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.
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.
We can prove, in Peano arithmetic, that there exist entities capable
of counting. 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.
So arithmetic assures the prior existence of entities capable of
counting.
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.
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.
Bruno
http://iridia.ulb.ac.be/~marchal/
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