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Full disclosure: I haven't entirely finished the book yet, so I'm hoping that this isn't an obvious question.

When the Sphere visited A Square, he was aware of A Square's dream about Lineland and seemed familiar with its content. The Sphere stated:

Do you not remember - for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain - do you not remember, I say, how, when you entered the realm of Lineland, you were compelled to manifest yourself to the King, not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or section of you? In precisely the same way, your country of Two Dimensions is not spacious enough to represent me, a being of Three, but can only exhibit a slice or section of me, which is what you call a Circle. (From Chapter 16, "How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland")

How did he know that? He merely vaguely states that he "sees all things." Most of the other things the sphere describes would be easy to learn if you really were 3D (rather than flat), but this is an obvious exception.

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Most of the other things the sphere describes would be easy to learn if you really were 3D (rather than flat), but this is an obvious exception.

Why?

It may seem obvious to us that dreams are a non-physical phenomenon and wouldn't be visible to someone living in a higher dimension, but I have some possible counter-arguments to that.

  1. How do we know that our thoughts wouldn't be visible to someone who could see every atom of our brain spread out as though on a map?

    Imaging of the brain is still very much an ongoing topic of research in every field from medicine to mathematics, and very little is truly understood about how things like thoughts, emotions, and dreams actually work.

    Try to imagine, from our three-dimensional perspective, what it would be like for a four-dimensional being to look 'down' on our world. Imaging of the brain would suddenly be a trivial problem, as they'd not only be able to see the exact structure of our brain, but also be able to see every single part of it at the same time, laid out as if on a map. They could see our bones and nervous system, perhaps even the electrical impulses travelling along conduits inside our body.

    Are you really sure that that level of physical knowledge of someone's brain wouldn't be enough to determine their thoughts? At least for a higher-dimensional observer who'd made a study of lower-dimensional beings - they probably wouldn't simply be able to read off mental images from physical interactions, but an expert in lower-dimensional medical studies might well be able to deduce a surprising amount from the precise details of those electrical signals inside the brain.

  2. How much was known about the structure of the brain when Flatland was written?

    Not much, I imagine. See, for example, the following chronology of brain imaging (source):

    timeline

    I mentioned above that we don't know much even today about how things like thoughts and dreams actually work. In 1884 we knew even less. Today we think of thoughts as being the effect of electric impulses in the brain; perhaps at that time many people imagined actual images, imprinted somewhere in our heads, of what we see in our "mind's eye". If there really is such physical evidence anywhere in our bodies, it would be easily visible to a higher-dimensional being.

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  • This doesn't really answer the question. If it isn't explained how the dream was known, just say that. It shouldn't be necessary to invent some pseudoscientific explanation especially when that sort of pseudoscience wasn't present in the story. – user111 Dec 15 '17 at 16:00
  • @Hamlet Even if it isn't explained explicitly in the text how the dream was known, it can still be useful/interesting to think about how it could have been known, within the spirit of the story. I may be biased, but I do think this answer increases the reader's appreciation of the story more than a simple "we dunno" would have done. (You may think it "doesn't really answer the question", but apparently the OP thought it was good enough to accept.) – Rand al'Thor Dec 15 '17 at 16:03
  • If you think this explanation is in the spirit of the story, I strongly encourage you to cite some quotes or give an explanation for why this is the case. (I don't think this explanation is in the spirit of the story, given that the story never once discusses neuroscience or any similar type of science, but I would be willing to be convinced otherwise). – user111 Dec 15 '17 at 16:25

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