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Kategorie: Philosophy

The question of probability is of great importance above all because it could be a basis for deciding between determinism and indeterminism ("determinism or indeterminism). If the probability of quantum mechanics can only be interpreted indeterministically, as many physicists believe, determinism is refuted. 

What is probability

General
  • Probability refers to the degree of expectation as to how often or whether a future event will occur. If the probability is 1, one expects that the expected event will necessarily occur; if the probability is 0, the occurrence is certainly not to be expected. With a probability of 0.5, for example, the expectation is that the event will occur 50% of the time and will not occur 50% of the time.
  • Basically, two different forms of probability are distinguished: objective and subjective probability, both of which are usually determined. In the field of quantum phenomena, however, there also seems to be an indeterminate objective form of probability based on strict or absolute chance. It corresponds to various intuitions around the notion of free will ("Quantum Mechanics and Free Will), but it is disputed whether an indeterminate probability can be thought without contradiction at all (below: "two ways to interpret quantum mechanics).
Determinate probability
  • subjective probability
    • Future
      • Subjective probability is often also called "epistemic" probability, meaning that it is a probability that is "dependent on knowledge" or that "concerns knowledge". It usually relates to the future because our knowledge of it is always limited, since there are so many variables involved between the present and the future that future events can never actually be predicted with absolute certainty. For example, we may be firmly convinced that the sun will also rise tomorrow, but we cannot know this with absolute certainty because we do not have all the information at our disposal. So, strictly speaking, the probability is never 1 with regard to the future; as in the case of the sun, it can theoretically only be almost 1, but in fact we assume that we can speak of absolute certainty here and can dispense with probability.
      • I am perhaps similarly certain that I will get up tomorrow. Subjectively, this probability also seems to me to be almost 1, but it could be that I die in an accident today.
      • But there are also events whose probability I consider only relatively probable. For example, I consider the probability that I will eat pizza tomorrow to be perhaps 0.3. Since I often eat pizza, the probability is not entirely small, but I am definitely not sure either. However, if I already knew how my day would develop and whether a colleague would call me at noon to take me out for pizza, I would assess the probability differently. So this form of probability is definitely subjective and depends directly on my knowledge.
    • Past
      • Subjective or epistemic probability calculations can also be applied to the past. For example, I don't have the information, the memories, of how often I ate pizza the week before last. However, based on my experience, I can estimate the probability that I ate pizza the week before last. The subjectively assessed value is neither 1 nor 0. However, this does not mean that I only probably ate a pizza, but only that I lack the information about it, that it is just a matter of epistemic probability.
    • Cube
      • Epistemic probability also exists in the game of dice. Even if dice seem to behave randomly, it is (theoretically) possible to calculate exactly which number will be on top for each throw. The more I know about the throw (e.g. strength of the throw, weight of the dice, air resistance, nature of the ground, etc.), the more precisely it can be calculated which number will be on top after the throw. With increasing knowledge, the probability thus approaches "0" or "1" more and more. The probability is epistemic.
  • Objective probability
    • From subjective to objective probability
      • In retrospect, I can then say with certainty how great the probability was in reality (i.e. objectively): namely always zero or one. Did the sun rise, did I get up, did I eat a pizza, did the "six" lie on top - in retrospect there is an objective answer to this. With increasing knowledge, subjective probability cancels itself out and becomes objective probability, whereby this form of objective probability is always "0" or "1".
      • Objective probability again comes in two varieties, both of which can be well demonstrated with the dice game.
    • in relation to a single variable
      • If you only consider a single throw, the course of each individual throw can (theoretically) always be calculated exactly. The objective probability for the number that is on top is therefore always "0" or "1". Either the number is on top or it is not. The initial conditions before the throw determine the exact course of the throw and this is completely independent of the subject or the knowledge of the subject.
    • in relation to many variables
      • However, if we consider not one roll but many rolls, another form of objective probability can be noted. Thus, for an unmanipulated six-sided die, under normal circumstances, the objective probability that a specific side will be on top after a roll is 0.17 or 1/6. While the probability of which number will be on top after a roll is always "0" or "1" for an individual roll, this is not true when considering multiple rolls.
      • A probability calculated in this way applies regardless of whether one knows what a cube actually is. In this respect, it is not epistemic, i.e. not dependent on knowledge, and it is also not subjective, since it also applies independently of a (human) subject. Even if one has calculated every single one of a hundred dice exactly, the probability that a specific number will be on top is about 1/6.
      • This "approximately" is part of the determined objective probability. Without knowing the exact framework conditions, one can estimate the future on the basis of the objective probability and will be pretty much right. However, if one has calculated each individual throw, it can be calculated that the probability distribution will be slightly different, that the six, for example, will be up slightly less often than 1/6.
      • The objective probability applies regardless of whether I can calculate the individual throws or not. In the dice game, therefore, there is both a subjective and an objective probability.
Indeterminate probability
  • classical physics and quantum mechanics
    • The determinate concept of probability corresponds to the understanding of classical physics. The throw of the dice, the rising of the sun, my decision to eat pizza can be explained causally. For all these phenomena, there are causes that we simply may not know, all these events are determined. However, there are many physicists who believe that a determinate concept of probability is no longer sufficient for quantum physics.
    • This is because in the field of quantum phenomena, in contrast to the classical cube, it is fundamentally impossible to calculate exactly where the quantum object will appear or when, for example, a radioactive isotope will decay. In principle, it is not possible to have sufficient knowledge (or this knowledge does not even exist) to calculate more than the probability of the location or the time of the decay.
  • indeterminate probability in relation to a single variable
    • In classical physics, the objective probability with respect to a single variable is always "0" or "1". Other probabilities in relation to individual variables are always subjective or epistemic. In quantum mechanics, however, objective probabilities that are not "0" or "1" should also exist for individual variables. These individual quantum objects would therefore not be determined.
  • Determined probability in relation to many variables
    • What is astonishing about this finding, however, is that the determined objective probability remains for a set of quantum objects. For there is no question that the objective probability with which a set of quantum objects will appear or decay somewhere can be calculated exactly. Analogous to the dice, it can be calculated that the probability for a quantum object to appear at a certain place is, for example, 1/6. There are places where the objective probability is large and places where it is almost negligible. In the case of radioactive decay, the half-life can be calculated analogously, i.e. the time it takes until (approximately) half of the isotopes have decayed.

Two ways to interpret quantum mechanics

The fact that for individual quantum objects basically no more than the probability of their location can be calculated (cf. indeterminate probability above) is undisputed ("Heisenberg's uncertainty principle"). However, there are various ways of interpreting this fact. 

Indeterministic interpretations
  • Objective probability in relation to individual "quantum objects
    • Indeterministic interpretations of quantum mechanics assume that the location of quantum objects is actually indeterminate and that this finding objectively describes reality. Quantum objects are not determined, they exist quasi only as (objective) probability. What this means, however, is not entirely clear. As Albert Einstein once remarked, the statement that "a single particle behaves probabilistically is as meaningless as saying it has a temperature" (Byrne 2012, p. 131). A single "particle", whether it is a quantum object or not, cannot be in one place and 0.6 in another with a probability of 0.4. Nor can it be in ten different places with a probability of 0.1 each. A "particle" is either in a certain place or it is not.
    • In quantum physics, therefore, one is reluctant to speak of "quantum object", since "quantum phenomena" show properties of particles and waves at the same time. However, probability waves are certainly conceivable, but "wave-particle dualism" is a contradictory idea (Schrödinger's cat). However, the most important truth criterion of all natural science, including physics, is the conviction that contradictions cannot exist. If there is a contradiction anywhere, this means that the theory is wrong, perhaps proves itself as a heuristic, but does not describe the real events. If one accepts this premise ("premises"), then this must also apply to interpretations of quantum physics. The concept of objective probability can be applied without contradiction only to a set of "particles", but if it is a matter of subjective or epistemic probability, then the probability is based on a lack of knowledge and the interpretation leads to determinism. 
  • Objective probability in relation to many "quantum objects
    • For a large number of "quantum objects", the objective probability can be calculated, as is also the case classically. It is possible to calculate the objective probability of how often the number "2" will be at the top of the dice, it is possible to calculate the objective probability with which a "quantum object" will be located at a certain place during a measurement. In order to explain this empirical finding, one can now assume that individual "quantum objects" "behave" probably or absolutely randomly - or causally and deterministically. If one does not want to give up the freedom from contradiction, the "behaviour" of the "quantum objects" cannot be explained with probability, as we have shown above. But can the empirically measurable objective probability of a set of "quantum objects" perhaps be traced back to an absolutely random "behaviour" of the individual "quantum objects", as is claimed in many books on quantum mechanics?
    • This can obviously be ruled out. Objective probability is something very regular and is thus quasi the counterpart of absolute chance. Absolute chance means that there is no cause and thus no reason why one probability should be more probable than another. "If individual "quantum objects" behaved absolutely randomly, this would also apply to a large number of "quantum objects". No objective probability could be calculated, but every probability distribution would be exactly equally probable. If there were no cause for the fact that a quantity of "quantum objects" probably "behaved", one would have to speak of a "miracle", which, however, would occur completely regularly. What happens without a cause, what happens absolutely by chance, cannot be explained in principle, since every explanation presupposes causality.
  • Evaluation of indeterministic interpretations
    • Indeterministic interpretations of quantum mechanics must therefore accept the contradictoriness in the realm of quantum phenomena and thus give up their most important criterion of truth and additionally accept the existence of (fundamentally!) inexplicable miracles. There is no other way to explain the empirically measured objective probability of a set of "quantum objects". It is therefore very surprising that indeterministic interpretations are very popular in physics and are hardly questioned - as Albert Einstein, for example, did throughout his life.
    • But not only the objective probability of a set of "quantum objects" behaves analogously to a deterministic objective probability, other elements in the field of quantum phenomena are also clearly deterministic (cf. e.g. "Falkenburg: Quantum physics is also deterministic). Thus, indeterministic interpretations are not indeterministic at all, but dualistic conceptions ("determinism or indeterminism). Dualisms, however, are extremely frowned upon in science, since they give rise to serious problems, especially around interaction. In this case, it is particularly a question of how indeterministic processes in the realm of individual quantum objects that function without a cause can produce a deterministic form of probability - and the whole deterministic processes of classical physics. This is the so-called measurement problem. 
Deterministic interpretations

Deterministic interpretations explain the impossibility of calculating more than the probable location of a single quantum object, for example, by the fact that quantum mechanics is not complete. While indeterministic interpretations assume that no further causes exist, Bohmian mechanics, for example, assumes that there are parameters or causes that are fundamentally hidden. They exist, but they cannot be determined in principle. This trick in turn explains the behaviour of the individual "quantum objects" as epistemically probable and is thus deterministic. 

However, since these hidden parameters or variables cannot be detected in principle, it can only be a speculation that can neither be proven nor disproven. Excitingly, however, Bohmian mechanics can be calculated and it leads to the identical results as, for example, the most widespread interpretation of quantum mechanics, the "Copenhagen interpretation.

Whether Bohmian mechanics is the "true Jacob" is not up for discussion here. However, the fact that deterministic interpretations are possible should be enough, after the explanations of indeterministic interpretations, to finally put the focus more on those.

Conclusion

Indeterministic interpretations of quantum mechanics that assert indeterministic probability are only conceivable if contradictions can exist in reality (which makes it precisely inconceivable), if probability can be inexplicably based on absolute chance and if they can explain the interaction between indeterministic elements and deterministic elements also required by such interpretations. Since deterministic interpretations cannot be ruled out so far (Scarani: possibility of determinism), they are fundamentally preferable to indeterministic interpretations, since indeterministic probability cannot be formulated without contradictions.