Benacerraf-style arguments that numbers couldn't be any particular collection of abstract entities (say, some particular set-theoretic construction) because there is a multitude of other constructions that could play the same role will fail if numbers play an explanatory role in the world. And one can imagine metaphysical views on which they play even a physical explanatory role. For instance, charge and mass play an explanatory role, indeed perhaps a causal one, in the world. But a Platonist could think that to have a charge or mass of x units (in the natural respective unit system) is to be charge- or mass-related to the number x. In other words, such determinables are relations, whose second relatum must be a number, and their determinates are cases of that relation for a fixed second relatum.
Now, one can still construct a relation to some set-theoretic isomorph of the numbers that structurally functions just like charge. For instance, if f is an isomorphism from the abstracta relata of charge to some abstract Ss, then we could say that a is related by charge* to y, where y is one of the Ss, precisely when a is related by charge to an x such that y=f(x). But there will be a matter of fact as to whether it is charge or charge* that explains the motion of particles. Surely they both don't—that would be a bogus case of overdetermination.
The point generalizes to other cases of Platonic entities that play an explanatory role—not necessarily a physical one—in the world. For instance, propositions might explain the co-contentfulness of sentences. An isomorph of the system of propositions could play some of the same roles for us, but it would not in fact explain the co-contentfulness of sentences. Compare this case. There is an isomorphism between legal US voters and some set of social security numbers. We can then construct a relation voting* between numbers and candidates such that n votes* for c if and only if the voter with social security number n votes for c. But while one could use facts about voting* to organize our information about elections, it is facts about voting—an action performed by persons, not social security numbers—that in fact explain election outomes.
That said, I think this approach will still tell against the standard set-theoretic constructions of numbers in two ways. First, it will tell against any particular construction. For how likely is it that this construction is the right one? Second, it will tell against anything like the set-theoretic constructions being the numbers. For it seems really unlikely that having a charge of three units is anything like a matter of being related in some way to the set {∅, {∅}, {∅, {∅}}}. So this approach is most plausible if numbers are some kind of sui generis entities.
But, on the other hand, the Benacerraf argument could apply against Platonic entities that play no explanatory role but are merely introduced for our convenience of expression. On some views, possible worlds are like that.
2 comments:
"There is an isomorphism between legal US voters and some set of social security numbers. We can then construct a relation voting* between numbers and candidates such that n votes* for c if and only if the voter with social security number n votes for c. But while one could use facts about voting* to organize our information about elections, it is facts about voting—an action performed by persons, not social security numbers—that in fact explain election outomes."
There are two simple alternative explanations on election results. The first one is that one of my coworkers always admonishes everyone on the team to "Vote early and vote often." The other explanation is that dead people are often very conscientious about their civic duty to vote. We can even add a third explanation and it has something to do with hanging chads . . .
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