The Q-device
In a blog post dated 2010 Mar 17, mathematician and philosopher Alexander Pruss expressed some interesting ideas regarding probability and countably infinite samples in an attempt to show an absurdity which would, in his judgment, lend support to the claim that the existence of what he and others call "actual" infinite collections of physical objects is impossible.[1] He suggests a set of hypotheses which, taken together, appear to violate our intuition regarding probability. While I do not believe this constitutes evidence against actual infinities, I find the argument interesting in another way which I shall discuss here. In particular, I maintain that we ought not assign probability values under certain conditions whereby the probability measurements in question are insufficiently interpreted. Paul Castell calls this position abstention,[2] and Pruss's ideas yield an opportunity to give an example of how we might find reason for adopting it.
Pruss asks us to consider the following: Suppose that a random process bestows the unobservable property Q on a subset of a population with total size N, where each person has the constant probability p ∈ (0,1) of being granted Q. Let K denote the total number of people who are actually given property Q; then we can expect the value of K to be approximately pN. So, consider Mary, a member of that population. With only the approximation pN for K at her disposal, she would correctly conclude that the probability she has Q is precisely p. Indeed, that is true practically by hypothesis. However, if she were to discover the actual value of K, then that knowledge would supersede the estimate pN. No longer would she say that the probability she has Q is p; instead she would assign herself a probability of K/N for having Q.
Now suppose we alter this hypothetical scenario by considering not a finite population of size N, but rather a countably infinite population. In that case, what is the probability that Mary, if she is a member of that population, has Q? We might be tempted to say that the probability is p; indeed, Pruss insists that this answer is "obvious," although ultimately contradictory, since he invites us to infer from p that there are infinitely many members of the population with Q, and infinitely many without. However, I believe his judgment is too hasty, for reasons I shall presently expound.
As described thus far, the scenario is deceptively ambiguous. What exactly does it mean that each person in the population is bestowed Q with probability p, especially if that population is infinite? This point is not at all clear from Pruss's description, but I suggest that we need to know exactly what we are measuring with the value p before we can draw any conclusions from it. We can endow p with any one of a number of particular meanings: For example, we could consider what would happen if the same process which acts on the infinite population instead acted on a population of finite size N, interpreting p as the anticipated value of K/N for that case---and here I simply mean the value which we should think most reasonable, not the formally-defined expected value. Alternatively, we could hypothesize that the process in question is drawn out over an infinite period of time, and never actually completes; then we may interpret p as the anticipated value of the limit of kn/n as n approaches infinity, where kn is the number of people endowed with Q after n have been passed over by the process.
However, I prefer the following interpretation: Suppose there is a Q-device which passes over a denumerable population with such rapidly increasing speed that the device assigns either Q or no-Q to every member in a finite time interval. In particular, suppose that it only takes the device half the time to assign to some member of the population Q or no-Q as it did the previous person. So, if t is the time it takes for the device to assign the first member of the population with Q or no-Q, then the total time required by the Q-device to pass over the entire population is ∑t/2i=2t, where i=0,1,2.... In this way, we may interpret p as the propensity of the device to assign each person with Q, that is, the anticipated value of the limit of kn/n as discussed in the previous example. One may notice that the possible existence of such a Zeno-reminiscent device remains dubious. I freely acknowledge this difficulty, and I do not claim it is realistic or physically possible. It is merely a hypothetical construct for the present conceptual exercise.
We turn now to what I regard as the central matter: What is the probability that, being part of a denumerable population after the Q-device has completely passed over it, Mary has Q? Before we can answer this question, we must consider what it means to talk about this probability. For ease of notation, let M denote the event whereby Mary has Q, given that the device has passed over the whole population. Then the probability of interest is P(M), and we ask what P(M) is actually measuring. This does not seem to be immediately clear. If only the population had finite size N, we could interpret P(M) as the anticipated value of the ratio of correct guesses to N, should every person in the population guess whether or not he has Q. However, this interpretation will not do, since the population of interest is infinite.
Many of us might be tempted to give P(M) in a folk interpretation. For example, we could imagine a set of trials where Mary attempts to guess whether or not she has Q, then time is rewound and Mary guesses again, repeated as needed. Of course, it is naive to talk about rewinding time, and presumptuous to suppose Mary's guess should be different from trial to trial even if it were somehow conceivable. Such interpretations are inappropriate for a consistent analysis.
We might wish instead to interpret P(M) as the anticipated value of G/S, where a subset of finite size S is chosen randomly, without privileging one member over another, from the population, and G is the number of people in that subset to correctly guess whether or not they have Q. However, such a definition turns out to be incoherent; for it is impossible to assign a uniform distribution to a denumerable sample space. In other words, one cannot select elements from an infinite set without privileging certain elements over others, and so it is nonsense to suggest that we can choose a finite subset of the population in that manner.
Although these examples hardly exhaust the myriad of possible approaches to providing P(M) with some meaningful interpretation, I believe they help illustrate the difficulties involved. While I have no cause to claim it is impossible, I nevertheless remain highly skeptical that we will ever find an appropriate definition for P(M) which would satisfy both mathematical convention and intuition. In any case, we learn an informative lesson from this exercise, that, at least under certain circumstances, we ought not leave probability measurements ambiguous or otherwise undefined. In these situations, we cannot always draw conclusions about or assign values to those measurements until such time they are endowed with sufficient meaning.
This difficulty is the primary motivation behind the abstention position. Castell, who I should add is not at all concerned with the sort of exercise I have outlined here, offers a number of reasons for opposing it, responses to which I shall save for another occasion, except to mention that I find his arguments unpersuasive if insightful. For the present time, I wish only to have shown that the Q-device provides a helpful illustration of how insufficient meaningfulness of probability measurements may lead us to prefer abstention.
Notes:
[1] Pruss, Alexander. ``Another argument against actual infinity,'' 2010 Mar 17. http://alexanderpruss.blogspot.com/2010/03/another-argument-against-actual.html
[2] Castell, Paul. "A Consistent Restriction of the Principle of Indifference," The British Journal for the Philosophy of Science, Vol. 49, No. 3 (Sep., 1998), pp. 387-395.
Pruss asks us to consider the following: Suppose that a random process bestows the unobservable property Q on a subset of a population with total size N, where each person has the constant probability p ∈ (0,1) of being granted Q. Let K denote the total number of people who are actually given property Q; then we can expect the value of K to be approximately pN. So, consider Mary, a member of that population. With only the approximation pN for K at her disposal, she would correctly conclude that the probability she has Q is precisely p. Indeed, that is true practically by hypothesis. However, if she were to discover the actual value of K, then that knowledge would supersede the estimate pN. No longer would she say that the probability she has Q is p; instead she would assign herself a probability of K/N for having Q.
Now suppose we alter this hypothetical scenario by considering not a finite population of size N, but rather a countably infinite population. In that case, what is the probability that Mary, if she is a member of that population, has Q? We might be tempted to say that the probability is p; indeed, Pruss insists that this answer is "obvious," although ultimately contradictory, since he invites us to infer from p that there are infinitely many members of the population with Q, and infinitely many without. However, I believe his judgment is too hasty, for reasons I shall presently expound.
As described thus far, the scenario is deceptively ambiguous. What exactly does it mean that each person in the population is bestowed Q with probability p, especially if that population is infinite? This point is not at all clear from Pruss's description, but I suggest that we need to know exactly what we are measuring with the value p before we can draw any conclusions from it. We can endow p with any one of a number of particular meanings: For example, we could consider what would happen if the same process which acts on the infinite population instead acted on a population of finite size N, interpreting p as the anticipated value of K/N for that case---and here I simply mean the value which we should think most reasonable, not the formally-defined expected value. Alternatively, we could hypothesize that the process in question is drawn out over an infinite period of time, and never actually completes; then we may interpret p as the anticipated value of the limit of kn/n as n approaches infinity, where kn is the number of people endowed with Q after n have been passed over by the process.
However, I prefer the following interpretation: Suppose there is a Q-device which passes over a denumerable population with such rapidly increasing speed that the device assigns either Q or no-Q to every member in a finite time interval. In particular, suppose that it only takes the device half the time to assign to some member of the population Q or no-Q as it did the previous person. So, if t is the time it takes for the device to assign the first member of the population with Q or no-Q, then the total time required by the Q-device to pass over the entire population is ∑t/2i=2t, where i=0,1,2.... In this way, we may interpret p as the propensity of the device to assign each person with Q, that is, the anticipated value of the limit of kn/n as discussed in the previous example. One may notice that the possible existence of such a Zeno-reminiscent device remains dubious. I freely acknowledge this difficulty, and I do not claim it is realistic or physically possible. It is merely a hypothetical construct for the present conceptual exercise.
We turn now to what I regard as the central matter: What is the probability that, being part of a denumerable population after the Q-device has completely passed over it, Mary has Q? Before we can answer this question, we must consider what it means to talk about this probability. For ease of notation, let M denote the event whereby Mary has Q, given that the device has passed over the whole population. Then the probability of interest is P(M), and we ask what P(M) is actually measuring. This does not seem to be immediately clear. If only the population had finite size N, we could interpret P(M) as the anticipated value of the ratio of correct guesses to N, should every person in the population guess whether or not he has Q. However, this interpretation will not do, since the population of interest is infinite.
Many of us might be tempted to give P(M) in a folk interpretation. For example, we could imagine a set of trials where Mary attempts to guess whether or not she has Q, then time is rewound and Mary guesses again, repeated as needed. Of course, it is naive to talk about rewinding time, and presumptuous to suppose Mary's guess should be different from trial to trial even if it were somehow conceivable. Such interpretations are inappropriate for a consistent analysis.
We might wish instead to interpret P(M) as the anticipated value of G/S, where a subset of finite size S is chosen randomly, without privileging one member over another, from the population, and G is the number of people in that subset to correctly guess whether or not they have Q. However, such a definition turns out to be incoherent; for it is impossible to assign a uniform distribution to a denumerable sample space. In other words, one cannot select elements from an infinite set without privileging certain elements over others, and so it is nonsense to suggest that we can choose a finite subset of the population in that manner.
Although these examples hardly exhaust the myriad of possible approaches to providing P(M) with some meaningful interpretation, I believe they help illustrate the difficulties involved. While I have no cause to claim it is impossible, I nevertheless remain highly skeptical that we will ever find an appropriate definition for P(M) which would satisfy both mathematical convention and intuition. In any case, we learn an informative lesson from this exercise, that, at least under certain circumstances, we ought not leave probability measurements ambiguous or otherwise undefined. In these situations, we cannot always draw conclusions about or assign values to those measurements until such time they are endowed with sufficient meaning.
This difficulty is the primary motivation behind the abstention position. Castell, who I should add is not at all concerned with the sort of exercise I have outlined here, offers a number of reasons for opposing it, responses to which I shall save for another occasion, except to mention that I find his arguments unpersuasive if insightful. For the present time, I wish only to have shown that the Q-device provides a helpful illustration of how insufficient meaningfulness of probability measurements may lead us to prefer abstention.
Notes:
[1] Pruss, Alexander. ``Another argument against actual infinity,'' 2010 Mar 17. http://alexanderpruss.blogspot.com/2010/03/another-argument-against-actual.html
[2] Castell, Paul. "A Consistent Restriction of the Principle of Indifference," The British Journal for the Philosophy of Science, Vol. 49, No. 3 (Sep., 1998), pp. 387-395.
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