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No, not really. Symmetries of an n-gon (a 2-dimensional figure with n sides) are just permutations of the n numbers representing the sides. A cognitive metaphor is to call certain types of symmetries “rotations” or “reflections”, because we can’t really describe all symmetries through a metaphor like that. The symmetries are not literally operated by picking up your figure and flipping it over. On the other hand, mathematical abstractions of dimensionality not cognitive metaphors. A “hole” is a meaningful topological abstraction in various dimensions that corresponds to what we consider a colloquial “hole” in 3-dimensional space. If I tell you a Klein bottle is a bottle that has only one surface (no inside or outside) without any “hole”, I’m not making a metaphor, I’m describing a 4-dimensional object as closely as I can describe a mug or a vase in 3-dimensional space. Just because I’m not defining it through pure mathematics and we can’t immediately visualize it (in entirety) doesn’t mean it is a cognitive metaphor, because it perfectly corresponds to the actual concept of dimensionality (instead of being a clever figurative description of it). Similarly, if we geometrically represent a black hole we can clearly see the curvature of space (from one angle), leading down to an event horizon, within which is a singularity. This representation corresponds to rolling a coin at an angle down a curvature, watching it spin at the bottom a bunch of times, then fall in. In fact, a shuttle falling out of orbit down to the event horizon would look just like this as it entered the whirl zoom of a black hole, before the unstable orbit failed and it just fell in. Even further, there is a very cogent, topological sense of what “within” means here. Just as you can fall into a manhole in the middle of the street, you can fall into the event horizon of a black hole. If we represent a street as a 2-dimensional plane, a manhole in the street is the topological space in the plane that has no points - a mathematical hole. In three dimensions, a jelly donut has a topological hole, and the hole is also 3-dimensional (the missing points would be representable as 3-dimensional vectors). Information theoretically speaking, the inside of a black hole’s event horizon does not exist - it’s strictly an absence that we can only reason about heuristically. It would be a cognitive metaphor for me to say that we can only reason about the inside of the event horizon in the same way we can reason about the middle of a donut that tastes like nothing because there is “no donut”. But it is not a cognitive metaphor to call the black hole a “hole”, because fundamentally and literally it acts like one. The singularity inside the black hole is not a hole, but that’s different. Our common sense of what it means to be a hole is literally described by the mathematical abstraction of a hole. When mathematical abstractions are actually just literal abstractions of what we already understand, they’re not cognitive metaphors. If you’d like an example of a cognitive metaphor insead of a pure abstraction, look at string theory. While our common sense of a hole literally corresponds to its 3-dimensional topological abstraction, our common sense of what we call a “string” has no precise mathematical correspondence. A string in the common sense of the word is a 3-dimensional object that is very long and thin. A string in the mathematical/physical sense is a one dimensional object, like a line. To call it a string is to invoke an intuition of something like a small thread that in some sense seems “barely” 3-dimensional. |
There's your metaphor. If you didn't need to qualify it that way, you could perhaps get away with saying otherwise—but with the qualification, your telling us about a different domain in which something has the same 'structure' as a (physical) hole. That's exactly how analogies/metaphors work: two things which are analogous share the same structure but have representations of those structures in differing domains.
There's a simpler way of seeing that it's metaphor still, though. You have chosen some characteristics of holes and arbitrarily decided that they are the ones which define it—but if we want to say it's literally a hole and not metaphorically, then all the attributes which familiar physical holes have should apply. For instance, there should be an interior surface, and things inserted into it should be retractable. Your description treats the singularity and event horizon as two distinct objects, which provides a kind of solution to the second—but it seems like those two things aren't as readily separable as, for instance, if we had a hole in the ground and it was filled with a powerful acid: in that case it's clear which is hole and which is thing filling it. Perhaps I'm mistaken, but I'd bet that the way in which the singularity 'fills' a black hole cannot be anything more than metaphorically.