Integers

Infinity

entirelyuseless Metaphysics, Science, Uncategorized , , , , , , , , , , ,

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.

Discount Rates

entirelyuseless Ethics, Mathematics, Probability, Uncategorized , , , , , , , , , ,

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.