Determinism

A billiard player strikes the cue ball. Given the exact angle, force, spin, friction of the cloth and elasticity of the balls, the rest of the shot seems settled the instant the cue makes contact. A good physicist with perfect measurements could, in principle, write down where every ball will stop.

Now extend that thought to everything: every atom in the room, every neuron in the player's brain, every photon from the lamp above the table. If the whole world works like the billiard table, then the state of the universe at one moment, together with the laws of nature, fixes everything that will ever happen, including the player's decision to take the shot in the first place.

That is determinism. It is one of the oldest ideas in philosophy and one of the most practically important questions in physics. It also raises a question that most people care about more personally: if everything I do was fixed before I was born, in what sense do I choose anything?

Three questions are often blurred together: what determinism claims, whether the physical world is deterministic, and whether determinism would rule out free will. The first has a precise answer, the second is genuinely open, and the third is much less obvious than it first appears.

What It Would Mean

Laplace's demon

The classic image comes from Pierre-Simon Laplace. In 1814, in the introduction to an essay on probability, he imagined an intellect that knew, at one instant, every force acting in nature and the position of every body. If this intellect were vast enough to analyse all that data, Laplace argued, nothing would be uncertain for it: the future and the past would both lie open to its view, like a calculation. ^[1] Later writers called this imagined being Laplace's demon.

Laplace's point was not that such a being exists. It was that probability, for him, measured human ignorance, not any looseness in nature. The world itself, he thought, runs on rails.

Causal determinism

Philosophers now state the idea more carefully. In Carl Hoefer's formulation for the Stanford Encyclopedia of Philosophy, causal determinism is the thesis that the way the world is at one time, together with the laws of nature, fixes a unique way the world can be at all other times. ^[2]

Three features of this definition matter:

• It is about the world, not about knowledge. A deterministic world could be completely unpredictable in practice. Determinism is about whether the future is fixed, not whether anyone can know it.
• It depends on the laws. Whether our world is deterministic is a question about which physical theory is true.
• It is not fatalism. Fatalism says certain outcomes will happen whatever anyone does. Determinism says outcomes happen because of what came before, including what people do. ^[2] A determined decision to study still causes passing the exam.

The Formal Description

Physicists describe a system by its state, a list of numbers such as the positions and velocities of all its parts. The set of all possible states is the state space. A physical law then specifies how the state changes over time. In classical mechanics, this is typically a differential equation of the form

where $\mathbf{x}$ is the state and $\mathbf{F}$ says how fast each part of it changes. A theory is deterministic when every initial state leads to exactly one history. Formally, there is a map $\Phi_t$ that sends the state at time $0$ to the state at any time $t$:

In words: two worlds that agree completely at one moment and obey the same laws agree at every moment. That is the precise content of determinism. Mathematicians have theorems (such as the Picard–Lindelöf theorem) guaranteeing this uniqueness when $\mathbf{F}$ is sufficiently smooth. When it is not, determinism can fail even in classical physics.

Is Classical Physics Deterministic?

It is often said that Newtonian physics is the model of a deterministic theory. That is mostly true in practice, but not strictly true in principle. ^[2]

Norton's dome

In 2008 the philosopher John Norton described a simple case. ^[3] Imagine a frictionless dome shaped so that the height below the summit at a distance $r$ (measured along the surface) is

where $g$ is the acceleration due to gravity. A ball placed exactly at the top, at rest, obeys Newton's second law in the form

One solution is that the ball stays at the top forever: $r(t) = 0$. But another family of solutions also satisfies the same law with the same starting conditions:

for any time $T$. The ball can sit still for an arbitrary time and then spontaneously roll off in any direction, with no cause. Because $r^{1/2}$ is not smooth at $r = 0$, the uniqueness theorem does not apply.

Whether the dome is physically meaningful is debated. Critics argue that it relies on idealizations, such as a perfectly shaped surface and an exactly placed ball, that no real system could satisfy. The example still teaches something important: determinism is a property of specific equations, not a built-in feature of physics. Similar failures appear in Newtonian systems of point particles that can escape to infinity in finite time. ^[2]

Chaos is not indeterminism

The most common confusion about determinism concerns chaos.

In 1963 the meteorologist Edward Lorenz studied a simplified model of atmospheric convection with three variables. ^[4] Restarting a computer run from rounded-off numbers, he found that tiny differences in the starting values grew until the new run bore no resemblance to the old one. The equations were perfectly deterministic, yet long-term prediction was impossible.

This behaviour is called sensitive dependence on initial conditions. In a chaotic system, a small initial error $\delta_0$ grows roughly exponentially:

where $\lambda > 0$ is the Lyapunov exponent, a measure of how quickly nearby trajectories separate. Every improvement of our measurements by a factor of ten buys only a fixed extra amount of prediction time, about $(\ln 10)/\lambda$. ^[5]

Chaos therefore limits prediction, not determination. A chaotic system still has exactly one future for each exact present. It is simply that no finite measurement can pin down the present precisely enough to calculate that future far ahead. Laplace's demon, with infinite precision, would not be troubled by chaos; human forecasters are. ^[5]

This distinction matters for the free-will debate. Unpredictable behaviour, whether of weather or people, is not evidence that determinism is false.

Relativity and Spacetime

Special relativity is deterministic in the relevant sense: the state of fields and particles on a suitable slice of spacetime determines what happens elsewhere. General relativity complicates matters. Some solutions have regions that are not determined by any initial data, or contain closed timelike curves where the idea of "initial state" breaks down. ^[2]

Relativity has also been used to argue for a stronger thesis. In 1966 C. W. Rietdijk argued that, since observers moving relative to one another disagree about which distant events are "now", future events must already be real, and so the future must be fixed. ^[6] That argument belongs to the debate over the block universe. Most philosophers now distinguish the question of whether future events exist from the question of whether they are determined by earlier ones.

Quantum Mechanics: Where the Question Becomes Open

Quantum mechanics is where the deterministic picture meets its most serious challenge, but the challenge is interpretive rather than settled.

A deterministic equation with random outcomes

The basic equation of quantum mechanics, the Schrödinger equation,

is deterministic. Here $\psi$ is the wavefunction, which encodes the state of the system; $\hat{H}$ is the Hamiltonian, which represents its energy; and $\hbar$ is the reduced Planck constant. Given $\psi$ now, the equation fixes $\psi$ at every later time.

The trouble appears with measurement. The wavefunction usually assigns probabilities to several outcomes, and experiments yield one definite outcome at random, with frequencies given by the Born rule: the probability of an outcome is the squared magnitude of the corresponding part of $\psi$. Whether that randomness is fundamental depends on how one interprets the theory.

Three interpretations

Copenhagen and related views. In the family of views associated with Bohr and Heisenberg, measurement outcomes are genuinely indeterministic. The wavefunction is a tool for predicting results, and there is no deeper fact that fixes which result will occur. ^[7] On collapse-based versions, the physical world is fundamentally indeterministic.

Many-worlds. In 1957 Hugh Everett proposed dropping collapse altogether. ^[8] The wavefunction of the whole universe always evolves according to the Schrödinger equation. When a measurement occurs, the universe branches, and every outcome happens in some branch. The theory is fully deterministic at the level of the universal wavefunction; the apparent randomness reflects the fact that an observer cannot know in advance which branch they will find themselves in. ^[9]

Bohmian mechanics. In 1952 David Bohm showed that one can keep the wavefunction and add actual particle positions, guided by the wavefunction along definite trajectories. ^[10] The theory is deterministic: the randomness of outcomes comes from ignorance of the initial particle positions, much as in classical statistical mechanics. The price is that the guidance is non-local: what happens to one particle can depend instantly on distant ones. ^[11]

All three reproduce the same experimental predictions for standard laboratory tests. Physics currently does not tell us whether the world is deterministic.

What Bell's theorem does and does not show

In 1964 John Bell proved that no theory in which outcomes are fixed by local "hidden variables" can reproduce all the predictions of quantum mechanics. ^[12] Experiments since then, including a loophole-free test in 2015 using electron spins separated by 1.3 km, have confirmed the quantum predictions. ^[13]

Bell's theorem is often summarized as "hidden variables are impossible". That is not what it shows. It rules out local deterministic theories. Non-local deterministic theories, such as Bohmian mechanics, remain consistent with every experiment. ^[11] A more radical escape, called superdeterminism, denies that experimenters' choices of measurement settings are independent of the systems measured. It is logically possible but regarded by most physicists as speculative, since it seems to undercut the assumptions behind all experimental science.

The Free Will Problem

Why determinism seems threatening

Most people believe they sometimes act freely, and that free action is required for moral responsibility, praise, and blame. The worry about determinism can be put as a short argument, known in modern form as the consequence argument and associated with Peter van Inwagen: ^[14]

1. If determinism is true, our actions are consequences of the laws of nature and events in the remote past.
2. We have no control over the laws of nature or over what happened before we were born.
3. Therefore, we have no control over the consequences of those things, including our present actions.

Those who accept this argument are incompatibilists: they hold that free will and determinism cannot both be true. Incompatibilists then split into two camps.

• Hard determinists (and the related hard incompatibilists) accept that free will, in the sense needed for deep moral responsibility, does not exist. Some, such as Derk Pereboom, argue that this need not make life meaningless and would justify a less retributive approach to punishment. ^[15]
• Libertarians (in the metaphysical, not political, sense) hold that we do have free will, so determinism must be false, at least for human choices. Some locate freedom in indeterministic processes in the brain; others posit agent causation, in which a person, not merely an event, originates an action. ^[16]

Libertarianism faces a well-known difficulty, often called the luck objection. If a choice is not determined by the agent's prior reasons and character, and instead depends on an undetermined event, it is unclear how that makes the choice more the agent's own. Randomness does not obviously add control. ^[16] This is why quantum indeterminism, even if real, does not automatically rescue free will.

Compatibilism: free will inside a determined world

The other major tradition denies that determinism conflicts with free will at all. Compatibilists argue that the incompatibilist has the wrong idea of freedom. ^[17]

The core compatibilist move goes back to Thomas Hobbes and David Hume. In his Enquiry Concerning Human Understanding (1748), Hume argued that the relevant kind of liberty is simply the power of acting according to the determinations of one's own will: if you choose to stay, you can stay; if you choose to go, you can go. ^[18] Its opposite is not causation but constraint: being imprisoned, coerced, or physically forced. A person whose action flows from their own desires and reasoning acts freely, even if those desires have causes.

Twentieth-century philosophers refined this in two important ways.

Frankfurt's cases (1969). Harry Frankfurt challenged the principle that a person is responsible only if they could have done otherwise. ^[19] Imagine that Jones decides, for his own reasons, to do something. Unknown to him, a neuroscientist has a device ready to force the same decision if Jones shows any sign of wavering. Jones never wavers, and the device never activates. Jones could not have done otherwise, yet he seems fully responsible, because the action came from his own reasoning. If that is right, the ability to do otherwise, which determinism seems to rule out, may not be what responsibility requires.

Frankfurt's hierarchy (1971). Frankfurt also proposed that what makes a will free is its structure. ^[20] Humans have first-order desires (to smoke, to rest) and second-order desires about which first-order desires they want to be moved by (to want to quit). A person has free will when the desires that move them are the ones they endorse on reflection. An addict who wishes not to want the drug lacks this freedom; a person whose actions align with their considered values has it. None of this depends on whether the desires were caused.

Dennett's naturalism. Daniel Dennett argued in Elbow Room (1984) and later works that the free will worth wanting is a capacity of certain evolved systems: the ability to model the future, weigh options, respond to reasons, and avoid anticipated harms. ^[21] A chess program that looks ahead and avoids a trap is deterministic, yet it genuinely "avoids" the trap in a way a rock does not. On this view, determinism does not mean that deliberation is idle. Deliberation is one of the causes through which the future is determined.

Playing with the idea

Compatibilism invites a shift in perspective. The worry "my choice was determined" imagines the causes of the choice as something outside me, pushing me along. But if determinism is true, among the causes of my choice are my own beliefs, values, memories, and reasoning. A determined choice is not necessarily a choice determined against me; it can be a choice determined by me.

Consider the billiard player again. Determinism says the player's decision to take the shot was fixed by prior states. But those prior states include years of practice, a judgement about angles, a desire to win, a moment of concentration. Remove them, and the shot does not happen. On the compatibilist reading, that dependence is exactly what it means for the shot to be the player's own.

Incompatibilists reply that this relocates the problem rather than solving it: the player did not choose the practice, the talent, or the upbringing that produced their values, and ultimately those trace back to events before they existed. ^[14] Whether that regress undermines responsibility is the heart of the modern debate. In a large 2009 survey of professional philosophers, compatibilism was the most common view, accepted or leaned towards by a majority of respondents, but it is not a consensus, and the question remains open. ^[28]

Certainty note. The positions above are philosophical interpretations. No experiment can establish that compatibilism is correct, because the disagreement is partly about what "free" should mean.

The Libet Experiments

In the 1980s, neuroscience appeared to enter the debate directly.

The readiness potential

In 1965, Kornhuber and Deecke discovered that a slow build-up of electrical activity over the brain's motor areas precedes voluntary movements by up to a second or more. ^[22] They called it the Bereitschaftspotential, or readiness potential.

In 1983 Benjamin Libet and colleagues asked participants to flex their wrist whenever they felt the urge, while watching a fast-moving clock dot and noting its position at the moment they first became aware of the intention (the time "W"). ^[23] Averaged over many trials, the readiness potential began several hundred milliseconds before the movement, while W came only about 200 ms before the movement. The brain activity appeared to start roughly 350 ms before people reported deciding.

Many readers took this as evidence that the brain "decides" before the conscious mind, and that conscious will is an after-the-fact story. Libet himself was more cautious: he suggested that consciousness might still "veto" an action in the final fraction of a second. Later fMRI work by Soon and colleagues reported that patterns of brain activity could predict which of two buttons a person would press several seconds before their reported decision, though only at modestly above chance accuracy (around 60%). ^[24]

Reinterpretation: Schurger's accumulator model

In 2012, Aaron Schurger, Jacobo Sitt, and Stanislas Dehaene proposed a different explanation of the readiness potential. ^[25] They modelled spontaneous movement as a process in which neural activity drifts and fluctuates randomly, gradually accumulating, until it crosses a threshold that triggers action. Because experimenters average brain activity time-locked to the movement, the average of many random fluctuations that happened to cross the threshold looks like a slow, steady ramp. The ramp, in other words, may be partly an artifact of averaging, not the signature of a decision made long in advance.

On this model, the "decision" to move, in the sense of the threshold being crossed, happens shortly before the movement, not hundreds of milliseconds earlier. The model also made a testable prediction: if participants are interrupted at random moments with a demand to move immediately, faster responses should be preceded by the same kind of negativity. The authors reported data consistent with this. ^[25]

Later work strengthened the case against the strong reading. A 2019 study found that the readiness potential, present when people made arbitrary, meaningless choices, was absent or much reduced when they made deliberate choices that mattered to them, such as which charity would receive a donation. ^[26] A 2021 review by Schurger and colleagues concluded that the readiness potential is better understood as reflecting the build-up of background neural activity than a committed, unconscious decision to act. ^[27]

What the experiments do and do not show

• Established: measurable brain activity precedes spontaneous movements, and precedes the reported moment of conscious intention.
• Contested: whether that activity represents a decision, or just fluctuations that make movement more likely.
• Limited scope: the tasks involve arbitrary, trivial movements with no reasons behind them, which is almost the opposite of the reasoned choices that the free-will debate is about. Reports of the timing of an inner intention are also known to be imprecise.
• Not shown: that determinism is true, or that free will in any philosophical sense is an illusion. Even a complete neural explanation of a choice would be exactly what compatibilists expect; they locate free will in the kind of process that produces the action, not in the absence of a physical cause.

Limitations and Open Problems

We do not know whether the world is deterministic. Classical physics is mostly deterministic but has exceptions. Quantum mechanics has deterministic and indeterministic interpretations that agree on all current experiments. ^[2,11] A future theory of quantum gravity might settle the question, or change it.

Determinism is not predictability. Chaos, measurement limits, and the sheer complexity of brains make detailed prediction of human behaviour impossible in practice, whatever the underlying laws.

The free-will question is partly conceptual. Whether determinism threatens free will depends on what free will is. That is a question physics cannot settle alone.

Neuroscience has not delivered a verdict. The Libet-style findings are real, but their interpretation has shifted substantially, and they concern a narrow class of actions.

Why the Question Persists

Determinism matters because it sits at the junction of physics and self-understanding. In physics, it forces precision about what a theory really says: whether its equations have unique solutions, whether its randomness is fundamental, whether unpredictability reflects ignorance or nature. The distinction between chaos and indeterminism, and the fact that quantum mechanics can be read either way, are among the clearest examples of how the same equations can support very different pictures of reality.

In human terms, it matters because our practices of praise, blame, punishment, and self-improvement rest on assumptions about agency. If compatibilists are right, those practices are consistent with a law-governed universe, and the task is to understand which kinds of causal histories make an action one's own. If incompatibilists are right, then either physics must leave room for something beyond determined events, or our ideas of responsibility need rethinking.

Many physical laws are deterministic, others are formulated probabilistically, and unpredictability alone proves neither case. Current theories accommodate a fully determined universe, a fundamentally random one, or one whose description changes character between levels. Physics has not decided which picture is ours, much less whether the answer changes what it means to choose.

References

[1] Laplace, P.-S. (1814; English translation 1951). A Philosophical Essay on Probabilities (trans. F. W. Truscott & F. L. Emory). New York: Dover Publications. https://archive.org/details/philosophicaless00lapl

[2] Hoefer, C. “Causal Determinism.” Stanford Encyclopedia of Philosophy (first published 2003; substantively revised 2023). https://plato.stanford.edu/entries/determinism-causal/

[3] Norton, J. D. (2008). “The Dome: An Unexpectedly Simple Failure of Determinism.” Philosophy of Science, 75(5), 786–798. https://doi.org/10.1086/594524

[4] Lorenz, E. N. (1963). “Deterministic Nonperiodic Flow.” Journal of the Atmospheric Sciences, 20(2), 130–141. https://doi.org/10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2

[5] Bishop, R. “Chaos.” Stanford Encyclopedia of Philosophy (first published 2008; substantively revised 2024). https://plato.stanford.edu/entries/chaos/

[6] Rietdijk, C. W. (1966). “A Rigorous Proof of Determinism Derived from the Special Theory of Relativity.” Philosophy of Science, 33(4), 341–344. https://doi.org/10.1086/288106

[7] Faye, J. “Copenhagen Interpretation of Quantum Mechanics.” Stanford Encyclopedia of Philosophy (first published 2002; substantively revised 2024). https://plato.stanford.edu/entries/qm-copenhagen/

[8] Everett, H. (1957). “‘Relative State’ Formulation of Quantum Mechanics.” Reviews of Modern Physics, 29(3), 454–462. https://doi.org/10.1103/RevModPhys.29.454

[9] Vaidman, L. “Many-Worlds Interpretation of Quantum Mechanics.” Stanford Encyclopedia of Philosophy (first published 2002; substantively revised 2026). https://plato.stanford.edu/entries/qm-manyworlds/

[10] Bohm, D. (1952). “A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables. I.” Physical Review, 85(2), 166–179. https://doi.org/10.1103/PhysRev.85.166

[11] Goldstein, S. “Bohmian Mechanics.” Stanford Encyclopedia of Philosophy (first published 2001; substantively revised 2025). https://plato.stanford.edu/entries/qm-bohm/

[12] Bell, J. S. (1964). “On the Einstein Podolsky Rosen Paradox.” Physics Physique Fizika, 1(3), 195–200. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[13] Hensen, B., et al. (2015). “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres.” Nature, 526, 682–686. https://doi.org/10.1038/nature15759

[14] Vihvelin, K. “Arguments for Incompatibilism.” Stanford Encyclopedia of Philosophy (first published 2003; substantively revised 2022). https://plato.stanford.edu/entries/incompatibilism-arguments/

[15] O’Connor, T., & Franklin, C. “Free Will.” Stanford Encyclopedia of Philosophy (first published 2002; substantively revised 2022). https://plato.stanford.edu/entries/freewill/

[16] Clarke, R., Capes, J., & Swenson, P. “Incompatibilist (Nondeterministic) Theories of Free Will.” Stanford Encyclopedia of Philosophy (first published 2000; substantively revised 2021). https://plato.stanford.edu/entries/incompatibilism-theories/

[17] McKenna, M., & Coates, D. J. “Compatibilism.” Stanford Encyclopedia of Philosophy (first published 2004; substantively revised 2024). https://plato.stanford.edu/entries/compatibilism/

[18] Hume, D. (1748). An Enquiry Concerning Human Understanding, Section VIII, “Of Liberty and Necessity.” Project Gutenberg edition. https://www.gutenberg.org/ebooks/9662

[19] Frankfurt, H. G. (1969). “Alternate Possibilities and Moral Responsibility.” The Journal of Philosophy, 66(23), 829–839. https://doi.org/10.2307/2023833

[20] Frankfurt, H. G. (1971). “Freedom of the Will and the Concept of a Person.” The Journal of Philosophy, 68(1), 5–20. https://doi.org/10.2307/2024717

[21] Dennett, D. C. (1984; new edition 2015). Elbow Room: The Varieties of Free Will Worth Wanting. Cambridge, MA: MIT Press. https://mitpress.mit.edu/9780262527798/elbow-room/

[22] Kornhuber, H. H., & Deecke, L. (1965). “Hirnpotentialänderungen bei Willkürbewegungen und passiven Bewegungen des Menschen: Bereitschaftspotential und reafferente Potentiale.” Pflügers Archiv, 284, 1–17. https://doi.org/10.1007/BF00412364

[23] Libet, B., Gleason, C. A., Wright, E. W., & Pearl, D. K. (1983). “Time of Conscious Intention to Act in Relation to Onset of Cerebral Activity (Readiness-Potential).” Brain, 106(3), 623–642. https://doi.org/10.1093/brain/106.3.623

[24] Soon, C. S., Brass, M., Heinze, H.-J., & Haynes, J.-D. (2008). “Unconscious determinants of free decisions in the human brain.” Nature Neuroscience, 11(5), 543–545. https://doi.org/10.1038/nn.2112

[25] Schurger, A., Sitt, J. D., & Dehaene, S. (2012). “An accumulator model for spontaneous neural activity prior to self-initiated movement.” Proceedings of the National Academy of Sciences, 109(42), E2904–E2913. https://doi.org/10.1073/pnas.1210467109

[26] Maoz, U., Yaffe, G., Koch, C., & Mudrik, L. (2019). “Neural precursors of decisions that matter—an ERP study of deliberate and arbitrary choice.” eLife, 8, e39787. https://doi.org/10.7554/eLife.39787

[27] Schurger, A., Hu, P., Pak, J., & Roskies, A. L. (2021). “What Is the Readiness Potential?” Trends in Cognitive Sciences, 25(7), 558–570. https://doi.org/10.1016/j.tics.2021.04.001

[28] Bourget, D., & Chalmers, D. J. (2014). “What do philosophers believe?” Philosophical Studies, 170(3), 465–500. https://doi.org/10.1007/s11098-013-0259-7