Deryk Barker writes in reply to me:

Many branches of mathematics have
been invented/discovered decades before any real world application was
discovered.

Deryk misunderstands the location of the "deep roots in the real world"
I ascribe to the systems of symbol manipulation underlying both mathematics
and music.  They are in the human brain, which EVOLVED in the kind of
world that the animal carrying it lives in.  This at once explains why
aspects of pure mathematics find, many years after their development,
seemingly unexpected practical applications, as Deryk pointed out.  This
phenomenon itself would be a staggeringly improbable coincidence---were
it not for mathematic's roots in the world itself via the human brain.

I agree entirely with Bill Pirkle's comments on the relationships between
music and mathematics and architecture.  I also agree with him that:

>...  math and music are both a subset of a third "thing" but I'm not
>sure what to call it---'meta-zen-philosophy' perhaps.

Whatever we call it, it resides in the human brain AND is thereby related
to the world in which that brain was selected for in evolution.  This
should not come as a surprise---unless one imagines that the human brain
and its workings appeared out of nowhere, abracadabra.

My hunch (no more than that) is that Bill's "meta-zen-philosophy" has
to do with statistical operations---thus placing it, by way of probability
theory, at the juncture of mathematics and epistemology.  Somehow, these
operations (which start by telling us how to process and "make sense of"
sensory inputs) entrain music, aesthetics, grammar, and baseball.  I
also agree that we are a long way from grasping even the shape of the
elephant.  By the way, I am sure that elephants also have some of the
"meta-zen philosophy" in their brains too.  I know of at least two
elephants who paint, although none who write music.

Cheers/// JON GALLANT   ([log in to unmask])