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Tim,
First, thanks for your Nature article suggestion in
relation to degeneracy.
And you're perfectly right, I was to "degeneracy"
in the sense suggested by Edelman, Tononi and Sporns (PNAS
publication).
---
At the moment, I've the same problem as you in
trying to understand the realization of the model constructed p.251 in
LI.
I do not know what this F
used to define b^ is. And I do not understand why : "we
want b^-1 = beta"
Moreover, could we close the diagram working on f
or A instead of b? Can you think of an argument why we should rather focus on
b?
Rosen says "I have since repeated this formal
argument many times in previous work and need not repeat it here".
Unfortunately I do not know which these previous
works are. It's a pity that LI, at such a crucial
step, is not self-contained anymore.
It is really a surprise how Rosen construction
p.251 comes with no justification.
Moreover I was not able to find the 1959 BMB
article you refer to.
So I will try to find the "Quantum genetics"
article you refer to in Found. of Math. Biophysics, 1972.
I would be vey interesting to know more about this
"bi-dual" story.
Are these two articles the "previous works" Rosen
speaks about?
Is there any chance you or someone on the list has
a digitalized version of them?
Judith : have you ever thought of a "complete
works" CD, as was recently done with René Thom's works? (see : http://www.ihes.fr/~cdthom/)
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If I am not wrong, Rosen construction implies, in
categoric terms, morphisms being also objects of the category. This, from what I
know is not possible in the SET category Rosen seems to be working in. For
instance it might be that we should work in Cartesian Closed Categories (for a
definition you can look p.17 in ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/2Constr.ps.gz)
Do you know if this apsect has already been
discussed by Rosen or his readers?
---
Happy new year to all people interested in
theoretical biology!
----- Original Message -----
Sent: Thursday, January 13, 2005 3:42
AM
Subject: Re: Why four categories of
causation?
Arno,
AG:
> For instance, in the famous
diagram of fig.
> 10.C.6 in LI there is an interesting double relation
between Phi,
> B and F:
>
> 1) B is material cause (Phi
being efficient cause and F outcome)
> 2) B is efficient cause (F
material cause, Phi outcome)
>
> So the 'really' interesting
point, I think, is: what does this
> relation between B and F look like,
apart from the definition of
> these various causal relations? There
MUST be some kind of
> ambiguous process going on, that can be
interpreted as material
> or efficient relation, but how do these two
relations mix up in
> their physical realization? It is in answering
such questions, I
> think, that Aristotle's causal system may not be
sufficient here.
The questions about the realization of this model has
haunted me for about as long as I have known of it. I still have difficulty
fully understanding the replication mapping Beta insofar as its meaning in
physical terms. I believe B as an efficient cause in ch. 10 of LI is related
to Rosen's comments waaay back in ch.6 on p. 157 about evaluation maps. There
he talks about how one could equally write s(f) for f(s). (I know it
should be an s with a circumflex over it, but this is the closest symbol
I could find in Outlook.) The example he uses for the physical interpretation
of the evaluation map is that of a meter: the meter value is caused
by s forcing the meter (which is designed to respond in a predetermined
manner) such that a certain meter value results, so s is more naturally
seen as the efficient cause rather than s being considered an "input" (i.e., a
material cause) to f.
In a similar way, if we have the repair mapping
F:B ® H(A,B)
then the evaluation map b Î B is
b:Fb ® H(A,B)
So, as I read this, a metabolic output b
is to be considered as a forcing upon the process F, thereby causing a certain set of mappings H(A,B).
The inverse of the evaluation mapping is
b-1:H(A,B) ® Fb
But, since Fb
is the mapping B ® H(A,B), then the inverse of
the evaluation mapping can be rewritten as
b-1:H(A,B) ® H[B ®
H(A,B)]
which is the desired replication mapping b.
I can understand how, using the meter analogy,
some 'input' b to F can be actually be the forcing
agent, causing F to produce a certain set of
mapping H(A,B). Of all that is unclear to me, the most unclear is the physical
interpretation of the inverse mapping. In that case, there is a set of
mappings H(A,B) which are forced (by b-1)
to produce a certain set of repair components. If H(A,B) is the set of
mappings which include the metabolic 'enzymes' (such as f), then
this seems to me to say that the set of 'enzymes' (or perhaps better, the
creation of 'enzymes'?) are being forced in such a way as to result in a
certain set of repair components.
It would probably help if I somewhat
understood "the embedding of a vector space into its second dual". Rosen
refers to the evaluation map as an abstract version of this [Found. of Math.
Bio. 1972, Vol. 2 p. 235 and BMB Vol. 21,p. 116, 1959].
Any ideas or thoughts?
Regards,
Tim
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