Hi Mirek,

Welcome to the list,

Le 13-août-07, à 16:54, Mirek Dobsicek a écrit :

> Hello Bruno !
>
> I am a freshman to this list and it seems to me that some kind of a
> 'course' is going to happen.


Let us say that I try to give some information linking my (already old) 
work and the main discussion on this list.

In the case you know french the most extensive description of what I 
have done is in "Conscience et Mécanisme", or in some preceding papers.
In English my last papers could perhaps help, you can find them here:

http://iridia.ulb.ac.be/~marchal/publications.html



> I checked a couple of last messages and it
> looks interesting. Please, would you mind to repeat what is
> approximately the starting point of your explanations and where do you
> aim? Hopefully, I'll be able to follow.


The starting point of this list is the idea that "everything exist" 
could be an easier explanation than any assumption of the kind 
"something exists".
My own starting point is what I called the "indexical digital mechanist 
thesis", which is called also "computationalism", and which is a 
digital version of Milinda-Descartes mechanism thesis. When Milinda 
asked Nagarjuna, what is the nature of a person, Nagarjuna answered by 
an informal exposition of the mechanist thesis.
What I show (or try to show) is that once we take the idea that "I am a 
machine" seriously enough, then this entails a reversal between physics 
and "intensional number theory", or "mathematical computer science" or 
(as I can justify) "machine theology", and this in a sufficiently 
precise way so that the physics extracted from the computationalist 
"machine theology" can be compared with the empirical facts.





Lennart Nilsson wrote:

> Bruno wrote:
>> I don't think Church thesis can be grasped
>> conceptually without the understanding that the class of programmable
>> functions is closed for the diagonalization procedure.
>
> This is something I never grasped but would love to understand.


Thanks for saying; it is indeed a key point, which, I realize now,  is 
rarely understood, although in my opinion Emil Post has seen this point 
in the 1920, and Judson Webb has written a genuine book in the 1980 
(ref in my thesis).

But at the computability meeting in Siena 2007, I have heard that some 
people still believe that Church thesis could be viewed as a definition 
(of computable function), which, as I hope being able to explain, is 
complete  nonsense.  Actually, when Church did propose what he did 
consider as a definition, Stephen Kleene makes clear it has to be a 
thesis (i.e. an hypothesis). I will come back on this.

I will have to come back on Cantor. Meanwhile people can consult my old 
post on diagonalization.
Actually you can search for diagonalisation (with a s) for my older and 
more basic posts, and then search for diagonalization (with a z) for 
some more recent one.

Well, the list archive are no so easy to search in: here is my older 
"diagonalisation" post, send to George Levy on the list, the 21 
augustus 2001, shortly after the sudden death of James Higgo.

-------------------------------- copy of my first diagonalzation 
post----------------------------
                 in memory of James,


Hi George, Hi People,

I guess most of you know the famous proof by diagonalisation
  of the uncountability or non enumerability of the reals.

To my knowledge diagonalisation appears in the work of
  Dubois-Reymond, but it is Cantor who first used it for proving
  that the set of reals is bigger, in some sense, than the set of
  natural numbers N, or the set of integers Z or the set of
  rational numbers Q.

Here I want recall Cantor proof, and then I want to show you
  a weird, similar but false diagonalisation reasoning.

The  correction of that reasoning will give a shortcut to Godel's
  incompleteness result, which is itself a step toward G and G*.

In this post I prove Cantor theorem, and then I give you the
  similar but wrong proof. I will let you search the error.

I hope you see that Q is countable. There are simple
  drawing proof of that. But without drawing, it is enough
  to realise that a rational numbers like -344/671 is described
  by a finite string in the alphabet {O, 1, 2, 3, ... 9, -, /}.
  And finite strings can be ordered by length, and those
  with the same length which remains can be ordered by some
  chosen alphabetical order. I call this order (on string) the
  *lexicographic* order.

Actually we will be interested by the functions of N to N.
  N is the set of natural numbers (positive integers).

============================

So here is a variant of Cantor theorem:

1. Theorem: The set of functions from N to N is NOT countable.

(Note: if you know Cantor proof, just skip it and go to 2. below).

Proof: (by absurdum and diagonalisation)

 I recall that a function from N to N is just an assignment for each
   natural number (called the argument or input) of a natural numbers,
   called the output or value.

 exemples:
   -the constant function:

   input   0   1   2   3   4   5   6    ...
     output  1   1   1   1   1   1   1    ...

-the identity function:

   input   0   1   2   3   4   5   6     ...
     output  0   1   2   3   4   5   6     ...

-the factorial function:

   input   0   1   2   3   4    5     6     ...
     output  1   1   2   6   24   120   720   ...

-an "arbitrary" function:

   input   0   1   2   3   4    5     6            ...
     output  10  0   0   3   56   1   35465439087    ...

Of course in this last case, the  "..." has only sense
  in Plato heaven or in God's Mind.

Suppose now (our absurd hypothesis) that the set of all
  function from N to N is countable.
  This means that we can count or enumerate the functions.
  So we would have a sequence of functions, each associated
  to a natural number such that all functions appear in that
  sequence. We would have a sequence:

     f_0  f_1  f_2  f_3  f_5  f_6 ...

of all functions (from N to N, but this will not be repeated).

Let us put those functions with their value in a matrix:

    input   0   1   2   3   4   5   6      ...

f_0         4   7   0   0   0   19  5      ...
  f_1         8   6   6   2   1   3   49     ...
  f_2         66  36  5   2   4   5   8      ...
  f_3         1   2   3   2   1   3   2      ...
  f_4         1   2   3   4   5   6   7      ...
  f_5         10  0   0   3   56  1   356    ...
  f_6         0   10  7   2   2   35  0      ...

..                 ....                  ...

Now we will get a contradiction. Indeed we pretend that
  all functions belongs to the sequence f_0  f_1  f_2 ...
  and this makes the matrix well defined (in Plato heaven
  or Cantor paradise, we don't ask that the matrix can be
  algorithmicaly generable).

But here is a function, certainly well defined in Plato
  Heaven, which, by definition, will not appear in the
  matrix. It is the function g which send n on f_n(n) + 1.

             g(n) = f_n(n) + 1

In particular g(0) = f_0(0) + 1 = 4+1 = 5;
                g(1) = f_1(1) + 1 = 6+1 = 7;
      g(2) = 6;  g(3) = 3; g(4) = 6, g(5) = 2; g(6) = 1, ...

You see we just change the "diagonal value" so as to be
  sure that g is different from f_0 on the value O
  (it is f_0(0) + 1), g is different from f_1 on the value 1, etc.

Would have g belong to the list f_0  f_1  f_2  f_3  f_5 ...
  A number k would exist such that g = f_k, but then, applying
  g on k gives g(k) = f_k(k), but remembering the definition
  of g, we have also g(k) = f_k(k) + 1. Contradiction.

The proof does not depend of the choice of the matrix, so that
  we have just shown that the set of functions cannot be put
  in an exhaustive sequence. N^N is not countable.
  (A^B is a notation for the set of function from B to A).

==============================

2. A paradox ?

I will say that a function f is computable if there is
  a well defined formal language FL in which I can explained
  non ambiguously how to compute f on arbitrary input (number).
  I must be able to recognise if an expression in my language
  is a grammaticaly correct definition of a function, that is
  syntax error must be recoverable.
  All expression in the language must be of finite length, and
  the alphabet (the set of symbols) of the language is asked to
  be finite.
  Now all finite strings, a fortiori the grammatically correct
  one, form a countable set. Just use the lexicographic order
  defined above.

So the set of function computable with FL is countable.
  A synonym of countable is *enumerable*, and is used by
  computer scientist, so I will also use it.

So I can enumerate the functions computable with FL, that is
  I can put them in sequence like f_0, f_1, f_2, f_3, ...
  going through all functions computable in or with FL.

Let us consider the function g defined again by

                   g(n) = f_n(n) + 1

Now, g is not only well defined, but is even computable. To
  compute it on n, just search in the enumeration f_i the nth
  function, apply it to n (this gives an answer because the f_i
  are computable) and add 1.
  So, there is a number k such that g = f_k, but then, again

                g(k) = f_k(k) = f_k(k) + 1

But f_k(k) is a well defined number (f_k is computable), and
  no numbers can be equal to "itself + 1". Contradiction.

Could the set of computable functions in FL be uncountable, after
  all, or is g not expressible in FL, or what?

Where is the error?   I let you think a little bit. You can
  ask question. (Notably in case of notational problem).

Bruno


--------------------------------



http://iridia.ulb.ac.be/~marchal/

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