[FRIAM] Can empirical discoveries be mathematical?

[Jon Zingale]

Nick writes:

"""
I think I am starting to know the answer just by being badgered by you
guys.  I can from relativity theory predict that during a solar eclipse a
distant star will pop out from behind the sun at T= xxxx =/- xxxx sec.  I
can observe empirically that, indeed, the star popped out within that time
range.  Thus I have arrived at the same proposition both by mathematical
and empirical means.  Is that oK?
"""

and in particular:
"Thus I have arrived at the *same* proposition"

In some cases, it is fine to speak about strict equivalence, though in
others it might be better to consider what is preserved between the two
models (one model with the privilege of being designated *real*). The
mathematical model agrees pretty well with the other on some things but not
others, and not necessarily in a pedantic or trivial way. It is the case
that one can apply a small force to the planet mercury as well as perform a
corresponding manipulation in the mathematical theory to arrive at
agreeable results between the two models. In this sense, the mathematical
model (Newton's laws, say) is *faithful* to the real. On the other hand,
the mathematical model may have no *full* relation to the real. That is,
there may be more to discover about the real that has no counterpart
(actually nor effectively) in the mathematical model. Granted a full and
faithful relation between models, I see no harm in pronouncing a functorial
equivalence, or what DaveW might call a *dead metaphor*.

"""
To be a bit more pedantic, you have discovered that t-shirts transform
under the SO(3) _representation_ of the rotation group.  If you were not a
mathematician or a physicist, you would say “I had “the” group of
rotations; what is there to represent?”  But a mathematician would tell you
that there are many representations of the rotation group, all having the
same algebra, yet different formal constructs...
"""

Changing the image slightly, consider a favorite melody[𝄞]. One *may*
crudely choose to represent this melody in any of 12 keys by translating
about a piano. These choices all give grounds for the melody. But if jazz
music taught us anything, ultimately not even the notes need be the same,
somehow the melody can be found to transcend pitch, mood, and rhythm to a
significant degree. For one such instance, we could instead look to
arpeggios, those collections of notes which illuminate an implied harmonic
moment. At each moment along the melody we can imagine substituting our
original for another in the class, and not only can we find the melody
preserved by often with pleasant results.

[𝄞] I woke up with "My Favorite Things" in my head this morning, and not
necessarily anyone's particular rendition. I considered playing John
Coltrane's version but then finally settled on Julie Andrew's version.
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