Determinism in classical physics is an idealization and an approximation.

Determinism is an emergent property that shows up for large numbers of elementary particles, which as individuals or in small numbers are more accurately described with indeterministic quantum physics. All the laws of physics are statistical laws. All are the consequence of averaging over the irreducible quantum indeterminacy of the elementary particles.

Macroscopic (phenomenological) physical laws are arbitrarily accurate in the limit of infinite numbers of particles. But it is beyond the possibility of experimental accuracy to "prove" the idea of perfect determinism. Nevertheless, we can "prove" that small numbers of elementary particles exhibit indeterminate behavior, to within the highest level of experimental accuracy achieved in modern physics.

Isaac Newton knew his mathematical laws were an approximation. Ludwig Boltzmann, arguably the first scientist to appreciate the atomic nature of matter, also knew it.
He said that the assumption of perfect determinism "goes beyond experience." His colleague at the University of Vienna Franz Exner said the same thing, as did Exner's student Erwin Schrödinger, before he reversed his views to agree with Einstein, Max Planck, and others to reject quantum indeterminacy.

Here is Boltzmann, for example.

Since today it is popular to look forward to the time when our view of nature will have been completely changed, I will mention the possibility that the fundamental equations for the motion of individual molecules will turn out to be only approximate formulas which give average values, resulting according to the probability calculus from the interactions of many independent moving entities forming the surrounding medium — as for example in meteorology the laws are valid only for average values obtained by long series of observations using the probability calculus. These entities must of course be so numerous and must act so rapidly that the correct average values are attained in millionths of a second.

(Part II, Chapter VII, The Kinetic Theory of Gases, 1898 (1964), p.449, tr. Stephen G. Brush)

And here is Franz Exner.

This significantly restricted version of the law of causality as a purely empirical proposition forces yet another question of fundamental importance: If certain conditions are met, and the lawful flow of phenomena takes place naturally in all its phases, is the state of the system then predetermined (voraus bestimmter) at every moment? Or it is random and is it only the average state that is determined over a period of time?

This is the classic Laplacian view denied.

If there were for a gas a certain moment when all the speeds and directions of the molecules were given, can we then say that the location and speed for each individual molecule is determined in a subsequent moments? It is certain that an affirmative answer goes beyond our experience.

(Lectures on the Physical Principles of Natural Science, 1922, p.677)

Finally, consider Erwin Schrödinger. In his early career, Schrödinger was a great exponent of fundamental chance in the universe. He followed his teacher Franz S. Exner, who was himself a colleague of the great Ludwig Boltzmann at the University of Vienna. Boltzmann used intrinsic randomness in molecular collisions to derive the increasing entropy of the Second Law of Thermodynamics.

Most nineteenth-century physicists, mathematicians, and philosophers believed that the chance described by the calculus of probabilities was actually completely determined. For them, chance was subjective and an epistemological problem, not objective and ontological.

(Adolphe Quételet and Henry Thomas Buckle are important examples, very likely influenced by the views of Immanuel Kant.) The "bell curve" or "normal distribution" of random outcomes was itself so frequently observed that it seemed to imply deterministic laws governing individual events. Statistics and probability were thought to be epistemological questions. We simply lack the knowledge necessary to make exact predictions for these individual events. Pierre-Simon Laplace saw in his "calculus of probabilities" a universal law that determines the motions of everything from the largest astronomical objects to the smallest particles.

On the other hand, in his inaugural lecture at Zurich in 1922, Schrödinger argued that the evidence did not justify our assumptions that physical laws are deterministic and strictly causal. His inaugural lecture was modeled on the inaugural lecture of Exner in Vienna in 1908.

"Exner's assertion amounts to this: It is quite possible that Nature's laws are of a thoroughly statistical character. The demand for an absolute law in the background of the statistical law — a demand which at the present day almost everybody considers imperative — goes beyond the reach of experience. Such a dual foundation for the orderly course of events in Nature is in itself improbable. The burden of proof falls on those who champion absolute causality, and not on those who question it. For a doubtful attitude in this respect is to-day by far the more natural."

Several years later, Schrödinger wrote

"Fifty years ago it was simply a matter of taste or philosophic prejudice whether the preference was given to determinism or indeterminism. The former was favored by ancient custom, or possibly by an a priori belief. In favor of the latter it could be urged that this ancient habit demonstrably rested on the actual laws which we observe functioning in our surroundings. As soon, however, as the great majority or possibly all of these laws are seen to be of a statistical nature, they cease to provide a rational argument for the retention of determinism.

"If nature is more complicated than a game of chess, a belief to which one tends to incline, then a physical system cannot be determined by a finite number of observations. But in practice a finite number of observations is all that we can make. All that is left to determinism is to believe that an infinite accumulation of observations would in principle enable it completely to determine the system. Such was the standpoint and view of classical physics, which latter certainly had a right to see what it could make of it. But the opposite standpoint has an equal justification: we are not compelled to assume that an infinite number of observations, which cannot in any case be carried out in practice, would suffice to give us a complete determination.

Despite these strong arguments against determinism, just after he completed the wave mechanical formulation of quantum mechanics in June 1926 (the year Exner died), Schrödinger began to side with the determinists, including especially Max Planck and Albert Einstein, and against those professing statistical interpretations of quantum mechanics, Max Born, Werner Heisenberg, and Niels Bohr.

When small numbers of atoms and molecules interact, their motions and behaviors are indeterministic, governed by the rules of quantum mechanics.

Werner Heisenberg's principle of indeterminacy (mistakenly called "uncertainty," as if the problem is epistemic/subjective and not ontological/objective) gives us the minimum error in simultaneous measurements of position x and momentum p,

where h is Planck's constant of action. To see how "adequate" determinism emerges for large numbers of particles, note that the momentum p = mv, the product of mass and velocity, so we can write the indeterminacy principle in terms of velocities and positions as

When large numbers of microscopic particles get together in massive aggregates
( h / m → 0 ), the indeterminacy of the individual particles gets averaged over and macroscopic "adequately" deterministic laws "emerge." The positions and velocities of large massive objects can be "determined" beyond our ability to measure.

Determinism is an emergent property.

The "laws of nature," such as Newton's laws of motion, are all statistical in nature. They "emerge" when large numbers of atoms or molecules get together. For large enough numbers, the probabilistic laws of nature approach practical certainty. But the fundamental indeterminism of component atoms never completely disappears.

So determinism "emerges" today from microscopic quantum systems as they become a part of larger and more classical systems. But we can says that determinism also emerged in time. In the earliest years of the universe, large massive objects did not yet exist. All matter was microscopic.

We can now identify that time in the evolution of the universe when determinism first could have emerged. Before the so-called "recombination era," when the universe cooled to a few thousand degrees Kelvin, a temperature at which atoms could form out of sub-atomic particles (protons, helium nuclei, and electrons), there were no "macroscopic objects" to exhibit deterministic behavior.

The early universe was filled with positive ions and negatively charge electrons. The electrons scattered light photons, preventing them from traveling very far. The universe was effectively opaque past very short distances. Then the charged particles combined to form neutral atoms (hydrogen and helium) the photons suddenly could "see" (travel) to enormous distances. The universe first had the transparent sky that we take for granted today (on cloudless nights).

Those 3000 degree K photons have cooled as a result of the universe expansion and now appear to us as the 3 degree K "cosmic microwave background" radiation left over from the big bang. We are looking at a moment in time when "classical" objects obeying apparently deterministic causal laws did not yet exist.

• When the water in a turbulent cell far from equilibrium is convected upward by the heat below, it drags along most of the water molecules that compose it. This is Ilya Prigogine's prime example of a "dissipative structure" exhibiting emergent "order out of chaos."

• When a ribosome assembles 330 amino acids in four symmetric polypeptide chains (globins), each globin traps an iron atom in a heme group at the center to form the hemoglobin protein. This is downward causal control of the amino acids, the heme groups, and the iron atoms by the ribosome. The ribosome is an example of Erwin Schrödinger's emergent "order out of order," life "feeding on the negative entropy" of digested food.
  
  When 200 million of the 25 trillion red blood cells in the human body die each second, in each of 200 million new cells 100 million hemoglobins cell must be assembled. With the order of a few thousand bytes of information in each hemoglobin, this is 10 thousand x 100 million x 200 million = 2 x 10²⁰ bits of information per second, a million times more information processing than today's fastest computer CPU.

• When a ribosome produces a protein that does not fold properly, a chaperone enzyme, shaped like a tiny trash can, opens its lid and captures the protein. It then closes the lid and squeezes the protein. Upon release, the protein then frequently folds properly. If it does not, the chaperone captures it again and disassembles it back to its amino acids. The chaperone is an emergent that is in no way the result of "bottom-up" processes from its amino acid components. It is also an example of biological error detection and correction.

• When a single neuron fires, the active potential rapidly changes the concentration of sodium (Na+) ions inside the cell and potassium (K+) ions outside the cell. Within milliseconds, thousands of sodium-potassium ion channels in the thin lipid bilayer of the cell wall must move billions of those ions from one side to the other. They do it with emergent biological machinery that exerts downward causation on the ions, powered by ATP energy carriers (feeding on negative entropy). Random quantum indeterministic motions of the amino acids drive them near the pump opening, and quantum collaborative forces capture them in a lock-and-key structure.

• When many motor neurons fire, innnervating excitatory post-synaptic potentials (EPSPs) that travel down through the thalamus and the spinal cord and cause muscles to contract, that is as literal as downward causation gets in the body.

• When the emergent mind decides to move the body, that mental causation is realized as downward causation.

• Who saw this first? Consider the great Latin poet, philosopher, and scientist
  Titus Lucretius Carus, who described the action of the mind thus:

  Therefore when the mind so bestirs itself that it wishes to go and to step forwards, at once it strikes all the mass of spirit that is distributed abroad through limbs and frame in all the body. And this is easy to do, since the spirit is held in close combination with it. The spirit in its turn strikes the body, and so the whole mass is gradually pushed on and moves...

  Again, there is no need to be surprised that elements so small can sway so large a body and turn about our whole weight. For indeed the wind, which is thin and has a fine substance, drives and pushes a great ship with mighty momentum, and one hand rules it however fast it may go, and one rudder steers it in any direction; and a machine by its blocks and treadwheels moves many bodies of great weight and uplifts them with small effort.

  (De Rerum Natura, Loeb Library, trans. W.H.D.Rouse, p.345)

• When the helmsman turns the wheel of a great sailing ship, he has downward causal control over all the matter of that great ship.

• When an emergent philosopher rearranges and communicates ideas, verbally in lectures, or as written words in a published paper, or as the bits of information in a computer memory, this is "information out of order," ultimately dependent on the body digesting food, feeding on the energy (ATP) with negative entropy ("order out of order"), but in no way controlled "bottom-up" by the molecules of body or food material, or by the energy consumed.

Abstract information is neither matter nor energy, yet it needs matter for its concrete embodiment and energy for its communication. Information is the modern spirit, the ghost in the machine.