According to open future views, the proposition that in 2015 a fair and indeterministic coin lands heads has some probability but is not true. However, that proposition is apt to become true in 2015. So the probability of the proposition isn't the same as the probability of the proposition being true, since it's certainly not true now, but might well become true in 2015.
So far so good (or bad). Suppose God promises you that from 2015 onward, every year, a fair and indeterministic coin will be tossed. Now let Q be the proposition that there are infinitely many years after 2014 during each of which a fair and indeterministic coin lands heads [I screwed up in the original formulation of Q and wrote "that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads"; Alan Rhoda's response targets my screwup; see my response to him below]. Now note that on open future views Q can never possibly become true. For on any date, the proposition requires for its truth that there will be infinitely many fair and indeterministic heads results still past that date, and on open future views a proposition that requires an undetermined future event won't be true.
So, open future views have to say that it's impossible for Q to ever be true. But a proposition such that it's impossible for it ever to be true should get probability zero. But the probability that of the infinitely many coin tosses, infinitely many will be heads is 1 according to classical probability theory. So open future views should be rejected.
Here's another argument in the same vein. Suppose I know I will have an eternal afterlife, and I promise you that I will freely pray for you every day, ad infinitum, starting November 1, 2014. On open future views, the object of my promise is a proposition that can never be true. But it's clearly a bad thing to promise something that can never be true. Yet what I promised wasn't a bad thing to promise. So open future views are false.
One might even have the direct intuition that one could keep the promise. That intuition is incompatible with open future views.
92 comments:
There's another interesting argument in the neighborhood. Any proposition that could not be true, one should conclude, has to be false. But open theists must claim that Q is not false, either. So, some propositions p such that ☐~p are not false on open theism. But ☐~p and p is not false is incoherent.
Piling on ...
If OF views are true, one cannot make true the propositions whose contents constitute eternal promises. (E.g. "I will pray for you every day forever.") Ought implies can. So if you make such a promise, you have no obligation to keep it.
I think the biggest problem here is that you're leaving out Mike said: The OF proponent doesn't think Q is ever false either. It just doesn't have truth value. So, while saying "a proposition which can't ever be true should get a probability 0" seems right; it just ignores the symmetrical statement that "a proposition which can't ever be false should get a probability of 1". The truth is that the OF proponent doesn't assign either probability, but rather treates Q as having no truth value at all; akin to other vacuous statements like "Santa Clause's hairline is receding".
If God promises that there will be a coin toss every year from now on, then He has committed Himself to causing that to be the case from now on. However, he hasn't said how the coin toss will turn out, and so it may turn out either way. Since it has not yet turned out either way, it is meaningless to say that Q is false OR true. Heads at some point is highly probable, but probability just gives us epistemic warrant to expect it; it doesn't assign truth values.
As for your promise: So long as there never comes a day when you do not pray for me, you are fulfilling your promise. The big thing about OF is that temporal becoming is a real and objective feature of the world in addition to mere being. So while it never becomes true that you prayed every day of eternity (even typing out "it is true that you do pray every day of forever" seems like total nonsense to me, since "is" and "do" are in the present tense... but then I'm an OF guy!)... it can nevertheless be true on any given day that you are fulfilling your promise to not let a day go by without praying for me.
Mike: It's rather sneaky of you to only put the ☐ on one part, and not the other, despite the fact that the OF proponent thinks Q being false is equally as impossible as Q being true. The right thing to say is that it lacks truth value altogether. As I said in my comment to Pruss: the statement "Q is true" is in the present tense.
Heath: You do have the obligation each day to not fail to pray. For the OF proponent, your promise means nothing more than "I will never let a day go by when I do not pray for you"; which you can be in the process of fulfilling forever.
Heath:
Nice point. We can also make the point without reference to a general ought implies can principle. It seems to be a part of the normativity of promises that a promise to do something that can never be done has no force.
Michael:
Actually, there are two kinds (at least) of OF views. On one version, future contingents are neither true nor false. On another, they are all false (Alan Rhoda, I think, defends this). My formulation was meant to be neutral between the two views.
"The truth is that the OF proponent doesn't assign either probability, but rather treates Q as having no truth value at all; akin to other vacuous statements like 'Santa Clause's hairline is receding'."
But if "Santa Claus's hairline is receding" is nonsense, it doesn't have a high probability. And Q has a high probability.
"Heads at some point is highly probable, but probability just gives us epistemic warrant to expect it; it doesn't assign truth values."
This comes perilously close to saying: "I know that p isn't true, but I have epistemic warrant for p."
I'm not sure what you mean. Adding ☐ to Q does not affect my argument at all. Go ahead an add it. My argument is that, for any proposition p, if (☐~p & p is not false) is incoherent. That argument is not affected by the assumption that ☐p. It does offer another argument, viz., (☐(p is not false) & ☐~p) is incoherent: that is, it is incoherent to urge that a proposition p could not be true and could not be false.
Here's a different (but related) line of argument.
1. If something is probable enough, it's assertible.
2. If P is assertible, and one is rational, then ~(P is true) is not assertible.
3. If OF, then there will be cases where ~(P is true) will be assertible even when one is rational and P is has as high probability as one wishes.
4. So, ~OF.
And another argument:
1. If Q is a sentence that can never be true, then either Q is nonsense or Q is a necessary falsehood.
2. If Q is nonsense, then P(Q) is undefined.
3. If Q is a necessary falsehood, then P(Q)=0.
4. So if P(Q)>0, then Q can at some time be true.
But in my example P(Q)=1, and yet if OF, then Q can never be true. And that's impossible.
Mike:
"Any proposition that could not be true, one should conclude, has to be false"
This seems to beg the question against a number of non-bivalence views.
For instance, suppose you take a non-bivalence view of the original liar sentence. "This sentence is false" then expresses a proposition p that is neither true nor false. Moreover, it is not a contingent matter that p is neither true nor false. Thus, p not only isn't true, but couldn't be true. But we shouldn't conclude it's false.
Or suppose that you take the view that all propositions expressed by sentences of the form "a is P" are neither true nor false when "a" fails to refer. Then consider the proposition expressed by "Sherlock Holmes is a square circle." This proposition cannot be true. For in worlds (if there are any) in which Sherlock Holmes exists it's false. In worlds in which Sherlock Holmes doesn't exist, it's neither true nor false. But in no world is the proposition true. But, according to the view, it's not actually false.
I agree with you, of course, in the end, but that's because I accept bivalence.
Michael:
" For the OF proponent, your promise means nothing more than 'I will never let a day go by when I do not pray for you'; which you can be in the process of fulfilling forever."
How can you be in the process of fulfilling something where the fulfilling of it is metaphysically never possible? Can you be in the process of squaring a circle?
Hey Dr. Pruss!
So then you are saying that there is an integer N such that N+1=Infinity? And God knows that number since He sequentially flipped a coin an infinite number of times?
I didn't intend to say anything like that. Did I do so somewhere nonetheless?
Dr. Pruss,
You begin by saying:
"Suppose God promises you that from 2015 onward, every year, a fair and indeterministic coin will be tossed. Now let Q be the proposition that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads."
You conclude by saying:
"But the probability that of the infinitely many coin tosses, infinitely many will be heads is 1 according to classical probability theory."
I don't see any claim of there being such an integer.
How do you move from a finite state affairs to an infinite one?
Where do I do that?
Sorry, my email alerts are on the fritz...
Pruss: First off, you're right; I should have clarified that my own OF view is that such statements lack truth value, rather than all being false. However, I don't think I said "Santa Claus' hairline is receding" is nonsense. I think it's vacuous in the sense that no such person happens to exist. It's still a meaningful sentence. And a person could probably make an argument on probabilities that, if Santa Claus does exists, then it is very likely (as a humanoid male) that his hairline is indeed receding. It's a conditional statement. Likewise, I don't think the future exists. But, when I say something is highly probable to happen, perhaps I'm making a conditional statement of that kind. If the state of affairs described by a particular future-tense scenario were to exist, then there is a high probability that that state of affairs would also contain X. As the world changes and progresses, if we arrive at a point where lots of coin flipping has happened, then there's a good chance some of those were heads.
As for "I know p is not true, but I have epistemic warrant for p", perhaps thinking of it as a conditional takes care of that problem too? What I'm really saying is "I know the antecedent of the conditional doesn't currently hold, but I still have strong epistemic warrant for the conditional itself". I don't know. This seems an interesting way of addressing the problem.
Or perhaps we could look at it in something akin to your own view of modality (I just finished "Actuality, Possibility, and Worlds" a couple of weeks ago). Perhaps statements of future probabilities have to do with our epistemic warrant to expect certain outcomes given nothing more than the currently-existing powers and dispositions of things. When I say "X is likely to happen", what I really mean is "the things which have powers to cause X are also strongly disposed to do so", or something like that.
In either case, Q doesn't seem like it has a very high probability at all, since for Q to ever become true would mean that we have actually arrived at the end of an infinite series. I don't even think the probability of Q is high at all; I think it's zero, since it requires us to arrive at the end of an infinite series. However, a modified Q that just says "at some point in the yearly coin flipping, there will be heads" can be very probable in one of the senses I've sketched above, I think.
Pruss: If the "fulfilling" just consists of building a track record, and not of actually finishing anything, then you are fulfilling your promise every day. No one who is promised the daily prayer is actually expecting an infinity of prayers to get completed. They are just expecting that at any point in the future, if someone were to check, it would be true that no previous day had lacked prayers from you.
Dr. Pruss you say about proposition Q:
"Now let Q be the proposition that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads."
And you say:
"For on any date, the proposition requires for its truth that there will be infinitely fair and indeterministic heads results still past that date,"
I would say:
For on any date, it seems to me that proposition Q only requires for it's truth an infinite number of years still past that date and during each of these years the result of a coin toss being a fair and indeterministic head.
Mark:
Sure. But I don't see the problem.
Michael:
"Q doesn't seem like it has a very high probability at all"
There is a probability theory argument for it. :-) For instance, by the law of large numbers, with probability 1, the frequency of heads tosses in any infinite sequence of fair tosses converges to 1/2, which of course requires infinitely many heads. Such limit laws are central to probability theory.
"They are just expecting that at any point in the future, if someone were to check, it would be true that no previous day had lacked prayers from you."
On your view, the sentence following "expecting that" has no truth value.
In an infinite sequence, yes, but no such infinite sequence ever actually comes into being on the Open Future view. It's potentially infinite, but never actually infinite, and so I don't see the analysis applies. Sure, it could be true that the limit being approached is 1/2, but we are always dealing with a finite sequence (you can't build an infinite sequence from successive additions of finites, can you?).
"On your view, the sentence following "expecting that" has no truth value."
To an Open Future theorist, my sentence means "as the world continues to progress and change, there it will never be the case that a day has passed without you praying for me". That is what you are promising me. And your promise (even on the common sense notion of a promise) expresses nothing more than your strong intention; it is not meant to express any actual truths about future-tense realities.
Interesting argument, Alex. Very clever. You will of course not be surprised to find out that I don't think it works. I penned a detailed response to it on my blog (http://alanrhoda.net/wordpress/?p=435), but the short response is that I think you conflate
(a) the chance of Q's becoming true
with
(b) the chance of Q's having become true by any finite number of years after 2015
On open futurism the chance of (a) must be non-zero, while the chance of (b) must be zero. But since there's no contradiction between (a) and (b), there's no problem here for open futurism.
Alan:
I am very sorry: I screwed up the central sentence of my post. I just edited the post to say what I really meant to say and I put in brackets my original misformulation. I am sorry that you wasted time attacking the misformulation. (There is some evidence in other parts of my post that what I now have is what I meant to include when I originally posted, but still the fault is mine.)
With the original formulation, the probabilities just don't work out.
But in the new formulation they do work out.
Now, Q is equivalent to the proposition that infinitely many years starting 2015 have a fair and indeterministic coin toss landing heads.
It's a consequence of the law of large numbers, for instance, that given that there will be infinitely many fair and independent coin tosses, the probability that infinitely many of them will be heads is one.
So here's what's going on. Standard probability theory says P(Q)=1. (Some people say P(Q)=1-infinitesimal. They're wrong, but it shouldn't affect the argument.) But on open future views, it is impossible that Q is ever true.
How could a proposition have probability equal to 1 and yet be necessarily never true?
Thanks for the quick reply, Alex! I'll think carefully about your revision and see if I can come up with a good reply, probably not by tonight though. Best wishes.
Alex, I posted a reply to your reformulation on my blog.
Short answer: You've got two different reformulations of Q. In one case P(Q) = 0. In the other P(Q) = 1. In neither case is the open futurist committed to anything absurd (so far as I can see).
Cheers!
Thanks, and I responded there. And I've reworded my phrasing in the post once again to avoid the ambiguity. :-)
Pruss: What would be wrong with considering all talk of futures as talk of fictions or stories, within which the rules of probability and such all hold, and so we can make statements about the probability in the fiction of an already-infinite future (which can never actually arrive, but this is fiction) and say that the probability of infinitely many heads in such a fiction is 1. God's promise boils down to limiting the number of fictions which He will ever allow to be actualized (it is a statement of intent; albeit an un-thwartable intent).
Now, an actually infinite amount of time having actually already past is an impossible state of affairs anyway. But, even if it were possible, God (given His promise) would not permit it to elapse without their being a fair coin toss in each year, and in this fictional state of affairs it is certain that an infinite number of coins have landed heads.
What am I missing?
If it's possible for there to be such a time, maybe this would work.
But if it's not possible for there to be a time after infinitely many times, this is problematic. For in a fiction where impossible things happen, it's hard to avoid making everything happen.
Hi Alexander, Thank you for this challenging post.
First, I did not read every, so sorry if I repeat anything.
Second, here is some clarification of language:
P1. Given a beginning, never ending years is potentially infinite and never actually infinite.
P2. A fair coin toss by definition has an indeterministic outcome while the probability distribution is .5 heads and .5 tails.
P3. If *n* is the number of fair coin tosses and *n* approaches infinity, then the number of heads approaches infinity.
P4. An open futurist can say that P3 is true while no particular fair coin toss is predictable.
Peace,
Jim
Thanks! I don't see how an open futurist can affirm P3. After all, it is quite open that all the coin tosses will be tails, and so it's not TRUE (given open future) that the number of heads will approach infinity (given open future).
Actually, if *n* is the number of fair coin tosses and *n* approaches infinity, then the number of heads approaches infinity and the number of tails approaches infinity. Also, the probability of all tails approaches 0.
Do you insist that if a series of events has a probability that approaches 0 then that series of events is possible?
All-tails has zero limiting probability but it's no less likely than any other particular infinite sequence. Surely it's possible.
Let us look at it another way. P3 is true according to binomial probability. Are you claiming that binomial probability is invalid?
I want to add that your OP already rejected the possibility of an open future based on binomial probability :-)
Classical probability assigns probability zero, but that doesn't mean it says this doesn't happen. Classical probability assigns zero to each precise point on a target that you're randomly throwing a dart with a perfectly defined tip at, but of course you're going to hit some point or other.
P3 is consistent with classical probability and with the existence of one actual potentially infinite series of fair coin tosses. Unless you disagree with the previous sentence with a valid reason, then I see no reason why P3 is inconsistent with an open future.
P3 cannot be *guaranteed* to be true.
P3 entails that we don't get all tails.
But we have no guarantee that we don't get all tails. All we have is that the probability of all tails is zero.
Suppose you say that it's guaranteed that you won't get all tails. Then by the same token, it's guaranteed for ANY other mathematically infinite sequence that you won't get that sequence. (After all, TTTT... is not any the less likely than some random looking THTT....) But then it's guaranteed that you won't get ANY particular sequence. And that's absurd.
Think about this in terms of free will. Suppose each day for eternity you independently choose whether to dance a jig. Your reasons for dancing a jig are pretty much balanced--P(dance jig on day n) is somewhere between 40% and 60% no matter what happened the day before. But surely it's within your freedom to dance the jig always or to dance the jig never. So it's not guaranteed that you will dance the jig infinitely often--how could probability theory force you to do that? However, given the above assumptions, P(dance jig infinitely often) = 1.
Whoa:
I agree with you that "P3 cannot be *guaranteed* to be true."
I disagree with you that "P3 entails that we don't get all tails." After all, I agree with your potentially infinite version of the lottery paradox.
Unless I accept your false assumption about P3, which you already demonstrated is false, then you provided no valid reason to accept that P3 is inconsistent with a open future.
If the number of heads approaches infinity (as P3 says), the number of heads will be greater than zero. If it's greater than zero, then it's not all tails.
"If the number of heads approaches infinity (as P3 says), the number of heads will be greater than zero."
No, that is invalid. If the number of heads approaches infinity (as P3 says), the number of heads becoming greater than zero has a probability that approaches 1. That probability is never actually 1 while it approaches 1.
Please remember that my first list of objections includes that you confused actual infinity with potential infinity. I hope this does not take forever to clarifiy this....
Surely: "The number of heads approaches infinity" implies "Eventually there will be at least one heads." If not, I don't understand what "approaches" means.
I hope this link helps:
http://www.mathsisfun.com/calculus/limits.html
By the way, your recaptcha just got much harder to read.
The standard definition of a sequence s(n) approaching +infinity as n goes to +infinity is this: for all real M, there is an N such that for all n>N we have s(n)>M.
Let's apply this definition. Let h(n) be the number of heads among the first n tosses. Let M=0. If h(n) approaches infinity, there is an N such that for all n>N we have h(n)>0. In particular, h(N+1)>0. But if h(N+1)>0, then obviously we don't have all-tails among the first N+1 tosses.
I am having trouble following your standard deviation (SD) scenario and might need it broken down more. But the conclusion is contradictory to a possibility. For example, the first 10^100 fair coin tosses in a potentially infinite number of fair coin tosses could possibly be tails while P3 is still true. You can also replace "10^100" with any counting number while P3 is true. Do you agree with my example in the previous two sentences? And does my previous example agree with your SD example?
Also, your SD scenario contradicts your (November 15, 2014 at 12:41 AM) statement: "All-tails has zero limiting probability but it's no less likely than any other particular infinite sequence. Surely it's possible."
I see no contradiction here.
I agree that any finite number of tails could occur and P3 still be true.
Do you also agree that P3 is consistent with an open future?
No, since P3 makes a causally contingent claim about the future.
I think we have some deep misunderstanding here.
Yes, I see that we have a misunderstanding. I will work on resolving this by breaking down key points in our dialogue.
You say:
1. P3 is true.
2. P3 is false if the future is open.
I gather that this means that P3 is true in the case of a settled future and P3 is false in the case of an open future.
You say that P3 is false in the case of an open future "since P3 makes a causally contingent claim about the future."
I gather that a "causally contingent claim" is the same as a "contingent claim."
I suppose that P3 is a hypothetical causally contingent claim or a hypothetical contingent claim, which I suppose is the same. For example, the "If" at the beginning of P3 indicates that "*n* is the number of fair coin tosses and *n* approaches infinity" is hypothetical.
Here I divide P3 into two categories: (1) an open future, (2) a settled future.
P3. If *n* is the number of fair coin tosses and *n* approaches infinity, then the number of heads approaches infinity.
P3o. If *n* is the number of fair coin tosses and *n* approaches infinity while the future is open, then the number of heads approaches infinity.
P3s. If *n* is the number of fair coin tosses and *n* approaches infinity while the future is settled, then the number of heads approaches infinity.
I fail to see how P3o is false while P3s is true.
Hey Mr. Goetz!
You say:
P1. Given a beginning, never ending years is potentially infinite and never actually infinite.
I was wondering how the sequential years can never be infinite and yet you claim they are potentially infinite?
Thanks for your time.
Hi Mark, potential infinity versus actual infinity is an established concept that is easy to find on philosophical web sites, for example:
"The potential infinite is a group of numbers or group of “things” that continues without terminating, going on or repeating itself over and over again with no recognizable ending point."
http://sites.middlebury.edu/fyse1229pisapati/mathematical-work/potential-infinite-v-actual-infinite/
Consider the elapse of seconds or any measurement of time in a flat universe that expands forever. The number of elapsed seconds is always a finite number while the elapse of seconds never ends.
Actually, I think the concept of a potential infinite is rather dubious.
1. If for every natural number n, there actually are at least n Fs, then there actually are infinitely many Fs.
2. If for any F there is a later F, where "later" is a transitive and irreflexive relation, then for every natural number n, there actually are at least n Fs.
3. The "later than" relation for times is transitive and irreflexive.
4. For any day, there is a later day.
5. So, for every natural number n, there actually are at least n days (2-4).
6. So, there are actually infinitely many days. (1,5)
The same goes for any other example I can think of of a potential infinite.
Hi Alexander, I need more context for your model to properly understand your previous comment in this thread. In this actual infinity, are you assuming A-theory of time or B-theory of time?
I don't see any premise for which it matters.
Well, the concept of potential infinity involves a process. For example, in the case of days, potential infinity refers to the process that is the passage of time, so this is relevant only in the case of presentism.
Pruss: If you say "for any day there IS a later day", then you are presupposing a non-presentist view. There is no "later day" on most forms of A-theory, which is probably why Goetz asked you that. The future is potentially infinite only in the sense that there is nothing limiting the number of days that may elapse as time goes on, and so the limit that is being approached is infinity. But, the later days do not actually exist yet.
Now, to respond to your response to me: For in a fiction where impossible things happen, it's hard to avoid making everything happen. I think my point was that, for the sake of the fiction, we treat it as possible that an infinite future is actually reached. As such, we can assign a value to the number of heads. Really, I think, whenever we work with transfinites, we are working out a fiction. There are not actually infinitely many things (certainly not uncountable infinities of things), but we can still talk about what would be the case if (per impossibile) they were true. From what you've written before, I didn't think you'd have a problem with conditionals that have per impossibile antecedents. At any rate, I don't see how they open up the possibility of anything else, since they are acknowledged as impossible, but the conditional can still be true.
Hey Mr. Goetz,
I found the website,
http://sites.middlebury.edu/fyse1229pisapati/mathematical-work/potential-infinite-v-actual-infinite/
that you were pointing me to. Very nice. So looking at the examples of potential infinite that they give:
"The obvious example is the the grouping of natural numbers. No matter where you are while listing or counting out natural numbers, there always exists another number to proceed the one before. Also, a geometric line with a starting point could extend on without end, but could still be potentially infinite because all one would have to do is add on more length to a finite length to allow it to extend."
So now you have said:
"P1. Given a beginning, never ending years is potentially infinite and never actually infinite."
So comparing our years to these examples we would want to say that the number of our years grows larger but never approaches infinity?
Michael:
If there is no tomorrow, there is no potential infinity of future days. For a potential infinity may not be as "big" as an actual infinity, but it had better be bigger than zero. And if there is no tomorrow, or the day after tomorrow, and so on, then there are zero future days, not a potential infinity of them.
As for per impossibile conditionals, I suspect they are too finicky a thing--too sensitive to changes of context, for instance--to form the grounding of a discipline with as high an epistemic status as mathematics.
James:
I think the infinite sequence involved in P3 is distracting. A simpler case might do.
Suppose I spin an indeterministic spinner with a perfectly defined tip, and the possible spinner outcomes are uniformly distributed over the circumference of a circle. Let Z be a specific point on that circumference.
Classical probability theory says that the probability that the spinner will land at Z is zero. Let R be the proposition that the spinner will land at a point other than Z. Then P(R)=1. Or at worst (though I dispute this option) P(R)=1-infinitesimal. So we are near certain of R. But according to open future views, R isn't true. How can one be near certain of something that on one's metaphysics isn't true?
Michael:
Actually, it looks like some people use "potential infinite" in a way that doesn't imply it's bigger than zero. It just implies that any finite length is possible.
If that's the use of "potential infinite" then my little argument with days doesn't work. But note that this use is problematic for the future days on presentism. For on presentism there can't be any future days, since the future is necessarily nonexistent.
For instance, suppose you take a non-bivalence view of the original liar sentence. "This sentence is false" then expresses a proposition p that is neither true nor false. Moreover, it is not a contingent matter that p is neither true nor false. Thus, p not only isn't true, but couldn't be true. But we shouldn't conclude it's false
Alex, those sentences (in the Liar) do not express propositions at all. I'm not troubled by it being contingently true that a proposition does not take true or false as a value. I cannot understand how a proposition be might necessarily not false or have the property of being necessarily not true and not have the property of being false (or, at least possibly false). The fact that some proposition is not true is a function of semantic decision to use it this way or that. Surely it is possible to use it in a way that gives it a determinate truth value.
There are not actually infinitely many things (certainly not uncountable infinities of things), but we can still talk about what would be the case if (per impossibile) they were true.
Michael,
This is an interesting thing to say. Here are some obvious candidates for "actual infinites". The set of natural numbers is actually infinite, the class of all propositions is actually infinite, the set of space-time points are uncountably infinite, the set of all possible worlds (whether you think of them as fictions, maximal propositions, concrete worlds) are uncountably infinite, the actual universe might well be infinitely large (we certainly don't know a priori that it isn't). Probably, no human being will succeed in counting all the members of an infinite set, but see Dretski here (http://www.jstor.org/discover/10.2307/3326723uid=3739920&uid=2&uid=4&uid=3739256&sid=21104572692461). But it's not obvious that no human being has the property of possibly counting such a set. Here's a suggestion: multi-location seems at least possible. Take any human being that has the modal property of being possibly multiply located. Go to a world at which that human being is located at infinitely many points. Count.
One last point on counting to infinity. Here's a little argument that it is possible (though certainly no one will do it).
1. S can count to infinity iff. for every finite number n, S can count to n.
2. S cannot count to infinity iff. there is some finite number n such that S cannot count to n. From (1)
3. There is no finite number n such that S cannot count to n. Fact
4. :./ S can count to infinity. (2), (3)
Hi Alexander,
First, classical probability breaks down with the actual infinite as does any division, which is a reason for limits approaching infinity or zero. For example, infinity is not a real number while the set of all real numbers is an infinite set. Also, the set of all rational numbers and the set of all integers are infinite sets.
I've done spinner and circle thought experiments myself. That experiment is logically possible but nomologically impossible because a Planck length (1.6 x 10^-35 meters) according to quantum uncertainly is smallest measurable length, not that any current technology comes close to measuring a Planck length. And please remember that any physical spinner is ultimately composed of vibrating elementary particles.
Also, the density of real numbers excludes the definition of two consecutive real numbers such as a smallest positive real number next to 0. Likewise, the real number line has no definition for consecutive points.
Apart from all this, your nomologically impossible scenario is true or false regardless if the future is open or settled, unless there is something that you have not yet explained.
Mike: I don't think numbers (or any other mathematical objects) actually exist as objects, so they are not counter examples to my statements about actual infinities. As for your argument about counting to infinity, the first premise seems to me to be obviously false. The ability to count to any finite number doesn't bring you the slightest bit closer to counting to actual infinity. To use the old "counting man" thought experiment, imagine that a man is approaching you, and you hear him mutter "-3, -2, -1, 0" and then he sighs. You ask him what he has been doing, and he says "counting all the negative numbers... I just finally finished". Obviously, he will have to have been counting for an infinite amount of time, and so the question is why didn't he finish yesterday? An infinite amount of time had passed at that point. Indeed, an infinite amount of time had passed 100 years prior to your meeting him. 1,000 years. A million. So, the idea of counting to infinity, and actually completing the task, is incoherent.
Pruss: I don't think "potential infinity" is meant to have any size. It's just a placeholder to say that you will never come to the end. It's like the "limit" concept in basic calculus. You're not meant to actually subdivide forever, and then arrive at that limit. It's considered impossible (and would lead to 0/0 in any case, which is undefined). The limit is just there to tell you what you are approaching, the more you subdivide. Likewise, the potentially infinite future is a non-existent "limit" that is approached, but can't actually be reached. It doesn't exist in any sense. I can make per impossibile conditional statements about it; but that's all.
As for mathematics: I certainly don't think it's enough to just say "it's all conditionals with impossible antecedents". I think we discover mathematical truths because there are specific logical consequences to any set of axioms and theorems. We do not invent these; discover them. But that doesn't require that there are any actual objects corresponding to the numbers, sets, etc, any more than there needs to actually be an object called "Wednesday" in order to make true statements about what happens every Wednesday.... Anyway. Sorry if I spill my guts a lot in these correspondences. I normally don't have anyone to talk about these sorts of topics with.
Cheers.
The set of natural numbers is our smallest infinite set. There are no numbers in the naturals that are not finite. So, if there is no finite number n such that you cannot count to n, then you can count to infinity. This is a simplified, bloggy, version of Dretski's argument. And it's pretty difficult to show that there is some finite number that no one could count to.
Denying that numbers exist is bizarre, but it's not a way to avoid the problem. Suppose numbers are fictional objects. There are still infinitely many of those useful fictional objects. Going fictional helps only with the epistemology. You can't do math without all of the, infinitely many, fictional objects.
Mike:
What if it gets more and more difficult to count the higher n is, so that you become less and less likely to succeed?
Alex,
It depends on what's being asserted when it's claimed that 'it's impossible to count to infinity'. I took that claim as a restricted de re modal claim, viz., that there are no human beings who have the property of possibly counting to infinity. So, my claim is that (being very explicit now) there is some human being x who is such that possibly x counts to infinity. So, the counterpart of x who does the counting would be one who does not tire easily, who lives an infinitely long time, etc. Of course, you may deny that any human being has such a counterpart. I think that's all good, since what is being claimed is something stronger anyway. They are saying that it is impossible that anything counts to infinity. All we need now is some possible being in some world that counts to infinity. For my part, I'd immediately urge that God can do it; but there is no need to go that far up the hierarchy of beings.
Recognizing that infinity may come in different sizes, I would want to start counting a small infinity. For instance what about time? It began about fourteen billion years ago and is now at the point where it grows sequentially one year every year like a potentially infinite series of whole numbers. An hour is composed of i guess finite units of the present but how many present moments are in an hour?
Mike: You're missing my point. Even if someone could count to any finite number (though I agree that that would become less and less plausible as the numbers got bigger), they would always have some further finite number to count to. Counting is a sequential activity, and no amount of sequential additions is going to take you from talking about finite sums to talking about infinite ones. You just never cross that threshold, which is specifically why potential infinity is used to define the "limits" in mathematics. No matter how long someone counts, they will always be on some finite number.
Counting is a sequential activity, and no amount of sequential additions is going to take you from talking about finite sums to talking about infinite ones.
I don't think I'm missing the point; Craig made that point long ago and has since made it over and over again. I'm really familiar with it. I think you might be underestimating the argument. The natural numbers include only finite numbers. Every number in the set is finite. The naturals is our smallest infinite set. It follows straightforwardly that you cannot count all of the naturals only if there is some finite number n to which you cannot count. Put positively, if you can count to every finite number n, then you can count all of the naturals. But then you can count to infinity. But there is no argument--no argument I know of--which even suggests that there is some finite number to which, and past which, we cannot count. So there is no argument that shows that we cannot count all of the finite numbers in the naturals. But then there is no argument which shows that we cannot count the naturals. So no argument that we cannot count all of the elements in the smallest infinite set of numbers.
Mike:
I like your argument, but let me try this potential counterargument against you. I am not convinced by it myself:
As long as you have non-zero probability of success, then you have the ability to do something. But when the probability of success is zero assuming you keep on trying.
Suppose that each time you count a number, you have a 50% chance of losing your voice. Suppose you're determined to count as far as possible. Then for any finite number n, the probability that you'll count to n is 1/2^(n-1). But the probability that you'll count all finite numbers is zero. So you can count to each finite number but can't count them all.
I meant that I'm not convinced by the counterargument. I like your argument, and it neatly dovetails with my attempts to fix my account of modality.
I should point out that there is some quantifier ambiguity in the argument, but it does not affect the conclusion. Here are the options.
(1) ~☐(Ex)(Vy)((x is greater than all the counted naturals y).
This just says that it's not necessary that some uncounted number x is greater than all the counted numbers y. I take (1) to be obviously true, but less than the argument needs.
(2) ~☐(Vy)(Ex)(x is greater than all the counted naturals y)
This says that it's not necessary that, for all the counted numbers y, there is some greater uncounted number x.
(2) is what the argument needs, and it is less obvious than (1). But (2) is false only if there is some finite number n such that for no world w is n counted to in w. But that's not credible. For every finite number n there is some world w such that n is counted to in w. Indeed, you can map the worlds in which each natural is counted to 1 to w1, 2 to w2, all the way up. So there are worlds in which the naturals are counted. But that is equivalent to saying that it is possible to count to infinity.
He is arguing that from
(a) I can’t count all the natural numbers
(b) All the natural numbers are finite numbers
We can infer
(c) There is some finite number to which I cannot count
Which of course seems wrong, and suggests that we should abandon the first premise.
But suppose we talk about ‘naming’ numbers rather than ‘counting to’ them, because counting to a number n is just naming all the numbers 1..n sequentially. Now intuitively,
(a’) I can’t name all the natural numbers
is still true but
(c’) There is some finite number I cannot name
is even more obviously false. That suggests the inference from (a’) to (c’) is not what Mike’s original argument makes it out to be, and that Alex’s 50% strategy for defeating it is on the wrong track, because it won’t work for naming.
The obvious thing to say is that I can name ANY number, but not ALL of them. This means that Mike’s inference is invalid. I am not sure how to represent this formally (and I am bothered that I am not sure) but the idea is simple. “I can’t grade all these papers in an hour” doesn’t mean that there is some paper that takes longer than an hour to grade but that the collective set of papers can’t be graded in the time allotted. If we argue
(a”) I can’t grade in an hour an infinite set of papers
(b”) All these papers are three pages long
(c”) Therefore, there is some three-page paper I can’t grade in an hour
the fallacy is obvious.
Preface to my previous comment, cut off by Blogger for some reason:
"It seems to me one should reject Mike’s argument but for a rather subtle reason."
Heath,
You seem to be attributing to my argument the claim in (1) above, which I am not using to support my claim that you can count to infinity. I took the time to point this our above. What I am using is (2).
(2) ~☐(Vy)(Ex)(x is greater than all the counted naturals y)
This says that it's not necessary that, for all the counted numbers y, there is some greater uncounted number x.
That is, I am claiming that it is false that, necessarily, for any number you count to, there is some greater number or other that you did not count to. I understand why someone might think that (2) is false. They might be thinking that in order to count to infinity you have to arrive at some possible world where someone utters the words "...infinity, there I did it, I counted to infinity!". But that is not what it means to count to infinity. There is no such number "infinity". Craig gives the deeply misleading impression that this is what must happen in order to count to infinity, and that gives his argument some misleading force.
In order to count to infinity you have to count all the finite numbers. You of course do not reach the number "infinity", since there is no such number. But then how do you count to infinity? Consider an countably infinite set of possible worlds all numbered w0, w1, . . .,wn. Let these worlds overlap with respect to a "counter" who counts to 1 in w1, counts to 2 in w2, 3 in w3...all the way up. It is easy to map the naturals to worlds and a counter in this way. If the "counter" counts to 1 in w1, 2 in w2, 3 in w3, and so on upward, and the possible worlds are infinite in cardinality, then the counter has counted to infinity. He has counted to infinity though there is no possible world in which he says "infinity! there I did it!!". There is no possible word in which he does not count to a finite number!. Rather, he has counted all of the finite numbers. It follows from this that it is true in every world that he can count to infinity, since in each of infinitely many possible worlds wn he has counted to the finite number n.
I think the argument can be made even stronger. I could concede the intuition that (2') is true and depend on (3) alone for my claim.
(2') ☐(Vy)(Ex)(x is greater than all the counted naturals y)
This says necessarily, for all the counted numbers y, there is some greater uncounted number x.
(2') just concede what I did above that in every world the counter counts to a finite number. But (3) guarantees that the counter nonetheless counts to infinity. Again quantifying over the naturals.
(3). ☐(Vx)(there is some world in which the counter has counted to x).
This says that every finite number is counted to in some world or other.
That accommodates the intuition that in no world does the counter say "infinity, there I counted to infinity" (i.e. he counts all and only finite numbers). And it is nonetheless true that there is no finite number that is not counted in some world.
Mike:
Here's a counterexample that seems to work. Suppose that there is no afterlife, but God offers me as long a finite life as I want. I just have to specify the number of years. He'll even give me extra time to specify the length of life, as long as I spend that time doing nothing else than specifying that number (in particular, I'm not allowed to count).
So, for any finite number n, I can count to n. How? I first specify to God how long it will take for me to count to n, and then I count to n.
But I cannot count through all the natural numbers, because that would require me to have eternal life, and in the story God hasn't offered me that option.
I think I can make Alex’s counterexample simpler, at least as an analogy. Suppose God offers me an arbitrarily long life, after which I will die. I can specify any natural number and God will let me live that number of years. So there is a possible world W1 in which I live 1 year, a world W2 in which I live 2 years, etc. So for any (and every) natural number, it is possible that I live that long.
But from the fact that it is possible that I live any length of time, it does not follow that I am possibly immortal. And from the fact that I can count to any finite number, it does not follow that I can count to infinity.
Alex,
Let me alter your example slightly. Suppose there is a world in which there is a countably infinite number of times, t1, t2, t3, t4,...,tn. Now suppose it is true that, for each finite time tn, you will count to tn. There is no tn in the sequence of times such that you do not count to tn at some point. This too may be a wish God has granted to you.
CLAIM: If for every finite time tn in the sequence you count to that time (1 for t1, 2 for t2, n for tn), then you will count to infinity.
Suppose you did not thereby count to infinity. Then there would be some finite time tn such that you did not count to tn. But by hypothesis that is false. Therefore, you counted to infinity.
Yes, counting to every finite time is counting to infinity. The illusion that counting to infinity entails reaching the point where you say "...infinity, there I did it!!" really needs to be resisted. It is a badly misleading representation of what's required. All of the naturals are finite, count all the finite numbers and you have counted to infinity.
Agreed: if you count to every finite number, you count to infinity.
But you cannot infer from:
For all n: Possibly: you count to n.
To:
Possibly: For all n: you count to n.
This discussion reminds me of the difference between consistency and omega-consistency.
Hi Alexander, I do not want to assume anything, so I ask:
Do you still think that any of this discussion disproves the possibility of an open future?
Also, if your answer is yes, then do you think it also disproves the possibility of a closed future?
Alex,
I hope I didn't commit an elementary quantifier switch fallacy. I think I conceded that there is no world where it is true that every finite number has been counted. That is (2') above. What I claimed was that there is no finite number to which S does not count in some world. That shows, I'm urging, that S can count to infinity though there is no world in which he does. You would think that there is something mistaken about this claim only if you think that everything that happens, happens in a world. Imagine that S is a modal continuant, if that will make the case easier to think about. The modal continuant S counts to infinity. Now ask: is it really necessary to think of S as a modal continuant to reach the conclusion that (i) S counted to infinity and (ii) there is no world in which S counts to infinity? I think the answer to that is no.
Mike:
I really think you're missing a fundamental point. For every finite number, there is always a larger one. There is no finite number which does not have a number larger than it. Therefore your statement, " ~☐(∀y)(∃x)(x is greater than all the counted naturals y)", is false. It is necessarily true that, in any case of counting, no matter how far you've gotten, there is still a number you have not yet counted.
Now you try to re-word the case in terms of worlds. You say "☐(∀x)(there is some world in which the counter has counted to x)." But, even if that's true, there is always another number that that person did not count to. The point is that, for any person to actually accomplish the task of counting to infinity, they do indeed have to be able to say "... infinity! There! I finished." If they can't truthfully say that, then there is no sense in which they have counted to infinity. In every world, I can always say to the counter, "add 1".
Saying you can count to ANY finite number does not mean you can count ALL of them. Heath and Pruss are using intuitive cases to get at that.
We're reaching the point of diminishing utility in this discussion. Alex has already agreed, if you count to every finite number, then you have counted to infinity. See the post above (Nov. 19, 4:44). I will repost here.
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Blogger Alexander R Pruss said...
Agreed: if you count to every finite number, you count to infinity
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It's just confused to say that you must say "infinity, there I've reached it". You seem to think that infinity is a number, and it isn't. You seem to think there is an infinite number in the naturals, and there isn't.
I think I can and have made some progress in the discussion with Alex; but there's otherwise too much work for too little pay.
Mike:
I'm not saying infinity is a number. I'm saying that accomplishing a task has to include the ability to truthfully say "I just did X". If you can't ever truthfully say "I just finished counting to infinity" then you can't ever get to the point where you have counted to infinity. Whether Pruss or anyone else says a sentence including the phrase "count to every finite number", it's still a totally incoherent phrase. There is no highest number, so it doesn't mean anything to say that you've counted every finite number. It's like saying "if you could catch the horizon, you would indeed fall of the edge of the Earth".
Ok, Michael, thanks.
Alex,
If for every finite number x, there is a world in which S counts up to x, and there are infinitely many worlds, then for every finite number y, S counts up to y in some world. But then there cannot be a finite number that S does not count up to in some world. As you get closer and closer to the infinite limit on worlds, you get closer and closer to counting every finite number. At the limit, you've done it.
Compare a plant that is one centimeter taller in each successive world in an infinite sequence of worlds. At the limit the plant has grown by an infinite number of increments. I'm pretty sure that's uncontroversial.
Compare an infinite sequence of worlds that increase by one turp of evil per world. At the limit we have a world that includes infinitely many turps of evil.
These are all the same argument essentially. None is especially controversial.
I am afraid that we're not making progress. I suggest a bit of silence on this thread, unless there is something radically new to say.
ah, too bad. thought we were right there. ok.
The argument at the heart of this blog post is a central part of a paper coming out in Faith and Philosophy.
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