A long walk

Alice has lived forever in a universe with an infinite road that has a beginning and no end, and is marked every mile. Every day of her life, by an irresistable longing, she has followed these rules:

1. If somehow she’s not on the road, she goes to mile zero on the road.

2. If she is on the road, she walks a mile in the endless direction of the road.

3. Besides the movement required by 1 and 2, she stays in place.

Where is Alice now? Nowhere! There is no possible present scenario compatible with the rules. She can’t either be at mile zero or off the road, because if two days ago she was on the road, she would now be on the road now be past mile zero, and if two days ago she wasn’t on the road, she would now be on the road past mile zero. She can’t be at mile n, because then she would have to have been off the road some time back, which violates the previous argument.

The rules are all coherent. A person who has to follow these rules seems to be possible. Yet the story is impossible. What went wrong? The neatest explanation seems to be that a (causally connected) infinite past is impossible.

(This is inspired by a recent infinite past Grim Reaper story I read from Rob Koons.)

A cosmological argument from the Hume-Edwards Principle

The Hume-Edwards Principle (HEP) says:

1. If you’ve explained every item in a collection, you’ve explained the whole collection of items.

This sounds very plausible, but powerful counterexamples have been given. For instance, suppose that exactly at noon, cannonball is shot out of a cannon. The collection C of cannonball states after noon has the property that each state in C is explained by an earlier state in C (e.g., a state at 12:01:00 is explained by a state at 12:00:30). By the Hume-Edwards Principle, this would imply that C is self-explanatory. But it plainly is not: it requires the cannon being fired at noon to be explained.

But I just realized something. All of the effective counterexamples to the Hume-Edwards Principle involve either circular causation or infinite causal regresses. We can now argue:

2. HEP is necessarily true.

3. If circular causation is possible, counterexamples to HEP are possible.

4. If infinite causal regresses are possible, counterexamples to HEP are possible.

5. So, neither circular causation nor infinite causal regresses are possible.

6. If there is no first cause, there is a causal circle or an infinite causal regress.

7. So, there is a first cause.

Similarly, it is very plausible that if infinite causal regresses are impossible, then causal finitism, the thesis that nothing can have an infinite causal history, is true. So, we get an argument from HEP to causal finitism.

Dialectically, the above is very odd indeed. HEP was used by Hume and Edwards to oppose cosmological arguments. But the above turns the tables on Hume and Edwards!

Objection: Not every instance of causal regress yields a counterexample to HEP. So it could be that HEP is true, but some causal regresses are still possible.

Response: It’s hard to see how there is sufficient structural difference between the cannonball story and other regresses to allow one to deny the cannonball story, and its relatives, while allowing the kind of regresses that are involved in Hume’s response to cosmological arguments.

Final remark: What led me to the above line of thought was reflecting on scenarios like the following. Imagine a lamp with a terrible user interface: you need to press the button infinitely many times to turn the lamp on, and once you do, it stays on despite further presses. Suppose now that in an infinite past, Alice was pressing the button once a day. Then the lamp was always on. Now I find myself with two intuitions. On the one hand, it seems to me that there is no explanation in the story as to why the lamp was always on: “It’s always been like that” just isn’t an explanation. On the other hand, we have a perfectly good explanation why the lamp was on n days ago: because it was on n + 1 days ago, and another button press doesn’t turn it off. And I found the second intuition pushing back against the first one, because if every day’s light-on state has an explanation, then there should be an explanation of why the lamp was always on. And then I realized this intuition was based on somehow finding HEP plausible—despite having argued against HEP over much of my philosophical career. And then I realized that one could reconcile HEP with these arguments by embracing causal finitism.

An Aristotelian argument for a causal principle

Start with these assumptions:

1. Laws of nature are grounded in the powers of things. (I.e., Aristotelian picture of laws.)
2. Space can be infinite.
3. Newtonian physics is metaphysically possible.

There is a somewhat handwaving argument that if (1)-(3) are true, then an object cannot come into existence ex nihilo for no cause at all, and hence we have a causal princople.

Here's why. Say that a gridpoint in a Newtonian three dimensional space is a point with coordinates (x,y,z) where x,y and z are integers (in some fixed unit system).

Given (1)-(3) and assuming that objects can pop into existence ex nihilo, it should possible to start with a universe of finite total mass and then for a Newtonian particle of equal non-zero mass to simultaneously pop into existence at all and only those gridpoints (x,y,z) where z is positive, with nothing popping into existence elsewhere. Here's why. At each gridpoint, the object should be able to pop into existence. But objects that pop into existence causelessly at one location in space would be doing so in complete oblivion of what happens at other gridpoints. There should be total logical independence between all the poppings into existence. If so, then any combination of poppings or non-poppings should be able to happen at the gridpoints, and in particular, it should possible to have particles of equal mass pop into existence at the gridpoints with positive z-coordinates but nowhere else. But if this happened, then each particle would experience an infinite force in the direction of the z-axis (this follows from Newton's shell theorem and some approximation work), which would result in an infinite acceleration, which is absurd.

A relativistic version of this argument would require that spacetime can be infinite, so we could arrange the particles popping into existence along a single backwards light-cone.

There is a more general point here. The above example will remind regular readers of an argument I recently gave for causal finitism. I think many paradox-based arguments for causal finitism can be turned into arguments for causal principles in something like the above way. If this is right, this is very cool, because we can get both premises of a Kalaam Cosmological Argument out of the paradoxes then.

From a past-infinite causal sequence to a paradoxical lottery: A cosmological argument

Infinite fair lotteries are well-known to be paradoxical. Let's say that an infinite fair lottery is played twice with tickets 1,2,3,.... Then whatever number wins first, you can be all but perhaps certain that in the next run of the lottery a bigger number will win (since the probability of any particular number winning is zero or infinitesimal, so the probability that the winner is a member of the finite set of numbers smaller than or equal to the first picked number is zero or infinitesimal). So as you keep on playing, you can be completely confident that the next number picked will be bigger than the one you just picked. But intuitively that's not what's going to happen. Or consider this neat paradox. Given the infinite fair lottery, there is a way to change the lottery that makes each ticket infinitely more likely to win. Just run a lottery where the probability of ticket n is 2^-n (which is infinitely bigger than the zero or infinitesimal probability in the paradoxical lottery)

What makes the infinite fair lottery paradoxical is that

1. there is a countable infinity of tickets

and

2. each ticket has zero or infinitesimal chance of winning.

Let's stipulate that a lottery is "paradoxical" if and only if it satisfies (1) and (2).

Suppose now that a past-infinite causal sequence is possible (e.g., my being caused by my parents, their being caused by theirs, and so on ad infinitum). Then the following past-infinite causal sequence is surely possible as well. There is a machine that has always been on an infinite line with positions marked with integers: ...,-3,-2,-1,0,1,2,3,.... Each day, the machine has tossed a fair coin. If the coin was heads, it moved one position to the right on the line (e.g., from 2 to 3) and if it was tails, one position to the left (e.g., from 0 to -1). The machine moved in no other way.

We can think of today's position of the machine as picking out a ticket from a countably infinite lottery. Moreover, this countably infinite lottery is paradoxical. It satisfies (1) by stipulation. And it's not hard to argue that it satisfies (2), because of how random walks thin out probability distributions. (And all we need is finite additivity for the argument.)

So if past-infinite causal sequences are possible, paradoxical lotteries are as well. But paradoxical lotteries are not possible, I say. So past-infinite causal sequences are not possible. So there is an uncaused cause.

From the Principle of Sufficient Reason to causal finitism

Causal finitism says that nothing can have infinitely many things in its causal history. Suppose causal finitism is false. Then surely the following scenario is possible:

1. There has always been a lamp.
2. There is a backwards infinite sequence of flips of the lamp switch at times t₋₁>t₋₂>t₋₃>....
3. Each time the switch was flipped, the lamp state flipped between on and off.
4. All lamp state changes are explained by switch flips, and are not overdetermined by any other causes.

Call this story the "Reversed Thomson's Lamp" (RTL). If RTL holds, the Principle of Sufficient Reason is false. To see this, suppose that the lamp is on right after t₋₁. Then it's on right after t_−n for all odd n. But there is no explanation why it's on right after t_−n for all odd n; it could have instead been on right after t_−n for all even n. It would be circular to explain its being on after the odd-numbered times in terms of its being off after the even-numbered ones, since by the same token its being off after the even-numbered ones would be explained by its being on after the odd-numbered ones. If one had overdetermination, one could explain things in terms of the overdetermining causes, but that would go against (4).[note 1]

So if the Principle of Sufficient Reason is necessarily true, RTL is impossible, and hence causal finitism is true.

Forming an infinity step-by-step

Suppose I am assigned to write a book with infinitely many pages. Luckily for me, the past is infinite. Then I could have accomplished the task on a leisurely schedule by having written a page every decade.

But the odd thing is that then the task will then have always been finished! For in every past, present or future decade, it is true that I've already written infinitely many pages.

Yet the task of writing a set size of book is surely a paradigm of a task done step-by-step. And it is a central feature of doing a task step-by-step that the task wasn't always completed. So the task of writing an infinite book wasn't always completed, and yet it was always completed, a contradiction.

Hence, the task of writing an infinite book a page at a time is impossible.

The best response I can think of is to deny that writing a book of a given size a page at a time is always a task done step-by-step. It's true for finite sizes, but need not be true for infinite ones. Maybe the problem is that it's not a possible task (this thought is inspired by Richard Gale's comments on Zeno paradoxes).

A better way to run Kalaam arguments?

Crucial to the Kalaam argument is showing that the universe has only a finite past. A standard approach is:

1. If the universe has an infinite past, there is an actual infinite.
2. If an actual infinite is possible, Hilbert's Hotel is possible.
3. Hilbert's Hotel is impossible.
4. So, the universe doesn't have an infinite past.

Apart from the real challenge, which is defending (3), there are at least three distracting difficulties here.

First, one needs to defend the often not very clear distinction between a potential infinite and an actual infinite.

Second, holding on to the argument while believing in an infinite future afterlife requires a very particular theory of time: the growing block theory. For consider the two alternatives: eternalism and presentism. On eternalism, future infinites are no less actual than past ones, and so an infinite future is just as much a problem as an infinite past. On presentism, neither past nor future infinites are actual, and premise (1) fails.

Third, the conditional in (2) is dubious. Not all actual infinites are of the same sort. An actual infinity of past events does not seem to be close enough to Hilbert's Hotel—a present infinity of rooms—to allow inferring the possibility of the Hotel from the possibility of the past infinity.

Here is an alternative:

5. If the universe has an infinite past, a simultaneous infinity of objects is possible.
6. If a simultaneous infinity of objects is possible, Hilbert's Hotel is possible.
7. HIlbert's Hotel is impossible.
8. So, the universe doesn't have an infinite past.

Advantages: We don't need any murky distinction between actual and potential infinites, just the much clearer notion of a simultaneous infinite.[note 1] An infinite future is not an issue, and any theory of time can be plugged in. Finally, the move from a simultaneous infinity of objects to Hilbert's Hotel in (6) is much more plausible than the corresponding move in (2). For if a simultaneous infinity of objects is possible, surely they could be shaped like rooms.

The one disadvantage, of course, is that (5) needs an argument. Here it is. If an infinite past is possible, surely it's possible that at infinitely many past times an object came into existence that has not yet ceased to exist. But if that happened, there are now infinitely many objects, a simultaneous infinity.

Still, one might worry. How can we argue for the possibility of an object coming into existence at infinitely past times, given an infinite past? Well, we can imagine a law of nature on which when two particles collide they are annihilated into four particles, with correspondingly smaller individual mass-energy, and we can imagine that these particles by law cannot otherwise disappear. We can then suppose that there have always been such particles, and that during each of a past infinity of years there was at least one collision. Then that very plausibly possible story implies that there is a present infinity of particles.

I think the difficulties in arguing for (5) are far outweighed by the advantages of the simultaneous infinity formulation of the argument.

No event can irreducibly depend on infinitely many things

I am now thinking the following principle is likely to be true:

• (NoInfDep) No event can irreducibly depend on infinitely many things.

This might be true even if we drop "irreducibly", but the "irreducibly" makes me more confident. When A causes B in such a way that B is determined to cause C, then I say that C reducibly depends on B. No additional information is given about how C came about by describing B than is already implicit in a sufficiently rich description of A. In other words, reducible dependence is dependence on a merely and totally instrumental cause, one that doesn't make any contribution truly of its own. We can then see the debate between compatibilists and incompatibilists as a debate on whether an agent's free actions have to irreducibly depend on something in the agent.

Why think NoInfDep is true? The general line of argument is this. There are a number of paradoxes that NoInfDep rules out. Now in the case of each paradox, there is a narrower modal principle that could rule out the paradox, but the narrower principle is ad hoc in a way that NoInfDep isn't, and so our best explanation as to why the paradoxes are ruled out is (1).

Here are the paradoxes I currently have in mind:

1. Thomson's Lamp
2. Grim Reapers
3. Coin sequence guessing
4. Infinite fair lotteries resulting from infinitely many fair coin tosses (see the discussion in one of my comments of the paradoxicality)
5. Satan's Apple and some other decision-theoretic paradoxes (e.g., the game where we have dollar bills numbered 1,2,3,... and you start with dollar bill #1, and in each round you give me your lowest numbered bill, and I give you two bills with higher numbers; at the end you have nothing)
6. Realizations of the Banach-Tarski Paradox and maybe even things relating to nonmeasurable sets.

There are other solutions for some of these. (I've never been that impressed by Thomson's lamp, but it's a freebie here.) But NoInfDep provides an elegantly uniform solution. Moreover, NoInfDep expresses the intuition that there cannot be a "completed infinity" without committing one to a dubious presentist or growing block ontology. Indeed, NoInfDep shows that the "no completed infinity" intuition goes beyond considerations of time: it is about dependency.

I want to say something about the Banach-Tarski case. The paradox there is purely mathematical. But to realize this paradox in real life--to actually decompose a solid ball (if there were such a thing) into two of equal size--you would need to make something like a choice function, which would require infinitely many data points, and those would require, I suspect, irreducibly infinitely many events to generate.

And now we have the Kalaam argument.

Another argument against an infinite past?

I wonder if this very neat argument can't be used to provide another Grim Reaper style argument against an infinite past? The argument nicely fits with the intuition that Grim Reaper induces in me, namely that no event can have an infinite number of events in its causal history.

Koons on grim reapers and Kalaam arguments

I've been telling people about this paper by Rob Koons, defending a grim-reaper based Kalaam cosmological argument, but I didn't realize it was online.

Actual infinity

I am occasionally asked what I think about actual infinities.  Mainly because of a Grim Reaper argument, but also because of the bob argument here, I do think there is a problem about certain kinds of actual infinities.  But not all of them.  The Grim Reaper argument, as well as the bob argument, only tells against one kind of actual infinity: an actual infinity of causal influences on a single event.  So while one could conclude that actual infinities are impossible, a more conservative move is to conclude that it is not possible for an infinite number of causal influences to be in the past history of any event.

This limited anti-infinitism means that I do not need to worry about arguments in favor of actual infinities such as the following:

• The actual existence of an infinite future.
• The actual existence of mathematical infinities.
• The intuitively very plausible possibility of a simultaneous infinity of objects.  At least, it's plausible to me.

In fact, my worry is not so much about an actual infinite as such, but about infinitely many causal influences coming together.  

Initially, when I came to the Grim Reaper argument against an infinite past, I found the anti-infinitist conclusion very counterintuitive.  But now it has become clear to me that it's not: for it's not that counterintuitive to think that absurdities can result if an infinite number of causal influences can work together.  My pro-infinitist intuitions were based on non-causal mathematical considerations.  But I can, if I wish, retain those intuitions.  

In fact, I can even retain the intuition that there could have been an infinite past, as long as this does not imply a backwards-infinite causal chain.  Consider, for instance, a world that consists of a multiverse of universes.  The first universe is one year old.  The second universe is two years old.  The third is three years old.  And so on.  The world as a whole has an infinite past.  But as long as there isn't the wrong kind of causal dependence between the universes, such an arrangement need not imply the kind of infinity in causal influence that my arguments lead me to think is problematic.

This strengthens the Kalaam argument by showing that the premises can be weakened: the Kalaam argument only needs the kind of causal anti-infinitism that I now cautiously accept.

From the Grim Reaper paradox to the Kalaam argument

A Grim Reaper (GR) timed to go off at t₀ is an entity which does the following at exactly t₀. If Fred is not alive at t₀, the GR does nothing at t₀. If Fred is alive at t₀, the GR instantaneously annihilates Fred. (If instantaneous action is not logically possible, one can complicate the situation by allowing shorter and shorter time intervals for these actions.) The GR Paradox then is this scenario. Fred is alive at 11:00 am today, and that he does not die today unless killed by a GR and he does not get resurrected today. There are infinitely many GRs, timed to go off in a staggered way at the respectively times t₁,t₂,... where t_n is equal to 11:00 am + 1/n minutes. Well, by 11:02 am, Fred is certainly dead, since it is impossible that he survive a time at which a GR is timed to go off. But when was he killed? He wasn't killed by the 11:00 am + 1 minute GR, because if he were alive just before 11:01 am, then he would have been alive at 11:00 am + 1/2 minute, when another GR went off, and he can't survive a GR going off. It seems that none of the GRs could have killed him, because before each, there was another. So we have a contradiction: he both was and was not killed. Somebody has suggested that Fred is killed by the mereological sum of all the GRs, but that's mistaken in the present setting because the GRs check if Fred is already dead before they do anything, so in the present setting, none of them actually do anything—and if they don't do anything, how can they kill Fred?

The Kalaam argument needs the premise that there couldn't be a backwards infinite sequence of events. Here is an argument for this:

1. If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible.
2. If Hilbert's Hotel were possible, the GR Paradox could happen.
3. The GR Paradox cannot happen.
4. Therefore, there cannot be a backwards infinite sequence of events.

Actually, one could make steps 1 and 2 into a single step, but this is more fun, and, if it works, establishes the interesting corollary that Hilbert's Hotel couldn't exist.

Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result.

Argument for (2): If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes. And that would result in the GR Paradox.

The argument for (3) was already given at the beginning of the paper.

For about two years, I've smelled this argument coming, but I think my vanity has kept me from seeing it. I still have to confess that I have a really hard time accepting the corollary that Hilbert's Hotel couldn't exist—that corollary seems extremely counterintuitive to me. I wish I had some good way out.

On the other hand, establishing a major premise of an argument for the existence of God is a very happy outcome.

More on the Kalaam argument

Here is another counterexample to the claim that it is impossible to form a concrete and actual infinity through successive addition. Consider a gamma-widget. A gamma-widget is a stochastic critter that, once it is produced, produces at least one offspring. How many offspring it produces and how quickly is random. For every n>0, a gamma-widget has probability 2⁻ⁿ of producing exactly 2²ⁿ offspring in exactly 2⁻ⁿ years. The offspring are exact copies of the parent (this is a way in which the gamma-widget differs from the beta-widget). They, too, are gamma-widgets, and hence produce offspring. All the probabilities are independent.

Here is a mathematical fact (at least, it is a fact, if my scribbled calculations are right): There is a non-zero probability that if a gamma-widget comes into existence, it will be an ancestor of infinitely many gamma-widgets a year later. What has non-zero probability is also possible. Therefore, it is possible for this to happen. Here is another fact: The probability that there is a finite number n such that n years later there are infinitely many gamma-widgets is one. What is almost certain (i.e., has probability one) is a fortiori possible. Therefore, it is possible to have an infinite number of gamma-widgets arise through successive addition.

(If one is worried that the addition is not successive because multiple offspring are produced at the same time, we can stagger the productions in some way, and then not count the ones whose productions happen to overlap.)

Does this argument, and that in my previous post, destroy Craig's Kalaam argument? I think not yet. For there two ways that an infinity prima facie can be built up by successive addition. In one way, there is a first addition but no last one. In the other way, there is a last addition but not first one. What my examples show is that the first way is possible—there is a first addition but no last one. But I have not shown that there is no possibility of successive addition of the other sort.

However, the latter possibility follows from a modification of the case. Suppose that each gamma-widget that comes into existence before the year 2020 randomly and uniformly chooses a time during 2020, and completes a mug at that time. Then there is, almost surely, a subset of the mugs that has a last addition but no first one.

Craig can get out of these arguments by supposing that time cannot be infinitely divisible. Or he might just deny the possibility of my widgets. But what grounds would there be for that denial? Consider a variant on the gamma-widget, the delta-widget. It is just like the gamma-widget, except that it has probability 2⁻ⁿ of generating n² offspring in 2⁻ⁿ years. The delta-widget is just like the gamma-widget, except it is much less prolific: n² instead of 2²ⁿ. I am pretty sure (I haven't checked) that with probability one, delta-widgets do not have an infinite population explosion. So Craig can't object, it seems, to delta-widgets except by denying the infinite divisibility of time. But if delta-widgets are possible, why not the more prolific gamma-widgets?

Maybe my calculations are wrong. But I am pretty sure they're not. If they are wrong, replacing 2²ⁿ with an even faster growing sequence should fix things.

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.