerged as I read your interesting piece. And since your initial response indicated you intended to follow up when you were better rested, I waited to pen my response. So, let me begin with a few things inspired by your two responses.
I am very much an advocate of multidisciplinary scholarship and have been since I began graduate school, first in linguistics and then in theoretical computer science. And I am an advocate of multi-paradigmatic approaches to a given issue, each paradigm offering its own unique opportunity to circumambulate the issue in its own way. Take alchemy, for example, an area that I have been working with for the last several years. Serious scholarship on alchemy, say the Graeco-Egyptian alchemical tradition, requires rigorous scholarship in history of religion, history of science, linguistics, knowledge of ancient languages such as Greek, Coptic, Syriac, and then its translation into Arabic, history of the first 3 or 4 centuries in Roman Egypt, archeology, etc. No single person could hope to be an expert in all that is required here. Even more important, there is the very real danger that a single person, even if somewhat proficient in all of these areas, will (unconsciously?) arrange the data culled from the different disciplines in a way that is far too personal (e.g., a sort of synthesis into a unified whole which mirrors oneself more than representing the tradition more objectively). This is why, in addition to my being an advocate of multidisciplinary scholarship, I am also an advocate of collaboration among experts of different disciplines, even (and especially) if this leads to areas of disagreement. It is critical, in my opinion, not to necessarily strive for a unified view, but instead to strive for a consciousness of areas of agreement and areas of disagreement as it applies to a single discipline such as Graeco-Egyptian alchemy.
I also want to mention something about the unconscious. For various reasons, I am working on an effort to reconsider the unconscious. And in that work, I recently came across a short paper by Dr. Sonu Shamdasani entitled Questioning the Unconscious published in 2017 in the Annals of the New York Academy of Sciences. I believe it was a conference he attended. And in that paper, he discusses different models of the unconscious and that there is great doubt whether any model will garner universal agreement. Now, in my technical past, developing models (stochastic/mathematical-based and simulation-based) was part of what I used to do and teach – models of hard real-time networks, of consumer electronics, of interconnected automobile safety systems, etc. And this gave me a perspective on some of the advantages and disadvantages of developing a model. As you know, a model is an abstraction, one which has been consciously developed with certain intentions in mind and informed by a set of goals. Such abstraction is done by modeling only those details which are important to the set of goals, and this differentiates the model from that which it is modeling. The reasons abstraction is done include faster development time (prototypes are exceedingly difficult to build and the cost is high), mathematical tractability, and reduction in computer simulation time (i.e., amount of time it takes to get answers).
Now, a major advantage of models is, quite simply, that you can *play* with them. You can see the push-pull of the components that make up the model, discover its interdependencies, examine the relationship of change of behavior to differing inputs, reveal areas of stress, etc. It also provides a way to develop a far more nuanced understanding of that which you want to create. Perhaps most importantly, this model serves as an object through which one’s imagination can take flight, but flight which is not entirely free, but necessarily tied in some ways to the real world. Thus, one can see this as being a kind of controlled imagining.
It is in this way that I would like to use complexity science and category theory – as models for certain concepts that Jung developed. Crucially, these models are not intended to be the real thing. I am not creating an archetype out of math. But, math can reveal certain properties that Jung might have seen in the archetype or in synchronicity. Take acausality, for example. In complexity science, one can create simple models where higher complex structures emerge from more basic interconnected components. Such emergence cannot at this time be explained causally. Now, it may be possible that in such a model, there are multiple complex structures that emerge that somehow have connection to one another, but there is no cause discernible for such a connection. Such may be an area of exploration in order to get a better understanding of the acausality that connects inner and outer events in a synchronicity. Again, the model is not synchronicity – it just exhibits an aspect of it, namely acausality.
I should also say something about the Natural numbers (1, 2, 3, …) here. As you know, Jung and Pauli came to see the Natural numbers as being archetypes of order. Furthermore, they saw that the order embodied by the Natural numbers (an order which is explored in exceedingly rigorous ways in mathematics in areas like Number theory) is acausal and an example of what they called acausal orderedness (nuclear decay and synchronicity are other examples). Because the natural numbers are the most basic of archetypes and because there is a tremendous amount of knowledge of the Natural numbers as it relates to their orderedness, it makes a lot of sense to try to understand archetypes through trying to understand the Natural numbers and how they function, especially as they, as with all archetypes, govern both psyche and matter (physis). Jung would hand off this work to his greatest collaborator, Dr. Marie-Louise von Franz, who would go on to publish one of her masterpieces Number and Time. One might consider, then, the Natural numbers and their use in the world to be models which we can investigate and develop a deeper understanding of archetypes. Archetypes such as the anima/animus are far too difficult to work with in this way.
Now, an example of working with mathematics which can reveal greater nuance into how certain things function in depth psychology is the study of the psychological typology function of intuition in the context of formal mathematics. I wrote a piece entitled Reflections on Intuition Based on the Ramanujan/Hardy Collaboration in Formal Mathematics (1914-1920) in which I explored mathematical intuition and, in part, why it is sometimes wrong. Now, with less formal uses of intuition, when it presents possibilities that are not real or are unrealizable, we normally do not know why it has done so. But when mathematical intuition suggests a possibility, we can evaluate whether that possibility is consistent mathematically. And if we find that it is not consistent, we can better determine why intuition gave the wrong possibility, and it often comes down to a mistaken set of assumptions held by the individual. This is what I discovered in doing the research for my note. So, again, intuition in the context of mathematics is quite limited, but it serves as a model to provide greater nuance into understanding how intuition functions overall. It also provides instruction on how the individual can collaborate with the unconscious so that the language of the unconscious is clearer. I have found that when the individual has done the leg work, meaning when they have exhausted themselves in doing the analysis, the reflection, etc., the response by the unconscious is often stronger and of far greater clarity and relevance.
Returning to your post and whether the archetype is necessarily deduced, I do wonder about one thing. What was it that inspired Jung to pursue something beyond that which is directly experienceable? Why did he not settle for archetypal images and symbols? Why did he feel the need to go further and to hypothesize the unknowable psychoid archetype? These questions really have nothing to do with the correctness of the concept of the archetype. It has more to do with the experience Jung had in developing the concept – what it was that drove him. For example, was it part of his experiences during his confrontation with the unconscious from 1913 to 1932 that inspired him to search beyond the image? Or, did it somehow lend itself intellectually – did it account for important necessary dynamics in his model of the unconscious? I would very much love to explore this line of thought. And with the Black Books, the Red Book, and the 2018 book The Art of C. G. Jung, we are in a far better position to do so than we were, say, 15 years ago.