Abstract
This chapter is for those who neither want to bother with details nor have time for the expositions and the explanatory rants of previous parts; or as a teaser to go deeper into the subject and to want to know more.
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This chapter is for those who neither want to bother with details nor have time for the expositions and the explanatory rants of previous parts; or as a teaser to go deeper into the subject and to want to know more.
1 Provable Unknowables
First and foremost, from the rational, scientific point of view, there is and never will be anything like “absolute randomness.” Knowledge of absolute randomness, if it “exists” in some platonic realm of ideas, is ineffable and thus strictly metaphysical and metamathematical, as it is blocked by various theorems about the impossibility of induction (cf. Chap. 7, p. 35ff), forecasting (cf. Chap. 6, p. 29ff), and representation (cf. Sect. A.4, p. 173ff). Any proof of such theorems, and thus their validity, is only relative to the assumptions made.
Claims regarding “absolute randomness” – and, for the same reasons, “absolute determinism” – in physics should therefore be met with utmost skepticism. Such postures might serve as a heuristic principle, a sign-post, but they do not signify anything beyond the contemporary, most likely transient (some might even say spurious), worldview, as well as the personal and subjective preferences of the individual issuing them. Like all constructions of the mind and society, physical theories are suspended in free thought – an echo chamber of sorts.
Whoever trusts a physical random number generator has to trust the assurances of the physical authorities that it indeed performs as claimed – in this case, that it produces random numbers. The authorities in turn base their judgement on personal inclinations [68, p. 866] and in metaphysical assertions [589]; as well as on their trust on the theories and models of functioning of such devices. Theories and models are considered trustworthy if they satisfy a “reasonable” and “meaningful” catalogue of criteria; but never more than that.
One such reasonable criterion is the requirement that it should at least in principle be possible to locate the (re)source for randomness or indeterminism. Unfortunately both in quantum mechanics, as well as for classical systems, there is no such consolidated agreement about the physical resources of randomness. Therefore, whenever a physical random number generator is employed, one has to bear in mind the insecure, and means relative, performance of this device. It is not that, pragmatically and for all practical purposes, it would not be usable. But it could fail in particular circumstances one has little idea about, and control of.
2 Quantum (In)Determinism
There are three classes or types of quantum indeterminism: complementarity (cf. Sect. 12.3 and Chap. 19), value indefiniteness (often, referred to as contextuality after the realist Bell; cf. Sect. 12.9.8.7, p. 97ff), as well as single measurement outcomes and events; all of them tied to the quantum measurement problem (cf. Sect. 12.10, p. 118ff). Thus quantum random number generators are subject to some form of the quantum measurement problem, which lies at the heart of an ongoing debate – a debate which has been declared (re)solved or superfluous by various self-proclaimed authorities for a variety of conflicting reasons. Alas, quantum mechanics, despite being immensely useful for the prediction and comprehension of certain phenomena, formally operates with an inconsistent set of rules; in particular, pertaining to measurement. As has already been pointed out by von Neumann, the assumption of irreversible measurements contradicts the unitary deterministic evolution of the quantum state. (Inconsistencies, even in the core of mathematics, such as in Cantor’s set theory, should be rather a reason for consideration and prudence but not cause too much panic – after all, as noted earlier, those constructions of our mind are suspended in our free thought.)
Some supposedly “active” elements such as beam splitters are represented by perfectly deterministic (unitary, that is, distance preserving permutations, such as the Hadamard gate) evolutions (cf. Chap. 11 and Sect. 12.5). Therefore they cannot be directly identified as quantum resource for indeterminism.
The measurement process in quantum mechanics appears to be related to entanglement and individuation: in order to be able to know from each other, the measurement apparatus has to acquire knowledge about the object; and in order to do so, the former has to interact with the latter. Thereby entanglement in the form of relational properties of object and apparatus is created. Because of the permutativity (one-to-one-ness) of the entire process (resulting in a sort of zero-sum game) these relationally definite properties (or, by another term, statistical correlations) come at the price of the indefiniteness of the individual, constituent parts – the original object as well as the measurement device are in no definite individual state any longer. If one forces individuality upon them (by some later measurement on the individual parts), then the outcome cannot be totally (but may be partly) pre-determined by the state of the constituent parts before that measurement. Thereby it may be justified to say that “the measurement creates the outcome which is indeterminate before.” But this is a rather trivial statement expressing the fact that the outside environment with its supposedly huge number of degrees of freedom, in particular also the measurement device, has contributed to the outcome.
The author’s impression is that Bohr and his followers may never have understood the true reason for value indefiniteness: the scarcity and constancy of information encoded into the quantum state; and the entanglement across the Heisenberg cut between object and measurement device. This scarcity also shows up in “static” Kochen–Specker type theorems [6, 314, 401] expressing the fact that only a single maximal observable or context is defined at any time.
3 Classical (In)Determinism
Classical (in)determinism depends on its definition, and on the assumptions made. One of these assumptions is the existence of the continuum – not only as formal convenience but as a physical entity. Almost all elements of a continuum are random (cf. Sect. A.2, p. 171ff). Any computable form of evolution “revealing” the algorithmic information content “buried” in a single supposedly random real physical entity (e.g., initial values) corresponds to a form of deterministic chaos (cf. Chap. 18, p. 141ff). If the assumption of the physical existence of the continuum is dropped in favour of constructive, computable entities, then what remains from these indeterministic scenarios is the high sensitivity of the system behaviour on variations of initial states.
Another form of model-induced classical indeterminism is due to representation and formalization of classical physical systems in terms of differential equations. In such cases the question of uniqueness of its solutions is crucial. Nonunique solutions indicate indeterminism. However, the requirement of Lipschitz continuity guarantees uniqueness in many cases which appear to be indeterministic (due to the possibility of weak solutions) without this property (cf. Sect. 17.4, p. 137ff).
4 Comparison with Pseudo-randomness
Should one prefer physical (re)sources of randomness over mathematical pseudo-random? Of course, “anyone who considers arithmetical methods of producing random digits is \(\ldots \) in a state of sin” [553, p. 768]. And yet, some desired features of randomness can be formally certified even for such computable entities.
For instance, take Borel normality; that is, the property that every subsequence of length n occurs in a “large” b-ary sequence with frequency \(b^{-n}\). Almost all real numbers are normal to a given base b; in particular, all random sequences are Borel normal [102]. Yet, individual (even computable) numbers are hard to “pin down” as being normal; and no well-known mathematical constant, such as e or \(\log 2\), is known to be normal to any integer base. Also the normality of \(\pi \), the ratio of the circumference to the diameter of a “perfect” (platonic) circle, remains conjectural [26], although particular digits are directly computable [25]. Von Neumann’s paper [553] quoted earlier contains a way to eliminate bias (and thus establish Borel normality up to length 1) of a binary sequence (essentially a partitioning of the sequence into subsequences of length 2, followed by a mapping of \(00,11 \mapsto \emptyset \), \(01 \mapsto 0\), \(10 \mapsto 1\)); but only if this sequence is generated by independent physical events. Physical independence may be easy to obtain for all practical purposes, but difficult in principle.
On the other hand, Champernowne’s number 0.12345678910\(\ldots \), obtained by concatenating the decimal representations of the natural numbers in order, as well as the Copeland–Erdos constant 0.2357111317192329\(\ldots \), obtained by concatenating the prime numbers in order, are both Borel normal in base 10. So, if Borel normality suffices for the particular task, then it might be better to consider such carefully chosen pseudo-random numbers (cf. Ref. [110] for comparisons with certain quantum random sources).
There exist situations which are perplexing yet not very helpful for practical purposes: Chaitin’s \(\varOmega \) (cf. Sect. A.6, p. 176ff) is also Borel normal in any base, and additionally it is provable random. Algorithms for computing the very first couple of digits of \(\varOmega \) [109, 111] exist; alas the rate of convergence of the sum yielding \(\varOmega \) is so bad (in terms of time and other computational capacities worse than any computable function of the d-ary place) it is incomputable.
5 Perception and Forward Tactics Toward Unknowns
Whatever one’s personal inclinations toward (in)determinism may be – one might characterize our situation either as an ocean of unknowns with a few islands of preliminary predictables; or, conversely, as a sea of determinism with the occasional islands or gaps of an otherwise lawful behaviour – every such inclination remains strictly means relative, metaphysical and subjective. Maybe such preferences says more about the person than the situation; because a person’s stance is often determined by the subconscious desires, hopes and fears driving that individual. Choose one, and choose wisely for your needs; or even better, “if you can possibly avoid it [211, p. 129],” choose none, and remain conscious about the impossibility to know.
Let me finally quote the late Planck [410] concluding that [409, p. 539] (see also Earman [186, p. 1372]) “\(\ldots \) the law of causality is neither right nor wrong, it can be neither generally proved nor generally disproved. It is rather a heuristic principle, a sign-post (and to my mind the most valuable sign-post we possess) to guide us in the motley confusion of events and to show us the direction in which scientific research must advance in order to attain fruitful results. As the law of causality immediately seizes the awakening soul of the child and causes him indefatigably to ask “Why?” so it accompanies the investigator through his whole life and incessantly sets him new problems. For science does not mean contemplative rest in possession of sure knowledge, it means untiring work and steadily advancing development.” Footnote 1
Planck also emphasized the joy of and the motivation from the unknown [411]: “We will never come to a completion, to the final. Scientific work will never cease. It would be bad if it stopped. For if there were no more problems one would put his hands in his lap and his head to rest, and would not work anymore. And rest is stagnation, and rest is death – in a scientific sense. The fortune of the investigator is not to have the truth, but to gain the truth. And in this progressive successful search for truth, lies the real satisfaction. Of course, the search for itself is not satisfactory. It must be successful. And this successful research is the source of every effort, and also the source of every spiritual enjoyment. When the source dries up, when the truth is found, then it is over, then one can fall asleep mentally and physically. But that is taken care of, that we don’t experience this, and therein persists our happiness.” Footnote 2
Notes
- 1.
In German [410, p. 26]: “\(\ldots \) das Kausalgesetz ist weder richtig noch falsch, es ist vielmehr ein heuristisches Prinzip, ein Wegweiser, und zwar nach meiner Meinung der wertvollste Wegweiser, den wir besitzen, um uns in dem bunten Wirrwarr der Ereignisse zurechtzufinden und die Richtung anzuzeigen, in der die wissenschaftliche Forschung vorangehen muss, um zu fruchtbaren Ergebnissen zu gelangen. Wie das Kausalgesetz schon die erwachende Seele des Kindes sogleich in Beschlag nimmt und ihm die unermüdliche Frage “warum ?” in den Mund legt, so begleitet es den Forscher durch sein ganzes Leben und stellt ihm unaufhörlich neue Probleme. Denn die Wissenschaft bedeutet nicht beschauliches Ausruhen im Besitz gewonnener sicherer Erkenntnis, sondern sie bedeutet rastlose Arbeit und stets vorwärtsschreitende Entwicklung, nach einem Ziel, das wir wohl dichterisch zu ahnen, aber niemals verstandesmäßig voll zu erfassen vermögen”.
- 2.
German original: “\(\ldots \) zum Abschluss, zum Endgültigen, werden wir nie kommen. Das wissenschaftliche Arbeiten wird nie aufhören – es wäre schlimm, wenn es aufhören würde. Denn wenn es keine Probleme mehr gäbe, dann würde man die Hände in den Schoß legen und den Kopf zur Ruhe und würde nicht mehr arbeiten. Und Ruhe ist Stillstand, und Ruhe ist Tod – in wissenschaftlicher Beziehung.
Das Glück des Forschers besteht nicht darin, eine Wahrheit zu besitzen, sondern die Wahrheit zu erringen. Und in diesem fortschreitenden erfolgreichen Suchen nach der Wahrheit, da liegt die eigentliche Befriedigung. Das Suchen an sich befriedigt natürlich noch nicht. Es muss erfolgreich sein. Aber dieses erfolgreiche Arbeiten, das ist dasjenige, was den Quell jeder Anstrengung und auch den Quell eines jeden geistigen Genusses darstellt. Wenn der Quell versiegt, wenn die Wahrheit gefunden ist, dann ist es zu Ende, dann kann man sich geistig und körperlich schlafen legen. Aber dafür ist gesorgt, dass wir das nicht erleben, und darin besteht unser Glück”.
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Svozil, K. (2018). (De)briefing. In: Physical (A)Causality. Fundamental Theories of Physics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-70815-7_22
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