Life in Three Dimensions
"Were you ever an isosceles triangle?"Since we've had a lot of very serious and technical matters from theology recently, I thought we'd go to the other end of the University and spend some time with the math department. Of course this gets me in trouble right away: first because of the tendency of certain Chestertonians to think that the ACS and its blogg, and - in fact - ALL Chestertonians - are supposed to be "lit'ry" people, and shun the technical and mathematical. But then a quiet little Greek scholar sticks her head up and points out that the "Great Commission" in Mt 28:19 has the verb maqhteusate or mathêtusate which means "make disciples" - make LEARNERs. The ancient Greeks called this hated subject "The Learning" because it was learned. And some years ago, while I was at the unnamed school doing my doctorate, and we heard about the talking doll that said "Math is hard", I spent some time looking into the matter. The answer, as you might expect, is in Aquinas (his commentary on Boethius), and the answre is that Math is easy - as far as its "class" of knowledge is concerned. Obviously, it can be that any given aspect of math might be easy- or hard - for any given person. But that is not the same thing at all.
[Gabriel Gale asks this in "The Yellow Bird" in GKC's The Poet and the Lunatics
There is something beyond expression moving to the imagination in the idea of the holy fugitives being brought lower than the very land; as if the earth had swallowed them; the glory of God like gold buried in the ground. Perhaps the image is too deep for art, even in the sense of dealing in another dimension. For it might be difficult for any art to convey simultaneously the divine secret of the cavern and the cavalcade of the mysterious kings, trampling the rocky plain and shaking the cavern roof. Yet the medieval pictures would often represent parallel scenes on the same canvas; and the medieval popular theatre, which the guildsmen wheeled about the streets, was sometimes a structure of three floors, with one scene above another.
[GKC "Bethlehem and the Great Cities" in New Witness, December 8, 1922; reprinted in The Spice of Life 139]
Christ was not only born on the level of the world, but even lower than the world. The first act of the divine drama was enacted, not only on no stage set up above the sight-seer, but on a dark and curtained stage sunken out of sight; and that is an idea very difficult to express in most modes of artistic expression. It is the idea of simultaneous happenings on different levels of life. Something like it might have been attempted in the more archaic and decorative medieval art. But the more the artists learned of realism and perspective, the less they could depict at once the angels in the heavens and the shepherds on the hills, and the glory in the darkness that was under the hills. Perhaps it could have been best conveyed by the characteristic expedient of some of the medieval guilds, when they wheeled about the streets a theatre with three stages one above the other, with heaven above the earth and hell under the earth. But in the riddle of Bethlehem it was heaven that was under the earth. [GKC The Everlasting Man CW2:305]
However interesting it may be to explore epistemology - the science of knowledge, it is a bit more than I want to write about. I have enough writing to do just now, and I want to get into something fun. It all started with last week's discussion of light - and the Lepanto novena. I say the "Luminous" mysteries, the "Mysteries of Light" - which explore the various scenes of the Public Ministry of Jesus. But one could just as easily call them the "Mysteries of Water" because water enters into each of them in a special way.
So I thought I might talk about water - thinking about the Baptism of our Lord at the Jordan, I imagined that every baptism since then contained a molecule of water which had touched Jesus.... and I started to wonder how many molecules that would be.
Well, one mole of water is just 18 grams - and a mole contains about 6.02e23 molecules - that's a computer way of writing a BIG number: 602,000,000,000,000,000,000,000 or 602 sextillion. That's plenty of molecules.
An aside about the word "mole" - this is also called "Avogadro's number" - it is really just a number with a "special name" like dozen, gross, or score... which reminds me of a limerick. But I will spare you. (If you are desperate, see here.)
But not having any nice imagery for such a vast number, I decided to look at the smaller one - 18 grams. Now that's not very big at all - but how big is it? I began to work it out, and found myself using a specialised form of mathematics called "dimensional analysis" - I wonder if it is even taught any more. It's a way of managing equations, say of physics or chemistry, or any real-world problem, so that the units are correct. Often someone is playing with miles per hour and needs the speed in feet per second - or something like that... dimensional analysis is just algebra applied to dimensions. Just to give you the answer, 18 grams of water is less than an ounce - about a shot-glass full.
Now, "dimension" comes from a Latin word meaning "measure": length, time, weight, temperature, and so on... Most of the time, we think there are "three" dimensions, sometimes called x, y, and z - left and right, in and out, up and down. It's when we talk about the fourth and other dimensions that things can get complicated. And then there are the fractional dimensions... at first, it sounds like a joke, like trying to go upstairs by exactly three and a half steps - but labelling a train platform "Nine and Three Quarters" starts sounding quite respectable to such geometers!
And perhaps you've thought I've lost the Chestertonian view? Not at all! In fact, the rest of this posting is pure GKC. You need to get the three dimension down pat, so I can proceed into other dimensions in a future posting. Please read carefully, and think about what you read; there MAY be a quiz next time...
Press here to enter the Chesterton Dimension...
The tendency of mankind to split up everything into three is hard to explain rationally. It is either false and a piece of superstition; or it is true and a part of religion. In either case it cannot be adequately explained on ordinary human judgment or average human experience. Three is really a very uncommon number in nature. The dual principle runs through nature as a whole; it is almost as if our earth and heaven had been made by the Heavenly Twins. There is no beast with three horns, no bird with three wings; no fish with three fins and no more. No monster has three eyes, except in fairy tales; no cat has three tails, except in logic. Sages have proved the world to be flat and round and oblate and oval; but none (as far as I know) have yet proved it to be triangular. Indeed, the triangle is one of the rarest shapes, not merely in the primal patterns of the cosmos, but even in the multifarious details of man's civilisation. There are three-cornered hats, certainly, and three-cornered tarts; but even taken together they scarcely provide the whole equipment of civilisation. Three-cornered tarts might be monotonous as a diet; as three-cornered hats would certainly be inadequate as a costume. The tripod was certainly important in pagan antiquity; but I cannot help thinking that its modern representative, the three-legged stool, has rather come down in the world. Evolution and the Struggle for Life (if I may mention such holy things in so light a connection) seem to have gone rather against the tripod; and even the three-legged stool is not so common as it was. Victory has gone to the quadrupeds of furniture: to the huge, ruthless sofas, the rampant and swaggering armchairs. It seems clear, therefore, that there is nothing in common human necessities, just as there is nothing in the structure and system of the physical world, to impregnate man with his curious taste for the number three. Yet he shows it in everything from the Three Brothers in the fairy-tale to the Three Estates of the realm; in everything from the Three Dimensions to the Three Bears. If the thing has a reason, it must be a reason beyond reason. It must be mystical; it may be theological.
[GKC ILN Dec 10 1910 CW28:643-4]
Mathematicians still go cracked over the mysterious properties of the number Nine; on hot days you can hear their heads going pop on all sides like chestnuts. Inventors still run about with little machines for Perpetual Motion. Philosophers still argue about the Fourth Dimension, without having the faintest reason to suppose that there is any such thing. These things are parts of the divine energy of man, because they are Games. But the disaster is this, that by calling our worst sins and tragedies by the name of "problems" we hazily remind people of these everlasting amusements - such as squaring the circle; and so make them content with slowness, with pedantry, with idleness, and with sterility. The reformer thinks himself as swift as Achilles if he goes nearly as fast as a tortoise. ...while the philosopher lives in the fourth dimension, the other three dimensions are closing in in meaner rooms and darker prisons around others of the children of men; and it takes a great deal longer to square the circle than to square the politicians. [GKC ILN Nov 25 1911 CW29:194]
The truth is that Professor Einstein has indeed revealed a kind of relativity which he did not intend to reveal. It is a relativity more relative, in Hamlet's sense of the word, than his own. Whatever be the merits of his own scientific theory, he has let out a secret about all scientific theories - or rather, to speak more justly, about the way in which all scientific theories may become scientific fashions. And that is by simply ceasing to be scientific.
And the importance of Einstein and his relativity in this relation is that, in his case, there cannot be anything scientific in the fashion, whatever there may be in the theory. In this case at least, if in this case for the first time, the public is quite certainly talking about what it does not understand. In the biological and psychological cases it may at least have been talking about what it imperfectly understood. It would not be very satisfactory for a biologist or a psychologist to be not so much a theory as a name, and not so much a name as a joke. It would not satisfy a biologist to be applauded in connection with the antics of a pantomime elephant. It would not have pleased Darwin that the Missing Link should appear only in the place of the pantomime cat. But at least it might be argued that men recognised the reference because they recognised the idea. At least it might be argued that the Darwinian idea does apply to elephants and does apply to cats. The popular impression of Darwinism was doubtless very dim and confused, as it is still. For most people it amounted to the notion that men were descended from monkeys; for many people it included the notion that men ought to scramble and fight each other like monkeys. This was not Darwin, but it was Darwinism. It was an idol more enormous, more evident, more solid, and perhaps more permanent than the idea which it misrepresented. The scientific thesis of natural selection was quite serious and thoughtful, and has been largely abandoned by scientific men. The fashionable legend was quite anarchical and absurd, and it is still firmly maintained by multitudes of unscientific men. But if the legend was a caricature of the theory, there was something in the theory to caricature. Darwin did say something about men and monkeys, as well as about cats and elephants; and the something could be popularised, if only in a pantomime. There is something to laugh at in the idea of a man who is half a monkey. There is nothing to laugh at in the idea of relativity. Men did make an image of the Missing Link; though it was an illogical image, because he was missing. They do not make an image of the Fourth Dimension, even an illogical image, because it is missing from imagination as well as experience. If they cheer and laugh at the mere word, it is not only because it is a word, but actually because it is a nonsense word. [GKC ILN Apr 15 1922 CW32:356-7]
...for a medieval man, his Paganism was like a wall and his Catholicism was like a window. No discussions of degree or relativity can get over the difference between a wall and a window. It is more even that a difference of dimension or of plane; it is very near to one of negative and positive. Anyhow, just as a very white sheet of paper looks black if held up against the sun, so any wall looks dark against any window. It may be a whitewashed wall, but it will not be as white as the dullest daylight.
... [Chaucer] had set up, as part of the structure of his own mind a sort of lower and larger stage, for all mankind, in which anything could happen without seriously hurting anybody; and an upper stage which he kept almost deliberately separate, on which walked the angels of the justice and the mercy and the omniscience of God. This was a sort of cosmic complexity, which was supported by the dual standpoint of his morality and philosophy, but which belonged in any case to his individual temperament. It was a temperament especially English; but it is not quite fair to infer in the usual fashion, that it was therefore merely illogical. At bottom, it was no more illogical than the three
dimensions are illogical. It depended on whether he was thinking along one line; or in the flat, as in broad farce; or of the solid images of virtue. [GKC, Chaucer CW18:354, 358]
It is queer that there have been so many philosophical fancies about The Fourth Dimension, in a world in which so many people have not yet discovered The Third. For in that spirit of antic allegory we may say that the modern materialistic world has been in two dimensions and very flat; rather like the Loves of the Triangles or those fishes on the floor of the sea which are almost as flat as figures in geometry. For most periods and civilisations, except the modern period in our civilisation, have really had something which may be best described as a third dimension; a third dimension of depth. It was also, as in the mathematical parallel, a third dimension of height. One way of putting it is to say that people had more of an inner life; but it was an inner life that sank into the abysses and ascended to the sky. We commonly cover it with the name of religion; but it must here be used in a wider sense than anything that is commonly meant by Christianity. Indeed, one of the most obvious forms of it is commonly called Paganism. It was the sense that something was present in the most material actions of men, which was not material but mystical.
If two friends were drinking wine together, there was also a third friend present, for whom wine was actually poured out; the god who had given wine to the human race. To him the vine was sacred; and the vine remained sacred long after the god was rather vague. But it gave to the very act of drinking a ritual character, which was ultimately a religious character. ... I have used the figure of the Loves of the Triangles; but perhaps, oddly enough, it is often the tragedy of this modern love that it is not a triangle. I am aware that there is a threadbare and rather shabby theme that is vulgarly identified with the triangular figure; but there is something deeper and more dignified which deserves much better to be called by the title of the Eternal Triangle. It is the third thing with which the lovers are united at the wedding, as the friends were united over the wine-cup. It is that third dimension of something deeper and more divine which increases all that is most happy and human. We say that it takes two to make a quarrel; and where they are really only two, they probably will quarrel. We say that two is company and three is none; and we shall have gone much deeper into the deepest realities before we discover what even the heathens knew: that three is company and two is none. [GKC ILN June 1, 1935 - thanks to Frank Petta and my mother]
There is perhaps nothing so perfect in all language or literature as the use of these three degrees in the parable of the lilies of the field; in which he seems first to take one small flower in his hand and note its simplicity and even its impotence; then suddenly expands it in flamboyant colours into all the palaces and pavilions full of a great name in national legend and national glory; and then, by yet a third overturn, shrivels it to nothing once more with a gesture as if flinging it away “...and if God so clothes the grass that to-day is and to-morrow is cast into the oven - how much more...” [Mt 6:28-30] It is like the building of a good Babel tower by white magic in a moment and in the movement of a hand; a tower heaved suddenly up to heaven on the top of which can be seen afar off, higher than we had fancied possible, the figure of man; lifted by three infinities above all other things, on a starry ladder of light logic and swift imagination. Merely in a literary sense it would be more of a masterpiece than most of the masterpieces in the libraries; yet it seems to have been uttered almost at random while a man might pull a flower. But merely in a literary sense also, this use of the comparative in several degrees has about it a quality which seems to me to hint of much higher things than the modern suggestion of the simple teaching of pastoral or communal ethics. [GKC The Everlasting Man CW2:332-3]