[FOM] Baby arithmetic

[Peter Smith]

I surely can't be alone in this! Towards the end of my first-level 
logic course [for philosophers], I've introduced first-order 
numerical quantifiers (E_1x), (E_2x) ... in the usual way; then we 
show that the likes of

[(E_2x)Ax & (E_3x)Bx & -(Ex)(Ax & Bx)] --> (E_5x)(Ax v Bx)

is a first order theorem, and I armwavingly say "That kinda says two 
things and another three things make five things -- or as they put it 
in the kindergarten, two and three makes five.  So logic here seems 
to touch baby arithmetic. Come back in the third year to my Logic and 
Arithmetic course to find out more."

Then a couple of years later, the kids come back, and off I go: 
here's first/second order Peano arithmetic, etc. etc. (and later we 
talk about Fregean logicism and neo-logicism, etc. etc.) But I 
confess I never really join things up -- i.e. I don't really discuss 
how much  baby arithmetic can be treated as kinda baby logic in some 
sort of disguise, or how best to do this, and the limits of this sort 
of construction. I know of Bostock's book from way back of course, 
but that is set in an idiosyncratic framework. Can anyone suggest 
good/useful references to explore?

Peter S.
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Dr Peter Smith
DoS in Philosophy and HPS
Jesus College
Cambridge CB5 8BL, UK
http://www.phil.cam.ac.uk/Smith
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