FOM: Evolution and reason; math and the brain

[Neil Tennant]

Joe Shipman asked "do S and T imply PA?"

Origin of S:
	
	There is a system S of logic such that:
	whatever the laws of physics, chemistry etc. might be in any
	possible world in which reasoning creatures could evolve, their
	reasoning would have to obey the laws of S.

Origin of T:

	There is a system T of theorizing about numbers such that:
	whatever the laws of physics, chemistry etc. might be in any
	possible world in which reasoning creatures could evolve, their
	theorizing would have to be consistent with T (i.e.
	extendible so as to include T, upon suitable translation).

The best answer I can give to Joe's question is "I sure hope so!".
The difficulty would lie in giving a logicist derivation of the recursion equations for multiplication.  I think the recursion equations for addition should follow from the principles

	If there are exactly n Fs and exactly m Gs and nothing is both F and G
	then there are exactly (n+m) (F or G)s

	There are exactly n Fs iff the number of Fs = n


Neil Tennant