Monday, January 16, 2006
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Philosophy. Straight. With ice and lemonade, a cherry and a slice of lemon.
When I Am Not Being A Philosopher
I play and write folk music. Click 'play' to hear a bit.
You can use the 'forward' and 'back' buttons to move between songs.
I play and write folk music. Click 'play' to hear a bit.
You can use the 'forward' and 'back' buttons to move between songs.
2 comments:
Very nice paper Carrie, clearly-written and interesting. Of course, I have a couple of misgivings. Firstly, a very small point about the way you've expressed yourself. On page 9 you write:
'In the case of Russell's paradox, what was intended is better captured by the axioms generated by the iterative conception of set.'
The talk of axioms generated by the IC is a little awkward, since IC doesn't obviously generate (on the only reading that naturally suggests itself to me) all of the axioms of ZFC; not choice, replacement or extensionality, and not infinity without some possibly suspect assumptions about the heirarchy (Boolos makes those assumptions; Roy Cook resists making then in 'Iteration One More Time' (65-6)).
Of course, I'm not suggesting you get remotely involved in that controversy. Quite the opposite; I think it might be prudent to pick a phrase that doesn't, even potentially, carry any implications for that debate. (For example: 'what was intended is better captured by the axioms of standard iterative set theory'.)
More generally, there's some awkwardness in the comparison between Fitch and Russell. The idea that we've mistaken a wolf-in-sheep's-clothing claim for a codification of a thought we wanted to capture might be plausible in the case you're interested in, but with the analogy you seem to be making the claim that the following is a diagnosis of why one might be surprised or 'baffled' by the Russell paradox (first full paragraph of p9); we've mistaken naive comprehension for a codification of the thought underlying the axioms of standard iterative set theory. But this doesn't look all that plausible as a diagnosis. Naive comprehension looks like it captures exactly the (sadly mistaken) thought that every property determines a set; the suggestion that this wasn't the the claim early set-theorists intended to make looks (at best) overly charitable. And the axioms of iterative set theory don't try to capture an innocent thought about sets mis-expressed in naive comprehension, but rather offer a radically different conception of sets - one which proponents of IC claim is both consistent and non-arbitrary (in a way that, for example, a limitation of size characterisation of the sets is not). That's quite a different project from the one your diagnosis suggests proponents of IC are engaged in.
Perhaps a better (though obviously not uncontentious) comparison would be to the Sorites paradox. The inductive premise for a particular predicate can seem like nothing other than a statement of the vagueness of the expression in question, but then it seems such expressions inherently give rise to paradox. Notions like Patrick's epistemtic tolerance, Heck's quasi-tolerance, etc., are attempts to characterise the claim we really wanted to make in calling a term vague, and which don't support Sorites reasoning.
Thanks for that Aidan. Thanks also to those who've emailed me about this paper. I'm now rewriting and the final version will be much improved by these comments!
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