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How is a spectrum supposed to not have a total ordering? To me saying sth is a spectrum always invokes an image of being able to map to/represent the property as an interval (unbounded or bounded) which should always give it a total ordering right?
How is a spectrum supposed to not have a total ordering?
I'm pretty sure a spectrum is always totally ordered. You can't say "this point on the spectrum holds no relation to that point", because then it's not a spectrum.
Only 1D spectrums. In 2D spectrums there's only a maximum relative to a specific 1D projection
It all comes down to definitions. First off, Totally Ordered is a property of the function that compares two elements not the set you are talking about. most sets have total orderings (if the axiom of choice is true then all sets have a total ordering). With Fields and vectorspaces there is the concept of a totally ordered Field which is essentially when the total ordering is compatible with it's field operations (e.g the set of complex numbers has many total orderings, but the field of complex numbers is not an ordered field).
So it really depends on how we define the sexuality spectrum. So long as it's simply a set then it has a total ordering. But if we allow us to add and multiply the gays then depending on how we define those functions it could be impossible to order the gay field.
Also a total ordering doesn't mean that there is exactly 1 maximal element (it would need to be a strict total ordering to have that property), so we can all be the gayest.
There could also be an elite group of the gayest people on earth. Or it could just be 2 gay lords.
I think that same as you can't get an arbitrarily accurate measurement of length, since at sub-atomic scales you can no longer perfectly define where exactly an object ends (and eventually you'll reach planck scales and quantum foam and the task becomes even more impossible), it might not be possible to measure gayness with enough accuracy to decide between the most gay contenders. Let's be real - gayness is probably extremely fuzzy if you look into it closely.
Gay field... A word I never imagined will encounter in my life
The color wheel, for instance
A colour wheel is not even partially ordered, I don't think. There is a relation between some colours on the wheel but it's not an ordering.
You can impose a partial ordering on it. HSL uses a hue angle. If we assume full saturation and lightness and pick an arbitrary direction to be positive, there's your partial ordering. But not total ordering, there is no most clockwise color.
I feel like this doesn't qualify as an ordering relationship, because of the circular nature of the wheel: for any two elements a and b (a ≠ b) on the wheel it's both true that a is further clockwise than b and b is further clockwise than a (just keep rotating). This violates the antisymmetry property that an ordering relation should have.
You can fix it by establishing some point on the wheel as "least clockwise" (essentially unfolding it into just a straight line) but that immediately establishes a total ordering.
Measure acute only. Idk how to deal with 180° though
A spectrum can have multiple dimensions, but that might not matter if you're only comparing in a single dimension?
Ig thats where most of my confusion comes from, to me saying sth is a "spectrum" always evokes sth along the lines of
gay <--------------------> straight(ie one dimensional) with things mapping into this interval. But ig if you also include more than one axis in your meaning of "spectrum" there wouldn't be as straight forward of an ordering for any given "spectrum". + Like @[email protected] said technically even the 1 dimensional spectrum can have more than one order and the "obvious" one is just obvious because we are used to it from another context not because its specifically relevant to this situation.Gynophilia, androphilia, romantic attraction, sexual attraction etc. absolutely makes everything complicated yeah. And then there's cultural stuff and minor personal preferences. There's no real end to how many axis you could legitimately argue for including in a sexuality chart.
Total ordering doesn’t mean that the order is strict though. You can have multiple individuals with the same level of gayness.
Total ordering means all elements are comparable (=< would be a suitable relation), not that all elements have their individual rank (< relation).
A spectrum implies that the set is totally ordered but not necessarily strictly ordered.