The physical realization of this mathematics is described in statistical mechanics and non-equilibrium thermodynamics by Prigogine and the Brussels-Austin group.
Ken > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] On Behalf Of Roger Frye > Sent: Sunday, August 10, 2008 7:24 AM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] Rosen, and mapping > > I agree with Russell and Carl, but a couple of mathematical > examples might help. > > Consider the mapping (i.e. arrow) from a pair of factors to > their product. There is not a unique reverse mapping from > the product to the factors. Also, if the factors are > positive, consider the mapping from them to their individual > logarithms; then a mapping from that pair to their sum. The > logarithm and anti-logarithm provide a two directional arrow > between the sum and product, allowing sums of logarithms to > be used in place of multiplication. > > Andrew Wiles summarized the problem of Fermat's Last Theorem > as knowing that there were arrows in one direction between > elliptic curves, modular forms and galois fields, but needing > to show that one of the arrows could be reversed for the > particular elliptic curve that represented a^n+b^n=c^n for n>5. > -Roger > > On Aug 9, 2008, at 9:14 PM, Russell Standish wrote: > > > The standard language of maps (aka functions) over sets > will give you > > want you want. Category theory is not needed. > > > > On Sat, Aug 09, 2008 at 08:58:02PM -0600, Nicholas Thompson wrote: > >> Roseners, and anybody else vaguely interested in category theory. > >> > >> Rosen seems to be interested in situations in which A maps > to B but > >> not all the values in B can be generated by the mapping. > >> > >> this is a lot like the Intension and the Extension of an > utterance. > >> I say with assurance that Mrs. Vanderbilt wished to sail on the > >> Titanic. In this case, Mrs Vanderbilt's "wanting" is a function > >> (mathematical sense) that maps from her wants to a subset of the > >> properties of the Titanic. All the properties of the Titanic > >> constitute (in philosophic lingo ) it's extension. The > subset, the > >> "image" of Mrs Vanderbilt's wanting , constitutes the intension of > >> her utterance, "I want to sail on the Titanic." Among the > titanic's > >> attributes, but outside that image, is the property "hit > an iceberg > >> in the North Atlantic and sank." > >> > >> I guess the question is whether there is a less tortured > mathematics > >> than category theory that would allow one to talk about > these things. > >> > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
