[FOM] Remedial mathematics?
martin at eipye.com
Wed May 25 21:22:38 EDT 2011
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
>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
>but I think it is important not only to be open for but also try to push such
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.
Professor Emeritus, Courant-NYU
Visiting Scholar, UC Berkeley
eipye + 1 = 0
More information about the FOM