Drawing the Line. An essay on Flatland and phil. of mathematics.

For anyone curious about theoretical mathematics, alternate dimensions or the quirky old novel Flatland. This essay was done with less time to spare than I would have liked, so I apologize for the confusing syntax and grammar to any native English speakers. I only hope it's understandable enough.

To the enlargement of the imagination.”

-A. Square

A popular topic of discussion within the more science-fictional sounding part of mathematics and physics over the last decades has focused on alternative dimensions. People wonder if they are real, how many they are, how they would appear to us, if there exist unfamiliar beings there, if familiarity with a higher dimension gives us more control of the lower ones, and so the questions go on ad nauseam. In this paper I will list the classic trilogy of philosophical questions and take steps, some careful and others more adventurous, towards theorizing answers or approximations towards them. The broad, philosophical questions are, of course, ontological; “do these dimensions and the beings within exist, and in what way?”, epistemological; “can we know and learn about these dimensions and their inhabitants? Why, why not, and how?”, and ethical; “do we have duty or reason to come to conclusions regarding these dimensional matters, and, whatever the answers to the ontological and epistemological questions are, what should we do with the knowledge gained?”.

After a brief introduction of the issue I intend on approaching and contemplating in this paper, I will list in order the main events of Edwin Abbott Abbott's 1884 mathematical novella, Flatland; A Romance of Many Dimensions, and afterwards elaborate on how the events are relevant to the bigger picture of the matter at hand, or how they may lead us to creative hypotheses for answers.¹

The reader may be warned that I will not go deeply into practical matters of mathematics in this paper, and much of the elaborations within may seem made to complicate the matter more than help to seek solutions. I will refrain from making big claims easily disputed by elementary knowledge of mathematics and physics, but imaginative, mathematical and physical theorizing will abound. We are dealing with a still very theoretical field of science.

To better prevent confusion over seemingly repetitious phrases, concepts and terms to follow, I will go through some elementary definitions to begin with. When speaking of dimensions in the context of our issue here, I mean the axes by which points in space are drawn to generate spatial perspective and even reality.

  • The first dimension is quite simply any line drawn between two points in space, except there is nowhere else in 'space' to go but further along the same line, infinitely. That line we will call axis x. For context of the further explanations, let us say that x stretches infinitely to west and east.

  • The second dimension has west and east, but lines stretching north and south, along the y axes (at a 90° angle to x), can intersect that line at any angle. That way any kind of flat geometrical shapes can be formed. A square, for example, is a one-dimensional line on x of any given length, where each of its infinite amount of points are stretched along y, leaving in their wake a 'trace' until it is of equal lengths along x and y.

  • The third dimension, along with lines stretching every way along x and y, has lines intersecting them outwards and inwards. A ready square, from the previous definition, could be stretched outwards in its entirety, along the z axis, leaving again a trace until it reaches as far out along z as it does x and y. Thus we would have created a cube, or even a solid.

This is of course unnecessarily complicated for anyone who has basic understanding of simple geometry. But these are the dimensions we are used to considering in our day-to-day calculations, and it goes without saying that we live within the third dimension, made of solids and traveling freely along all three axes, x, y and z. But any child will ask themselves “what comes after 'three'?” one time or another. And so have super-scientists also done, thankfully.

  • The fourth dimension moves things into a more abstract sounding territory, or actually outside territory as such. The common, popular theory is that to us in the third dimension, the fourth dimension is time, to put it so simple as to amount to barbarism. What we did with the line to make a square, and the square to make a cube, was to stretch each point in the figure along a new axis lying at 90° angle from the previous axes each time. How would we stretch a cube along an entirely new axis? Somehow time seemed most plausible, to move us into 'hyperspace'. To accurately picture in one's mind's eye a four-dimensional cube, or a hypercube as it has been dubbed, is humanly impossible. But attempts have been made, and will get further explained later in the paper.

There is no doubt that any sane and willing person may think about dimensions alternative to our own, given a certain knowledge of the accepted theories on how each particular dimension likely appears and functions. We would hardly then accept that the same person is thinking in that same dimension, even though metaphorically one can be said to be thinking in one or two dimensions in certain instances.

A simple but functional comparison would be when a person is asked to imagine themselves walking across a famous plaza in a city they've never visited, but about which they have read enough to theoretically be able to navigate around. They may be wrong about certain functional or aesthetic details of the actual plaza (assuming it even exists outside the person's imagination or the information they use for this visualization) but they know enough for the imagined experience to feel possible and the thought experiment is a success. It is hopefully clear that this person has thought about this location and not, in fact, in it.

Some may argue that a plaza, full of tourists and pigeons, surrounded by charming architecture and bathed in warm sunlight is not comparable to a higher or lower alternative dimensional reality. But the fact is that in both cases we imagine ourselves entering a reality which we are able to make sense of, at least theoretically, and even exist in, however different or absurd the surroundings might seem to us. The more obviously possible space (the plaza) and the more absurd, theoretical space (alternative dimension) need to meet similar standards for us to consider them real in any relevant sense. By 'real', I mean existing and/or possible.

Equally, we may wonder whether a thought experiment involving a being from an alternative dimension visiting our own is a matter of a logically possible scenario or simply a leap of imagination. It should be no more absurd than a scenario involving extraterrestrial lifeforms or even ghosts (given we imagine the possibility of anything scientifically explicable and non-fantastic considered as a 'ghost', unlikely as that may seem).

The hypothetical situations which we might find us in as either hosts or visitors in scenarios as these may seem like the stuff of children's books, or new age brochures about ascending to higher states of consciousness and such, and have less to do with questions of mathematical and physical matters.

Luckily, in the late 19 century, an English schoolmaster, theologian and Shakespearean academic wrote a novella of mathematical fiction which paints a picture of just such an event.

In Abbott's story we accompany the protagonist A. Square (who happens to be nothing but a square) through his introduction to the third dimension by a spherical, solid visitor. On the way we also visit the one dimensional world and an imagining of what a dimensionless one would look like. Furthermore, the possibility of a fourth dimension, while only alluded to in the story, will play a larger role in the context of this paper.

One of the reasons Flatland is as intriguing and inspiring as it is may be that it asks the reader to, first, put themselves in the shoes of a two-dimensional being by describing in detail how it views its own reality, and then imagine the experience of having their world turned, quite literally, inside out. The perspective we are asked to appropriate from the beginning is completely two-dimensional. A. Square is nice enough to explain what that entails.

Envisioning Flatland as a horizontal surface, like a map, may help to prevent confusion in the following descriptions. Flatland is home to geometrical shapes of varying multitude of angles and sizes, but none of them have any height, only width and length, although they have an idea of 'up', as in 'going north', since they have the four directions and an unexplained, weak gravitational pull towards south. Despite the gravity, Flatlanders move freely along any of the north, south, east, west axes. The gravity does serve some purpose of context for them, as the roofs of their pentagonal houses always face north or 'up', and the rain falls south or 'down'.

Rain, and humidity, are actually key to their perception and understanding of their surroundings, since the fog allows a Flatlander to judge the angle of any corner he faces by how abruptly its sides fade into obscurity. This technique is necessary for them as they don't have the 'thickness' to fit two eyes on their bodies and therefore only have one, so they are not blessed with the same amount of depth perception as we are.

Just as with the unexplained gravity, the Flatlanders' thickness is a concession the author has to make for the story to be better imaginable to the reader. We can imagine a Flatlander, such as A. Square, as a royalty on a playing card with even sides and 90° angles, and a single eye on one corner. Were we to lay him on the edge of a desk and lower ourselves so that we were looking straight along the desk's surface, Square would appear to us as a thin line. In Flatland, one is thus perceived as an almost infinitely thin playing card of a sort. There are a few further matters of perception explained by Mr. Square, but they serve mainly to explain social aspects and hierarchy within Flatland's society.

Square tells of a dream or a vision he has, wherein he is transported do a place where he sees nothing but a straight line which seems to stretch on forever north and south. A segment of the line is introduced as the king of Lineland. The king and Square start arguing about matters of reality, as Square appears to the king as a voice out of thin air, seeing as his perspective is only along the line. Each of his eyes, one on each end, sees only his neighbors, one on each side. In short, all he ever sees appears as a dot to him.

We can crudely imagine Linelanders as an apparently infinite row of infinitely thin wagons in a train, each one destined to stay in the same place in the row and no windows on the sides, only back and front.

Because of their limited use of vision, Linelanders have evolved an extreme precision of hearing. By hearing, they judge the distance between and size of each other, similar to how the human ears calculate the direction of sound, where the minute time between the sound reaching each ear indicates the angle.

Square does what he can to try and explain 'left' and 'right' to the king, but is met with frustration and mounting anger. Eventually Square goes as far as to move into the king's line of sight which understandably startles the king immensely, as he has never known anything other than that same dot in front of him, with the same color and in the same distance his whole life. At that moment, the king loses his temper and attacks Square, who is startled awake back in Flatland.

At this point, the third dimension finally invades. After sending his son off to bed for supposing there might be such a thing as 3 to the power of 3 (a punishable heresy in Flatland) Square hears a voice which we would consider coming from above, but to him it seems to echo from everywhere and inside himself at once.

The voice announces that Square is wrong, and that 3³ has “an obvious Geometrical meaning”. To Square's amazement, he sees a figure forming and expanding before his eye; a dot which quickly spreads into a line, getting ever wider. What he is perceiving is Sphere, a ball shaped three-dimensional solid figure, lowering itself into the two-dimensional plane. But given the infinitely thin field of Square's vision, he only sees Sphere as an infinitely thin slice at a time; a cross section.

We can imagine pushing an inflated balloon into a body of water. A single point must first touch the surface, and the widening rest follows. What Square would see is whatever circumference of the balloon the surface of the liquid touches at each moment.

To add even further to Square's confusion, this figure has no discernible edges or angles to speak of. In Flatland, even the roundest of circles seem round simply because of their high number of angles. So, in fact, to Square the reality of the situation seems to be that a perceptible but impossibly, almost unthinkably 'beautiful' being has stepped into his reality by some way which he can not understand or imagine. Square is startled and antagonistic at first and refuses to listen to or believe Sphere's (to him) nonsensical talk of 'above'. After the realization of Sphere's pure circularity, Square's mood becomes more and more reverent.

Sphere claims that he comes from a world above and below Flatland; not from south or north, but different. He has come down into this plane after observing Square's surroundings from above, which allowed Sphere to see inside whatever Square and the other Flatlanders would consider 'solid'. Square finds this claim, understandably, absurd. To prove his point, Sphere lists the exact location of each member of Square's household around his home and even admits to having witnessed and observed Square's dreams and intimate thoughts about Lineland. Flatlander's of course have nothing covering their 'sides', and are therefore exposed from above.

After a brief but failed attempt for Square's understanding, using a Socratic interrogation about the Flatlanders' actual height, or thickness, 'infinitesimal' as it is, Sphere resorts to describing himself to reveal the truth of the matter.

“You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, on, or in, the top of which you and your countrymen move about, without rising above it or falling below it.

I am not a plane Figure, but a Solid. You may call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere – which is my proper name in my own country – if he manifest himself at all to an inhabitant of Flatland – must needs manifest himself as a Circle.”

Square is still unable to fathom this information, so Sphere appeals to Square's love of geometry and arithmetic to lead him towards understanding, going through the evolution from point to line to square and beyond. But this is, of course, blasphemous talk in Flatland, and Square attacks the heretic alien.

Sphere slips up and disappears to dive into the family's safe and bring Square valuables from within, without the safe being opened. A simple transaction from Sphere's perspective, but unimaginable to Square unless by some trickery. Tired and disappointed by this square he chose as the messiah of the second dimension, Sphere pokes at Squares insides and finally whips him up into the third dimension.

Square experiences something of a religious revelation:

“There was a darkness; then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space: I was myself, and not myself.”

“I looked, and, behold, a new world! There stood before me, visibly incorporate, all that I had before inferred, conjectured, dreamed, of perfect Circular beauty.“

Square considers himself now blessed with a divine gift, since only God should be able to see all, but Sphere comments that if that is indeed the only requirement, then every petty soul of his own world is a god in Flatland. After this forced and shocking progress of Square's awareness, the education continues.

To further explain solids, Sphere builds a cube out of fabricated squares, similar to Square himself. He lays one on top of another until they form a shape with six equal sides and eight corners, like a deck of square playing cards. Square is amazed that he himself can be considered an originator of such a complicated object.

But, after all this, a strange idea has formed with Square. He begins asking about dimensions of beings even more perfect than Sphere; beings which might easily look within Sphere and all other solids; the home of a some figure like a cube but with sixteen terminal points rather than the 'divine' eight. Sphere brushes his ideas off as 'inconceivable', even though there were indeed reports within the third dimension of figures appearing and disappearing, and whoever claimed the existence of such beings was considered a heretic. Square pushes on and is eventually cast back down into Flatland by an infuriated Sphere.

Saddened by his punishment, but still intrigued by what he has learned, Square goes to bed in his home, and dreams yet again. Now of moving through blank space, next to Sphere, towards a “bright but infinitesimally small Point”. Sphere calls this place Pointland, “the lowest depth of existence, […] the Abyss of No dimensions”. This point exist in total solipsism; not only ignorant and unaware of the surrounding world, but incapable of understanding any experience as different than his own. Whatever is said to him, he assumes to be his own clever, creative thought. There is nothing else to learn from this non-dimension.

Square awakes in his bed, excited about the prospect of delivering the prophecy and message revealed to him. But, sadly, he is dismissed as a lunatic by the flat society and imprisoned for life, and the story ends there.

This has been a rather elongated summary of a short story, but I have saved any serious thoughts and questions which rise during the reading, until now. I hope to approach the more significant ones in order with as little confusion or chaos as possible.

A general question to pose might be; what do we talk about when we talk about alternative dimensions? This would be the ontological consideration, that is, whether they can be said to exist. I fully admit that within the context of Flatland, a metaphor for the dimensional issue, we presume more than should be allowed in serious mathematical thought, including sentient yet infinitely thin beings with language common to beings of every other dimension, allowing for communication.

But, however possible or not it would be to communicate with a square, a line, a point and so on, the question of the level of their existence must be a valid one, just as with beings or the possible experiences exclusive to higher dimensions than ours. It feels easy enough to imagine beings of lower dimensions and consider oneself aware that of course they wouldn't speak our language, and that this is only an experiment; the beings would still be able to exist in some basic manner of the way we imagine them. But being able to picture something before our mind's eye is of course no sign that there must be a possibility involved. Famous enough are the examples of a candle staying lit in vacuum, or a bar of gold floating in regular water.

Said candle, the vacuum, the gold and the water can indeed be said to exist, but where would we find the square we seek to communicate with? Is it enough for me to draw a sequence of a dot (to signify an infinitely small point), a line, a square and a cube for all of these to be considered as existing? Is there some sort of Platonic existence I allude to by presenting these constructions?

One common, if a bit playful point made in discussions of dimensions is one of shadows. We can observe that a cube may be held and oriented so that it casts a completely square shadow. A square may be held so that its shadow is nothing but a thin line. A line, if pointed straight towards a light source will simply cast a dot. What could it be, then, that casts a cube as a shadow? It is absurd to think about in the same context as looking at shadows cast on flat surfaces.

That may be because we are still trying to picture it within our own capabilities. Although Abbott most likely did not have time in mind as the imagined fourth dimension, long before anything like the theory of relativity was put forth. But it is a common enough idea today. It is then proposed that an object in the fourth dimension appears as its own entire lifespan as a whole. A person would appear as herself from conception until death in one and the same moment. A construction perceptible to us might look like a long sausage, with a person's life extending through it. Rather than talking about shadows, we might imagine that a cross section of an object of each dimension looks like an object from the next dimension beneath. So, a person in our three-dimensional space appears to us as a cross section of her entire four-dimensional lifespan. You cut the sausage anywhere, and the person appears in the wound at the appropriate age depending on where along the sausage it was cut.

The way for us to imagine and talk about these dimensions seems to always require some dose of absurdity, but is that not the necessary evil until science can present further proof or definitions? To move past theses and guesses, and towards the possibility of actual experience and empirical evidence, we will have to devise ways of perceiving other dimensions. But how might that be possible?

The question moves us towards epistemological considerations; can we ever know these things? Would it come down to some external, technological equipment? We can imagine some sort of complex spectacles. Or is it possible that the human being could evolve the ability to perceive higher dimensions? Perhaps something along the lines of how we advanced in cognitive abilities through evolution; from cells to primitive lifeforms all the way to primates and now (but not 'at last', since we may have a long way to go still) to human beings.² A subtext of this is the mind/body problem. If we can at all experience other dimensions, the possibility should lie in either matter or cognition, or both, whether that means that here are different ways to achieve the goal on the side of mind vs. body or it is fallacious to consider them as separate paths. I will not delve into the mind/body problem too deeply, but touch on it if needed.

First, the physical matter, which we would utilize to help the human body to achieve perception of the fourth dimension. (I will, humbly, only attempt to theorize higher dimensions one at a time.) There would need to be a way to develop and construct some sort of technology by which we would see through to the fourth dimension. However that is to be done, it would practically work as a looking glass. With it we see. Without we do not, even though we feel aware of what we know to exist beyond our senses.

Second, the case of purely cognitive advances. In this case the human body would evolve to a stage where perception of the fourth dimension is actually a natural part of the senses. Strict training could perhaps help to push the process along over a stretch of generations, given that the right kind of focus and training had been discovered and developed. Some might even argue for the case of meditation or some sort of medication to alter the state of the brain's chemistry, and hence the mind. That does indeed involve matter, as the first option did, but the matter of the human body and brain itself. The prior option needed technology and equipment alien to the body to allow for the perception to take place.

Notice that I do not speak of 'going to' other dimensions, since they do not exist in different, unreachable locations where different rules apply. They are here, around us, at all times. I do not have the ability to notice shifts of sub-atomic structures in my hands as I type, or the radio waves passing around and through me, and I would have no reason to suppose either were real if I were not familiar with the supporting scientific evidence. Likewise, I can not perceive higher dimensions 'happening' around me, or interact with lower dimensions, even though I can easily construct and abstract the relevant geometrical forms on any piece of paper.

An overarching problem between the characters in Flatland is that varying dimensions are homes to creatures of varying perspectives and conceptions of space and reality, but again, they share a common language to illustrate these problems even further. Can we find a similar situation within our own present reality? That is, could we communicate with a being whose personal library of experiential concepts are remarkably different from ours? We could come at least theoretically close to it by researching human beings able to communicate perfectly despite always having been blind.

No matter how long I would discuss matters of experience which I have gathered by sight, whether it be painting, natural scenery, cinema, geometry or what have you, I could never be certain that I and the blind person are visualizing similar things. In general, one can not really be certain of shared thoughts or opinions with other people, but it is damn likely that two people looking with perfect eyesight at the same painting can describe at least similar colors. With the blind person, on the other hand; even though they can have experience of everything I mentioned, by smell, hearing and touch; whatever they create within their mind's eye must only be a cross section, a 'shadow', of what I try and describe to them.³

The difference between this case and the situation in Flatland, though, is that a blind person most likely has an idea of what they are missing since they regularly communicate with fellow seeing humans. They don't have the idea pushed on them that there is in fact nothing to see. To do so would be a cruel prank, indeed.

If the blind are missing out on the full experience of three dimensions, what of those who might claim to see glimpses beyond it? Since the science of dimensions has not been blessed with much concrete evidence yet, it is easy for many to claim 'dimensional interference' to explain their superstitious believes or pseudo-scientific ventures. However, only the disingenuous swindlers should be dismissed, since there is ever the slightest possibility that some individuals genuinely experience events which we can only describe as 'supernatural', and use terms such as 'ghosts'. A genuine 'ghostly apparition' might be at last explained as a higher dimensional being appearing as a three dimensional, humanly perceptible object, for a split second. Remember Sphere's phrasing in the story, where he explains to Square how he reveals a two dimensional part of himself for Square to be able to comprehend him. Those people, who happen to have these experiences are, in our times and reality, doomed to suffer ridicule from any skeptic, simply because of how easy it is to point out dozens of fakers for each possibly true case. Further advances might, if nothing more, help us to finally tell the liars from the sincere.

Quantum physics has notoriously suffered a kind of occult abuse, spawning a sub-strata of new age science called 'quantum mysticism', and even post-modernist schools of philosophy have had their reputation smudged by careless, amateur misuse of scientific concepts. Dimensional science is indeed in danger of becoming familiar enough to exploit long before any proper proof surfaces and spreads among the misinformed public. The rollback of misinformation related to quantum physics has only recently begun, as the risks have materialized over the last decade. The reader may wonder, how is this in any way relevant to the question of the possibility of alternative dimensions? For the answer I will turn at last towards ethics; a field most people seem to think the mathematician does not have to worry about at any stage in his profession. What is or could be our ethical duty when it comes to alternative dimensions?

Two perspectives of this ethical part come to mind at first; how would it be best for us to handle this knowledge vis-a-vis going after discoveries and keeping misinformation at bay, and later, should the reality of extra-dimensional beings become clear, what would be our ethical (or even civic) duty towards them? The first question relates to our duty towards ourselves, and the latter to our duty towards these dimensions or rather the mutual duty between us and them.

Regarding our own duties within, let us say, the third dimension, it must be said that most likely information will be misused however carefully and clearly it is distributed amongst the public, academic circles and those in between hoping to gain from new ways of selling their ideas or beliefs as plausible to the public by seeming more academic than they in fact are. Astrology's prosperity even in recent years despite progresses within astronomy being made constantly is one example that comes to mind. Not everyone will be persuaded, no matter the amount or clarity of the evidence put forth. As a quote often appropriated to Abraham Lincoln states: "You can fool all the people some of the time and some of the people all of the time, but you cannot fool all the people all the time".

Still, I do find comfort in my suspicion that those fooled, by snake-oil promises of higher dimensions professionally covered in scientific jargon, will veer far enough into mysticism to stay out of the way of actual scientific progress. They may congregate and attempt to reach dimensions through mystic means at their own expense and comfort. I am not aware of cases where astronomy or physics have in modern times been held back by popular beliefs ingrained by gurus and life coaches.

The duty to still chase after further progress and confirmation of the existence or non-existence of these other planes of reality, by proper means, should still be seriously considered, since there is at least strong enough suspicion and theories to warrant investigation. No one can yet say for sure what knowledge may lie within, and how it might benefit us. This field may not be fully comparable to the investigation and research done by NASA and others to mine space for information, since the planets are of course already perceivable by any seeing person, but there was a time when even the skies were believed to hold nothing but gods and fire.

When it comes to our duty towards these possible higher dimensional inhabitants, the matter is more complicated, seeing as if they exist, they exist around and amongst us already. Higher or lower dimensions are, again, not alternative places, any more than the 'possible worlds' we may mention in talk of alternative, possible timelines exist somewhere far out of reach. These dimensions are our own world seen with better or at least different sense organs, capable or perceiving what we may only theorize about, for now.

I doubt that any serious mathematician considers himself obligated to respect two-dimensional geometrical shapes as persons, but at most as aesthetically pleasant characters in the context of his passion, maybe as an artist respects and admires the different characteristic hues his paints create on the canvas.

But would we not expect respect from creatures of higher dimensions, who have likely been aware of us, or at least our theoretical existence around them? Maybe they have never been directly aware of us, but only abstractions and constructions of us, used in formulas and examples such as squares, triangles and such do for us on paper. A four dimensional person might find it weird but curious to think that a cross-section of her would look like a thirteen year old Nigerian boy, or a middle-aged investment banker, without ever touching on the reality of the dimensional matter. I can not begin to speculate what sort of calculations would involve constructions of 'simple' three dimensional beings, but a square might think the same about himself when theorizing about the existence of us.

However, if we assume that we have been an integral part of the world of the four-dimensional beings, and them fully aware of us, it would be curious to find out if they have some more control over our surroundings than we might have thought. We pretend to alter lines and squares, and maybe we actually are doing so and not only abstracting. Could it then be that events that we find chaotic and uncontrollable are in fact controlled or at least touched on by these higher beings, for their own comfort or survival? Weather, disease and aging, for example, are things we try our best to figure out and even fight, but time makes it difficult.

Would we then ask that they consider and respect us enough to trouble themselves and save us? Perhaps it would be as unlikely as human beings lending a sympathetic ear after suddenly getting a distress call from the society of rodents to stop lab experimentation, now that they had finally found the way to make themselves understood to us.

Should communications and relations be established between the third and fourth dimensions, nothing in our livelihood would necessarily change, but only our perspective towards how we have already been living. Like realizing that your pillow is full of bed mites, or that there is or is not a God, nothing in your behavior needs to change if you were not bothering either the mites or God before. But you can easily find ways to change your perspective and better your behavior with either knowledge at hand. In the case of us coming to terms with being aware of higher beings, some might even choose to consider them as divine beings, and would of course be free to do so within safe limits.

Whatever ethical duties would seem appropriate would entirely depend on the reality of the situation. The fourth dimension may well be empty of any extra conscious being. In that case, it will then be completely up to us to decide what to do with the gained knowledge and opportunities, and the ethical questions involved will be elaborated on our already established ethical code to take care that discoveries and technologies are used for good and gain of all.

All in all, this has been a matter of interdimensional communication; whether we theorize talking to a sentient geometrical shape, a ghost, our own dog or even our own god. Without appealing to religion and faith, there should at least be a plausible scientific explanation to how this might be possible. Dimensions may just be where to look for these opportunities, even if only to decide where an actual, genuine case for extra dimensions can be made. Furthermore, we would be able to recognize arguments which use less than properly explained scientific phenomenon to account for imagined objects of faith or even superstition. Just as the terms and concepts of quantum physics have been better introduced to the common knowledge and education of the public, arming people better against misinformation used for ill gains, interdimensional sciences must not fall prey to disingenuous gurus but be saved and used for all the progress it has potential for.

So far I have elaborated on the issue of dimensions, more specifically the second and fourth, since they are 'closest' to us. Without any concrete answers, since the field has not moved so far as to sincerely prove much, I attempted to highlight questions regarding the existence of these dimensions (ontology), how we might know or perceive them (epistemology), and how we might have to consider behaving towards them or their inhabitants, should they be proven real (ethics). There is no solid conclusion to this paper, but my hope is to have the reader realize that there is something more to consider there than simply thought experiments and mind games, although they, too, can be an integral part of coming up with theories on how to progress towards any discoveries. Edwin A. Abbott was just such a thinker, who, through playful imagination, sparked ideas and theories within mathematicians of each generation to come.

Notes

  1. Edwin Abbott Abbott (20 December 1838 – 12 October 1926), an English theologian and schoolmaster, originally wrote this novella (under the pseudonym 'A Square') as a satire, ridiculing the hierarchy of Victorian England. In the story, Flatlanders are judged and placed in society according to the number of their angles, the more the supposedly wiser, and anyone found with irregularities is deemed inhuman and unworthy. Each individual's aspiration is to have a healthy child, since each child is born with one more angle than it's father, and hope for their child to succeed. Women are completely without angles. They are lines, and therefore with no room for thought. They are emotional and dangerous for that reason.

  2. One is reminded of Kant's talk of space and time. To him, time and space were as tools for us to perceive and understand the world outside ourselves. We judged distances by space, and duration between events by time, for example. Space and time are therefore a kind of a filter, or spectacles, which we observe and make sense of the world through. Thus, and upgrade to those Kantian spectacles is in some sense what might be needed to reach the fourth dimension.

  3. The reader is hopefully familiar with the famous Molyneux's problem. It is a thought experiment which Molyneux (an Irish philosopher in the 17 century) proposed to John Locke (English, 17 century), wherein a blind person is asked to lay their hands on a sphere and a cube, and when he suddenly gains sight is asked to distinguish between the two objects by sight alone. This was to raise the question of how ideas were formed in the mind, through the senses or any other way.

Bibliography

Abbott, Edwin Abbott. Flatland: A Romance of Many Dimensions. New York: Dover Publications, 1992.

Crilly, Tony. "Why Are Three Dimensions Not Enough?" The Big Questions. London: Quercus Plc, 2011. 114-23.

Copyright; Gestur H. Hilmarsson, 2012

3 Comments
3 Comments
Posted by Penzilneck

For anyone curious about theoretical mathematics, alternate dimensions or the quirky old novel Flatland. This essay was done with less time to spare than I would have liked, so I apologize for the confusing syntax and grammar to any native English speakers. I only hope it's understandable enough.

To the enlargement of the imagination.”

-A. Square

A popular topic of discussion within the more science-fictional sounding part of mathematics and physics over the last decades has focused on alternative dimensions. People wonder if they are real, how many they are, how they would appear to us, if there exist unfamiliar beings there, if familiarity with a higher dimension gives us more control of the lower ones, and so the questions go on ad nauseam. In this paper I will list the classic trilogy of philosophical questions and take steps, some careful and others more adventurous, towards theorizing answers or approximations towards them. The broad, philosophical questions are, of course, ontological; “do these dimensions and the beings within exist, and in what way?”, epistemological; “can we know and learn about these dimensions and their inhabitants? Why, why not, and how?”, and ethical; “do we have duty or reason to come to conclusions regarding these dimensional matters, and, whatever the answers to the ontological and epistemological questions are, what should we do with the knowledge gained?”.

After a brief introduction of the issue I intend on approaching and contemplating in this paper, I will list in order the main events of Edwin Abbott Abbott's 1884 mathematical novella, Flatland; A Romance of Many Dimensions, and afterwards elaborate on how the events are relevant to the bigger picture of the matter at hand, or how they may lead us to creative hypotheses for answers.¹

The reader may be warned that I will not go deeply into practical matters of mathematics in this paper, and much of the elaborations within may seem made to complicate the matter more than help to seek solutions. I will refrain from making big claims easily disputed by elementary knowledge of mathematics and physics, but imaginative, mathematical and physical theorizing will abound. We are dealing with a still very theoretical field of science.

To better prevent confusion over seemingly repetitious phrases, concepts and terms to follow, I will go through some elementary definitions to begin with. When speaking of dimensions in the context of our issue here, I mean the axes by which points in space are drawn to generate spatial perspective and even reality.

  • The first dimension is quite simply any line drawn between two points in space, except there is nowhere else in 'space' to go but further along the same line, infinitely. That line we will call axis x. For context of the further explanations, let us say that x stretches infinitely to west and east.

  • The second dimension has west and east, but lines stretching north and south, along the y axes (at a 90° angle to x), can intersect that line at any angle. That way any kind of flat geometrical shapes can be formed. A square, for example, is a one-dimensional line on x of any given length, where each of its infinite amount of points are stretched along y, leaving in their wake a 'trace' until it is of equal lengths along x and y.

  • The third dimension, along with lines stretching every way along x and y, has lines intersecting them outwards and inwards. A ready square, from the previous definition, could be stretched outwards in its entirety, along the z axis, leaving again a trace until it reaches as far out along z as it does x and y. Thus we would have created a cube, or even a solid.

This is of course unnecessarily complicated for anyone who has basic understanding of simple geometry. But these are the dimensions we are used to considering in our day-to-day calculations, and it goes without saying that we live within the third dimension, made of solids and traveling freely along all three axes, x, y and z. But any child will ask themselves “what comes after 'three'?” one time or another. And so have super-scientists also done, thankfully.

  • The fourth dimension moves things into a more abstract sounding territory, or actually outside territory as such. The common, popular theory is that to us in the third dimension, the fourth dimension is time, to put it so simple as to amount to barbarism. What we did with the line to make a square, and the square to make a cube, was to stretch each point in the figure along a new axis lying at 90° angle from the previous axes each time. How would we stretch a cube along an entirely new axis? Somehow time seemed most plausible, to move us into 'hyperspace'. To accurately picture in one's mind's eye a four-dimensional cube, or a hypercube as it has been dubbed, is humanly impossible. But attempts have been made, and will get further explained later in the paper.

There is no doubt that any sane and willing person may think about dimensions alternative to our own, given a certain knowledge of the accepted theories on how each particular dimension likely appears and functions. We would hardly then accept that the same person is thinking in that same dimension, even though metaphorically one can be said to be thinking in one or two dimensions in certain instances.

A simple but functional comparison would be when a person is asked to imagine themselves walking across a famous plaza in a city they've never visited, but about which they have read enough to theoretically be able to navigate around. They may be wrong about certain functional or aesthetic details of the actual plaza (assuming it even exists outside the person's imagination or the information they use for this visualization) but they know enough for the imagined experience to feel possible and the thought experiment is a success. It is hopefully clear that this person has thought about this location and not, in fact, in it.

Some may argue that a plaza, full of tourists and pigeons, surrounded by charming architecture and bathed in warm sunlight is not comparable to a higher or lower alternative dimensional reality. But the fact is that in both cases we imagine ourselves entering a reality which we are able to make sense of, at least theoretically, and even exist in, however different or absurd the surroundings might seem to us. The more obviously possible space (the plaza) and the more absurd, theoretical space (alternative dimension) need to meet similar standards for us to consider them real in any relevant sense. By 'real', I mean existing and/or possible.

Equally, we may wonder whether a thought experiment involving a being from an alternative dimension visiting our own is a matter of a logically possible scenario or simply a leap of imagination. It should be no more absurd than a scenario involving extraterrestrial lifeforms or even ghosts (given we imagine the possibility of anything scientifically explicable and non-fantastic considered as a 'ghost', unlikely as that may seem).

The hypothetical situations which we might find us in as either hosts or visitors in scenarios as these may seem like the stuff of children's books, or new age brochures about ascending to higher states of consciousness and such, and have less to do with questions of mathematical and physical matters.

Luckily, in the late 19 century, an English schoolmaster, theologian and Shakespearean academic wrote a novella of mathematical fiction which paints a picture of just such an event.

In Abbott's story we accompany the protagonist A. Square (who happens to be nothing but a square) through his introduction to the third dimension by a spherical, solid visitor. On the way we also visit the one dimensional world and an imagining of what a dimensionless one would look like. Furthermore, the possibility of a fourth dimension, while only alluded to in the story, will play a larger role in the context of this paper.

One of the reasons Flatland is as intriguing and inspiring as it is may be that it asks the reader to, first, put themselves in the shoes of a two-dimensional being by describing in detail how it views its own reality, and then imagine the experience of having their world turned, quite literally, inside out. The perspective we are asked to appropriate from the beginning is completely two-dimensional. A. Square is nice enough to explain what that entails.

Envisioning Flatland as a horizontal surface, like a map, may help to prevent confusion in the following descriptions. Flatland is home to geometrical shapes of varying multitude of angles and sizes, but none of them have any height, only width and length, although they have an idea of 'up', as in 'going north', since they have the four directions and an unexplained, weak gravitational pull towards south. Despite the gravity, Flatlanders move freely along any of the north, south, east, west axes. The gravity does serve some purpose of context for them, as the roofs of their pentagonal houses always face north or 'up', and the rain falls south or 'down'.

Rain, and humidity, are actually key to their perception and understanding of their surroundings, since the fog allows a Flatlander to judge the angle of any corner he faces by how abruptly its sides fade into obscurity. This technique is necessary for them as they don't have the 'thickness' to fit two eyes on their bodies and therefore only have one, so they are not blessed with the same amount of depth perception as we are.

Just as with the unexplained gravity, the Flatlanders' thickness is a concession the author has to make for the story to be better imaginable to the reader. We can imagine a Flatlander, such as A. Square, as a royalty on a playing card with even sides and 90° angles, and a single eye on one corner. Were we to lay him on the edge of a desk and lower ourselves so that we were looking straight along the desk's surface, Square would appear to us as a thin line. In Flatland, one is thus perceived as an almost infinitely thin playing card of a sort. There are a few further matters of perception explained by Mr. Square, but they serve mainly to explain social aspects and hierarchy within Flatland's society.

Square tells of a dream or a vision he has, wherein he is transported do a place where he sees nothing but a straight line which seems to stretch on forever north and south. A segment of the line is introduced as the king of Lineland. The king and Square start arguing about matters of reality, as Square appears to the king as a voice out of thin air, seeing as his perspective is only along the line. Each of his eyes, one on each end, sees only his neighbors, one on each side. In short, all he ever sees appears as a dot to him.

We can crudely imagine Linelanders as an apparently infinite row of infinitely thin wagons in a train, each one destined to stay in the same place in the row and no windows on the sides, only back and front.

Because of their limited use of vision, Linelanders have evolved an extreme precision of hearing. By hearing, they judge the distance between and size of each other, similar to how the human ears calculate the direction of sound, where the minute time between the sound reaching each ear indicates the angle.

Square does what he can to try and explain 'left' and 'right' to the king, but is met with frustration and mounting anger. Eventually Square goes as far as to move into the king's line of sight which understandably startles the king immensely, as he has never known anything other than that same dot in front of him, with the same color and in the same distance his whole life. At that moment, the king loses his temper and attacks Square, who is startled awake back in Flatland.

At this point, the third dimension finally invades. After sending his son off to bed for supposing there might be such a thing as 3 to the power of 3 (a punishable heresy in Flatland) Square hears a voice which we would consider coming from above, but to him it seems to echo from everywhere and inside himself at once.

The voice announces that Square is wrong, and that 3³ has “an obvious Geometrical meaning”. To Square's amazement, he sees a figure forming and expanding before his eye; a dot which quickly spreads into a line, getting ever wider. What he is perceiving is Sphere, a ball shaped three-dimensional solid figure, lowering itself into the two-dimensional plane. But given the infinitely thin field of Square's vision, he only sees Sphere as an infinitely thin slice at a time; a cross section.

We can imagine pushing an inflated balloon into a body of water. A single point must first touch the surface, and the widening rest follows. What Square would see is whatever circumference of the balloon the surface of the liquid touches at each moment.

To add even further to Square's confusion, this figure has no discernible edges or angles to speak of. In Flatland, even the roundest of circles seem round simply because of their high number of angles. So, in fact, to Square the reality of the situation seems to be that a perceptible but impossibly, almost unthinkably 'beautiful' being has stepped into his reality by some way which he can not understand or imagine. Square is startled and antagonistic at first and refuses to listen to or believe Sphere's (to him) nonsensical talk of 'above'. After the realization of Sphere's pure circularity, Square's mood becomes more and more reverent.

Sphere claims that he comes from a world above and below Flatland; not from south or north, but different. He has come down into this plane after observing Square's surroundings from above, which allowed Sphere to see inside whatever Square and the other Flatlanders would consider 'solid'. Square finds this claim, understandably, absurd. To prove his point, Sphere lists the exact location of each member of Square's household around his home and even admits to having witnessed and observed Square's dreams and intimate thoughts about Lineland. Flatlander's of course have nothing covering their 'sides', and are therefore exposed from above.

After a brief but failed attempt for Square's understanding, using a Socratic interrogation about the Flatlanders' actual height, or thickness, 'infinitesimal' as it is, Sphere resorts to describing himself to reveal the truth of the matter.

“You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, on, or in, the top of which you and your countrymen move about, without rising above it or falling below it.

I am not a plane Figure, but a Solid. You may call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere – which is my proper name in my own country – if he manifest himself at all to an inhabitant of Flatland – must needs manifest himself as a Circle.”

Square is still unable to fathom this information, so Sphere appeals to Square's love of geometry and arithmetic to lead him towards understanding, going through the evolution from point to line to square and beyond. But this is, of course, blasphemous talk in Flatland, and Square attacks the heretic alien.

Sphere slips up and disappears to dive into the family's safe and bring Square valuables from within, without the safe being opened. A simple transaction from Sphere's perspective, but unimaginable to Square unless by some trickery. Tired and disappointed by this square he chose as the messiah of the second dimension, Sphere pokes at Squares insides and finally whips him up into the third dimension.

Square experiences something of a religious revelation:

“There was a darkness; then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space: I was myself, and not myself.”

“I looked, and, behold, a new world! There stood before me, visibly incorporate, all that I had before inferred, conjectured, dreamed, of perfect Circular beauty.“

Square considers himself now blessed with a divine gift, since only God should be able to see all, but Sphere comments that if that is indeed the only requirement, then every petty soul of his own world is a god in Flatland. After this forced and shocking progress of Square's awareness, the education continues.

To further explain solids, Sphere builds a cube out of fabricated squares, similar to Square himself. He lays one on top of another until they form a shape with six equal sides and eight corners, like a deck of square playing cards. Square is amazed that he himself can be considered an originator of such a complicated object.

But, after all this, a strange idea has formed with Square. He begins asking about dimensions of beings even more perfect than Sphere; beings which might easily look within Sphere and all other solids; the home of a some figure like a cube but with sixteen terminal points rather than the 'divine' eight. Sphere brushes his ideas off as 'inconceivable', even though there were indeed reports within the third dimension of figures appearing and disappearing, and whoever claimed the existence of such beings was considered a heretic. Square pushes on and is eventually cast back down into Flatland by an infuriated Sphere.

Saddened by his punishment, but still intrigued by what he has learned, Square goes to bed in his home, and dreams yet again. Now of moving through blank space, next to Sphere, towards a “bright but infinitesimally small Point”. Sphere calls this place Pointland, “the lowest depth of existence, […] the Abyss of No dimensions”. This point exist in total solipsism; not only ignorant and unaware of the surrounding world, but incapable of understanding any experience as different than his own. Whatever is said to him, he assumes to be his own clever, creative thought. There is nothing else to learn from this non-dimension.

Square awakes in his bed, excited about the prospect of delivering the prophecy and message revealed to him. But, sadly, he is dismissed as a lunatic by the flat society and imprisoned for life, and the story ends there.

This has been a rather elongated summary of a short story, but I have saved any serious thoughts and questions which rise during the reading, until now. I hope to approach the more significant ones in order with as little confusion or chaos as possible.

A general question to pose might be; what do we talk about when we talk about alternative dimensions? This would be the ontological consideration, that is, whether they can be said to exist. I fully admit that within the context of Flatland, a metaphor for the dimensional issue, we presume more than should be allowed in serious mathematical thought, including sentient yet infinitely thin beings with language common to beings of every other dimension, allowing for communication.

But, however possible or not it would be to communicate with a square, a line, a point and so on, the question of the level of their existence must be a valid one, just as with beings or the possible experiences exclusive to higher dimensions than ours. It feels easy enough to imagine beings of lower dimensions and consider oneself aware that of course they wouldn't speak our language, and that this is only an experiment; the beings would still be able to exist in some basic manner of the way we imagine them. But being able to picture something before our mind's eye is of course no sign that there must be a possibility involved. Famous enough are the examples of a candle staying lit in vacuum, or a bar of gold floating in regular water.

Said candle, the vacuum, the gold and the water can indeed be said to exist, but where would we find the square we seek to communicate with? Is it enough for me to draw a sequence of a dot (to signify an infinitely small point), a line, a square and a cube for all of these to be considered as existing? Is there some sort of Platonic existence I allude to by presenting these constructions?

One common, if a bit playful point made in discussions of dimensions is one of shadows. We can observe that a cube may be held and oriented so that it casts a completely square shadow. A square may be held so that its shadow is nothing but a thin line. A line, if pointed straight towards a light source will simply cast a dot. What could it be, then, that casts a cube as a shadow? It is absurd to think about in the same context as looking at shadows cast on flat surfaces.

That may be because we are still trying to picture it within our own capabilities. Although Abbott most likely did not have time in mind as the imagined fourth dimension, long before anything like the theory of relativity was put forth. But it is a common enough idea today. It is then proposed that an object in the fourth dimension appears as its own entire lifespan as a whole. A person would appear as herself from conception until death in one and the same moment. A construction perceptible to us might look like a long sausage, with a person's life extending through it. Rather than talking about shadows, we might imagine that a cross section of an object of each dimension looks like an object from the next dimension beneath. So, a person in our three-dimensional space appears to us as a cross section of her entire four-dimensional lifespan. You cut the sausage anywhere, and the person appears in the wound at the appropriate age depending on where along the sausage it was cut.

The way for us to imagine and talk about these dimensions seems to always require some dose of absurdity, but is that not the necessary evil until science can present further proof or definitions? To move past theses and guesses, and towards the possibility of actual experience and empirical evidence, we will have to devise ways of perceiving other dimensions. But how might that be possible?

The question moves us towards epistemological considerations; can we ever know these things? Would it come down to some external, technological equipment? We can imagine some sort of complex spectacles. Or is it possible that the human being could evolve the ability to perceive higher dimensions? Perhaps something along the lines of how we advanced in cognitive abilities through evolution; from cells to primitive lifeforms all the way to primates and now (but not 'at last', since we may have a long way to go still) to human beings.² A subtext of this is the mind/body problem. If we can at all experience other dimensions, the possibility should lie in either matter or cognition, or both, whether that means that here are different ways to achieve the goal on the side of mind vs. body or it is fallacious to consider them as separate paths. I will not delve into the mind/body problem too deeply, but touch on it if needed.

First, the physical matter, which we would utilize to help the human body to achieve perception of the fourth dimension. (I will, humbly, only attempt to theorize higher dimensions one at a time.) There would need to be a way to develop and construct some sort of technology by which we would see through to the fourth dimension. However that is to be done, it would practically work as a looking glass. With it we see. Without we do not, even though we feel aware of what we know to exist beyond our senses.

Second, the case of purely cognitive advances. In this case the human body would evolve to a stage where perception of the fourth dimension is actually a natural part of the senses. Strict training could perhaps help to push the process along over a stretch of generations, given that the right kind of focus and training had been discovered and developed. Some might even argue for the case of meditation or some sort of medication to alter the state of the brain's chemistry, and hence the mind. That does indeed involve matter, as the first option did, but the matter of the human body and brain itself. The prior option needed technology and equipment alien to the body to allow for the perception to take place.

Notice that I do not speak of 'going to' other dimensions, since they do not exist in different, unreachable locations where different rules apply. They are here, around us, at all times. I do not have the ability to notice shifts of sub-atomic structures in my hands as I type, or the radio waves passing around and through me, and I would have no reason to suppose either were real if I were not familiar with the supporting scientific evidence. Likewise, I can not perceive higher dimensions 'happening' around me, or interact with lower dimensions, even though I can easily construct and abstract the relevant geometrical forms on any piece of paper.

An overarching problem between the characters in Flatland is that varying dimensions are homes to creatures of varying perspectives and conceptions of space and reality, but again, they share a common language to illustrate these problems even further. Can we find a similar situation within our own present reality? That is, could we communicate with a being whose personal library of experiential concepts are remarkably different from ours? We could come at least theoretically close to it by researching human beings able to communicate perfectly despite always having been blind.

No matter how long I would discuss matters of experience which I have gathered by sight, whether it be painting, natural scenery, cinema, geometry or what have you, I could never be certain that I and the blind person are visualizing similar things. In general, one can not really be certain of shared thoughts or opinions with other people, but it is damn likely that two people looking with perfect eyesight at the same painting can describe at least similar colors. With the blind person, on the other hand; even though they can have experience of everything I mentioned, by smell, hearing and touch; whatever they create within their mind's eye must only be a cross section, a 'shadow', of what I try and describe to them.³

The difference between this case and the situation in Flatland, though, is that a blind person most likely has an idea of what they are missing since they regularly communicate with fellow seeing humans. They don't have the idea pushed on them that there is in fact nothing to see. To do so would be a cruel prank, indeed.

If the blind are missing out on the full experience of three dimensions, what of those who might claim to see glimpses beyond it? Since the science of dimensions has not been blessed with much concrete evidence yet, it is easy for many to claim 'dimensional interference' to explain their superstitious believes or pseudo-scientific ventures. However, only the disingenuous swindlers should be dismissed, since there is ever the slightest possibility that some individuals genuinely experience events which we can only describe as 'supernatural', and use terms such as 'ghosts'. A genuine 'ghostly apparition' might be at last explained as a higher dimensional being appearing as a three dimensional, humanly perceptible object, for a split second. Remember Sphere's phrasing in the story, where he explains to Square how he reveals a two dimensional part of himself for Square to be able to comprehend him. Those people, who happen to have these experiences are, in our times and reality, doomed to suffer ridicule from any skeptic, simply because of how easy it is to point out dozens of fakers for each possibly true case. Further advances might, if nothing more, help us to finally tell the liars from the sincere.

Quantum physics has notoriously suffered a kind of occult abuse, spawning a sub-strata of new age science called 'quantum mysticism', and even post-modernist schools of philosophy have had their reputation smudged by careless, amateur misuse of scientific concepts. Dimensional science is indeed in danger of becoming familiar enough to exploit long before any proper proof surfaces and spreads among the misinformed public. The rollback of misinformation related to quantum physics has only recently begun, as the risks have materialized over the last decade. The reader may wonder, how is this in any way relevant to the question of the possibility of alternative dimensions? For the answer I will turn at last towards ethics; a field most people seem to think the mathematician does not have to worry about at any stage in his profession. What is or could be our ethical duty when it comes to alternative dimensions?

Two perspectives of this ethical part come to mind at first; how would it be best for us to handle this knowledge vis-a-vis going after discoveries and keeping misinformation at bay, and later, should the reality of extra-dimensional beings become clear, what would be our ethical (or even civic) duty towards them? The first question relates to our duty towards ourselves, and the latter to our duty towards these dimensions or rather the mutual duty between us and them.

Regarding our own duties within, let us say, the third dimension, it must be said that most likely information will be misused however carefully and clearly it is distributed amongst the public, academic circles and those in between hoping to gain from new ways of selling their ideas or beliefs as plausible to the public by seeming more academic than they in fact are. Astrology's prosperity even in recent years despite progresses within astronomy being made constantly is one example that comes to mind. Not everyone will be persuaded, no matter the amount or clarity of the evidence put forth. As a quote often appropriated to Abraham Lincoln states: "You can fool all the people some of the time and some of the people all of the time, but you cannot fool all the people all the time".

Still, I do find comfort in my suspicion that those fooled, by snake-oil promises of higher dimensions professionally covered in scientific jargon, will veer far enough into mysticism to stay out of the way of actual scientific progress. They may congregate and attempt to reach dimensions through mystic means at their own expense and comfort. I am not aware of cases where astronomy or physics have in modern times been held back by popular beliefs ingrained by gurus and life coaches.

The duty to still chase after further progress and confirmation of the existence or non-existence of these other planes of reality, by proper means, should still be seriously considered, since there is at least strong enough suspicion and theories to warrant investigation. No one can yet say for sure what knowledge may lie within, and how it might benefit us. This field may not be fully comparable to the investigation and research done by NASA and others to mine space for information, since the planets are of course already perceivable by any seeing person, but there was a time when even the skies were believed to hold nothing but gods and fire.

When it comes to our duty towards these possible higher dimensional inhabitants, the matter is more complicated, seeing as if they exist, they exist around and amongst us already. Higher or lower dimensions are, again, not alternative places, any more than the 'possible worlds' we may mention in talk of alternative, possible timelines exist somewhere far out of reach. These dimensions are our own world seen with better or at least different sense organs, capable or perceiving what we may only theorize about, for now.

I doubt that any serious mathematician considers himself obligated to respect two-dimensional geometrical shapes as persons, but at most as aesthetically pleasant characters in the context of his passion, maybe as an artist respects and admires the different characteristic hues his paints create on the canvas.

But would we not expect respect from creatures of higher dimensions, who have likely been aware of us, or at least our theoretical existence around them? Maybe they have never been directly aware of us, but only abstractions and constructions of us, used in formulas and examples such as squares, triangles and such do for us on paper. A four dimensional person might find it weird but curious to think that a cross-section of her would look like a thirteen year old Nigerian boy, or a middle-aged investment banker, without ever touching on the reality of the dimensional matter. I can not begin to speculate what sort of calculations would involve constructions of 'simple' three dimensional beings, but a square might think the same about himself when theorizing about the existence of us.

However, if we assume that we have been an integral part of the world of the four-dimensional beings, and them fully aware of us, it would be curious to find out if they have some more control over our surroundings than we might have thought. We pretend to alter lines and squares, and maybe we actually are doing so and not only abstracting. Could it then be that events that we find chaotic and uncontrollable are in fact controlled or at least touched on by these higher beings, for their own comfort or survival? Weather, disease and aging, for example, are things we try our best to figure out and even fight, but time makes it difficult.

Would we then ask that they consider and respect us enough to trouble themselves and save us? Perhaps it would be as unlikely as human beings lending a sympathetic ear after suddenly getting a distress call from the society of rodents to stop lab experimentation, now that they had finally found the way to make themselves understood to us.

Should communications and relations be established between the third and fourth dimensions, nothing in our livelihood would necessarily change, but only our perspective towards how we have already been living. Like realizing that your pillow is full of bed mites, or that there is or is not a God, nothing in your behavior needs to change if you were not bothering either the mites or God before. But you can easily find ways to change your perspective and better your behavior with either knowledge at hand. In the case of us coming to terms with being aware of higher beings, some might even choose to consider them as divine beings, and would of course be free to do so within safe limits.

Whatever ethical duties would seem appropriate would entirely depend on the reality of the situation. The fourth dimension may well be empty of any extra conscious being. In that case, it will then be completely up to us to decide what to do with the gained knowledge and opportunities, and the ethical questions involved will be elaborated on our already established ethical code to take care that discoveries and technologies are used for good and gain of all.

All in all, this has been a matter of interdimensional communication; whether we theorize talking to a sentient geometrical shape, a ghost, our own dog or even our own god. Without appealing to religion and faith, there should at least be a plausible scientific explanation to how this might be possible. Dimensions may just be where to look for these opportunities, even if only to decide where an actual, genuine case for extra dimensions can be made. Furthermore, we would be able to recognize arguments which use less than properly explained scientific phenomenon to account for imagined objects of faith or even superstition. Just as the terms and concepts of quantum physics have been better introduced to the common knowledge and education of the public, arming people better against misinformation used for ill gains, interdimensional sciences must not fall prey to disingenuous gurus but be saved and used for all the progress it has potential for.

So far I have elaborated on the issue of dimensions, more specifically the second and fourth, since they are 'closest' to us. Without any concrete answers, since the field has not moved so far as to sincerely prove much, I attempted to highlight questions regarding the existence of these dimensions (ontology), how we might know or perceive them (epistemology), and how we might have to consider behaving towards them or their inhabitants, should they be proven real (ethics). There is no solid conclusion to this paper, but my hope is to have the reader realize that there is something more to consider there than simply thought experiments and mind games, although they, too, can be an integral part of coming up with theories on how to progress towards any discoveries. Edwin A. Abbott was just such a thinker, who, through playful imagination, sparked ideas and theories within mathematicians of each generation to come.

Notes

  1. Edwin Abbott Abbott (20 December 1838 – 12 October 1926), an English theologian and schoolmaster, originally wrote this novella (under the pseudonym 'A Square') as a satire, ridiculing the hierarchy of Victorian England. In the story, Flatlanders are judged and placed in society according to the number of their angles, the more the supposedly wiser, and anyone found with irregularities is deemed inhuman and unworthy. Each individual's aspiration is to have a healthy child, since each child is born with one more angle than it's father, and hope for their child to succeed. Women are completely without angles. They are lines, and therefore with no room for thought. They are emotional and dangerous for that reason.

  2. One is reminded of Kant's talk of space and time. To him, time and space were as tools for us to perceive and understand the world outside ourselves. We judged distances by space, and duration between events by time, for example. Space and time are therefore a kind of a filter, or spectacles, which we observe and make sense of the world through. Thus, and upgrade to those Kantian spectacles is in some sense what might be needed to reach the fourth dimension.

  3. The reader is hopefully familiar with the famous Molyneux's problem. It is a thought experiment which Molyneux (an Irish philosopher in the 17 century) proposed to John Locke (English, 17 century), wherein a blind person is asked to lay their hands on a sphere and a cube, and when he suddenly gains sight is asked to distinguish between the two objects by sight alone. This was to raise the question of how ideas were formed in the mind, through the senses or any other way.

Bibliography

Abbott, Edwin Abbott. Flatland: A Romance of Many Dimensions. New York: Dover Publications, 1992.

Crilly, Tony. "Why Are Three Dimensions Not Enough?" The Big Questions. London: Quercus Plc, 2011. 114-23.

Copyright; Gestur H. Hilmarsson, 2012

Posted by Venatio

That is alot of text, Flatland is the land of imagination that Sheldon from the Big Bang Theory was talking about right?

Here's a clip

Posted by Penzilneck

@Venatio: That's exactly right. I was surprised and happy to find that clip while doing a bit of extra research for the essay. Glad to see that Flatland isn't absolutely too obscure for the odd pop-culture reference.