On 08 Nov 2012, at 01:38, Stephen P. King wrote:
On 11/7/2012 12:44 PM, Bruno Marchal wrote:
On 07 Nov 2012, at 17:13, Stephen P. King wrote:
On 11/7/2012 9:41 AM, Bruno Marchal wrote:
Arithmetic explains why they are observers and how and why they
make theories.
Dear Bruno,
This is a vacuous statement, IMHO. Absent the prior existence of
entities capable of counting there is no such thing as Arithmetic.
Your belief to the contrary cannot be falsified!
In the same sense that "2+2=4" cannot be falsified.
Dear Bruno,
You are missing my message. "2+2=4" is universally true only
because there does not exist a counter-example that can be agreed
upon by 3 or more entities. You continue to only think from the
point of view of a single entity. My reasoning asks questions from
the point of view of many entities
But even with that criteria, your point will not go through, as it is
obvious that nobody will find any counterexample to the fact that
observer exists in arithmetic (assuming comp). In the same "sense"
that 2+2=4.
It just mean that you have never grasp the proof of Gödel's
incompleteness theorem, which does that, almost in passing. This is
proved in all good introduction text to logic.
Ha! OK, but you are wrong.
If you stiudy and grasp by yourself Gödel's prrof, you should see why
the prior existence of comp-counting entities is a theorem of
arithmetic, and have the same reality as 2+2=4. You would not have the
remark above.
We can prove, in Peano arithmetic, that there exist entities
capable of counting.
If we follow the orthodox interpretation of Tennenbaum theorem
there can only be one entity that can count; the standard model of
PA. I am thinking of the uncountable infinity of non-standard models
of PA that have a condition imposed on them: each thinks that it
alone is the standard model as it cannot know that it is non-
standard. In this way we can get an infinity of standard models of
PA instead of just one. I think that this gives us a modal logical
form of general relativity!
By Gödel's completeness those entities exists in the standard model
of arithmetic (arithmetical truth), and in all model, and thus
also in all non standard model.
Yes, but each model must be able to truthfully believe that it
alone is the standard model; it cannot see the constant that defines
it as non-standard from inside itself.
It is confusing to use models for modeling thinking entities.
So arithmetic assures the prior existence of entities capable of
counting.
Sure.
Ah? But it is a key point that you seem to avoid in many of your post.
You have the Matiyasevich's book. A (more complex) proof is given
there. It is more complex as it proves this for a much more tiny
fragment of arithmetic.
It is impossible for me to write my ideas in Matiyasevich's
language. :_(
I am not sure why. I have taught computer science to disabled people,
and unless severe disability, it is always possible to overcome any
technical limitations.
Forgive me, but I need to let others explain my argument as I have
run out of patience with my inability to form sentences that you
will understand. This article (which I cannot asses completely due
to the paywall) seems to make my claim well: http://www.springerlink.com/content/052422q295335527/
Nomic Universals and Particular Causal Relations: Which are Basic
and Which are Derived?
"Armstrong holds that a law of nature is a certain sort of
structural universal which, in turn, fixes causal relations
between particular states of affairs. His claim that these nomic
structural universals explain causal relations commits him to
saying that such universals are irreducible, not supervenient upon
the particular causal relations they fix. However, Armstrong also
wants to avoid Plato’s view that a universal can exist without
being instantiated, a view which he regards as incompatible with
naturalism. This construal of naturalism forces Armstrong to say
that universals are abstractions from a certain class of
particulars; they are abstractions from first-order states of
affairs, to be more precise. It is here argued that these two
tendencies in Armstrong cannot be reconciled: To say that
universals are abstractions from first-order states of affairs is
not compatible with saying that universals fix causal relations
between particulars. Causal relations are themselves states of
affairs of a sort, and Armstrong’s claim that a law is a kind of
structural universal is best understood as the view that any given
law logically supervenes on its corresponding causal relations.
The result is an inconsistency, Armstrong having to say that laws
do not supervene on particular causal relations while also being
committed to the view that they do so supervene. The inconsistency
is perhaps best resolved by denying that universals are
abstractions from states of affairs."
I don't assume nature. And comp refutes naturalism, that's the point.
You are thinking too literally about that I am writing. comp
assumes Platonism, no?
No. Just arithmetical realism. Platonism is a consequence of comp.
Comp itself is neutral. It is just the belief that our brain can be
replaced by a computer (so we assume a Turing complete physical
reality, but tis agnostic on the precise nature of it (primitive or
not).
Why can we not see Platonism and Naturalism as just opposite or
"polar" views on Reality? Platonism looks Top-down and Naturalism
looks from the bottom-up.
Because it violates the facts derived from comp. Nature or the
physical world is only a surface of a "non natural" (but arithmetical)
volume.
Bruno
http://iridia.ulb.ac.be/~marchal/
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