[FOM] Remedial mathematics?

[Walt Read]

To extend the analogy a little, the result of the work on non-standard
(non-Euclidean) models of geometry was a recognition of the other
models as equally valid. Eventually it was considered reasonable that
even ``the" universe might better be modeled by one of the
non-standard models. Do we see N the same way? Is there a unique thing
denoted by ``the natural numbers", accessible to us through intuition
or however, with the non-standard models of PA being essentially
artifacts of the formalization process? Or do the non-standard models
have equal claim as models of ``the natural numbers"? Might we at some
time in the future see one of the non-standard models as better suited
to our understanding as we learn more about natural numbers? The
latter view would make N something of a work-in-progress and would fit
well with the ``analogous (to some degree) with physical theory" view
of FOM. But it would seem to be a very different view than most people
have traditionally held. What is the position of classic FOM on this?

-Walt

On Wed, May 25, 2011 at 6:22 PM, Martin Davis <martin at eipye.com> wrote:
> On May 24, 2011 Andrei Rodin wrote:
>
>>In my sense in order to not "believe in N" one doesn't need to be a
>> sceptic.
>>Would you call a sceptic someone who did not believe in Euclid's 5th
>> Postulate
>>before the non-Euclidean geometries were well established? Actually the
>>brightest mathematicians didn't believe it and continued to work hard on
>> it. In
>>fact the discovery of the non-Euclidean geometries didn't require one to
>> throw
>>away any essential part of the earlier practice - albeit it required to
>> give up
>>certain philosophical ideas about the geometrical space. The situation with
>> N
>>may turn to be similar. I admit that for the moment this is a sheer
>> speculation
>>but I think it is important not only to be open for but also try to push
>> such
>>possible developments.
>
> Not sheer speculation at all. What happened with non-Euclidean geometry in
> the 19th century could be described in contemporary terms as follows: it was
> discovered that the axioms of Euclidean geometry with the parallel postulate
> omitted have non-standard models, that is, interpretations different from
> the "intended" interpretation, the "standard" that one had in mind in
> formulating the axioms. Well, something similar has been found about N, and
> precisely by means of f.o.m. research. PA has non-standard models, models
> other than N. This is the subject of a vast amount of research, and one can
> also ask about such models for a whole range of formal systems weaker or
> stronger than PA.
>
> Martin
>
>
>
>
> Martin Davis
> Professor Emeritus, Courant-NYU
> Visiting Scholar, UC Berkeley
> eipye + 1 = 0
>
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