Thursday, April 2, 2009

Infinite numbers of objects

Here is an argument that it is possible for an infinite number of objects to come into existence by successive addition, contrary to William Lane Craig. I am not sure how far I find this argument convincing, but it seems to me to be pretty strong. It's inspired by an idea of Wes Morriston.

An alpha-widget is an entity that has the following property. As soon as an alpha-widget x is made, it spends a year playing the violin, and doing nothing else. At the end of that year, x makes an almost-duplicate of itself in half of the time in which x itself was made: the almost-duplicate is just like x, except that if x has the information that it was made in t units of time, the almost-duplicate has the information that it was made in t/2 units of time. And alpha-widgets are never destroyed.

I now claim that it is possible for an alpha-widget to be made over the period of a year. If it were made, there would be a potentially infinite but never completed sequence of alpha-widgets coming into existence: the first at the beginning of year zero, the second at year 1.5, the third at year 2.75, and so on. Since the spacing between alpha-widgets is always more than a year, because of that year of violin-playing, we do not here have any counterexample to Craig.

It would be difficult, I think, for Craig to object to this possibility. It doesn't violate his strictures against actual infinities. It does require laws of nature different from those of our world, or perhaps divinely mediated miracles, in order to overcome speed of light limits on the production speed of an alpha-widget.

But now consider a beta-widget. Recall that an alpha-widget would first play the violin for a year, and then would make an almost-duplicate of itself in half the time that it itself was made. A beta-widget does the same thing, but in opposite order: it first makes the almost-duplicate in half time, and then plays the violin for a year. Since the alpha-widget is not doing anything in that year of violin playing other than playing the violin (it's not, for instance, setting up a production line for its almost-duplicate), there is no reason to suppose that it would be harder for God to make a beta-widget in a year than to make an alpha-widget in a year. (And, yes, God can make things non-instantaneously if he so chooses.)

So, it should be logically possible to make a beta-widget in a year. But if a beta-widget is made in a year, then half a year later, it makes another beta-widget. That one, then, makes another in a quarter of a year. And so on. By the time two years are up, we have an infinite number of beta-widgets produced by successive addition.

I should, of course, note that while I think there are problems in the Kalaam argument's a priori argument for the finite age of the universe, its a posteriori argument may well be fine, and in any case there are other cosmological arguments that work just fine.

28 comments:

Dannac said...

Does the distinction of an actual infinite versus a potential infinite need to be made, as per Craig?

Matthew said...

I believe there are some problems with this.

First, it assumes that time is infinitely divisible (like Wes' argument).
But since the question is "Can there be an actual infinite?", this is question-begging.

Second, if we add the Thomson's Lamp "Is it on or off?"-paradox to it and we have to say "Well, God can decide if it's on or off" and the possibility of this is important for an argument for the existence of God, atheists aren't better off.
They have to invoke God to answer an argument for his existence.

Alexander R Pruss said...

1. I don't think a distinction between an actual and a potential infinite helps Craig here. The argument concludes to the possibility of an actual infinite.

2. The claim that time is infinitely subdivided would beg the question. The claim that time is infinitely subdividable does not.

3. The atheist has to say that the story underdetermines whether the lamp is on or off, and so either the lamp's being on or off has no cause (atheists don't generally mind such a possibility) or it has some cause not in the story. Consider the following story: "George makes a lamp." Now ask: "Is it on or off?" The answer is: "The story is not sufficient to answer the question." Likewise in the Thomson's lamp case--the story is not sufficient to answer the question. We may fallaciously think the story has described everything, but it hasn't.

Mattie said...

Alex (if I may),

Regards your third point, couldn't one conceivably suggest that the set of facts we are usually given in Thomson's lamp experiments are exhaustive? So when it comes to whether or not the lamp is on or off, we know everything that is explanatorily relevant to the final stage of the lamp?

Matthew said...

If time is infinitely subdividable, it contains an infinite amount of smallest time amounts. So we would have an infinite here.

The problem with Thomson's lamp is this.

Let's say, it's on at the beginning and that is the value 1 (if it's off, the value is -1). When it's turned off, add -2. If it's turned on, add 2.

If this goes on forever you get:

1 -2 + 2 - 2 + 2 ... =
1 - (2 - 2) - (2 - 2) ... = 1 - 0 - 0 ... = 1

1 - 2 + 2 - 2 + 2 ... =
1 - 2 + (2 - 2) + (2 - 2) ... = -1 + 0 + 0 ... = -1

Contradiction?

Alexander R Pruss said...

Mattie:

"couldn't one conceivably suggest that the set of facts we are usually given in Thomson's lamp experiments are exhaustive" - One could suggest it, but could one prove it?

Matthew:

That time is infinitely subdividable means this. If t0 is prior to t2, then there could be a time between t0 and t2. It does not follow that there is a time between t0 and t2.

As for your infinite sums, 1-2+2-2+2... is a divergent series, so it's not surprise that different ways of grouping things give different results.

Maybe you're imagining that beside the process there stands a calculator, and you alternately enter "-2" and "+2" into it. And then you ask what you'd see on the calculator at the end. Again, the right answer is: "The story fails to say." Lots of options are possible. Maybe it gives you 1. Maybe it gives you -1. Maybe it gives you 4848.112. Maybe the calculator blows up. Maybe the calculator sprouts wings and flies away. All of these options are consistent with the story.

Perhaps you will say: "Given the laws of nature operating in the story, the calculator can't sprout wings and fly away. Nor can it blow up or give 4848.112. It must give -1 or 1." Sure, you can add that qualification on the laws of nature to the story. Now you've reduced the range of what the story allows. But it still equally allows -1 as 1. So the story is still insufficient to specify.

Suppose you add to the story: "And the laws of nature are such that it deterministically follows from the previous results what the calculator outcome is." I say: "The story is still undeterdetermined as to what the answer is. I could posit a set of deterministic laws of nature consistent with the story on which the answer is +1. And I could posit a set of laws of deterministic laws of nature consistent with the story on which the answer is -1."

Matthew said...

Interesting thoughts. However, I don't think the Thomson's Lamp paradox is solvable.

Alexander R Pruss said...

Challenge: Formulate the Thomson's lamp argument as a valid argument from some premise like "There is a lamp which switches states at 10:00, 10:30, 10:45, 10:52.5, etc." to an explicit contradiction. Then we can talk about which premise, if any, is false. :-)

James said...

Alex: That time is infinitely subdividable means this. If t0 is prior to t2, then there could be a time between t0 and t2. It does not follow that there is a time between t0 and t2.

I'm a bit confused as to the possible versus actual distinction here. For what does this possibility hinge on? That is, if there could be a time between t0 and t2, then what circumstances could prevail such that there wouldn't be such a time?

Alexander R Pruss said...

The theory of an infinitely divisible but not infinitely divided time is Aristotle's. On Aristotle's view, time is correlate with change. Thus, if there are no states of change between t0 and t2, there is no moment of time between t0 and t2. There is just a smooth flow from t0 to t2, without intermediate times. Now if there were a change--if, say, some object changed direction, or something like that, between t0 and t2, then there would be a time there.

Craig himself writes: "I agree with G. J. Whitrow when when he writes, 'although the hypothesis that time is truly continuous has definite mathematical advantages, it is an idealization, and not an actual characteristic of physical time.' This position does not imply that minimal time atoms, or chronons, exist; time, like space, is infinitely divisible, in the sense that division can proceed indefinitely, but time is never actually divided, neither does one arrive at an instantaneous point." (Theism, Atheism and Big Bang Cosmology, p. 29)

My and Wes's arguments start with time being infinitely divisible, which Craig grants, and proceed to the conclusion that it is possible that time be infinitely divided, which Craig denies.

James said...

I see (I think). So Craig's objections to time's being infinitely divided have to do with the impossibility of an actual infinite's existing rather than the nature of time itself?

Drew Mazanec said...

Wasn't an argument like this made in the book: Zeno's Paradoxes?

James said...

Yes, but you need to prove that it's possible to infinitely divide space in order to refute Craig's claims (who, I think, would claim that it's possible to infinitely divide space but that it's impossible that space actually be infinitely divided). In which case the issue turns on the same kinds of issues that Alex's time-based illustration turns on.

larryniven said...

Would this particular instance differ from Wes's in that it assumes there's an actually infinite amount of matter? I haven't given this too much thought, so I could probably be talked out of this, but it seems like there could only possibly be an infinite number of self-replicating objects if there was infinitely much matter to begin with. That, though, would beg the question - right?

The difference, I think, is that the other guy has God creating the matter directly (or am I misremembering that post?), whereas this has matter-composed objects creating more matter-composed objects. In the first case, there needn't be any matter at all cause God can do miracles, but in the latter case there aren't any miracles (except, maybe, the creation of the first beta device).

Alexander R Pruss said...

Larry:

I hadn't thought of that. Thank you for the objection.

I don't know if Bill Craig would object to an infinite amount of undifferentiated matter. Maybe it would all count as one object for him, and we would just be snipping bits of it off?

If that doesn't work, I can make one of three moves:

1. Suppose a world with continuous, infinitely divisible matter, and suppose that the successive generations of widgets are smaller and smaller.

2. Suppose that God has given the widgets the power of producing matter ex nihilo. I know that there is a theistic tradition that only God can produce things ex nihilo, but I do not know that it is doctrinally binding, and I am not sure I am convinced by the arguments for that tradition.

3. The widgets are sapient, and before they begin their work, God promises them to create matter for them in whatever quantities they like. So, then they just ask God for the ingredients they need as they need them.

Alexander R Pruss said...

Which Zeno's Paradoxes book was that?

Drew Mazanec said...

The one edited by Wesley Salmon.

Dannac said...

Please excuse my naivety...
If we were to run your widget function, would it not, then, be true that we would never actually reach t=2? The gap is unbridgeable.

Alexander R Pruss said...

That would be weird. Suppose that the experiment happens in another galaxy (maybe with the widgets getting smaller and smaller, or maybe with them getting piled up in the direction facing away from earth, and with God adjusting the gravity so it doesn't affect us). We never notice it on earth. What happens to us, then, as t=2? Does time stop? Do we all die? Surely, we reach t=2, then t=3, and so on. And then if we go and visit that galaxy, we'll see an infinite number of widgets.

larryniven said...

Yeah - I feel like those three might work. Can't say as I'm too expert in Craig's position here, though, so I was really just tossing the idea out there. Happy to help, though!

Drew Mazanec said...

Would this work under a reductionist view of time, as Aristotle and Craig believe?

If time does not exist independently of events but is dependent on them, then is it coherent to say that an infinite number of events occurred in a finite amount of time?

Alexander R Pruss said...

As long as the length of an interval of time is not defined by the number of events happening during that interval, I think it might be consistent with a reductionist view of time.

There is another way in which this might not be inconsistent with a kalaam argument. The kalaam argument requires us to rule out the possibility of an infinite causal regress of the form "A1 caused by A2 caused by A3 caused by ...". These constructions of mine do not give infinite causal regresses.

(I used to think that infinite causal regresses are possible, as long as there is something outside the regress to cause the regress as a whole. I no longer believe this, though I wouldn't go so far as to say I disbelieve it. The Grim Reaper paradox continues to really puzzle me, and I think it may be a paradox about causal regresses.)

Edward said...

"That would be weird. Suppose that the experiment happens in another galaxy (maybe with the widgets getting smaller and smaller, or maybe with them getting piled up in the direction facing away from earth, and with God adjusting the gravity so it doesn't affect us). We never notice it on earth. What happens to us, then, as t=2? Does time stop? Do we all die? Surely, we reach t=2, then t=3, and so on. And then if we go and visit that galaxy, we'll see an infinite number of widgets."

I don't think this really answers the objection as it assumes that the process you have outlined is actually possible. I would take your argument above as a reductio ad absurdum showing that the beta-widget process you have in mind is in fact impossible. The process is set up in such a way that it can never reach t=2. This is clear, it seems to me, from the very nature of the process and the potentially infinite divisibility of time. First you have 1/2, then 1/4, then 1/8, etc..... Never in the process do you ever have anything but a finite fraction. But a series of fractions with finite denominators is a finite series of fractions. And since the number of widgets corresponds to the number of fractions, there will always be a correspondingly finite number of widgets. Moreover, since there will always be more time to subdivide, one can clearly never reach t=2. But the fact that we do indeed reach t=2 shows that this conclusion is false. Hence, the process itself is impossible.
It seems to me that you must show how your process reaches t=2 when in fact it is clear that it merely approaches it asymptotically.

Another point to consider. Either the beta-widget creation process takes some time or it is instantaneous. If it takes time, then clearly there will only be a finite number of widgets corresponding to the finite subdivision of the time. If it is instantaneous, then one need not bother with the process at all. One could simply posit the creation of an infinite number of widgets in one moment. Since the creation of one widget takes no time, then the creation of an infinite number of widgets also takes no time. But this would merely assume the possibility of an actual infinite which is what you are trying to prove. It seems to me that your process makes the same assumption but simply spreads it out over a time interval.

Ed

Alexander R Pruss said...

Since I'm trying to show, inter alia, that time can be infinitely subdivided (or at least, that's an obvious consequence of the conclusion of the argument), to assume that it can't be begs the question.

What do you make of the fact that it shouldn't be any harder for God to make one beta-widget than for him to make alpha-widget? And once you have one, that's all you need for the argument.

Edward said...

"Since I'm trying to show, inter alia, that time can be infinitely subdivided (or at least, that's an obvious consequence of the conclusion of the argument), to assume that it can't be begs the question.

What do you make of the fact that it shouldn't be any harder for God to make one beta-widget than for him to make alpha-widget? And once you have one, that's all you need for the argument."

Dr. Pruss,
In all honesty I do not believe that you have shown that time is actually infinitely subdividible, nor do I think that I have assumed that it is not. In your original argument you simply stated that once we get to t=2 we will have an infinite number of beta-widgets. That statement requires proof and you have not given it. I will now proceed to prove that whether we reach t=2 or not, there will never be an infinite number of beta-widgets. For my proof, I will assume only the process you have given us and the continuity of time (i.e., that time is continuously divisible). Here is the proof:

Every beta-widget is produced at some time t prior to t=2. This is evident from the very nature of the process given. Hence, the difference between t and 2, i.e., 2-t is always a finite amount of time. But every finite amount of time is a part of the whole time and there will exist some number n such that the whole time is equal to n times (2-t). This means that there will always be a finite number of such intervals of time that make up the whole time. But the number of beta-widgets produced corresponds to the number of such time intervals. Hence, the number of beta-widgets produced is always finite.
Now, it seems to me that what you must do is to show that the above argument is wrong. You must also show that your process produces and actually infinite number of beta-widgets and not simply state that it does.

Ed

Alexander R Pruss said...

"Hence, the difference between t and 2, i.e., 2-t is always a finite amount of time." -- I take it that "always" here means "while the process is running". And if so, then your argument only establishes that while the process is running, the process is not completed. But of course I grant that.

If a beta-widget is produced in one unit of time ending at t=0, then either there will be no t=2--time will never reach that point--or there will be infinitely many beta-widgets at t=2. That follows from the definition of a beta-widget.

So, you have to either deny the possibility of a beta-widget, or else say that producing a beta-widget will bring time to a standstill for everybody. (That's an option I hadn't considered originally, and I am grateful for your comments for making me see this needs to be considered.)

Now it would be really weird if someone's making a beta-widget in another galaxy somehow brought time to a standstill for us. What would it feel like for us not to reach t=2? After all, our lives, and the events in them, wouldn't be affected by the beta-widget production, as long as the duplicates didn't extend in our direction.

Edward said...

"I take it that "always" here means "while the process is running". And if so, then your argument only establishes that while the process is running, the process is not completed. But of course I grant that."

Dr. Pruss,
If you grant that during the process of beta-widget production there are always a finite number of widgets, then it must be the case that at any time prior to t=2 there are only a finite number of widgets. Hence, if there are an infinite number of widgets at t=2, an infinity of widgets must have all been produced in the instant that t=2. But this violates the process whereby succeeding beta-widgets are produced in half the time. Either that, or you must consider an instant to be a length of time, with all succeeding widgets being produced in zero time -- the zero continually being half of the previous zero. But if this were the case, it would be no different than the mere assertion that an actually infinite number of objects can exist and, indeed, can be produced in an instant.

Ed

Alexander R Pruss said...

Or else there are infinitely many instants. Or else there are no instants, but only intervals of time, infinitely many of them.