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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