On Fri, Aug 15, 2008 at 8:57 AM, Nicholas Thompson < [EMAIL PROTECTED]> wrote:
> Thanks, Russell. > > Is your comment differ with Ken's or is it Ken's in another language. > > For a former english major, the LANGUAGE is everything. > > The operation of functional composition, taking *f: A -> B* and *g: B -> C*and composing them to get *gf: A -> C*, is qualitatively different from the *inner* entailments which only involve sets and mappings. The *inner* entailments were summarized as *f => (a => f(a))* which reads that *f* is the efficient cause and *a* is the material cause of *f(a)*. This gives us an element, *f(a)*, as a consequence of a mapping and an element, *f* and *a*. The *outer* entailments speak to the causes of mappings and sets. So *F => (f, g => F(f,g)) *says that functional composition is the efficient cause and the functions *f *and *g* are the material cause of the function * gf.* And, the example left for the reader to work out, *C => (a, b => C(a,b))* says that the cartesian product is the efficient cause and the elements *a* and *b* are the material cause of the element *a x b.* In the first case we get a mapping as a consequence of composition and two mappings, in the second case we get an element as a consequence of cartesian product and two elements. No functors were deployed in the construction of these paragraphs. At the end of section 5H (p 130) Rosen notes: We can formally do a great deal with the modes of inner and outer entailment inherent in any category. In particular we can concatenate them to form, and characterize, arbitrarily complicated abstract block diagrams from the sets and mappings in any particular category. In fact, the totality of abstract block diagrams that can be formed in this way constitutes a new category [ ... ] as a (free) monoid A~s stands to its set of A of generators [...]. Baez and Stay in "Physics, Topology, Logic and Computation: A Rosetta Stone" are essentially applying the same "arbitrarily complicated abstract block diagrams" formalized as various subclasses of "symmetric monoidal categories". By now there is an extensive network of interlocking analogies between physics, topology, logic and computer science. They suggest that research in the area of common overlap is actually trying to build a new science: *a general science of systems and processes*. So they agree that physics, logic, and computation are pretty much the same thing, that arbitrarily complex block diagrams are the key, and that a general science of systems and processes is the goal. -- rec --
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