🚀 go-pugleaf

In article <1992Nov20.140159.4770@sun0.urz.uni-heidelberg.de>, gsmith@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
> In article <361@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
> 
> >This would be a valid argument if uncountable had an absolute definition.
> >I think uncountable is only meaningful relative to some formal system. 
> 
> "Uncountable" means no one-to-one relation with the integers
> can be given.  This does not refer to a formal system.

Unless you have a formal system in which to determine what constitutes
a one-to-one relationship you are not doing mathematics. You are doing
philosophy. All mathematical proofs are derivable from within a formal
system even if the proof is given informally. 

> >There are plenty of examples of sets uncountable in one system that
> >are countable in stronger systems.
> 
> If they are countable in the stronger system, that means that
> a one-one map can be found in it, which means that these sets
> are countable.

You believe that the real numbers have an absolute meaning. I do not.
That is a philosophical difference between us that we will not resolve
by argument. I do not think countable is meaningful except in the
context of a formal system. You do. Everyone is entitled to his own
opinion. You are not entitled to call your philosophical opinion mathematics
even if many mathematicians hold the same opinion. Many but not all
mathematicians agree with you, but most recognize that this is an
issue of philosophy of mathematics.

> >The question of whether uncountable has an absolute definition is a
> >philosophical one that I expect we have different opinions on.
> 
> I say "uncountable" means what mathematicians mean by it.  This is
> more a linguistic question than a philosophical one.

Mathematics is what is done in the context of formal systems or can
be formalized. Philosophy of mathematics involves the interpretation
and meaning of those systems. When you talk about uncountable as having
an absolute meaning you have gone beyond what can be formalized.
Mathematicians can usually reach a consensus if a proof is valid in
a particular formal system. They will *never* reach a consensus about
what is the correct philosophy of mathematics.

Paul Budnik