FOM: Reply to Vorobey on Peano-Dedekind arithmetic

[Neil Tennant]

In an earlier posting I raised the question

"What theory of arithmetic would be excogitable by a disembodied Cartesian
soul in a universe with no physical objects in it at all?"

and claimed that the answer is: exactly Peano-Dedekind arithmetic. 

Anatoly Vorobey then asked me to "focus on that and explain why that
answer seems so obvious to [me]."

I didn't claim that the answer was *obvious*. But it *is* forced upon
one by purely logical considerations. For a proof, I refer Vorobey to
my book 'Anti-Realism and Logic', Clarendon Press, Oxford, 1987, in
which I gave detailed derivations of the the Peano-Dedekind axioms for
successor arithmetic from some simple principles that I claimed were
meaning-determining for 0, s( ), #x(...x...) and N( ). Had the answer
been "obvious", I would not have bothered to give all the
derivations. (Indeed, nor would Frege have bothered, on his own
approach---unhappily based on an inconsistent theory of classes, but
guided by fruitful insights nonetheless, as has since been revealed
through the work of various neo-Fregean logicists.)

The derivation of Peano-Dedekind (successor) arithmetic in my book
differs from those of other neo-Fregean logicists such as Crispin
Wright ('Frege's Conception of Numbers as Logical Objects', Aberdeen
University Press, 1983) because it does not make use of Hume's
Principle (i.e. that #xFx=#xGx iff there is a 1-1 correspondence
between the F's and the G's). I call my own theory 'constructive
logicism', because it avoids the ontological commitment that is
induced by Hume's Principle to cardinal numbers for every F, no matter
how large the extension of F might be.  My own theory generates
commitment only to the natural numbers. Moreover, it is developed
using only intuitionistic relevant logic.

To my earlier question "Could someone please explain how the mere
presence of some physical objects could show that Peano-Dedekind
arithmetic is somehow incorrect?", Vorobey replies, somewhat
disingenuously to my mind,

> I think you are attacking a strawman here. Noone said presence
> of physical objects could show the Peano-Dedekind incorrect (and
> what could "incorrect" mean in this context?); only
> that Peano-Dedekind arithmetic, useful as it is in itself,
> may not be the appropriate generalization of our intuitive concepts
> of counting physical objects.

Well, it strikes me that this last characterization of the situation
("Peano-Dedekind arithmetic ... may not be the *appropriate
generalization of our intuitive concepts of counting physical
objects") is just another way of saying "Peano-Dedekind arithmetic is
incorrect". Moreover, the burden of explication of the notion of
incorrectness is properly on the finitists or feasible
arithmeticians. After all, it is they who are concerned to deny the
universal applicability of ordinary arithmetic in the light of certain
(altogether rum) considerations of fickle physicality and physical
finitude.

If Vorobey's way of putting matters does not amount to the momentous
claim that Peano-Dedekind arithmetic is incorrect, then we are owed an
explanation as to why this is so. And if that explanation is
convincing, then clearly there is no debate to be had; we shall all
have been talking right past each other.

Neil Tennant