Positive utility in an infinite universe

post by casebash · 2016-01-29T23:40:45.491Z · LW · GW · Legacy · 13 comments

Content Note: Highly abstract situation with existing infinities

This post will attempt to resolve the problem of infinities in utilitarianism. The arguments are very similar to an argument for total utilitarianism over other forms which I'll most likely write up at some point (my previous post was better as an argument against average utilitarianism, rather than an argument in favour of total utilitarianism).

In the Less Wrong Facebook group, Gabe Bf posted a challenge to save utilitarianism from the problem of infinities. The original problem is from by a paper by Nick Bostrom.

I believe that I have quite a good solution to this problem that allows us to systemise comparing infinite sets of utility, but this post focuses on justifying why we should take it to be axiomic that adding another person with positive utility is good and on why the results that seem to contradict this lack credibility. Let's call this the Addition Axiom or A. We can also consider the Finite Addition Axiom (only applies when we add utility into a universe with a finite number of people), call this A0.

Let's consider what other alternative axioms that we might want to take instead. One is the Infinite Indifference Axiom or I, that is that we should be indifferent if both options provide infinite total utility (of the same order of infinity). Another option would be the Remapping Axiom (or R), which would assert that if we can surjectively map a group of people G onto another group H so that each g from G is mapped onto a person h from H where u(g) >= u(h), then v(H) <= v(G) where v represents the value of a particular universe (it doesn't necessarily map onto the real numbers or represent a complete ordering). Using the Remapping Axiom twice implies that we should be indifferent between an infinite series of ones and the same series with a 0 at one spot. This means that the Remapping Axiom is incompatible with the Addition Axiom. We can also consider the Finite Remapping Axiom (R0) which is where we limit the Remapping Axiom to remapping a finite number of elements.

First, we need to determine what are good properties of a statement we wish to take as an axiom. This is my first time trying to establish an axiom so formally, so I will admit that this list is not going to be perfect.

• Uses well-understood and regular objects, properties or processes. If the components are not understood well, it is highly likely that our attempt to determine the truth of a statement will be misguided.
• An axiom close to the territory is more reliable than one in the map because it is very easy to make subtle errors when constructing a map.
• Leads to minimally weird consequences.
• Extends included axioms in a logical way. If the axiom is an extension of a simpler alternative axiom, then it should be intuitive that the result would extend to the larger set; there should be reasons to expect it to behave the same way.

Let's look first at the Infinite Indifference Axiom. Firstly, it deals purely with infinite objects, which are known to often behave irregularly and results in many problems in which there is no consensus. Secondly, it exists in the map to some extent (but not that much at all). In the territory, there are just objects, infinity is our attempt to transpose certain object configurations into a number system. Thirdly, it doesn't seem to extend from the finite numbers very well. If one situation provides 5 total utility and another provides 5 total utility, then it seems logical to treat them as the same as 5 is equal to 5. However, infinity doesn't seem to be equal to itself in the same way. Infinity plus 1 is still infinity. We can remove infinite dots from infinite dots and end up with 1 or 2 or 3... or infinity. Further, this axiom leads to the result that we are indifferent between someone with large positive utility being created and someone with large negative good being created. This is massively unintuitive, but I will admit it is subjective. I think this would make a very poor axiom, but it doesn't mean it is false (Pythagoras' Theorem would make a poor axiom too).

On the other hand, deciding between the Remapping Axiom and Addition Axiom will be much closer. On the first criteria I think the Addition Axiom comes out ahead. It involves making only a single change to the situation, a primitive change if you will. In contrast, the Remapping Axiom involves Remapping an infinite number of objects. This is still a relatively simple change, but it is definitely more complicated and permutations of infinite series are well known to behave weirdly.

On the second criteria, the Addition Axiom (by itself) doesn't lead to any really weird results. We'll get some weird results in subsequent posts, but that's because we are going to going to make some very weird changes to the situation, not because of the Addition Axiom itself. The failure of the Remapping Axion could very well be because mappings lack the resolution to distinguish between different situations. We know that an infinite series can map onto itself, half of itself or itself twice, which lends a huge amount of support to the lack of resolution theory.

On the other hand, the Addition Axiom being false (because we've assumes the Remapping Axiom) is truly bizarre. It basically states that good things are good. Nonetheless, while this may seem very convincing to me, people's intuitions vary so the more relevant material for people with a different intuition is the material above that suggests the Remapping Axiom lacks resolution.

On the third criteria, a new object appearing is something that can occur in the territory. Infinite remappings initially seem to be more in the map than the territory, but it is very easy to imagine a group of objects moving one space to the right, so this assertion seems unjustified. That is, infinity is in the map as discussed before, but an infinite group of objects and their movements can still be in the territory. However, when we think about it again, we see that we have reduced the infinite group of objects X, to a set objects positioned, for example, on X = 0, 1, 2... This is a massive hint about the content of my following posts.

Lastly, the Addition Axiom in infinite case is a natural extension of the Finite Addition Axiom. In A0 the principle is that whatever else happens in the universe is irrelevant and there is no reason for this to change in the infinite case. For the Remapping Axiom, it also seems like a very natural extension of the finite case, so I'll call this criteria a draw.

In summary, if you don't already find the Addition Axiom more intuitive than the Remapping Axiom, the main reasons to favour the Addition Axiom are 1) it deals with better understood objects, 2) it is closer to the territory than the map 3) there are good reasons to suspect that Remapping lacks resolution. Of these reasons, I believe the the 3rd is by far the most persuasive; I consider the other two more to be hints than anything else.

I only dealt with the Infinite Indifference Axiom and the Remapping Axioms, but I'm sure other people will suggest their own alternative Axioms which need to be compared.

Increasing a person's utility, instead of creating a new person with positive utility is exactly the same. Also, this post is just the start. I will provide a systematic analysis of infinite universes over the coming days, plus an FAQ conditional on sufficient high quality questions.

 

 

 

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