Abstract
Since we used the word “determinism” in the previous chapters, it might be good to pause for a moment and to define what this word means and what it implies.
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Notes
- 1.
The reason why one need to specify both positions and velocities and not positions alone is due to a property of Newton’s laws, that we will not go into.
- 2.
This expectation will be justified trough the law of large numbers in Sect. 3.4.1.
- 3.
To define the frequency with which some series of symbols occurs in a sequence of results, one counts the number of those series of symbols in that sequence and one then divides that number by the total number of results.
- 4.
Thus, every finite series of results of length n will occur a fraction of the time equal to \(\frac{1}{2^n}\). This of course makes sense only for an infinite random sequence. Since, in practice, every sequence has a finite length, the definition given here has to be considered as an idealization.
- 5.
This intelligence is often referred to as the “Laplacian demon”. (Note by J.B.).
- 6.
There are several caveats that should be made here, in order to be rigorous, but it would go far beyond the scope of this book.
- 7.
Nevertheless, one can argue that the development of a scientific world view has led to an increased skepticism with respect to the doctrine of free will and that has led to changing attitudes with respect to education and to criminal law: “rewards and punishments” have been increasingly viewed as a practical matter rather than a question of principle based of distinguishing true merit and true guilt.
- 8.
For example, the distinguished physicist Nicolas Gisin links the supposed lack of determinism in quantum mechanics to free will, see [90, 119].
- 9.
We will come back to the mind-body problem in Sect. 11.5.1.
- 10.
The philosopher Colin McGinn has developed the interesting idea that the problem of “free will” may lie beyond the limits of human understanding [123].
- 11.
In case our observations systematically deviate from what one would expect on the basis of this assignment of probabilities, namely that all faces fall with equal frequencies, we will suspect the coin or the dice to be biased and then revise our probabilities.
- 12.
This notion will be important in Sect. 8.4.2 when we discuss the de Broglie–Bohm theory.
- 13.
We shall not use that formula, but we give it for the interested reader: writing \(n=2m\), since n is even, that number of sequences is: \(\frac{n \times (n-1)\times (n-2) \dots \times (m+1)}{1\times 2 \times 3 \dots \times m}= \frac{n!}{m!^2}\), where, for a number m, m! is a shorthand notation for \(1\times 2 \times 3 \dots \times m\).
- 14.
Again, for the interested reader, that number is: \(\frac{n!}{k! (n-k)!}\).
- 15.
Which is equal to \(2\times 2\times 2 \dots =2^n\). We shall not be precise about what “large”, “more or less” and “almost equal” mean here. For a mathematician, n “large” means that one considers a limit where n tends to infinity, “more or less” means that the frequencies of outcomes where the coin falls heads is very close to \(\frac{1}{2}\), and “almost equal” means that the equality becomes exact in the limit where n tends to infinity.
- 16.
Again, this notion becomes precise in the (idealized) limit where the number of results tends to infinity.
- 17.
In fact, one way to verify that the chosen probability distribution was correctly chosen is through this coincidence between the frequencies of events in most of the sequences of results and that probability. For example, if a coin is biased, one will observe a deviation between the observed frequencies and the assigned probabilities, and this should lead us to revise those probabilities.
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Bricmont, J. (2017). “Philosophical” Intermezzo I: What Is Determinism?. In: Quantum Sense and Nonsense. Springer, Cham. https://doi.org/10.1007/978-3-319-65271-9_3
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DOI: https://doi.org/10.1007/978-3-319-65271-9_3
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