Synchronicity, Causality, Complexity, and the Brouwer Fixed-Point Theorem

Scientific Essay, 2018

8 Pages


Synchronicity, Causality, Complexity, and the Brouwer Fixed-Point Theorem

© 2018

M. Goldberg

Carl Jung[2] defined synchronicity[1] as an acausal connecting principle, and a ‘meaningful coincidence.’ He first introduced the concept in the 1920s. In 1952, he published the paper “Synchronizität als ein Prinzip akausaler Zusammenhänge" (Synchronicity – An Acausal Connecting Principle), which included a monograph by Wolfgang Pauli[3]. Jung theorized that temporally coincident occurrences of ‘meaningful’ events are acausal, and not bound by the axiom of causality (AOC), the principle that all effects have causes and all causes produce effects.

In his book Synchronicity [2] Jung tells the following story of a synchronistic event: ‘My example concerns a young woman patient who, in spite of efforts made on both sides, proved to be psychologically inaccessible. The difficulty lay in the fact that she always knew better about everything. Her excellent education had provided her with a weapon ideally suited to this purpose, namely a highly polished Cartesian rationalism with an impeccably "geometrical" idea of reality. After several fruitless attempts to sweeten her rationalism with a somewhat more human understanding, I had to confine myself to the hope that something unexpected and irrational would turn up, something that would burst the intellectual retort into which she had sealed herself. Well, I was sitting opposite her one day, with my back to the window, listening to her flow of rhetoric. She had an impressive dream the night before, in which someone had given her a golden scarab — a costly piece of jewelry. While she was still telling me this dream, I heard something behind me gently tapping on the window. I turned round and saw that it was a fairly large flying insect that was knocking against the window-pane from outside in the obvious effort to get into the dark room. This seemed to me very strange. I opened the window immediately and caught the insect in the air as it flew in. It was a scarabaeid beetle, or common rose-chafer (Cetonia aurata), whose gold-green colour most nearly resembles that of a golden scarab. I handed the beetle to my patient with the words, "Here is your scarab." This experience punctured the desired hole in her rationalism and broke the ice of her intellectual resistance. The treatment could now be continued with satisfactory results.’

In another example of synchronicity: ‘The French writer Émile Deschamps claims in his memoirs that, in 1805, he was treated to some plum pudding by a stranger named Monsieur de Fontgibu. Ten years later, the writer encountered plum pudding on the menu of a Paris restaurant and wanted to order some, but the waiter told him that the last dish had already been served to another customer, who turned out to be de Fontgibu. Many years later, in 1832, Deschamps was at a dinner and once again ordered plum pudding. He recalled the earlier incident and told his friends that only de Fontgibu was missing to make the setting complete – and in the same instant, the now-senile de Fontgibu entered the room, having got the wrong address.

The phrase ‘acausally-connected events’ refers to the simultaneous or near-simultaneous occurrence of events which appear related in a meaningful way, yet seem to have no causal connection. Synchronicity, a concept first introduced by Carl Jung, holds that events are "meaningful coincidences" if they occur with no discernable causal relationship.

This paper proposes that synchronicity is:

- not acausal, but causal, and a consequence of the axiom of causality (AOC).
- an emergent phenomenon arising from the complex interactions of causal events.
- the result of a deterministic system modeled by a universal cellular automaton.
- bound by the Principle of Computational Irreducibility (PCI).
- a consequence of the Brouwer Fixed-Point Theorem, BFPT[5, 7].

The Axiom of Causality (AOC) implies that all effects have causes, and all causes have effects. Every ‘cause-effect’ pair or vector arises from an antecedent cause and, in turn, produces an immediately subsequent effect which is also a cause. There are no causes without effects or effects without causes; the system is closed—every vector comes from a vector and goes to a vector. It is proposed that these vectors can interact with one another in very complex ways which can be modeled by a universal cellular automaton such as Wolfram #110[4]. Moreover, because the system is ‘closed’ these vectors can be represented as a vector field on the N-1 surface of an N-dimensional closed surface (e.g. a sphere)—the Brouwer Fixed-Point Theorem (BFPT).

In a universal cellular automaton such as Wolfram #110[4], the Principle of Computational Irreducibility (PCI) teaches that no ‘shortcut’ equation can tell us the state of the cellular automaton at some future time merely by ‘plugging in’ a value (TFUTURE). Instead, one must ‘run’ the cellular automaton to see its future state. The cellular automaton is deterministic, but a priori indeterminable. Furthermore, for a given ‘state’ of the cellular automaton, it is not generally possible to trace the pathway by which that state was reached, even though it was reached deterministically. To do so would violate the PCI.

In the case of synchronicity we have many causes and effects, including all those causes which can produce effects on memory, and dreams. The mind, in turn, is the cause of other effects on the world, including other minds. In other words, simple immediate (direct cause-effect) vectors, AB, CD, EF (instantaneous ‘world lines’) can interact with one another in the most complex ways. The word ‘immediate’ means occurring together over a ‘short’ period of time where Δt, the time magnitude of the vector, is infinitesimally small.

All of these simple immediate cause-effect relationships can be seen as the elements of a ‘rule’ defining the computations of a universal cellular automaton where these vectors interact in complex ways.

The Principle of Computational Equivalence (PCE)[4] holds that simple rules, such as Wolfram #110, can lead to complex behavior. A multitude of interactions among these vectors as elements in a complex cellular automaton can result in unexpected causal relationships.

However, the Principle of Computational Irreducibility (PCI)[4] does not allow us to determine by what circuitous computational routes a particular state was reached. The causal relationship is cryptic. This seems peculiar because we expect a deterministic system to be transparent, yet universal computational systems can produce results that are deterministic yet a priori indeterminable. Consequently, events can be related causally, but we are fundamentally unable to determine that they are.

The Brouwer Fixed-Point Theorem (BFPT)[5, 7] relates to the Axiom of Causality (AOC) because both are related to closed systems. Accordingly, the vectors mentioned above can be seen as a set of vectors parallel to the surface of a sphere (see Figure 1 below). The BFPT results in a confluence of these vectors at some region in space and time where these vectors form a WHORL. A related theorem, called the Hairy Ball (Hedgehog) Theorem (HBT)[6] states that one cannot comb a hairy ball smooth (see Figures 2 and 3 below). In terms of vectors, this means that two or more causally-related vectors can appear together in a WHORL such that they might seem to be acausally-connected events.

Every hairy ball has a whorl or cowlick. Vectors ‘circling the whorl’ will occupy the same region of space Δ(X, Y, Z) and the same interval of time Δ (T). The delta sign, Δ, can be smaller or larger depending how far apart the vectors are on the ‘spiral galaxy shape’ of the whorl—either on the outer edges of the whorl or tighter in towards the center.

Suppose many of these vectors circling the whorl are causally-related, but the Principle of Computational Irreducibility prevents our ‘calculating’ the complex chain of causality that connects them. Accordingly, the confluence of these vectors may be seen by a conscious mind as acausal synchronistic events.

There is nothing mysterious about synchronicity. ‘Synchronicity’ is merely a convenient term that arises because the PCI precludes ‘calculating’ the complex causal chain connecting events that end up in the whorl created by the Brouwer Fixed-Point Theorem. Arcane, ‘new age,’ and occult interpretations of synchronicity are seen to be superfluous. A particular human brain with certain memories might interpret these events as an example of synchronicity, even though the events are not directly one-to-the-other causally related except in the mind of the agent (‘meaningful occurrence’) who observes the events.

To others, without the same memories, the circumstance would not be considered meaningful. Synchronicity is observer specific, and only exists if there are conscious minds able to assign meaning to these events. If Jung’s patient that day had not had a dream about a golden scarab, then when the beetle rapped at the doctor’s window, there would have been no synchronicity of events noted. Similar events involving one or more people over many years depend on the memories of those involved, and would not be considered synchronicity by others who do not have those connecting memories.

Referring to Figure 3 below, vectors on the outer rim of the whorl and in the same space, but separated in time by a large ΔT, might lead one to interpret the events incorrectly not only as synchronistic, but also that one event, such as a dream, had foreseen or anticipated a ‘future’ event.

On the other hand, vectors on the outer rim of the whorl occurring at the same time, but separated in space might be interpreted as synchronistic events separated by distance. ‘Minds,’ including memories and dreams, are part of the world, and are both causes and effects. Minds are able to interact with one another immediately or over various periods of time (seconds to millennia) through communication channels such as direct contact, the internet, shared myths, books, and genetically-determined synaptic connections and neural structures.

The vectors can be represented as parallel to and swirling around on the surface of the sphere.

illustration not visible in this excerpt

Figure 1 (Authors own work)

Figure 1 above illustrates the Brouwer Fixed-Point Theorem, BFPT[5,6] as a set of short (‘direct’) cause-effect vectors lying parallel to the surface of a sphere, and forming a whorl or region of confluence. Each vector or element of synchronicity that is part of the whorl may be the result of a causal chain in the past, and could well have occurred at different places, times, and under other circumstances and not have been perceived as an example of synchronicity, but because they are crowded together in the whorl might be interpreted as synchronistic events. The presence of vectors representing ‘mind’ results in the word ‘meaningful.’ If the ‘ball’ in Figure 1 above is oriented to a 3-D coordinate system (X,Y,T), then the whorl represents a region on the ball where the vectors occupy the same region or patch of space and time.


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Synchronicity, Causality, Complexity, and the Brouwer Fixed-Point Theorem
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synchronicity; causality; complexity; cellular automata; principle of computational irreducibility; Brouwer fixed-point theorem; archetypes; collective unconscious; precognition.
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MD Dr. Marshall Goldberg (Author), 2018, Synchronicity, Causality, Complexity, and the Brouwer Fixed-Point Theorem, Munich, GRIN Verlag,


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