This thread was prompted by the Atheism and Could You Believe discussions.
I would like your help with a problem that a friend of mine is having. My friend is Mr. A Square, of Flatland. Perhaps you’ve read the book that details his adventures? In brief, A.'s world consists of only two spatial dimensions: length and width. Notwithstanding the lack of a third dimension, the inhabitants of his world have a fully developed society - better, perhaps, than our own, because we have Pauly Shore. They have no sense of a third dimension, and in their world, “up” and “north” are synonyms.
In any event, A. lived a rather mudane life there until he was contacted by a three dimensional creature, who, to demonstrate the existence of a third dimension, pulled A. up out of his world and through a multitude of others, using a direction that A. never knew existed.
Upon his return to Flatland, he tried in vain to convince his fellows that there are dimensions other than theirs. When asked to describe such dimensions, he was reduced to inanely sputtering, “Up - but not north.” This was, as far as his brtehern were concerned, a nonsensical concept.
So he has turned to me. (Never you mind how he and I communicate) I, in turn, appeal to you. Is there a way Mr. Square can demonstrate the truth of the third dimension to creatures that cannot possibly use their senses to perceive it?
Does a third dimension exist?
Is the third dimension relevant?
If A. Square is unable to use the third dimension to manipulate objects in Flatland, then it has no relevance to its inhabitants. His experience is similar to an epiphany. A. Square is naturally convinced by the evidence of his own senses, but since no other inhabitant can share his experience, it is useless for a shared scientific understanding of the third dimension.
However, if A. Square is indeed able to manipulate object in the third dimension, he can readily show that it exists, by changing a well-defined object to its mirror image (possible only through rotation in the third dimension), or by moving something inside or outside of a given closed boundry.
Against stupidity the very gods / Themselves contend in vain.
At present, A. is unable to use the third dimension at all. He can recall what the experience was like, but cannot, on his own, duplicate the movement. I agree with you that, insofar as the rest of Flatland is concerned, the existence of a third dimension is irrelevant.
Would it be fair to describe A.'s conviction that the third dimension exists rational or irrational, given that it is irrelevant to the remainder of Flatland?
Mr. Square should remain calm in order to avoid being rammed by triangles.
All he needs to do to describe the third dimension is to state that it is at a right angle to all directions, just as the second dimension is at right angles to all directions in the first dimension.
Like may simple and complete answers, that doesn’t really help.
We three dimensional beings can examine a model of a tesseract which is a three dimensional “Shadow” of a four dimensional object. This may help us understand the difficult scope of the problem awaiting Mr. Squares explanation.
Scylla, he tells me he’s tried that calm and rational explanation. The reactions he’s gotten have been varied.
Some people dismiss the entire concept as nonsense. Others, more erudite, admit that mathematically, what he is describing is possible, but deny that it actually exists. “We are intellectually honest,” they say, “and admit that such a thing is theretically possible. But the chances of it being actually real are so close to 0% that it is, for all practical purposes, simply not true.”
This irritates A. no end, because he knows the third dimension exists. But he is stymied in his efforts to convince others. Moreover, he is really wounded by those in his world that dismiss him as irrational and weak-minded for believing as he does.
A’s “knowledge” of a third spatial dimension hinges solely upon his perceptual experience. Now, assuming that this extraordinary orthogonal dimension did indeed exist, it seems obvious that it is imperceptible to those of A’s world. Were this not the case, then it would simply be necessary to point and say, “look”.
So, A’s claim now hinges upon direct perceptual evidence for the (to him) imperceptible. This necessarily implies that his perceptions were altered during teh perios in which he experienced his “new” dimension. The agency and mechanism of this altered perception is unknown, perhaps unknowable. Is there, therefore, any reason to suspect that the altered perceptions were valid?
A, of course, claims “I saw it!”
Others reply, “Do you still see it?”
“No, but I know it is there!”
“How?”
Here, of course, A has no rational reply. Even given his conviction that the revalations of his altered perceptions was true, he has no way of verifying that the condition he saw then still hold. He cannot convince others using rational argument because there quite literally is no rational argument to support his case.
He might be correct, but not even his personal conviction is based upon rationality. It is based upon his trust in a distorted (from his norm) sensory experience.
The best lack all conviction
The worst are full of passionate intensity.
*
Perhaps Mr. Square should consider that the sphere that showed him the third dimension didn’t really do him any favors since he’s not sticking around long enough to show his evidence to others.
Maybe he should ask “why am I so special that the sphere chose me to show the third dimension to?”
Maybe he should ask “Was it possible that I ate the worm when I polished off that triangle of tequila last night?” “did I halucinate?”
Maybe he should realize that without evidence an admission of possibility is the best response he’s likely to get.
Why does Mr. Square feel the need to share, nay inflict his revelations on others. If he did indeed have a wondrous experience, why can’t he just enjoy his enlightenment?
Is it possible that it was a bunch of BS, and that it is his own inadequacies and lack of inner resources attempting to give his life greater meaning by imagining a 3rd dimension?
Does he feel compelled to share his delusion under the belief that if he can convince others than maybe that will somehow dispel his own nagging doubts?
Will this information be of any use since the third dimension is not accessible except to three dimensional beings? No? Then it’s only an interesting idea without practical application. Why is he bothering?
He should go home and have sex with his triangle and try to forget about such things if he doesn’t have the ability to integrate them into his character in a constructive fashion.
I think that Mr. Square should write an allegorical book called “Lineland” where he can show how Ms. Segment would percieve a flatland.
Actually I’d generally agree with Spiritus. There is no way for Square to present a rational argument based on a knowingly distorted perception. (well, the mathemeticians would believe him, but thats about it)
The only way to unambiguously demonstrate the 3rd dimension to the unbelieving Flatlanders is to contact the three dimensional creature and have him make an appearance in Flatland. This would be very startling to the Flatlanders, as the only thing they would see would be the edge of the plane of intersection between 3DMan and Flatland. As 3DMan moves through the plane of Flatland, he would suddenly appear out of nowhere and appear to change shape in a way they had never experienced before. If enough Flatlanders saw this new phenomenon, it would surely cause a disruption in their belief system, and perhaps force the adoption of a completly new cosmology that included the “miraculous” appearance of 3DMan.
However, if 3DMan refused to visit Flatland, or didn’t appear to enough credible witnesses, then Mr. A. Square would justifiably be labelled a crackpot and be taunted unmercifully.
TT
“It is better to know some of the questions than all of the answers.”
–James Thurber
In one sense, this is undoubtedly true. How much more convincing his descriptions would be if the Sphere were simply waiting at his command… “OK, that octagon north of me - pull him out next! He’s been particularly vocal about how crazy he thinks I am!”
However, for reasons which make perfect sense to the Sphere, he’s not hanging around to do the bidding of A. Square. And, since you and I both know the third dimension to exist, perhaps those reasons are not incomprehensible to us, either.
For instance, it’s obvious that (if the world of A. Square truly exists) there is the possibility of an infinity of similar two-dimensional worlds, never intersecting, to exist. The Sphere may be spending all his time helping out A. Prime or A. Double-Prime. He may have lost interest. He may be experimenting to see if it’s possible to convince others in A.'s world without his further intervention.
He may ask those things, all right. He may well begin to doubt the validity of his experience. “It’s just an undigested bit of beef,” he may say, unknowingly quoting Scrooge trying to dispel the memory of Marley’s visit.
But – again, you and I know that there is such a thing as the third dimension – after all, we live in it. So I am reluctant to advise him to start believing that he imagined the whole thing, when I know that he didn’t. Are you suggesting that the solution to his problem is to dismiss the experience, even when you know it to be based on reality?
I think he feels that he’s glimpsed something that transcends the mundanity of ordinary, two-dimensional existence, and he feels called upon to share it. If you were contacted by a four dimensional creature, and shown how to move through a fourth spatial dimension – would you enjoy your enlightenment in silence, or discuss it with others?
Again – he may worry about such things, but you and I both know beyond any reasonable doubt that there is, in fact, a third dimension. So for you to suggest that he’s delusional is… well… unfair.
Again I ask: if you were pulled through the fourth dimension, and had a chance to look at a figure that consisted of eight cubes, all on the outside, and suddenly it all clicked into place that the fourth spatial dimension was not simply an imaginary construct of use to n-dimensional geometry - but instead was real, and inhabited by creatures much like us (except, of course, that they too are lacking Pauly Shore; it’s axiomatic there can only be one Pauly Shore in the entire pan-verse) – would your return to our dimension consist only of going home to your wife and forgetting such things?
My friend A. is concerned that even if 3DMan (aka the Sphere) makes such an appearance, all that will happen is he’ll gain the accpetance of those who see the phenomenon personally. Even if all the witnesses carefully write down what they saw - the shape changing you describe - he’s afraid that in a few years their accounts will be dismissed as overactive imaginations, madness, or explained away as natural (2D) phenomenon.
It is easy to demonstrate multiple dimensions with mathematics.
Sure, the masses might not understand math beyond simple addition, but Mr. Square shouldn’t care what the masses think. Especially as long as his sultry circle and loyal rhomboid stick by his side. He should enjoy life, buy some merlot and go square his circle
Yet to be reconciled with the reality of the dark for a moment, I go on wandering from dream to dream.
Being a rational, compassionate and ethical three-dimensional being I would offer to accompany Mr. Square and demonstrate the 3rd dimension to any naysayers.
IF I revealed myself and failed to follow through then I would not be compassionate, rational and ethical three dimensional being, would I?
If I was unable to do so, and it were within my power I would bitchslap the sphere for being cruel.
If I could not assist Mr. Square, I would try to convince him that there was no 3rd dimension, since he seems to lack the inner resources to handle the knowledge. Truth must be tempered by compassion don’t you know.
If I was Mr. Square I would share my information with anyone interested, but not throust it out there in a vain attempt to forcefeed to uninterested parties. Perhaps I would use the knowledge gained to become the Greatest Flat Theoretical Mathematician of all time.
Maybe nothing much would change and my life would be quietly enriched because of the experience.
Mathematics, of course, prove nothing about the actual universe. It is trivial for me to create an equation of arbitrary dimensionality. That in no way implies that those dimensions represent a one-to-one correspondence with the physical dimension.
Bricker:
Shall I assume that Mr A did not find my analysis comforting?
The best lack all conviction
The worst are full of passionate intensity.
*
First of all, a few quibbles. Flatland would have three dimensions: two spatial, one temporal. What A. Square would have to convince others of is not the existence of 3 dimensions, but of four dimension. But I will take (and use) n-d to mean n spatial dimensions.
Also, how could A. Square possibly be shown a three dimensional world? Surely a brain that evolved in a two dimensional world would have no reason to devote any mental resources towards comprehending a three dimenisional world. And of course his eyes, since they are two dimensional, would be incapable of seeing a three dimensional object (our eyes, in order to see three objects, create 3d images within the eye. A 3d image would not fit in a 2d eye). So A. Square could not possibly be shown a 3d world, and so his belief that he has is irrational.
But suppose just for the sake of argument that there is some procedure that a 3d being could perfrom to show A. Square a 3d world, and suppose that this procedure were somehow incapable of being replicated by A. Square.
Would A. Square be able to prove the existence of the 3rd dimension? Yes, quite easily. It is quite obvious that an infinite number of dimnsions exist. The only question is: are there any universes in which more than two dimensions have any relevance? Now things get difficult for Mr. Square. If all the evidence he has is “Well, I’ve been to this other universe, but I can’t get back and no one else can go either” than he’s got a problem.
Is his belief in the other universe irrational? No, if he saw the universe, then it is certainly rational to believe in its existence. What would be (highly) irrational would be to tell others about his experience, and expect people to simply automatically believe him just on his say-so.
TT posted:
Why? If you suddenly saw an object appear (seemingly) out of nowhere, get bigger, then get smaller, then disappear, would you suddenly believe in a 4d universe ? Or would you believe that you just saw an object appear, get bigger, get smaller, and then disappear? Even if you determined that the object did indeed appear out of nowhere, that would just prove that there’s another universe (or place of some sort). It wouldn’t prove that the other universe is 4d.
Bricker:
Umm, isn’t it pretty much a gven from the way the OP was worded that Mr. Square was not shown how “to move through a fourth spatial dimension” or even a third such dimension? If he were able to do that, he would not need your help in convincing others of the reality of his experience. That or else he really isn’t too bright.
I’d probably mention that I’d had a certain expperience, and perhaps people can get something out of it, but I wouldn’t expect people to automatically believe that I saw what I thought I saw.
Why would they do so? If the event were properly recorded, and accurately and promptly reported by a large number of corroborating witnessess, why would it be dimissed as madness? It certainly would be better than nothing, wouldn’t it? I think that Mr. Square is just looking for excuses.
Now, I think that something needs to be said: I think that it is quite obvious, and most if not all readers have determined, that you are using this as some sort of parable or analogy or such for religious belief. Everyone’s been playing along and pretending that they don’t see the resemblance, but I think that this needs to be brought out into the open. I do not believe the analogy is valid, but if you simply want to discuss Flatland I will not go into detail as to what those objections are. And if we are just talking about Flatland, and not religion, then Scylla’s response was, I believe, inappropiate in that it was more directed at religion than at the Flatland situation. However, if you insist on restricting this discussion solely to the issue of Flatland, then I should point out that any statements I make, or indeed any conclusion that are drawn, about Flatland should not be extended to religion.
This whole thing reminds me of something from Raymond Smullyan’s 5000 BC and Other Philosophical Fantasies. In the book he proposed this thought experiment:
Imagine a planet that has an unusual phenomenon in which every object of a particular shape has the same color. All spheres are red, all cubes are green, etc. If an object changes shape it changes color on this world. This is all perfectly natural to the natives; to them “red” and “spherical” mean the same thing; in fact there is no need for them to invent two words for this.
Suppose some natives of this world leave in a spaceship and land on Earth. What would they think? They see red spheres and green cubes, but also green spheres, and blue cubes. How would they describe what they saw? “I saw objects that were both green and spherical.” “What does that mean? If an object is spherical it is red by definition. You are telling me something is both X and not-X.”
The natives who didn’t leave are forced to believe that the travelers are either mistaken about what they experienced, lying, mentally impaired, or perhaps gained some kind of mystical insight that the non-travelers don’t have. The travelers, however, realize that all that’s happened is they became aware of a “color perception”, separate from “shape perception”. Further, on returning to their world, they would recognize the “redness” and “sphericalness” of an object as being two different things; they would probably believe that they always had both perceptive senses, and were made aware of them rather than “gained” them.
Smullyan used this example in a chapter about the mind-body problem, but I think it is similar to Mr. Square’s problem. I wonder of the travellers would have an easier job than he would, of convincing the non-believers.
Do you then consider there to be something called a “physical dimension” which exists in the real world and is completely separate from mathematics? As far as I’m concerned, the term “dimension” is strictly a mathematical term, not a physical term, and so the existence or non existence of a dimension is a matter than is determined rationally rather than empirically.
Of course, we are now getting into a disagreement about the meaning more than the nature of “dimension”. Perhaps it should be clarified just what Mr. Square is trying to prove to his companions. Is he trying to prove that a 3rd dimension is a valid concept? Or is he trying to prove that it actually has a physical interpretation? If he’s trying to show the latter, well, he’s wrong. Just because there’s some 3d universe somewhere, that doesn’t change the fact that his universe it 2-d, and a 3rd dimension would have no physical interpretation (at least not a third spatial dimension). If he’s trying to prove that there is some universe where a 3rd dimension does have a physical interpretation, then he’s asking for people to accept the existence of an entirely different universe just because he says so.
Time is not considered a dimension. If you want to be testy you could say that we live in three spatial dimensions and one temporal.
Bricker mentions that A. Square is a denizen of Flatland, which to most would be sufficient to convey that we are talking about a place with only two spatial dimensions. The accepted practice both in speech and mathematics is that if you are talking about multiple dimensions you are referring to spatial dimensions unless you specifically state otherwise.
Not only are you nitpicking, you are incorrect.
You said:
“And of course
his eyes, since they are two dimensional, would be incapable of seeing a three
dimensional object (our eyes, in order to see three objects, create 3d images within
the eye. A 3d image would not fit in a 2d eye). So A. Square could not possibly be
shown a 3d world, and so his belief that he has is irrational.”
A moment’s thought will show how silly this idea is.
Have you ever watched TV?
It’s this wonderful device that represents 3 dimensional images on 2 dimensional surface. (2 spatial dimensions that is )
Then you said:
" It is quite obvious that an infinite number of dimnsions exist."
Oh really? Care to elaborate? Since it’s neither obvious, provable nor probably true I’m just going to assume that’s a Wag.
Then you said:
“If you suddenly saw an object appear (seemingly) out of nowhere, get bigger,
then get smaller, then disappear, would you suddenly believe in a 4d universe ?”
If somebody said they were about to materialize objects from nothingness, make them change shape and mass and then make them disapear. If they allowed me to examine their evidence, and explained they were doing this by moving objects through the forth dimension, and then IF THEY ACTUALLY DID IT. Well, yes. I would seriously consider the possibility of a 4th dimension.
Another thing:
Bricker’s story is not literally true. It is a parable. That is, an illustrative story designed to prove a point or stimulate thought. It does not have to be literally true or scientifically valid in order to do so.
The man a Mr. Abbot I believe (could be wrong it was twenty years ago) who wrote the book Flatland upon which the OP is based was rather careful to be mathematically and scientifically accurate in the framework he designed for his little story (there are flaws though, none of which you picked up on.) Are you going to argue that a square is an unlikely organism to evolve? It has nothing to do with the point of the whole thing. SHeesh!
You think there is a secret subtext to this thread. Why my dear Ryan, whatever do you mean?
Well of course there is &%*%^%*(#@%^#ing subtext!!!
Bricker said so in the OP for crying out loud!
Did you read it?
I think I have the right to respond to it any way I see fit within the bounds of the rules of this bored, my own judgement ang good taste.
If you find it objectionable, well tough shit.
Maybe I’m being harsh, but your half-cocked indignation and marginal understanding of the concepts involved is ruining what seemed like a rather interesting discussion.
)