Riddled with gaps: How the Old One plays at dice
I think there are strong arguments to support the idea that chaotic evolution in classical (non-quantum) physics is not only unpredictable in practice, but also undetermined in principle, with plenty of room for divine action.
If nature is undetermined in principle (the future is not uniquely determined by the present), then divine action can take place without violating natural laws.
By “divine action” I mean not only God’s action, but also the action of apparently “supernatural” technologies used by hyper-advanced civilizations in the cosmos (hopefully including future humanity). In other words, intelligent life might learn and use God’s tricks.
Some readers said that they prefer calling G “the Cosmic Mind” or something like that. I think it’s pretty much the same thing, but here I’ll follow Einstein and call G “the Old One.”
In a letter to Max Born, Einstein said that quantum nondeterminism “is not yet the real thing.”
“The theory says a lot, but does not really bring us any closer to the secret of the ‘old one’. I, at any rate, am convinced that He is not playing at dice.”
Einstein wanted a fully deterministic universe. Born disagreed, and pointed out that even classical physics is nondeterministic, because computing the future with absolute certainty would require specifying the initial conditions (the present) with the infinite precision of a real number. In his 1954 Nobel Prize lecture, Born said:
“As a mathematical tool the concept of a real number represented by a nonterminating decimal fraction is exceptionally important and fruitful. As the measure of a physical quantity it is nonsense… concepts which correspond to no conceivable observation should be eliminated from physics… the determinism of classical physics turns out to be an illusion, created by overrating mathematico-logical concepts.”
According to Born, since every conceivable observation has a margin of uncertainty, classical physics should be formulated in terms of statistical distributions.
Chaos theory brings Born’s approach to the surface. In fact, even in classical (non-quantum) physics, many mathematical models exhibit deterministic chaos. These models are deterministic (the future is uniquely determined by the present), but strongly sensitive to initial conditions. Small initial differences are amplified exponentially in time, which makes prediction impossible in practice.
The weather is nonlinear, dissipative, chaotic, and impossible to predict in practice. This is often illustrated with the butterfly effect: A flap of a butterfly’s wings “could lead to a tornado that would not otherwise have formed, [or] equally well prevent a tornado that would otherwise have formed” (Edward Lorenz in “The Essence of Chaos”).
The chaotic behavior of a dynamical system can be intuitively visualized with phase (state) space portraits and trajectories. A highly recommended book is “Dynamics: The Geometry Of Behavior,” by Ralph Abraham and Christopher Shaw, a masterpiece of “visual math” with plenty of illustrations and no conventional formulas. For a more conventional introduction, try “Chaotic Dynamics: An Introduction Based on Classical Mechanics”, by Tamás Tél and Márton Gruiz.
The trajectory of a real-world (dissipative and nonlinear) dynamical system in its phase space eventually reaches an attractor, which can be geometrically simple or strange as a fractal. All trajectories that start in the attraction basin of an attractor eventually reach the attractor.
What if the system starts exactly on the boundary between two different attraction basins? It seems that the system, unable to choose between the two attractors, could only make a random choice.
But (using Born’s argument against him) one could reply that a point exactly on the boundary is a mathematical abstraction: The starting point must be on either one or the other side of the boundary, and in principle we can always specify the starting point accurately enough to be in either one or the other basin.
Things start to become more complex when the boundary is fractal. In this case, the precision needed to determine which attraction basin contains the starting point increases with the fractal dimension of the boundary. The system will wander chaotically for some time near the boundary, and eventually reach the attractor that corresponds to the starting point.
But chaotic behavior can be even more complex, and undetermined not only in practice, but also in principle. Enter ultra-fractal “riddled basins,” for which nothing short of the infinite precision of a mathematical real number will work.
Ultra-fractal riddled basins
A riddled basin is a “basin of attraction [with] the property that every point in the basin has pieces of another attractor’s basin arbitrarily nearby,” explains the review paper “Fractal structures in nonlinear dynamics” (2009), by J. Aguirre et al.
In other words, a set of intermingled riddled basins can be thought of as a space-filling “fat fractal” boundary between different attraction basins.
A system that starts on the boundary stays on the boundary, but the boundary is space-filling and contains (in the sense of extending arbitrarily close to) the attractors.
The starting point can even “be arbitrarily close to one of the attractors and still end up going to the other attractor eventually” (“The End of Classical Determinism,” by John Sommerer).
Every neighborhood of the starting point, no matter how small, contains points that will eventually reach different attractors.
Therefore, no matter how accurate is the specification of the starting point, the attractor that the system will eventually reach is undetermined.
This is worth repeating in boldface: Regardless of the accuracy of the starting point, the attractor that the system will eventually reach is undetermined.
Riddled basins, which have been found in many dissipative systems described by simple maps and differential equations, “show that totally deterministic systems might present in practice an absolute lack of predictability,” note Aguirre at al. See also the book “Transient Chaos: Complex Dynamics on Finite Time Scales” (2011), by Ying-Cheng Lai and Tamás Tél.
I suspect that the fractal depth of riddled basins might be widespread in real-world, dissipative dynamical systems, and be the rule rather than the exception. If so, chaotic evolution is really nondeterministic in principle.
Nature “knows” (or, the Old One knows) the starting point of the system as an infinitely precise real number. But we can’t know the starting point with infinite precision, and any finitely precise starting point in a riddled basin contains the possibility of different outcomes. For any finitely precise starting point, only the probability of different outcomes can be known.
What about quantum physics?
One could think that this argument is wrong because quantum physics, not classical physics, seems to be fundamental. But, if we switch to quantum physics, we must deal with another source of nondeterminism: The (apparently random) collapse of quantum states upon observation.
There are conceptual parallels between riddled basins in dynamical systems and quantum physics: In both cases, an initial state contains (can be considered as a “superposition” of) different possible outcomes. After the discovery of riddled basins in 1992 some scientists (e.g. Tim Palmer in ”A local deterministic model of quantum spin measurement”) have suggested dynamical models for quantum collapse.
While no solid theory has emerged so far, I have the impression that this research program is worth pursuing. Analogies between dynamical systems with riddled basins and quantum systems could lead to physical (but still nondeterministic) models for quantum behavior. See for example the draft paper “Entanglement, symmetry breaking and collapse: correspondences between quantum and self-organizing dynamics,”by Francis Heylighen.
The physics of divine action
In “The Law of Causality and its Limits” (1932), Philipp Frank distinguished between two conceptions of divine action. According to the first, a higher power intervenes in the world by violating natural laws.
“The other, I should like to say more ‘scientific’, conception is that it is not in the character of natural laws that they predetermine everything. Rather they leave certain gaps. Under certain circumstances they do not say what definitely has to happen but allow for several possibilities; which of these possibilities comes about depends on that higher power which therefore can intervene without violating laws of nature.”
Riddled basins, fractally riddled with holes that belong to other basins, suggest that natural laws are riddled with gaps through which the Old One can act subtly and steer the universe with elegance, without violating any law.
We only know rational numbers (numbers with arbitrary but finite precision). But the Old One (or a sufficiently advanced intelligence) knows real numbers, and uses them to play at dice. The Old One doesn’t throw the dice randomly, but accurately places the dice on the table face up, slipping divine action beneath the laws of nature.
This is a little contribution to the conception of divine action through nondeterministic gaps in classical physics. Other conception of divine action rely on quantum nondeterminism, or strong emergence (downward causation).
As noted above, future theories might blur the distinction between classical and quantum nondeterminism, perhaps through “sub-quantum ether” models, but the nondeterministic gaps would likely still be there.
In “Randomness and Undecidability in Physics” (1993) Karl Svozil suggests that randomness in physics might be a signature of mathematical undecidability in the Gödel sense. See also Svozil’s recent open access book “Physical (A)Causality: Determinism, Randomness and Uncaused Events” (2018).
In the reality-as-simulation picture, it makes a lot of sense for the Old One to design a world riddled with gaps that allow intervention via a programming interface, without changing the source code.
Magic butterflies operated by the Old One & company might be flapping their invisible wings to subtly induce tornadoes in the fabric of reality. Perhaps one is near you. It’s interesting to note that a butterfly is also a symbol of rebirth and renewal.
Cover picture from Wikimedia Commons, riddled basin images produced with a visualization tool by Takashi Kanamaru and J. Michael T. Thompson, based on “Riddled Basins” (J.C. Alexander et al., 1992).
