The previous lesson stated determinism precisely enough to ask whether it is true, and the honest answer from physics is that nobody knows, for reasons more interesting than a shrug.
This lesson is the one place in the course where the subject matter is empirical. It is worth doing carefully, because the two most common moves in popular discussion, that classical physics proved determinism and that quantum physics refuted it, are both wrong, and a reader who knows why is protected against a great deal of nonsense in both directions.
Classical mechanics, and the cracks in it
Newton's laws are second-order differential equations: give the positions and velocities of every particle at one instant, and the equations return the trajectory forwards and backwards in time. That is determinism in the sense of the previous lesson, and it is why Laplace could write his sentence in 1814 with confidence.
The confidence was slightly too high, and the exceptions are worth knowing because they show how delicate the property is. In 2003 John Norton described a dome, shaped so that a particle resting exactly at its apex is in equilibrium, and yet Newton's laws permit the particle to remain there for any length of time and then spontaneously slide off in any direction, with no force applied and no violation of the equations. The dome exploits the fact that the standard existence-and-uniqueness theorem for differential equations requires a smoothness condition that the surface fails at exactly one point.
Worse, in 1992 Zhihong Xia proved that a system of five point masses under Newtonian gravity can send a particle to spatial infinity in finite time, which by time-reversal means that a particle can arrive from infinity at an unpredictable moment. Nothing in the initial state announces it. Classical mechanics, taken literally, is therefore not quite deterministic, and it is saved only by conditions on the shape of surfaces and the finiteness of forces that are physically reasonable and not part of the laws themselves.
None of this matters for human beings and none of it delivers free will. It matters as a warning: determinism is not something a theory wears on its face, and reading it off a set of equations takes more care than it looks.
The two dynamics of quantum mechanics
Quantum mechanics is famously said to have destroyed determinism, and the truth is more specific. The theory has two rules for how states change, and they are not alike.
The first is the Schrödinger equation, which governs an isolated system's evolution. It is deterministic, linear, and time-reversible: the state now plus the equation gives the state at any other time, exactly as Newton's laws do. Nothing in this half of the theory threatens Laplace.
The second is the measurement rule. When a measurement occurs, the outcome is one of the eigenvalues of the measured quantity, with probabilities given by Born's rule of 1926, and the state jumps to match. This half is irreducibly probabilistic. A single silver atom in a Stern-Gerlach apparatus goes up or down, the theory gives only the probabilities, and nothing in the prior state distinguishes the atoms that will go up from those that will go down.
The awkwardness is that the theory does not say what a measurement is. Measuring devices are made of atoms and should obey the first rule, so the second rule looks like an extra postulate applied by hand when a system gets large or complicated. That is the measurement problem, and it is not a puzzle at the fringes: whether the world is deterministic turns entirely on how it is solved.
What Bell's theorem does and does not rule out
The obvious repair is to say that the probabilities reflect our ignorance, exactly as Laplace said of probability generally. If each silver atom carried a hidden property fixing which way it goes, quantum mechanics would be an incomplete statistical description of a determined world. Einstein, Podolsky and Rosen argued in 1935 that the theory is incomplete in something like this way.
In 1964 John Bell showed that the repair has a testable consequence. If the outcomes of measurements on two separated particles are fixed by properties carried locally by each particle, then correlations between the results must satisfy an inequality. Quantum mechanics predicts violations of it. This is the rare case where a metaphysical question about the completeness of a theory has an experimental answer.
The experiments have been done, and done to exhaustion. Alain Aspect's group produced the first convincing violations in 1982. Loophole-free versions arrived in 2015: a group at Delft, led by Ronald Hanson, ran the test with entangled electron spins in diamond separated by 1.3 km, far enough that no signal at light speed could connect the measurement choices to the distant outcomes, and reported a violation with a p-value of 0.039 on 245 trials, while photonic experiments at NIST and in Vienna reported violations with vastly smaller p-values the same year. In 2018 a group used light from quasars billions of years old to choose the measurement settings, pushing any conspiracy between the settings and the particles back into the deep past.
Now the crucial point, routinely misreported. Bell's theorem rules out local hidden variables. It does not rule out hidden variables, and it does not establish indeterminism. David Bohm's theory of 1952 is a fully deterministic hidden-variable theory in which particles have definite positions at all times and are guided by a wave that acts nonlocally; it reproduces every prediction of standard quantum mechanics. It is unpopular for other reasons, chiefly its awkward relationship with relativity, but it is not refuted by any experiment, and it is deterministic in Laplace's exact sense.
The interpretations disagree, the predictions do not
Line up the serious options and the situation becomes clear.
The Copenhagen approach and its descendants take the measurement rule at face value: real chance, and determinism is false. The spontaneous collapse theory of Ghirardi, Rimini and Weber, published in 1986, replaces the vague measurement rule with a precise stochastic one in which each particle's wavefunction localises at random at a fixed low rate, so that a macroscopic object collapses almost instantly; it is genuinely indeterministic and, unlike Copenhagen, makes predictions that differ slightly from standard quantum mechanics and are being tested. Bohmian mechanics is deterministic. The Everett or many-worlds view keeps only the Schrödinger equation, deletes the measurement rule entirely, and is therefore the most deterministic theory on the list: the universal state evolves unitarily, and the appearance of chance comes from an observer's ignorance about which branch they are in.
These make the same predictions for every experiment yet performed. So the answer to "is the world deterministic?" is currently a function of an interpretive choice that the evidence does not force, and anyone who tells you physics has settled it is reporting their favourite interpretation as a result.
Example. A commentator writes that Bell's theorem proves the universe is not deterministic, so Laplace was refuted in the laboratory. What is the mistake, and what would a Bohmian say?
The mistake is dropping the word "local". Bell's inequality is derived from local hidden variables, meaning that each particle's outcome depends on properties it carries and on the setting of the nearby apparatus, and it is that conjunction the experiments refute. Give up locality and determinism survives untouched: in Bohmian mechanics every particle has a definite position at every moment, the trajectories are fixed by the guiding equation, and the statistical predictions match quantum mechanics exactly because the initial positions are distributed according to the Born rule. The Bohmian's reply is therefore that the experiments refuted locality, which is what Bell himself said they showed, and that determinism was never on the scaffold.
Now you. Bell tests assume that the experimenters' choices of measurement setting are statistically independent of the hidden state of the particles. What position denies this, and why is it uncomfortable for everyone?
Answer
Superdeterminism, defended by Gerard 't Hooft among others. If the settings are correlated with the particles' hidden properties, because both were fixed by a common past, the derivation of the inequality fails and local hidden variables survive. It is uncomfortable in two directions. For the physicist, it undermines the practice of experiment itself, since any correlation might be an artefact of the initial conditions rather than a discovery, and it is why the 2018 test drove the setting choices back to quasar light emitted billions of years ago rather than merely to a random number generator. For our subject, it is a reminder that the assumption being made in these experiments is called the free-choice assumption, and that Conway and Kochen's free will theorem of 2006 has the same conditional shape: if experimenters' choices are not functions of prior information, then neither are the particles' responses. Physics keeps needing a version of the very thing this course is trying to evaluate.
Does any of it reach a brain?
Suppose the world is indeterministic at the smallest scale. The libertarian needs that indeterminism to be present where decisions are made, which means inside a warm, wet organ at 310 kelvin. That is a physical question with a fairly hard answer.
The relevant comparison is between quantum length scales and the sizes of the structures that carry a neural signal. A particle in thermal equilibrium has a typical momentum , and its de Broglie wavelength is , which measures how spread out its quantum state is. For a sodium ion, mass about 23 atomic mass units, at body temperature, that wavelength is about 0.03 nm. The narrowest part of a potassium channel, the selectivity filter mapped by Roderick MacKinnon's group in 1998, is about 0.3 nm across, and a neuron is roughly ten micrometres wide. An ion whose quantum spread is ten times smaller than the hole it passes through behaves, for every purpose that matters to a cell, like a small classical ball.
Max Tegmark made the argument sharply in 2000. He estimated decoherence times for the candidate quantum degrees of freedom in the brain, including the microtubules proposed by Penrose and Hameroff, and got figures in the region of seconds and below, against neural dynamics operating on timescales of seconds and slower, a gap of ten orders of magnitude or more. Hagan, Hameroff and Tuszynski replied in 2002 with corrected estimates around seconds, which narrows the gap without closing it, and the dispute is not resolved. What is not in dispute is that no quantum effect in the brain has been measured, whereas quantum effects in biology have been measured elsewhere, in photosynthetic energy transfer and in enzymatic tunnelling.
Note the honest qualification. The same calculation for an electron gives a wavelength of about 6 nm, larger than the 5 nm thickness of a cell membrane, which is why electron tunnelling in proteins is real, routine and essential to respiration. The claim is not that brains are classical objects. It is that the specific structures carrying decisions, ions crossing channels and vesicles fusing with membranes, are far too heavy and far too warm for coherence to survive.
Example. Repeat the wavelength calculation for a potassium ion, mass 39 atomic mass units, at 310 K, and say what it shows.
Thermal energy first: J. The mass is kg. Then kg m/s, and m, that is 0.023 nm. It is smaller than the sodium figure, as it must be, since a heavier particle at the same temperature has more momentum and a shorter wavelength. Against a 0.3 nm filter, the ion is localised to about a thirteenth of the aperture, so its passage is a matter of electrostatics and geometry rather than of interference.
Now you. Neurons are noisy: the opening of an ion channel is a random event, and a vesicle often fails to release when an action potential arrives. Does that noise give the libertarian what they need?
Answer
No, on two counts. First, the noise is thermal and statistical rather than quantum: a channel gate is a protein rattling around in a bath at 310 K, and its randomness is the same kind a classical gas has, entirely compatible with underlying determinism. Second, and more important, even if the noise were irreducibly quantum it would still be noise. An agent whose decision is nudged by an undetermined channel opening has not thereby gained control over the decision; something outside their reasons has intervened in it. That objection is the luck problem, it is the central difficulty for libertarian theories, and the lesson on libertarianism takes the best attempts to answer it seriously.
Example. Grant for the sake of argument that each ion channel's opening is genuinely undetermined. Estimate how much of that randomness survives at the level of a neuron's decision to fire.
Very little, and the estimate is a one-line calculation. Independent random events average out with relative fluctuations of order , where is the number of contributing events. A patch of membrane bringing a neuron to threshold involves on the order of channel openings, giving a relative fluctuation of , or one percent. A decision that recruits populations of neurons is averaging over vastly more, and the corresponding figure is . So microscopic indeterminacy does not vanish, but it is suppressed by the square root of a very large number every time the signal is passed up a level. A libertarian needs not merely noise but a mechanism that amplifies it, which is why Kane's account in a later lesson appeals to chaotic sensitivity: chaos is the only known way to run this calculation backwards.
Now you. Does that calculation refute libertarianism?
Answer
No, for two reasons worth keeping separate. The averaging argument assumes the events are independent, and neural systems are full of mechanisms, including recurrent excitation and near-threshold dynamics, that couple them and can amplify a small difference into a different outcome; this is exactly what sensitive dependence means, and it is the reason weather is unpredictable despite averaging over vastly more molecules than a brain has neurons. The second reason is that the calculation addresses only whether indeterminism reaches the level of behaviour, which is the libertarian's second requirement. Even if it does, the third requirement remains, and it is the hard one: showing that the indeterminism yields control rather than noise. A libertarian who wins the physics argument has not yet started on the philosophical one.
What the physics settles
Three conclusions, and they are narrower than either camp usually wants.
Determinism is not established. Classical physics is deterministic only under conditions it does not itself guarantee, and quantum physics divides on the question along interpretive lines that no experiment yet distinguishes.
Determinism is not refuted either. Bohmian mechanics and the Everett interpretation are deterministic, empirically adequate and taken seriously by working physicists, and Bell's theorem refutes locality rather than determinism.
And, most importantly for what follows, the debate does not turn on the answer. The incompatibilist argument of the next two lessons works just as well against near-determinism, where the past and the laws fix the chances, since chances fixed before your birth are no more up to you than outcomes fixed before your birth. Meanwhile the compatibilist never needed the question answered at all. It is a curious feature of this subject that the empirical question everyone reaches for first is close to irrelevant to it.
There is one further reason not to lean on physics, which the next lesson develops. Arguments that freedom is impossible do not need determinism in the first place. Some of the oldest of them use nothing but the assumption that statements about the future are already true, or that somebody already knows what you will do, and they reach the same conclusion with no laws of nature anywhere in sight. They fail, but seeing exactly how they fail sets the standard that a good argument against free will has to meet.