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
