[FOM] Baby arithmetic

Peter Smith peter.smith at phil.cam.ac.uk
Thu Sep 12 05:19:13 EDT 2002

```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.
--
_________________________________________________________________________
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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