Abstract
The term determinism has often been used in fields other than physics, such as psychology and sociology, causing some bewilderment.
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Notes
- 1.
- 2.
For instance, Popper (1992) argued that a determinist must be a reductionist, although a reductionist is not necessarily a determinist; while others identify reductionism with determinism.
- 3.
We shall see how determinism refers to ontic descriptions, while predictability (and in some sense chaos) has an epistemic nature.
- 4.
The idea of causality explicitly enters Laplace’s as well as our reasoning. Indeed, the strict notion of “causality” leads to considerable difficulties from epistemological and ontological points of view. This does not concern us. In its evolution, classical mechanics has developed a principle of legal determinism, in which the notions of cause and effect are not explicitly invoked. The idea was anticipated by Kant, who stated that the geschiet (the effect) presupposes an antecedent (worauf) from which it follows according to a rule. The adjective “causal” is still used in the same sense.
- 5.
We stress the importance of identifying the state vector \( \mathbf{X} \) which fully describes the phenomenon under investigation. For instance, in classical mechanics, \( \mathbf{X} \) is given by the positions and velocities of particles. This is a fundamental step which took a long time to be understood. For instance, in Aristotelian physics only the positions were considered.
- 6.
From the conference Does the progress of Physical Science tend to give advantage to opinion of Necessity (or Determinism) over that of the Contigency of Events and the Freedom of the Will?, cf. Campbell and Garnett (1882) Chap. XIV.
- 7.
In brief, van Kampen’s argument is the following. Suppose the existence of a world A which is not deterministic and consider a second world B obtained from the first using the following deterministic rule: every event in B is the copy of an event occurred one million years earlier in A. Therefore, all the observers in B and their prototypes live the same experiences despite the different natures of the two worlds.
- 8.
We consider systems whose phase space is bounded.
- 9.
Equation (5.1) holds for infinitesimal distance. Because the phase space is bounded, the distance between the two trajectories cannot grow forever and reaches its maximum in a finite time.
- 10.
It is worth stressing how dramatically chaos affects our predictions. Because of the logarithm in 5.2, increasing the predictability time \(T_p\) by a factor 5 increases the required precision of the initial conditions by five orders of magnitude, e.g. from metre-order precision to micrometre-order precision. For all relevant phenomena this is and will forever remain impossible to be achieved. This is why our local weather forecast are restricted to 5–7 days predictions (roughly speaking the time given by the Lyapunov exponent) and one cannot hope to greatly improve on that by making more accurate measurements of the initial conditions.
- 11.
In the pre-computer age, numerical computations relied on tabulated numbers for, e.g. logarithmic and trigonometric functions.
- 12.
In philosophical language the classical trilemma of Agrippa: if we are asked to prove how we know something, we can provide a proof or an argument. Nonetheless, a proof of the proof can be then asked and so on, leading to an infinite process which never ends.
- 13.
For the sake of simplicity, we restrict ourselves to the case of discrete-time dynamical systems, but continuous systems may be treated analogously.
- 14.
- 15.
Consider the following two \(m\)-sequences, produced by tossing a fair coin:
$$\begin{aligned}&01010101010 \ldots 010101\nonumber \\&01001010110 \ldots 101001 \end{aligned}$$One finds that the first sequence is compressible, while the second appears to be stochastic, in spite of the fact that both occur with probability \(2^{-m}\). This shows that algorithmic complexity, which characterises a single sequence, and information, which amounts to a probabilistic notion, are conceptually different.
- 16.
To be rigorous, this is true for particles interacting through a potential, i.e. in all cases of physical interest.
- 17.
Cited in Dyson (2009).
- 18.
From the conference Does the progress of Physical Science tend to give advantage to opinion of Necessity (or Determinism) over that of the Contingency of Events and the Freedom of the Will?, see Campbell and Garnett (1882).
- 19.
The letter is reprinted in Popper (2002).
- 20.
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Chibbaro, S., Rondoni, L., Vulpiani, A. (2014). Determinism, Chaos and Reductionism. In: Reductionism, Emergence and Levels of Reality. Springer, Cham. https://doi.org/10.1007/978-3-319-06361-4_5
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