Can you do it (unproblematically)?
(I am interested, because I am looking at suggestions about how to measure a person's freedom. Roughly, it is suggested that this be done by counting the actions that the person is free to do. But as far as I can see, everyone (even in nasty police states) has an infinite range of actions that they are free to do.
To illustrate: Jones is free to have a game of speed "air-chess" (like air guitair, but with chess) with himself for twenty minutes, then perform an action of his choosing (all of course, with his left pinkie at a precise level of contraction that is of his choosing: and to top it off, he can pick a few numbers at random between 0 and ... lets leave the upper bound).
The problem: we want to say that Jones with freedom of speech has more freedom than Jones without freedom of speech. But this involves saying that one infinite set (of free actions) is larger than another.
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8 comments:
Here’s a start:
There is a standard and quite simple way to rank infinite sets but I don’t think it will be useful to you. The idea is to say that two sets are the same size iff there is a bijection (a one-one and onto function) from one set to the other. For example, the set of letters a,b,c has the same number of elements as the set of numbers 1,2,3 because I can define a bijection between them, namely, a goes to 1, b goes to 2, c goes to 3. This concept of ‘having the same size’ also works for infinite sets. For example, the set of natural numbers is the same size as the set of even natural numbers because I can define the following bijection between the two sets:
Let n be a natural number, then B(n)=2n.
Here the function B(_) is one-to-one and onto (this is trivial to verify). More interestingly, it turns out that on this concept, the prime numbers, the natural numbers, the integers, the rational numbers (Q), the set Q x Q of pairs of rational numbers all have the same number of elements! This is counterintuitive. Consider the fact that the set of natural numbers is the same size as the set of rational numbers. Clearly all natural numbers are in the set of rationals, and clearly there are infinitely many rational numbers that aren’t natural numbers. Yet these sets are the same size! The proof of this, due to Cantor, is incredibly neat and compelling (http://en.wikipedia.org/wiki/Countable_set). Cantor also proved another really important fact about the size of sets. He showed that the set R of real numbers is bigger than all the other sets I just mentioned. It is also easy to show that the set of all subsets of natural numbers is as big as the reals, and that any interval in the reals such as the interval from 0 to 1 is as big as all of the reals. After the reals you get can arbitrarily large infinities of sets that only logicians and set theorists are interested in (check wikipedia on ‘cardinal number’ for details).
Anyway, I don’t think this will be relevant to you for the following reason. Any person could play count the real numbers just as well as they could play count the natural numbers. Similarly, they could play ‘count the elements in the set’ for any set, no matter how huge an infinity of elements it contains.
So you might instead consider a more familiar measure of infinite sets. Think about the difference between the set of real numbers R (visualized as a line) and the set of all pairs of real numbers RxR (visualized as the standard x,y plane). Both sets are infinite, and both in fact have the same size of infinity by the above definition. But in one set you can move in two dimensions rather than one. Of course, there is no reason to stop at 2 dimensions. We can consider the set of n-tuples of real numbers, RxRx……xR, i.e. the set of points in n-dimensional space, where n can be any natural number. This set again has the same size as the set R, but this time you can move in n different dimensions. [Let me know if you want a more thorough mathematical explanation of this.] Maybe you can look at freedom in this way. Humans may have infinitely possible activities but they only have some large finite number of classes or dimensions of activities. For instance, you might select playing a classical instrument as a particular dimension. Clearly, there are infinitely many music pieces you could play, infinitely many weird techniques you could invent for your instrument, etc. etc. But you might say that all of these different musical activities lie in the same dimension. As you play different pieces you merely move between points on a single axis, since the activity of playing one piece is so similar to playing another. In order to move in a different dimension, you would have to do a significantly different activity, like raising a family. (We even speak in these terms, e.g. “Taking up dancing added a whole new dimension to my life!”). Admittedly, this is a vague proposal. How does one determine what the finite list of activities is that constitute the finite number of dimensions? I have a couple of ideas. Some ethical theorists (email if you want names) think there is a list of objectively good human activities (appreciating beauty, love of family and friends, developing the intellect, etc.). These lists are presumably finite, as they appear in non-infinitely long books, so you might look at some and pick those as your dimensions.
Another idea would be to look at the activities people actually do in their lives (counting only substantial things—you couldn’t count things like ‘biting his nails’). You would get a really long, but finite list. You could then cull a large part of list, only selecting things that people spend more than one hour on per week (or some other cut-off point). The activities you have left over would then be your dimensions. My guess is that if you did this for the UK compared to North Korea you would get many more activities for the UK. North Korea may have some unique activities (being tortured, doing stuff with the communist party, singing the national anthem all the time in school, etc.) but these would be fewer and less varied than the activities of Brits.
Another idea would be look at lists of activities that people would like to do. Presumably lots of Chinese families would like to have more than one child. Lots of Russians might want to visit the US, etc. etc. If you polled adults and asked for feasible activities, you would get a long but finite list of activities.
Thank you for your comment!
The problem in a more succinct fashion (which I think is a powerful and pertinent one, yet absent from any literature that I have read on the topic):
1) Comparisons of freedom are made by comparing the size of the sets of free actions available to different people (or the same person under different circumstances)
2) The set of free actions available to a person is always infinite
3) The size of infinite sets cannot be compared
4) Comparisons of freedom can be made
A. You explain (very interestingly) that 3 is false. But you also suggest the following, which resurrects the problem:
2a) The set of free actions available to a person always contains an arbitrary large infinity of elements
3a) The size of sets containing an arbitrary large infinity of elements cannot be compared.
You justify 2a) by claiming:
5) that anyone is always free to play ‘count the elements in the set’ for a set containing an arbitrarily huge infinity of elements.
But 5) only shows that anyone always has one particular action available, naming the counting option that you mention (which would take a very, very, very (infinitely) long time). Thus 5) does not entail 2a).
Better would be 5a) that anyone is always free to play ‘pick a random element in the set’, where the set contains an arbitrary large infinity of elements. Playing this game creates the quantity of available actions that we are looking for.
But is 5a) true? I wouldn’t know where to start if asked to pick a random element from a set with an arbitrarily large infinity of elements.
More broadly: I’m not entirely convinced that 2) is true (the set of free actions available could just always be very, very, very large). If it is, we could do with a succinct demonstration of that fact. And further, to have some way of demonstrating what type of infinity we are dealing with. Calling out random numbers is not a way to do this, because there is a limit on how big a number must be before it would take us our whole lives to utter the syllables needed to express them (even with abbreviations like 9^9^9^9^9). Does the same problem apply when we think of numbers?
But, if we are dealing with some variant of infinity, then I cannot see how any sort of government intervention could alter the set of free actions available to a person to the extent that the infinity of elements within the set changed type. Thus no comparison is possible (because the type of infinity is the same both before and after intervention).
B. You suggest a modification of 1) to:
1a) Comparisons of freedom are made by comparing the size of the sets of dimensions of free actions available to different people (or the same person under different circumstances)
The problem here is that in addition to the contentious area: what does it mean for an action to be free?, you get the disputes in deciding how to group the actions.
But I think this may be a promising route to take, and I’ll give it some more thought.
C. Consider the implications of rejecting 4)
It would still be possible to say that a law attacks freedom in the sense that it renders unfree to perform actions that we were previously free to do, but paradoxically it wouldn’t leave us any less free.
We could say: A is free to do things that B is not free to do, and that A is free to do everything that B is free to do. But this would not imply that A had more freedom than B.
Having more free actions (and thus more freedom) might be thought to be a moral good, but how do you get them? Does adding a new free action to the set already available increase the number within that set? Are we then all locked onto the same level of freedom? If so, freedom would be a useless concept in analysing differences in actual and possible welfare levels.
The key question at this point is what type of infinity we are talking about. We could say that Bob has an infinite number of actions available to him if (A) Bob is free to do more different actions that he could possibly perform in a 120 year lifetime. That is, if Bob started to perform all of his available actions in sequence, then he would die before finishing the sequence. I picked the number 120 as a rough upper bound of human lifespan. The problem with (A) is that freedom may be less dependent on the set of actions which is physically possible (possible according to the physical laws which restrict human lifespan to approx. 120 years) and more dependent on the set of actions which is imaginable or is metaphysically or epistemically possible. For example, Bob might feel more free, in a way which makes a substantial difference to his life, if he believed that in the possible world most like the actual world in which Bob lives 200 years, he had the freedom to have a 200-year game of Monopoly. Bob might come to know about such non-actual possibilities by making inferences from his actual possibilities. There is no actual legal restriction preventing Bob playing a 200 year game of Monopoly, and no reason to suppose that in a world with a 200+ year-old Bob there would always be some such legal restriction. [I leave aside the question of whether the possible worlds here are epistemic (i.e. worlds that Bob thinks are possible but which might actually be self-contradictory) or metaphysical (i.e. logically possible ways that the actual world could be)].
So we might modify (A) and say that Bob has an infinite number of actions available to him if (B) Bob is free to do more different actions that he could possibly perform in a lifetime of any possible finite length (e.g. 200, 5000, or 10^10^10 years). But even this might be too restrictive. Some physicists believe that the universe is infinite in size (which has interesting consequences for consequentialism (http://www.nickbostrom.com/ethics/infinite.pdf)). Some people think that think that time will go on infinitely or at least that it is a possibility that time will go on infinitely. So some people's freedom may be affected by whether or not they have (C) more actions available to them than they'd be able to perform in an infinite life.
So now we have three ways in which Bob can be free to perform an infinite number of actions. The first question is whether any humans actually are free to perform an infinite number of actions (in the senses (A), (B) and (C)). I say yes. Consider the action of shaking your hand up and down (not as in a handshake) as you do when you want to shake water off it (when your hand is wet). If you do this multiple times, then each shake is slightly different (and recognizably so) because of how the mechanics of our bodies works. So even if you an infinite amount of time, you could just keep shaking your hand, knowing that the action will be slightly different each time. The fact that in practice you would get tired of doing this is not important. You could switch after some time to just repeating the words 'hello world' again and again. Each utterance of the word would be slightly different, because of the nature of the human voice box. These examples are not entirely preposterous. Some people enjoy hearing or playing the same piece of music many thousands of times, and they often justify this by saying things like 'it is a different experience every time I hear/play it'. And this is plausibly not a delusion. If someone is playing a piano, then the experience will depend on the temperature and humidity in the room, the tuning and age of the piano, the way in which the person hits the keys, and the general state of mind of the person playing.
The key objection to this example is that, say, the different utterances of hello world should not count as different actions. You might say that people would not see these as distinct actions, and so they shouldn't be counted as such. Similarly, I'm sure that many people would not count the different varieties of the game 'shout a member of this set of numbers' as different actions. So again we come to the problem not of counting the number of actions people are free to do, but of saying what constitutes a distinct actions. We also have the problem of how objective/subjective freedom is. For example, suppose there are two actions that are objectively different (saying 'hello world' on two different days). If Bob doesn't think they are different actions, can we still say that Bob is free to perform two distinct actions?
The examples I've given above make clear that humans will be free to perform an infinite number of actions (in the sense of (A), (B), and (C)). I don't see how the things thought of as restricting a person's freedom (governments, jail, kidnappers) would affect the number of actions available to a person (it would always be infinite in the above sense). I'm sure there are other ways to think about some having an infinite number of actions. For instance, you might say that Bob having an infinite number of actions means Bob having an infinite set S, where S is the set of all actions that Bob could perform at present. More specifically, S is the set of all possible worlds which are identical to the actual world except in the fact that Bob performs a distinct action in each of them. On this conception of infinite, I'm not sure if the 'hello world' action would give you infinitely many distinct actions. But here you could use the game of 'shout (or think about) a random element of a set'. An easy way to do this would be to have a computer generate a pseudo-random number from an infinite set. You can do this for the real numbers, giving you uncountably many possible actions. I don't know if you can do this for sets bigger than the real numbers, but I don't see why not. So you just press a button on your keyboard and then shout the number that comes up on the screen.
Having more free actions (and thus more freedom) might be thought to be a moral good, but how do you get them? Does adding a new free action to the set already available increase the number within that set? Are we then all locked onto the same level of freedom? If so, freedom would be a useless concept in analysing differences in actual and possible welfare levels.
It is clearly false that we all have the same level of freedom. I like to think of this sort of analysis of concepts as being like fitting a hypothesis to data. Suppose you do a test in which you ask individuals to judge whether Mr. X is more free than Mr. Y. If there is lots of agreement (without people conferring) on this question, then it is clear that the different individuals are carrying out some common mental computation when they think about freedom. What you are trying to do is describe this computation in simpler terms. [I presume you're not trying to come up with a normative concept that is somewhat like the pre-existing notion of freedom.] In trying to describe a computation, you come up with hypotheses that are meant to explain the data you have. As in science, your data is sacred: you do not bend the data to fit your hypotheses. I take it as a datum, a good piece of data, that it is false that we all have the same level of freedom. So any hypothesis that entails the negation of that datum will be ipso facto refuted.
To add to my last point, it may be the case that with freedom there just isn't a common computation going on in the minds of different people. Berlin famously distinguishes two sorts of freedom. Maybe there are really two (or more) computations that people perform when considering questions of freedom. Even worse, it may be that there isn't actually a common computation. Maybe if you asked people questions about freedom you'd get no pattern in the answers, because everyone means something different when they use that word.
In any case, my basic point holds. It is a scientific sin to be influenced by your hypothesis when looking at the data: you should always look at whether the data falsifies the hypothesis and not whether you can massage the data to fit the hypothesis. In philosophy, it's really hard to avoid massaging the data to fit the hypothesis. For instance, one hypothesis is that knowledge is justified true belief. If you get attached to that hypothesis, it's easy to tell yourself that supposed counterexamples to that hypothesis (cases where there is justified true belief and no knowledge) are actually cases of knowledge according to your intuitions. But this is a sin! Your intuitions about what is and what is not knowledge are your data: they give you facts about knowledge that enable you to decide whether a hypothesis is correct or incorrect. If these intuitions are clouded by the hypothesis you're trying to test, then you're in trouble. You want your pristine intuitions, the intuitions that you'd have if you hadn't heard your hypothesis but still understood the concept of knowledge.
I can't really tell whether you (Will) are violating any of these recommendations, but they are important things to keep in mind when looking at something like freedom.
A criticism of something I said in my first comment:
The problem with looking at lists of activities that people actually perform or lists of activities that people say they want to perform is that people may feel more free if they have actions available to them that they would never want to perform. Bob might say something like the following:
"I never take cocaine and I don't expect I will ever want to take. It is also very hard for me to imagine ever being in a situation where I would want to take it, even if I were quite a different person from who I am. Moreover, I don't want any of my friends to ever take it. Yet I think I am more free for having the option of taking cocaine."
How to make sense of such a statement? One idea is that Bob thinks that there is a miniscule but non-zero chance that he will change his ways and be interested in cocaine, and so he wants to have that option if he does. Another is that Bob just thinks that having restrictions put on him is bad, regardless of whether the restrictions make any actual or possible difference to his life.
1) I have no strong views on what the meaning of freedom is. Rather, I am simply exploring one idea that is sometimes tied to the term freedom, and seeing if it has any major defects (e.g. being useless for comparisons of levels of freedom).
2) “ We could say that Bob has an infinite number of actions available to him if (A) Bob is free to do more different actions that he could possibly perform in a 120 year lifetime. “
I don’t like this statement, for three reasons:
A) Suppose that the average duration of an action is 5 seconds. Then (12*60*24*365*120)+1 of these actions could not possibly be performed in 120 years. But (12*60*24*365*120)+1 is not an infinite number! This argument works for an average duration of an action and any lifespan.
B) There is a problem of how we divide composite actions into basic action units. E.g. “Walking to London” can be decomposed into daily action plans, which can be decomposed into steps, and then into muscular contractions etc.
C) I cannot perform all of the actions available to me now in sequence. After I have done the first action, some actions will no longer be available. E.g. If I am in Chichester now, I can walk from Chichester to any other town. But as soon as I walk to another town, I can no longer walk from Chichester to London without first returning to Chichester. To put it differently, sets of available actions are circumstance specific. Performing an action will change circumstances.
D) The only way in which I can see that lifespans are relevant to freedom is through placing restrictions on very long composite actions – things that would take a lifetime + to do, e.g. read every text on freedom ever written.
3) I suggest that we distinguish between:
i) X being free (objective)
ii) X believing that he is free (subjective)
iii) X feeling free (this is a quite different phenomenon, concerning a subject’s affective state) it may, or may not, coincide with objective and/or subjective freedom
4) “The first question is whether any humans actually are free to perform an infinite number of actions (in the senses (A), (B) and (C)). I say yes. Consider the action of shaking your hand up and down (not as in a handshake) as you do when you want to shake water off it (when your hand is wet). If you do this multiple times, then each shake is slightly different (and recognizably so) because of how the mechanics of our bodies works. So even if you an infinite amount of time, you could just keep shaking your hand, knowing that the action will be slightly different each time.”
You say that if I kept shaking my hand forever, I would get a different movement each time. Perhaps. But what you need to say is that at any given time, I have any of these infinite hand-shakes available to me. The issue is about the set of available actions, not about the possibility of going through life (or eternity) without performing the same action twice.
I sort of agree with you that the number of available actions is always infinite, but I’d like to have a clear demonstration.
5) What about this as a demonstration that the set of available actions is finite?
I. We have a finite number of muscles
II. There are a finite number of commands that can be given to each muscle
(From I and II) III. There are a finite number of muscle-commands.
IV. An action is a combination of muscle-commands.
V. The product of any two finite sets is also finite. (I’m not sure if ‘product’ is the right word. What I mean is the operation on {a,b,c} and {d,e,f] by which you get {(a,d), (a,e), (a,f) etc.
(therefore) VI. The set of available actions is finite
I guess that the weak point is II. If that turns out to be false, then we have infinite available actions. Maybe it’s time to e-mail a bio-physicist…
There are a finite number of commands that can be given to each muscle
This assumes that we have a discrete set of muscle commands: to give a crude example, the set of commands for my eyelids would be {open, half-open, closed}. But it seems more intuitive that the actions of my muscles vary continuously (not discretely), and that my muscle commands would also vary continuously. Are there only a finite number of ways that I can move my eyelids? Or a finite number of ways that I can move my index finger?
This seems a bit like the question of how many different space points there are. Space doesn't seem to be discrete: between any two points in space there seems to be another point. So space is infinitely divisible, and so there are infinitely many points in space at which I could stand (or better: at which my centre of gravity could stand). So if I am standing in my living room, there are infinitely many points in space to which I could move my centre of gravity.
This gets into thorny issues of physics and metaphysics. I know quantum mechanics and information theory made an impact on how people thought about whether space is continuous or discrete.
I also emailed a bio-physicist. (Well, mainly a bio-chemist, but he also knows a huge amount of physics.)
I'll first say a few things about suitable orderings of infinite sets, since that's the question in the post, then I'll add a couple of comments about the conception of freedom you're working with.
1) Clearly, looking at the size of the infinite sets will not help you. You could curtail most of the possible actions available to a person without the resulting set being smaller. (Compare the natural numbers with the primes.) We ideally want what mathematicians call a total order of the sets of available actions: a relation defined on the set of these sets which is transitive, antisymmetric and total (the first two of these are familiar from further maths; a relation is total iff for all x,y in the domain, xRy or yRx). An example of a total ordering is the relation 'greater to or equal than' defined over the reals. Unfortunately, there is no such relation on the set of sets of possible actions which will do what you want. (Why? Not because the set is too large. If you're really interested, Will, email me.)
We can get something -- a partial order -- simply by invoking subsethood. Let us say that a person is at least as free with option set A as with option set B if B is a subset of A. In other words, if every option in B is in A, then the person can't be any less free if A is the set of their options. They can do anything in B. If they also have an option not in B, they are more free than they would be if B were their option set.
However, this doesn't handle the interesting cases. What if A contains some options not in B, and B contains some options not in A? Neither is a subset of the other, so we're stuck. We can cover some of these cases by adopting more complex definitions of the 'at least as free as' relation, but almost all of them (I use this phrase in the technical sense) will remain unclassified -- that is, our definition almost never determines whether someone with arbitrary option set X is freer than someone with arbitrary option set Y.
My answer is mostly negative, then. I hope what I've written is of some interest nonetheless.
2) This approach to freedom resembles that developed by Amartya Sen. Is that where you got it from? Sen has provided the necessary mathematical underpinnings for his theory, which does not provide for infinite sets of options. Sen's theory is a pragmatic one, really. It requires that a set of relevant options (or dimensions of freedom) be specified. The idea is that most important freedoms in a given context can be listed; Sen's theory then allows quantitative statements to be made about them in the context of a given population.
Attempting to generalise to infinite option sets seems ill-advised to me. Clearly, an account of freedom where one selects a small sample of actions for consideration as one sees fit, more or less, is not philosophically adequate. But you cannot accept the generalised form of this account without conceding that, in almost any case we ever consider, there is no answer to the question whether one person is more or less free than another.
There are many other problems with thinking of freedom in this way. One of them you've mentioned already, more or less: there is no obvious way to distinguish actions, and consequently no obvious way of counting them: in jargon, we have no individuation criteria for actions, or indeed for events. There are other objections which are deeper and more serious, but I'll save those for conversation.
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