Re: Fw: Paper by Landauer and Bellman

[Judith Rosen <***>]

The following is a somewhat different response to the paper that tried to show a "flaw" 
in the mathematics my father used to describe certain aspects of Complexity. Aloisius 
Louie is one of my father's PhD students from Dalhousie and a friend of mine. He agreed 
to analyze the paper and give me his conclusions. He also gave me permission to post his 
analysis to the list. I hope subscribers find it of interest:

-----Forwarded Message-----
From: "Dr. Aloisius H. Louie"
Sent: Feb 20, 2004 1:30 PM
To: Judith Rosen <***>
Subject: Re: Fw: Paper by Landauer and Bellman

Dear Judith the jetsetter:

        Another trip!  Wow!  Anyway, below is my response to the
Landauer-Bellman paper.  They missed the point entirely.

        Landauer and Bellman are confused between necessity and
sufficiency.  The Rosennean statement is that "a cell is a material
structure that realizes an (M,R)-system".  This is a statement of
necessity.  It means that if C is a cell, then THERE EXISTS an
(M,R)-system S such that C realizes S.  It does NOT mean that if
one comes up with an arbitrary (M,R)-system (or worse, something

that only vaguely resembles an (M,R)-system), there has to be a
cell that realizes it.  "If C, then S" (S is necessary for C),
is not "if S, then C" (S is sufficient for C).


        So the example of an (M,R)-system that Rosen usually gives,
in the category Ens of sets, is realizable by a cell.  Be mindful that
this still does not mean that EVERY (M,R)-system in Ens has a cell
that realizes it.  It only means that the (M,R)-system that the cell
realizes can possibly come from Ens -- an (M,R)-system from Ens

is only a possible candidate.  The beauty with the category

Ens is that there are no further mathematical structures imposed
on its objects, and the (M,R)-system idea already works.  We don't
need the category Top of topological spaces and continuous functions,
for example, so that (M,R)-systems only need to be algebraic, not
analytic (in the mathematical sense of the word).  Less is more with
Ens: a Zen-like "no states, no environment, no recursion".  This is

why the "equational formulation" is nonsense.

        The first "counterexample" that the L-B paper gives is in
the category of formal power series, and they show that replication
entailment does not work here.  There is nothing wrong with this exercise.
It simply shows that this one particular construction in the category of
formal power series does not result in an (M,R)-system that a cell realizes.

In fact, without the recursive entailment, it is not even a real
(M,R)-system.  There is no difficulty in constructing arrow
diagrams that do not quite work out as (M,R)-systems.  The existence
of such mathematical objects (e.g. all the supposed "counterexamples"
in the L-B paper) certainly does not disprove Rosen's

"necessity statement".  It only shows that not all arrow diagrams

are (M,R)-systems.


                Aloisius



--
Dr. Aloisius H. Louie
Mathematical Biologist





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