Chaos, Dimensionality and Free Will (Part 2)

Edward Lorenz in 1961 wrote down a set of three coupled non-linear differential equations to model the atmosphere. The belief at the time was that once you write down a set of differential equations then provided initial conditions you can always exactly find the parameters of the system at any point in time however long afterwards. Small errors shouldn't deviate you much from the true results. [The Lorenz system]

But to Lorenz's surprise, small changes in the initial conditions cascaded into huge differences in the outcome of the simulations. Resulting in a butterfly like pattern, which he termed a strange attractor- 

Lorenz Butterfly - Bounded and Finite, yet unpredictable.

The trajectories neither stabilize nor diverge to infinity, they strangely orbit a finite region without any specific periodicity. The term 'Butterfly Effect' comes from here- small changes leading to huge outcomes.

This had huge consequences not just mathematically but physically too. The differential equations themselves are exact, but our measurement of the initial conditions will always have errors - however small will cause the predictions to go haywire. 

One must understand that despite the highly unpredictable nature, the equations governing it are still deterministic- hence in principle, chaos is deterministic and it exists in classical physics. [See Double Pendulum]

Double Pendulum is a simple example of chaos in Classical Mechanics. Two pendulums starting very close to each other, diverge away chaotically. 

[Animated in Manim Python library using GPT 5.2, music is self-composed]

The real-life unpredictability comes due to errors from our end. Computers have finite memory hence they must approximate the numbers, and measuring tools will always have a least count below which they are unreliable and so on. 

But the errors introduced due to quantum mechanical laws are truly random and unavoidable, and thus quantum fluctuations that were earlier thought to be negligible for practical purposes could no longer be ignored as chaos would amplify these and give totally different outcomes. 

So everything macroscopically chaotic is heavily impacted by quantum physics and so, effective determinism and predictability breaks. 

Important Technicality- In the fundamental theory of QM, time evolution is unitary, due to which chaos in the usual sense is not possible, but it is an active area of research to study analogue quantum systems. [ See ETH & Quantum Chaos]

However, the point I'm trying to make here is that, in the effective classical descriptions, like Lorenz model for weather prediction, quantum mechanical errors due to the Uncertainty principle can be introduced to the initial conditions which would change the entire trajectory of the system. This can be treated as a semiclassical approach as the macroscopic equations are still governed by fluid dynamics which is a branch of classical physics.

Role of Dimensions

Chaos as mathematicians proved exists ONLY in three dimensions and above, for continuous differential equations. 

The Poincaré-Benedixon theorem forbids chaos in 2D systems. Here, as the region R is 2D all trajectories (red line) either stabilize to a fixed point or form an orbit.

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So if space was two dimensional (x, y, t) and its dynamics was modeled by continuous* differential equations then chaos would be impossible - determinism for all practical purposes would still exist and the fluctuations would all be irrelevant. That is, minds of beings living in 2d space can be easily read, if they existed.

It is very important to understand that because we live in a 4d space-time, the equations that govern the dynamics of particles allow for chaos and so quantum mechanical fluctuations influence large scale dynamics.

True indeterminism is thus a byproduct of Quantum physics, dimensionality of our universe and chaos.

*Discreteness of space-time is still a major question in Quantum Gravity. 

To summarize everything discussed so far, 

CM is Classical Mechanics; QM is Quantum Mechanics.

So does Free Will exist after all?

Unfortunately, not conclusive yet- so you can still blame your procrastination on the weather ;)

Our brain, as it exists in 4d space-time is definitely impossible to predict exactly, even approximately for a long enough time. Though unpredictability of decision making is a necessity for free will, it's not a sufficient condition. 

The underlying laws are still governed by Quantum mechanics, and we have no control over which particle goes where inside our brain.

This is where two schools of argument emerge - Materialists and Dualists.

Materialists believe that psychology is reducible to neuroscience and our mind is a consequence of biophysics inside the brain. While Dualists are of the opinion that mind and matter are two separate entities. Matter is physical, governed by laws while mind is metaphysical and emergent.

According to materialists, even though our decisions are indeterministic it is a consequence of the chemical processes inside our brain. We have no control over these chemical reactions when we take decisions, so free will is just an illusion.

But for Dualists, mind is an emergent phenomenon, something that cannot be reduced to fragments. The whole is greater than the sum of parts. This also has some scientific backing as many natural phenomena are emergent. Cellular automata is a huge class of toy models that mathematicians and physicists study that have very simple rules but are impossible to track and predict. An example is Conway's game of life, popularly known as a zero-player game. The complex patterns that emerge in this game are emergent, impossible to reduce to the sum of its parts. It's also 'Turing complete' which means it can perform any computational task including simulating itself. Many argue that consciousness too is an emergent phenomenon and cannot be reduced to motion of mere atoms.

It is not yet clear what is the definite answer, and so, the debate for free will, shall continue until we have a clear understanding of what consciousness is and how brain facilitates intelligence.

Note: This article is fully written by me without any AI help and it expresses my own views on the subject at the time of writing. Wherever images/code generated by AI have been used, due credit has been given.