FOM: numbers

[Neil Tennant]

Moshe' Machover's response to Martin Davis in connection with my proposed
adequacy condition on a theory of natural numbers was right on the mark.
The philosophical and technical interest (hence, one can assume?:) foundational
interest of the condition is that it prompts an investigation of how *modest*
a theory can meet it. Modesty can come from being *logically elementary* in
some suitable sense, and *ontologically parsimonious*.
Sure, one can opt for equivalence classes (or sets within them as their
'representatives') if one likes; but then one buys into all the set-theoretic
machinery, which (for natural numbers) is to take a hammer to a walnut.
By contrast, it seems to me that the absolutely minimal theory meeting the
proposed condition is one that involves grafting *just the numbers themselves*
onto one's existing ontology (if it does not already contain the numbers).
The numerals refer to numbers; and so do some of the expressions of the form
#xF(x) [the number of Fs] which denote a number n just in case there are
exactly n Fs. 
In my own investigation of what I called 'constructive logicism', I found that
the adequacy condition could be met by using a "schematic second order" logic
in which all proofs were both constructive and relevant. It is also a free
logic, cateriing for the possibility that a term of the form #xF(x) might be
non-denoting (because, say, there are infinitely many Fs and one has
constructivistic reservations about committing oneself to "the number of Fs"
in such a case).
That is not to say that one cannot *extend* the theory to take account of
infinite numbers if one wishes; it is only to say that one can deal with
*natural* numbers with a logic and a mathematical ontology falling way short
of what the modern mathematician, so well versed in ZF, takes for granted.
It is a salutary exercise to step back and examine just how much excess baggage
might be lurking in the Bourbaki bandwagon when it comes to natural numbers!
Neil Tennant