Thursday, July 11, 2002

The Volokh brothers complain about the familiar idiom about comparing apples and oranges. But Sasha misstated (and then corrected) the definition of a well-ordered set. Here is a correct definition. The reals are ordered, but not well-ordered.

To an economist, all commodities can be compared by looking at their values in the marketplace. To a mathematician, any two elements of a set can be compared by using an ordering on the set. Comparing apples and oranges is ambiguous unless the ordering is specified. The people who say you just cannot compare apples and oranges are plainly wrong. But neither of the people the Volokhs are criticizing were saying that.

Being a mathematician, I side with the mathematician's view. If someone says that comparing vegetarian and meat-eating diets is like comparing apples and oranges, then that means that any comparison depends on which ordering is in use and there is no canonical ordering. Being a vegetarian often comes with lifestyle choices and moral worldviews that are not shared by most meat-eaters. Their belief that vegetarianism is superior is based on an ordering that the meat-eaters would not choose. So the comparison is not just a matter of objectively determining which diet is healthiest.

So I think that apples and oranges can be compared, but still it often makes sense to say that a comparison is like comparing apples and oranges. In mathematical terms, it means that the ordering is not defined. Or to an economist, it means that it depends on the utility function.

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