Parsons’s Mathematical Thought: Sec. 7

Back in Sec. 1, Parsons says “Roughly speaking, an object is abstract if it is not located in space and time and does not stand in causal relations.” In the last section of the first chapter, he returns to question of characterizing abstract objects, and suggests a distinction among them between pure abstract objects (e.g. pure sets) and those which “have an intrinsic relation to the concrete” — Parsons calls the latter quasi-concrete.

As a paradigm example of the quasi-concrete, Parsons takes the example of sentence types: “what a sentence [type] is is a matter of what physical inscriptions are or would be its tokens”. (Actually, just as an aside, I suppose we might wonder whether sentence types might be a counter-example to the claim that abstract objects lack temporal location. We might ask: did the sentence type “the cat is on the mat” really exist in 2000 BC before anyone spoke English?)

But how should we generalize from this case? Parsons writes “What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete ‘representation'”. The scare quotes are there in Parsons — and you can see why. Should we really say, for example, that a sentence token is a representation of its type? Your first response might be: the token isn’t about the type, so isn’t a representation of it. But, reading on, it becomes clear that Parsons doesn’t mean representation but representative. And then, yes, we might say that the token is a representative of the type. Parsons also writes “Although sets in general are not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership.” (no scare quotes!). Again, we might say the spoon in my coffee cup is a representative of the set of cutlery (though not a representation).

How clear is the idea of “having a concrete representative”? You might have supposed that the Earth’s equator is a candidate for belonging with sentence types as tangled with the concrete. But does the equator have a concrete representative? Could it? What about that old Fregean example, the direction of a line. Of course there can be physical lines with that direction; but it doesn’t seem quite natural to me to say a particular line is a representative of the direction. (We might say the equator or a direction could have a representation, painted on the ground!)

Parsons’s discussion here thus seems to me to be rather undercooked. To be sure, it is plausible to say that some abstract objects are more purely abstract than others, but I don’t think he has given a sharp characterization of the phenomenon.

But let’s go, for the moment, with his notion of the quasi-concrete. Then he raises the question, are numbers quasi-concrete? We might be tempted to say yes, suggesting that the number five, for example, has the concrete representatives like: ||||| . Parsons makes two Fregean points against this. First, to take that block as representative, we have already to take it as a set or sequence of strokes (rather than as a single grid, for example). So the representative here is not strictly concrete but itself quasi-concrete. Perhaps then we can say that numbers are quasi-quasi-concrete (meaning they have quasi-concrete representatives). But second, that can’t be the whole story, as numbers can number anything, including the purely abstract. (Parsons says he is going to return to talk about this in Chapter 6, so I’ll say no more for the moment.)