# Infinity

I discussed this topic previously, but without coming to a definite conclusion. Here I will give what I think is the correct explanation.

In his book Infinity, Causation, and Paradox, Alexander Pruss argues for what he calls “causal finitism,” or the principle that nothing can be affected by infinitely many causes:

In this volume, I will present a number of paradoxes of infinity, some old like Thomson’s Lamp and some new, and offer a unified metaphysical response to all of them by means of the hypothesis of causal finitism, which roughly says that nothing can be affected by infinitely many causes. In particular, Thomson’s Lamp story is ruled out since the final state of the lamp would be affected by infinitely many switch togglings. And in addition to arguing for the hypothesis as the best unified resolution to the paradoxes I shall offer some direct arguments against infinite regresses.

Thomson’s Lamp, if the reader is not familiar with it, is the question of what happens to a lamp if you switch it on and off an infinite number of times in a finite interval, doubling your velocity after each switch. At the end of the interval, is it on or off?

I think Pruss’s account is roughly speaking correct. I say “roughly speaking” because I would be hesitant to claim that nothing can be “affected” by infinitely many causes. Rather I would say that nothing is one effect simultaneously of infinitely many causes, and this is true for the same reason that there cannot be an infinite causal regress. That is, an infinite causal regress removes the notion of cause by removing the possibility of explanation, which is an intrinsic part of the idea of a cause. Similarly, it is impossible to explain anything using an infinite number of causes, because that infinity as such cannot be comprehended, and thus cannot be used to understand the thing which is the supposed effect. And since the infinity cannot explain the thing, neither can it be the cause of the thing.

What does this imply about the sorts of questions that were raised in my previous discussion, as for example about an infinite past or an infinite future, or a spatially infinite universe?

I presented an argument there, without necessarily claiming it to be correct, that such things are impossible precisely because they seem to imply an infinite causal regress. If there an infinite number of stars in the universe, for example, there seems to be an infinite regress of material causes: the universe seems to be composed of this local portion plus the rest, with the rest composed in a similar way, ad infinitum.

Unfortunately, there is an error in this argument against a spatially infinite world, and in similar arguments against a temporally infinite world, whether past or future. This can be seen in my response to Bertrand Russell when I discuss the material causes of water. Even if it is possible to break every portion of water down into smaller portions, it does not follow that this is an infinite sequence of material causes, or that it helps to explain water. In a similar way, even if the universe can be broken down into an infinite number of pieces in the above way, it does not follow that the universe has an infinite number of material causes: rather, this breakdown fails to explain, and fails to give causes at all.

St. Thomas gives a different argument against an infinite multitude, roughly speaking that it would lack a formal cause:

This, however, is impossible; since every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental.

By this argument, it would be impossible for there to be “an infinite number of stars” because the collection would lack “a species of multitude.” Unfortunately there is a problem with this argument as well, namely that it presupposes that the number is inherently fixed before it is considered by human beings. In reality, counting depends on someone who counts and a method they use for counting; to talk about the “number of stars” is a choice to break down the world in that particular way. There are other ways to think of it, as for example when we use the word “universe”, we count everything at once as a unit.

According to my account here, are some sorts of infinity actually impossible? Yes, namely those which demand an infinite sequence of explanation, or which demand an infinite number of things in order to explain something. Thus for example consider this story from Pruss about shuffling an infinite deck of cards:

Suppose I have an infinitely deep deck of cards, numbered with the positive integers. Can I shuffle it?

Given an infinite past, here is a procedure: n days ago, I perfectly fairly shuffle the top n cards in the deck.

This procedure is impossible because it makes the current state of the deck the direct effect of what I did n days ago, for all n. And the effect is a paradox: it is mathematically impossible for the integers to be randomly shuffled, because any series of integers will be biased towards lower numbers. Note that the existence of an infinite past is not the problem so much as assuming that one could have carried out such a procedure during an infinite past; in reality, if there was an infinite past, its contents are equally “infinite,” that is, they do not have such a definable, definite, “finite” relationship with the present.

# Perfectly Random

Suppose you have a string of random binary digits such as the following:

00111100010101001100011011001100110110010010100111

This string is 50 digits long, and was the result of a single attempt using the linked generator.

However, something seems distinctly non-random about it: there are exactly 25 zeros and exactly 25 ones. Naturally, this will not always happen, but most of the time the proportion of zeros will be fairly close to half. And evidently this is necessary, since if the proportion was usually much different from half, then the selection could not have been random in the first place.

There are other things about this string that are definitely not random. It contains only zeros and ones, and no other digits, much less items like letters from the alphabet, or items like ‘%’ and ‘\$’.

Why do we have these apparently non-random characteristics? Both sorts of characteristics, the approximate and typical proportion, and the more rigid characteristics, are necessary consequences of the way we obtained or defined this number.

It is easy to see that such characteristics are inevitable. Suppose someone wants to choose something random without any non-random characteristics. Let’s suppose they want to avoid the first sort of characteristic, which is perhaps the “easier” task. They can certainly make the proportion of zeros approximately 75% or anything else that they please. But this will still be a non-random characteristic.

They try again. Suppose they succeed in preventing the series of digits from converging to any specific probability. If they do, there is one and only one way to do this. Much as in our discussion of the mathematical laws of nature, the only way to accomplish this will be to go back and forth between longer and longer strings of zeros and ones. But this is an extremely non-random characteristic. So they may have succeeded in avoiding one particular type of non-randomness, but only at the cost of adding something else very non-random.

Again, consider the second kind of characteristic. Here things are even clearer: the only way to avoid the second kind of characteristic is not to attempt any task in the first place. The only way to win is not to play. Once we have said “your task is to do such and such,” we have already specified some non-random characteristics of the second kind; to avoid such characteristics is to avoid the task completely.

“Completely random,” in fact, is an incoherent idea. No such thing can exist anywhere, in the same way that “formless matter” cannot actually exist, but all matter is formed in one way or another.

The same thing applies to David Hume’s supposed problem of induction. I ended that post with the remark that for his argument to work, he must be “absolutely certain that the future will resemble the past in no way.” But this of course is impossible in the first place; the past and the future are both defined as periods of time, and so there is some resemblance in their very definition, in the same way that any material thing must have some form in its definition, and any “random” thing must have something non-random in its definition.

# Discount Rates

Eliezer Yudkowsky some years ago made this argument against temporal discounting:

I’ve never been a fan of the notion that we should (normatively) have a discount rate in our pure preferences – as opposed to a pseudo-discount rate arising from monetary inflation, or from opportunity costs of other investments, or from various probabilistic catastrophes that destroy resources or consumers.  The idea that it is literally, fundamentally 5% more important that a poverty-stricken family have clean water in 2008, than that a similar family have clean water in 2009, seems like pure discrimination to me – just as much as if you were to discriminate between blacks and whites.

Robin  Hanson disagreed, responding with this post:

But doesn’t discounting at market rates of return suggest we should do almost nothing to help far future folk, and isn’t that crazy?  No, it suggests:

1. Usually the best way to help far future folk is to invest now to give them resources they can spend as they wish.
2. Almost no one now in fact cares much about far future folk, or they would have bid up the price (i.e., market return) to much higher levels.

Very distant future times are ridiculously easy to help via investment.  A 2% annual return adds up to a googol (10^100) return over 12,000 years, even if there is only a 1/1000 chance they will exist or receive it.

So if you are not incredibly eager to invest this way to help them, how can you claim to care the tiniest bit about them?  How can you think anyone on Earth so cares?  And if no one cares the tiniest bit, how can you say it is “moral” to care about them, not just somewhat, but almost equally to people now?  Surely if you are representing a group, instead of spending your own wealth, you shouldn’t assume they care much.

Yudkowsky’s argument is idealistic, while Hanson is attempting to be realistic. I will look at this from a different point of view. Hanson is right, and Yudkowsky is wrong, for a still more idealistic reason than Yudkowsky’s reasons. In particular, a temporal discount rate is logically and mathematically necessary in order to have consistent preferences.

Suppose you have the chance to save 10 lives a year from now, or 2 years from now, or 3 years from now etc., such that your mutually exclusive options include the possibility of saving 10 lives x years from now for all x.

At first, it would seem to be consistent for you to say that all of these possibilities have equal value by some measure of utility.

The problem does not arise from this initial assignment, but it arises when we consider what happens when you act in this situation. Your revealed preferences in that situation will indicate that you prefer things nearer in time to things more distant, for the following reason.

It is impossible to choose a random integer without a bias towards low numbers, for the same reasons we argued here that it is impossible to assign probabilities to hypotheses without, in general, assigning simpler hypotheses higher probabilities. In a similar way, if “you will choose 2 years from now”, “you will choose 10 years from now,” “you will choose 100 years from now,” are all assigned probabilities, they cannot all be assigned equal probabilities, but you must be more likely to choose the options less distant in time, in general and overall. There will be some number n such that there is a 99.99% chance that you will choose some number of years less than n, and and a probability of 0.01% that you will choose n or more years, indicating that you have a very strong preference for saving lives sooner rather than later.

Someone might respond that this does not necessarily affect the specific value assignments, in the same way that in some particular case, we can consistently think that some particular complex hypothesis is more probable than some particular simple hypothesis. The problem with this is the hypotheses do not change their complexity, but time passes, making things distant in time become things nearer in time. Thus, for example, if Yudkowsky responds, “Fine. We assign equal value to saving lives for each year from 1 to 10^100, and smaller values to the times after that,” this will necessarily lead to dynamic inconsistency. The only way to avoid this inconsistency is to apply a discount rate to all periods of time, including ones in the near, medium, and long term future.

# Quantum Mechanics and Libertarian Free Will

In a passage quoted in the last post, Jerry Coyne claims that quantum indeterminacy is irrelevant to free will: “Even the pure indeterminism of quantum mechanics can’t give us free will, because that’s simple randomness, and not a result of our own ‘will.'”

Coyne seems to be thinking that since quantum indeterminism has fixed probabilities in any specific situation, the result for human behavior would necessarily be like our second imaginary situation in the last post. There might be a 20% chance that you would randomly do X, and an 80% chance that you would randomly do Y, and nothing can affect these probabilities. Consequently you cannot be morally responsible for doing X or for doing Y, nor should you be praised or blamed for them.

Wait, you might say. Coyne explicitly favors praise and blame in general. But why? If you would not praise or blame someone doing something randomly, why should you praise or blame someone doing something in a deterministic manner? As explained in the last post, the question is whether reasons have any influence on your behavior. Coyne is assuming that if your behavior is deterministic, it can still be influenced by reasons, but if it is indeterministic, it cannot be. But there is no reason for this to be case. Your behavior can be influenced by reasons whether it is deterministic or not.

St. Thomas argues for libertarian free will on the grounds that there can be reasons for opposite actions:

Man does not choose of necessity. And this is because that which is possible not to be, is not of necessity. Now the reason why it is possible not to choose, or to choose, may be gathered from a twofold power in man. For man can will and not will, act and not act; again, he can will this or that, and do this or that. The reason of this is seated in the very power of the reason. For the will can tend to whatever the reason can apprehend as good. Now the reason can apprehend as good, not only this, viz. “to will” or “to act,” but also this, viz. “not to will” or “not to act.” Again, in all particular goods, the reason can consider an aspect of some good, and the lack of some good, which has the aspect of evil: and in this respect, it can apprehend any single one of such goods as to be chosen or to be avoided. The perfect good alone, which is Happiness, cannot be apprehended by the reason as an evil, or as lacking in any way. Consequently man wills Happiness of necessity, nor can he will not to be happy, or to be unhappy. Now since choice is not of the end, but of the means, as stated above (Article 3); it is not of the perfect good, which is Happiness, but of other particular goods. Therefore man chooses not of necessity, but freely.

Someone might object that if both are possible, there cannot be a reason why someone chooses one rather than the other. This is basically the claim in the third objection:

Further, if two things are absolutely equal, man is not moved to one more than to the other; thus if a hungry man, as Plato says (Cf. De Coelo ii, 13), be confronted on either side with two portions of food equally appetizing and at an equal distance, he is not moved towards one more than to the other; and he finds the reason of this in the immobility of the earth in the middle of the world. Now, if that which is equally (eligible) with something else cannot be chosen, much less can that be chosen which appears as less (eligible). Therefore if two or more things are available, of which one appears to be more (eligible), it is impossible to choose any of the others. Therefore that which appears to hold the first place is chosen of necessity. But every act of choosing is in regard to something that seems in some way better. Therefore every choice is made necessarily.

St. Thomas responds to this that it is a question of what the person considers:

If two things be proposed as equal under one aspect, nothing hinders us from considering in one of them some particular point of superiority, so that the will has a bent towards that one rather than towards the other.

Thus for example, someone might decide to become a doctor because it pays well, or they might decide to become a truck driver because they enjoy driving. Whether they consider “what would I enjoy?” or “what would pay well?” will determine which choice they make.

The reader might notice a flaw, or at least a loose thread, in St. Thomas’s argument. In our example, what determines whether you think about what pays well or what you would enjoy? This could be yet another choice. I could create a spreadsheet of possible jobs and think, “What should I put on it? Should I put the pay? or should I put what I enjoy?” But obviously the question about necessity will simply be pushed back, in this case. Is this choice itself determinate or indeterminate? And what determines what choice I make in this case? Here we are discussing an actual temporal series of thoughts, and it absolutely must have a first, since human life has a beginning in time. Consequently there will have to be a point where, if there is the possibility of “doing A for reason B” and “doing C for reason D”, it cannot be any additional consideration which determines which one is done.

Now it is possible at this point that St. Thomas is mistaken. It might be that the hypothesis that both were “really” possible is mistaken, and something does determine one rather than the other with “necessity.” It is also possible that he is not mistaken. Either way, human reasons do not influence the determination, because reason B and/or reason D are the first reasons considered, by hypothesis (if they were not, we would simply push back the question.)

At this point someone might consider this lack of the influence of reasons to imply that people are not morally responsible for doing A or for doing C. The problem with this is that if you do something without a reason (and without potentially being influenced by a reason), then indeed you would not be morally responsible. But the person doing A or C is not uninfluenced by reasons. They are influenced by reason B, or by reason D. Consequently, they are responsible for their specific action, because they do it for a reason, despite the fact that there is some other general issue that they are not responsible for.

What influence could quantum indeterminacy have here? It might be responsible for deciding between “doing A for reason B” and “doing C for reason D.” And as Coyne says, this would be “simple randomness,” with fixed probabilities in any particular situation. But none of this would prevent this from being a situation that would include libertarian free will, since libertarian free will is precisely nothing but the situation where there are two real possibilities: you might do one thing for one reason, or another thing for another reason. And that is what we would have here.

Does quantum mechanics have this influence in fact, or is this just a theoretical possibility? It very likely does. Some argue that it probably doesn’t, on the grounds that quantum mechanics does not typically seem to imply much indeterminacy for macroscopic objects. The problem with this argument is that the only way of knowing that quantum indeterminacy rarely leads to large scale differences is by using humanly designed items like clocks or computers. And these are specifically designed to be determinate: whenever our artifact is not sufficiently determinate and predictable, we change the design until we get something predictable. If we look at something in nature uninfluenced by human design, like a waterfall, is details are highly unpredictable to us. Which drop of water will be the most distant from this particular point one hour from now? There is no way to know.

But how much real indeterminacy is in the waterfall, or in the human brain, due to quantum indeterminacy? Most likely nobody knows, but it is basically a question of timescales. Do you get a great deal of indeterminacy after one hour, or after several days? One way or another, with the passage of enough time, you will get a degree of real indeterminacy as high as you like. The same thing will be equally true of human behavior. We often notice, in fact, that at short timescales there is less indeterminacy than we subjectively feel. For example, if someone hesitates to accept an invitation, in many situations, others will know that the person is very likely to decline. But the person feels very uncertain, as though there were a 50/50 chance of accepting or declining. The real probabilities might be 90/10 or even more slanted. Nonetheless, the question is one of timescales and not of whether or not there is any indeterminacy. There is, this is basically settled, it will apply to human behavior, and there is little reason to doubt that it applies at relatively short timescales compared to the timescales at which it applies to clocks and computers or other things designed with predictability in mind.

In this sense, quantum indeterminacy strongly suggests that St. Thomas is basically correct about libertarian free will.

On the other hand, Coyne is also right about something here. While it is not true that such “randomness” removes moral responsibility, the fact that people do things for reasons, or that praise and blame is a fitting response to actions done for reasons, Coyne correctly notices that it does not add to the fact that someone is responsible. If there is no human reason for the fact that a person did A for reason B rather than C for reason D, this makes their actions less intelligible, and thus less subject to responsibility. In other words, the “libertarian” part of libertarian free will does not make the will more truly a will, but less truly. In this respect, Coyne is right. This however is unrelated to quantum mechanics or to any particular scientific account. The thoughtful person can understand this simply from general considerations about what it means to act for a reason.