---
title: "Free Will and Determinism"
description: "Reading Robert Sapolsky's Determined, I reflect on free will, causality, and the resolution of the world we understand."
date: "2026-10-24T09:32:00.000Z"
categories: ["essay"]
source_url: "https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/"
---

# Free Will and Determinism

I recently read Robert Sapolsky's *Determined*. When I saw the headline questioning the existence of free will, it sounded close to what I had been thinking, so I picked it up right away.

What is free will? Before thinking about that, I want to consider the world we are part of.
Our world is built on causality. If an event or phenomenon occurs, there must be a cause that brought it about, and that cause must have another cause of its own.
If we trace those causal relationships endlessly back, where do they converge?

## Divided into Ticks

Here, I imagine one possibility.
Even for phenomena we describe probabilistically in quantum mechanics, could there be more fundamental states and causal relationships underneath that we do not yet know?
The probabilities we observe might arise not from fundamental randomness in the world itself, but from our inability, as part of that world, to fully know its underlying state.

Assuming such a world, let's divide the world we exist in into tiny units of time—ticks, like in a game.
The state at the next tick follows from the state at the current one. Without the previous tick, there would be no next tick.

Of course, this isn't a scientific claim that the actual universe is made of discrete ticks. Still, I think it is a valid analogy for understanding causality. Let's assume that tracing those ticks all the way back converges at the initial point of the Big Bang.
If all variables and the transition rules of this giant state machine are given, the next tick would also be uniquely determined. What I mean here is not that we can actually calculate the future,
but that the next state is theoretically fixed. The tick after that would follow in the same way, and what we call the future could be seen as already determined in a chain.
If these underlying states and transitions are deterministic, then no matter how probabilistic the world we observe appears, its future is already determined at an ontological level.

Humans are nothing special—just another phenomenon of the universe—so we cannot defy the rules of this world.

What causal history has made us who we are?
Looking at ourselves simply as individual objects, what makes us who we are right now—the biological conditions given at birth, the experiences we go through while growing up, learning and socialization, and the connections between neurons that have changed along the way—is all part of a causal chain continuing from previous states.

## The Black Box of Consciousness

Then what is human consciousness?

We often think of consciousness as something noble and special. Perhaps it only looks like magic because we haven't yet understood its concrete mechanisms and operations well enough.

Viewed at a biological and chemical level, the brain is one giant biological neural network where countless neurons are intricately connected. Experiences and stimuli gradually change the strengths of connections between neurons—their synaptic strengths—and how those neurons respond. Those changes, in turn, affect our next actions and thoughts.
Of course, we cannot equate it simply with the weights of an artificial neural network, but it is similar in that the results of learning are reflected in physical structure and state.

We experience the output of this system as a conscious choice. Yet we still cannot consciously observe all the computations leading up to that choice.
Perhaps we believe free will lives inside that black box.

In the Appendix to Part I of his *Ethics*, Spinoza explains that people regard themselves as free because they are conscious of their desires but unaware of the causes that determine those desires.[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)
Yet opacity is only part of the explanation. It does not, by itself, explain free will. Something more is needed to account for the feeling of free will.
Let's try a thought experiment.

## A Duplicated Me

Suppose there are hundreds of automata with the exact same structure and state.
If they receive the same input in the same environment, they will all produce the same output. No matter how many times we repeat it, the result will be the same.

Let's replace those automata with humans.
This time, suppose the universe itself is duplicated in a completely identical state.
Not only are the observable environment and quantum states the same, but even the underlying ontological states we do not yet know are identical.
If the same causal rules operate in that state, the actions and thoughts produced by the humans in each universe will also be the same.

Then can we say those automata, or those humans, act with free will?
If making a decision itself counts as free will, then perhaps we can. But if they all display identical thoughts and actions, where in that choice is a will uniquely my own?

Free will is supposed to originate from me, yet in that environment there is nothing that distinguishes "me" from the other individuals. What meaning does "I" have there?

In such an environment, I'm not sure it even makes sense to discuss free will. Since everyone produces the same result, there is no uniqueness. Here, my will is also another's will. The action belongs not to me as an individual, but to the type of being I am. Does free will, then, require an awareness of uniqueness to exist?

This time, suppose these automata spend a long time responding to stimuli, learning, and changing in different environments. Since the stimuli and experiences each one faces are different, their internal states will gradually diverge. After passing through countless ticks like that, they will become systems that produce different outputs even for the same input.

What matters here is that this difference was not created non-deterministically.
Why they ended up in different environments, what experiences they had in those environments, and how their internal structures changed as a result can all be traced back through the causal chain of previous ticks.
Uniqueness itself is a deterministic outcome.

What we observe in reality are individuals producing different outputs. However, it is hard to know what causal history those individuals went through to reach their current state.
Isn't it because it is so complex, and because different results appear, that we accept them as each person's independent choice?

To an observer, these varied outputs can be represented as a probability distribution. But that does not mean each output was chosen without a cause.

Of course, this isn't to say that uniqueness is free will. Just because a unique result appears doesn't mean we necessarily perceive it as acting freely.
I just think that part of what we feel as free will might stem from this deterministically created uniqueness and the process of recognizing those differences.

Uniqueness is not a property an individual possesses in isolation; it is a difference from another individual—a delta. If the copies are completely identical, the delta is zero, and as their histories diverge, the delta grows.
In reality, no two people have exactly the same history, so there is no case where the delta is zero. The duplication experiment is only a limiting case that shows what happens as the delta approaches zero.

Understanding the self through its relationships with others also brings to mind G. W. F. Hegel's account of recognition (*Anerkennung*). Hegel explains self-consciousness through its relation to another self-consciousness that recognizes it.[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/) This is not the same concept as the differences between individuals I am discussing here, but I see a connection in the difficulty of explaining the self as an isolated individual.

The conditions for freedom do not stand in opposition to determinism; they are its product.

## The Word "Freedom"

Then what do we mean by freedom here in the first place?
Is it complete randomness? But how does a randomly generated result become a free choice?

If we define freedom as a choice independent of causes, it is a concept that can hardly hold in a deterministic world. (Because that would be like magic that cannot be explained by the rules of the world!)
On the other hand, if we define freedom as acting according to one's own judgment and intention, that does not necessarily contradict the fact that the intention itself is also a causal outcome.

In the end, it becomes difficult even to define what we call freedom.
If we know that everything is actually determined and we are only walking that path, can we judge ourselves to be free?

Of course, whether the world's future is determined and whether we can calculate it are separate questions. We ourselves are part of that giant state machine, with no view from outside it that would let us observe all its underlying states. What we can observe and understand is always limited, and within those limits, the future can still be described probabilistically.
Even so, that probability does not necessarily mean the world is fundamentally non-deterministic.

Then why do humans want to believe they have free will?
When even defining free will is not easy, why do we wish for it to be a part of us?

Is it out of a desire to have our uniqueness guaranteed? Perhaps at the base lies a biological drive to be an entity distinct from others and to pass that uniqueness on to future generations.

Or perhaps resistance to oppression is expressed as a desire for freedom.
Humans learn common norms and behavioral patterns through socialization, but each person's causal history is different. In situations where unique desires formed that way clash with the demands of society, we might become more strongly conscious of the concept of freedom.

Might the brain's computational limits also play a role in our instinctive attachment to the concept of free will?

To understand the cause of an action by tracing every previous tick would be biologically far too costly. So our brains follow the causal chain only so far, then rely on the internal models we have learned to make inferences.

We share a foundation learned through socialization, but each of us builds on it through our own experiences. We use those models to estimate the world probabilistically, much as an autoregressive Transformer model predicts the next token from the context given so far, rather than simulating all the physical laws of the world.

Our brains look for patterns. Because we cannot observe all the ways the complex neural network inside us produces its next output, we attach the name "free will" to that ambiguous black box; because we cannot know all of another person's history, we attach labels of "good and evil" to the actions right in front of us.
In effect, we process complex causal relationships through a kind of lossy compression.

Of course, this too is only speculation. The origins of such psychological and cognitive mechanisms belong to evolutionary biology and neuroscience, so it is hard to jump to conclusions. Perhaps later, when we understand the concrete logic of how the human brain operates much more deeply than we do now, we will be able to explain what mechanism gives rise to the longing for free will.

## Deterministic Resolution

What meaning is there in my claiming that free will cannot exist in a deterministic world? I don't think it is a particularly helpful claim.
So rather than jumping to conclusions about whether free will exists, I am more curious about why we created such a concept and came to long for it.

Then in a world where everything is deterministic, what does it mean for us to deliberate and make judgments?

Being deterministic means being explainable, not being dragged through life by something else.
Our thoughts already occupy a layer above that underlying process, so they are both products of causality and participants in it. What I am thinking now is, of course, determined by the past. Yet from my position as someone living through this, I cannot step outside it and take a view from beyond the system. As a participant, I keep creating further causes.

Maybe free will felt complicated because I was thinking about it across different layers at once.

What does living in a deterministic world mean for the part we play in what happens next? Is my desire for free will still meaningful? What meaning should I give it?

What matters to me, then, is increasing what I call our "deterministic resolution": the level of detail at which we understand the causal processes shaping our lives.
In an earlier post, [Cognitive Prison](https://yongseok.me/blog/en/congitive_prison/), I talked about how, because we cannot understand all the variables in the world, each of us interprets the world through an internal physics engine built from our own limited samples of the world.

As I mentioned above, there is a limit to the variables we can account for and the computations we can perform. I doubt we are making full use of even that limited capacity. As I touched on in another short piece, we want to build an explainable world. For me, that means understanding the causal relationships we can identify in greater detail. Expanding what we can make sense of, and seeing and reasoning more clearly about the conditions given to me—and to us—isn't that part of what we can do in this deterministic world?

Alongside judging what is right or wrong, shouldn't I ask how much of the world I understand when I make that judgment?
How much more of the past ticks can we understand? What variables were compressed and omitted, and what causal mechanisms produced the current state?

To be carried along without knowing the causes is bondage; to understand those causes adequately is freedom, even within the same necessity. Freedom is a matter of degree, and understanding causes raises that degree. I call that resolution.

## In Closing

Thinking this far, it also feels a bit hollow.
The claim that everything in a deterministic world is a causal outcome feels, in a way, all too obvious. And that fact alone does not produce any concrete message about what we ought to do.

Maybe I am laying out this logic just to show off my own unique thoughts and sound a little smart.
Even that desire would be the result of my experiences and learning up to now.
In a way, it's a funny story. By the logic of this essay, that desire is itself a wish to have my uniqueness recognized—and that uniqueness is a product of determinism.

Still, I don't think these reflections only have meaning if they reach some final conclusion.

The world is complex, and the world we understand is always limited. So rather than being certain that our current judgment is the complete answer, don't we need the effort to understand more context?
That could be someone else's actions, a social phenomenon, or my own thoughts and judgments.

As the resolution of our understanding increases, causal relationships that were previously invisible will come into view. And when we understand new causal relationships, the judgments we made before may change as well.

Of course, raising our resolution will not give us the answers to every problem.
But if we become able to understand a wider context, couldn't we at least ask different questions than before?

Asking questions, talking, and revising our judgments like that will also be part of what constitutes the next tick of the world.

In the end, what I find important is not reaching some completed truth, but the ongoing process of raising the resolution of the world I understand.
Because we are already living inside it.

Perhaps what we need is simply to keep talking and to keep practicing.

## Before the Addendum

This essay began by assuming that the world is ontologically deterministic. Modern physics does not guarantee that assumption. Quantum mechanics describes measurement outcomes in terms of probabilities, and its predictions are experimentally very accurate. But whether a probabilistic description means the world is fundamentally non-deterministic is a separate question. The addendum below follows interpretations that held on to that question without giving up determinism—de Broglie–Bohm theory, 't Hooft's cellular automaton interpretation, and Palmer's invariant set theory—and considers what they gave up and what they retained in the face of Bell's theorem. It also revisits which of these ideas comes closest to the ticks and state machine imagined in the main text.

## Addendum: Automata Universe and Deterministic Quantum Interpretations

In the mainstream Copenhagen interpretation of quantum mechanics, individual measurement outcomes following wave function collapse are interpreted as fundamentally probabilistic. However, physicists like Einstein, who found it hard to accept such "effects without causes," proposed alternative interpretations suggesting that deterministic causality might still operate beneath quantum phenomena.

Einstein famously expressed his view on the probabilistic interpretation of quantum mechanics by saying, "God does not play dice."

What mattered to him was not only determinism, but also the principle of locality—the idea that when two physical systems are far apart, what happens on one side cannot influence the physical reality of the other faster than light.

In 1935, Einstein, together with Boris Podolsky and Nathan Rosen, published the so-called [EPR paper](https://doi.org/10.1103/PhysRev.47.777)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/).
The problem raised in this paper is related to what we today call quantum entanglement.
The original paper dealt with position and momentum, but it is easier to think about using spin.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

Suppose two entangled particles are far apart. If the total spin of the two particles is 0, measuring their spins along the same direction will yield spin-up on one side and spin-down on the other.
Each measurement outcome is probabilistic, but once you know the outcome on one side, you can predict the outcome on the other side with certainty.

Yet if the two particles are sufficiently far apart, it is hard to think that a measurement on one side instantaneously changes the physical state on the other.
Then didn't the other particle already possess some physical property that determined its outcome even before it was observed?

This is the starting point of the EPR argument. Einstein thought that the wave function of quantum mechanics is not a complete description of nature, and that there could be physical information we have not yet grasped.

This approach later developed into the question of local hidden variables:
the idea that if particles carried information determining their outcomes from the start and did not influence each other once separated, quantum entanglement could be explained.

However, there was a critical problem with this idea.

In 1964, physicist John Stewart Bell showed through [Bell's theorem](https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/) that there is a mathematical limit to the correlations that any theory satisfying conditions like local causality and statistical independence of measurement settings can produce.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

Simply getting opposite results along the same direction is not a problem; one could still think the particles carried pre-determined information.

The problem lies in the correlations that appear when the two particles are measured along different directions. When we compare outcomes obtained across various combinations of measurement directions, a limit emerges on the correlations that local hidden variables can explain.
In the representative CHSH inequality, the upper bound for such local models is 2, whereas quantum mechanics predicts up to about 2.83 under specific conditions.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

And in actual experiments, violations of Bell's inequality have been repeatedly observed. In 2015, a [Bell test](https://doi.org/10.1038/nature15759)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/) closing major experimental loopholes was conducted, and this line of research led to the [2022 Nobel Prize in Physics](https://www.nobelprize.org/prizes/physics/2022/press-release/)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/).

Ultimately, the approach Einstein had hoped for—explaining all correlations of quantum phenomena solely through local information carried by particles in advance—became difficult to reconcile with experimental results.

That does not mean determinism itself was disproven.
Bell's argument involves multiple assumptions together. Depending on which assumptions you keep and which you give up, different physical interpretations remain possible.

Nor does violating Bell's inequality mean information can be transmitted faster than light. Local causality at issue here and the possibility of signal transmission in relativity need to be distinguished.

### Interpretations That Keep Determinism

A representative example is the [de Broglie–Bohm theory](https://plato.stanford.edu/entries/qm-bohm/)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/).
Proposed by Louis de Broglie in 1927 and developed by David Bohm in 1952, this theory holds that particles have actual positions even before measurement, and the wave function deterministically guides their motion.

In standard quantum mechanics, the wave function is used to calculate the probability of finding a particle at each position. In de Broglie–Bohm theory, by contrast, the actual position of the particle exists as a variable alongside the wave function.

The wave function evolves according to the Schrödinger equation, and the particle's position evolves according to a separate guiding equation. Therefore, if the wave function and the initial positions of all particles are given, their subsequent motion is also determined.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

In the double-slit experiment as well, each particle actually passes through one of the two slits, but because the wave function guides the particle's path, collecting the arrival positions of many particles produces an interference pattern.
The probabilities we observe are explained as arising because we do not know the exact initial positions of these particles.

However, this theory does not preserve the locality that Einstein considered so important.

Earlier, the EPR argument supposed that distant particles do not instantaneously influence each other and that measurement outcomes could be determined by physical properties each particle carried on its own.

In de Broglie–Bohm theory, however, entangled particles are described by a single joint wave function. This wave function does not describe each particle independently, but takes the positions of multiple particles together as variables.

Because of this, the motion of one particle can depend on the actual position of another particle far away. Furthermore, the setting of a measurement device on one side can non-locally participate in the process that determines the motion of the particle on the other side.
In other words, while the outcomes of the particles are determined, the process of determination does not depend solely on each particle's immediate surroundings.

Of course, this does not mean we can use such non-local dependence to send desired information faster than light.
Ultimately, de Broglie–Bohm theory is an approach that keeps determinism at the cost of giving up the requirement of local causality.

There is also a slightly different approach.

Gerard 't Hooft, the 1999 Nobel Laureate in Physics, proposed the [Cellular Automaton Interpretation of Quantum Mechanics](https://arxiv.org/abs/1405.1548)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/).
This approach explores the possibility that beneath the world we describe with quantum mechanics lie much more fundamental deterministic states.

't Hooft starts from the point that even a deterministic system can be expressed in the mathematical language of quantum mechanics.

For example, consider an automaton with several distinguishable states that transition to the next state according to fixed rules.
If we represent each state as a basis vector in a Hilbert space, the state transition can be expressed as a permutation operator that reorders the basis vectors. Because such a permutation operator is also a type of unitary operator, it can be handled using the mathematical formalism of the Schrödinger equation.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

That is, the actual system has a single definite state at every moment and transitions deterministically, yet we can still use the mathematical tools of quantum mechanics to describe it.

't Hooft calls these underlying actual states ontological states, and interprets the superposition states we use in quantum mechanics as template states used to describe them.

Then does the wave function we use necessarily represent the fundamental reality of the world as it is? Or could it be a mathematical representation we use to handle underlying deterministic states?

What is particularly interesting is that he explores this possibility through a cellular automaton where discrete states transition according to rules.
It reminds me of the tick and state-machine structure I imagined in the main text.

Of course, the fact that a deterministic automaton can be expressed in the mathematical formalism of quantum mechanics is different from the claim that the actual universe is such an automaton.

't Hooft himself does not claim that just any automaton can explain real quantum phenomena. The challenge remains to find a fundamental system capable of reproducing the particles, interactions, and relativistic symmetries we know.
Nor has it been proven that the actual universe is a giant automaton updating every Planck time.

Even so, the approach of describing a deterministic system in the language of quantum mechanics was fascinating in itself. While it is not the exact same concept as the lossy compression I discussed in the main text, it prompts similar thoughts in that it distinguishes between the model we use and the underlying state of the world.

### Superdeterminism and Invariant Set Theory

Another approach is superdeterminism.

Deriving Bell's inequality relies on the assumption of measurement independence—that there is statistical independence between the measurement direction chosen by the experimenter and the hidden variables of the particle. Superdeterminism questions this assumption. Not only the particle being measured, but also the experimental apparatus and the observer's choice belong to the causal history of the same universe; what guarantee is there that the two variables are completely independent?

Of course, sharing a common past does not automatically break statistical independence. To explain actual quantum phenomena, one must provide a concrete mechanism for why such correlations arise. Because of this, superdeterminism faces criticism that the required correlations amount to overly contrived "fine-tuning" or are impossible to test, and it is not widely accepted in mainstream physics.

In response, Sabine Hossenfelder and Tim Palmer argued in a 2020 [paper](https://arxiv.org/abs/1912.06462)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/) that such criticisms do not rule out superdeterminism as a whole and that testability can be explored through concrete models—a representative example of which is Invariant Set Theory, developed by Palmer.

In [The Invariant Set Postulate](https://arxiv.org/abs/0812.1148)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/), published in 2009, Palmer starts not by asking how states change according to equations as in conventional dynamics, but by geometrically defining which states in the universe's state space are physically possible in the first place.

In chaos theory, the states of a nonlinear system following deterministic rules sometimes converge over time into a specific region and trace trajectories only within it. This is called an attractor, and among them are "strange attractors" with fractal structures whose self-similar patterns repeat even when magnified.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

Palmer proposed the hypothesis that the state space of the entire universe is likewise not a continuous space where every combination of variables can be arbitrarily realized, but lies on a fractal invariant set within which the system remains as time passes.

The nature of a fractal structure is easy to grasp by thinking of the Cantor set, made by endlessly removing the middle third of a line segment. Infinitely many points remain, but at the same time, countless gaps—regions outside the set—exist everywhere.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

Therefore, even if two states appear very close to each other in state space, one may lie on the invariant set while the other falls into a gap. On the surface, only a single variable has changed slightly, yet one is a physically permitted state while the other is a state that cannot be realized in the universe.

This geometric constraint undermines the counterfactual assumption presupposed by Bell's inequality. We easily imagine, "What if the hidden state of the particle had been kept the same, and I had only set the angle of the measurement apparatus differently back then?" In Invariant Set Theory, however, a combination where only the measurement setting variable is arbitrarily altered from an actually occurring state falls into a fractal gap (outside the invariant set), making it a state that can be imagined logically but cannot be reached physically.

Yet a natural question follows: if changing the measurement angle ever so slightly renders a state physically impossible, isn't that the very kind of contrived "fine-tuning" superdeterminism was criticized for?

To resolve this, Palmer introduced the p-adic metric in a [2015 study](https://arxiv.org/abs/1507.02117)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/) to redefine distance on a fractal structure. In the p-adic metric, which defines the difference between two numbers by the degree to which it is divisible by powers of a specific prime, two states that look very close in ordinary Euclidean distance are actually very far apart if one of them lies outside the invariant set. On Palmer's account, this is not fine-tuning, but a consequence of the geometry he proposes for the underlying state space: a state that seems nearby in ordinary distance may be far away under that geometry.

Furthermore, in a [2020 study](https://doi.org/10.1098/rspa.2019.0350)[Citation (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/), he applied this abstract geometric model directly to the Bloch sphere, which represents spin states in quantum mechanics. By discretizing angles on the sphere into finite bit strings and connecting them to trajectories on a fractal invariant set, he developed a concrete model that mathematically derives the violation of Bell's inequality while preserving both locality and determinism.

[ConceptNote (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)

Invariant Set Theory is not identical to the tick-based state machine I imagined in the main text, in that it focuses on geometric constraints of the universe's state space rather than the discretization of time.

Still, its view that state transitions in the universe do not move freely among arbitrary possibilities, but occur deterministically only along trajectories permitted by underlying rules, resonates strongly.

Of course, whether the fractal structure Palmer assumes actually exists in the universe, and whether experimental evidence distinguishing it from standard quantum mechanics can be found through constraints arising from finite state representations, remains an open question.

Even so, simply encountering a pattern of thought that reframes "the possibility of choosing otherwise"—long treated as the basis of free will—as a geometric constraint of state space was fascinating enough.

### Back to Deterministic Resolution

What I came to think about again while looking into these theories is that asking whether the world is deterministic and asking how much of that world we can understand are two different questions.
The former is an ontological question about how the world actually exists, while the latter is an epistemological question about how far we can observe and predict that world.

Even if a deterministic foundation exists, that does not mean an observer can know all of its underlying states.
Conversely, the mere fact that we explain a phenomenon probabilistically does not allow us to conclude that probability is necessarily an ultimate property of the world.

Nor is it proven that probability is simply the result of compressing variables we do not know. At the very least, current quantum mechanics and experiments do not settle that conclusion for us.

### References

[References (interactive original)](https://yongseok.me/blog/en/%EC%9E%90%EC%9C%A0%EC%9D%98%EC%A7%80%EC%99%80-%EA%B2%B0%EC%A0%95%EB%A1%A0/)