“Free will” consists of the ability of a person to determine their own future actions by some entirely internal process (which can observe, but is not controlled by, external factors); where “person” is defined as a collection of stuff such that the collection of stuff that makes up you has no overlap with the collection of stuff that makes up me and neither of us have any overlap with the collection of stuff that makes up (say) Barack Obama, or Trevor Noah, or Jacob Zuma.
Do you understand “is not controlled by” in such a way that having “free will” is inconsistent with (1) purely deterministic physics and/or (2) purely deterministic+random physics? (On the face of it your definition makes free will inconsistent with #1 but not with #2, but I can e.g. imagine a definition that restricts those “external factors” to, say, the state of the world outside one’s body in at most the last year, in which case “free will” might be compatible with outright determinism.)
I don’t think that free will can be reconciled with purely deterministic physics—free will implies that, in exactly the same situation, with each and every particle in exactly the same space, I can still choose whether to purchase those biscuits or not.
On the other hand, my decision whether or not to purchase those biscuits is not exactly random, either. There are a number of factors that go into it—in fact, considering force of habit, quite a few of my decisions are extremely predictable. So I’m not sure that random physics is entirely reconcilable either.
OK. So, do you consider that you actually have good evidence for the existence of free will in this sense? If so, what is that evidence?
The obvious alternative hypothesis, which seems to me to explain all the evidence I know of just as well, is that at the level of physics there’s nothing but determinism and maybe randomness, but it looks different to us because we can’t see all the details. We think “I could have done otherwise in the exact same situation” because we have seen ourselves and others do different things in very similar-looking situations, we can imagine making a different decision in what feels like the same situation, etc.; but we don’t get to observe the exact states of all the particles that compose us and the world around us, and what we think of as “the same situation” may actually be quite different in its details. We also don’t get to observe the mechanisms that lead to our making whatever choices we do, so those choices feel like opaque black-box miracles to us. No magical contra-causal free will is required for things to look this way to us.
I did think I had a good argument for free will (given the existence of God), but TheAncientGeek has punctured that. (I had a second argument as well, but I’m waiting to see whether TheAncientGeek has any comment on that one).
Aside from that, all I’ve really got is that:
(a) What I do feels like free will; that may be an illusion.
(b) What other people do is consistent enough to suggest that their actions are being guided by individual, similarly free-willed minds.
...both of which are fairly weak evidence, if anything.
It’s clear that if you put someone in very similar situations and ask them to make a choice, over time they will converge to making a certain choice a certain percentage of the time. That could easily be the same percentage of the time that would be predicted by deterministic physics plus e.g. quantum uncertainty, so I don’t see any reason in principle why your account of free will could not be consistent with everything happening according to the laws of physics, if there is randomness in the laws of physics.
As for the feeling, if a deterministic chess computer had feelings, it would have to have the feeling that it could make any move it wanted, because if it didn’t feel that way, it couldn’t consider all the possibilities, and it can’t decide on a move without considering all the possibilities. This doesn’t prevent chess computers from being deterministic, so it might not prevent you from having a feeling like that, even if your actions are in fact deterministic.
It’s clear that if you put someone in very similar situations and ask them to make a choice, over time they will converge to making a certain choice a certain percentage of the time.
No, it’s not clear at all. If ask me to make choices in similar situations, first I might humor you, then I’ll get bored and start fucking around with the system, and then I’ll get really bored and stop cooperating with you. There won’t be much of a convergence over time.
then I’ll get bored and start fucking around with the system, and then I’ll get really bored and stop cooperating with you. There won’t be much of a convergence over time.
The problem with that model seems to be that as time goes on, the situation in which you are put in becomes increasingly dissimilar to the original one, just because of we’ve added memories of having had to make this choice x number of times before. If we could run the experiment so that you always felt like it was the first time you were in this situation, perhaps by putting the same kind of decision in different contexts and spreading them out over time and with various distractions, do you think you’d still deviate in the same way?
I know I’m going back from territory to less practical abstraction here, but I think this kind of difficult-to-collect data would be more revealing for this question.
Most of my point is that you can not. Among I things, I change over time.
As a practical example, I drink beer. Various kinds of. My beer preferences do not converge over time. Instead, they wander over different styles, different hoppiness/maltiness/etc., even different breweries. I have no idea what kind of beer I will like in, say, a year, but it probably will be different from what I like now.
Showing that something works in a toy model does not show that the same thing works in actual reality.
Sure, I totally agree with you—in real life, we can really put a person in exactly the same situation twice. If we could, this whole free will argument would be a lot easier to solve.
That said, I do think the toy models are useful. Pretending we can do this experiment gives an answer to the problem I’ve never managed to pick a hole in (and tbh getting other people’s input on it is the hidden motivation for entering this discussion):
If we could let you choose a beer, then rewind the universe—including all particles, forces, and known and unknown elements of cognition anyone might postulate such as souls and deities back to their starting position—then let it go again, there are only really two things that could happen:
1) you choose the same beer because that’s what the universe was leading up to or
2) you choose a different beer despite the fact that all parameters of the universe known and unknown are the same.
The first outcome would suggest determinism; the second randomness, or at least independence from all variables which we consider “self” such as personality, memory and perhaps souls and things, since they were all rewound with the universe. I’d be really interested to hear of any third option anyone can think of!
As you say, showing this in a toy model isn’t the same as showing it in actual reality; but when the actual experiment is impossible, one is arguing about abstract concepts anyway, and one has a lot of difficulty imagining outcomes not encompassed in the model I’m not sure we can do much better.
I suspect that if we take the average of e.g. the bitterness of the beers that you have been drinking, it has already converged to an average, and future developments will probably not change that average much, even if there are some years when you drink sweet beers and some years when you drink bitter beers.
Perhaps, although I don’t see how you can know that unless you have been making measurements, or unless it has definitely been going in the direction of getting more and more sweet, or more and more bitter.
In any case, since beer does not differ an infinite amount in sweetness and bitterness, it won’t be easy to stop that average from converging sooner or later.
Um, if I’m swinging from Lambics to Stouts with excursions into IPAs and Belgian Trappists, do you really think I converged on a particular bitterness?
it won’t be easy to stop that average from converging sooner or later.
The concept of convergence does not apply to the “average position”. It always exists.
You are probably thinking of statistical estimation with uncorrelated errors. That is not the case here, you are not estimating some unobserved parameter.
But that doesn’t mean my taste in beer will converge to some value. All it means is that the average of history of my beer wanderings will be somewhere around the middle of the range—an observation which is quite useless for the free will debate.
The general point I was making is that there is nothing about free will, even if by definition it means you have more than one option in the same physical situation, which gives us a reason to expect a pattern different from determinism with the addition of some randomness. So unless someone can show how those patterns would be different, there isn’t any special reason to suppose that our actions couldn’t correspond entirely to the laws of physics, without that meaning we don’t have free will.
a pattern different from determinism with the addition of some randomness
That’s heavily underspecified. Most everything can be fit into a pattern of “determinism with the addition of some randomness”.
In any case, you started with a specific claim that the choices will converge. Outside of the toy-model setup I didn’t think it was necessarily true and I still don’t think so.
I meant that the changing average of the choices will converge, in the way I expect to happen in the beer case. I still think this will happen under all normal circumstances.
I’m not seeing what that says about free will. If you pick out a selection of numbers from one to a hundred, and you keep going, then the more numbers you pick out the less effect each new number will have on the running average.
I just don’t see how this leads to “free will can be explained by deterministic physics plus randomness”.
I don’t think it proves that. I think it suggests that it may be the case. We already know that deterministic physics plus randomness will result in statistical patterns like that. I am just saying that free will is going to result in statistical patterns as well.
As far as I know, those could be the same patterns, which would mean that free will would be consistent with deterministic physics plus randomness. That is not a proof. I am just saying I don’t know of any specific reasons to think those patterns will be different. Do you have reasons like that?
The thing is, the “pattern” you’ve picked up is a pattern that every series of numbers follows—that every series of numbers must follow. Any hypothesis will result in the same pattern—if I think that free will is controlled by the number of cookies eaten in Western Australia every second Tuesday, and I try to see if it follows the pattern of the average converging to a value as more samples are added, I’ll find the same pattern.
If “having the same pattern” is to have any sort of predictive power at all, then that pattern must be a pattern that the data can possibly not have under another hypothesis.
I’m mystified by this discussion. entirelyuseless seems to be saying that if you look at the history of choices someone has made in similar situations, they will show some kind of convergence, which I see no reason at all to believe. And CCC seems to be saying that every sequence of numbers shows this pattern, which I not only see no reason to believe but can refute. (E.g., consider the sequence consisting of one +1, two −1s, four +1s, eight −1s, etc. The average after 2^n-1 steps oscillates between about −1/3 and about +1/3.) And none of this seems to have anything much to do with free will; in so far as the “libertarian” notion of free will makes sense at all, it seems perfectly consistent with making choices that average out in the long run, at least with probability → 1 or something of the kind, and determinism is perfectly consistent with not doing so, since e.g. the sequence of numbers I just described can be produced by a very simple computer program (at least if it either has infinite memory or has at least a few hundred bits and isn’t observed for longer than the lifetime of the universe).
Well, that isn’t what “converges” means in mathematics (it means there’s a particular value towards which one gets and stays arbitrarily close), but with that definition it is indeed tautologously true that wandering around within a bounded region yields “converging” averaged values. (But not if the region can be unbounded. Easy counterexample: One +1, two −2, four +4, eight −8, etc.)
Well, that isn’t what “converges” means in mathematics (it means there’s a particular value towards which one gets and stays arbitrarily close), but with that definition it is indeed tautologously true that wandering around within a bounded region yields “converging” averaged values.
If the steps get smaller sufficiently slowly, there are counterexamples similar to yours with +1s and −1s. It’s difficult to get convergence short of explicitly imposing convergence.
Confinement to a bounded region implies accumulation points (points to which some subsequence converges), but there can be any number of those. They can even be dense in the space.
There are counterexamples to convergence-in-the-usual-sense, and my example was one of them.
If one takes CCC’s statement of what s/he meant by “convergence” absolutely at face value, then there are not only counterexamples but trivial counterexamples; e.g., begin 0,0,1; the first step moves the rolling average by 0 and the second doesn’t.
What I took CCC as actually meaning was that the differences converge-in-the-usual-sense to zero, or equivalently that one can make them small enough by waiting long enough. That’s straightforwardly true because if the diameter of your region is D then after n steps your average can’t move by more than D/n per step.
What I took CCC as actually meaning was that the differences converge-in-the-usual-sense to zero, or equivalently that one can make them small enough by waiting long enough. That’s straightforwardly true because if the diameter of your region is D then after n steps your average can’t move by more than D/n per step.
Just to confirm—yes, that’s what I’d meant. Which it trivially true, yes; that is why I couldn’t understand why entirelyuseless was attaching any special significance to it.
What I took CCC as actually meaning was that the differences converge-in-the-usual-sense to zero, or equivalently that one can make them small enough by waiting long enough.
That’s what CCC said in his last post, but it’s not a useful property for showing what was originally claimed by entirelyuseless, who said:
I meant that the changing average of the choices will converge, in the way I expect to happen in the beer case.
This doesn’t follow from shrinking steps, because:
That’s straightforwardly true because if the diameter of your region is D then after n steps your average can’t move by more than D/n per step.
The sum of that over all n is divergent, so the average can move around anywhere if you wait long enough. In fact, for any number of passes of averaging, that will still be true: if the underlying sequence goes somewhere and sticks there long enough, the average will eventually get there. Then the underlying sequence can go somewhere else, and so on.
To put this in more concrete terms, with an example of a property that for many people shows no long-term stability in their lifetime, consider physical location (relative to the geocentric frame). A person may live for years in one place, then years in another town or another continent, and make such moves at various times in their life. For such a person, there is no useful concept of their average location. In the case of the original example, a person’s taste in beer can make just as drastic changes, on top of which, the world changes and new beers are created, the space of the random walk changes as the walk is being made.
entirelyuseless’s original claim:
The random walk doesn’t converge. But the average position does.
I agree. (Did something I said give a contrary impression? I thought I’d said right at the outset that the original claims of both entirelyuseless and CCC are wrong.)
I have already said that I agree that it is mathematically possible to prevent the average from converging, just that this is not likely to happen in real life. In RichardKennaway’s comment, “goes somewhere and sticks there long enough” means progressively longer periods of time, and so is not realistic.
I’m not saying that people’s choices converge in the sense of getting closer to a particular value, but that the average converges. You are right to say that this is not a necessary property of every sequence of values (and CCC was mistaken.)
You say that there is no reason to think I am right about this, but your proposed sequence of numbers suggests that I am, namely by showing that the only way the average won’t converge is if you purposely choose a sequence to prevent that from happening. Suppose you offer someone chocolate or vanilla ice cream once a week. I think there are very good reasons to think that the moving average would begin to change slower and slower very quickly, and would basically converge after a while. This would happen unless the person used a sequence like the above: namely, unless he chose a sequence with the explicit intention of preventing convergence.
I agree that someone can have this intention, and that this would not refute determinism. In that sense you could say that the whole discussion is irrelevant. But the relevance, from my point of view, is that it makes the question more concrete. The basic point is that you can, if you want, define free will so that it is not consistent with determinism. But then it will be consistent with determinism plus randomness, unless you propose some prediction which is not consistent with the second. And no one had done that. So no one has even suggested a definition of free will which would be inconsistent with being produced by some form of physical laws.
So no one has even suggested a definition of free will which would be inconsistent with being produced by some form of physical laws.
Given that our universe clearly does operate on some form of physical laws, if anyone were to provide such a definition of free will, it should be trivial to show that it’s not how our universe works.
I think there are very good reasons to think that the moving average would begin to change slower and slower very quickly, and would basically converge after a while.
Not if the person’s preferences are changing gradually over time. That is a real thing that really happens.
(For the avoidance of doubt: I agree that any notion of free will it’s credible to think we have is consistent with physicalism.)
I agree that a person’s preferences can change over time but it will not have the effect of an average that goes back and forth without his preferences changing back and forth, but remaining stable at the extremes for longer and longer periods of time (much like your sequence). This is not a likely thing to happen in real life.
Anyway I also agree that the particulars of this are not that important to my point.
Basically I am saying that deterministic physics plus randomness can produce any possible pattern, as you’re noting. So it can also produce the pattern produced by free will. Or do you have some idea of what free will would do which is different from deterministic physics plus randomness? If so, I haven’t see it suggested yet.
Basically I am saying that deterministic physics plus randomness can produce any possible pattern, as you’re noting. So it can also produce the pattern produced by free will.
Yes, I agree. It can. Deterministic physics alone can, if you have a long enough list of rules.
As for the feeling, if a deterministic chess computer had feelings, it would have to have the feeling that it could make any move it wanted, because if it didn’t feel that way, it couldn’t consider all the possibilities, and it can’t decide on a move without considering all the possibilities.
...I’m not seeing this. It can consider all the possibilities even if it knows that it must play the possibility with the highest odds of winning—in fact, knowing that means that it must consider all the possibilities in order to calculate those odds, surely?
Even if it knows that it must play the move with the highest odds of winning, as far as it knows when it starts considering, that could be any of the moves.
But yes that would be knowledge that its move is objectively deterministic. This would not necessarily prevent it from feeling like it could make any move it wanted, just like people who believe themselves subject to deterministic physics still feel like they can do whatever they want.
But the chess computer doesn’t have to know what is determining its moves, in which case it will be even more likely to feel that it can make whatever move it wants.
Well, yes, feeling like it has freedom doesn’t really prevent it from not having freedom; but I don’t see how the feeling of freedom makes any difference at all. Why shouldn’t the chess computer feel constrained?
I agree that the feeling doesn’t make any difference. That’s what I’m saying: whether it feels constrained or not, it may or may not be deterministic. Those are two different things. The same is true for us.
With what definition of “free will”?
“Free will” consists of the ability of a person to determine their own future actions by some entirely internal process (which can observe, but is not controlled by, external factors); where “person” is defined as a collection of stuff such that the collection of stuff that makes up you has no overlap with the collection of stuff that makes up me and neither of us have any overlap with the collection of stuff that makes up (say) Barack Obama, or Trevor Noah, or Jacob Zuma.
Do you understand “is not controlled by” in such a way that having “free will” is inconsistent with (1) purely deterministic physics and/or (2) purely deterministic+random physics? (On the face of it your definition makes free will inconsistent with #1 but not with #2, but I can e.g. imagine a definition that restricts those “external factors” to, say, the state of the world outside one’s body in at most the last year, in which case “free will” might be compatible with outright determinism.)
I don’t think that free will can be reconciled with purely deterministic physics—free will implies that, in exactly the same situation, with each and every particle in exactly the same space, I can still choose whether to purchase those biscuits or not.
On the other hand, my decision whether or not to purchase those biscuits is not exactly random, either. There are a number of factors that go into it—in fact, considering force of habit, quite a few of my decisions are extremely predictable. So I’m not sure that random physics is entirely reconcilable either.
OK. So, do you consider that you actually have good evidence for the existence of free will in this sense? If so, what is that evidence?
The obvious alternative hypothesis, which seems to me to explain all the evidence I know of just as well, is that at the level of physics there’s nothing but determinism and maybe randomness, but it looks different to us because we can’t see all the details. We think “I could have done otherwise in the exact same situation” because we have seen ourselves and others do different things in very similar-looking situations, we can imagine making a different decision in what feels like the same situation, etc.; but we don’t get to observe the exact states of all the particles that compose us and the world around us, and what we think of as “the same situation” may actually be quite different in its details. We also don’t get to observe the mechanisms that lead to our making whatever choices we do, so those choices feel like opaque black-box miracles to us. No magical contra-causal free will is required for things to look this way to us.
I did think I had a good argument for free will (given the existence of God), but TheAncientGeek has punctured that. (I had a second argument as well, but I’m waiting to see whether TheAncientGeek has any comment on that one).
Aside from that, all I’ve really got is that:
(a) What I do feels like free will; that may be an illusion. (b) What other people do is consistent enough to suggest that their actions are being guided by individual, similarly free-willed minds.
...both of which are fairly weak evidence, if anything.
It’s clear that if you put someone in very similar situations and ask them to make a choice, over time they will converge to making a certain choice a certain percentage of the time. That could easily be the same percentage of the time that would be predicted by deterministic physics plus e.g. quantum uncertainty, so I don’t see any reason in principle why your account of free will could not be consistent with everything happening according to the laws of physics, if there is randomness in the laws of physics.
As for the feeling, if a deterministic chess computer had feelings, it would have to have the feeling that it could make any move it wanted, because if it didn’t feel that way, it couldn’t consider all the possibilities, and it can’t decide on a move without considering all the possibilities. This doesn’t prevent chess computers from being deterministic, so it might not prevent you from having a feeling like that, even if your actions are in fact deterministic.
No, it’s not clear at all. If ask me to make choices in similar situations, first I might humor you, then I’ll get bored and start fucking around with the system, and then I’ll get really bored and stop cooperating with you. There won’t be much of a convergence over time.
The abstraction is not the territory.
The problem with that model seems to be that as time goes on, the situation in which you are put in becomes increasingly dissimilar to the original one, just because of we’ve added memories of having had to make this choice x number of times before. If we could run the experiment so that you always felt like it was the first time you were in this situation, perhaps by putting the same kind of decision in different contexts and spreading them out over time and with various distractions, do you think you’d still deviate in the same way?
I know I’m going back from territory to less practical abstraction here, but I think this kind of difficult-to-collect data would be more revealing for this question.
Most of my point is that you can not. Among I things, I change over time.
As a practical example, I drink beer. Various kinds of. My beer preferences do not converge over time. Instead, they wander over different styles, different hoppiness/maltiness/etc., even different breweries. I have no idea what kind of beer I will like in, say, a year, but it probably will be different from what I like now.
Showing that something works in a toy model does not show that the same thing works in actual reality.
Sure, I totally agree with you—in real life, we can really put a person in exactly the same situation twice. If we could, this whole free will argument would be a lot easier to solve.
That said, I do think the toy models are useful. Pretending we can do this experiment gives an answer to the problem I’ve never managed to pick a hole in (and tbh getting other people’s input on it is the hidden motivation for entering this discussion):
If we could let you choose a beer, then rewind the universe—including all particles, forces, and known and unknown elements of cognition anyone might postulate such as souls and deities back to their starting position—then let it go again, there are only really two things that could happen: 1) you choose the same beer because that’s what the universe was leading up to or 2) you choose a different beer despite the fact that all parameters of the universe known and unknown are the same.
The first outcome would suggest determinism; the second randomness, or at least independence from all variables which we consider “self” such as personality, memory and perhaps souls and things, since they were all rewound with the universe. I’d be really interested to hear of any third option anyone can think of!
As you say, showing this in a toy model isn’t the same as showing it in actual reality; but when the actual experiment is impossible, one is arguing about abstract concepts anyway, and one has a lot of difficulty imagining outcomes not encompassed in the model I’m not sure we can do much better.
Within the toy model, yes. In actual reality, you still don’t know.
The trivial third option is to drink wine :-P
On a bit more serious note, if you set up the problem so that the outcomes are X and not-X, there could be no third option.
I suspect that if we take the average of e.g. the bitterness of the beers that you have been drinking, it has already converged to an average, and future developments will probably not change that average much, even if there are some years when you drink sweet beers and some years when you drink bitter beers.
Empirically speaking, you are wrong.
Perhaps, although I don’t see how you can know that unless you have been making measurements, or unless it has definitely been going in the direction of getting more and more sweet, or more and more bitter.
In any case, since beer does not differ an infinite amount in sweetness and bitterness, it won’t be easy to stop that average from converging sooner or later.
Um, if I’m swinging from Lambics to Stouts with excursions into IPAs and Belgian Trappists, do you really think I converged on a particular bitterness?
Random walk, even if bounded, does not converge.
The random walk doesn’t converge. But the average position does.
The concept of convergence does not apply to the “average position”. It always exists.
You are probably thinking of statistical estimation with uncorrelated errors. That is not the case here, you are not estimating some unobserved parameter.
I mean your average position on any day taken as the average of all the values up to that day.
As days increase indefinitely, this changing average will converge (e.g. to the central value.)
But that doesn’t mean my taste in beer will converge to some value. All it means is that the average of history of my beer wanderings will be somewhere around the middle of the range—an observation which is quite useless for the free will debate.
The general point I was making is that there is nothing about free will, even if by definition it means you have more than one option in the same physical situation, which gives us a reason to expect a pattern different from determinism with the addition of some randomness. So unless someone can show how those patterns would be different, there isn’t any special reason to suppose that our actions couldn’t correspond entirely to the laws of physics, without that meaning we don’t have free will.
That’s heavily underspecified. Most everything can be fit into a pattern of “determinism with the addition of some randomness”.
In any case, you started with a specific claim that the choices will converge. Outside of the toy-model setup I didn’t think it was necessarily true and I still don’t think so.
I meant that the changing average of the choices will converge, in the way I expect to happen in the beer case. I still think this will happen under all normal circumstances.
I, um...
I’m not seeing what that says about free will. If you pick out a selection of numbers from one to a hundred, and you keep going, then the more numbers you pick out the less effect each new number will have on the running average.
I just don’t see how this leads to “free will can be explained by deterministic physics plus randomness”.
I don’t think it proves that. I think it suggests that it may be the case. We already know that deterministic physics plus randomness will result in statistical patterns like that. I am just saying that free will is going to result in statistical patterns as well.
As far as I know, those could be the same patterns, which would mean that free will would be consistent with deterministic physics plus randomness. That is not a proof. I am just saying I don’t know of any specific reasons to think those patterns will be different. Do you have reasons like that?
The thing is, the “pattern” you’ve picked up is a pattern that every series of numbers follows—that every series of numbers must follow. Any hypothesis will result in the same pattern—if I think that free will is controlled by the number of cookies eaten in Western Australia every second Tuesday, and I try to see if it follows the pattern of the average converging to a value as more samples are added, I’ll find the same pattern.
If “having the same pattern” is to have any sort of predictive power at all, then that pattern must be a pattern that the data can possibly not have under another hypothesis.
I’m mystified by this discussion. entirelyuseless seems to be saying that if you look at the history of choices someone has made in similar situations, they will show some kind of convergence, which I see no reason at all to believe. And CCC seems to be saying that every sequence of numbers shows this pattern, which I not only see no reason to believe but can refute. (E.g., consider the sequence consisting of one +1, two −1s, four +1s, eight −1s, etc. The average after 2^n-1 steps oscillates between about −1/3 and about +1/3.) And none of this seems to have anything much to do with free will; in so far as the “libertarian” notion of free will makes sense at all, it seems perfectly consistent with making choices that average out in the long run, at least with probability → 1 or something of the kind, and determinism is perfectly consistent with not doing so, since e.g. the sequence of numbers I just described can be produced by a very simple computer program (at least if it either has infinite memory or has at least a few hundred bits and isn’t observed for longer than the lifetime of the universe).
...I had understood “converges” to mean that each successive sample moves the rolling average by a smaller and smaller amount.
Well, that isn’t what “converges” means in mathematics (it means there’s a particular value towards which one gets and stays arbitrarily close), but with that definition it is indeed tautologously true that wandering around within a bounded region yields “converging” averaged values. (But not if the region can be unbounded. Easy counterexample: One +1, two −2, four +4, eight −8, etc.)
If the steps get smaller sufficiently slowly, there are counterexamples similar to yours with +1s and −1s. It’s difficult to get convergence short of explicitly imposing convergence.
Confinement to a bounded region implies accumulation points (points to which some subsequence converges), but there can be any number of those. They can even be dense in the space.
There are counterexamples to convergence-in-the-usual-sense, and my example was one of them.
If one takes CCC’s statement of what s/he meant by “convergence” absolutely at face value, then there are not only counterexamples but trivial counterexamples; e.g., begin 0,0,1; the first step moves the rolling average by 0 and the second doesn’t.
What I took CCC as actually meaning was that the differences converge-in-the-usual-sense to zero, or equivalently that one can make them small enough by waiting long enough. That’s straightforwardly true because if the diameter of your region is D then after n steps your average can’t move by more than D/n per step.
Just to confirm—yes, that’s what I’d meant. Which it trivially true, yes; that is why I couldn’t understand why entirelyuseless was attaching any special significance to it.
That’s what CCC said in his last post, but it’s not a useful property for showing what was originally claimed by entirelyuseless, who said:
This doesn’t follow from shrinking steps, because:
The sum of that over all n is divergent, so the average can move around anywhere if you wait long enough. In fact, for any number of passes of averaging, that will still be true: if the underlying sequence goes somewhere and sticks there long enough, the average will eventually get there. Then the underlying sequence can go somewhere else, and so on.
To put this in more concrete terms, with an example of a property that for many people shows no long-term stability in their lifetime, consider physical location (relative to the geocentric frame). A person may live for years in one place, then years in another town or another continent, and make such moves at various times in their life. For such a person, there is no useful concept of their average location. In the case of the original example, a person’s taste in beer can make just as drastic changes, on top of which, the world changes and new beers are created, the space of the random walk changes as the walk is being made.
entirelyuseless’s original claim:
cannot be salvaged.
I agree. (Did something I said give a contrary impression? I thought I’d said right at the outset that the original claims of both entirelyuseless and CCC are wrong.)
I have already said that I agree that it is mathematically possible to prevent the average from converging, just that this is not likely to happen in real life. In RichardKennaway’s comment, “goes somewhere and sticks there long enough” means progressively longer periods of time, and so is not realistic.
I’m not saying that people’s choices converge in the sense of getting closer to a particular value, but that the average converges. You are right to say that this is not a necessary property of every sequence of values (and CCC was mistaken.)
You say that there is no reason to think I am right about this, but your proposed sequence of numbers suggests that I am, namely by showing that the only way the average won’t converge is if you purposely choose a sequence to prevent that from happening. Suppose you offer someone chocolate or vanilla ice cream once a week. I think there are very good reasons to think that the moving average would begin to change slower and slower very quickly, and would basically converge after a while. This would happen unless the person used a sequence like the above: namely, unless he chose a sequence with the explicit intention of preventing convergence.
I agree that someone can have this intention, and that this would not refute determinism. In that sense you could say that the whole discussion is irrelevant. But the relevance, from my point of view, is that it makes the question more concrete. The basic point is that you can, if you want, define free will so that it is not consistent with determinism. But then it will be consistent with determinism plus randomness, unless you propose some prediction which is not consistent with the second. And no one had done that. So no one has even suggested a definition of free will which would be inconsistent with being produced by some form of physical laws.
Given that our universe clearly does operate on some form of physical laws, if anyone were to provide such a definition of free will, it should be trivial to show that it’s not how our universe works.
Not if the person’s preferences are changing gradually over time. That is a real thing that really happens.
(For the avoidance of doubt: I agree that any notion of free will it’s credible to think we have is consistent with physicalism.)
I agree that a person’s preferences can change over time but it will not have the effect of an average that goes back and forth without his preferences changing back and forth, but remaining stable at the extremes for longer and longer periods of time (much like your sequence). This is not a likely thing to happen in real life.
Anyway I also agree that the particulars of this are not that important to my point.
Basically I am saying that deterministic physics plus randomness can produce any possible pattern, as you’re noting. So it can also produce the pattern produced by free will. Or do you have some idea of what free will would do which is different from deterministic physics plus randomness? If so, I haven’t see it suggested yet.
Yes, I agree. It can. Deterministic physics alone can, if you have a long enough list of rules.
...I’m not seeing this. It can consider all the possibilities even if it knows that it must play the possibility with the highest odds of winning—in fact, knowing that means that it must consider all the possibilities in order to calculate those odds, surely?
Even if it knows that it must play the move with the highest odds of winning, as far as it knows when it starts considering, that could be any of the moves.
But yes that would be knowledge that its move is objectively deterministic. This would not necessarily prevent it from feeling like it could make any move it wanted, just like people who believe themselves subject to deterministic physics still feel like they can do whatever they want.
But the chess computer doesn’t have to know what is determining its moves, in which case it will be even more likely to feel that it can make whatever move it wants.
Well, yes, feeling like it has freedom doesn’t really prevent it from not having freedom; but I don’t see how the feeling of freedom makes any difference at all. Why shouldn’t the chess computer feel constrained?
I agree that the feeling doesn’t make any difference. That’s what I’m saying: whether it feels constrained or not, it may or may not be deterministic. Those are two different things. The same is true for us.