Nachdenklichkeiten, Welterklärungen, Zehnte Klasse Leistungsstufe

Free will, the halting poblem and inferential agency.

Beware: This is a fragment. It was my first attempt to tackle the problem of free will; I wrote it roughly 1,5 years ago. It was only after that that I came to write the already published criticism to Sabine Hossenfelders blog and to some physical objections against free will in general. However, since I am currently writing a completely new (shortened and streamlined AND even german) piece about free will, I decided to publish this fragment as is, with all it’s flaws and fallacies…

0. Introduction.

Free will is often discussed as the conundrum of how our decisions could possibly be free on one hand while on the other hand we assume that everything in the physical world (including our brains) is determined by the laws of physics (i.e. not by „us“). Here, I shall give reasons as to why this way of posing the question of free will is largely misguided. I propose that the problem of free will should not be a debate about the special (and seemingly weird) kind of causation that might be at operation in volitional acts, but rather about the inner workings, the experience of and the interpretetation of the decision-making processes in our brains. In particular it seems that the ability to decide on the decideability of a decision is key to the „freedom“ in free will. And this in turn bears much resemblance to the halting problem in computational theory.

Fundamental aspects of these ideas have been put forward already in the 1960s by Donald Mackay and others but it was not until the 2013 paper of Seth Lloyd („A turing test for free will“) that free will has been explicitly connected to the halting problem. In this paper, Lloyd shows that regardless of the determinacy of the underlying world (i.e. regardless of the physics of our brain), a decider who is capable of self-reference can make decisions, which are fundamentally unpredictable to both, the decider and a possible outside observer of the decider.(fn) The fastest way to arrive at decisions, would therefore be to live through all the decision-making efforts it takes; generally, there is no shortcut (and no other author than the decider) to this process. Lloyd, however left out the reasoning as to why considerations about computability and predictability of computational outcomes would help to clarify the problem of free will. Here, I want to fill in some reasons, why the focus of the discussion about free will should be at least for the freedom-part on the computational routines used by the brain to exercise free will, rather than the causal nature of the brains physics.

Another gap in Lloyds argument pertains the type of unpredictability, he refers to. The algorithmic (i.e. logic) unpredictability of a deciders computations mainly precludes that another algorithm could be used to arrive faster at the final decision of the decider to predict her decissions. It does, however, not preclude the same algorithm on a faster computer from arriving earlier at the decision. You can simply think of the faster computer as the exact same brain in a slightly better condition (nutrition, temperature etc.). Such a brain could predict the outcome of its slightly less optimal doppelganger because it would (viewed as a deterministic turing machine) perform just the same operations — only a bit faster. Does the possibility of a faster doppelganger rule out free will? Are doppelgangers at all logically consistent with free will? I want to also give reasons, why the decisions of a decider would nonetheless be free in any sensible meaning of the word even if there was a doppelgangery decider.

Lastly, I want to refine the notion of free will that Seth Lloyd’s ideas point to. I hold that his ideas specify mostly the „freedom“-part, but fall short on the „will“-part. Therefore, I beg to differ from his (intentionally provocative) idea of any self-referential computer to be a bearer of free will. Since the notion of a will is tightly bound to some sort of self-awareness, I would pled to reserve the concept of free will to creatures/artifacts which can be both, free and the bearer of a will.

I. Is free will about fundamental properties of reality?

Let us start with some simple observations about different ways to decide. Free will is often conceived as the question about the causation of (and causation within) decisions: But what kind of causation are we looking for? And what kind of decisions are we looking at? The latter question is the easier one. Most philosophers would agree that the decisions should be relevant to the decider (i.e. the result matters for her in some way) and there should be no outer coercion; and furthermore, the decision should be made on rational grounds (i.e. the decider is allowed to ponder over her decision). Good examples for exercises of free will would be the choice of a job, the choice of school one sends her children to (provided, the financial situation allows for a free choice), but also smaller choices, like what restaurant to chose on that evening or what present to chose for christmas.

And while these are all perfectly valid (1st world) examples of free will at operation, there are also decisions, for which it seems less clear if they indeed are an exercise of free will: What about, for example, scratching an itching mosquito bite? While some may hold that one can chose freely at any moment to scratch or not to scratch, others may counter that it can be rather difficult to resist the desire to scratch, and that in fact, the persistent desire to scratch would violate a necessary prerequisite for free choice. Are the boundary conditions of free will met in itching the mosquito bite (obviously the argument can be accentuated by turning the itching into pain etc.)? Though it might seem to be only a discussion about meeting the conditions of free will, it does indicate the direction of the first point I whish to make: There are inbetween-decisions, i.e. decisions, which are not really free, but also not really unfree or purely subconscious. (Actually, we make these kind of decisions every minute. Maybe you chose to sit differently, while reading this. Or you chose to brew a quick coffee, to make for a more convenient read etc.)

And if your brain can make free and unfree decisions, as well as decisions, which are a bit free and a bit unfree (i.e. in between), then it is hard to see, how free will could be linked to (or conflict with) fundamental properties of reality like determinism or chanciness or mind/body-dualism and so forth. Because if the brain (more specific, the decision making processes it performs) seems to be capable of producing all shades of freedom, it would imply that the brain is somehow capable of exemplifying the underlying fundamental causal nature of itself in different ways. However, since the brain is always the same factual piece of biomachinery, it is rather implausible that it sometimes displays one way of causation and sometimes another. Instead, it seems a matter of which decision-making program is invoked by a certain decision-demanding situation. It’s the properties of these decision-making programs that matter, and the causation of their calling. (fn)

But before we turn to these processes, let me strengthen the above point by turning shortly to the question of how free will is brought about during individual developement, i.e. on the way from the new born to the young adult. Looking at the developement of children is (among other things) to look at the development of free will. Again, it leads to the observation that freedom of will comes gradually, as we gain the ability to refrain from our immediate urges and start to establish and later refine a decision making machinery. The ability to exercise free will evolves in several steps, like e.g. the ability to evoke a self-containing world-model (i.e. becoming conscious through the establishment of an „I“ which in turn can be the bearer of a free will); the ability to compute and take into account long-term consequences of ones actions; the ability for ethical reasoning and social responsibility and so on. It is only after all these abilities (call them „computing modules“ in terms of brain-descriptionary language) have been developed that we would concede a person being able to exercise free will; and this reflects in the agreement that we hold only adult persons to be fully accountable for their actions. In fact, it is an important (if not the singlemost important) criterion for adulthood to seemingly be able to freely and rationally decide, i.e. to bear a fully developed free will.

Like with inbetween-decisions (scratching mosquito bites and the like), the fact that free will depends on the existence of a refined decision-making machinery (consisting of different computing modules) strongly promts that free will is begging the question of understanding these computational modules, their interplay and their invokation, rather than the question of how the physics of the brain could possibly be reconciled with the freedom of our will. After all, the physics of the brain stays in all likelihood always the same. All that changes from childhood to adulthood and from subconcsious to free decisions — all that changes, are the actions that the brain performs to arrive at its decisions. (I have deliberately put the last sentence in terms of a brain-story instead of an I-story or decider-story to signify that the brain produces both, the „I“ and its decisions. I will argue below that this does nothing to refute, but on the contrary, permits freedom of will.) Therefore, I hold that its okay to stick to what is called „naturalism“, i.e. the view that we do not need some additional ingredient to physics in order to explain the causal nature of free will.

II. The „freedom“ in „free will“.

Okay. Let us become more specific on the freedom in free will; and whilst the concept of „will“ needs be problematized, too (and seems actually to be the deeper problem once you mulled over it for a while), it is mostly the „freedom“-aspect that gives the headaches when you start to wonder about free will. As suggested above, there are good reasons to look at the decision-processes in our brains to come to a better understanding of what this freedom pertains to. That is to say, this decision-process has some remarkable properties, such as:

  1. it can decide on the decideability of a decision-problem (i.e. it can determine its own halt)
  2. it can dynamically assess both, the difficulty and the importance of a decision
  3. it can actively seek to both, extend and reduce the informational basis, on which decisions are made (i.e. adapt the input of its computations)
  4. it consists, in all probability, of different semiautonomous sub-computations, pertaining different aspects of the decision, some of them on quite different timescales; only a few of these sub-computations come to the conscious mind of the decider (which itself, after all, seems to be just one of many sub-computations, that the brain performs)
  5. the subjective experience of hitting a decision-problem / going through the effort of decision-making / having made a decision has a longer tail into the past and future of the conscious decision-process, which helps both, to bring the matter to the conscious part of the decision-making processes in the first place and to stabilize the experience of having made a decision afterwards
  6. Decision-making is often embedded in a social context; not only indirectly such that many decisions pertain our social life, or that social interactions have provided us with much of the machinery for free decision-making (like language), but also in the very specific context, that within the process of decision-making we often seek other peoples opinions, thereby extending and paralleling the computing power of the decision-process and making it a matter of not only internal but also of external debate.

While the above aspects do not seek to give a complete account on the decision-making machinery in our brains, they where chosen because they all contribute in their own way to what might be called „openness“ of a decision, i.e. to the fact that making decissions is a computing-effort with unknown outcome (I will discuss the question, to whom this outcome is unknown in the next chapter). Point one in the above list means that decisionmaking is self-referential (in the Lloydian sense and with the consequences he pointed out); point two means that the computing-algorithm itsef may change during computation (as a result of the computation!); point three means that also the input of the algorithm may change during computation (again, as a result of the computation); point four and five indicate that not one big, but different smaller computations are performed (in all probability on quite different timscales), each of which may change the final outcome, thereby also contributing to some unpredictability and „openness“; and point six hints that even the number and kind of „computers“ and „algorithms“ may change during computation.

Apart from the overarching point one, point two and three are arguably most important reasons for the „freedom“ in free will. Both points entail that envoking „rational pondering“ when hitting a decision-demanding situation is not just a prerequisite or condition of free will but already a large part of it. What rational pondering really means (in terms of computing) is to adjust the computing algorithm and its data input until the whole situation makes for a decideable problem. And by doing so, our brain puts in a gap between the initial decision-demanding situation and our decision.

This gap is (in my view) key to the freedom in free will. Before we look at its far reaching consequences, let us first particularize on its causation and inner working. As already pointed out, our brains make many decisions subconsciously; these are decisions, which, upon hitting the decision-demanding problem, it finds very likely to be decideable. In such cases, our brain acts very much as an unfree decision-automaton; it recieves some input query and (after subconsciously judging the importance and difficulty of the problem), spits out one of the the learned solutions to the problem. Examples of these decisions may be to reorder your hair, to sit down while waiting etc. If, however, a decision can not be made fully automatically, the question is brought to the conscious mind, which entails envoking a much larger (energetically costlier, and probably slower) computational machinery than with subconscious (largely pre-fab) decisions. There is, in other words, another computation tied to every decision making, which reports on the difficulty and the importance of the decision-demanding problem (e.g. problems posed by other people are almost always important enough to not be automatically decided). It is this other (meta-)computation, which prevents human brains from being mere decision-automata. It causes that every decision-making process can be dynamically extended and escalated to some higher computing-instance or delegated to or split into subcomputations etc. In other words, the second computation puts a gap into operation between decision-demanding and deciding; a gap, which is dynamically extendeable, depending on the outcome of the preceding computing/deciding effort.

If a problem seems hard or important or both, it will not be automatically decided by our brain; instead, the initial decision making machinery is extended to a degree which can’t be known beforehand. In all probability, this extension entails some inner experimenting, i.e. the brain produces an ad hoc-model which is hugely data-reduced with respect to the real problem and therefore (hopefully) decision-ready; however, if the brain still can not decide, the model will be refined or drastically changed; other aspects of the problem are fed into it, while some superfluous are left out and so forth. And this is already quite a removal from initial instincts or urges. This is already a great deal of freedom.

Now, if the brain still can’t decide the matter (i.e. the decision on the decideablity is not unambigous), the brain can seek more external data to make the problem decideable. One would then, for instance, read on the matter. Or ask for other peoples opinions. Or learn something new, like eg. look at different clinics to decide the one which the decider would give birth to her child etc. This, essentially, brings the decision making machinery to halt. It does not, however, stop in the sense, that it renounces the decision altogether; it just does not come to a conclusion until its knowledge suffices to „make up ones mind“.

Are we slaves to our brains? Yes and no. It is the brains remarkable propensity to decide when a matter seems to be undecideable. In that case, it just stops its calculations and demands for a re-iteration, a change of the available information and so forth. We are slaves to the brain insofar as it forces us to make an important decision either sufficiently unimportant or decideable. But it does not force the decision itself upon us.

And while these are all very crude decriptions, we can use them to order these ideas in terms of „I-stories“ vs. „brain-stories“ and to refine our notions of freedom.

Lets stick a bit with the above points and particularize on some of them. Point one is adressed in Lloyd’s paper and might be arguably the most important one. It implies that the decision-making process is computationally open in two fundamental senses: (1) in the Lloydian sense that the fastest way to arrive at a decision is to make it (i.e. there is no computational shortcut) and (2) even more important, that _free_ decisions are those decisions, to which another (higher) decision-making process is attached, which decides about the halting of the first process (i.e. when a matter is „through“, or „ripe“ for decision). This suggests that the human brain has adapted to different sorts of dicision situations: Those, which are already known to come to halt (ie. they surely will yield a definite result) and others, which are not only harder in that they might require more computing to decide, but harder in the sense that it seems unpredictable if they will result a clear decision at all.

I suppose this second process to be part of our concsious experience of deciding, but the details of this will be left to a more detailed understanding on the neuro-computational machinery behind decision-making. However, I believe one important aspect of the „freedom“ in free will, is the ability (and the felt effort that the exercise of this ability is) to postpone a decision and delegate it to all sorts of sub-computations until the decider decides that the brain has come to a solution. To put it more catchy, one aspect of freedom in free will lies in the _ability not to decide_ until an agent feels the matter to be decideable. I other words, the decision-machinery puts a gap between the situation which calls for a decision and the decision itself. It makes room (time) for computation (rational pondering). It may postpone or dismiss initial urges or simple solutions in favor of allocating more computing power to a problem until it finds it sufficiently decideable. This gap can be legitimately called „freedom“ because it frees the decider not only from being slave to its immediate urges (like animals or small kids) but also opens the possibilities to arrive at different outcomes and to broaden the computational basis of a decision at will. This is not just „talk of freedom“ but a real freedom, because it immensely broadens the possibilities to behave and to find solutions.


I want to further argue that all these different aspects of the decision-process work together to make for a _real_ opennes of our decissions. Here, the word _real_ signifies that it is not merely the subjective experience of a decider not to know the outcome of his decision-making in advance, but an objective property of the brain processes to arrive at outcomes which are in all sensible meanings of the word, open. (I will look closer on the differences between the I-story and the brain-story of decision-making in the next chapter.)

It is, however, only one aspect of the freedom in free will. Equally important is the Lloydian finding that a decider can mostly not know what the outcome of his decision-making will be before she has actually made it. This is not so much important on in an objective world of „how things are“, but in a subjective world (i.e. for „me“ as a decider and my subjective experience). Although it may well be that an recursive decision-making algorithm is also objectively unpredictable it matters much more to the decider that the subjective process needs to be fully completed before she gets to know the answer.

Deciding, prediction testing and consciuousness

Bayesian machines

What does it mean to „update my knowledge“?

Now I don’t consider humans to be particularly rational animals but as far as rationality goes in the animal kingdom, still the most rational animal there is. And, as it stands, rationality seems to be one source of his freedom. The arrival of rationality between a stimulus and action is one important source of freedom; and to some extent also the „freedom“ in free will.

III. To whom are decisions open?

This chapter will be about the reality of the openness of decisions. And lets assume that there are three different levels, on which decisions can be open:

  1. On the lowest level, openness it could be only subjective. It’s only the decider, who does not know the outcome of the decision-making processes and therefore she has to go through all the efforts it takes to make her decision. There might be, however, the possibility that somebody else can get to know the decisions of the decider in advance.
  2. The second level would be openness of one deciders decisions to all other persons as well; lets call this openness „for all practical purposses“ (fapp). In this case not only the decider but anybody in the world would be unable to know the result of a deciders decision, regardless the effort they make.
  3. The third level of openness would be openness to even an all-knowing hypothetical being, like Laplaces demon or god (or your mum and dad, for that matter). Let’s all this openness „in principle“. If the decision-making machinery of our brains would be open on even this level, one could declare the decisions of a decider to be open on all accounts whatsoever.

There are two different discussions connected to the above levels of opennes: One discussion about which level of openness would be sufficient to fullfill the requirements of free will and another discussion about which of the above levels of openness is reached in decision-making processes in human brains.

IV. Is openness == freedom?

I hope, I could convince the reader that the decision-processes in our brain are fundamentally open in many different ways. But is opennes necessairy and sufficient for the „freedom“ in free will? The necessity-part is easy: Ofcourse it is necessary for the experience of being free in ones decisions to not know the outcome of thinking about it beforehand. It would also constitute a paradoxon: Knowing the outcome of decisionmaking while at the same time knowing that without taking the effort of the computing the decision, there will be no decision at all. WTF?

But what about the sufficiency? Does it suffice not to know what ones effort to decide a matter will result in? Isn’t it rather trivial to say something like „predicting the future is impossible, therefore we are essentially free“? Does the „freedom“ in free will not pertain to some kind of autonomous agency in the procedure of predicting the future and making decisions?

I think, there are some misconceptions hidden in these questions.

First of: Retrodiction is possible. This is actually not an argument against, but in favor of free will. The brain decides what happened, even in a (almost) fully determinate world!

Free machines? Are self-referential machines bearers of free will in the radical sense of Lloyd? I would beg to differ. Machines can be free in a Lloydian sense, but they are not bearers of a will. A will has to do with self awareness of the computing machine in question. (<< read Mackay again)

Informationally (computationally) openness vs. physical continuity

Lets reiterate: Who is the bearer of free will. Is it the brain, or the „I“ which is produced by the brain?

The I-story vs. brain-story repeats itself on a grander level when thinking about the determinacy of the future of the universe; i.e. THE FUTURE. The question about the future of the universe can be posed either in terms of a passive, objective story (ie. To which degree of certainty it is possible to know what will happen next? == universe as a brain) or in terms of a subjective story (ie. does the universe know, what it will do next? == universe as a decider)? The interesting thing is, that — although it seems to be just two slightly diffrent ways to put one and the same questions — it invokes two very different lines (and fomalisms) of reasoning. While the objective story in all probability invokes some talk of phase space (or hilbert space, for that matter), the subjective story invokes talk about information, Bayesian inference and computeability. And while there are some well known links between the two ways of describing the evolution of a system (causing its own future), there are some huge metaphysical differences to be pointed out.

Some misconceptions within the free will-debate

„Fundamental determinism“ (and its refutation by the 3-body problem)

One claim of determinism is, that the future of the universe (and therefore the future of anyone) is calculatable at least in principle; it’s only a lack of knowledge that prevents us from doing so. Here, I shall briefly argue, that this idea is somewhat extreme, if not fundamentalistic. It is also, as it seems to me, plain wrong.

Here is why. Let’s see, how this deterministic prediction of the universe could be done „in principle“. First of, one should be able to assemble all the necessairy information about some state of the universe to feed it into the predictive computation. Now, this information is in all probability HUGE. It contains at least something like all the positions and momenta of all particles in the accessible universe (lets assume, the universe would be ultimately made out of particles). One would have to devote quite an amount of the universe just to store all that information and even more, to process it. Seems like something on the order of ten to the power of 80 bits (plus minus some orders of magnitude). But lets say, there would be a clever way to compress all the required data so that a comprehensible chunk of matter could be used to store and process that information. Okay. But then we also need to make shure that this chunk of matter (our storage + computer) is only very weakly connected to the rest of the universe, so that computations, performed on it, would not perturb the rest of the universe. In fact, no information would be allowed to leak backwards (i.e. from the computer to the universe). Otherwise, we would run into an infinite regress, because the state of our prediction-computer itself would be needed to feed in its own computation, which in turn brings us to the halting problem (i.e. a computer that is asked to predict itself), which has been proved to be an unsolvable problem. So, to sum up the requirements for being „in principle“ able to predict the future of the universe, we need quite a large chunk of matter, that is connected to the rest of the universe such, that it can gain all required information about it without interfering with it and without telling the universe the results of its computations. Sounds familiar to you? An all-knowing entity, quite outside the universe we live in, but able to predict everything within it, while at the same tim being unable to reveal its knowledge to the universe? Most people would call this a „god“. So, basically, saying that predicting the world is possible in principle, is just a more advanced way of posing that in principle there can be something like a god. Now, the funny thing is, that the people, who hold this strong kind of determinism, are usually on the opposite of being a theist.

Okay. But that is still not an argument. It just reveals some of the fundamentalism hidden within the notion of „in principle predictability“ of the universe. Here comes the argument. And it draws on the very old knowledge of the 3-body problem. I am not going to repeat the problem here, but what it basically entails is, that already with three bodys (say spheres), that interact via simple power-law potentials, it is impossible to predict their positions and momenta for all future, even if you knew their positions (and momenta) at a given instant with infinite precision. The really important word here is „infinite“. Already for a tiny universe, consisting of a mere three objects, it is impossible to predict all it’s future, simply because it requires infinite knowledge (i.e. infinitely large storage), which, even in a loosely attached infinite computing-universe, would take infinitely long to write and read (and to process), which, in turn, prevents it from being able to predict anything. So. Already assembling the knowledge required to predict the future of three bodies (say in a closed spacetime topology) is an infinitely large task. And that has nothing to do with quantum mechanics and Heisenbergs uncertainty principle (which puts a limit to the accuracy of information we could gain and use for making predictions anyways). So, to conclude, there is not even „in principle“ the knowledge (information) attainable to predict the universes future. It’s just not there; if you tried to somehow extract it, it would take forever (therefore leaving no time for predictions). Nobody, not even an all knowing god, could „in principle“ predict the future of the universe. One has to just accept, that even in a fully deterministic universe, the future in real universes (i.e. containing slightly more than just three interacting particles) is principally underdetermined. And if you don’t want to look like a fundamentalist, you better do so, too.

Laplace’s Demon.

Okay. If the laws of physics do not allow for perfect pre- and retrodictions, what could determinism pertain to at all? This is mostly (without going into detail at all) to refine the idea of determinism: What the laws of physics (be it quantum mechanics, relativity, thermodynamics or classical mechanics) – what those laws really entail is not some form of Laplace’s demon. Never have been. Instead, they all posit the laws of physics to be either continuous (with respect to time) or at least do very small jumps only in phase space (or hilbert space, for that matter). The universe (as we know it) does not perform large jumps, for whatever reason. It’s just not the case. And that entails that two universes which are similar in all aspects at one moment will not be all too different in the moment after that moment (or have been all tot different from that moment before). This idea of a smooth metric is the true core of determinism. It does, in fact, not entail any inferences about the long term future (or past) of a system, but only about some „prediction horizon“ („retrodiction horizon“), the reason being that the information, which is needed to obtain an infinitely large prediction horizon is („in principle“) not extractable from the universe. It is, as I said already, not there. Not even in principle (if you accept infinity as the ultimate refutation of „in principle“).

Question: Can this be proven or even calculated? Would it be possible to treat the prediction horizon as the question of how two neighbouring points in phase space can be told apart as a function of their distance in time“?

If so, then the whole problem could be treated similar to the derivation of Abbes or Rayleighs resolution limit. Is there a point spread function in phase space? Try it!


— end of fragment —


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