For several years now I have been reading this paper by Kevin Knuth and John Skilling, about the foundations of measure theory, probability, and MaxEnt. I still only 70% understand it, but the idea is interesting and appeals to my sensibilities. Kevin has a much gentler essay about it, and I’ve also had a go at using it to explain why addition would appear in formal arguments about ethics.
The main result of Knuth and Skilling, which gets recycled throughout their paper, is that addition is the unique (up to regrading – a caveat I shall ignore) way of assigning a value to a thing that is composed of a combination of other things, when there is also an ordering (one thing is bigger/better/has a higher number associated with it [whatever the interpretation], than another).
When something is made up of a combination of , , and , the value associated with the combination is the sum of the values of , , and on their own:
Here is a function that takes some kind of object (whatever , , and are) and returns a real number, and I have assumed , , and are disjoint (don’t worry about it). The main properties you need are associativity (it doesn’t matter in what order you combine things):
and order (if thing is bigger than thing , then thing “ combined with ” is bigger than thing “ combined with “):
So far, all this is very abstract. But applications can be super trivial. For example, if I have a bag with 3 apples, a bag with 10 apples, and a bag with 14 apples, the total number of apples is 3 + 10 + 14 = 27. These numbers are consistent with the underlying ordering (which collections of apples are bigger than others) and the associative symmetry of combining collections of apples. I could even have the same three bags of apples, and apply the sum rule to different concepts about them. For example, the sum rule also applies to the masses of the apples, and (usually) the prices of the apples.
The other day I was wondering whether this had unforeseen applications. I came up with a sports example that works — until it doesn’t. What if I want to select the best possible cricket team, and I have some numbers that describe how good each player is. Maybe something like this (I’m using arbitrary made-up quality numbers here):
We can combine the players to build a team. The combination operator (set union) is associative — combining Smith with Brewer and then adding Steyn gives the same result as combining Brewer and Steyn and then adding Smith. That’s one of the properties needed for the sum rule. The other is order, which seems plausible here. If one player is better than another, the combinations formed from the better player are better than the combinations formed from the worse player.
So we could measure the quality of the Smith-Brewer-Steyn team by doing addition, getting the result 95.6 + 1.9 + 90.4 = 187.9. The theory would work, for example, by showing that taking me out of a team and putting in virtually anyone else who’s even played cricket at high school level will result in a better team.
But the addition theory of team selection breaks down eventually. Consider, say, a team composed of 11 Glenn McGrath clones, each with a quality of 93. The total team quality would be 1023. He was among the best of all time at bowling, but was bad at batting. This team would also score about 80 runs per innings and lose most matches as a result. Why doesn’t the theory work? The associative property still holds. It must be order that breaks down. And it does.
Let be a partial cricket team composed of 8 McGrath clones. We could combine with yet another McGrath clone. Or we could combine it with an okay batsman with a quality of 50. The order property says that since McGrath is higher quality than the okay batsman,
it should always be better to select yet another McGrath clone than it is to select an okay batsman
This is false. Associativity but no order implies no sum rule. When you don’t have to worry about the balance of the team, the order property is more true, and so you’d have sum rule behaviour at the beginning of the selection process, but it breaks down as you select more and more players.