Swapping the Order of Summation
First published: 6th July 2026
As a utilitarian, you may try to maximise the total utility of all qualia-experiencing beings.
But there's another way. Instead, you could try to maximise the total utility provided by all qualia-providing objects.
For example: if you had some humans (who experience qualia) and some cakes (which we assume for now do not), you could think about the total utility in these two ways.
First, let's look at the usual way. We could let each human have a utility function which specifies how much utility that human will get from eating any given cake. For the sake of simplicity, suppose for now that these values are independent of how many cakes that human has already eaten. If each human eats cakes , we can then take the total utility and try to maximise over all possible cake-assignments subject to the constraint that each cake is eaten by at most one person.
So far so good.
But how does the cake feel about all this?
Let's define an indicator function called "eats" . Then we can write our total utility sum as . What does this get us? Well now we can swap the order of summation. Let's do that.
. We can interpret each term in the outer sum as the joy provided by cake .
The utilitarian who tries to maximise the joy experienced by all humans (the first expression we had for ) must necessarily agree about all states of the world and all decision problems as the utilitarian who tries to maximise the joy provided by all cakes (the second expression for ). We will call this second viewpoint reverse-sum utilitarianism.
How could we interpret the morality of reverse-sum utilitarians? They might be assuming that cakes have qualia, humans don't, and cakes want to be eaten . This will lead them to exactly the same actions as the normal utilitarians, who assumes that humans have qualia, cakes don't, and humans like eating cake.
Where does this break down?
In a cake-abundant world, it may be that all humans have all the cake they desire, and the creation of extra cakes would be completely neutral to the normal utilitarian (ignoring the cost of producing them). Equally, in this world, the destruction of an uneaten cake is completely neutral to the normal utilitarian. This would mean that in a human-abundant (cake-scarce) world our reversed-sum utilitarian is indifferent to the birth and death of hungry humans.
Maths in the real world
There are sometimes difficulties with swapping the order of summation. It's fine when all of your sums are finite, but when they are not, some really surprising things can happen. There are even sums that can take any real value at all if you are allowed to change the order of the terms. Do we live in a world with infinitely many objects that could potentially provide joy? If yes, what are we supposed to do with a total utility function that isn't even well-defined?
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