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 HÃ¶lder'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**

- Ethics with this property have also been called "discontinuous" or having a "lexical" priority.
- 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.