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Boris's post had a phrase that I need to address, unrelated to his
original post's question:
I'd like to submit to Rosen readers what might appear at first sight as a
rather technical "detail". But it is not only a math problem because it is the
difference between relational models and (analytic=synthetic) models that is put
into question.
I think this needs some clarification. So I'm posting this
excerpt:
On page 153 of Life, Itself:
Robert Rosen wrote: "As we shall see, an examination of
modeling within mathematics reveals two quite different approaches to modeling.
One of them, which I shall call 'analytic', is intimately tied to the idea of
direct (Cartesian) product; in the context of natural systems, this approach is
tied to the notion of efficient cause. The other, which I shall call
'synthetic', is tied rather to the idea of direct sum and to the notion of
material cause. In the Theory of Categories, these two ideas turn out to be dual
to each other in a precise sense, but they are generally very different. Only in
very special situations do they happen to be equivalent in any sense, and these
special situations all inherently involve some kind of linearity.
Nevertheless, it will turn out that the modeling enterprise itself, both in
science and mathematics, has been tacitly predicated on the coincidence of
the analytic and the synthetic approaches, i.e., on the equivalence of direct
products and direct sums.
In fact, the difference between direct sum and direct
product, between synthetic and analytic models, is also closely allied to the
difference between syntactic and semantic. In this light, we can already see the
profound effects of supposing from the outset, however tacitly, that they
coincide.
In a sense, it is the thrust of this entire work that this
hypothesis of analysis = synthesis must be dropped. Above all, it must be
dropped if we are to do biology, and hence a fortiori, it must be dropped if we
are even to do physics. By dropping it, we enter a new realm of system, which I
call "complex", and which in a certain sense needs to have no synthetic models
at all. The distinction between relational and Newtonian models of natural
systems will become crucial here, because as we shall see, the former extend to
the realm of complex systems, while the latter cannot."
Judith |