Peter,
Everything is fine. You should understand the reasoning by using only
the formal definition of "arithmetical realism", 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).
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, it is also the discovery by the formalisms
and by the machines of their own limitations. And of the rich geometry
and topology of those limitations.
Bruno
On 07 Feb 2011, at 17:06, Bruno Marchal wrote:
On 06 Feb 2011, at 22:20, 1Z wrote:
On Feb 5, 7:43 pm, Bruno Marchal <[email protected]> wrote:
Computationalism needs Church thesis which needs AR (Arithmetical
Realism).
Nope, just AT (arithmetic truth).
Actually, comp needs only, for the ontology, the quite tiny complete
Sigma_1 truth.
As I have stated many times, it doesn;t matter in the least
how many or few immaterial objects you attribute existence to.
It's like saying pixies exist, but only a few
What?
It is always better to make a theory precise.
Please don't put metaphysics where there is only
religion
Believing in what is not proven is religion. I can
argue for anti realism.
I argue in favor of nothing. That's philosophy. You force me to be
explicit on this; I do science. I am a logician, and I show that
rational agent believing in comp believe that ... etc. I don't know
about the truth.
(saying yes to the admittedly betting doctor).
Saying yes to the doctor will not guarantee your
immaterial existence if there is no immaterial existence.
But there is immaterial existence. I recall you that I say in the
ontological context that something exist if Ex (bla-bla-bla x) is
true in the standard model of arithmetic. I use the standard meaning
of existence of numbers, etc.
AR/Platonism is a separate assumption to yes Dr.
I have drop out AR. You need AR (in which everyone believes except
the ultrafinitists and the bad faith philosophers) to understand the
term "digital" used by the doctor.
<And with comp,
it is math, indeed, even (full, above Sigma_1 arithmetic.
Arithmetical realism is what you need to apply the excluded middle
in
computer science and in arithmetic.
The excluded middle is a much of a formal rule as
anthing else. Formalists can apply it, so it is compatible
with anti realism.
The theory admits a formal study. You don't act like a formalist at
all. The term "Formalism" makes not an atom of sense without
arithmetical realism. In philosophy arithmetical realism is the
weaker of all possible realism, except again for the ultrafinitists.
If you are formalist and anti realist on the numbers you are in
contradiction, or, once and for all, just replace numbers by the
following formal expression 0, s(0), s(s(0)), etc. + the axioms I
just sent to Andrew, etc.
AUDA provides more than a formalism, it provides an arithmetisation,
which is a *weakening* of formalism, made possible by AR. Gödel
already exploited this.
To understand the fundamental
consequences of Church thesis you need to accept that some program
computes function despite we have no means to know if it is total or
partial, or that a program will stop or not.
And I can accept that by positing LEM as a formal rule. I don;t
have to posit an immaterial Plato's heaven
I make clear that the immaterial Plato Heaven for the machine is
just the truth of arithmletical proposition. It is an non
arithmetical notion, union of the entire Kleene-Mostowki
arithmetical hierarchy, and well know non controversal mathematical
object in the field of logic.
I use the "immaterial Plato heaven" terminology, either as poetical
shortcut, or as a point in the arithmetical representation of some
term in Plotinus theory.
I explained this already to you, but you keep adding metaphysical
stuff which don't exist.
Only ultrafinitist denies AR.
Wrong. Anti realists deny it. I have pointed this out many
times. You think the only debate is about the minimal
set of mathematical objects, and that is not the only debate. Anti
realists
can accept a maximal set of objects, with the proviso that their
existence is fictive and not real existence
I am agnostic on all notion of existence. All, except my own
consciousness here and now. I suggest a theory, and derive
consequences in that theory.
AR+, the idea that we don't need more than AR, in this setting, is a
consequence of the math. From 'outside' the tiny effective universal
sigma_1 complete set is enough. from inside, even mathematicalism is
not enough (it is more 'theologicalism').
The ontological status of
mathematical
objects is a area of contention in metaphysics, and not
straightforwardly
proven by mathematics itself.
With comp, you don't need more than the part on which almost
everybody
agrees: arithmetical realism.
Anti realists do not agee on the real existence of any
part. There are no pixies at all, not just a few pixies.
If you believe in prime numbers, and if you are patient and good
willing, I can explain that there are all universal numbers, and why
assuming comp that's enough and that's necessary to solve the white
rabbit problem. And that postulating physical laws miss the
epistemological existence of the qualia.
The basic ontology is not important. If you take less than a
universal system (like numbers, combinators, ...) you don't have
enough for comp, if you take more you miss the qualia.
No problem with a formalist interpretation of all this. Actually
S4Grz formalize at the meta-level what the machine can uderstand to
be non formalisable (like consciousness).
The engineers, the scientists, most
philosophers.
Except for Thorgny Tholerus I never met an ultrafinitist. You don't
have to decide if numbers are idea of the mind or sort of angel in
Plato Heaven. With comp the very idea of number will itself be a
number, a sort of second order number, relative to universes
(universal numbers).
ULtrafinitism has nothing to do with it. For formalists
no number exists. They have no prejudice about any kind,
If comp is false because, according to you 7 does not exist, then it
is your problem.
I told you that I am working in a theory, which is neutral on those
question. Formalist have normally less problem with a tiny sigma_1
arithmetic than with the real or complex numbers.
I just don't believe you don't believe in seven. You confuse
'immaterial' with 'inexistent'.
If you don't believe in seven, do you still believe in the formal
expression s(s(s(s(s(s(s(0))))))) ?
http://en.wikipedia.org/wiki/Philosophy_of_mathematics
And you cannot refute an argument by anticipating a refutation.
So if
you have a refutation of MGA you should present it.
See Colin Klein;s refutation of Maudlin's Olympia.
We have already discussed this and Colin Klein does not touch the
movie graph argument.
Then you had better stop saying the MGA and Olympia are equivalent
The movie graph *argument*, is not attacked by Klein. Klein attacks
the Olympia *argument*.
When I say that Olympia and MGA are equivalent, I am talking about
the conclusion, not about the arguments leading to the conclusion.
Bruno
http://iridia.ulb.ac.be/~marchal/
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