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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.)

Kristian Berry
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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 '23 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. – Kristian Berry Jul 23 '23 at 00:24
  • The reason I chose the word "coamorphous" is because the general principle is the same for both *B* and (***C***: *B*), that they are not well-ordered. I admit that "choiceless continuum" might be the better phrasing in public text, but on the other hand, I hope the aesthetic of my terminology would elicit the intuitions of the reader, to solve part of the text by their interpretation of it, apart from any intent to praise or condemn it: just to "recite" it and then try to find ways to apply it (and if the final output is undefinable, then the abyss of the continuum will consume our theories). – Kristian Berry Jul 23 '23 at 00:29
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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 '23 at 01:23
  • @causative I did preface the construction by saying to use a multiverse standpoint in set theory, where if you go to a lot of the formal texts nowadays, essays of theirs, where a lot of things turn out to be rather fluid (metaphors of mice and geology and machines switching axioms on and off), you find that you might not want to try to understand the absolute entirety of that multiverse, but you are here just focused on two worlds where in one omega/aleph-zero is the only infinity axiomatized, whereas in the other an amorphous cardinality is an axiom too (or even instead). – Kristian Berry Jul 23 '23 at 01:48
  • Actually, "coamorphous" can be understood as a higher-order property, and what it means is that it is the universal (class) complement of all amorphous sets (by type), or we might write *V* / {amorphous sets}, etc. I know it's not perfectly clear in the OP, but I think some of our pre-theoretical intuitions about continuity need to be assessed more carefully, neither dismissed nor agreed with *per se*, but at least we need to test our theories of infinitesimals partially in terms of a theory of intuition. – Kristian Berry Jul 23 '23 at 01:53
  • If there was ever a case where having an AI stored rendition of someone's consciousness would be helpful, I think we could use Mr. Ramanujan now. – Scott Rowe Jul 23 '23 at 02:09
  • "*Think of a hyperplane slicing through super-hyper continua not at the point of here-now.*" - Heinlein, " The Number of the Beast" – Scott Rowe Jul 23 '23 at 02:10
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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 '23 at 02:41
  • @causative I didn't quite intend to prove things, I just used some basic alternatives among the axioms to stipulatively define a set of two set worlds, one being a logical complement of the other. The justification of this is whatever justification is provided by the multiversal standpoint, then. It says, "You're free to study whichever worlds you wish, in any way, so long as your theory is not trivialized by incoherence and self-inversion." The result here is a thought experiment about the perceptible structure of spacetime, not a strict proof (which was not substantively intended). – Kristian Berry Jul 23 '23 at 02:54
  • Incidentally, the attribute of "coamorphous" for the choiceless continuum is not meant such that the only other choiceless type is the properly amorphous. The choiceless continuum is just a *sample* of the multiplicity of choiceless sets (hence the citation of the Reinhardt problem in the OP, I should emphasize). *V* is the universe of ZFC/consistent extensions or permissible variations (and the standards of permission are very permissive, it turns out...). – Kristian Berry Jul 23 '23 at 03:04
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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 '23 at 03:07
  • I have to say that I always love Kristian's question. I hardly understand a word of them, but I admire the thought that must go into them. – Marco Ocram Jul 23 '23 at 06:52
  • @causative there are a lot of areas of math that I don't know much about yet. But I do know a *lot* about the multiverse standpoint. Over on the MathOverflow, one of the professors/researchers in this domain compared the image of math as building a sturdy house on a fixed foundation as different from the multiverse standpoint in the sense that this is more like having a house-boat going down a river. And *V* just is used as the marker for the universe, it's how they characterize Reinhardt cardinals for example. – Kristian Berry Jul 23 '23 at 09:51
  • @causative [Hamkins' essay about instabilities in the cumulative hierarchy](https://arxiv.org/abs/2208.07445) is illuminating in this connection. One example of the "randomness" he constructs, concerns the question, "How many inaccessible cardinals are there?" He notes that it is logically possible to define an *n*-valued function from higher arithmetic that spits out a finite number of inaccessibles, but the selection of that function is somewhat arbitrary. [Hamkins' multiverse essay](https://arxiv.org/abs/1108.4223) talks about switching from world to world easily, too. – Kristian Berry Jul 23 '23 at 09:54
  • I should note that in my notes on my computer, I **am** going through these ideas much more rigorously. The derivation of infinitesimals is slower than it is for Robinson's hyperreals or Conway's surreals, with the traditional definitions analyzed closely and each property classically attributed to infinitesimals being defended (or displaced) by careful reasoning. But presenting my entire theory of infinitesimals for the sake of a philosophical question about qualia would be even worse overkill than my post already is. – Kristian Berry Jul 23 '23 at 10:01
  • @KristianBerry I think you are trying to say that "coamorphous" just means "not amorphous" and by V / (amorphous sets) you actually just mean the set difference, which should be written with a backslash, V \ (amorphous sets), not an actual quotient. The problem then becomes that the reals are not an amorphous set regardless of whether you take the axiom of choice or not, contradicting what you were saying about Charlie and Kripke. The axiom of choice just has nothing to do with amorphous sets. – causative Jul 23 '23 at 15:06
  • @causative "coamorphous" means "ill-ordered but not amorphous," yes. The background assumption is that amorphous sets are the relatively most simple "ill-ordered" sets (unless there's a simpler example I don't know/recognize), so even though they're incommensurable with the continuum, they have an ambient **thematic** relationship with the continuum. It's less a substantial distinction *per se* and just a generalization over the two samples, with ill-ordered Reinhardt sets. "Choiceless" might be the simplest word usage but "coamorphous" has moments where it might flow better (it seems to me). – Kristian Berry Jul 23 '23 at 16:59
  • @KristianBerry You know the real numbers without choice do have the usual total order, right? The one where 3 < 3.07 etc. I wouldn't call them "ill-ordered" just because they don't have a well order. A well ordering of the real numbers would be something very weird and unintuitive. – causative Jul 23 '23 at 18:01
  • @causative (I can't believe it's not prompting the chat; my apologies, moderators...) I actually try to avoid the phrases "ill-founded" and "ill-ordered," because their connotations are "there's something *wrong* with those things." I prefer "parafounded" and then "choiceless" in my notes on these topics. But since I have hundreds of pages of notes at this point, I can't expect myself to post all the details of the overarching argument/reasoning, because that wouldn't be useful for the SE network (it would be too long of a question). – Kristian Berry Jul 23 '23 at 18:52

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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.