### Why Classical Utilitarianism is the only (Archimedean) Ethic

Probably the most famous graph in ethics is this one of Parfit's:

He's constructing a series of worlds where each one has more people, but those people have a lower level of welfare. The question is whether the worlds are equivalent, i.e. whether it's equivalent to have a world with a huge number of barely happy people or a world with a small number of ecstatic individuals.

Classical utilitarianism answers "Yes", but some recent attempts to avoid unpleasant results (such as the "repugnant conclusion") have argued "No". For example, Parfit says:
Suppose that I can choose between two futures. I could live for another 100 years, all of an extremely high quality. Call this the Century of Ecstasy. I could instead live for ever, with a life that would always be barely worth living. Though there would be nothing bad in this life, the only good things would be muzak and potatoes. Call this the Drab Eternity. I believe that, of these two, the Century of Ecstasy would give me a better future.

The belief that the "Century of Ecstasy" is superior to the "Drab Eternity", no matter how long that eternity lasts, has been called "Non-Archimedean" by Arrhenius, in reference to the Archimedean Property of numbers, which says roughly that there are no "infinitely large" numbers.1 Specifically, a group is Archimedean if for any $x$ and $y$ there is some $n$ such that $$\underbrace{x+x+\dots+x}_{\text{n times}}>y$$
The following remarkable fact is true:
Classical Utilitarianism is the only Archimedean ethic.
This means that if we don't accept that the briefest instant of a "higher" pleasure is better than the longest eternity of a "lower" pleasure, then we must be classical utilitarians.

### Proof

First, define the terms. As always, we assume that there is some set $X$ which contains various welfare levels. There is an operation $\oplus$ which combines welfare levels; the statement $x\oplus y=z$ can be read as "A life with welfare $x$ and then welfare $y$ is equivalent to having a life with just welfare $z$."2 It is assumed that this constitutes a group, i.e. the operation is associative and inverses and an identity exist.

In order to make decisions, we need some ranking; the statement $x>y$ means "The welfare level $x$ is morally preferable to $y$." We require $>$ to agree with our operation, i.e. if $x>y$ then $x\oplus c > y\oplus c$ for all $c$.

With the stipulation that our group is Archimedean, this reduces to a theorem of Hölder's, which states that all Archimedean linearly ordered groups are isomorphic to a subgroup of the reals under addition, i.e. classical utilitarianism. The proof is rather involved, but a fairly readable version can be found here.∎

### Discussion

In order to be useful, non-Archimedean theories can't just say that there is some theoretical amount of welfare which is lexically superior - this level of welfare must exist in our day-to-day lives. Personally, when comparing a brief second of happiness on my happiest day to years of moderate happiness, I would choose the years. This leaves me with no choice but to accept classical utilitarianism.

Footnotes
1. Ethics with this property have also been called "discontinuous" or having a "lexical" priority.
2. Unlike in past blogs where I used $\oplus$ to be a population ethic, here I define it in terms of intra-personal welfare to fit more in line with Parfit's quote.

### Group Theory and the Repugnant Conclusion

A fundamental question in population ethics is the tradeoff between quantity and quality. The world has finite resources, so if we promote policies that increase the population, we do so at the risk of decreasing quality of life.

Derek Parfit is credited with popularizing the importance of this problem when he pointed out that any population ethic which obeys some seemingly reasonable constraints must end up with what he called "the repugnant conclusion" - the conclusion that a world full of miserable people is better than a sparsely-populated world full of happy people. Since Parfit, there have been a range of theories seeking to preserve our intuitions about ethics while still avoiding this conclusion.

One discovery of abstract algebra is that we can understand the limitations of systems based solely on the questions they are able to answer, even if we don't know what the answers are.

Here, I'll consider any system capable of answering a question like "Are two people who each live 50 years morally equivalent to one person who lives 100 years?" (Again, we don't require that the answer be "Yes" or "No", but merely that there be some answer.) For notational ease, I use the symbol $\oplus$ to be the "moral combination", e.g. the above question can be written $$(50\text{ years})\oplus(50\text{ years})=100\text{ years?}$$ Such a system I will call a "moral group" and require that it obey a few standard requirements. These are:

1. Any two people can be replaced with one who is (significantly) better off
2. There is some level of welfare which is "morally neutral", i.e. a person of that welfare neither increases nor decreases the overall moral desirability of the world.
3. For any level of welfare, no matter how high, there is some level of welfare which is so negative that the two cancel out

With this definition, we have an impossibility theorem:

Theorem: In any "moral group", the repugnant conclusion holds.

Proof: Suppose that $x$ is a welfare level that is better than "barely worth living". Formally, say that there must be some $y$ where $0 < y < x$, i.e. it's possible to be worse off than $x$ and still have a "life worth living". We'll show that a world with just $x$ is morally equivalent to a world with two people who are both worse off than $x$. Repeating this ad infinitum leads to the conclusion that a world with a few happy people is equivalent to a world with a large number of people whose lives are "barely worth living."

Choose some $y$ between $0$ and $x$ (one exists, by the definition of $x$). Note that $x=y\oplus z$ where $z=y^{-1}\oplus x$, so we just need to show that $z<x$. Since $y>0$, $y^{-1} < 0$ because if it weren't then we'd have $y^{-1} > 0$; adding $y$ to both sides results in $0>y$ which contradicts the assumption that $y>0$. Therefore $y^{-1} \oplus x < x$, or to write it another way: $z < x$. So $x=y\oplus z$, with $y$ and $z$ both worse than $x$.

This means that for any world with people $x_1,x_2,\dots$ of high welfare, there is an equivalent world $y_1,y_2,\dots$ with more people, each of whom have lower welfare. By adding some person of low (but still positive) welfare $y_{n+1}$ to the second world, it becomes better than the first, resulting in the repugnant conclusion.∎

### Algebra and Ethics

Symmetry is all around us. The kind of symmetry that most people think of is geometric symmetry, e.g. an equilateral triangle has rotational symmetry:

I've rotated the triangle by 1/3 of a rotation, but it remains the "same", just with a "relabeling" of the points. Hence this rotation is a symmetry of the triangle.

Ethical positions generally express another type of symmetry; when someone argues for "marriage equality" what they mean is that the gender of partners is merely a "relabeling" that keeps the important aspects like love and commitment the same. Symmetries in pain processing between humans and other animals has lead thinkers like Richard Dawkins to declare that species is merely a relabeling, and that causing pain to a cow is "morally equivalent" to causing pain to a human, calling our eating practices into question.

In 1854 Arthur Cayley gave the first modern definition of what mathematicians call a "group", and showed that groups are essentially permutations, thus establishing the theory of groups as the language of symmetry. Despite the importance of groups to symmetry and the importance of symmetry to ethics, I'm not able to find any ethical works based on group theory. So I hope to give what may be the first ever group-theoretical proof of ethics.

"Group-like" Ethics
I'm going to be concerned with questions like "is having two people, each of whom live 50 years, equivalent to having one person who lives 100 years?" I don't require that this question be answered either "yes" or "no", but only that the question has some answer.

So that this post doesn't take up a huge amount of space, I'm going to define the symbol $\oplus$ to mean "moral combination" and $=$ to mean moral equivalence, so the statement "two people, each of whom live fifty years, is equivalent to one person living 100 years" can be written as $$(50 \text{ years})\oplus(50 \text{ years})=100 \text{ years}$$ There are many different ways to define $\oplus$. For example, we might care only about the worst-off person - in this case $(50 \text{ years})\oplus(50 \text{ years})=50 \text{ years}$ as the worst-off person on the left-hand side of the equation has the same length of life as the worst-off person on the right. Alternatively, we might point out that quality of life degrades as you get older, so in fact maybe $(50 \text{ years})\oplus(50 \text{ years})=150 \text{ years}$ since the two young people get so much more joy out of their life. The World Health Organization follows this model and weights lives like this:

According to their formula, old age is so awful that $(40 \text{ years})\oplus(40 \text{ years})=125 \text{ years}$ and one person would have to live for thousands of years to be equivalent to two 50 year lifespans.

In addition to requiring that statements like $(50 \text{ years})\oplus(50 \text{ years})$ have some answer, I will also require that there is an "identity", i.e. there is some quality of life such that adding a person with that quality of life doesn't change the overall value of the world. This is a reasonable assumption because:
1. Sometimes increasing the population is a good idea, i.e. there is some $y$ such that $x\oplus y > x$
2. Sometimes increasing the population is a bad idea, i.e. there is some $z$ such that $x\oplus z < x$
3. By the intermediate value theorem, there must therefore be some value which I'll call $0$ such that $x\oplus 0 = x$

Any ethical system which has an operation like $\oplus$ I will call "group-like" (although observant readers will note that I'm making fewer assumptions than what groups require - technically this is a "unital magma").

"Utilitarian-like" Ethics
The classic definition of "utilitarianism" is to look only at happiness and to define $\oplus=+$, e.g. two people with five "units" of happiness is equivalent to one person with ten units of happiness.

There are a plethora of "utilitarian-like" ethical theories which define $\oplus$ as being sort of like addition, but not really. For example, negative utilitarians would first discard any pleasure, and look only at the pain of each individual before doing the addition. Prioritarians wouldn't completely disregard pleasure, but they would weight helping those in need more strongly. The Sen social welfare function weights income by inequality before doing the addition. And so on.

I will describe an ethical system as "utilitarian-like" if it is equivalent to doing addition with some appropriate transformation applied first. Formally, utilitarian-like operations are of the form $x\oplus y = f(x)+f(y)$.

The Theorem
With these definitions in mind, we can state our theorem:
The only ethical system which is both group-like and utilitarian-like is classical ("Benthamite") utilitarianism.
Observant readers will notice that my examples in the "group-like" section were different than the examples in the "utilitarian-like" section. This theorem proves that this is not an accident.

Proof: $x\oplus 0 = f(x)+f(0)$ so $x = f(x)+f(0)$ or to rewrite it another way, $f(x)=x - f(0)$ where $f(0)$ is some constant. This means that all group-like and utilitarian-like functions are equivalent, just shifted slightly. To use a formal definition of "equivalent", the homomorphism $\phi(x) = x + f(0)$ can be easily seen via the first isomorphism theorem to be an isomorphism $(\mathbb{R},\oplus)\to(\mathbb{R},+)$.

Discussion
The reason why Prioritarians et al. fail to be group-like is something I haven't seen discussed much in the literature: a lack of an identity element.

For example, suppose $x\oplus y = f(x)+f(y)$ where $$f(x) = \left\{ \begin{array}{lr} 2x & x < 0\\ x & else \end{array} \right.$$ This is a negative utilitarian-type ethics which weights suffering (i.e. negative experience) more strongly.

Consider a few possible worlds in which we add someone of utility 2:

1. $-1\oplus 2 = 0$
2. $-2\oplus 2 = -2$
3. $-3\oplus 2 = -4$

In the first case, adding someone of utility two improves the world. In the second, it keeps the world the same and in the third it makes the world worse.

That negative utilitarianism requires this isn't immediately obvious to me, and I believe it to be a non-trivial result of using group theory.

Conclusion
We might view negative utilitarianism or prioritarianism as a form of "pre-processing". For example, we might say that painful experiences affect utility more than positive ones. But when it comes to comparing utility to utility, it must be "each to count for one and none for more than one" with all the counter-intuitive results that implies.