On 07 Feb 2011, at 23:58, 1Z wrote:
On Feb 7, 6:29 pm, Bruno Marchal <[email protected]> wrote:
Peter,
Everything is fine. You should understand the reasoning by using only
the formal definition of "arithmetical realism",
You reasoning *cannot* be both valid and ontologically
neutral because it has ontological conclusions.
Wrong. It is enough it has ontological premise.
.which is that a
machine is arithmetical realist if she believes in the axiom of
elementary arithmetic *with* (the realist part) the principle of the
third excluded middle (allowing non constructive reasoning, as
usual).
What machine? Show me one!
See my papers. Read a book on logic and computability. Boolos and
Jeffrey, or Mendelson, or the Dover book by Martin Davis are excellent.
It is a traditional exercise to define those machine in arithmetic.
Recently Brent Meeker sent an excellent reference by Calude
illustrating how PA can prove the existence of universal machine (or
number). I will search it.
And I encourage you to interpret all this, including my thesis in
purely formal term. AUDA shows, notably, that this is possible.
You might also read the book by Judson Webb, which has been recently
republished and which shows the positive impact of Gödel on both
formalism and mechanism. Actually Webb argues that formalism and
mechanism are basically the same philosophy, or the same type of
philosophy. And I do follow him on that. A machine is before all a
form. A digital machine is a form which can be described locally
(relatively to a universal number) by a number. Webb call the kind of
AR used here: finitism.
And with AUDA you get a conversation with a machine, and a quasi
correct explanation why she is not a machine? How could a formalist
not love that ....
Gödel is not just the discovery of the provability limitations of
formalisms and machines,
Godel has no impact on "game playing" formalism.
?
(Well the more usual critic in our context is that Gödel has *only*
impact on "game playing" formalism).
I was just saying that Gödel's second incompleteness theorem is a
theorem in Peano arithmetic, about Peano arithmetic. Or by Peano
Arithmetic, about Peano arithmetic.
Bruno
http://iridia.ulb.ac.be/~marchal/
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