Source: http://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/howwebelieve.htm
How We Believe: The Search for God in an Age of Science
By Michael Shermer
W. H. Freeman and Company.
Friday, February 11, 2000
Abbott's surrealistic story begins in a world of two dimensions, where the inhabitants—geometrical figures such as lines, triangles, squares, pentagons, hexagons, and circles—move left and right, forward or backward, but never "up or down." Looking at a coin you can see the shapes within the circle, much like you could see the inhabitants of Flatland from Spaceland looking down; but if you turn the coin on its side, the interior disappears and you only see a straight line. This is what all geometrical shapes look like to Flatlanders.
One day a mathematician Square in Flatland encounters a stranger that mysteriously changes sizes from a point, to a small circle, to a big circle, back to a small circle, and finally vanishes altogether. Since Flatlanders do not arbitrarily grow and shrink in size, the Square is confused. The stranger explains that he is not a single circle changing sizes but "many circles in one," and to prove his three-dimensional nature to the Square he employs logic and reason: "I am not a plane figure, but a solid. You 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 made in your plane a section which you, very rightly, call a Circle."
The Square still does not understand, so the stranger, a Sphere, turns from example to analogy:
Sphere: Tell me, Mr. Mathematician, if a Point moves Northward, and leaves a luminous wake, what name would you give to the wake?
Square: A straight line.
Sphere: And a straight line has how many extremities?
Square: Two.
Sphere: Now conceive the Northward straight line moving parallel to itself, East and West, so that every point in it leaves behind it the wake of a straight line. What name will you give to the figure thereby formed? We will suppose that it moves through a distance equal to the original straight line. What name, I say?
Square: A Square.
Sphere: And how many sides has a square? How many angles?
Square: Four sides and four angles.
Sphere: Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward.
The problem, of course, is that "upward" has no meaning for a two-dimensional being who has never experienced the third dimension of "height." The Square is still confused, so the Sphere walks him through a clear-cut proof: If a point produces a line with two terminal points and a line produces a square with four terminal points, then the next number is 8, which the Sphere explains makes a cube—a six-sided square in Spaceland. This he further proves with logic: If a point has zero sides, a line two sides, a square four sides, then the next number is 6. "You see it all now, eh?" says the Sphere triumphantly. Not quite. For the dimension-challenged Square, reason is not revelation: "Monster, be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish."
With failed reason the Sphere, in a throe of frustration, reaches into Flatland and yanks the Square into Spaceland, whereupon he instantly transforms into a cube. Revelation! But then a thought occurs to the Cube. If the Sphere is many circles in one, there must be a higher dimension that "combines many spheres in one superior existence, surpassing even the solids of Spaceland.... [M]y lord has shown me the intestines of all my countrymen in the land of two dimensions by taking me with him into the land of three. What therefore more easy than now to take his servant on a second journey into the blessed region of the fourth dimension?" But the Sphere will not hear of such nonsense: "There is no such land. The very idea of it is utterly inconceivable." So the Cube, with a touch of ersatz innocence, recalls the Sphere's mathematical arguments, noting the Sphere's impatience with the Cube's impertinence:
Cube: Was I not taught below that when I saw a line and inferred a plane, I in reality saw a third unrecognized dimension, not the same as brightness, called "height"? And does it not now follow that, in this region, when I see a plane and infer a solid, I really see a fourth unrecognized dimension? . . . [A]nd besides this, there is the argument from analogy of figures.
Sphere: Analogy! Nonsense: what analogy?
Cube: ... [I]n one dimension, did not a moving point produce a line with two terminal points? In two dimensions, did not a moving line produce a square with four terminal points? In three dimensions, did not a moving square produce ... a cube, with eight terminal points? And in four dimensions shall not a moving cube—alas, for analogy, and alas for the progress of truth, if it be not so—shall not, I say, the motion of a divine cube result in a still more divine organization with sixteen terminal points? Behold the infallible confirmation of the series, 2, 4, 8, 16; is not this a geometrical progression?
The Sphere, now fit to be tied, will have nothing to do with this bohemian heresy, so he promptly thrusts the Cube back into Flatland where he becomes, once again, a lowly two-dimensional square. The story closes with the Square in prison, locked up after he attempted to explain to his fellow Flatlanders what divine dimensions he had experienced: "Prometheus up in Spaceland was bound for bringing down fire for mortals, but I—poor Flatland Prometheus—lie here in prison for bringing down nothing to my countrymen."
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This article was introduced in the 1st intensive session where we discussed "What's the moral of the story".
Well, part from this article, there's another well-known article that was discussed alongside with this... that is, the story about blind man trying to figure out how an elephant looks like.
Indeed, 4 key points were drawn from the above story:
- Perspectives
- Beliefs
- Mindset - openness and being receptive to new ideas and changes
- Responsibility and Knowledge
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