Tim: You wrote: "William Sit and someone from Rutgers had what seemed to me to be an amazingly clever generalization. I don't know what happened with that."
-- I don't think what Li Guo (Rutgers, Newark) and I proposed is, at least mathematically speaking, anything like "amazingly clever generalization". Moreover, we did not invent the concept of operads (which is I think what you are referring to; for explanation of what an operad is, see http://en.wikipedia.org/wiki/Operad_theory). What you read about was from an NSF proposal that was not funded, in which Guo and I proposed to base the algebra hierarchy in Axiom on the idea of operads. Various algebraic structures, like Clifford or Grassmann algebras, would then be descendents. The operads concept is I believe closely related to lambda-calculus and so should not be difficult to implement in Lisp. In the grand scheme of things, it would be also useful for the proviso project, which would require real-time manipulation of the Axiom compiler (in a way more involved than what Tim recently described, for example, there has to a new embedded language to make conditionals and indefinite iterations native). Many years ago (in the 1980's), another researcher from Rutgers (New Brunswick), Joseph Johnson, had a Theory of Universes that was built on the idea of partially defined functions as first-class objects (he did not use that term, but I think it is a close description of the objective) and this purely algebraic theory works across several categories like complex analysis, algebraic geometry, differential geometry, and differential algebraic geometry. Unfortunately, he had a stroke and his work was not completed. To summarize this is what is roughly Johnson's own work: functions are just objects in which you can plug in anyting that makes sense and then evaluate (like physicists do, without worrying about domains of definition and singularities explicitly). In some ways, there is a built-in algebra for domains of definition that handles provisos transparently and automatically. William -- William Sit Department of Mathematics..Email: wy...@sci.ccny.cuny.edu City College of New York................Tel: 212-650-5179 New York, NY 10031, USA.................Fax: 212-862-0004 Home page: .......http://scisun.sci.ccny.cuny.edu/~wyscc/ _______________________________________________ Axiom-developer mailing list Axiom-developer@nongnu.org http://lists.nongnu.org/mailman/listinfo/axiom-developer
