No, hasty generalization is the case, as there are variables that can make this untrue. Namely, that the statement doesn't take into account unmentioned spaces greater than everything in the verse, which we do not know can even exist.
You acknowledge that the Hitler example qualifies as such. Why? What is the variable that can make it untrue here? Taken literally, there is none. The only answer is that the person who made that statement didn't make it with such precise attention as to take such extreme circumstances into account for the general rule. That the statement is not to be taken that literally. Same with a transcend dimensions statement. If an author says a character transcends dimensions, it's a big jump to assume this is meant to include extreme cases like higher cardinals or generally anything we don't know to exist. Eve if the statement is reliable, that doesn't mean it is to be taken literally to this extent.
You are confusing a mathematical statement, with a statement in fiction. Heck, even a mathematician would raise an eyebrow at "above dimensions" and ask you of a dozen definitions for what the hell that is supposed to mean, as for a mathematician such statements are way to imprecise to work with.
What physics is concerned: The laws of nature decide everything. They don't just decide what doesn't exist, but also what does exist in physics. And I'm not necessarily talking about real-life laws of nature, but fictional ones too. If higher dimensions exist, those are also part of physics and laws of nature. And if they don't, that's also a part of it. In reality, nothing exists that is not governed by physics. If it exists, it exists because physics makes it so. Point is: If there is no part of physics that ever caused the creation of a higher dimensional space, then there is none. And it is not implausible, that within physics it's impossible to create one, as physics includes no mechanism for such a thing. In fact, it is imaginable, that there is a law explicitly making it impossible. We don't know for sure, but you can exclude the consideration of physical laws of reality putting a limit on the size of it, meaning they are one of those variables that can make the statement false. Hence the hasty generalization fallacy applies.
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Namely, that the statement doesn't take into account unmentioned spaces greater than everything in the verse, which we do not know can even exist" is a point that doesn't really properly address the reasoning I've made because it presupposes that the dimensions which physically exist (or can exist) in the setting are all that is relevant when it comes to indexing statements like this, which is something that I contested in my post. More specifically, I said:
Not to mention that, if a character is described as beyond dimensionality altogether, then that's not necessarily restricted only to dimensions that exist in the physical. If a verse has only 4 dimensions and then a structure encompassing it is described as exceeding dimensionality entirely, do you really think that, in-verse, that structure could be modelled as a 5-dimensional space? If so, why? And how wouldn't that just contradict the idea that it is supposed to be above dimensionality?
As far as I can tell, you didn't address this (Though I suspect that the tangent about mathematicians and imprecise statements in the third paragraph was supposed to have been the answer. If it was, do elaborate more on it, since I don't understand how it works as a response at all), so I'll posit the question again: Do you really think that, if a verse has only 4 dimensions, and then a structure encompassing those four is described as surpassing dimensionality entirely, the structure in question can be appropriately modelled as a 5-dimensional space? Does it make sense to say that its size is on par with R⁵, even though it would exceed measure entirely? Does it make sense to say that it exceeds R⁴ in the same way R⁵ does? If so, why?
The answer is that it doesn't, because that would be taking the structure and applying to it properties that it lacks (In this case, volume), and whose foundational principles it does not participate in at all. It doesn't really matter if the dimensions in question are theoretical or physically existent, because neither notions would really apply to it regardless. So, for example, if you tried to make a measure-theoretic statement about it, you'd be unable to even write down anything because there isn't even a starting point to begin discourse about such a thing from within the framework's language. It doesn't even exist in that context, functionally speaking. It's a non-thing. In other words you might say that, even if it
weren't possible for those higher dimensions to physically be brought to existence, it wouldn't really matter because of the nature of it alone.
(Inconsistent theories don't count for this argument, by the way. As you probably know, inconsistent theories have no models, which is to say there is no structure satisfying the statements in them, which is to say that they're not actually describing anything. For instance if it turns out an inaccessible cardinal doesn't exist, then there's no set that satisfies the formula describing one, and as such the formula itself might as well be air)
I can ground these questions with a scenario from a verse I've been wanting to index for a while, even. I won't get into specifics, but in that verse, there is a structure that is explicitly described as being unable to be equated to, charted out, of explained by, any model of spacetime, even when the characters attempted to include higher dimensions into the equation, due to the transcendent nature of it. Do you believe it makes sense to say the structure in question is just, itself, equivalent to something that's a couple dimensions higher? If not, what exactly distinguishes it from non-qualifying cases?
Now apply this point:
Mathematical concepts are not like this: We don't wave our hands around and conjure these things by magic, we establish primitive notions and instruments, and then work our way up, building larger and more complex things out of the foundation they provide. What this results in is a state of affairs where larger things depend on smaller, more basic ones, and are defined in terms of them. When we consider R⁵, all we're thinking about is the real number line multiplied by itself five times over (R x R x R x R x R), which is to say that 5-dimensional space is defined in terms of 1-dimensional space. To have 5 or 91 or 81,519 dimensions, we first need to define 1 dimension. This is a degree of uniformity and dependence between members of a group that is not present in other cases where a hasty generalization is being made, so, for all intents and purposes, when you transcend one dimension, you already transcended them all.
This transcendence I speak of, of course, being on a fundamental level. This is an important distinction to make because when we talk about all these fancy terms like "transcendence" and "qualitative superiority," all we really mean is "This thing has uncountably infinitely more power than this other thing," which is really not superiority over the lesser thing in any fundamental sense. Compare a 3-D object that has length, height and breadth with a 2-D object that has only length and height. The 3-D object isn't "above" the dimensions of length and height, it's just infinitely larger than something that has only those dimensions and no third one, and were you to remove 2/3 of its dimensions you'd find it is now reduced to a 1-D object.
And it's not hard to see why it doesn't fall under a hasty generalization.
(And ontop of that you didn't answer a question I made: What of verses where the structure of reality is, in fact, not dictated entirely by physics? What of verses where mathematical structures are the basis of what exists?)
Also, now that I think about it, the Hitler example you made is a bad one, anyway, because by using it you're trying to make a point that generalized statements don't necessarily reflect the view and intent of the person who wrote them down, but the issue is that you're trying to use out-of-verse factors (Authorial intent) to try and make a conclusion about in-verse factors (The scope of the statement), acting as if the former is what ultimately matters, and not the latter, which is futile when the former is completely unknowable at the end of the day.
So, if a book says that "Murder is bad," then, sure, you can take this to mean that all murder is bad according to it. As for the deeper intent of the person who wrote it, though? That's a mystery. That person could've thought that murder, even if bad, is ultimately a necessary evil in certain cases. They could've indeed been against killing Hitler. They could even be a person who condones bad acts (Acknowledging something is bad doesn't mean you're against it, or specific instances of it). We just don't know, and this intent, being so cloudy, is thus a non-factor. When all we have is the text, we go by the text (And before you say something like "But the text doesn't mention that this thing is above aleph-many dimensions," the rest of this post is dedicated entirely to explaining why it doesn't need to, so, if that's on your mind, don't)
Now, magic is the same game. A fictional work is not only governed by the laws of physics. There are also magical laws and forces. What you are wrong about is the idea that a force would need to suppress the existence of higher dimensions. That's because you go in with the weird idea that if they exist if nothing prevents them. That's wrong. It's not that higher dimensions need to be prevented by something, it's that they need to be caused by anything to begin with. If nothing can cause a aleph_1 space to exist, then it doesn't and can't.
So, if say a god creates everything in a verse, then if there is a limit on its power as for how large structures it can create, then that is also the limit on how large the verse can be, due to the rules governing the supernatural powers of the verse.
Or, other example, maybe there are magical rituals that can create higher dimensions, but those could be governed by occult laws that prevent the creation of any too large. E.g. maybe you can use the ritual as often as you like, but due to the inherent mechanics of magic layering it infinite times at once to pass from finite to infinite dimensions is not possible. Since that's just how magic works. And due to that nobody could create more dimensions, so there are none. Then it would be rightful to say that a character that transcends all (possible) dimensions is just aleph_0 dimensions level, as in the verse having more is impossible by its internal logic. Hence, we have another variable that can turn the generalization false. That makes 3.
Bolded tidbit is not what my point is. Even if there is no physical mechanism that'd condition the material existence of higher dimensions, those higher dimension could still be a thing in the abstract, the logical, which segues into the argument above. Physical possibility isn't necessarily relevant here. Unicorns don't exist, sure (I wonder what one tastes like, though...), but the idea of a unicorn does.
Aren't we forgetting one teensy-weensy, but ever so crucial little, tiny detail? We see that stuff happening. This isn't about knowing author intent, this is about putting a minimal amount of effort into interpreting what is actually supposed to be going on in story.
If a character blows up a mountain we don't know how many tons of TNT the author had in mind, but we know the character blew up a mountain.
If we just have a statement "this character blows up a mounatin", know what we do in calculations? We take the smallest mountain possible and calc that.
And if the statement is just "this character blow up a hill"? Then we don't calculate it at all, because we can't tell what is going on with enough certainty to make any estimate.
You can't just take all statement to it's most literal conclusion because "we don't know what is meant". You have to try to properly interpret them. If the statement is vague, you go to the lowest end of what that vagueness can describe. Just because you can't tell whether the author meant the smallest mountain, doesn't mean you go and rank the character by a bigger one. You look through all legitimate ways the statement could be meant and interpret it in the lowest reasonable way.
So, the point is, amongst all reasonable interpretations, many lower than what you propose are contained. The statement is probably not supposed to be interpreted the way you propose. But even if we assume that authors are mysterious beings with unknowable thoughts, we would still not take it like that, because other reasonable interpretations are still available.
You don't win this debate by proving that your interpretation is not categorically excluded. You only win if you can show that it is with reasonable certainty correct. And you will have a hard time of that if anyone with half an understanding of the subject knows that your interpretation is not the intention, even if nobody knows which other interpretation is the true intention.
I would say it is about author intent, yes. Whenever you mention "what is actually supposed to be going on," you, like it or not, try to invoke the intention of the person who wrote the story. If your point is actually "This statement is satisfied by much lower tiers, and as such we will choose something around that range, instead of the highest possible takeaway from it," then you're more than free to bring it up (I am doing it myself, after all. I'm not proposing that being above dimensionality is automatically Tier 0, but High 1-A), but the point about what authors know or don't know is wholly unrelated to that and not even necessary for its formulation, as you yourself agreed to when you said "
But even if we assume that authors are mysterious beings with unknowable thoughts, we would still not take it like that, because other reasonable interpretations are still available." I think it borders on a non-sequitur, actually.
And then, of course, there's the fact I'm arguing that "one dimension higher" is not a rating that satisfies statements about things being above dimensionality, to begin with, so, yeah. Those paragraphs don't actually do much to address what I said. I never, once, attempted to argue anything you're accusing me of.
No, higher tiers should be harder to get because extraordinary claims require extraordinary evidence. That works in real life that way and just as much in fictional analysis. I expect more evidence for a High 1-A character being Higher 1-A, than for a Wall level character being Wall level. If your argument that a character should have a 10% higher rating has a 10% chance of being wrong that's ok. If your argument that a character being infinite levels of infinity higher than it's ranking has a 10% chance of being wrong that's concerning. The more wrong you potentially are, the more certain you should be that you are not. If my character Bob has a 5% chance of being the strongest character in fiction, but otherwise in 9-B, and my character Karl has a 5% chance of being 9-B, but otherwise is 9-C, I am much more inclined of approving Karl to have a 9-B ranking, but not Bob
So, it isn't about "difficult to get". They can be easy to get, by just having good really clear evidence. Like, as you mentioned, being explicit about cardinals. However, if you want a Low 1-C character to be High 1-A based on a single argument, than that argument better be rock solid. That it would make sense for a statement to mean High 1-A isn't enough. You need to know that it actually did. The error is too large for vague guesswork.
The Sagan Standard is a good saying but it's not really what I'd call a good principle to base your stance on. This being due to the fact that, when you get down to it, it's not really clear that something like "If a character is above dimensionality, then they are above aleph-many dimensions" is an extraordinary claim. In fact, whether it is one or not is exactly what the above discussion is all about, so if you want to mention it, that's fine, but don't expect it to have any actual weight. Without something substantiating why it is that the claims I am making are extraordinary, it's just a platitude that rings quite hollow. At most it will make your argument seem more rational to some onlookers, but that is really it.
In short what you're saying doesn't mean much when my whole argument is that existing above dimensionality is already "good really clear evidence" on par with being explicit about cardinals. I acknowledge that this doesn't necessarily invalidate your arguments done in the paragraphs above, sure, but that's not the point here: The point is that you are acting like "High 1-A should be held to higher standards of evidence" is something that directly attacks my stance, when it isn't, since my whole argument is that a specific kind of statement does, in fact, hold up to standards of evidence. So you may as well not have brought that up at all, in my view.
That aside, I find it amusing that you doubt the existence of singularities, as they arise from the formalism of a theory with much evidence like general relativity (although we have measured them by now), but don't doubt the existence of higher dimensions of which we know no mechanism which would indicate their physical existence (or non-physical existence in any tiering relevant fashion).
This seems to be a misunderstanding of my point, and a false equivalence ontop of that. "Nothing in this world can damage this character" is a statement that relies on things that physically exist in the verse, because something that doesn't exist cannot be said to be "in this world" under ordinary definitions. Being above dimensionality doesn't necessarily rely on the dimensions that physically exist in the setting, like I said up there, and which you yourself acknowledge to a degree when you allow characters to scale above "possible" dimensionalities. That's the point I've made that you, it seems, have left unanswered.
Even if the verse had actually presented a black hole, it would still be assumptive, because it stretches the statement to no end. Like, I bet you in Dr. Who someone has at some point created a black hole on Earth and yet we wouldn't give a character stated to be immune to everything that was ever on Earth a High 3-A durability rating, just because there was technically is a thing with High 3-A destructive power at some point 10 years ago in a different storyline. Even there we would probably expect some closer evidence of that black hole being included.
And why is that, exactly? To all of those things. If you translate that line of reasoning to the case of dimensionality, then the argument becomes "Even if the verse actually presented a space with X number of dimensions, it would still be assumptive to assume that a character described as above dimensionality would be superior to it." So, for example, you'd have to make it possible for there to be cases where a character is "beyond-dimensional" to a 2-D structure but not so to a 3-D structure, even if 3-D structures exist elsewhere in their verse, because right now you are saying that, if a statement says that a character is beyond a class of things, we won't treat them as above all things in that class even if the verse presents said things as existing physically.
Needless to say, this is an absurdity, because you are positing a fundamental difference between spaces of different dimension that makes it so you can exist beyond the nature of one without doing the same to the other, which is very much incorrect. For an n-dimensional space, all the dimensions that comprise it are interchangeable, they are the exact same dimension, ultimately. Dimensions have no ordinal positions (There is no such thing as the "first" dimension, and neither is there a "second" or "fourth" dimension. It makes no sense to consider length as coming before height, for instance, and neither does it make sense to consider height as coming before depth), and the distinction between them breaks down entirely when you examine it closely, too.
For example, take a cube. It has length, height and breadth. Now, remove the length and the breadth of it. What you have left is just a line segment standing on the vertical, laid over the y-axis, which... is literally just a length. Now remove its length and its height, and all you have now is its breadth, which ultimately is just a line segment laid over the z-axis. Once again, literally just a length. So you see here that there quite literally is no intrinsic distinction between one dimension and another. They're all the exact same.
As such, I repeat what I said up there: When you transcend one dimension, you already transcended all of them. If you are above dimension in relation to a 4-dimensional space, then you are also above it in relation to a 30-dimensional space, because all the dimensions that comprise the 30-D space are just additional lengths that are in no way different from the ones comprising the 4-D space. So if you transcend "length" (Or "measure," for the more mathematicaly inclined) in general, then you're above all those other lengths as well, not just a single one.
