Re: Why four categories of causation?

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

then the evaluation map b Î B is

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

But, since F_b 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?