Last time, I blogged about McFarlane's criticisms of Wright and Hale - here's what I think of their response, published in the same journal.
W+H say that the difference between stipulating the axioms of PA vs. stipulating Hume's principle is that the former stipulation fails their requirements by being "arrogant". So, what exactly is supposed to be arrogant about this stipulation:
A(x) x=x iff there is an object 0, and .... (the axioms of PA)
that doesn't also apply to all the instances of the schema below?
the Fs are equinumerous with the Gs iff the number of Fs = the number of Gs
I confess that I'm not sure exactly what W+H's answer to this, even after multiple readings of the paper. But here are some things I see in the paper that don't seem to work as explanations for why Hume's principle escapes arrogance:
1. Specification of truth conditions for all atomic sentences:
One criterion that comes up is that the latter collection of stipulations gives truth conditions to all atomic sentences in the language of numbers, in terms which someone who doesn't yet understand number talk can understand.
But then I don't see why the advocate of just stipulating the PA axioms couldn't break their stipulation down into a similar infinite series of stipulations, each of which is equated to a logical truth as follows and so on for all the countably many atomic sentences derivable from PA e.g.
A(x) x=x iff 0 is a number
A(x) x=x iff 1 is a sucessor of 0
You might worry that we can't intend to make any such stipulation, but note that both series of axioms will be recursively axiomatizable, (we aren't enumerating all the truths of arithmetic, just all the ATOMIC truths). So it's hard to see how we could be capable of intending all the instances of W+H's schema, but not the PA schema.
2. RHS can generate understanding of the new terms
Somehow W+H think that saying 'Let 'the numbers' name to some cannonical collection of objects which relate to each other in such a way that there's a "0 object" it stands in the successor relation to other objects etc' would not suffice to let someone who did not previously have the concept of number understand it, whereas hume's principle does.
But I don't buy the argument for this. All W+H say is that a) something else, namely the ramisfication of the PA stipulation, would have to be couched in second order logic and hence presuppose something like understanding of the numbers and b) the PA stipulation just adds to the ramsification by giving labels to the particular objects that stand in the relations stipulated by the ramsification.
They then seem to conclude that making the stipulation of the axioms of PA cannot suffice for understanding. This would be a plausible argument if they first showed that the ramsification doesn't suffice to express the concept of number, and then argued that the straight version doesn't add anything. But what they actually argued was that the ramsification would be incomprehensible to someone who didn't already have something more powerful than the concept of number. So, the fact that the straight stipulation of the PA axioms doesn't *add anything* to the ramsification, doesn't suffice to show that it couldn't be used to give someone understanding of the concept of number.
(which basically asserts that there are things satisfying the PA axioms using the language of second order logic).
W+H's argument seems to be exactly analogous to saying:
The stipulation 'a bachelor is a unmarried man' doesn't suffice to introduce someone to the concept of bachelorhood, because it doesn't add anything to 'a bachelor is an unmarried man who is either a happy bachelor or a sad bachelor' and that statement cannot be used to introduce someone to the concept of bachelorhood.
And this is surely not a good argument.
p.s. I don't think the notion of 'what it takes to gives someone a concept who doesn't previously have it' is well defined - like psycholgoically people can know related concepts, be susceptable to conditioning in various ways etc. Just giving some people one example of gouchery would suffice to make them understand the word, whereas you could say any number of things to a rock, and this wouldn't teach the rock the concept.