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
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.
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.
So arithmetic assures the prior existence of entities capable of
counting.
Sure.
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. :_(
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? 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.
Bruno
http://iridia.ulb.ac.be/~marchal/
--
Onward!
Stephen
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