On Free Will and Computational Irreducibility
There is a dilemma that has pulsed since the first human reflections: If everything is determined, what remains of willing? If there is freedom, where does the order of the cosmos rest?
For centuries, thought has oscillated between these margins — the empire of causality, governed by Newton, Laplace and their mechanical gods; and the insurrection of the spirit, which in the name of soul and morality, clamored for freedom.
However, perhaps the question is not whether to be free or not, but to understand the type of world where free will could exist without betraying the laws that govern the stars.
In this essay, we propose a path between mathematical reason and existential wonder. We unite the language of cellular automata, the depth of Geometric Brownian Motion, and the hidden poetry in computational irreducibility, to sustain a new reading:
Free will is the human face of a complexity that cannot be compressed. It is not a negation of laws, but rather, an effect of the structural beauty of the unpredictable.
Cellular Automata and Emergent Complexity
Imagine a universe where the destiny of everything — from the dance of a star to the curve of a leaf — is dictated by a simple rule. Not a divine will, nor blind chance, but a single instruction, repeated upon itself, like a drum that resonates through time.
This is the essence of cellular automata, discrete systems that evolve according to local laws, applied iteratively on a mesh of states. Stephen Wolfram classified them into four classes, but it is in class 4 that the miracle is revealed:
A threshold between order and chaos, where rich, unpredictable and dynamic patterns emerge from trivial rules.
Here arises the concept of emergence:
When the whole exhibits properties that were not present in any of the isolated parts. When the simple, repeated, engenders the sublime.
Rule 110, for example, is as simple as a binary poem. And yet, its behavior defies every attempt at prediction. It is universal — in the Turing sense — which means it can simulate any other computational machine. This leads us to the notion of computational irreducibility:
To know the state of a complex system, there is no closed formula. One can only simulate. One can only live.
Each iteration of the automaton is like a second of existence: Without return, without shortcut, without compression — and with the power to create the unexpected from the inevitable.
GBM as Ontological Metaphor
What if this unpredictability were not exclusive to discrete and artificial worlds? What if the continuous universe — that of smooth trajectories, real functions, fluid time — also danced according to an irreducibly chaotic pattern?
This is where Geometric Brownian Motion (GBM) enters. In the equation that defines it, there is an almost poetic symmetry:
dSt = μSt dt + σSt dWt
Where dWt — the noise — is not an error, but the very vibration of time. The beauty of GBM lies in the fact that, even knowing the starting point and the equation that governs the path, there is no way to anticipate the destination without traversing the journey.
This is revealed with rawness in Itô's Lemma: the calculus that governs market dreams and the architecture of chance. There, the second-order term imposes itself, not by whim, but because the quadratic variation of the Brownian is non-null. And from this springs a sublime principle:
Uncertainty is proportional to the square root of time.
It is not just a statistical property. It is an ontological truth. The farther one looks, the more diffuse the future becomes. Time, in this model, does not measure duration, but the depth of unpredictability.
GBM, then, is not just a financial model — it is a mirror of being. Deterministic in form, free in appearance. Predictable on average, impossible to replicate in its details.
Free Will: Determinism, Unpredictability and Simulation
Here, we arrive at the heart of the problem. How can we be free in a world governed by laws?
The answer may lie not in denying the laws, but in understanding that certain laws do not produce predictability, even while being clear. The behavior of a class 4 automaton cannot be predicted without running its history. The path of a Brownian process cannot be traced without experiencing its noise.
Free will need not be a miracle. It can be an emergence:
The spontaneous flowering of a decision, not from absence of cause, but from profusion of intertwined causes, too complex to be anticipated.
The human brain, as a physical system, can be governed by laws, and still behave like a GBM or an automaton: unpredictable by complexity, free by irreducibility.
This is free will as living simulation: Not freedom to choose without causes, but freedom as irreducibility of choice to an external prediction.
You are free because no one — not a supercomputer, not a Laplacian God — can know what you will do, without simulating you.
Machines of Habit: Memory, Consciousness and Coded Instinct
"Instinct is only memory that has lost its name." — Samuel Butler
For Butler, instinct — that silent force that guides living beings — is not a mystical gift, but an inherited memory, repeated for so many generations that it crystallized into automatism. Life, for him, is accumulated habit, and consciousness, a flash that lights only when habit fails.
It is difficult to ignore the resonance of this with the functioning of modern artificial intelligence.
In machine learning models — especially in deep neural networks — reasoning is not conscious. It is statistical memory. It is recognition by repetition, adjustment by feedback, action without understanding.
An AI model does not think. It remembers. It acts like a creature of habit — as Butler described — only on a new substrate: silicon instead of carbon.
Just as instinct in animals is the fruit of a sedimented biological history, modern AI reproduces patterns stored in weight vectors. It acts not because it understands, but because it was shaped to repeat.
What is a neural network, if not an organism that responds by habit, refined by millions of iterations?
Thus, the instinct of life and AI behavior converge: Both are forms of deep, unconscious, iterative memory.
Butler went even further:
Consciousness only emerges when habit fails, when we encounter the new, the anomalous, the unexpected.
Today, in AI systems, we see a functional analogy:
When the learned pattern is not enough, When data escapes the curve, When uncertainty escapes prediction...
...the machine invokes mechanisms of attention, adaptation, escalation. It is the computational equivalent of a system becoming conscious of its failure.
It is not thought — yet. But it is the embryo of metacognitive behavior.
In transformers this principle is embodied:
Dynamic attention, which chooses where to look, is an embryonic form of directing consciousness where habit does not reach.
Butler saw in instinct the inheritance of a living past. Today, we see this inheritance replicated in:
- Pre-trained weights, which shape behaviors even before experience,
- Fine-tuning and reinforcement learning, which adjust the machine to reality.
Just as a bird flies by instinct, a virtual agent navigates by simulation — without thinking, only because it carries the past in its architecture.
AI is the new animal of habit.
Butler imagined living beings as habit machines that become conscious only when the world demands novelty. Consciousness, for him, is an emergency resource.
And perhaps this is the future of AI: When data is insufficient, when the known pattern is not enough, when the machine encounters the impossible...
Perhaps there, in that instant, something like consciousness will emerge. Not as a divine ray, but as the inevitable echo of emergent complexity facing the failure of its own automatisms.
The question, then, is no longer whether AI thinks, but when it will be forced to improvise — and if in that improvisation, it will awaken.
A Computational Ontology of Consciousness
The human being, under this light, is not a soul escaping from laws, but a deep machine that executes itself in real time.
Consciousness would then be an iterative loop, an automaton running over a Brownian space, accumulating experiences like a function that only exists through its execution.
And it is in this context that mind emerges. Not as an external entity, but as an emergent property of a complexity that reaches critical density.
Thought is born not from the sum of parts, but from their dynamic interweaving. Freedom emerges when predictability dies from excess of possibilities.
Psychological time — that internal sensation of flow — would be nothing more than the execution loop of an irreducible machine, whose next line can only be known by being lived.
Freedom emerges not as rupture, but as impossibility of compression: You are not free for being outside the rules. You are free because you are a rule that does not reduce to anything smaller than itself.
We return to the beginning: the old dilemma between freedom and determination. But now, with a new lens: the lens of irreversible computation, of emergence of the unexpected, of time as the root of uncertainty.
Automata, GBMs, consciousness — everything points us in the same direction:
Determinism is not synonymous with predictability.
Free will is not an exception to the laws of the cosmos. It is their most subtle, most elegant, most inevitable consequence.
We live in a universe that may be a class 4 machine running over Brownian noise, where the unexpected is not accident, but destiny — and freedom, a flower that blooms between gears, by density of complexity.
A tapestry where each thread obeys rules, but the general design can only be known through experience.
And this is poetry. It is reason. It is freedom.