We argue that the ordinary commutative-and-associative algebra of
spacetime coordinates (familiar from general relativity) should perhaps
be replaced, not by a noncommutative algebra (as in noncommutative
geometry), but rather by a Jordan algebra of Hermitian operators
(leading to a framework which we term "Jordan geometry"). We present the
Jordan algebra (and representation) that most nearly describes the
standard model of particle physics, and we explain that it actually
describes a certain (phenomenologically viable) extension of the
standard model: by three right-handed (sterile) neutrinos, a complex
scalar field phi, and a $U(1)_{B-L}$ gauge boson which is Higgsed by $\phi$.
We then note a natural extension of this construction, which describes
the Pati-Salam model of unification. Finally, we discuss a simple and
natural Jordan generalization of the exterior algebra of differential
forms.