This paper aims to demonstrate how a rigorous definition of determinism can be formulated and applied to a formal axiomatization of a physical theory. Examining an example of Newtonian classical particle mechanics (CPM) formalized in set-theoretical axioms by Patrick Suppes, the paper shows why definitions of determinism based on the temporal development of physical systems—such as those proposed by Suppes (2002) and Montague (1974)—fail to assess determinism correctly. It is argued that, to deliver a meaningful verdict, the definition of determinism cannot be applied to the entire framework of CPM itself, but rather to its particular theories : definitions of specific physical systems that incorporate the force equation. Therefore, the verdict of determinism or indeterminism typically depends on the uniqueness (or non-uniqueness) of solutions to the differential equations governing the system’s behavior. The paper concludes with a demonstration of how a formally correct definition of determinism can be used to show the indeterminism of the theory of Norton’s dome stated within the framework of CPM. Although the example concerns pre-relativistic Newtonian mechanics, the crucial philosophical distinction between frameworks and theories extends beyond this case and suggests that assessing the determinism of any physical theory requires special caution to distinguish it properly from its framework.
This paper aims to demonstrate how a rigorous definition of determinism can be formulated and applied to a formal axiomatization of a physical theory. Examining an example of Newtonian classical particle mechanics (CPM) formalized in set-theoretical axioms by Patrick Suppes, the paper shows why definitions of determinism based on the temporal development of physical systems—such as those proposed by Suppes (2002) and Montague (1974)—fail to assess determinism correctly. It is argued that, to deliver a meaningful verdict, the definition of determinism cannot be applied to the entire framework of CPM itself, but rather to its particular theories : definitions of specific physical systems that incorporate the force equation. Therefore, the verdict of determinism or indeterminism typically depends on the uniqueness (or non-uniqueness) of solutions to the differential equations governing the system’s behavior. The paper concludes with a demonstration of how a formally correct definition of determinism can be used to show the indeterminism of the theory of Norton’s dome stated within the framework of CPM. Although the example concerns pre-relativistic Newtonian mechanics, the crucial philosophical distinction between frameworks and theories extends beyond this case and suggests that assessing the determinism of any physical theory requires special caution to distinguish it properly from its framework.
The purpose of this paper is to give a brief survey the implications of the theories of modern physics for the doctrine of determinism. The survey will reveal a curious feature of determinism: in some respects it is fragile, requiring a number of enabling assumptions to give it a fighting chance; but in other respects it is quite robust and very difficult to kill. The survey will also aim to show that, apart from its own intrinsic interest, determinism is an excellent device for probing the foundations of classical, relativistic, and quantum physics. The survey is conducted under three major presuppositions. First, I take a realistic attitude towards scientific theories in that I assume that to give an interpretation of a theory is, at a minimum, to specify what the world would have to be like in order for the theory to be true. But we will see that the demand for a deterministic interpretation of a theory can force us to abandon a naively realistic reading of the theory. Second, I reject the “no laws” view of science and assume that the field equations or laws of motion of the most fundamental theories of current physics represent science’s best guesses as to the form of the basic laws of nature. Third, I take determinism to be an ontological doctrine, a doctrine about the temporal evolution of the world. This ontological doctrine must not be confused with predictability, which is an epistemological doctrine, the failure of which need not entail a failure of determinism. From time to time I will comment on ways in which predictability can fail in a deterministic setting. Finally, my survey will concentrate on the Laplacian variety of determinism according to which the instantaneous state of the world at any time uniquely determines the state at any other time. The plan of the survey is as follows. Section 2 illustrates the fragility of determinism by means of a Zeno type example. Then sections 3 and 4 survey successively the fortunes of determinism in the Newtonian and the special relativistic settings..
Abstract Uses a historical approach to explore the extent to which quantum theory mandates indeterminism. Drawing on important examples in the history of physics, it discusses explicitly the crucial distinction – often overlooked in the philosophical literature – between indeterminacy of physical quantities and causal indeterminism. It pursues at a more sophisticated level, issues raised in Ch. 2 concerning determinism and indeterminism in modern science. It concludes that quantum indeterminacy is an objective feature of the world, but that the jury is still out regarding quantum indeterminism.
Abstract In Norton’s dome, a singular solution of the Newton/Hamilton equations represents a particle starting to move without cause. Similar indeterminism occurs for forces that are not the gradient of a potential, i.e. Newtonian but not Hamiltonian. An exactly solvable quantum partial counterpart displays no trace of the Norton phenomenon. A completely different context is chaotic classical motion of gas molecules in a box, whose walls are penetrable because gravity cannot be screened. The gravity of a single electron at the limit of the observable universe renders the motion of the gas molecules unpredictable after a few nanoseconds. But motion is still determined by the initial conditions. However, the much larger perturbation from the gravity of the computers or ‘Laplace’s demon’ calculating the motion of the molecules introduces self-reference, making it impossible to follow the motion and rendering the determinism irrelevant to physics for such chaotic systems. For the quantum gas molecules, perturbation from the distant electron vastly exceeds the energy level separation for the idealised impenetrable box, rendering individual energies physically insignificant. Calculations are not difficult, and the combination of science and philosophy deserves to be introduced to physics students.
Classical physics is traditionally characterized as deterministic and founded on the real number continuum, while quantum mechanics is distinguished by its indeterminism and fundamental uncertainty relations. This apparent dichotomy, however, obscures a deeper continuity. We argue that both frameworks embody a common ontological principle: the physical prohibition of absolute precision. In classical physics, this prohibition is implicit, manifesting in information-theoretic bounds, the non-physicality of infinite-precision initial conditions (chaos theory), and the breakdown of determinism in singular configurations (Norton's dome). These phenomena reveal that the classical reliance on mathematical real numbers with infinite information content is an unphysical idealization. Quantum mechanics, by contrast, does not merely reflect this prohibition; it provides its precise quantitative expression. The Heisenberg uncertainty principle σxσp≥ℏ/2 and the non-commutative structure of observables are the exact mathematical formulation of the fact that physical information is fundamentally finite. The Planck constant ℏ is the universal scale that quantifies the minimal phase space volume—the “atom” of information—below which no further refinement is possible. Thus, quantum mechanics is not a separate theory but the completion of classical physics: it supplies the missing quantification of the principle that nature admits no infinite information density, and consequently, no absolutely precise physical quantities.
Most modern physical theories are formulated in terms of differential equations. As a result, if we know exactly the current state of the world, then this state uniquely determines all the future events - including our own future behavior. This determination seems to contradict the intuitive notion of a free will, according to which we are free to make decisions - decisions which cannot be determined based on the past locations and velocities of the elementary particles. In quantum physics, the situation is somewhat better in the sense that we cannot determine the exact behavior, but we can still determine the quantum state, and thus, we can determine the probabilities of different behaviors - which is still inconsistent with our intuition. This inconsistency does not mean, of course, that we can practically predict our future behavior; however, in view of many physicists and philosophers, even the theoretical inconsistency is somewhat troubling. Some of these researchers feel that it is desirable to modify physical equations in such a way that such a counter-intuitive determination would no longer be possible. In this paper, we analyze the foundations for such possible theories, and show that on the level of simple mechanics, the formalization of a free will requires triple interactions - while traditional physics is based on pairwise interactions between the particles.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.