**Summary**: I sketch an argument that population ethics should, in a certain technical sense, be similar to addition. I show that a surprising theorem of Hölder's implies that this means that we should be total utilitarians.

Addition is a very special operation. Despite the wide variety of esoteric mathematical objects known to us today, none of them have the basic desirable properties of grade-school arithmetic.

This fact was intuited by 19th century philosophers in the development of what we now call "total" utilitarianism. In this ethical system, we can assign each person a real number to indicate their welfare, and the value of an entire population is the sum of each individuals' welfare.

Using modern mathematics, we can now prove the intuition of Mills and Bentham: because addition is so special, any ethical system which is in a certain technical sense "reasonable" is equivalent to total utilitarianism.

The most basic premise is that we have some way of ordering individual lives.

We don't need to say how much better some life is than another, we just need to be able to put them in order. We might have some uncertainty as to which of two lives is better:

In this case, we aren't certain if "Medium" or "Medium 2" is better. However, we know they're both better than "Bad" and worse than "Good".

In the case when we always know which of two lives is better, we say that lives are totally ordered. If there is uncertainty, we say they are lattice ordered.

In either case, we require that the ranking remain consistent when we add people to the population. Here we add a person of "Medium" utility to each population:

The ranking on the right side of the figure above is legitimate because it keeps the order - if some life X is worse than Y, then (X + Medium) is still worse than (Y + Medium). This ranking below for example would fail that:

This ranking is inconsistent because it sometimes says that "Bad" is worse than "Medium" and other times says "Bad" is better than "Medium". A basic principle of ethics is that rankings should be consistent, and so rankings like the latter are excluded.

The most obvious way of defining an ethics of populations is to just take an ordering of individual lives and "glue them together" in an order-preserving way, like I did above. This generates what mathematicians would call the free group. (The only tricky part is that we need good and bad lives to "cancel out", something which I've talked about before.)

It turns out that merely gluing populations together in this way gives us a highly structured object known as a "lattice-ordered group". Here is a snippet of the resulting lattice:

This ranking is similar to what philosophers often call "Dominance" - if everyone in population P is better off than everyone in population Q, then P is better than Q. However, this is somewhat stronger - it allows us to compare populations of different sizes, something that the traditional dominance criterion doesn't let us do.

Let's take a minute to think about what we've done. Using only the fact that individuals' lives can be ordered and the requirement that population ethics respects this ordering in a certain technical sense, we've derived a robust population ethics, about which we can prove many interesting things.

One obvious facet of the above ranking is that it's not total. For example, we don't know if "Very Good" is better than "Good, Good", i.e. if it's better to have welfare "spread out" across multiple people, or concentrated in one. This obviously prohibits us from claiming that we've derived total utilitarianism, because under that system we always know which is better.

However, we can still derive a form of total utilitarianism which is equivalent in a large set of scenarios. To do so, we need to use the idea of an

Here's that same ordering, except I've tagged each population with the total "utility" resulting from that embedding:

This is clearly not identical to total utilitarianism - "Very Good" has a higher total utility than "Medium, Medium" but we don't know which is better, for example.

However, this ranking

Due to a surprising theorem of Holder which I have discussed before, as long as we disallow "infinitely good" populations, there is always some embedding like this. Thus, we can say that:

If we accept the premises of this blog post, this intuition simply cannot be correct. If the inequitable society has greater total utility, it must be at least as good as the equitable one.

This fact was intuited by 19th century philosophers in the development of what we now call "total" utilitarianism. In this ethical system, we can assign each person a real number to indicate their welfare, and the value of an entire population is the sum of each individuals' welfare.

Using modern mathematics, we can now prove the intuition of Mills and Bentham: because addition is so special, any ethical system which is in a certain technical sense "reasonable" is equivalent to total utilitarianism.

### What do we mean by ethics?

The most basic premise is that we have some way of ordering individual lives.

We don't need to say how much better some life is than another, we just need to be able to put them in order. We might have some uncertainty as to which of two lives is better:

In this case, we aren't certain if "Medium" or "Medium 2" is better. However, we know they're both better than "Bad" and worse than "Good".

In the case when we always know which of two lives is better, we say that lives are totally ordered. If there is uncertainty, we say they are lattice ordered.

In either case, we require that the ranking remain consistent when we add people to the population. Here we add a person of "Medium" utility to each population:

The ranking on the right side of the figure above is legitimate because it keeps the order - if some life X is worse than Y, then (X + Medium) is still worse than (Y + Medium). This ranking below for example would fail that:

This ranking is inconsistent because it sometimes says that "Bad" is worse than "Medium" and other times says "Bad" is better than "Medium". A basic principle of ethics is that rankings should be consistent, and so rankings like the latter are excluded.

### Increasing population size

The most obvious way of defining an ethics of populations is to just take an ordering of individual lives and "glue them together" in an order-preserving way, like I did above. This generates what mathematicians would call the free group. (The only tricky part is that we need good and bad lives to "cancel out", something which I've talked about before.)

It turns out that merely gluing populations together in this way gives us a highly structured object known as a "lattice-ordered group". Here is a snippet of the resulting lattice:

This ranking is similar to what philosophers often call "Dominance" - if everyone in population P is better off than everyone in population Q, then P is better than Q. However, this is somewhat stronger - it allows us to compare populations of different sizes, something that the traditional dominance criterion doesn't let us do.

Let's take a minute to think about what we've done. Using only the fact that individuals' lives can be ordered and the requirement that population ethics respects this ordering in a certain technical sense, we've derived a robust population ethics, about which we can prove many interesting things.

### Getting to total utilitarianism

One obvious facet of the above ranking is that it's not total. For example, we don't know if "Very Good" is better than "Good, Good", i.e. if it's better to have welfare "spread out" across multiple people, or concentrated in one. This obviously prohibits us from claiming that we've derived total utilitarianism, because under that system we always know which is better.

However, we can still derive a form of total utilitarianism which is equivalent in a large set of scenarios. To do so, we need to use the idea of an

*embedding*. This is merely a way of assigning each welfare level a number. Here is an example embedding:- Medium = 1
- Good = 2
- Very Good = 3

Here's that same ordering, except I've tagged each population with the total "utility" resulting from that embedding:

This is clearly not identical to total utilitarianism - "Very Good" has a higher total utility than "Medium, Medium" but we don't know which is better, for example.

However, this ranking

**never disagrees**with total utilitarianism - there is never a case where P is better than Q yet P has less total utility than Q.Due to a surprising theorem of Holder which I have discussed before, as long as we disallow "infinitely good" populations, there is always some embedding like this. Thus, we can say that:

Total utilitarianism is the moral "baseline". There might be circumstances where we are uncertain whether or not P is better than Q, but if we are certain, then it must be that P has greater total utility than Q.

### An application

Here is one consequence of these results. Many people, including myself, have the intuition that inequality is bad. In fact, it is so bad that there are circumstances where increasing equality is good even if people are, on average, worse off.If we accept the premises of this blog post, this intuition simply cannot be correct. If the inequitable society has greater total utility, it must be at least as good as the equitable one.

### Concluding remarks

There are certain restrictions we want the "addition" of a person to a population to obey. It turns out that there is only one way to obey them: by using grade school addition, i.e. total utilitarianism.

[For those interested in the technical result: Holder showed that any archimedean l-group is l-isomorphic to a subgroup of (R,+). The proof can be found in Glass'

*Partially Ordered Groups*as Corollary 4.1.4. This article was originally posted here.]