*I could post a few more encores from my often rather rude blog posts about Murray and Rea’s *Introduction to the Philosophy of Religion*. But perhaps it would be better for our souls to to an altogether more serious book which I blogged about at length, Charles Parsons’ Mathematical Thought and Its Objects. I got a great deal from trying to think through my reactions to this dense book in 2008. But I often struggled to work out what was going on. Here, in summary, is where I got to in a series of posts about the book’s exploration of structuralism. I’m very sympathetic to structuralist ideas: but I found it difficult to pin down Parsons’s version. *

In his first chapter, Parsons defends a thin, logical, conception of ‘objects’ on which “Speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantiÞcation to make serious statements” (p. 10). His second chapter critically discusses eliminative structuralism. The third chapter presses objections against modal structuralism. But Parsons still finds himself wanting to say that “something close to the structuralist view is true” (p. 42), and he now moves on characterize his own preferred noneliminative version. We’ll concentrate on the view as applied to arithmetic.

Parsons makes two key initial points. (1) Unlike the eliminative structuralist, the noneliminativist “take[s] the language of mathematics at face value” (p. 100). So arithmetic is indeed about numbers as objects. What characterizes the position as structuralist is that we don’t “take more as objectively determined about the objects about which it speaks than [the relevant mathematical] language itself specifies” (p. 100). (2) Then there is “the aspect of the structuralist view stressed by Bernays, that existence for mathematical objects is in the context of a background structure” (p. 101). Further, structures aren’t themselves objects, and “[the noneliminativist] structuralist account of a particular kind of mathematical object does not view statements about that kind of object as about structures at all”.

But note, thus far there’s nothing in (1) and (2) that the neo-Fregean Platonist (for example) need dissent from. The neo-Fregean can agree e.g. that numbers only have numerical intrinsic properties (pace Frege himself, even raising the Julius Caesar problem is a kind of mistake). Moreover, he can insist that individual numbers don’t come (so to speak) separately, one at a time, but come all together forming an intrinsically order structured — so in, identifying the number 42 as such, we necessarily give its position in relation to other numbers.

So what *more* is Parsons saying about numbers that distinguishes his position from the neo-Fregean? Well, he in fact explicitly compares his favoured structuralism with the view that the natural numbers are *sui generis* in the sort of way that the neo-Fregean holds. He writes

One further step that the structuralist view takes is to reject the demand for any further story about what objects the natural numbers are [or are not]. (p. 101)

The picture seems to be that the neo-Fregean offers a “further story” at least in negatively insisting that numbers are sui generis, while the structuralist refuses to give such a story. As Parsons puts it elsewhere

If what the numbers are is determined only by the structure of numbers, it should not be part of the nature of numbers that none of them is identical to an object given independently.

But of course, neo-Fregeans like Hale and Wright won’t agree that their rejection of cross-type identities is somehow an optional extra: they offer arguments which — successfully or otherwise — aim to block the Julius Caesar problem and reveal certain questions about cross-type identifications as ruled out by our very grasp of the content of number talk. So from this neo-Fregean perspective, we can’t just wish into existence a coherent structuralist position that both (a) construes our arithmetical talk at face value, as referring to numbers as genuine objects, yet also (b) insists that the possibility of cross-type identifications is left open, because — so this neo-Fregean story goes — a properly worked out version of (a), together with reflection on the ways that genuine objects are identified under sortals, implies that we can’t allow (b).

Now, on the sui generis view about numbers, claims identifying numbers with sets will be ruled out as plain false. Or perhaps it is even worse, and such claims fail to make the grade for being either true or false (though it is, of course, notoriously difficult to sustain a stable, well-motivated, distinction between the neither-true-nor-false and the plain false — so let’s not dwell on this). Conversely, assuming that numbers are objects, if claims identifying them with sets and the like are false or worse, then numbers are sui generis. So it seems that if Parsons is going to say that numbers are objects but are *not* sui generis, he must allow space for saying that claims identifying numbers with sets (or if not sets, at least some other objects) are true. But then Parsons is faced with the familiar Benacerraf multiple-candidates problem (if not for sets, then presumably an analogous problem for other candidate objects, whatever they are: let’s keep things simple by running the argument in the familiar set-theoretic setting). How do we choose e.g. between saying that the finite von Neumann ordinals are the natural numbers and saying that the finite Zermelo ordinals are?

It seems arbitrary to plump for either choice. Rejecting both together (and other choices, on similar grounds) just takes us back to the sui generis view — or even to Benacerraf’s preferred view that numbers aren’t objects at all. So that, it seems, leaves just one position open to Parsons, namely to embrace *both* choices, and to avoid the apparently inevitable absurdity that \(\{\emptyset,\{\emptyset\}\}\) is identical to \(\{\{\emptyset\}\}\) (because both are identical to 2) by going contextual. It’s only in one context that ‘\(2 = \{\emptyset,\{\emptyset\}\}\)’ is true; and only in another that ‘\(2 = \{\{\emptyset\}\}\)’ is true.

And this does seem to be the line Parsons seems inclined to take: “The view we have defended implies that [numbers] are not definite objects, in that the reference of terms such as ‘the natural number two’ is not invariant over all contexts” (p. 106). But how are we to understand that? Is it supposed to be rather like the case where, when Brummidge United is salient, ‘the goal keeper’ refers to Joe Doe, but when Smoketown City is salient, ‘the goal keeper’ refers to Richard Roe? So when the von Neumann ordinals are salient, ‘2’ refers to \(\{\emptyset,\{\emptyset\}\}\) and the Zermelo ordinals are salient, ‘2’ refers to \(\{\{\emptyset\}\}\)? But then, to pursue the analogy, while ‘the goal keeper’ is indeed sometimes used to talk about now this particular role-filler and now that one, the designator is apparently also sometimes used more abstractly to talk about the role itself — as when we say that only the goal keeper is allowed to handle the ball. Likewise, even if we do grant that ‘2’ sometimes refers to role-fillers, it seems that sometimes it is used to talk more abstractly about the role — perhaps as when we say, when no particular \(\omega\)-sequence of sets is salient, that 2 is the successor of the successor of zero. Well, is this the way Parsons is inclined to go, i.e. towards a structuralism developed in terms of a metaphysics of roles and role-fillers?

Well, Parsons does explicitly talk of “the conclusion that natural numbers are in the end roles rather than objects with a definite identity” (p. 105). *But why aren’t roles objects after all, in his official thin ‘logical’ sense of object?* — for we can use “the linguistic devices of singular terms, predication, identity and quantification to make serious statements” about roles (and yes, we surely can make claims about identity and non-identity: the goal keeper is not the striker). True, roles are as Parsons might say, “thin” or “impoverished” objects whose intrinsic properties are determined by their place in a structure. But note, Parsons’s official view about objects didn’t require any sort of ‘thickness’: indeed, he is “most concerned to reject the idea that we don’t have genuine reference to objects if the ‘objects’ are impoverished in the way in which elements of mathematical structures appear to be” (p. 107). And being merely ‘thin’ objects, roles themselves (e.g. numbers) can’t be the same things as ‘thick’ role-fillers. So now, after all, numbers qua number-roles do look to be sui generis entities with their own identity — objects, in the broad logical sense, which are not to be identified with any role-filler — in other words, just the kind of thing that Parsons seems *not* to want to be committed to.

The situation is further complicated when Parsons briefly discusses Dedekind abstraction, though similar issues arise. To explain: suppose we have a variety of ‘concrete’ structures, whether physically realized or realized in the universe of sets, that satisfy the conditions for being a simply infinite system. Then Dedekind’s idea is that we ‘abstract’ from these a *further* structure \(\langle N, 0, S\rangle\) which is — so to speak — a ‘bare’ simply infinite system without other inessential physical or set-theoretic features, and it is elements of *this *system which are the numbers themselves. (Erich H. Reck nicely puts it like this: “[W]hat is the system of natural numbers now? It is that simple infinity whose objects only have arithmetic properties, not any of the additional, ‘foreign’ properties objects in other simple infinities have.”) Since the bare structure is all that is generated by the Dedekind abstraction, “it conforms to the basic structuralist intuition in that the number terms introduced do not give us more than the structure” (p. 105), to borrow Parsons’s words. But, he continues,

This procedure gets its force from the use of a typed language. Thus, the question arises what is to prevent us from later, for some specific purpose, speaking of numbers in a first-order language and even affirming identities of numbers and objects given otherwise.

To which the answer surely is that, to repeat, on the Dedekind abstraction view, the ‘thin’ numbers determinately do not have intrinsic properties other than those given in the abstraction procedure which introduces them: so, by assumption, they are determinately distinct from any ‘thicker’ object with such further properties. Why not?

So now I’m puzzled. For Parsons, does ‘the natural number two’ (i) have a fixed reference to a sui-generis ‘thin’ role-object (or Dedekind abstraction, if that’s different), or (ii) have a contextually shifting reference to a role-filler, or (iii) both? The latter is perhaps the most charitable reading. But it would have helped a lot if Parsons had much more explicitly related his position to an articulated metaphysics of role/role-filler structuralism. Elsewhere, he writes that “the metaphysical tradition is likely to be misleading as a source of ideas about the objects of modern mathematics”. Maybe that’s right. But then it is all the more important to be absolutely clear and explicit about what new view is being proposed. And here, I fear, Parsons’s writing falls short of that.

*Or so I thought now over seven years ago. I haven’t re-read Parsons’s text since. I would be very interested to get any comments from readers who worked their way to some **clearer understanding of his position. *