FOM: Goedel: truth and misinterpretations

[Torkel Franzen]

Vladimir Sazonov says:

  >It is 
  >greatest mistake to identify the formal statement Con(PA) 
  >with "Peano Arithmetic is consistent". But things like this 
  >are permanently happening here in FOM!

  Not only in FOM, but in mathematics. As you know, mathematicians in
general neither know nor care about the distinctions you emphasize,
but speak freely about the truth or falsity of the Riemann hypothesis,
the properties of algorithms, and so on, as do people in computer
science and other formal fields, without reference to any formal
theories. I would say that we have no grounds for thinking it is
possible to uphold in practice the distinction between mathematics and
"philosophy" that you urge.

  >It is completely unclear which way we could conclude (without 
  >making all the necessary distinctions!) that from some 
  >philosophical point of view there are arithmetical truths 
  >(IN WHICH SENSE, PLEASE?) which are unprovable (IN WHICH SENSE, 
  >PLEASE?). 

  Why, in the ordinary mathematical sense, of course. When we speak of
e.g. the possibility that Goldbach's conjecture is true but unprovable
in PA, we are speaking of the possibility that even if every even
number greater than two is the sum of two primes, there may be no
formal deduction in PA of the canonical formalization of this
statement.