I simply meant to challenge the conception of symmetry in the same way a circle with 0
area but finite radius is a line segment. A rigid paddle embedded in a rigid medium isn't
really a "paddle in a medium" ... it's a ... what? ... a sculpture that
invokes, metaphorically, a paddle in a medium. It could also invoke a Once and Future
King, depending on the shape of the paddle. [⛧]
But you hit the nail, anyway, with the discussion of frame-shifting to
Hamiltonians, position vs momentum eigenstates, and latent formal systems
groped at by folk physics (or folk anything).
E.g. at the last ThuAM, Stephen asked me if I was "spiritual". I tried to play the "define
your terms" game, which I normally reject because I believe terms can never be completely defined
(consistently, yes, completely, no). But in this case, I had no idea how to answer. Just prior to this
apparent tangent, though, Jon and I had been dancing around what I call "metaphysical commitments",
his to something similar to your (Eric's) latent formality and mine to the open-endedness of the universe. I
have no quantifiable evidence the universe is open, yet I still believe it. If that makes me spiritual, so be
it.
In some ways, it feels like those who posit formal systems as bases/foundations
for [ahem] reality believe the opposite, that the universe has an actual
boundary of some kind. (Not a metaphysical/degenerate limit, but an actual
boundary.) The formality feels closed, like an actual limit to the complexity
of the universe. The plethora of *useful* formal systems in what seems to be
our singular world is a kind of justificationist evidence for the
open-endedness of that world. But were we able to measure that set of formal
systems, e.g. N of them work 80:20 better than the rest of them in *this* world
- for all use cases, that would be evidence for me that there was a closure to
the universe (of some kind).
Degenerate constructs like a 1 group symmetry feel, to me, like metaphysical
commitments ... committed to in order to provide ethical justification for the
privileges and comforts we enjoy (like GPS satellites or projecting back in
time to the origins of the universe, etc.). It's analogous to a right-winger
arguing for barefoot women in the kitchen based on some metaphysical commitment
to a binary gender. We argue for degenerate cases because of the power the
non-degenerate cases give us. Show 1 group symmetry as the absurdity that it is
and the whole structure comes crashing down.
Anyway, thanks for entertaining my batshit confusion for even this long. I'm
sure Marcus has already screamed. 8^D
⛧ There's a common argument against sophistry that uses triviality in the same way I'm
trying to use degeneracy. But I don't think triviality is strong enough to get at
equivalence groups of formal systems (or logics). What I'm after are constructs like the
zero group which, unlike the empty set, are members of multiple systems, each of which
may have its "natural" application domain (use cases).
On 8/20/24 13:52, Santafe wrote:
Second inadequate reply, to Glen, unhappily similar to the first to Jon:
On Aug 19, 2024, at 23:37, glen <geprope...@gmail.com> wrote:
There's so much I'd like to say in response to 3 things: 1) to formalize and
fail is human, 2) necessary (□) vs possible (◇) languages, and 3) principle vs
generic/privied models. But I'm incompetent to say them.
So instead, I'd like to ask whether we (y'all) think a perfectly rigid paddle, embedded
in a perfectly rigid solid, with a continual twisting force on the handle, exhibits
"degenerative" symmetry? Of course, such things don't exist; and I hate thought
experiments. But I need this one.
I got lost here because I don’t know what “degenerative” symmetry is meant to
refer to. In context of your next para, I see a contrast between discrete
symmetries, such as the rotations that would preserve a crystalline unit cell,
versus continuous symmetries, which I need as a formal model to derive
restoring forces. Is “degenerative” somehow another term for the continuous
ones?
The question when a continuum model can be seen as a limit of discrete models
on finer and finer grains, and when one needs it to be an independent
construct, is interesting. It feels like it goes back to the Eleatics.
I have often thought that Zeno’s paradoxes nicely illustrate the things you
can’t do if you have a mechanics that mathematizes only positions. Hamilton
sweeps those limitations away by making momentum an independent coordinate in a
phase space, and in that way granting it status as an independent property of
objects from their positions (in classical mechanics). All the consequences
of Noether’s theorem, conservations, restoring forces, etc., are formulated in
terms of these independent and dual properties. With the advent of quantum
mechanics, their independence becomes even more foundational to the picture of
what exists, as a system in a momentum eigenstate is really in a completely
distinct state from one in a position eigenstate. The two are differentiated
in something like the way traveling waves and standing waves are differentiated
in various wave mechanicses.
Similarly, if the paddle+solid could only be in 1 of 2 states, rotation 0° and rotation 180°,
and would move instantly (1/∞) from one to the other, with `NaN` force at every other angle
and 100% force at the 2 angles. This seems like symmetry as well, but not degenerative. And we
could go on to add more states to the symmetry (3, 4, ...) to get groups all the way up to ∞,
somewhere in between where the embedding material becomes liquid, then gas, etc. and the
"symmetry" is better expressed as a cycle/circle. But I'm not actually asking
questions about 1D symmetry groups. My question is more banal, or tacit, or targeted to those
who think with their bodies. When all the other non-Arthur peasants try to pull Excalibur out
of the stone, my guess is they're not thinking it exhibits degenerative symmetry. And that
implies that normal language is not possible. It's impoverished, for this concept. Math-like
languages are necessary in the sea of all possible languages. The would-be King *must* use
math to describe the degenerative symmetry. (Missed opportunity in Python's Holy Grail, if you
ask me. "I didn't vote for you!”)
Here I end with the same one I ended the reply to Jon: I strongly bet that much of
what people think they believe for “Natural” reasons are actually learned beliefs
through formal systems. I don’t think farmers before Newton had a Cartesian and
Newtonian concept of space x time, or that they would have been bothered by
Einstein. I don’t think they would have cared about Einstein any more than they
cared about Newton. They had some ontology of “things", and the “places” that
things *occupy*. And probably an ontology of keeping appointments, which in a more
formal world might entail something analogous to a “theory of mind” construction
about what other people are doing somewhere else “at the same time” as you are doing
your thing here. But my default assumption would be that any of this only ever took
on the rigidities of a Cartesian system after the lived practice of Newtonian
mechanics had started to make some of its rigid entailments part of routine
experience. Then it became a struggle to let that go when Minkowskian geometry
required something different.
I don’t mean to be perverse and excessive in denying the implications of folk
physics: Probably, had farmers been dragged through it (strongly against their
will), they would have found QM’s notion that what we _should_ call a _thing_
can be characterized by “being at” multiple “places” more difficult than
Newton’s “thing at a single place”. But I’m not sure how much trouble it would
have been. Considering the worldviews people are proud to claim they hold in
various religious and superstitious traditions, the things asked from modern
physics seem relatively benign as imaginative lifts.
Would be nice to have something substantive to say about any of this, that
would deserve to last. But I don’t think I do.
Eric
--
ꙮ Mɥǝu ǝlǝdɥɐuʇs ɟᴉƃɥʇ' ʇɥǝ ƃɹɐss snɟɟǝɹs˙ ꙮ
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