Glen -
It's probably worth remembering that collections of spatio-
temporally located mathematicians will choose to use the "definitions"
that give them the amount of "traction" they want. They'll use
definitions that are sufficiently general as to cover the cases
they're most interested in, but specific enough to make theorem
statements and proofs appropriately concise and straightforward.
Also, in general, a "mathematical definition" is ordinarily just a
brief name for a collection of "axioms"; and, in various cases, there
is a "new name" when you add an additional "axiom" . . . so, for
example, a *semigroup* is a set with an associative binary operation.
A *monoid* is a semigroup with an identity element. A *group* is a
monoid with inverses.
The classic example of the process of "choosing the definition that
you like best" is the "definition" of a *ring* . . . Some
mathematicians require a *ring* to have a unit, others don't. So you
get slightly odd locutions like *rng* (called a "ring without unit" by
others), and "ring with unit" used by those who like the "more
general" version without the requirement of a unit.
(see, e.g., http://en.wikipedia.org/wiki/Pseudo-ring )
These days, most mathematicians are so comfortable with
associativity that they'll go ahead and include that as part of "the
definition" (of, e.g., a geometric algebra) . . . and then also they
won't have a bunch of theorems that start out, "Let A be an
associative geometric algebra . . ." rather than "Let A be a geometric
algebra . . ." (for example . . .)
It's possible that as, perhaps, String Theory gains traction (and
with it interest in exceptional Lie groups like, e.g., E8), the
insistence on including associativity in "the definition" will lessen,
but . . .
(see, e.g., http://en.wikipedia.org/wiki/E8_(mathematics) )
tom
On Oct 14, 2009, at 5:06 PM, glen e. p. ropella wrote:
Thus spake Owen Densmore circa 09-10-10 08:26 PM:
Has anyone read this?
http://math.ucr.edu/home/baez/octonions/conway_smith/
I've not read enough Conway and I'm not sure where to start!
So, is it fair to say that octonions are a geometric algebra, even
though they aren't associative? I think I remember reading somewhere
that they were considered a geometric algebra... perhaps in Hestenes
book or in Penrose's Road to Reality.
But wikipedia claims that a geometric norm _must_ be associative and
that a geometric algebra must be over a vector space with an
associative
norm. WTF? Is wikipedia oversimplifying? Or are octonions really
not
considered a geometric algebra despite their deep relevance to
geometry?
And, more importantly, why do my searches for Clifford fail in Adobe
Reader, but succeed in Evince, while reading the following file:
http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf
???
--
glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com
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Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
