Re: Torkel Franzen on truth

On Nov 28, 11:31 pm, "LordBeotian" <pokips...@xxxxxxxx> wrote:
"george" <gree...@xxxxxxxxxx> ha scritto

we instead DO know it by mere
consideration of the intuitive not-formalized arguments 1 or 2 above.

No, really, you don't know much of ANything, especially
not anything infinitary, THAT way.

I desagree. The majority of mathematicians usually talk about infinitary
things and are convinced about their arguments without even know what ZFC
is

But you said "arguments". "Argument" implies a chain of REASONING, of
logical INFERENCE. In other words, there are some axioms and rules
of inference being invoked to support these beliefs.

Axioms and rules of inference are connected only to *formal* reasoning.
Informal reasoning don't imply any defined set of axioms or rules, even if it
can if course be formalized in many different formal deductive systems.

You should also agree that we definitely don't need a reasoning to be framed
in a set of axioms and rules to be convinced. Of course it can help, but we
are convinced (for example) of infinity of primes or Pythagorean theorem long
before we hear about PA, Q, ZFC or Hilbert's axioms for geometry.

(and also knowing nothing about first order formal systems).

It's not like religion. It works whether you believe in it or not.

I know. The point is that even if you could be able to formalize the
reasoning of mathematicians in ZFC, this formalization is not really relevant
for their work and their beliefs.

The discrepancy is that the formalized proofs of PA's consistency have
zero cogency while the intuitive informal arguments - so I am told -
have 100% cogency. In this sense the claim tha that the intuitive
argument is formalizable is misleading.
.

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  • Re: Torkel Franzen on truth
    ... Informal reasoning don't imply any defined set of axioms or rules, even if it can if course be formalized in many different formal deductive systems. ... Of course it can help, but we are convinced of infinity of primes or Pythagorean theorem long before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. ... The point is that even if you could be able to formalize the reasoning of mathematicians in ZFC, this formalization is not really relevant for their work and their beliefs. ...
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  • Re: Torkel Franzen on truth
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