Re: Why four categories of causation?

Boris,

See interposed comments.

-----Original Message-----
From: ROSEN Forum [mailto:***On Behalf Of Boris Saulnier
Sent: Thursday, January 13, 2005 12:10 PM
To: ***
Subject: Re: Why four categories of causation?

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).

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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.

TG: I believe "F" is only meant for a general illustration of the idea of an evaluation map, not as a reference to a specific thing in the (M,R) model. I agree that it adds to the confusion.

And I do not understand why : "we want b^-1 = beta"

TG: If you follow my previous discussion, the inverse of the evaluation map gives the desired mapping needed for replication.

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?

TG: I suspect there are multiple possible ways to generate replication from within the basic (M,R)-system. Rosen does not argue that this particular method is necessarily the only one or the best one. I think Aloisius commented once that this method is simply the easiest one to generate formally. I would be interested in exploring those other possibilities. I don't think we could close the diagram using A, since A represents the raw inputs from the environment, which seems to me are not the kind of things which biologically directly induce replication.

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.

TG: I agree. The lack of the entire argument makes the (M,R)-system difficult to appreciate, which really makes the whole book difficult to appreciate, in my view. I speculate that it may be that he wanted to avoid the entire argument in order to maintain focus on the principle involved: 'closed to efficient causation' and the inability to represent such organization as mechanisms.

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?

TG: As far as I know, the two papers I mentioned contain the best discussions of the (M,R)-system in detail, especially replication and the evaluation map. I should have been more clear about the references. The specific references are:

1959. Rosen, R. , "A Relational Theory of Biological Systems II". Bulletin of Mathematical Biophysics, Vol 21:109-128

1972. Rosen, R., "Some Relational Cell Models: The Metaqbolism-Repair Systems", chapter 4 in Foundations of Mathematical Biology, Vol II Cellular Systems. (Rosen, R. ed., Academic Press)

("Quantum Genetics" is not one of those I referenced. That paper is in Vol I, and does not discuss the (M,R)-system, as I recall.)

Is there any chance you or someone on the list has a digitalized version of them?

TG: I do not. If necessary, I can write a post with the discussion regarding replication in the 1959 paper.

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?

TG: Hmmm....I am not sure. As I read in MacLane's Categories for the Working Mathematician, he writes that:

"Set is a cartesian closed category, with c^b = hom(b,c)." [p. 98]

A similar remark is in Basic Category Theory for Computer Scientists:

"The category Set is cartesian closed, with B^A = Set(A,B)." [p. 34]

So, I think the category Set already formally qualifies to be used for this purpose, but I am no expert at category theory.

Regards,

Tim