### 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.

1. I think there are actually two different kinds of "pre-processing". You're discussing what happens when comparing the utility between individuals, and trying to calculate the utility of something that adds or removes individuals.

But assigning each individual a single number as their utility assumes that the sum of suffering and happiness can be meaningfully combined on one axis. So every form of utilitarianism (including classic utilitarianism) needs to also perform a "pre-processing" step of (suffering + happiness = utility) to get the "within-individual" utility before being able to perform comparisons between individuals. And at least some of the differences between theories are actually better described as showing up at this stage, rather than the later one.

Now if one is a pure negative utilitarian, then one defines this earlier operation as (suffering + happiness = -suffering), and your later operation as f(x) = x. Assuming that suffering and happiness are both natural numbers, then the results of the first operation is the set of negative integers + 0, which means that the second operation has an identity element again. Of course, now the first operation is non-group-like.

But wait: there's no natural mapping from subjective sensations to natural numbers. How much worse is the pain from being burned alive than the pain of a mild itch? So we need one more pre-processing step where we construct such a mapping. And some people hold that different varieties of happiness are worth different amounts: "better to be an unhappy Socrates than a happy pig". So what we actually need is a mapping from a number of different sensations to a number of different quantities, and what we previously described as (suffering + happiness) is rather (suffering_1 + ... + suffering_n + happiness_1 + ... + happiness_n). Or perhaps just (quantity_1 + ... quantity_n).

Here it stops being clear whether even classical utilitarianism is any more "group-like" than any other form. Certainly if you are free to choose the mapping, you can find a way to make classical utilitarianism the only "group-like" one... but then if you e.g. just map all forms of happiness to a 0, you can also make negative utilitarianism "group-like".

1. Hey Kaj,

To clarify the goal: there are certain "unpleasant" conclusions of classical utilitarianism, e.g. utility monsters. One attempt around these is to weight utility in different ways. My point is that this doesn't resolve the underlying problem of utility monsters, because once embedded in $(\mathbb{R},+)$ ethics must be (isomorphic to) classical utilitarianism.

I have a follow up post on one application of this to the repugnant conclusion, and more are planned, so I hope to hear more from you on these results.

2. Thanks! I think your approach is intriguing, and it's also useful for me to see some applications of abstract algebra just when I'm supposed to be studying for an algebra exam :). I've added this blog to my RSS reader.

2. Why do you take as your definition of "group-like" x*y = f(x)+f(y) rather than x*y = f^{-1}(f(x)+f(y))? The latter seems much more natural to me.

1. g: I think your definition is good. But if we're trying to show an isomorphism between * and + it kind of begs the question to define x*y that way.

2. Oops: I wrote "group-like" but of course it was your definition of "utilitarian-like" that I was commenting on.

I don't see that one definition is any more question-begging than the other.

Perhaps the real question is: Why assume an addition operation that's "utilitarian-like" in either sense?

3. It does not beg the question, and your definition is actually incorrect. You have (x ⊕ y) ⊕ z = f(f(x) + f(y)) + f(z), but x ⊕ (y ⊕ z) = f(x) + f(f(y) + f(z)), which is not associative unless some very strict conditions apply to f. This does not capture the structure of utilitarianism. The input to f is a well-being level and the output is a utility, but this feeds the utilities back in as if they were well-being levels, which is a type error. In light of this, we can't really conclude the things about utilitarianism that you are concluding.

3. I am quite surprised by the unspoken assumptions, chief among them that ethics should be about (or, is adequately modeled by) a *universal* mapping of situations to a *complete* linear order relation that determines preference, mimicking money worth, while variance in ethics gets represented by disputes between such universal and complete valuations. As a (neither necessary nor sufficient, but elegantly self-contained) representative of my misgivings with that, I'll submit that such assumptions are in fact portrayed in Genesis as Eve's desire for the forbidden fruit: according to any interpretation that takes the literal name of the latter at face value - in contrast to the more usual ones.

4. Thanks for the post, Ben!

> 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.

I agree with the others that this seems wrong -- it's because you're not mapping backwards to the original input scale. For instance, suppose the function was f(x) = 2x for any x. Then

1 circle-plus 0 = 2
2 circle-plus 0 = 4

etc. So just adding the identity makes the world better!