On 17 Sep 2013, at 19:39, John Clark wrote:

On Mon, Sep 16, 2013  Bruno Marchal <marc...@ulb.ac.be> wrote:

>> So you are suggesting that a thing like broken glass is made of numbers

> ???? I was just saying that things are not made up of things. A broken glass is NOT made of number. That has no meaning at all. What happens is that addition and multiplication of natural numbers emulate dreams, which might be dream of a broken glass.

OK. How is that any different from saying broken glass is made of numbers?

It would be like saying that the relation between matter and energy (E = mc^2) is made of ink or of pixels.

>> don't tell me there is no such thing as a thing, that's just more gibberish.

> It is a matter of tedious, and not so simple, exercise to see that the computations exist in some definite sense when we postulate arithmetic. (This is done in good textbook, and very well done in Epstein & Carnielli, but also in Boolos & Jeffrey). Physical things then appears as stable percept

And concerning broken glass I said in my September 11 post "It must have stable properties of some sort or I wouldn't be able to identify it as a thing".

I agree. But a computation can provide stable things for another computations or subcomputations.
Then arithmetical truth is rather stable itself.

> by persons living those dreams.

OK. Therefore the physical universe and the physical things in it exist.

That makes sense. Just that such an existence is a first person plural construction. This exists for all universal system which can "run" different computations in parallel, and makes them interact.

>> Make up your mind! First you say everything is the process of "natural numbers" in "relative computations" and then you say "digital machines, which are defined in term of number relations" are an exception to this because what they do "is not a process". The sum of number relations is not a process?? None of this makes any sense to me.

> Some number relation defines some machines, or some programs, which are static entities. *Other number relations, involving the preceding one, defines computations, or processes,

Name a number relation that does not involve a computation or some other process!

It is difference between a number j used as a name for a program, like in the arithmetical relation phi_j(k) = r, and a number coding a computation, that is some sequence like phi_j(k)^1, phi_j(k)^2, phi_j(k)^3, phi_j(k)^4, phi_j(k)^5, phi_j(k)^6, phi_j(k)^7, ...

Here phi_i is an enumeration of the partial computable functions, k is a natural number input, and "^s" means the sth step of the computation.

> A machine, in that setting is basically one number, relative to some universal number.

Relative? A relation needs at least 2 things,

Yes. The two things are
1) the number playing the role of the machine (the j in phi_j(k)), and
2) the universal system (seen as a number, unless we start in the basic system assumed, like arithmetic, or the combinators) which computes phi_j(k).

You can look at the Matiyasevitch book for a nice implementation of arbitrary Turing machine *and* their computations (seen as something very dynamic) in the terms of Diophantine equations (since as very static). That can help. Providing examples is very long and technical, alas, but we will come back on this most probably.

and  some sort of computation with them.




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