Parsons’s Mathematical Thought: Secs. 1-4

Right, as promised, time to make a start commenting on Charles Parsons’s long awaited Mathematical Thought and Its Objects (CUP, 2008).

For those who haven’t had a copy in their hands, this is a pretty substantial volume (pp. xx + 378). Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years; but glancing ahead the material indeed seems to be reworked into a continuous book. The nine chapters are divided into 55 sections numbered continuously through the book, and those divisions will be very handy here: I’ll aim to comment on small groups of sections (from one to three or four) at a time. From what I’ve seen so far, the book needs and repays slow reading.

Chapter 1 is entitled Objects and Logic. And the claim to be defended is that “Speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification to make serious statements”. Thus construed, the idea of objects in general is loosened from ties with the idea of actuality (Kant’s Wirklichkeit) — where this has something to do with “act[ing] on our senses or at least producing effects which may cause sense-perceptions as near or remote consequences” (to quote Frege). Talk of objects is also loosened from ties with ideas of intuitability (whatever that Kantian idea comes to: things are left pretty murky at this stage, but then Parsons is going to talk a lot about intuition later in the book). Consequently, endorsing the logical conception of an object will “defuse too-high expectations of what the existence of objects of some mathematical type such as numbers would entail.” The suggestion is that those who are inclined to deny abstract objects, or find them puzzling, are illegitimately(?) imposing requirements on being an object that go beyond those captured in the logical conception.

Now, I’m entirely sympathetic to the Fregean line Parsons is following here. He says that “its most important advocates in more recent times are Carnap and Quine”. But I would have added Dummett’s name to the list, starting with his early paper on nominalism: and Dummett initiated the most sophisticated development of the Fregean line in the hands of Crispin Wright in his Frege’s Conception of Numbers as Objects, and then particularly Bob Hale’s Abstract Objects (neither of which Parsons mentions here).

I’m not sure, though, in quite what spirit Parsons is proposing “the view that the most general notion of object has its home in formal logic”.

Actually, as an aside, I’d remark that that surely isn’t the happiest way of summing up the view. After all, suppose we translate back from first-order logical notation into a disciplined core fragment of English — the sort of regimented English whose sentences are equivalent to the content of the logical wffs (and indeed the sort of English which we use in giving determinate content to the artificial language in the first place). Then here too we will find the core devices of singular terms, predication, identity and quantification. And the Quinean will presumably say that our commitments to objects are revealed equally well by rendering our theory of the world into the idioms of this disciplined core of ordinary language. Or if that’s not exactly right, because we can never quite discipline English enough (e.g. we can’t quite ensure that “It is not the case that …” always expresses propositional negation), then this is not, so to speak, a deep failing of the vernacular. Formal languages don’t magically do what ordinary language can’t do: they just do ordinary things like use singular terms and quantify in tidier ways. So turning to “formal logic” doesn’t really give us a different take on the general notion of object. Surely Parsons spoke better when he expressed the position he is proposing as the view that “speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification” to make serious, and indeed true, statements.

But to continue, as I said, I’m not sure in quite what spirit this view is being advanced. The fully Fregean line would be to insist that objects are what are referred to by singular terms in true sentences, and a singular term is whatever walks, quacks, and swims like a singular term in a disciplined way. We can’t first pick out a class of genuine objects and then locate the genuine singular terms as those that refer to them: it goes the other way about (e.g. from identifying true sentences by the appropriate mathematical criteria, via identifying the singular terms in those sentences by their compositional behaviour, to insisting that those singular terms functioning in truths refer to mathematical objects).

But suppose you rejected that line. You might still think, in a Quinean spirit, that such is the mess and conversational plasticity in our various ordinary ways of talking that to determine when we are committed to objects of one kind or another, the best thing to do is to see how things look when we regiment our claims into a well-understood disciplined core discourse of singular terms, predication, identity and quantification — the apparatus formalized in first-order logic.

A couple of Parsons’s remarks suggest the stronger and more contentious Fregean line. But then it is perhaps odd that he doesn’t more explicitly argue for it, and engage with the Dummett/Wright/Hale defence.