### Friday, December 23, 2005

## Who's the little Caesar?

It seemed time for a non-vagueness related post. Take Neo-Fregeanism in the philosophy of maths to be roughly the following three claims:

1. Hume's Principle suffices for the derivation of the Peano/Dedekind Axioms of arithmetic in full second-order logic. Hume's Principle has some nice epistemological status (analyticity perhaps), and second-order consequence preserves that status, so it is inherited by arithmetic. This is not, of course, how we learn arithmetic, but it does give a reconstruction of the arithmetical knowledge we have that vindicates its status as a priori knowledge of a body of necessary truths.

(Hume's Principle: the number of Fs = the number of Gs iff the Fs are equinumerous with the Gs.)

2. All it takes to establish the existence of a class of objects, the natural numbers say, is to show that the apparent singular terms purporting to refer to object of that sort are genuine singular terms, and feature in sentences which we are warranted in taking to be true (I'm glossing over some details here).

3. Hume's Principle suffices to introduce 'natural number' as a sortal concept.

The much discussed Julius Caesar Problem, intially at least, calls into question thesis 3. Grasp of a sortal concept is associated with grasp of both a criterion of identity and a criterion of application; that is, criteria that allow a subject to decide when items of the given sort are identical or distinct, and to decide whether a given item is of that sort or not. Hume's Principle is custom built to introduce a criterion of identity for the concept 'natural number'; Frege's worry with it (put into this terminology) was that it did not offer a criterion of application, since it fails to decide the truth-values of sentences of the form 'the number of Fs = Julius Caesar'.

There is now a whole family of Caesar problems facing the Neo-Fregean, not all of them attacking thesis 3, and it's fair to say that noone has played a bigger part recently in developing members of that family and assessing their potency and generality than Fraser MacBride. That said, I'm struggling to see the force behind his most recent version, the Julio Cesar Problem (introduced in a paper of that title, dialectica vol.59, 2005). Take the original Caesar problem in its most general form to be that of answering a demand for reassurance that terms drawn from different theories or domains of discourse (number theory and history in Frege's example) refer to distinct kinds of things. The Julio Cesar problem is that of providing an assurance that terms drawn from different languages refer to the same kinds of thing.

Here's the problem in brief. It is essential to the Neo-Fregean's second thesis that we be able to identify the genuine singular terms independently of knowing which of the purporting singular terms make reference to objects (otherwise the argument endorsed in 2 is circular). A suitable refinement of Dummett's suggestion that such terms can be identified by various tests to distinguish the inferential patterns singular terms generate from that of expressions of other categories would do the job here, but these tests seem language-specific. How is the radical translator, to take the extreme case, to identify the singular terms, given that he seems in no better position to pick out the quantifiers with which they interact?

MacBride argues that the problem is a real challenge to Neo-Fregean platonism, but that there is a solution as long as the Neo-Fregean is willing to realise that Cartesian certainty is beyond our grasp; we may not be certain that we have correct picked out the singular terms of a language, but we can make hypotheses using the following principle (from p49 of Evan's The Varieties of Reference):

(P) If S is an atomic sentence in which the n-place predicate R is combined with n singular terms t1. . .tn then S is true ifft1. . . t he referent of tn> satisfies R.

Evans pointed out that (P) could be treated as a simultaneous implicit definition of reference and satisfaction in terms of truth, and so could be used to acquire defeasible evidence about how to pick out the various constituents of sentences even when we lack any independent characterisation of singular terms and predicates. MacBride's suggestion is that the Neo-Fregean invoke (P) as part of an answer to the Julio Cesar problem.

I'm just failing to see the Julio Cesar problem as any kind of deep challenge to Neo-Fregeanism. Suppose we use Dummett-style tests to confirm that the Arabic numerals are genuine singular terms in English, and we agree (though note that I haven't presented the argument for this) that thesis 1 above shows that we are warranted in taking various familiar sentences of number theory featuring such expressions to be true. By thesis 2, we have established that numbers exist. What exactly is the problem we're meant to have uncovered here? MacBride takes Crispin Wright to be offering an earlier discussion of the Julio Cesar problem under a different name in Frege's Conception of Numbers as Objects, and Wright expresses the following worry (p63):

'If the best which we can do is to explain the notion of singular term piecemeal, for different languages, this parochialism must surely also infect the notion of object which the Fregean proposes to see as correlative to it. How then can there be such a thing as â€˜International Platonismâ€™, so to speak? Must we not recognise that the claim that the natural numbers are objects permits of no general defence, that we must first specify whether they are being presented as English objects, or German objects, or Hindi objects...'

Whatever Wright's worry is, I think it's a much deeper problem than Julio Cesar problem. That we cannot identify the singular terms in some utterly foreign tongue in no way seems to undermine the argument given above that we are warranted in taking numbers to be objects. Now, it does mean that our epistemological reconstruction of our arithmetical knowledge is language relative, and that is very likely cause for discomfort; discomfort that MacBride's solution to the Julio Cesar problem might help ease. However, the issues raised by the Julio Cesar problem fall way short of Wright's Quinean worry that we're in danger of introducing a relativity into the ontological commitments of number theory, and it's hard to see how MacBride's solution helps us avoid this worry. Wright is motivated not by an unreasonable and unrealisable demand for Cartesian certainty, but by a desire to steer clear of this form of Quinean ontological relativity. So says I.

1. Hume's Principle suffices for the derivation of the Peano/Dedekind Axioms of arithmetic in full second-order logic. Hume's Principle has some nice epistemological status (analyticity perhaps), and second-order consequence preserves that status, so it is inherited by arithmetic. This is not, of course, how we learn arithmetic, but it does give a reconstruction of the arithmetical knowledge we have that vindicates its status as a priori knowledge of a body of necessary truths.

(Hume's Principle: the number of Fs = the number of Gs iff the Fs are equinumerous with the Gs.)

2. All it takes to establish the existence of a class of objects, the natural numbers say, is to show that the apparent singular terms purporting to refer to object of that sort are genuine singular terms, and feature in sentences which we are warranted in taking to be true (I'm glossing over some details here).

3. Hume's Principle suffices to introduce 'natural number' as a sortal concept.

The much discussed Julius Caesar Problem, intially at least, calls into question thesis 3. Grasp of a sortal concept is associated with grasp of both a criterion of identity and a criterion of application; that is, criteria that allow a subject to decide when items of the given sort are identical or distinct, and to decide whether a given item is of that sort or not. Hume's Principle is custom built to introduce a criterion of identity for the concept 'natural number'; Frege's worry with it (put into this terminology) was that it did not offer a criterion of application, since it fails to decide the truth-values of sentences of the form 'the number of Fs = Julius Caesar'.

There is now a whole family of Caesar problems facing the Neo-Fregean, not all of them attacking thesis 3, and it's fair to say that noone has played a bigger part recently in developing members of that family and assessing their potency and generality than Fraser MacBride. That said, I'm struggling to see the force behind his most recent version, the Julio Cesar Problem (introduced in a paper of that title, dialectica vol.59, 2005). Take the original Caesar problem in its most general form to be that of answering a demand for reassurance that terms drawn from different theories or domains of discourse (number theory and history in Frege's example) refer to distinct kinds of things. The Julio Cesar problem is that of providing an assurance that terms drawn from different languages refer to the same kinds of thing.

Here's the problem in brief. It is essential to the Neo-Fregean's second thesis that we be able to identify the genuine singular terms independently of knowing which of the purporting singular terms make reference to objects (otherwise the argument endorsed in 2 is circular). A suitable refinement of Dummett's suggestion that such terms can be identified by various tests to distinguish the inferential patterns singular terms generate from that of expressions of other categories would do the job here, but these tests seem language-specific. How is the radical translator, to take the extreme case, to identify the singular terms, given that he seems in no better position to pick out the quantifiers with which they interact?

MacBride argues that the problem is a real challenge to Neo-Fregean platonism, but that there is a solution as long as the Neo-Fregean is willing to realise that Cartesian certainty is beyond our grasp; we may not be certain that we have correct picked out the singular terms of a language, but we can make hypotheses using the following principle (from p49 of Evan's The Varieties of Reference):

(P) If S is an atomic sentence in which the n-place predicate R is combined with n singular terms t1. . .tn then S is true iff

Evans pointed out that (P) could be treated as a simultaneous implicit definition of reference and satisfaction in terms of truth, and so could be used to acquire defeasible evidence about how to pick out the various constituents of sentences even when we lack any independent characterisation of singular terms and predicates. MacBride's suggestion is that the Neo-Fregean invoke (P) as part of an answer to the Julio Cesar problem.

Whatever Wright's worry is, I think it's a much deeper problem than Julio Cesar problem. That we cannot identify the singular terms in some utterly foreign tongue in no way seems to undermine the argument given above that we are warranted in taking numbers to be objects. Now, it does mean that our epistemological reconstruction of our arithmetical knowledge is language relative, and that is very likely cause for discomfort; discomfort that MacBride's solution to the Julio Cesar problem might help ease. However, the issues raised by the Julio Cesar problem fall way short of Wright's Quinean worry that we're in danger of introducing a relativity into the ontological commitments of number theory, and it's hard to see how MacBride's solution helps us avoid this worry. Wright is motivated not by an unreasonable and unrealisable demand for Cartesian certainty, but by a desire to steer clear of this form of Quinean ontological relativity. So says I.

Labels: Philosophy of Language, Philosophy of Mathematics