Points and Platonism

In this post I want to discuss the ontological status of points and its significance for semantic arguments for Platonism. No, not geometrical points; the points I’m talking about are the ones you gain or lose in playing certain kinds of games (including sports).

Suppose a Platonist argues as follows:

Let’s say that John has ten points in a game. Surely, then, these points must exist: Nothing is true of the non-existent, so if John really has ten points, there must be ten points which John has. Or consider the statement: “John is two points ahead of Fred”. This seems to assert a relation between the number of points John has and the number of points Fred has. If one thing stands in a relation to another they both must exist, so if “John is two points ahead of Fred” is true there must be points which each of them has. “And if one holds that points don’t exist,” the Platonist might add, “who should be the one to tell poor John and Fred that they both lose because they have no points?”

What is one to think of such reasoning? I have chosen this example because it resembles the sort of semantic arguments Platonists often use in other domains, in particular semantic arguments for properties and mathematical objects. In this case, however, the nature of the objects posited is particularly problematic. First, it seems difficult to conceive what these points would be: Besides trivially essential properties—properties such as being identical to itself, which are shared by everything that exists—and those such as being a point belonging to John, what properties would points have? This might not be so bad in itself, for there are more respectable abstract objects--sets, for example--which also seem to have no properties besides the kind mentioned above and those attributed to them by set theory. But one can ask more troubling questions. For example, if John loses a point and Fred gains one, is the point John lost the same point as the one Fred gained, or a different one? If questions of identity and difference make no sense for points, I think that is a good (though arguably not indefeasible) reason for rejecting such entities. But the plight of points is worse yet: Suppose John and Fred are playing a game where, so the rules prescribe, it is possible to have a negative number of points. We can imagine that they are playing a board game where you roll dice and move your piece the indicated number of spaces, and in which landing on a penalty box costs a player five points even when the number of points they have is less than five. The player who has the most points after twenty moves have been made wins. So, for example, if John has -25 points and Fred has -10 after twenty moves, Fred wins the game. Should the Platonist conclude from this example that, since points really exist, it is possible to have a negative number of something? Or should they say instead that, since there can’t be a negative number of anything, this game is somehow illegitimate?

As with other abstract objects, there are also epistemological problems with accepting the existence of points. If points are independently existing entities, how can we be sure we really gain or lose them in the manner the rules of the game prescribe? Could it be that, if the rules of the game say that move m is worth five points, one might only gain three points on executing move m because of some ontological glitch? For Humeans who reject necessary connections between distinct existences, it should appear suspicious that anything could guarantee that something we do causes us to stand in some contingent (or “external”) relation to independently existing abstracta.

If one finds the prospect of answering these bizarre questions distasteful, one will want to find some way of rejecting points while retaining the ability to make sense of our game-related talk and behavior. One might seek a nominalistic paraphrase of statements ostensibly about points, or perhaps try to give a Wittgensteinian account in which our talk of points is treated as moves in a language-game which is practically indispensable to our keeping score. Whatever one says here, I think it is clear that the semantic argument for points faces serious difficulties. If it can’t be made to work here, why should it fare any better with respect to other kinds of abstract objects? I wouldn’t say that other sorts of abstract objects can’t exist—on the contrary, I think some do. But if they do, it seems to me, we need far better arguments for believing in them.