Collective unconscious, archetypes, and mathematics: the TGD view

I couldn’t sleep and listened to Edward Frenkel’s latest talk (see this) about the basics of mathematics. Frenkel is doing a great service by increasing the public understanding of what deep mathematical thinking really is. The topic was the idea of an archetype and its generalization to the idea of a mathematical archetype.

A. Jung’s notions of collective consciousness and archetype

Collective unconscious and archetypes are Jung’s basic concepts. I would however speak more precisely and replace “unconscious” with “unconscious for me”. This could be regarded as the counterpart of the principle of relativity for the theory of consciousness. I would also do some nitpicking of the concept of “conscious-ness“. The “-ness” reflects the materialistic archetype, which states that consciousness is a property of a material system. Among other problems, this leads to conflict with free will.

In psychology, different archetypes appear in fairy tales, especially in dreams. They also populate popular media. Also mathematical concepts would be archetypes. That natural numbers are an archetype, as was suggested by Jung and, using different words, by many great mathematicians.

Jung associates three basic aspects with archetypes.

1. We can be under the spell of an archetype. The archetype looks completely self-evident to us and we need not be even aware of it. Therefore we cannot question it and ask if there could be other options for seeing the world.

A good example of this is the concept of the “shadow”. Most of us have an image of ourselves as a good and nice person in every way, even though we realize that we are not always able to be that. But this unconsciousness-to-us archetype has a powerful spell on us and controls us and to a high degree determines our perception of the world and our behavior in it.

Projection is another aspect of archetype. We are constantly faced with contradictions because the shadow of ourselves is reflected in the behavior of others. They experience that we are not always nice people and will express it, not always politely. We become aware of this and get irritated and blame the outside world for the mirror being crooked. We may even choose our company in such a way that our own shortcomings, which we do not want to be aware of, are highlighted, for example, in our friends. This easily happens to politicians in power. There is no need to mention names in the current situation of the world.

The comforting thing is that the archetypes can nevertheless be understood and integrated into one’s own personality. We can become aware of the shadow and become mature to admit that we are not always good guys and accept this as a part of our personality. After this, we can control our behavior and our shadow can no longer hold us in its power. We do not only react but consciously control our behavior. This does not mean that we start to “think positively”: we can be critical, we can raise our voices, and we can get angry about the problems of the real world. This is however a conscious choice, not a mere reaction. Disagreeing can become a pleasant art form instead of being a bloody battle.

B. Are also mathematical ideas archetypes?

Frenkel’s idea is that mathematical ideas are also archetypes. They are in everyone’s consciousness as intuitive archetypes. They are not necessarily understood precisely, even by a mathematician. The development of a new mathematical insight is not a computer program or a logically proceeding reasoning, but a semi-conscious process and mathematician just throws oneself into the flow and lets it go. The goal, however, is to formulate the visions precisely using existing or even new mathematical concepts.

Now too, it happens that these archetypes, as only partially understood mathematical ideas, are projected to be properties of the external world, even if one sees that there are problems. The growth of mathematical understanding means that the mathematician becomes aware of the archetype and is able to formulate and analyze it and also realizes that it is only one option among many.

Frenkel presents 2-dimensional Euclidean plane geometry as an example. Every layman has an intuitive sensory image of it as the surface of Earth. The projection in this case means that we once took it for granted that our physical world, the Earth, is a flat plane even though everyday observations could have demonstrated that this is not true. For instance, seeing how the ship disappears to the horizon makes clear that Earth is spherical.

Already in ancient times, mathematicians realized that there is a spherical geometry and Frenkel illustrated this with the concept of a rectangular coordinate system of whose existence which Descartes became first aware: the plane can be divided to squares of the same size and shape by a coordinate grid consisting of geodesic lines. For the sphere the small curvilinear squares defined by the counterparts of the coordinate grid approach points at the poles.

The process of awareness related to geometry has progressed. Riemann discovered the archetype of 2-D curved geometry. We can imagine 2-dimensional surfaces as surfaces in 3-D space. What was new was that Riemann abstracted surfaces into abstract geometries without any embedding as a surface in 3-D space that makes them visualizable. The notion of Riemannian Geometry generalizes to an arbitrary dimension.

The reason why this 2-D plane archetype prevailed for so long was that it could be thought of as a very familiar-to-us surface, the surface of Earth in 3-D space which looks flat to us. Sense perception offered a direct representation for it. Riemann’s insight was that geometry could be formulated abstractly without assuming a concrete representation as a surface. Note however that something is lost in the abstraction: this embedding adds the shape as seen by a perceiver living in 3-D space.

The development eventually led to general relativity and the concept of 4-D spacetime as an abstract (pseudo-)Riemmannian geometry, which was not represented as a surface in any higher-D space.

C. Is time mature to update the archetype of space-time of general relativity

General relativistic view of space-time has been the dominant archetype in theoretical physics and theoreticians have projected it into reality. Now it is becoming increasingly clear that something is badly wrong. JWST is constantly producing discoveries that do not fit into the cosmology of general relativity. It has been also known for about a century that the classical conservation laws of energy, momentum and angular momentum are strictly speaking lost in general relativity. This particular failure of the archetype has been forgotten.

For a long time, there has been a phase of questioning and the realization that there might be other options going on. I believe that this will eventually lead to an integration of the world view.

C.1 String theory did not make it

String theory drastically questioned the Einsteinian archetype and replaced spacetime with a 2-D surface in 10-D space. It was actually a return to the concrete, you could say. This new archetype was immediately projected into reality and superstring hegemony was born. The archetype produced some incredibly beautiful mathematics, especially the realization that conformal symmetry and algebraic geometry could be fundamental physics. However, the attempts to pull 4-D spacetime out of a hat in this picture failed: nothing was found in the hat. After 40 years of attempts it has been admitted that this archetype produced good mathematics but does not apply to physics.

C.2 What about TGD

Also TGD started by questioning this Einsteinian archetype: the key observation was that conservation laws of special relativity are lost. Perhaps sensory experience in which a sphere is a 2-D surface should not be abstracted to Einsteinian space-time geometry but to the idea about space-time as a 4-D surface.

We also want standard model physics and want to generalize Einstein’s geometrization program also to gauge fields. Perhaps the geometry of the spacetime surface is induced from H=M⁴×CP₂, which indeed geometrizes the standard model’s gauge fields and restores the lost symmetries of Minkowski space and, as an added bonus, predicts the standard model symmetries. This choice of H would be also the only mathematically viable option and physics would be unique from its mathematical existence.

This archetype turned out to bring new archetypes to the spot light of mathematical consciousness.

1. Perhaps the geometry of the space-time surface is algebraic, which would mean enormous predictive power. This led also to a possible physical analog of Langlands duality stating that there is a duality between geometry and number theory.
2. The general coordinate invariance, the basic principle of special relativity, forces holography so that space-time surfaces become analogs of Bohr orbits of particles identified as 3-D surfaces instead of point-like particles (the basic archetype in the consciousness of a particle physicist). Classical physics becomes exact part of quantum physics so that a generalization Bohr orbitology make a comeback to theoretical physics.
3. String models have conformal invariance as a basic symmetry: this is also a new archetype. In TGD it generalized led to holography= holomorphy vision and led to a reduction of field equations to purely algebraic equations: this simplification is gigantic.
4. This also led to a need to generalize the concept of geometry to the geometry of the world of all classical worlds (WCW). Ths geometry is infinite-dimensional and in infinite dimensions geometries are highly unique. Perhaps also the p-adic and adelic equivalents for space-time surfaces are possible and necessary for understanding cognition.

D. Mathematical consciousness in the TGD Universe

Frenkel started from conscious experience and a possible connection between mathematics and psychology and was led to collective consciousness and archetypes. What about the situation in TGD? What would the reality be in the TGD framework?

D.1 Ontology must allow both mathematical and subjective existence

In order to speak about conscious experience at all, we must challenge the materialist archetype, in which subjective reality and free will are regarded as an illusion and do not exist. One can also challenge the idea that there is only Platonia consisting of mathematical objects/ideas as archetypes.

In the TGD inspired ontology, both mathematical reality and subjective reality would exist. I call the mathematical reality quantum-Platonia. Mathematics is realized as mathematical objects. The basic requirement is that the system is internally consistent and consistent with the physics as we know it.

Space-time surfaces as the basic objects of WCW could be regarded as generalized numbers, which means a huge generalization of the archetype of the natural number, discovered also by Jung.

Subjective existence would not be given up. What is however new for a physicist is that there would be no separate physical world behind quantum-Platonia! Quite a loss for a physicist who thinks that he is gaining his monthly salary by studying the physical world, one might say! The physical world behind mathematics would be unnecessary.

D.2 Quantum Platonia as mathematical existence

Quantum Platonia would consist of the WCW as the geometry in the space of spacetime surfaces in 8-D embedding space H. Spacetime surfaces generalized to their p-adic variants and adelic variants would serve as correlates of cognition. Mathematically imaginable n-D spaces would correspond to algebraic extensions to p-adic number fields. Quantum states as counterparts of objective existence, would be (classical!) spinor fields in WCW. Perhaps one would identify these as archetypes.

D.3 Subjective existence is in quantum leaps

Subjective existence would be in quantum leaps, or more technically, state function reductions, between these quantum states. We would become aware of these mathematical archetypes as we make quantum leaps in the Hilbert space defined by the spinor fields restricted to an appropriate subspace of WCW.

The experiences related to these quantum leaps would however usually be something else than mathematical ideas: they give rise to what we experience every day. Only for a mathematician are they to a high extent mathematical observations and insights.

Number-theoretic evolution by quantum leaps would give rise to a universal evolution: spacetime surfaces would gradually become more algebraically complex in the sequence of quantum leaps.

D.4 Zero energy ontology and slight classical non-determinism makes it possible to learn and remember

Zero-energy ontology and a slightly broken classical determinism for the spacetime surfaces as Bohr orbits for particles as 3-surfaces (space = particle, this a new archetype too) would guarantee that we are able to remember and learn and our mathematical and other understanding develops. Also life and death can be understood: they correspond to the “ordinary” state function reductions changing the arrow of time. The counterpart of the Zeno effect giving rise to no state function reduction is a sequence of “small” state function reductions giving rise to the subjective existence.

The theories that we build of existence are doomed to be always incomplete because quantum as quantum measurements provide little information. But it is possible to get in touch with the collective levels of the hierarchy of consciousness, we have used to call this state a flow state. This makes it possible to become aware of the big principles and to identify the important bits.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/02/collective-unconscious-archetypes-and.html