Ok, so perhaps "an amazingly clever generalization" is possibly over-the-top
but the fact that it could form a basis for all of the algebra certainly
captured
my imagination. I did not know of Operad theory but I'll add it to the
study
queue. I still feel that what you and Li proposed was very elegant. I
really
liked the fact that Clifford and Grassmann algebras are subcases.
Tim
William Sit wrote:
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
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