Tuesday, February 07, 2012

The Free Will Function

Last year's FQXi conference was a memorable event for me. Not only because it doesn't happen all that often that my conversation partners abandon their arguments to hurry away and find some more of these pills against motion sickness. But also because I was reassured that I am not the only physicist with an interest in questions that might become relevant only so far into the future that the time spent on them rests precariously on the edge between philosophy and insanity.

Scott Aaronson (who blogs at Shtetl-Optimized) went ahead and gave a talk about free will (summarized by George Musser here), which in return encouraged me to write up my thoughts on the topic, though I've still hidden them in physics.hist-ph (which I previously didn't even know exists!):

So, here's the executive summary.

In my previous post on free will, I explained that I don't buy the explanation that in a deterministic universe, or one with a random element, free will exists as an emergent property. If you call something free will that emerges from the fundamentally deterministic or probabilistic laws that we know, then I can't prevent you from doing that, but there isn't anything free to your will anymore. If free will is as real as a baseball, then you have as much freedom in making decisions as a baseball has to change its trajectory in Newtonian mechanics, namely none.

You might seek comfort in the fact that it is quite plausible that nobody can predict what you are doing, but this isn't freedom, it's just that nobody is able to document your absence of freedom. If your "will" is a property of a system that emerges from some microscopic laws, its laws might be for all practical purposes unknown, but in principle they still exist. If time evolution is deterministic, any choice that you make now strictly followed from an earlier state of the universe. If time evolution has a probabilistic element, as in quantum mechanics, then choices that you make now must not necessarily follow from earlier times, but you didn't have any choice either because the non-deterministic ingredient was just random.

Needless to say, I have greatly simplified. Notably, I've omitted everything about consciousness and the human brain. Look, I'm a particle physicist, not a neurologist. The exact working of the brain and the question whether quantum mechanics is relevant for biological processes don't change anything about the actual root of the problem: There is no room for anything or anybody to make a decisions in the fundamental laws of Nature that we know.

One way out of this problem is to believe in what is known as "strong emergence." That would be if the laws of the macroscopic emergent systems (e.g. "you") do not follow from the microscopic laws. The only people I have met who managed to make sense of this idea are philosophers. There is presently no formal way to achieve such a behavior and there is no known example how this could work. (We discussed here a paper that made an attempt into this direction, but note that the assumption of an infinite rather than a large system is crucial for that to work.) But yes, finding an example for strong emergence would be a possibility. Just that I couldn't find one.

My paper is much simpler than that. In my paper I just pointed out that there exist time evolutions that are neither deterministic nor probabilistic, certainly not in practice but also not in principle. Functions that do that for you are just functions physicists don't normally deal with. The functions that we normally use are solutions to differential equations. They can be forward-evolved or they can't and that is exactly the problem. Yet, there are lots of functions which don't fall in this category. These are functions can can be forward evolved, yet you have no way to ever find out how. They are deterministic, yet you cannot determine them.

Take for example a function that spits out one digit of the number π every second, but you don't know when it started or when it will end. You can record as much output from that function as you want, you'll never be able to tell what number you get in the next second: π is a transcendental number; every string that you record, no matter how long, will keep reappearing. If you don't know that the number is π you won't even be able to find out what number the algorithm is producing.

The algorithm is well-defined and it spits out numbers in a non-random fashion that, if you'd know the algorithm, is perfectly determined. But even if somebody monitors all output for an arbitrarily long amount of time to an arbitrarily good precision, it remains impossible to predict what the next output will be. This has nothing to do with chaos, where it's the practical impossibility of measuring to arbitrary precision that spoils predictability: Chaos is still deterministic. The same initial conditions will always give the same result, you just won't be able to know them well enough to tell. Chaos too doesn't allow you to make a choice, it just prevents you from knowing.

But what if you'd make your decisions after a function like the one I described? Then your decisions would not be random, but they wouldn't be determined by the state of the universe at any earlier time either (nor at any later time for that matter). You need to have your function to complete the time evolution, which is why I call it the "Free Will Function."

This is far too vague an idea to be very plausible, but I think it is interesting enough to contemplate. If you would like to believe in free will, yet your physics training has so far prevented you from making sense of it, maybe this will work for you!

If you found this interesting, George Musser has storified the topic, so you can continue reading.

95 comments:

Uncle Al said...

If reality is a one-time pad,"free or determined" is not meaningful. It is inoperative to ask.

Perturbation theory specifically excludes emergence. Real world symmetry breakings are manually inserted. Epicycles indicate failed theory. If complexity successively emerges with scale, theory is a global heuristic - no TOE, perhaps no unification of geometry and probability.

Two classes of experiment: zero disorder and maximum disorder. The giggle is if they meet at the far side, the way population inversions have negative temperatures kelvin.

iya said...

"If you call something free will that emerges from the fundamentally deterministic or probabilistic laws that we know, then I can't prevent you from doing that, but there isn't anything free to your will anymore."

Technically you're correct, but it's not what actually matters when talking about free will. There's no higher self or soul which could decide something. All that exists is our body including our brain. Our brain is an advanced processor, running complicated functions that process inputs (senses) into outputs (actions), and that's what can be called "free will". Every brain is different and even your own brain is different from what it was yesterday and what it will be tomorrow.

Where it matters is in court, for example. By saying there is no free will, you're suggesting nobody is responsible for his actions, but since we can't be separated from our body and brain, which is the decision maker, we are responsible.

andrewg said...

Since one can find any arbitrary string of digits within pi, how is "consecutive digits of pi starting at a random digit" any different in principle from "random digits"? It's just a pseudorandom number generator with a random initial seed (the starting digit).

Bee said...

Hi Iya,

You probably didn't read my earlier post on free will. I recommend you do it know.

The argument that saying there is no free will is a socially difficult position doesn't bear any relevance to the question whether free will does or doesn't exist, just on the question whether one should talk about it.

I did however explain in my earlier post why the point you bring up is a non-issue. It doesn't matter for the solution of the problem whether somebody had any free will in committing a crime, you just have to ask the question differently. Instead of asking "what's the right punishment for this evil person" you should ask "what is the best thing to do to prevent further evilness." (Evilness meaning all further sociological consequences that are not beneficial.)

"Our brain is an advanced processor, running complicated functions that process inputs (senses) into outputs (actions), and that's what can be called "free will"."

If that's what you want to call "free will" there's nothing "free" about it. You should ask yourself, given input A, could there have been any other output but B? If not, what "freedom" did you (your brain/your processor) have in coming to a certain conclusion/performing a certain action? Best,

B.

Arun said...

Dear Bee,

This is excellent! disentangling all the philosophical mumbo-jumbo around free will/responsibility from the science of it.

Your free will function brings in mind a vision of a countable infinity of oracles, each of which has all the history of digits emitted of π emitted so far, and each making a prediction of the next digit. At each tick of the clock, an infinite number of these oracles are invalidated, but left behind are an infinite number of oracles not yet discredited.

--- As a practical matter, I'd ask the FQXi participants that when doing research, they constantly have to make decisions, do they feel happy that these decisions are all pre-determined.

Arun said...

iya: ". By saying there is no free will, you're suggesting nobody is responsible for his actions..."

Well, what is the point of arguing? Whether we hold someone responsible for their actions, or do not is a choice, and our choice is predetermined. We argue because we must, that too is predetermined :)

Bee said...

Hi Andrew,

Think about why it's called a pseudo-randomnumber generator rather than a randomnumber generator. Because it's not actually random. If you know the seed, the outcome is deterministic. If it was a true random number, there was nothing to know there. Best,

B.

Bee said...

Dear Arun,

Thanks for the kind words. Unfortunately, the example that I came up with isn't very satisfactory. (See how people get confused by its closeness to randomnumber generators?) I was hoping I'd find a non-trivial example of an evolution that is a combination of forward- and backward deterministic, but nothing came to mind. Best,

B.

andrewg said...

Pseudorandom number generators are only pseudo because the seed is not completely random - it must lie within the numerical range that the computer can address, reducing the search space for the output number strings. But pi has an infinite number of non-repeating digits, and we can start at any of them, so as we increase the allowed range of the initial seed, the search space for output strings approaches infinity and our pseudorandom generator approaches true randomness.

Also, I don't see how "they wouldn't be determined by the state of the universe at any earlier time either". What physical process is calculating the "free will" function? Is it not part of the state of the universe?

uair01 said...

In this discussion you seem to forget the undelying assumptions. You postulate a concept like "free will" without asking yourself "whose will?" or "what is the entity that is doing the willing?". If you try to answer those questions you will run into more problems. I don't think that physics can cope with the concept of a "person" or a "human being".

PTMR said...

Interesting ideas but I have the same objection Andrewg already stated.

The distinction between such a "free will pi function" and a "true random" function would be semantic only with no empirical way to tell them apart.

To make the equivalence more apparent lets turn things around - let's say you have a "true random" function and you make decisions according to it's output. Let's say you sampled the function n times during your lifetime. Now in the space of all mathematical functions there will be an infinite amount of completely deterministic functions whose consecutive output will match exactly your n samples so you could have just as well followed one of those and nobody would ever be able to tell the difference because there is no real difference between those two cases other then the semantics.

T. said...

Sabine, this is very interesting many thanks.

Looks to me like your transcendental free will function needs to be evolved from a universe that is non-deterministic? Because if it was then from its state at any time t, one could predict/retrodict the algotithm of the free will function for every being that will ever live/has ever lived ?

T. said...

Let me state that more clearly.
I'm assuming you agree that we are evolved beings, so your transcendental free will function still needs to be evolved.

Now even if this function is transcendental, if it has evolved in a universe governed by deterministic laws, then by knowing the (say) initial state of the universe one can predict exactly the algorithm of every free will function in the future. Whose outcome then cease to be unpredictable.

So if we have free will functions as you describe, then it seems to me that it is incompatible with a deterministic universe (as portrayed for instance in the many worlds interpretation of quantum mechanics).

Neil Bates said...
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Neil Bates said...

If the U is determined, then no FW, no matter what. If the U isn't determined, then viability of FW depends as a more subtle question. I think it's clear, to anyone not smothered under MWI, that a given state does not by necessity lead to a unique later state. Hence, FW is possible and to me, plausible but hard to make the case.

Well. how about an empirical reason to believe in FW? I argued in presentation at "Tucson 2000" conference on consciousness, that FW is revealed (simplified summary) in our ability to suddenly stop what we're doing, and to "freeze". That is a suppression of pre-existing trains of activity, it requires a powerful intervention of a "boss" state over the various producing sources of behavior. The capability, usually so easy, is evidence against the bottom-up constructive behavior theory known as "pandemonium."

I consider it quite possible for that to be an emergent property of something indeterminate yet at the same time, not a mere throwing together of separate random events. That requires some sort of wholeness, and at least we have some basis for such interconnectivity. (Iya, that is the relevant feature, the wholeness of that system, not whether there is "something else" supervening on top of it as well.)

Go ahead, complain about "woo" if you want, it's not near as woo-sy as the idea that decohering of superpositions somehow turns them into mixtures as some have said, arguing in rather post-modern ways. But I don't know for sure anymore than anyone. Yet I do reject that there cannot or we are sure there's no FW, which is not the same as knowing for sure there is.

BTW, as I have tried to convince Bee, math functions by necessity follow "determinism" in principle no matter how hard it is for someone to find out. The pi-digit "spitter" is determined regardless of whether someone can figure it out, that is just icing on the cake.

Arun said...

For the non-physicists, why not think about a Turing test for free will? If someone's behavior is indistinguishable from free will, then for practical purposes he/she has it.

Neil Bates said...

Arun, that's fascinating but what sort of behavior qualifies per se as properly "free will?"

Bee said...

Hi Andrew,

You can't send the search range to infinity unless it's a mathematical abstraction. Whether your random number generator can be really random depends ultimately on the fundamental laws that govern it, which is what I'm talking about.

"I don't see how "they wouldn't be determined by the state of the universe at any earlier time either". What physical process is calculating the "free will" function? Is it not part of the state of the universe?

Depends on what you mean with universe. It's not "in time" if that's what you mean. You're having a problem here because you are thinking of the function as something that is actually calculated by some machine or so. But it's just a function, it's a mathematical thing. You put in t and it spits out F(t) and that's it. As I said, it's not particularly plausible, but that's not the point. The point is that it's logically and mathematically possible. In contract to all other explanations of free will. Best,

B.

Plato said...

Just gathering some perspective.

Determinism/Indeterminism

Plato said...

"If experimenters have free will, then so do elementary particles."-John Horton Conway

Plato said...

Origins of CelLab-

A number of people at MIT began studying cellular automata other than Life during the 1970s. Probably the most influential figure there was Edward Fredkin. He had the idea for the two-state "Parity" rule under which EveryCell counts up the number of its firing neighbors and goes to 1 if the number is odd and to 0 if the number is even. The dimensionality of the space and the shape of the neighborhood are not crucial--in almost every situation, the Parity rule will make copy after copy of whatever starting pattern you give it. The copies are grouped into fractalized clusters. ¹ Although he himself holds no higher degrees, Fredkin was made a professor associated with the MIT Laboratory for Computer Science, and he directed a number of dissertations on Cellular Automata.

Bee said...

Hi PTMR,

Why do you guys get so hung up on the example with π? This is just something I've made up to show that functions with the property I explained a free will function must have do exist. I am certainly not saying that's a realistic or even remotely plausible example.

About your criticism that this is just semantics: if I had known any observable consequences, I wouldn't have dumped the paper in the philosophy section! All I said I'd do is that I offer you a way to think about free will that is really free and yet logically possible, and perfectly compatible with the idea that reality can be described by mathematics. And that's what I have done.

However, even though I can't deliver any concrete test, I don't think it is impossible to tell the difference between a random process and a free will function. Just because one cannot predict the time evolution of a system doesn't mean one can't tell anything about it. Imagine for example you have a theory of everything and you find in it some free will functions. Now you know what type of function it is. You might even know some more things, maybe something about spatial correlations, or conserved quantities, etc. That still wouldn't allow you to make a prediction, but that would be a description that's operationally much richer than just throwing the hands in the air and saying, it's random, period. Best,

B.

Bee said...

Hi uair01,

I'm a reductionist and while it is totally impractical, any "person" is merely a large selection of elementary particles that fulfill certain properties. Specifying these properties doesn't fall into the realm of physics, but it's entirely irrelevant to the question I am addressing. I'm not asking what properties you need to specify to get a meaningful definition of "person," I am asking what are the fundamental laws its constituents, and thereby itself (unless you believe in strong emergence) is governed by. Best,

B.

Bee said...

Hi T,

The free will function doesn't evolve. It just is. It plays the same role as a fundamental law of physics. Imagine you have a theory of everything and it has a master equation and that master equation describes, well, everything. The equation itself doesn't evolve because for that you'd need another equation, and so on. Is this equation itself then part of the universe? Depends on what you mean with universe. I don't see what prevents you from defining the universe either way. Best,

B.

Bee said...

Hi Neil,

Lack of determinism alone doesn't mean free will. You have to do more, which is what I've done in my paper.

Also, you apparently still haven't understood what a random variable is, otherwise you wouldn't still insist that "math functions by necessity follow determinism." You should have a conversation with Andrew, who seems to know more about the topic than you do. A true mathematical random variable is by definition not deterministic. It is a mathematical abstraction. Whether it exists in the real world depends ultimately on the fundamental laws. This is why (and I know I've said this several times before) your argument is ultimately circular.

I don't know how "freezing" shows free will. It's just a very non-linear behavior that we therefore rarely observe in non-animated natural objects like, say, a stone. My computer however is perfectly able to suddenly stop what it's doing and freeze. Should I conclude now that it has free will? Best,

B.

T. said...

Hi B.,

You say: "The free will function doesn't evolve. It just is."

You also answered to uair01: "I am asking what are the fundamental laws [a person]'s constituents, and thereby itself (unless you believe in strong emergence) is governed by.".

It seems to me that even defined in this way, as a disembodied mathematical formula floating in the space of all mathematical formula, the free will function still evolves, because it is the fundamental law that is governing a living system.

In other words, the fact that they are describing systems at an abstract level doesn't free the free will functions from ... being descriptions of these systems. If these systems have evolved, then so have the free will functions that describe them.

(Evolution then being reframed as a search in the space of free will functions, but that's not new in a sense, just replace "free will" by "fitness" and you fall back on the notion of an evolutionary landscape).

So in my view you still end up with free will functions that have evolved. Whether or not that is a problem for your approach is unclear to me though. Cheers,

Thomas

Bee said...

Hi Thomas,

You write that a "mathematical formula.. evolves" when it is a "fundamental law that is governing a living system." Leaving aside the question why it is relevant that the system "lives," I am just not using the word "to evolve" in this way. I mean evolution in the physical sense, something "evolves forward in time" if it has a time-dependence and something that gives you this time-dependence. This is not the case, for example, for a truly fundamental law (or constant), whether or not it describes a "living system" (whatever that is), and it's not the case for the free will function either. Best,

B.

Giotis said...

"Then your decisions would not be random, but they wouldn't be determined by the state of the universe at any earlier time either (nor at any later time for that matter)."


I don't understand what's the usage of this abstract function:-)

As far as I'm concerned you have arbitrary assigned properties to this abstract function which basically equates it with a God like metaphysical entity outside time and space who does not follow the fundamental laws nature.

If you make your decisions based on the will of this metaphysical entity this would have the same effect with your function.

Again your decisions would not be random (they will follow the will of the entity which is undetermined) and they wouldn't be determined by the state of the universe at any earlier time either.

Bee said...

Hi Giotis,

I don't know why you call something metaphysical just because it doesn't evolve in time. And yes, I have assigned arbitrary properties to that function. Clearly, that doesn't make a lot of sense and is not very useful either, but I was just trying to come up with a mathematical example to show what I mean. What you'd ideally want is to find a sensible theory that yields such functions, and tells you something about their embedding into the time evolution (e.g. so that they only become relevant for complex systems, is one thing you'd hope for). Best,

B.

T. said...

1. I think it is very relevant that the system lives because these are the only candidates to free will that we have yet observed. I thought your paper was describing living systems (in fact you quite explicitly make a disclaimer in your post that you have not factored in anything about neuroscience and that this might be a limit).

Whether or not it is necessary that a system be living to have free will is another, less important question in my opinion. The point is that all candidates to free will we know of have evolved.

2. Physical vs natural selection evolution: the free will function governs the physical evolution of a system in time. But the algorithms of the free will functions themselves must have (selectively) evolved, if they are to represent living systems. Natural selection on living systems must be mirrored by a (blind) search in your free will function space.


best,
Thomas

Bee said...

Hi Thomas,

You have entirely misunderstood what I said about neuroscience. I didn't say that this might be a limit, I said it's irrelevant to my argument.

There are no algorithms that *must* have evolved anything about any natural law, and not about the free will function either. It is really puzzling to me, at which point people started not only to think about the universe as a computer, but to actually believe that it cannot possibly be different.

What you are maybe trying to say is that some ways by which living systems could make use of free will have a selective advantage over other ways. Yes, sure. I haven't said anything about what is a good or not so good way to make use of free will.

Best,

B.

T. said...

Hi B.,

You do not confront the fact that the only systems we currently know of that can pretend to free will are living beings, and that they have evolved.

Why do you not see that this means your free will functions do not come out of the blue, contrarily to how (some) phycisists think about the physical laws that govern our universe ?

And the history of your free will functions can then inform us on what they do.

Best,
Thomas

andrewg said...

But it's just a function, it's a mathematical thing. You put in t and it spits out F(t) and that's it.

In other words, it's a hash function. All the information in the output is contained in the choice of function and the input. If the output is to be non-deterministic and non-random then the input must also have been. All you've done is obfuscate the question.

Bee said...

Hi Andrew,

I am not "obfuscating" the question. I have clearly explained in my paper that the determinism that I am concerned with, the one that does not allow you to have a free will, is a forward determinism in the time evolution. The determinism you refer to, in that the choice of the function and the input determines the output, is just not what is relevant to the question I have addressed. You could say that all I have done is to point out the distinction between both. Best,

B.

Bee said...

Hi Thomas,

It is entirely correct that I do not discuss "living beings" and I have no intention to. How one or the other subsystem comes to make use of a free will function, I don't know, maybe you have a suggestion. Of course the "free will function" does not come "out of the blue." I have said exactly the opposite, it doesn't come or go anywhere. You can however have systems that learn (evolve to) make or not make use of them. Presumably, you need to reach some level of complexity to be able to. But really, that's totally besides the point for my argument. Best,

B.

andrewg said...

If the input to F(t) is set by some nonphysical method, then it is a tap leaking information into the universe from elsewhere. The total amount of information within the universe must therefore increase monotonically with time. We're throwing out the second law of thermodynamics now?

Bee said...

I don't know what you mean is "nonphysical" about using some parameter as input. With regards to the universe, see above comment, depends on what you mean with universe. You can always increase the system so it's closed. In my understanding the universe is just all there is, so by that definition the free will function (and information it provides) is part of the universe. Though that notion of universe seems to have gone a little out of fashion lately. Best,

B.

T. said...

Hi B.,


Would you agree that the critical point in your paper is that although free will functions are deterministic, there is no way to tell what number they will return next if you do not have their algorithm available.

If so then the question of what one (timeless, ideal) observer could know about their algorithm becomes relevant. Would you not agree with that? i think this is the pivotal point in my argument.

Now you claim that your approach of free will operates at a level that goes beyond living systems (!). let's say this is true just for the sake of the argument, because I don't think you can discuss free will without discussing living systems (currently). Still, the free will functions of interest here describe material, physical systems.

But these subsystems have evolved (physical sense) according to some laws, and if you know the laws and the state of the entire system at some point, you then know something about the algorithms the free will functions that describe them are using (I've said that before, so to avoid repeating that last paragraph in the future I will refer to it as "paragraph A").

Best,
Thomas

Bee said...

Hi Thomas,

You are again putting words into my mouth that I never said. I never said that "my approach to free will operates at a level that goes beyond living systems." You are entirely misunderstanding what I am telling you. I am telling you, the free will function is a necessary ingredient to make sense of free will. It does not tell you anything about how or why or when "living systems" use them. But if there's no free will functions, there's nothing they can use to begin with.

"if you know the laws and the state of the entire system at some point, you then know something about the algorithms the free will functions that describe them are using"

That is a very vague statement. If you record all the output of a free will function, does that mean "you know something about the algorithm"? Arguably you know now that it has produced this or that output. But that doesn't mean that the way this output is produced depends in any way on the evolution of the system that uses the output. Best,

B.

Phil Warnell said...
This comment has been removed by the author.
Phil Warnell said...

Hi Bee,

An interesting piece and hypothesis which has so many twist and turns to it I really need to think it through. However just as an initial point of observation I think what many call free will is tied up with another fundamentally human quality, which is the ability to maintain hope regardless how inevitable undesirable outcomes may seem to be. That is I’m wondering if what you call a “free will function” couldn’t just as readily be called a “hope function”.


“Hope is important because it can make the present moment less difficult to bear. If we believe that tomorrow will be better, we can bear a hardship today.”

-Thich Nhat Hanh

Best,

Phil

andrewg said...

Alright then, I'll rephrase.

The "free will function" is an infinite, hidden reservoir of information which leaks out into the observable universe at some nonzero rate. To the naive observer, this will appear as a violation of the 2LoT.

Giotis said...

"What you'd ideally want is to find a sensible theory that yields such functions, and tells you something about their embedding into the time evolution"

Yes Bee this exactly what you need and since you can't possible have such theory then sorry but I don't see the science behind your proposal. It is closer to theology in my opinion.

Bee said...

Hi Andrew,

I agree except on the addition that the reservoir is "infinite." I don't even know what that means in this context.

B.

Bee said...

Hi Giotis,

As I already explained above, the intention of my paper is merely to show that it is logically possible to make sense of free will within the context of modern physics. I have not claimed that I know how to test this hypothesis. If I'd know how to do that, I'd not have said it's a philosophy paper. I don't know why you criticize me, and unconstructively so, for not having provided something I haven't claimed to provide. Best,

B.

andrewg said...

It is infinite in that it is a string of aleph-0 random digits, each containing finite information.

Fair enough, you've convinced me that F(t) is not forward-deterministic. But you have done so by showing that it is truly random.

T. said...

Hi B.,

I have put to you two very clear questions, which you haven't answered. It's a shame because i think we would make progress if you answered them.

You are again putting words into my mouth that I never said. I never said that "my approach to free will operates at a level that goes beyond living systems."

I never said you said that. You are misquoting me misquoting you. You said: "It is entirely correct that I do not discuss "living beings" and I have no intention to. " which to me is an even stronger statement.

I am telling you, the free will function is a necessary ingredient to make sense of free will.

I do hope you are saying as much, yes, otherwise i wouldn't see the point of the paper.

It does not tell you anything about how or why or when "living systems" use them. But if there's no free will functions, there's nothing they can use to begin with.

You fail to notice that I have extended the argument to non-living systems. And you do not adress my main point: that what an observer can learn about the nature of a free will function should matter in your approach.


If you record all the output of a free will function, does that mean "you know something about the algorithm"? Arguably you know now that it has produced this or that output. But that doesn't mean that the way this output is produced depends in any way on the evolution of the system that uses the output."

mm let me think about this one.

Well anyway, I don't want this argument to obscure my main reason for commenting: I think your paper is very interesting and thought-provoking, and what is more your post was crystal clear so it made the ideas accessible. looking forward to more. best,

Thomas

Giotis said...

"is merely to show that it is logically possible to make sense of free will within the context of modern physics"

Then I must have misunderstood your proposal because in my opinion you didn't show such thing. Could you explain how you were able to show that by merely postulating an abstract mathematical function completely unrelated to the physical world and its laws.

Neil Bates said...

@Bee: Yes, sure. I already implied that non determinism is a necessary but not sufficient basis for FW, wish I'd said so explicitly.

You wrote: "A true mathematical random variable is by definition not deterministic." - Sure, you can define a term by fiat but that doesn't mean it can even exist. Indeed, it is a mathematical abstraction - not an actual process that generates varying results in a given case.

Look at the use of so-called RVs. Their output is not given in an actual "run" but used in the abstract to make a frozen "set." It's not an actual sequence like 7, 3, 2, 6, 11, 4, ... and then another sequence the next time. There is no math entity that can actually produce that, not to be confused with the abstraction of just calling it "random" and then setting up the generalized set of output like "distribution of digits with weighted chance of getting a four" etc. That set doesn't create sequences in an any given applications.

"Whether it exists in the real world depends ultimately on the fundamental laws." But we can't effortlessly seque between "math" and "the real world." First, mathematics is bound by logic, there is no way to produce anything not pre-determined. IOW, no actual "mathematical muons" that are all "alike" but produce different results.

Your examples move around that, by having hidden features. But if they work off "math" then the result inexorably follows from the previous conditions, known to so-and-so or not. Don't "computations" have to produce a given result? Sure, if you have a *physical* intervention, no, but that just gets back to the difference between "math" versus "real stuff." Again, check with mathematical logicians or foundationalists.

Bee said...

Hi Andrew,

I haven't told you anything about how often all these functions are accessed during the lifetime of the universe. For all I have told you, there might be one instance in the history of the universe where you receive one bit. I fail to see how you can conclude the amount of information must be infinite (in particular since time might be finite).

Well, yes, the evolution is not deterministic because if you look at the string, it's truly random. Yet, as you said earlier, the input to the fwf determines the output, thus the evolution is not random either.

Btw, your suggestion with the reservoir has been very helpful to my own understanding, thanks. Best,

B.

Bee said...

Hi Giotis,

In my paper, I have first explained why there is no room for free will in the laws of physics, at least not if you want "free will" to have meaning. Then I have explained how you can make room for it without either using a notion of free will that is empty of meaning, or arguing that there are some mythical things that cannot be described by science or such. And that I have done by showing that there is a certain type of abstract functions that can be interpreted as free will functions, and I have explained why you can interpret them in this way. That does not mean they exist in fact, that does not mean that I know how to find out if something like this exists, I have merely said, look, you can do it, and you can do it in this way. Best,

B.

Bee said...

Hi Thomas,

I did answer all of your questions, you're just not willing to accept the answer. For all I can tell, the questions that you have asked me in this thread are

"Looks to me like your transcendental free will function needs to be evolved from a universe that is non-deterministic?"

I said it doesn't evolve.

"Because if it was then from its state at any time t, one could predict/retrodict the algotithm of the free will function for every being that will ever live/has ever lived ?"

I said it doesn't evolve.

"Why do you not see that this means your free will functions do not come out of the blue, contrarily to how (some) phycisists think about the physical laws that govern our universe ? "

I replied

"Of course the "free will function" does not come "out of the blue." I have said exactly the opposite, it doesn't come or go anywhere."

You further asked

"Would you agree that the critical point in your paper is that although free will functions are deterministic, there is no way to tell what number they will return next if you do not have their algorithm available."

The free will functions are determined, they are not deterministic in the way that I have defined and used the word.

I will not make another iteration of telling you that you are still misunderstanding me. I will just repeat what I have now told you several times once again in slightly different form: The question what is a living being is entirely irrelevant to the question I have addressed, namely, if you can have a time evolution that is neither forward deterministic nor random.

"You do not adress my main point: that what an observer can learn about the nature of a free will function should matter in your approach."

It matters of course for the idea of free will functions what we can learn about them. It does not matter however for the, quite simple argument, that was subject of the paper I was discussing here.

Glad to hear you like the paper. Best,

B.

Bee said...

Hi Thomas,

I did answer all of your questions, you're just not willing to accept the answer. For all I can tell, the questions that you have asked me in this thread are

"Looks to me like your transcendental free will function needs to be evolved from a universe that is non-deterministic?"

I said it doesn't evolve.

"Because if it was then from its state at any time t, one could predict/retrodict the algotithm of the free will function for every being that will ever live/has ever lived ?"

I said it doesn't evolve.

"Why do you not see that this means your free will functions do not come out of the blue, contrarily to how (some) phycisists think about the physical laws that govern our universe ? "

I replied

"Of course the "free will function" does not come "out of the blue." I have said exactly the opposite, it doesn't come or go anywhere."

You further asked

"Would you agree that the critical point in your paper is that although free will functions are deterministic, there is no way to tell what number they will return next if you do not have their algorithm available."

The free will functions are determined, they are not deterministic in the way that I have defined and used the word.

I will not make another iteration of telling you that you are still misunderstanding me. I will just repeat what I have now told you several times once again in slightly different form: The question what is a living being is entirely irrelevant to the question I have addressed, namely, if you can have a time evolution that is neither forward deterministic nor random.

"You do not adress my main point: that what an observer can learn about the nature of a free will function should matter in your approach."

It matters of course for the idea of free will functions what we can learn about them. It does not matter however for the, quite simple argument, that was subject of the paper I was discussing here.

Glad to hear you like the paper. Best,

B.

Giotis said...

Ok maybe I should read the paper and not just the blog post.

Anyway the thing I do not understand is the phrase "within the context of modern physics" that you said previously.

Where physics enters the picture? Why what you are saying is "within the context of modern physics" or physics in general? I don't see the connection.

Bee said...

Hi Andrew,

Sorry, you're right, I am wrong. The reservoir itself is infinite. I was talking about what one can obtain from the reservoir instead, which isn't necessarily infinite too. Best,

B.

Bee said...

Hi Giotis,

Oh, I simply mean that the laws of nature that we use in modern physics today are of a particular form. They are all differential equations, and then we have in addition the measurement prescription in qm, that makes for a non-deterministic ingredient. All I am saying it is conceivably possible that fundamental laws are just not all like that, and once you allow that there are ways to think of laws of nature that are not in conflict with free will, in the exact sense that I have explained in the first part of the paper. See, most of this is terminology, and once you've set on the terminology the point is pretty much trivial. Best,

B.

Arun said...

Neil Bates,
Arun, that's fascinating but what sort of behavior qualifies per se as properly "free will?"

Well, if free will doesn't result in observables distinguishable from absence of free will, then it is likely a meaningless concept.

----

BTW, the problem of free will seems to arise mostly in Christian theology. The agonizing over free will is mostly not there in Indian philosophical systems.

e.g., see
http://elisafreschi.blogspot.com/2012/02/free-will-in-indian-philosophy.html

Arun said...

Destiny/determinism is to be used only for retrodiction.

http://articles.timesofindia.indiatimes.com/2011-08-05/vintage-wisdom/29851056_1_karma-destiny-arrow

Quote: "My spiritual guru sat me down and told me that there is a role of both karma and free will in all aspects of life including in trying to save someone from the jaws of death. At the time of action, one should not think of one's destiny, because no one can predict it. 'Then what is the use of the theory of Prarabdha karma?" I asked Swamiji. I was told that the role of Prarabdha theory is only in explaining the results of the action. In spite of best efforts, if results could not be achieved, it can be attributed to destiny. The theory of destiny is useful because it helps a person in accepting results without being frustrated. At the same time it does not make a successful person egoistic, if he understands the role of destiny in his success."

End quote.

Notice, these ideas are not objective things sitting out there in the physical universe. They are rather ideas to help you approach life in a certain way.

Giotis said...

OK Bee then I should definitely read the paper first.

"They are all differential equations"

Not exactly:-) The quantum Hamiltonian constraint in LQC is a difference not differential equation. It is incremented by discrete steps and solved via a recurrence scheme.

Giotis said...

Arun:

"BTW, the problem of free will seems to arise mostly in Christian theology."

You are right and there is a reason for this of course.

Free will is essential for the monotheistic Christian theology because otherwise God would be responsible for all the bad things happening in the world. With Free will you can always blame the sinner...

Bee said...

Paul Davies, goes so far to say that the laws of nature themselves are Christian:

Taking Science on Faith

"[T]he very notion of physical law is a theological one in the first place, a fact that makes many scientists squirm. Isaac Newton first got the idea of absolute, universal, perfect, immutable laws from the Christian doctrine that God created the world and ordered it in a rational way. Christians envisage God as upholding the natural order from beyond the universe, while physicists think of their laws as inhabiting an abstract transcendent realm of perfect mathematical relationships.

And just as Christians claim that the world depends utterly on God for its existence, while the converse is not the case, so physicists declare a similar asymmetry: the universe is governed by eternal laws (or meta-laws), but the laws are completely impervious to what happens in the universe."

Bee said...

Hi Phil,

Indeed, hope matters, which brings up the question if not our hope that we have free will taints our willingness to accept or even consider certain descriptions of nature. One might speculate for example that determinism (and superdeterminism in particular) are not very popular, simply because they are, one might say, hopeless ;o) Best,

B.

Giotis said...

Bee I will reply to Paul Davies by quoting my wise ancestors:

"Ανάγκα και Θεοί πείθονται."

Which means: The Gods are driven by necessity.

Similar

"Ουδείς ανάγκης μείζον ισχύει νόμος."

Which means: No law could prevail over necessity

Plato said...

Arun said:Notice, these ideas are not objective things sitting out there in the physical universe. They are rather ideas to help you approach life in a certain way.

I guess objectification from inside could be looked "as if" we are much the same as the universe. We are made of "Star Stuff?"

So objectively "out there" can really mean, inside? What are ideas made up of then, and what are ideals.

So everything outside is objectified from the inside. You see? You are matter bound by such an exercise as to take on your own life, as to express where we came from, as to objectify outwardly that essence. But yet we are amidst it all? You are drawing the separation and the distinction?

So what logic shall be applied to the world you "live in?"

Best,

andrewg said...

Bee,

The output of the function is determined by its input, but only for a specific choice of function. Since we have no way to prefer a specific function (or equivalently a specific string of digits), the choice of function itself is random.

You're pushing the source of randomness back one step, but it is still there. I just can't see the gap between determinism and randomness that you're trying to fit the fwf through.

andrewg said...

Bee,

I'm beginning to thing this is a semantic argument. What's your definition of "random"?

PTMR said...

One such hidden reservoir of information is the universe beyond the cosmic horizon. So for example we could look for photons arriving from the very edge of observable universe and make decisions based on their properties. This would make such decisions non-random and non-deterministic in the sense that they cannot be predicted before hand even when one has access to all available information in the visible Universe.

The main assumption here being that we cannot reconstruct the universe beyond the horizon from the information contained in the visible portion which seems reasonable.

Personally though I believe in neither free will nor fundamental randomness (I believe in ignorance) and so I would still consider such decisions deterministic in the sense of being determined by laws of physics.

Arun said...

Plato,

"So everything outside is objectified from the inside."

Yes, a Vedantic truth.
:)

-Arun

Arun said...

andrewg,

We are ignorant of the transcendental number and of what position in the decimal representation the function is spitting out digits but ignorance != randomness.

-Arun

Arun said...

Hi Bee,

Yes, there is strongish argument that science (not as in "systematized knowledge of the world", but rather, as we think of it after Newton) could have arisen only in a Christian culture.

-Arun

Bee said...

Hi Arun,

Funny. I've always looked at that the other way round: People invented Gods exactly because there were laws of Nature which was what they were actually seeking to understand. While not every culture might have went along that path, I'd think that the wish to understand Nature is the underlying reason that would come trough either way, Christian or not. Best,

B.

Bee said...

Hi Andrew,

I actually think you got it right, it's not a semantic problem. Of course if the function itself is random you haven't gained anything. But as I was eluding to above, the idea is (forgive me for being a physicist ;o)) that one might have a candidate theory for the universe and all that, and it provides you candidates for such functions, so you could actually learn something about how free will works. Needless to say, this is very vague, and I don't know if it is even possible to make sense of this, but maybe it sheds light on what I was thinking of. Best,

B.

Bee said...

Hi PTMR,

If there's a time-slicing and a forward evolution in the universe that you are imagining, it doesn't matter, in principle (it arguably matters in practice), that there are horizons. As I wrote, what you gain this way is the comfort that nobody can predict what you're doing, but you still had no other choice. Best,

B.

andrewg said...

Bee,

But if the function is not set randomly, how is it set? If it is an inevitable consequence of the laws of the universe, then surely it is forward-deterministic?

Arun,

Probability is a measure of our ignorance, and randomness is a state of maximum ignorance. Something may appear random to us, but only because we are ignorant of its mechanism (e.g. chaos). Other things may be truly random in that we can never in principle know their mechanism, but because we can never rule out the possibility that the mechanism will be discovered tomorrow, the physical existence of a truly random process can never be proved.

Bee said...

Hi Andrew,

It's not a "consequence" of the laws, I am saying they could both arise from a common theory (in that sense, you could call the free will function a law too.) That theory would ideally tell you what properties the function has (say, it spits out only integers or the output range depends on the background curvature). The difference is that, in contrast to a law that is a (reasonably well behaved) differential equation, not every function allows you an extrapolation of the past. With some types of functions, the ones that I am talking about, you can sit there forever and measure and measure and you'd still not be able to predict what happens next. Best,

B.

Arun said...

Hi Bee,

We don't have much record of people inventing gods in ancient times, so it is difficult to understand their intentions. Here is a recent example (1960s) It definitely was not for trying to understand laws of nature.

-Arun

Arun said...

andrewg,

"Probability is a measure of our ignorance, and randomness is a state of maximum ignorance."

I find it very difficult to parse this statement, I think because you are using "randomness" in some unusual sense.

A physical theory like QM tells us what questions we cannot ask. In a sense the situation is no different from a child hearing of the electron and asking what color is it? It is a natural question, because everything the child sees has a color. When you answer that it has no color, the child may think you are ignorant. Saying that the electron has no color is not an expression of ignorance, it is a statement of the nature of the electron. Similarly, the uncertainty relations governing measurements of the electron are not statements of ignorance, it is stating the nature of the electron. If you think of it as ignorance, you might as well think of the answers to whether the electron has wings and what it had for breakfast as ignorance as well - add them to the set of things we'll never know.

Thus while we often deal with ignorance using probability, the presence of probability does not imply ignorance.

-Arun

andrewg said...

Bee,

Any function that is not completely random can be written in terms of one that is. Let's take an arbitrary random function to be one that produces a real number in the range [0,1]. We can bend and manipulate the output of this so that it produces only even integers, or a normal distribution, or anything we want. Any sort-of-random function like you describe is equivalent to a deterministic envelope function with a random input. You can derive the envelope function from theory, sure. But yet again we've just pushed the randomness back one step.

Arun,

Your analogy doesn't work. An electron has no colour because electron colour is a meaningless concept. Electron position is a meaningful concept, we just can't assign it a specific value until we perform a measurement.

Let me put it another way. If I toss a coin and look at it, but don't let you see, and I then ask you what the probability of it being heads is, you will answer 0.5. But if you ask me what the probability of it being heads is I will say 1 (or 0), because I've already looked at it and know that it is heads (or not). Both of us are correct, because probability is subjective. You are ignorant of certain facts that I am privy to, and this changes our perception of probability.

When I watch the results of a radioactive decay experiment, I cannot predict what the next result will be, save only for certain constraints. I am ignorant of the future because I do not know of any means in principle to calculate an accurate result in advance, so I describe the constraints as fully as I can using probability, and say the rest is "randomness".

Arun said...

andrewg,

Is the fact that matrix multiplication not commutative a matter of our ignorance? Or demanding that matrix multiplication should commute an expression of ignorance, like asking for the color of an electron?

Now, both momentum and position are represented in quantum mechanics by matrices. The fact that these matrices do not commute is hardly a symptom of ignorance. Rather, misguided by our classical intuition, we ask for things that don't exist, like the color of the electron.

-Arun

Eric said...

I wonder why it is so important to so many people, including Bee, whether there is free will or not. It essentially is a judicial distinction and not one that applies to physics. I think the more interesting question is what is at the root of this interest aside from the judicial relevance.

I suspect the real interest comes from the psychology side. We know everyone has a conscious and a subconscious side to their nature. If two identical twins are raised identically, or as close to identically as is humanly possible, then any differences in behavior would originate from the small biological differences between them, (they still have them), and from the unavoidable slight differences in what they experienced growing up.

But often we know intuitively that these small differences do not account for all the differences. The brain is a weighing machine. It weighs the burden of telling a lie with the benefit that might accrue from telling it. Individuals can train themselves to obfuscate and the easiest way to do that is to move those thoughts around the act of obfuscation into the subconscious. One could then argue that the person who repeated acts in ways that benefits him or herself to the detriment of others is deterministically not responsible for their actions because the matrix of emotions attracted to those antisocial actions have moved into a place where they don't have access to them.

But the fact remains that this person at some point in the suppression learning process he went through weighed his own feelings against the wellbeing of the persons he was hurting and chose to be selfish. The perpetrator at the First moment of choosing that path is responsible for his actions. If later the path becomes easy from subconscious suppression it does not extinguish his/her responsibility. So in a sense there is a branching point of responsibility that every person always retains.

I think the point I'm making is that everyone knows they have not acted perfectly in their life. The people who are not misogynistic learn from the negative feelings they initially had when doing it and the misogynistic people purposely trained themselves through repetition to suppress those feelings.

I can't help but think this perpetual search for a theory of no freewill is really the effect of a subconscious desire just to bury those feelings from the past. I'm not saying there is any psychopathic deeds that motivate that covering up but I think it issues from the same place.

Plato said...

Hi Eric,

I read your entry with interest.

I just started reading a book called, Snakes in Suits( When Psychopaths Go to Work)by Paul Babiak Ph.D.and Robert D. Hare, Ph.D.

It is an interesting read so far which brings me to the highlighted text you wrote:

"I can't help but think this perpetual search for a theory of no freewill is really the effect of a subconscious desire just to bury those feelings from the past. I'm not saying there is any psychopathic deeds that motivate that covering up but I think it issues from the same place."

I do not believe one could say what you are saying as to whether such subconscious is being used by the psychopath?

From what I read it appears to me that it is embedded in the genetics in some way?

Definitely not a expert but in context of freewill the psychopaths have a disregard\not part of their makeup for caring for a lot of things that a non psychopathic individual would.

I think Freewill has a greater need for how such a "forming ability" that could be granted to individuals at levels that would not be apparent for the subjective interrogation? Yet, it would be there amidst this ability of the subconscious mind to fabric the reality so as to discern it's schematic structure as a predictor of our moving forward now being aware( what do we call this sensitivity...introspection))?

Then so as to understand the mass forming characteristic(maybe like a Higgs) such abilities may beholding to those subatomic particles for some "mass gather attribute" of the objective reality?.

These then may be translated possibly to a chemical endocrinologist forming apparatus in the state of our being? A very fast messenger moving capability of our sensitive emotive developing environment.

Just some thoughts. All scientists would not believe this?:)

Best,

Eric said...

Hi Plato,
I'm no expert either but I still have my ideas. There are different ways to categorize the psychopaths or sociopaths. I actually think there is a small number of them who probably have something wrong with their brain physiology. The guy who shot about 20 people from a school tower in Texas about 40 years ago was later found to have a fast growing brain tumor that caused an abrupt change in his behavior. So it happens. But I think the genetic or biological component to the kind of behavior you are talking about is probably pretty rare. That is just my hunch.

The reason I say that is because I have personally known people growing up that changed as they grew up. They became more and more jaded and cynical - seemingly cut off from their feelings for other people whom they didn't know personally. In other words they became desensitized. In most cases I think it was a slow process, one step at a time, where they rationalized that anyone else would have done what they did had they been in their place. But there is always a sense of guilt in that desensitization.

The level of wrongs people commit can be all over the
spectrum, from really minor to substantial. But unless one has some form of brain damage the usual method of sublimating any feelings that arise from them would be to suppress them and move them to the subconscious. Some people are really good at it and others are very defensive about the most minor and inconsequential of wrongs. That would be where a really good theory of non free-will might come in handy.

andrewg said...

Arun,

Matrix multiplication has nothing to do with it. The propagation of the wavefunction, as described by unitary, non-commuting matrices, is deterministic. It is the collapse of the wavefunction that we do not understand and therefore call random.

Arun said...

andrewg,

Because the matrices representing dynamical variables (that are the analogs of the classical variables) do not commute, a state cannot be an eigenvector of all of them, and that is why we cannot know the values of all of the dynamical variables simultaneously, and that is the root of the "mystery" about QM. If everything commuted, that is in effect ℏ = 0 and we are back at classical mechanics.


-Arun

Kaleberg said...

This sounds like Scott Aaronson territory. Logic has lots of holes in it as anyone who ever studied the halting problem has learned. If you make a decision based on your making or not making a decision, you may wind up with an unpredictable decision, but a decision none the less.

I always found this kind of free will rather mystical. I think of free will as a social construct with implications in law, philosophy, literary analysis, religion and so on. In those areas, the concept of free will is pragmatic, so the underlying mechanisms may be deterministic or not.

Did Galileo recant because of his own free will? I'm guessing that in the eyes of the church, he did. I'm not sure of how a free will mystic would regard his decision, but I am curious.

Plato said...

Some believe that the alternative to determinism is randomness, and go on to say that “allowing randomness into the world does not really help in understanding free will.” However,
this objection does not apply to the free responses of the particles that we have described.
The Strong Free Will Theorem

andrewg said...

Arun,

a state cannot be an eigenvector of all of them, and that is why we cannot know the values of all of the dynamical variables simultaneously

You've skipped over a huge chunk of argument. Why are eigenvectors important? The reason we cannot know all the values simultaneously is because nature doesn't let us measure the mixed state directly. QM lets us calculate a mixed state, but measurement gives us an eigenstate instead. QM does not predict which eigenstate this will be. We are ignorant of the process nature uses to choose this state and so call it "random".

Arun said...

andrewg,

I'll repeat : Asking for something that is not there in nature and expecting an answer is ignorance, whether it be the color of an electron or the simultaneous exact values of momentum and position of an electron.

If the context of QM is confusing, then just consider a communications signal. It has the approximate characteristics of frequency and duration. A signal with an exact frequency has indefinite duration, and a signal with a definite duration can only be made from a spread of frequencies. Without any quantum mechanics, there is an uncertainty relation for frequency and duration.

That we cannot have both exact frequency and exact duration is not an expression of our ignorance, it is the result of demanding an answer to an inappropriate question.

andrewg said...

Why are you only allowing me to ask for a definite frequency or a definite duration? Why don't you let me ask for the shape of the waveform? You are keeping me ignorant by forcing me to ask inadequate questions. Similarly, nature keeps us ignorant by only letting us make inadequate measurements.

Plato said...

Quantum Gate?

Quantum logic gates are represented by unitary matrices.

Best,

Ulla said...

How does the Planck mass (or Planck scale) express free will? I would say it is hard to distinguish from uncertainty.

The Universe was bought forth by the main principle necessity, in old texts. What principle could this be? What happen in the state function reduction (the result of free will?) or what induce the state reduction, the compression? Minimum path length or quantization and superposition? So mass is eg. an expression of free will, seen in the randomness and uncertainty in matter? Fuzzy functions etc.

Don Foster said...

When you bring a pen to paper you are on the verge of leaving pragmatic physics behind. The momentum of the pen surely, but the whither of the pen is determined by such a multiplicity of meta-level, tidal inputs that physics becomes simply irrelevant.
While there is truth in the sentence, “Pigeons don’t like bananas”, it is a truth rarely relevant in the flux of some present moment.
Say, rather than a pen, you are on a surfboard. The determinant of whether you ride or take a tumble is finer than physics can effectively describe.
So, from an existential point of view, the notion that will is subsumed within physics, while perhaps true, is not pragmatic. Perhaps somethings are beyond the reach of physics as human enterprise.
Anyway, would any real proof be possible?

WNelson said...

This is a mind-twisting topic. In your paper you say "The free will function assigned to the agent one could loosely interpret as the agent’s character." But an agent's character precisely expresses that part of his or her will which is the least free, because it represents correlation of future decisions with past and is predictable.

Do you have some other examples besides the pi function? It would be very interesting to see a possible example that produces outcomes that are not statistically random.

Sabine Hossenfelder said...

Will,

I'll not fight with you over the use of the word 'character'. I used it to mean that here you can encode some kind of 'typical' behavior if you wish. I do not presently have any other example, no. I'm thinking about it though - it would be good and, I think, more convincing to have a continuous example. Best,

B.

Mohsin E. said...

Brilliant paper! I think looking for that "space" over which such a free-will function may be defined, and then defining that "space" should merit a focus of efforts.