A Pure Math Argument for Total Utilitarianism

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.

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.]