The norms of mathematical discourse and inquiry

David Lewis once (influentially) commented that it would be ludicrous to expect mathematicians to change their ways on the basis of philosophical arguments that mathematical objects don’t exist. Why he thought mathematical practices would have to be emended in the light of ontological facts about the existence of mathematical objects, I’m not sure.

Here’s a thought experiment to make explicit your own implicit commitments about this. Imagine that, instead of a philosopher, an infallible oracle told the world that mathematical objects don’t exist. Would mathematics professors be obliged to hand in their resignations? Would their discipline have been exposed as a sham?

I think the answer to these questions is a, very obvious, “no”, and I suspect that almost everyone would agree. But notice what that means. If we don’t accept that mathematical practices ought to change in light of word from an infallible oracle that mathematical objects don’t exist, then we must also accept that the norms governing mathematical discourse are not representational, in the robust sense of that word as pertaining to mapping, tracking or picturing how things stand with a domain of mathematical objects. The standards of correctness and incorrectness in mathematics do not derive from mathematical objects, but from standards internal to the game (or perhaps “game”) of mathematics itself.

Call this view normative nominalism. But if one is committed to normative nominalism (as a “no” answer to the above questions would reveal), then what could possibly be the motivation for platonism?

Wright on Deflationism

I’m reading Crispin Wright’s Truth and Objectivity for a reading group at the moment, where he sums up an argument against deflationism about truth in the following way (I quote at length):

The deflationist holds that “true”, although gramatically a predicate, denotes no substantial quality of statements, or thoughts, but is merely a device of assertoric endorsement, of use to us only because we sometimes wish so to endorse a single statement, referred to in a way which doesn’t specify it’s content, or batches of statements all at once. Apart from applications of those two kinds, it is, for the deflationist, a complete explanation of the truth predicate that it satisfies the Disquotational Schema. It is a consequence of this general conception of the role of the truth predicate that it can register no norm governing assertoric discourse distinct from warranted assertibility. Yet the central place assigned to the Disquotational Schema—and thereby to the Negation Equivalence—actually clashes with that consequence, for it follows that, while normative of assertoric discourse, and indeed coincident in (positive prescriptive) normative force with warranted assertibility, “true” is nevertheless potentially extensionally divergent from warranted assertibility—and hence has to be accounted as registering a distinct such norm. Since it’s compliance or non-compliance with a norm distinct from assertoric warrant can hardly be an “insubstantial”property of a statement, and since a uniform account is possible of what it is for any particular statement so to comply, deflationism collapses. (pp. 71–2)

For reference, the Disquotational Schema is:

(DS) “P” is T is and only if P

One can’t, without engaging in a kind of doublethink, say or believe things like ‘P and it is not warrantedly assertible that P’ for some proposition P, since you can rationally assert P if and only if it is warrantedly assertible (for you) that P. But truth can be used to contrast with warranted assertibility. Take (DS) and substitute ‘It is not the case that P’ for P:

(i) ‘It is not the case that P’ is T is and only if it is not the case that P.

From (i) and (DS) one can infer:

(ii) It is not the case that P if and only if it is not the case that ‘P’ is T.

And from (i) and (ii) you get:

(iii) ‘It is not the case that P’ is T if and only if it is not the case that ‘P’ is T.

But (iii) cannot be right if T means warrantedly assertible, so ‘true’ registers a norm distinct from warranted assertibility. Deflationism is the view that ‘true’ just is a devise of assertoric endorsement, and Wright thinks that a mere devise of assertoric endorsement couldn’t register a norm distinct from warranted assertibility, so deflationism must be false.

But there is a way to register the truth norm without using the truth predicate. One can say ‘P and it is not warrantedly assertible for S that P’ or ‘¬P and it is warrantedly assertible for S that P’ where S is some person other than yourself, and in doing so can register the truth norm without using the truth predicate. What is required is that one contrasts one’s own perspective with that of another. Registering the truth norm requires some kind of I-Thou contrast. If that’s the case then ‘true’ is not what, fundamentally, allows one to register the truth norm contrasting with the norm of warranted assertibility, and deflationism is off the hook.

I think this might tap into something deep about objectivity—more specifically, our ability to see the world as being objective or to conceptualise there being objective facts that outstrip our ability to know them—as it coheres with something Robert Brandom says about objectivity. Brandom (I won’t spell out the details here) also argues that conceptualising objectivity requires I-Thou relationships. This is made explicit in paradigmatically referential of-statements like ‘He believes of this criminal that he is an innocent man.’ Understanding “of” requires navigating between one’s own perspective and that of another. Since “of” is how we refer objectively to the world, talking (and hence thinking) about the world objectively requires navigating between one’s own perspective and that of another.