An extreme form of determinism is Laplacian determinism, so called because the physicist Pierre Simon de Laplace formulated it in his Philosophical Essay on Probabilities (1813). Laplace thought that probabilities were solely a measure of our ignorance, and that every event is precisely, causally determined. Laplace envisions a gigantic intellect or spirit. This spirit is able to know the position and state of motion of every particle in the universe and to perform very complicated and long calculations. This spirit, according to Laplace, is then able to predict every future event, including all human behavior and social changes (under the assumption that human behavior is physically caused). Laplace’s spirit could predict what you would be doing tomorrow morning or twenty years from now.

God would be an example of such a spirit, but, ironically, Laplace himself was an atheist. When asked by Napoleon why God did not appear in his celestial mechanics, Laplace said, “I have no need of that hypothesis.” Laplace thought that he had proven the stability of the solar system. He had not (Hanson, 1964). Since Newton had left it to God to occasionally intervene miraculously to readjust the planets, Laplace’s purported proof was the basis of his legendary comment to Napoleon about having no need of God in his celestial mechanics.

One of the great, unsolved problems of mechanics is to find a general solution to the problem of three or more bodies of arbitrary masses in arbitrary initial positions attracting one another by gravitational force. The exact calculation and prediction of the motion of the planets in relation to the sun (including the weaker attraction of the planets for each other) is an example of and motivation for this problem. In the late nineteenth century King Oscar II of Sweden and Norway became disturbed about the possibility that the solar system might fall apart, and, with goading from interested mathematicians, offered a prize in 1889 for a proof of stability, to be awarded on his birthday (Diacu and Holmes, 1996, pp. 23–7). The French mathematician and physicist Jules Henri Poincaré (1854–1912), in his work on this many-body problem of the long-term orbits of the planets, invented aspects of what is now called chaos theory. He showed that the solar system might be stable but that stable and unstable orbits are interwoven infinitely close to one another in an infinitely complicated braided pattern. Chaotic systems are deterministic but unpredictable. They can and do arise in classical mechanics and are independent of the Heisenberg considerations (see below).

They are mainly caused by non-linear equations (equations that have squares or powers of variables or higher order derivatives) that give rise to “sensitive dependence on initial conditions.” In an ordinary linear system, a slight shift in the initial positions of the objects yields a slight shift of the end positions. However, in a chaotic system, an infinitesimal shift in the initial conditions yields a big shift in the results. Newton’s laws of motion are expressed by non-linear equations. Since we cannot measure with infinitesimal accuracy but only with finite accuracy (even forgetting the Heisenberg principle, just sticking to Newton), we cannot predict chaotic systems, even though the math is deterministic. A common error in the “Science Wars” and of romantic or New Age objections to mechanism, even if justified on other grounds, is that they assume that since Newton’s laws are old fashioned, and that “linear” is boring, Newton’s equations are linear. Ironically, it is the newer and more exciting quantum mechanics that is linear (superposition principle), while the older, supposedly less exciting, Newtonian mechanics is non-linear.

Even more severe problems arise for Laplacian determinism from subatomic physics. According to Heisenberg’s uncertainty principle, it is impossible in principle to simultaneously measure exactly the position and momentum of a subatomic particle (Heisenberg, 1958). Other pairs of variables, such as the energy and the time interval of a process, and particle spin on different axes, also obey the relations. The mathematics of the uncertainty principle is built into the laws of quantum mechanics, which is the best theory we have of the structure of matter. According to Heisenberg’s principle, even Laplace’s demon couldn’t know the simultaneous position and motion of one particle, let alone of all of them. This is not a matter of our inaccuracy of measurement, but is built right into the equations of the theory. The operators differ in the value of the product, depending on the order in which one multiplies them (to use mathematical jargon, they are non-commutative). The difference between AB and BA, for these operators, equals the limit of accuracy, related to a fundamental constant of physics (Planck’s constant). Despite Heisenberg uncertainty at the atomic level, we can have statistical determinism in many systems, where we cannot predict exactly, but can predict within a range of error. Surprisingly, the teenage Heisenberg first learned of the idea of geometric structures of atoms from reading Plato, before he was disappointed by the crude materialism of the tinker-toy models in his chemistry text. His opposition to materialism was in part tied to the fact that he was defending his laboratory against Marxist revolutionaries, and read Plato’s theory of the creation of the universe in Greek for relaxation on the military college roof at lunchtime (Heisenberg, 1971, pp. 7–8).

Heisenberg later claimed to have been stimulated by Plato’s concept of the receptacle (matrix, or mother), a spatial principle leading to fuzziness or inexactness when the perfect forms shape real objects in space. Despite this early interest in Plato’s metaphysics, when Heisenberg first presented the mathematical skeleton of his quantum mechanics, he understood it in a strictly instrumentalist and positivist manner, claiming that the mathematics was merely a tool for making predictions, not a picture of reality. Later in life, Heisenberg understood his theory in terms of Aristotle’s theory of potentiality and actuality. For Heisenberg, the abstract, mathematical states are objective, but potential in nature, while the physical observations are subjective but actual, turning the usual notions of reality on their head.

Many people think that randomness occurs in the behavior of individuals (perhaps due to free will), but that statistical regularity of populations is true in the social sciences, such as sociology. The social sciences often in practice use statistics when they are attempting to be rigorously mathematical. Laplacian determinism would claim that probability is just a matter of our ignorance, our lack of exact knowledge.

Chaos theory and quantum mechanical indeterminacy suggest that for complicated systems of many particles, such as a human being or a society, such statistical methods may be necessary, and not replaceable with exact methods even in principle. It is perhaps on analogy to this situation that the notion of statistical determinism in the social sciences, called by Mesthene and Heilbroner “soft determinism,” is sometimes conceived.

Though the Heisenberg principle applies to the subatomic level, its operation is usually effectively so minuscule as to be unnoticeable in larger, many-particle systems. However, there are significant places where it has been suggested the subatomic indeterminacy may be “magnified” to have effects on the macroscopic level. These include certain biological mutations involving small changes (of a single atomic bond) of the genetic material that controls biological heredity (Stamos, 2001). Another, more speculative, example is in the firing of individual tiny dendrites or fibers in a neuron in the brain, where the chemical changes are on a small enough scale for the Heisenberg principle to be applicable (Eccles, 1953, chapter 8). More recently, physicist Roger Penrose has suggested that collapse of the quantum wave function in the microtubules of the brain is responsible for the non-mechanistic aspect of thought (Penrose, 1994, chapter 7).

Thus determinism has been challenged within physics from at least two directions. Chaos theory suggests that even the in-theory, deterministic mathematics of Newtonian mechanics can yield situations in which practical prediction is impossible. Quantum mechanics produces an even stronger, in principle, objection to universal determinism, in that the indeterminism is built right into the mathematical core of the theory. Free will is an older, more concretely human problem in relation to determinism.