Saturday, May 20, 2017

Quantum mechanics is another example of deductive reasoning

Objectivity of the truth is separate, unnecessary, and non-existent

For various psychological, metaphysical, and quasi-religious reasons, many people find it insanely hard to understand an extremely simple fact – namely that quantum mechanics allows you to reason to pretty much the same extent (when it comes to the applicability) as classical physics did before the birth of quantum mechanics; but it fundamentally rejects the idea that there are statements about Nature that are objective in character.

I say it's simple and it really is. The point is that the laws of a quantum mechanical theory are tools to produce lots of statements of the form
"IF... THEN..."
These two words, "IF" and "THEN", basically cover everything that you need to understand the basic character of quantum mechanics. You don't need 43 pages of rubbish about Jesus Christ, John Wheeler, and random statements by 150 philosophers and physicists.

Quantum mechanics requires that you know some assumptions – the propositions behind the "IF". Those are the latest measurements that you, an observer, did in the past. And it allows you to derive or calculate some conclusions – the propositions behind the word "THEN". Because of this structure, "propositions are derived from others", we may say that the reasoning is deductive.

Well, the conclusions sometimes include the word "probably" or "with the probability \(P\)" where \(P\) is a number. For this reason, it may sometimes be better to say that some of the quantum mechanical reasoning is inductive or abductive. But the differences between the adjectives inductive, abductive, and deductive are not what really matters.

What matters is that it's normal that there are some assumptions or inputs – the observations that have been made – and one shouldn't be surprised that there is no objective way to decide whether the assumptions are right. The truth value depends on the observer's perspective.

Let's assume that you are a third-grader again. You may be told\[

x+2 = 7

\] and asked to derive\[

2x = 10.

\] Your IQ surpasses that of most chimps so you know that the first equation implies \(x=5\) and it follows that \(2x=10\). Great. So there exist mathematical steps that allow you to go from the assumptions to the consequences. But what is important is that there is no "objective universal truth" saying that \(x=5\). The variable \(x\) may mean something else in a different context. It may be \(193,884\) or the monster group. The bulk of mathematics isn't the assignment of values to all letter variables – these values are variable which is why the variables are called variables. ;-)

The bulk of mathematics is the collection of rules that allow you to derive or prove some conclusions from some assumptions.

The variability of variables is sort of trivial but there are less trivial examples in axiomatic mathematics. For example, we will derive that there exist subsets of the interval \([0,1)\) of real numbers that are not measurable. How do we prove it? Well, divide the interval \([0,1)\) to equivalence classes \(C_i\) such as all the numbers in the equivalence class \(C_i\) for a fixed \(i\) differ by a rational difference, and all the numbers in classes \(C_i,C_j\) for \(i\neq j\) differ by an irrational number.

Now, assume the axiom of choice which allows you to pick one representative \(\lambda_i\) from each class \(C_i\). What is the measure \(m\) of the set of all these numbers \(\lambda_i\)? Well, there have been infinitely many classes and their measure was the same because they're shifted copies of each other. So \(m\) times infinity must be at most \(1\) because the measure is assumed/known to be subadditive. But that means that the measure \(m\) of each class of the representatives is \(0\). However, the addition of such sets over the rational numbers should reconstruct the interval. Because the addition of zeroes will never give you one, there is no way to assign the measures so that all the additive rules will work.

OK, I probably phrased this proof a bit differently than it's usual. The point is that one can prove the implication
Axiom of choice ... implies ... the existence of unmeasurable sets.

Or if you wish:

IF you assume the axiom of choice to be true, THEN you have to assume that the unmeasurable sets exist, too.
There have been other assumptions as well but the axiom of choice was the most "disputable one". An important lesson in axiomatic mathematics is that there is no "objective" way to decide whether the axiom of choice is true – whether you can collect the "set of representatives" from a "set of sets" even when it is an "infinite set of infinite sets". You may decide to believe the axiom of choice – and say e.g. that the selection of the representatives is analogous to the case of a finite number of representatives.

Or, you may be agnostic and say that one would need to do an infinite amount of work (and infinitely many decisions) to pick the infinitely many representatives which are in no way canonically given for the equivalence classes. So it's plausible that this infinite amount of work cannot be done. You may decide that the axiom of choice is false: there exist counterexamples of "sets of sets" for which the set of representatives cannot be chosen. If you take this perspective, you may keep on assuming that all subsets of the interval \([0,1)\) have a well-defined additive measure. You won't hit any contradiction because the axiom of choice is absolutely needed to prove the existence of unmeasurable sets.

Many "philosophers" among mathematicians prefer to consider the axiom of choice to be true – partly because some "desirable" proofs are more effective if you assume it. I slightly prefer to assume that the axiom of choice is false – because the measurability of all subsets of an interval looks like a much more natural, physical, "intuitively correct" axiom to me than an axiom that "an infinite number of decisions isn't a problem".

But all rational people should be aware of the fact that there can't exist any "objective proof" of the axiom of choice. (In experimental physics, we don't separately manipulate with infinitely many objects, so there can't be any experimental proof in one way or another, either. After all, it's obvious that we're doing pure mathematics so physical experiments just cannot be relevant.) It is an axiom so you need to assume it (or its negation) if you want to deduce something that depends on its truth value. Different people may assume different things about the validity of the axiom of choice. These people may use the same methods to prove statements in mathematics. But they may prove different things because they make different assumptions.

The case of quantum mechanics (and, to a large extent, even the classical statistical physics) is analogous. You may derive predictions about the future observations but you need some assumptions – the results of your recent past observations. You may analyze the dynamical laws (differential equations) for the probability amplitudes or operators and derive some probabilities that future observables have some values from the knowledge that the past observables had some values.

But the assumptions – the outcomes of the past observations – are not guaranteed or assumed to be objectively valid. There is absolutely no reason to believe this objective character of the outcomes. This objective character isn't assumed anywhere, the deductive reasoning and derivations work perfectly well without such an assumption. And indeed, in quantum mechanics, one may see that the assumptions are unavoidably observer-dependent. They are not objective in character.

But some people just can't get rid of the invalid assumption that their observations are fundamentally objective. They are so attached to this no longer valid assumption that was underlying classical physics that they want to fool themselves into believing that this assumption is "implied by everything around them". For example, you tell them that the more correct description of Schrödinger's cat is that "it is dead OR live", not "dead AND alive". And they protest (I've run to dozens of people who made the exact same protest): it can't be true, the word "OR" means that before the measurement, there is some objective answer to the question whether the cat is dead or alive.

However, if you are at least slightly rational, you must realize that such a statement – "the livelihood of the cat is objectively given before any measurement" – just doesn't follow and cannot follow from a very modest assumption, e.g. the assumption that you dared to use the preposition "OR". The word "OR" is a logical preposition. It is a way to construct a new composite statement, "A OR B", from two statements "A" and "B". It is a tool to deal with logical propositions. It has nothing whatever to do with the question whether the validity of "A" or "B" is objective in character.

To say that "the statement that 'the cat is dead or alive' implies that one of the two answers is objectively true" is exactly as idiotic as to say that the usage of the operator "OR" in the derivation of the unmeasurable set above implies that the validity of the axiom of choice must be unambiguous and objectively given. Well, it's obviously not. The axiom of choice may be assumed to be either true or false – this is analogous to the observer's assumptions about their observations – but whatever the assumption is, the tools of mathematics allow you to derive something out of these assumptions!

I have used the analogy with the axiom of choice because the axiom of choice also looks like an "objective question" analogous to the question "whether the Moon is out there when nobody looks". But if you think about it, the derivations of the statements "IF... THEN..." in quantum mechanics are much more analogous to the reasoning "if \(x+2=7\), then \(2x=10\)". In some sense, the only new thing in quantum mechanics is that the commuting observables such as \(x\) in the previous sentence are replaced with non-commutative but still associative ones.

So quantum mechanics allows you to derive e.g. that "if an electron is in the \(3s\) state of motion around a hydrogen nucleus, then the probability is \(p\) that after time \(\Delta t\), it will drop to \(2p\)". This "IF... THEN..." statement is relevant for an entity – an observer – who knows, however subjectively, that the electron was in the \(3s\) state to start with. However, quantum mechanics in no way guarantees that everyone else "has to know" the exact same things that "he knows". In practice, people may know almost the same things but in principle, it's very important to appreciate that knowledge is subjective (depending on the observer) in principle.

All of the wisdom of mathematics is hiding in the methods how you can prove some propositions assuming others – in the path from "IF..." to "THEN...". In the very same way, all the wisdom of quantum mechanics is hiding in the methods to derive the "THEN..." consequences (probabilistic predictions of outcomes of future measurements) from the assumptions (outcomes of measurements in the recent past). None of these things imply or implicitly assume that the outcomes of the measurements are reflections of any objective reality. This philosophical thesis isn't needed anywhere in the derivation or calculation of the predictions, it is separate from all the empirically proven beef of physics, and quantum mechanics indeed unambiguously rejects this assumption that the state of Nature may be assumed to be observer-independent.

Only the implications "IF... THEN..." are guaranteed to hold according to the quantum mechanical laws of physics. These laws have never said, don't need to say, and will never say that some of the assumptions about the initial state are objectively true and others are objectively false.

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