Tuesday, March 01, 2005

Multidimensional

Edwin A Abbott's Flatland: A Romance of Many Dimensions (available online) is one of the oddest little things I've ever read.

You see, on the one hand, Flatland did anticipate the mathematics of relativistic space. On the other hand, the book is pure social satire.

Flatland came out in 1884. The author was Edwin Abbott, a progressive Anglican clergyman. He believed we should use our minds to sort out the rising debate between science and religion.


There are many editions of Flatland, distinguished by their introductions and having a chiefly social, mathematical, or theological bent.

It's a very slim volume in two parts. The first reads like a travel report, with sections on, for example, climate and housing, inhabitants, women, painting, and priests, with some discussion of social structure and a few historical incidents thrown in.


"O brave new worlds, that have such people in them!" Posted by Hello

The second part is both more mathematical and philosophical in nature. The narrator, A. Square, glimpses Lineland (where sound is a major component of the inhabitants' relationship to the world), and then has a 3-dimensional visitor from Spaceland, which he also comes to be able to perceive.

The society of Flatland is founded on the principle that one's biology determines one's status. The inhabitants are regular geometric figures. The more sides you have, the higher in the social hierarchy. "Our whole social system is based upon Regularity, or Equality of Angles." A moral shock has the power of physical repercussions, throwing back a "criminal's" descendants to a lower class.

Women are essentially lines, at the bottom. From Flatland, on women:
One other word of warning suggest itself to me, though I cannot so easily mention a remedy; and this also refers to our relations with Women. About three hundred years ago, it was decreed by the Chief Circle that, since women are deficient in Reason but abundant in Emotion, they ought no longer to be treated as rational, nor receive any mental education. The consequence was that they were no longer taught to read, nor even to master Arithmetic enough to enable them to count the angles of their husband or children; and hence they sensibly declined during each generation in intellectual power. And this system of female non-education or quietism still prevails.

My fear is that, with the best intentions, this policy has been carried so far as to react injuriously on the Male Sex.

For the consequence is that, as things now are, we Males have to lead a kind of bi-lingual, and I may almost say bimental, existence. With Women, we speak of "love," "duty," "right," "wrong," "pity," "hope," and other irrational and emotional conceptions, which have no existence, and the fiction of which has no object except to control feminine exuberances; but among ourselves, and in our books, we have an entirely different vocabulary and I may also say, idiom. "Love" them becomes "the anticipation of benefits"; "duty" becomes "necessity" or "fitness"; and other words are correspondingly transmuted. Moreover, among Women, we use language implying the utmost deference for their Sex; and they fully believe that the Chief Circle Himself is not more devoutly adored by us than they are: but behind their backs they are both regarded and spoken of — by all but the very young — as being little better than "mindless organisms."

Our Theology also in the Women's chambers is entirely different from our Theology elsewhere.

Now my humble fear is that this double training, in language as well as in thought, imposes somewhat too heavy a burden upon the young, especially when, at the age of three years old, they are taken from the maternal care and taught to unlearn the old language — except for the purpose of repeating it in the presence of the Mothers and Nurses — and to learn the vocabulary and idiom of science. Already methinks I discern a weakness in the grasp of mathematical truth at the present time as compared with the more robust intellect of our ancestors three hundred years ago. I say nothing of the possible danger if a Woman should ever surreptitiously learn to read and convey to her Sex the result of her perusal of a single popular volume; nor of the possibility that the indiscretion or disobedience of some infant Male might reveal to a Mother the secrets of the logical dialect. On the simple ground of the enfeebling of the male intellect, I rest this humble appeal to the highest Authorities to reconsider the regulations of Female education.


In response, let it be noted:
Abbott was one of the leaders of the Women's Education Movement in Victorian England. He was regarded by the feminists of the day as an invaluable aid in their work to bring equal educational opportunity to women. . . I get frustrated when people reading Flatland superficially call Abbott a sexist. Abbott used satire to describe his society, which pained him so much, a society which treated women as though they were only one-dimensional. It’s a two-cultures allegory. Abbott’s sentiments are very clear. He talks about rationalists and intuitionists. All the rational goes into the two-dimensional men; all the intuition goes into the women. He found this disjunction unacceptable, and he took it to its logical conclusion in satire. His protagonist was totally limited by his own view of the world. A Square’s preconceptions had to be blasted away. They were — by his contact with a revelation of a higher order of existence.


For all the "detail" Abbott provides, there are many shortcuts. We never understand how people move through this world in any practical sense. For the purposes of his tale, the inhabitants need not be flesh and blood. He is clearly not interested in the physical aspects of existence in a world of different dimensions.

This article provides some background for the "analogy of dimensions," how the concept of a fourth dimension had evolved and to what extent Abbott would be aware of it.

In The Spirit on the Waters, he recounts the climactic visitation scene from Flatland where the hero, A Square, is confronted by the changing shapes produced in his two-dimensional universe by the passage of a being from the third dimension. He discusses the possible responses of the square, the most immediate of which might be to worship this being because of its mysterious God-like powers. Not so, says Abbott. Physical or intellectual powers do not automatically signify any of the moral and spiritual qualities we must demand of any object of our adoration. He concludes:

"This illustration from four dimensions, suggesting other illustrations derivable from mathematics, may serve a double purpose in our present investigation. On the one hand it may lead us to vaster views of possible circumstances and existence; on the other hand it may teach us that the conception of such possibilities cannot, by any direct path, bring us closer to God. Mathematics may help us to measure and weigh the planets, to discover the materials of which they are composed, to extract light and warmth from the motion of water and to dominate the material universe; but even if by these means we could mount up to Mars or hold converse with the inhabitants of Jupiter or Saturn, we should be no nearer to the divine throne, except so far as these new experiences might develop in our modesty, respect for facts, a deeper reverence for order and harmony, and a mind more open to new observations and to fresh inferences from old truths."

That last sentence just enlarges on the final phrase of Flatland's dedication, hoping that the experience of the dimensional exploration will contribute 'To the Enlargement of the Imagination and the Possible Development of that most rare and excellent gift of Modesty Among the Superior Races of Solid Humanity'.


For Abbott, visualizing a "higher dimension" is also metaphorical for knowing God.

To the Sphere visitor:
My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me — O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend — some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things...


In a final lesson, the Sphere further guides A. Square:
"I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions.

"Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn his lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy."


 Posted by Hello

Social Dimensions
A few weeks ago, China Miéville published a list of Fifty Fantasy & Science Fiction Works that Socialists Should Read. He stretches the definitiions to include Toni Morrison, Ayn Rand, and Oscar Wilde.

Of Mary Shelley's Frankenstein he notes:
Not a warning "not to mess with things that should be let alone" (which would be a reactionary anti-rationalist message) but an insistence on the necessity of grappling with forces one unleashes and the fact that there is no "innate" nature to people, but a socially-constructed one.


I've read about a quarter of the books on Miéville's list. Does that make me a quarter socialist?

The most interesting course I took in university, and undoubtedly the one with the most lasting impact, was Utopian and Dystopian Literature. In addition to introducing me to works (Zamiatin's We) and authors (for example, Doris Lessing) I might not otherwise have discovered or considered seriously, here I saw mathematicians and theologians mapping society's directions, trying to achieve a practical end to their efforts. Nowhere — not any literaure course, history, philosophy, or math — was it ever laid out more clearly to me humankind's pursuit of, and failure to obtain, an ideal.

Dimensions of the Universe
Roger Penrose has newly issued a "state-of-the-universe report," The Road to Reality: A Complete Guide to the Laws of the Universe, in which "he dismantles what is known about the nature of the universe and then puts it back together again."

The appearance of a new Penrose is a hybrid literary-scientific event. Physicists are eager to see what the matchless Sir Roger (he was knighted for his services to science) has come up with, while science fans, for whom physics is a spectator sport, hope to finally get the lowdown on how the universe turns.


The reviewer "came away stimulated, sometimes flummoxed, and with an encouraging sense one rarely gets from a science book: how very much there is we still don't know — even about Schrödinger's stupid pet trick."

Toward the end of the book, Penrose rejects superstring theory in favour of "a scheme that would construct space-time from abstract little somethings called twistors. A twistor is a higher-dimensional spinor. And a spinor? Something that ends up facing backward when it is rotated full circle in space. Penrose has played with twistors for over 40 years, he says, looking for the right fit."

Quantitative Measures
A recent meme askes which authors have you read more than ten books by. Here's my list, in order of the phases of my life in which they featured:

Carolyn Keene (the Nancy Drew books)
Agatha Christie
W Somerset Maugham
Michael Moorcock
Piers Anthony
Italo Calvino
Margaret Atwood (including poetry and children's books)
Umberto Eco (including nonfiction and children's books)
Lemony Snicket
Dr Seuss

Within the year, evidenced by those books in the stack beside my bed, I will be able to include:
Jose Saramago
Doris Lessing

The lists I've seen have a decidedly odd feel about them — featuring "genre" fiction, perhaps they say more about authors who produce tens of books than anything else.
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