FOM: Reply to Vorobey on Peano-Dedekind arithmetic

[Anatoly Vorobey]

You, Neil Tennant, wrote this on Mon, Oct 12, 1998 at 10:44:01AM -0400:

[about derivation of Peano-Dedekind axioms from simpler principles
by purely logical means]

> For a proof, I refer Vorobey to
> my book 'Anti-Realism and Logic', Clarendon Press, Oxford, 1987

Thank you for the reference. I will try to look at it later today.

> 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". 

What strikes you doesn't seem to strike me at all.

Your attitude towards the Peano-Dedekind arithmetic seems to be
rather demanding. On one hand, you claim that a disembodied soul in
an empty universe (nevermind the question of plausibility of this
gedankenexperiment) should formulate exactly the Peano-Dedekind
arithmetic. Mere presence of some physical objects, you say,
cannot therefore show that arithmetic incorrect. On the other hand, 
you are prepared, above, to equate correctness of this arithmetic (whatever
this means to you) with its being "the appropriate generalization
of our intuitive concepts of counting physical objects" - but *that*
concept definitely *is* dependent on "mere presence of physical
objects". 

To use a somewhat extreme example, consider a universe with
just two thousand basic objects. A mathematician somehow "living"
in such a universe would probably find counting objects
quite useful. It may be that such a mathematician will, in course
of his mathematical activity, arrive at the axioms of the
Peano-Dedekind arithmetic (after all, he's better off than
a disembodied soul in an empty universe). It well may be that he would find
them very interesting, and will proceed to study their
consequences in detail. I don't think that he would have
to consider the Peano-Dedekind arithmetic somehow "incorrect" - no
more than *we* would have to consider it incorrect upon
discovering that our universe is finite, or consider the Euclidian
geometry incorrect upon discovering general relativity. However,
it might be unwise for him to consider the Peano-Dedekind arithmetic the
appropriate generalization of his intuitive concepts of counting
physical objects (of which he only has two thousand). He might
prefer to use for *that* purpose a different axiomatic theory, one which
would recognize the obvious finiteness of his universe. It is
bizarre that *you* would consider such a choice on his part as
claiming the Peano-Dedekind arithmetic to be "incorrect", since
it is you who claims this arithmetic cannot be disproved by mere
presence of physical objects. 


> Moreover, the burden of explication of the notion of
> incorrectness is properly on the finitists or feasible
> arithmeticians. 

I am neither a finitist nor a feasible arithmetician. I am simply
concerned about what seems to me as your desire to use the big
scary word "incorrect" as a scarecrow against finitists/feasible
arithmeticians. They typically talk about a finite universe and
limitations on the concept of counting as a consequence, not
about "incorrectness" of Peano-Dedekind arithmetic (quoting Sazonov:
"Is the chess game correct or not?"); you are the one
who says they claim such incorrectness, so the burden is on
you to explicate this notion. 

> 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. 

I hope that the above may serve as such an explanation.

-- 
Anatoly Vorobey,
mellon at pobox.com http://pobox.com/~mellon/
"Angels can fly because they take themselves lightly" - G.K.Chesterton