  The role mathematics plays within physics has been of sustained interest for
physicists as well as for philosophers and historians of science. We explore
this topic by tracing the role the mathematical structure associated with SU(2)
has played in three key episodes in 20th century physics - intrinsic spin,
isospin, and gauge theory and electro-weak unification. We also briefly
consider its role in loop quantum gravity. Each episode has led to new physical
notions of a space other than the traditional ones of space and spacetime, and
each has had associated with it a complex and in places, contested history. The
episodes also reveal ways mathematical structures provide resources for new
physical theorizing and we propose our study as a contribution to a need Roger
Penrose has identified to develop a "profoundly sensitive aesthetic" sense for
locating physically relevant mathematics.
