On 17 Jan 2012, at 19:47, John Clark wrote:

On Tue, Jan 17, 2012  Bruno Marchal <marc...@ulb.ac.be> wrote:

"You don't need to assume them. They already exists at the natural numbers' epistemological level."

Then in addition to the natural numbers the non computable numbers are fundamental too.

But not primitive.
And are non computable real numbers fundamental? None occur in any theory. Only Omega and Post occurs in computability theory, and are not really constant but depend on the choice of a universal number. In physics and math all real constant seems to be gentle and computable (albeit often transcendent) like PI, e, gamma, etc.

"Just rememeber that when I use the term "number", I mean a natural number."

I have remembered and that's why I have a problem.

Are you aware that then natural numbers + addition + multiplication + a bit of logic is already Turing universal? I can use LISP programs if you prefer, or lambda expression, with abstraction and reduction. I use natural numbers becomes laymen meets them more often than LISP programs, or lambda abstraction.

"Together with the laws of addition and multiplication, they are. The rest is numbers dreams (themselves recovered by number relations, definable with addition and multiplication"

No they are not. Turing proved in 1936 that you can NOT come arbitrarily close to most real numbers using only the natural numbers and addition and multiplication

That's correct, but we don't need them at the ontological level. An idea like "all real numbers" belongs to the imagination of some relative natural numbers (see as a machine relatively to some universal numbers). You *can* assume them, but you don't have to assume them: it will change nothing in the laws of physics.

In fact the real numbers come already from physical observation, and is better to avoid reifying them to avoid treachery, and to simplify the machines interview.

"This comes from the fact that elementary arithmetic (on integers) is Turing universal."

But integers are very rare.

From the epistemology of natural numbers, you are correct. It is cleaner to put that kind of object in the epistemology. In the theory of everything on which all self-introspecting universal machine converge: there is only numbers (or combinators, lambda expression, etc.).

"You need to postulate the trigonometrical function to recover the natural numbers from the real."

And neither trigonometrical functions nor any other deterministic thing will help you get arbitrarily close to most real numbers, in fact such is the nature of infinite sets that if you picked a point at random on the real number line there is a 100% chance it will be non computable and a 0% chance it will be a natural number.

Real numbers does not need to be real, once you assume the comp hypothesis. The term "random" is notoriously hard to define. With comp, analysis and physics belongs to the natural numbers epistemology. There is no axiom of infinity in arithmetic.

"Someone doubting mechanism is not necessarily solipsist."

Why not?

Jacques Arsac is a french catholic who wrote a book against mechanism. He is not solipsist, and he doubts mechanism. One example is enough. Frankly why would a non mechanist be solipsist? How would you prove that statement? In which theory?

"very competent people can begin to believe that their are intelligent, and that's leads to stupidity."

It seems to me that both modesty and conceit leads to stupidity, if you're intelligent and you believe you are intelligent then your belief is true.

I will model "rational believability" of the ideally (arithmetically) correct machine by its provability predicate. The belief verifies that
1) elementary arithmetical axioms are believed, and
2) the beliefs are close for the modus ponens rule (i.e. if the machine believes A and believes A -> B, then the machine believes or will believe B). Such machines are consistent, but if they believe that, they become inconsistent. We have many true solutions x to Bx -> ~x. Some truth, when becoming axiom or theorem, leads to inconsistency. Computationalism is itself such a true but non provable proposition, even as axiom.


 John K Clark

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