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A properly defined free will is an established scientific fact

Sabine Hossenfelder has argued that

Free will is dead, let’s bury it.
That's how things may have worked according to Newton's physics but the quote above has been shown incorrect by the discovery of quantum mechanics – well, at least with some sensible definition of a "free will". It looks like Ms Hossenfelder has slept at least since 1925 or, if we insist on rigorous proofs, since 2006.

Arthur Schopenhauer

In classical physics, the future was determined by the past. Humans were bound states of some particles or other classical degrees of freedom so they evolved in the way fully determined by the initial state of the Universe and the laws of physics. Quantum mechanics prescribes similarly unambiguous equations for the evolution but what is being evolved are just probability amplitudes and the actual outcomes of observations are random.

Hossenfelder's understanding is that this randomness doesn't make people's will any more free than it was:
There is no free will in such a fundamental law because there is no “will” – there is just some randomness sprinkled over the determinism. In neither case do you have free will in any meaningful way.
Well, this may look like a reasonable attitude and from some perspective, it may even be one. But from more reasonable vantage points, it's not.

You know, a big problem hides in her previous sentence:
This randomness cannot be influenced by anything, and in particular it cannot be influenced by you, whatever you think “you” are.
But this sentence is incorrect. For a very reasonable and practical definition of "you" or "me", the sentence above is self-evidently false. What is the definition of "you" and "me" that we need? Well, we define "you" and "me" as the collection of particles and patterns within our bodies (including brains etc.).

However, here we're interested in a more "spiritual" definition of "me" and "you", something that is related to our decisions. The point is that we may define "you" and "me" as the collection of all the outcomes that Nature's random generator produces for observables located within our bodies (and especially brains).

You may and you arguably should identify yourself and your "soul" or "free will" with all the values of voltages between neurons or whatever is relevant in the brain that are picked from the spectrum of eigenvalues whenever we perceive something (we measure the voltage between the neurons, in my simplified but specific enough example). These random results in your brain are you!

With this understanding of "you" and your "soul", it is obviously true that you do influence the results of the random generator: they are the same thing!

How the "will" may be incorporated in physics

Why would I call these random outcomes of observations "free will"? Because they satisfy everything that the "free will" should satisfy. You may be missing something "more tightly" associated with your identity but the fact is that there's nothing else about your identity than
  • the collection of elementary particles and the patterns created by them, something that almost certainly distinguishes you from all other people; these bound states and patterns define the initial conditions or the precise "problems" for which quantum mechanics allows you to calculate the probabilities of answers
  • the sequence of the random, probabilistically predictable outcomes of measurements. Those that measure observables within your body may be said to be results of "your will" while the random generators associated with observables outside your body are "external influences".
The random outcomes of measurements within your body are "will" because these outcomes influence the world around you. According to the aforementioned definition of "you", it is therefore "you" who is influencing the world around your body! When the voltage between certain neurons has a certain value, you may move your finger and press a nuclear button and erase a city from the map, if I have to mention one spectacular application. This is an example of your "will".

Why is it the "will"? Philosophy defines will as
...faculty of the mind which selects, at the moment of decision, the strongest desire from among the various desires present. Will does not refer to any particular desire, but rather to the capacity to act decisively on one's desires.
Philosophers' definitions are usually self-contradictory and vacuous and this definition is just "slightly better" than the average. It doesn't say much because it reduces the word "will" to other words such as "desires". How does one know what a given person (or you) desires? We could get stuck in neverending complaints. But to simplify things, the definition above is compatible with the idea of a human as a machine that contains observables called "desire #1", "desire #2", and so on, and the eigenvalues of these observables actually say what the given human wants (or wants to do).

The notion of the "desire" is problematic but the "will" isn't the desire itself, as we hear, but the ability of the human to act "decisively" according to the most important desire. So the definition above is compatible with the following sequence of events: the human brain measures the voltage between some neurons, it gets one of the possible eigenvalues, and interprets the eigenvalue as a desire, and because of the interconnectedness of the neurons and muscles, it automatically controls the muscles in a way that is a function of this desire (eigenvalue). This is why the brain-muscle partnership represent the "will".

The desire was determined as a random outcome of a quantum measurement and I said that those should be components in the definition of "you". But do you have the will to change your desires? We could try to go in this direction but we wouldn't get anywhere (at most, we would be stuck in an infinite loop). Arthur Schopenhauer has understood the futility of such a search for the "deepest level of will" in his 1839 text "On the Freedom of the Will":
Der Mensch kann tun was er will; er kann aber nicht wollen was er will.

Man can do what he wills but he cannot will what he wills.
Man can do what he wants but he cannot want what he wants.
I have included both translations because "wants" may sound closer to common modern native English speakers' tastes. On the other hand, the verb "will" is closer to the topic we discuss, the noun "will", and it happens to coincide with the original German word for "want", namely "will", too. ;-)

This phrase is sort of clever and already when he was young, Albert Einstein got enthusiastic about this quote by Schopenhauer. The reason was that it showed that it was silly to spend too much time with questions "why" and "how this and that was decided" in this context. Even if you can do what you want, there's no clear way to determine or figure out what you want and decide whether it's you who decided what you want. It may still be some "other agent" that determines what you want and in this way, he or she or it or they decide what you do, too. You may also ask whether you have the free will to decide what you want to want and what you want to want to want, and so on, but we're not learning anything by adding the levels.

It makes sense to truncate this unproductive chain at the very beginning and just define the free will operationally. The voltages that fill your consciousness with certain sentences – and that tell the muscles what to do – should already be understood as your "will". According to quantum mechanics, the outcomes of the measurements of the neuron voltages are random and probabilistically predictable. The predictions may be almost the same as they are for your twin brother. But the fact that the random generator produces different outcomes for various measurements is a part of your identity that distinguishes you from your twin brother. Your broader identity isn't given just by your DNA etc. but also by your fate.

Now: is your will free? The political dimension...

We said that the "will" without adjectives (or with any adjectives) was the ability to act according to one's desires, and those were reduced to some observables linked to the neural cells' degrees of freedom. The desires are likely to be determined by the survival instinct we share with other animals or perhaps with some other, higher, more human instincts etc. But I don't want to study the evolutionary "details" here. The body is organized in such a way that the muscles etc. are controlled according to these neuron observables identified with the "desires" – and that's why we can call the whole mechanism the "will".

Is the will free? What does it mean for the will to be free?

The important answer is that we may define and we should define freedom or freeness of the will (and freedom in general!) as the absence of certain influences. What does it mean for us to be free to pay taxes that we consider right?

It means that the decision about how much you pay in taxes isn't determined by some degrees of freedom that exist outside your body – including the outcomes of measurements of observables that are geometrically separated from our bodies (or our brains). Are we free to pay the right taxes?

Well, mostly no. In principle, we could pay reasonable taxes but the IRS or a similar group of bullies would come and harass us. Most of us really don't want it – in almost all cases – which is why we don't pay the taxes we consider right but we pay the taxes that someone else decided to be right. Sadly, we are not really free to pay the right taxes because our acts are determined by external agents. We lack the freedom in millions of other situations and the number keeps on increasing in recent years – but this isn't supposed to be a libertarian essay.

Are we free to express our opinions about the sanity of the German politicians who continue their welcoming policies for the immigrants from the Muslim world? In my country, the answer is Yes. People may think and say that these welcoming policies are OK but they may also think and say (and they usually do) that the policies of Merkel and her collaborators are insane.

What makes us free is that our opinions or statements or acts are not determined by (i.e. not fully calculable/predictable from) external influences, rules, other people, institutions, governments, and so on. The freedom may be defined in this way. We may say that the freedom is defined negatively as the absence of certain causation or determinism.

Instead of parroting some government officials, we decide by our brains. The decision what we think is dictated by the current state of our brain and the patterns in it (the physical part of our identity); plus the results of the random but probabilistically predictable outcomes of measurements that take place within our brain (the spiritual/fate part of our identity).

Both parts of our identity were defined as some "boring" physical things – the initial conditions for the physics problems; plus the eigenvalue that is always predictable by the same statistical quantum mechanical laws. Maybe it looks too irreligious for you to call those things "you". But it's how physics reduces "you". "You" are the combination of the patterns in your brains plus the random decisions/measurements that shape your life – and that also affect how you influence the world around you.

(If this blog post were perfect, I would discuss not only the "Dirac choice", i.e. Nature's random generator deciding about the measured eigenvalue, but also about the previous "Heisenberg choice", i.e. observer's earlier choice of the interesting measured/perceived observable, i.e. the question to be probabilistically answered, and why the need for the "Heisenberg choice" implies that all the known data about the external world are always unavoidably subjective. This blog post avoids the subjectivity and the "Heisenberg choice" altogether. We may explain this fact by saying that I study the "freedom of other people and particles" as properties of observed, external degrees of freedom, and I evaluate everything from a particular observer's viewpoint, e.g. mine or yours, which you may identify with an "objective" viewpoint if you can't live without the latter.)

Punishing criminals for their bad intents

When we punish a criminal, we sometimes punish him only if he intended do something bad. If he were forced to do something bad, we may decide that he was innocent. A person may be innocent if his unfortunate act was caused by some psychiatric disease; and he may be innocent if he were forced to do a wrong thing because someone was pointing a gun to his head.

Why do we care about the intent i.e. about the question whether the act was decided by his free will?

Think about it. The answer is that the reason is that we want to find the first cause(s) of the bad act. We don't want to fight against the consequences or symptoms; we want to deal with the root of the problem. So if someone, Mr XY, were pointing a gun to Merkel's head when she was saying "herzlich wilkommen", it is this Mr XY who is the primary culprit of the ongoing migration wave. It wouldn't make too much sense to punish Ms Merkel because almost everyone would say "herzlich wilkommen" if a thug were pointing a gun to his or her head while saying "say herzlich wilkommen or you will die". Instead, we would think that something was wrong with the bodyguards or the system that allowed the thug to point the gun to Angela's head and that's what we would try to fix.

So the absence of rules that determine one's decisions as a function of external degrees of freedom is how we should define the freedom and the freedom of the will, too. Ms Hossenfelder wants to explicitly disagree with that:
But really you don’t have to bother with the details of these arguments, you just have to keep in mind that “indeterminism” doesn’t mean “free will”.
The "free will" may be defined in various ways (or the term may be completely avoided) but the absence of external data that fully determine one's behavior (i.e. indeterminism in this rather general sense) is a totally sensible definition, and perhaps the only sensible definition, we may give to the word "free will"!

Great. So what is the primary cause?

In classical, non-quantum physics, the behavior of all degrees of freedom was assumed to be fully determined by the initial state. Even the brains' and muscles' thoughts and movements were determined. Does it mean that murderers were innocent according to classical physics?

Yes and no. They were innocent according to the "primary cause". It may have been likely that like the hurricane in the Mexican Gulf, a man's decision to murder someone could have been influenced by the motion of the wings of a butterfly in Peking a week earlier, as a well-known metaphor in chaos theory says. And even if something "murderous" was present in a person's very brain, we could still say that it was the parents' fault because they gave her a particular combination of their DNA molecules. And there would be tons of other ways to blame someone or something else.

But this influence of the butterflies (external degrees of freedom) probably wouldn't be predicted to be a dominant factor creating the decisions even in classical physics. Even in classical physics, the people's decisions would be pretty much fully dictated by the state of their brains right before the decisions (which "is" the culprit/suspect). Even though the state of the brain before the fatal decision depended both on the degrees of freedom inside the brain and outside the brain (the external world) at previous times, it would still make sense to punish the culprit because the bad act indicates that the brain has some feature that may be dangerous in the future as well, due to some correlations between the properties of the brain in the present or recent past (which did lead to a bad outcome) and the future (which may lead to another bad outcome). Note that in classical physics, we punish a criminal because we're automatic machines that have evolved in a certain way, anyway, so we can't help. We have evolved as machines that (imperfectly) try to maximize our well-being or the society's well-being etc. Our desire to prevent future crimes; or to bring justice/revenge for the wrongdoing in the past is a part of this optimization process that evolution has imposed upon us.

In quantum mechanics, there's some extra randomness added to the decisions. May this randomness be a function of the past data?

The free will theorem

This leads me to the most rigorous incarnation of the concept of the "free will" in quantum mechanics. Ms Hossenfelder announced a challenge:
If you don’t want to believe that, I challenge you to write down any equation for any system that allows for something one could reasonably call free will. You will almost certainly fail.
Challenge accepted and I certainly succeeded. Well, two famous scholars succeeded a decade ago. Just search for the free will theorem at the Google Scholar. Legendary mathematician John Conway and Simon Kochen have published their
2006: The Free Will Theorem,
2008: The Strong Free Will Theorem
Google Scholar indicates that these papers have 219 and 119 citations at this moment, respectively. These are rigorous mathematical proofs (yes, with many equations) assuming three physical axioms and proving that all elementary particles have their own dose of the "free will" of the kind that pretty much agrees with my full intuitive discussion above – i.e. the freedom of not being determined by some data in the past light cone. The proof shows that the random decision "which face/eigenvalue the particle should show to the experimenters" was decided "now and here", at a given point of the spacetime, and it couldn't have been determined by some past data. So the random generator must be associated with the time and locus of the particle – it's the particle's free will.

Like a more clever cousin of Bell's theorem, the free will theorem eliminates some hidden-variable theories. But it also kills the objective Ghirardi-Rimini-Weber-style theories, among others. The theorem discusses some Bell-style inequalities but they are linked to spin-one particles.

The axioms in 2006 were called SPIN, TWIN, FIN. The last one was replaced in 2008 when they strengthened their theorem by weakening the last assumption (that's how it works in mathematics!) so the new axioms became SPIN, TWIN, MIN.

SPIN basically just means that the spin-1 particle measurements of "the axis" (not too different from \(j_z\)) has the usual three eigenvalues and corresponding orthogonal eigenstates associated with the axes \(x,y,z\). TWIN says that two spin-one particles may be entangled in a way guaranteeing the usual correlations in the measurements of "the axis". And FIN says that the information can't be sent superluminally. The weaker MIN axiom just says that the result of a measurement of a particle \(b\) can't be affected by the spacelike-separated decision of experimenter \(A\) who decided what kind of a measurement he wants to make. MIN is weaker because Conway and Kochen realized that they only need to ban the superluminal transmission between this pair of events (and its mirror). MIN may be derived as an easy consequence if you assume the experimenter's free will to press buttons of his choice; and the standard relativistic causality/locality.

At the end, the freedom of the measured eigenvalue of "the axis" for a given particle is shown not to be a function of any data in the past. This is basically "analogous" to the result of Bell's theorem that the measurements can't be dictated by local realist hidden variables except that the "free will theorem" is stronger and it was constructed by two folks who actually understand quantum mechanics as well as relativity correctly – which arguably wasn't the case of John Bell and his followers.

Note that the whole discussion and proof of the "free will theorem" completely avoided the term "probability". Conway and Kochen hoped that it would be enough to eliminate all the confusing arguments because the anti-quantum zealots only thrive because they invent controversial interpretations of the probability and the controversies disappear if the term "probability" is eliminated.

Needless to say, this hope wasn't realized. Even the "free will theorem" led to some controversies, especially because many people doing this stuff love to distort the meaning both of relativity and quantum mechanics (not just the origin of probabilities). So for example, in 2007, Conway and Kochen had to reply to Bassi+Ghirardi and Tumulka who had previously argued that their non-quantum fake theories claimed to reproduce the successes of quantum mechanics could have circumvented the "free will theorem".

Those claims were wrong. At the end, one could see that these non-quantum people not only refuse to even consider the observation-dependent quantum laws but they don't believe relativity in any form. So while the FIN or MIN axiom is assumed to be true in the "free will theorem" because they're consequences of relativity, Tumulka and similar folks conflate these axioms with the "Bell locality" which is, on the contrary, false in Nature. Needless to say, anti-quantum zealots won't ever be persuaded by a rational argument, however strong and loophole-free it is.

But let me return back to Ms Hossenfelder and the free will. She doesn't seem to be aware of the free will theorem but she tries to promote "superdeterminism" in quantum mechanics, a conspiracy theory claiming that the experimenters don't have the freedom of any sense to press buttons and a magical force may be forcing them to press buttons that make it look like the laws of physics are something even though the laws of physics are something completely different. Yes, if this were true and you wanted to know the "real truth about the Universe", you couldn't trust your experiments because they would mislead you. It's silly. Moreover, I am sure that you will agree that it would still be more important to learn the "apparent laws" that are needed to optimize our decisions if we assume that we do have some free will. Those would surely remain the most relevant laws of Nature for all practical purposes.

Superdeterminism is the ultimate conspiracy theory and I won't explain this fact again. She claims that about 3 people in the world are "working" on this theory. But one should mention that Google Scholar only shows a few papers on superdeterminism in quantum mechanics and none of them has over 3 citations.

Because this score is about 100 times poorer than the citation records of the "free will theorems", one is obliged to urge Ms Hossenfelder to tone down her arrogance at least by two orders of magnitude. Her superdeterminism babbling is worthless crackpottery but the free will may be given a reasonable definition which is compatible with the meaning of the "free will" in all other scientific, social, and political contexts. And with this sensible definition, the free will becomes a scientifically established fact and she has absolutely no evidence, results, or credentials to attack this important notion.

Bonus: a comment that is more than funny

The (deservedly) most upvoted comment under Hossenfelder's blog post said:
Hossenfelder: "I wish people would stop insisting they have free will."
Stor: How could they, if they have no free will! :)
It's a funny twist but if you think about it seriously, Stor has really shown that Hossenfelder's talk about the "free will" is self-contradictory. She wants to deny that anything that could be called the "free will" exists in Nature but her "wish" only makes sense if one assumes that the "offenders" actually have the free will. The free will is a problematic notion that is usually not used in fundamental physics because we may formulate laws without this word (and because we rarely discuss humans, their inner working, and cognitive and decision mechanisms) but one only creates contradictions if she wants to outright deny the existence of this concept.

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