[FOM] Indispensability of the natural numbers

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon May 17 19:28:14 EDT 2004

Timothy Y. Chow wrote:
> After some further reflection, I think I can state more precisely some 
> thoughts that I have partially expressed in my recent articles regarding
> the consistency of PA.  These ideas are a variation on an argument I heard 
> from Torkel Franzen 10+ years ago, that doubts about the natural numbers
> carry over to doubts about syntactic entities such as PA itself.

Yes, if you want to have a metatheory about syntactic entities.
Otherwise, as I wrote some time ago to FOM, there is no
visible problem at all. Mathematicians do not need to have
a metatheory. They just deduce their theorems (with the help
of some intuition, of course).

> If someone exhibits an explicit contradiction in PA, then I would argue
> that this indicates that there is something fundamentally incoherent about
> the mental concept of the natural numbers that I, along with presumably
> every sufficiently educated person, carry around in my head.

Whatever are our beliefs, the mental concept of the natural numbers
is something vague, anyway. It is only illusion of something solid.
Some variation of this mental concept will probably be quite sufficient
for new foundations if they will really appear and will be necessary
(*if* a contradiction will be found).

> An interesting property of the concept of the natural numbers is that it
> is rather "rigid"; that is, if you try to create a "small perturbation"  
> of the concept (e.g., by taking a nonstandard model of PA), you typically
> get something that is clearly *more* complicated than the natural numbers
> (e.g., nonstandard models of PA contain the naturals as an initial
> segment).  

Are you discussing some concepts of model theory which is usually
developed in the framework of ZFC, that is, in quite formal framework?
In this *formal* case natural numbers are really rather "rigid" as
is "rigid" any formal thinking.

If you mean anything else (platonistic?) then I have no idea how
to understand what are you talking about.

I would argue that this indicates that if our concept of the
> natural numbers is incoherent, then so is any halfway plausible infinitary
> substitute that we might propose.


> Next, I want to argue that if our concept of the natural numbers is 
> incoherent, then so is our concept of a formal system.  

See above.

Your further considerations (which I will not quote) are actually
about some AWFUL WORLD DISASTER if a contradiction would appear.
I do not agree with this. Mathematics will flourish as it flourished
the whole previous century after Russel's paradox. If, say, Bounded
Arithmetic (BA) would (even temporarily) survive after a contradiction
in PA, mathematicians will find a way how to reconsider in BA the most
important part of their work (say, that applicable in physics or
computer science). From your "catastrophic" point of view we will be
unable to use formal systems at all. I cannot accept your arguments.
The practical work with formal systems is quite manageable. School
children use formal rules (say, to add rational numbers) after some
training quite well. NO PROBLEM AT ALL! The problem would be only how
to change our intuition on natural numbers. The same problem on sets
was resolved quite efficiently (even if only temporarily) after
Russel's paradox. I see no essential difference.

Kind regards,

Vlkadimir Sazonov

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