[FOM] Remedial mathematics?

[Martin Davis]

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
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