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The SEP article on inverted qualia discusses this mostly as follows:

One of [Frege's] theses in The Foundations of Arithmetic is that arithmetic is “objective”, which he explains as follows:

What is objective…is what is subject to laws, what can be conceived and judged, what is expressible in words. What is purely intuitable is not communicable. To make this clear, let us suppose two rational beings such that projective properties and relations are all they can intuit—the lying of three points on a line, of four points on a plane, and so on; and let what the one intuits as plane appear to the other as a point, and vice versa, so that what for the one is the line joining two points for the other is the line of intersection of two planes, and so on with the one intuition always dual to the other. In these circumstances they could understand one another quite well and would never realize the difference between their intuitions, since in projective geometry every proposition has its dual counterpart; any disagreements over points of aesthetic appreciation would not be conclusive evidence. Over all geometrical theorems they would be in complete agreement, only interpreting the words in terms of their respective intuitions. With the word ‘point’, for example, one would connect one intuition and the other another. We can therefore still say that this word has for them an objective meaning, provided only that by this meaning we do not understand any of the peculiarities of their respective intuitions. (1884/1953, §26)

This is an inverted spatial qualia scenario. According to Frege, the inversion would not show up in the semantics of words: both Nonvert and Invert use the word ‘point’ with the same meaning, despite associating very different “intuitions” with it.

The question:

  1. Take a weakly multiversal standpoint about set theory and go to a local pair of axiomatized universes, one where there is the generalized powerset axiom and the choice axiom, the other where the natural powerset exists but is choiceless (because that axiom is omitted, there) and more acutely is not sufficiently well-ordered and so the choiceless continuum is coamorphous (properly amorphous sets cannot be sets of (all) reals and hence the continuum is not properly amorphous; however, in the choiceless light, it is shown to be thematically similar enough for us to refer to "coamorphism" as a relatively efficient term for the theme).

  2. Imagine two sentient unicorns, we'll call them "Kripke" and "Charlie": Kripke perceives spacetime as a well-orderable continuum (even if primarily or even essentially in abstracto) while Charlie's continuous perception is coamorphous. Particularly, for one unicorn the clearest/intuitive notion of infinitesimals is generalized from an ur-infinitesimal 1/A, when A is more or less abstracted from countable cardinality. The other starts from 1/B, which is an amorphous infinitesimal; helpfully, 1/A/A... to A-infinity comes out to 1/AA = 1/C (for the continuum), so letting CA be the well-ordered version, then CB is a coamorphous (power)set of the naturals. Accordingly, Kripke's viewpoint inherits the relevant topological possibilities of CA, whereas Charlie's has CB-relevant topological options.

  3. If spatial/temporal qualia could be so inverted, would the unicorns be able to tell that they had such inverted qualia, due to some (any?) topological "weirdness"(???) of the discrepancy between aleph-commensurable and aleph-incommensurable continua? (Being aleph-commensurable for a set means that e.g. the natural powerset has a version that is commensurable with the other alephs (and hence there is an aleph-continuum, after all, "somewhere"); or in a much more drastic case, per their transgression upon the choice axiom, then Reinhardt cardinals (an extremely large type of cardinal) are persistently suppressed or else there are sets whose cardinalities might not be properly well-orderable and so they would be rendered incommensurate in some way with the alephs.)

  4. Or worse, suppose that a generalized continuum of vagueness relations, including higher-order ones, i.e. a set of vague ranges whose cardinality is the continuum's, would make the unicorns have problems of vagueness in their interpretation of the physical world, such that each had their own kind of vagueness (on account of the choiceless discrepancy) which, as sets, are incommensurable, and hence we might think that the unicorns would then not be able to properly compare (mutually interpret) each other's perception anyway, and they would never really know the difference after all? (For example, a (co)amorphic multiset might have continuum-many copies of CB in it, so when Charlie's perception is "informed" by multiset "imagery," the set-metatheoretic ordering provided by this "information" is qualia-theoretically inaccessible to Kripke (in the sense that there are prequalia, or conditions that constitute qualia), who can put multisets of the continuum (the multicontinuum) into a "standard" well-order, so his understanding of the concept of copies at that level will essentially differ enough to where either unicorn's very notion of continuous identity will be qualia-theoretically inaccessible to the other.)

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    When you say anything about mathematics you have to be very precise and specific in your terminology. "Coamorphous" is not a term in mathematics (unlike "amorphous") and you don't define it, but you say it is a property of the reals without choice.
    – causative
    Jul 22, 2023 at 23:06
  • @causative one of the background ideas is that not only do both versions exist, but we can define transfinite irrationals from a relation like A/B, which since those two cardinals are incommensurable, the joint "object" {A + B} is fixed in such a state, like complex numbers have two "coordinates" per their real and imaginary characteristics. Whether anything interesting could be done with these, I'm not sure yet. But it seems like the basic characterization problem for infinitesimals leads us to allowing these "objects" because they help explain the proper infinitesimals themselves. Jul 23, 2023 at 0:24
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    So by "coamorphous" you literally meant nothing more than a continuum without the axiom of choice? You didn't say that. If you introduce a term in mathematics you must explicitly define exactly what it means. "Coamorphous" is a terrible word for that, by the way because the prefix "co-" in mathematics refers to a dual relationship, and you certainly haven't shown that the reals without the axiom of choice are in any way "dual" to an amorphous set. In fact, you haven't shown any relationship at all between amorphous sets and the reals without axiom of choice; they're completely unrelated.
    – causative
    Jul 23, 2023 at 1:23
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    @KristianBerry Saying "V / amorphous sets" would be meaningful here if you said what V was and defined what it means to take a quotient between V and "amorphous sets" and proved that "V / amorphous sets" was equal to the real numbers without choice. But, you did not say or define or prove any of those things, so it is not meaningful.
    – causative
    Jul 23, 2023 at 2:41
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    @KristianBerry You just don't understand how math works. You can't say "or permissible variations" when defining V, because you have not precisely defined what is a "permissible variation." It also makes no sense to say V is the "universe of ZFC"; if you mean the model of ZFC, ZFC has infinite possible models, not just one. And I don't think you have any precise idea what "V / (amorphous sets)" could mean. Math is not a field where you can just string together whatever words occur to you and hope it will work out.
    – causative
    Jul 23, 2023 at 3:07

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Mathematical Models and Qualia

The mathematical concepts we're discussing—well-orderable sets, coamorphous sets, and the like—are very high-level abstractions. They're tools we use to reason about the world, not direct descriptions of our experiences.

Qualia, on the other hand, are the raw "feels" of experience: the redness of red, the painfulness of pain. There's a significant gap between these high-level abstractions and these raw feels. This is part of the "hard problem of consciousness" that philosopher David Chalmers has described. It's not clear how we get from these abstract mathematical concepts to the raw feels of experience.

In the case of Kripke and Charlie, they might both use the same mathematical models to describe their experiences of the world, even if the raw feels of those experiences are different. Conversely, they might use different mathematical models but still have the same raw feels. This would depend on how exactly the mathematics relates to the qualia.

Detectability and Communication

In terms of detectability, it's not clear that Kripke and Charlie would be able to detect a difference in their qualia. They might be able to agree on all the mathematical descriptions of the world, and yet have different raw feels. On the other hand, if their raw feels led to different behaviors or reactions, then they might be able to infer a difference in their qualia.

Communication could potentially be a way for Kripke and Charlie to discover a difference in their qualia. If they tried to describe their experiences to each other, they might find that they use different words or concepts, or that they have different intuitive understandings of the same concepts. This could potentially be a clue that they have different qualia.

Vagueness, Interpretation, and Incommensurability

The idea of vagueness in this context is interesting. If Kripke and Charlie have different "kinds" of vagueness—in other words, if their experiences are vague in different ways—then this could potentially lead to communication difficulties. They might find that they can't fully understand each other's descriptions of their experiences, or that they can't make accurate predictions about each other's reactions.

Moreover, if their experiences are incommensurable in some way—as you suggest might be the case if they have different perceptions of the continuum—then this could potentially lead to even greater communication difficulties. They might find that they can't translate their experiences into each other's "language" at all.

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