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Positions in the Cage

Toward A Christian Experience of Space.

by Dr. Catherine Collins (Originally Dalzell)

I was on my way home, reaching for the key in my bag, and I happened to turn back and see the length ofthe street behind me. It was the moment between afternoon and night. Eastward, the twin rows of buildings were gathering their mass into the darkness. But their weight seemed almost to hover near the horizon, as if reaching toward the vast, translucent moon that had risen behind them. Westward, the sky was consumed by the colour we ascribe, wrongly, to fire, but which nature usually reserves for the smallest flowers. Here the buildings are smaller than in the eastward portion of the street. They had once been grand homes, and people still lived in them, in apartments, while the other direction contains only office blocks and government buildings. The line of the houses is low, interrupted by the branches of trees. The houses seemed to be fading into the street that night, and it was only the steeple of the church some blocks away that could still be seen against the burning sky.

I stopped to look, for in that moment the sky had suddenly become for me what it has been for most people in the past: a symbol and explanation of what is most important on earth. It seemed that the changeable and ordinary things of the inner city were being by swept into the night-time world, where their meaning and position would be fixed forever.

It is not hard to find an explanation for the fascination of the stars. The night sky always displays two attributes we would dearly like to find on earth: discernible pattern and predictable behaviour. It is as if time and space have been resolved into two components, so that the effect of each can be broughtinto focus. Space establishes a pattern in the stars, and time rotates that pattern without disturbing the relation of the parts to each other. Time does not disturb the pattern of space; space does not conceal the passage of time.

On earth and by daylight, things are much harder to understand. The relation of one thing to another changes so rapidly that one must admit shape and position to be more idealizations than facts. You can check this by trying to sketch some moderately quiet object: a person waiting at a bus stop, a sleeping dog, a tree on a windless day. You will soon discover that even something that seems to be at rest is always shifting. But at night pattern and progression are dislinguished in the stars; space and time are clear concepts then. It is easy to imagine that the world's insides are on view, like the workings of a great machine.

This being the case, it is not surprising that many have tried to pull the attributes of the heavens down to earth and, in the process, make out the earth to be more like the sky than it really is. Astrology is a classic attempt to bring the clarity of the heavens to earth. There is pattern in the sky; there must be pattern on the earth. And more: these two must be the same. The predictability of the planets is transposed into the predictability of human affairs.

The transposition is not quite perfect. Turn back from the cramped specifics of the horoscope and look once more at the stars. Something remains in the sky that the astrologers have not seized in their calculations, something less in the brilliance of the stars we can see than in the ifiuminated blackness in which we dwell. On earth, if a place is dark it is because it lacks light, and possibly space and air as well. But the black of night is never an absence; it seems to pose a question and to stand in judgement.

All of this is missing from astrology. One wonders, in fact, if it is about the actual at all. Some kind of substitution has taken place. First, a chart of images is stretched across the heavens to mark the location of the stars. This is then brought down to earth to mark the positions of men and women. The real sky and the real earth are replaced by the chart, which fails to express the mystery of one and the freedom of the other.

The mathematical physics of the modern era - the paradigm of science from Descartes to Einstein - also brought the stars down to earth, for another gain in clarity and prediction. Mathematical physics is undoubtedly more accurate than astrology, and certainly more useful but, like astrology, in bringing down to earth the regularity of the heavens, it leaves something behind. It pulls a mathematical chart over living and moving beings, and makes believe that it is only the mathematical world that is real. Mathematical space and mathematical time replace actual space and time.

I call this world the cage. In this brave new mathematical simulacrum of a universe, we live and move and have our being. Everything that exists and moves does so in the cage, with its motion and its existence defined by points in the cage. The cage itself is defined by the mathematical co-ordinates we must now call the three dimensions of space and the one dimension of time. And not only do things take their existence from the cage, but their relationships to each other are determined by and through the cage. First comes the cage. Second come the things which are parked in it and then relate to each other through their positions in the cage. Third come the relations between things, established through their positions in the cage. Consider, as the mathematicians say, a simple example. I place a small chair beside my writing table. Perhaps I want to sit down and write something, or maybe I want to sit and drink a cup of coffee. In the real world, table and chair relate to each other by the common purpose that has designed them for each other. I sit on the chair in order to make good use of the table. They relate also to the floor that supports them, and they relate to the other objects in the room by the overall impression that they give together. Is the room stylish? Is it elegant, or cluttered? Does it convey a peaceful impression, so that when I want to sit down at my table to write something I feel like concentrating my mind on the page? Or is it the sort of room that naturally draws one's attention out of the window, to see what is happening on the street below?

In the cage, these considerations are of no importance. The cage itself establishes the sort of relations that can exist between the inmates. As far as the cage is concerned, my table and my chair could both be reduced to point masses, lodged in space and time, somewhere and somewhen. Anything more is merely subjective. The cage is designed to convey the regularity and the predictability that we admire in the stars and the planets. Piece by piece, all of nature is brought into the scheme. First, space is replaced by mathematics, then time, then motion and, with it, all living things.

Time differs from space as much as music differs from art, but in the cage, time becomes a space-like entity, a dimension. So now, instead of existing in space and through time, we are pasted to the walls of the cage by means of a location "in" time and "in" space. One might almost imagine that the two are measured in the same way, so that a clock becomes simply an oddly shaped ruler.

A clock differs from a ruler because a clock has moving parts. It must have them if it is to measure the passage of time, which is only known through the change of things. Motion is the measure of time, so motion is a part of what measures time. Now motion, as we experience it, is a fundamental fact which can be reduced neither to space nor to time. It is one of those things to which the mind can allude, but never comprehend. And yet, in the cage, motion is comprehensible. Motion too, like space and time, is replaced by mathematics.

It is a sad feature of undergraduate instruction in mathematics that few people will be shocked by what I have just said; but we should be shocked to learn that mathematics can express motion, almost to the point of replacing it in the educated imagination. Mathematics, in itself, is static, and it talks about static things. It describes pattern and quantity. But motion is neither a pattern nor a quantity, but the transition from one pattern to another, and from one quantity to another. Mathematics can be expected to plot trajectories after the bird has flown, one might say, but not the flight itself.

It is the calculus that brings motion to a halt and fixes it in the bars of the cage. I have nothing against calculus, which is one of the truly great and original constructions of the modern age. It is beautiful, economical, and elegant. How it works is actually quite simple. There has always been a mathematical quantity associated with motion: if I travel a given distance in a given time, I can form the ratio and declare the result to be my average speed. Speed is not a motion, but it gives me a way of comparing moving things. Calculus begins with this ratio, and then proceeds to imagine diminishing distances and shorter spans of time. We then pass to the limit and assign a quantity to the instantaneous rate of change at each moment. Change has become a quantity and an inmate of the cage.

Where the imagination is called dangerously into play is when we "pass to the limit." The limit in question is what we get by taking ratios of shorter distances and shorter periods of time. It represents the speed at each point of time, and for every instant we can define a speed. This assumes that space and time are infinitely divisible, If we charted the trajectory against the walls of the cage, we would draw a smooth, continuous line, without gaps or corners. If there turned out to be a hole in the cage where the curve ought to pass, there would be no limit. If the line were porous, with gaps in it, so that distance was not defined at every moment, there would be no limit. If we could not talk of "every moment," there would be no limit.

Luckily, there is a sort of mathematical polyfilla known as the Real Numbers which makes all of this work out logically and mathematically, but real things behave differently. You can cut a loaf of bread only so thinly before it crumbles against the knife. You can only slice away so small a piece of iron before being left with a fragment that lacks the properties of the larger piece. Dissect a frog - it dies, and ceases to be a frog. But the lines and planes of mathematics can be endlessly subdivided, like the ultimate Russian doll, and only yield smaller and smaller planes. The real bars of a real cage might be filed away, and the iron converted to piles of dust, but nothing can file the bars of the mathematical cage. They are infinite, enduring, and indestructible - like God.

In certain circles, what I have called "the cage" is known as the "theory of absolute space and absolute tune." As such, it has long since been refuted, and from a number of angles. Most philosophers since Descartes have found its implications difficult to take. In the eighteenth century it became apparent to mathematicians that the notion of infinitesimal fractions lacked rigour and could lead, in the wrong hands, to proving that 0 equals 1. Calculus, redefined in the nineteenth century with more care where the infinite and the infinitesimal are concerned, has been converted from a metaphysical monster to a technological workhorse. Finally, Einstein convinced even the physicists that the universe is not a cage - or not that particular cage.

So why go into this now? Because the cage persists in the imaginations of many, and will continue to persist for as long as high-school physics classes continue to teach young people that the world is not truly as they see it, but rather as the diagrams in the text book report it to be. The cage will always be a danger to the literate, just as superstition will always be a danger to the uneducated. Indeed, the difficulty is not so much with one scientific theory or another as with a failure of the imagination that replaces nature itself with a description of nature. In this respect, Einstein is as much a dealer in cages as were his predecessors. The main advantage of his cage is that it is so complicated and so counter-intuitive that it screams aloud that it is only a mathematical construct. We are less likely to confuse it with the actual world. The old Newtonian model is more deceptive. It seems very close to normal experience, so one can easily forget that it has eliminated everything that makes that experience something experienced rather than depicted.

The cage persists; but we should not be too harsh on the sciences. The fault is with the imagination that substitutes mathematics for nature, not with mathematics itself. It may be that the cage existed in the imagination first, and that scientific theories were developed on that basis. Again, the error is not with the concepts, but with the enthusiastic use that has been made of them. A concept, after all, is like a window through which the world is perceived; the error is to make it the direct object of vision, as if it were an oil painting. The cage has many lovely pictures hanging on its walls, some of which resemble earth and some of which resemble the stars above; but to the extent that we make it a cage, it has no windows.

Clearly, the imagination of the seventeenth century was subject to invasion by the cage, but think how more liable we are to life behind bars now that the diagrams of the science books have, in a way, stepped out of the books and onto the city streets. The office towers of the inner city are like great chunks of geometry built in concrete among a Cartesian grid of streets. We have thought the cage, and consequently built the cage to resemble the thought. Modem industry is mathematics seen from the outside; conversely, mathematics is the language telling the machines what to do.

In fact, mathematics works better on machines than on nature. Open any texthook on mathematical modelling, whether directed to physics or statistics, and you will find that the classic examples are taken from technology: the inclined plane, the frictionless pendulum, the roulette wheel, the electric circuit, the cannon ball. It is rare to find examples from nature simple enough to be described well by the mathematics we currently understand. A simplified version of the orbit of the moon is one such case - but the moon does not reside on earth.

Terrestrial systems do not, as a rule, behave like machines. The movement is all wrong, for one thing. The waves move, the trees rustle, the sky is variable, and the ground subsides or crumbles underfoot. Cities, for the best of engineenng reasons, stay put. Things either belong to the background, like buildings, or rush about in the foreground, like cars and pedestrians. The result is to give an impression of a world in which things move against a fixed, permanent, and regular background - like that of the mathematical cage. This is convenient for us, but the world at large is simply not like that.

There are spiritual and psychological consequences to life in the cage. For the Christian, the cage is not a happy place to be. It is not easy, from the perspective of the cage, to believe in the existence and the power of God. It is not easy to believe in miracles or prayer. Many people are held back from a living faith largely by an inadequate experience of space, and by a deceitful imagination formed along the lines of the cage.

Space is fundamental. You can think of nothing without forming a mental image of some kind, even when you are thinking of something like God or justice that has no spatial habitation. Space, and pattern in space, is the first thing an infant learns to perceive. The experience of space is as natural as the look of a mother's face and the knowledge of one's own existence. So, if the experience of space is inadequate, one can experience nothing adequately, not even one's own existence.

This is true of space in the sense of extension and the relations between nearby objects, like my table and chair. It is also true of "outer space," a notion derived from observing the sky at night. The night sky is "space" in a particularly vivid way, because there is so much of it out there, and because we can view its extent and pattern without the impediment of countless living and moving things. Space is that sense that has always been a primary metaphor for God and his dwelling place; so again, false images in that area lead to false beliefs as well. "The heavens are telling the glory of God; and the firmament proclaims his handiwork" (Ps 19 [18]). Or they tell something else.

The cage proclaims the glory of God in muffled tones, partly because it is not God's handiwork, but ours. As a system of concepts, the cage is really a sort of machine, a calculating device, rather like a watch. And indeed, the cage is much like the impress of a watch on the inner mind: both emulate the motion of the sun and stars and produce a little model universe of their own. The cage is a mental construct for the description of moving things. it is self-contained. It never looks beyond itself for the source of its existence; there can be nothing beyond the cage that the cage knows of. It constitutes in itself the full extent of scientifically meaningful relations that can bind one object to another. It no more points to God than do the hands of my watch.

But nature itself and human life cannot be so contained and limited, nor are we. If prayer is possible, as Christian faith and practice maintain, then human life does not revolve around itself alone. Unlike the cage, it does tend beyond itself. And if we tend to God, the universe must tend there with us. "If these were silent, the stones would cry out" (Lk 19.40). If not, then we will make them cry out. Is that not what occurs when stones are built into a Church and metal formed into the pipes of an organ? If any small piece of the universe can be oriented toward God, then so must be the rest, for that small piece will pull the rest in its train. So whatever else may be true of the material world, it is not a self-contained network of mathematical concepts, where each part relates only to the rest like a perpetual motion machine.

If the cage were real and the universe truly self-contained, as the cage would have it, then the cage would be God. Indeed, whatever you define as eternal and immutable will be your god. If it has the attributes of God, then it is God. This is another reason why the cage is troublesome to Christians: it is a pseudo-god, which tends to drive out the real one. But for this very reason, the cage can be a comfort. To the extent that it resembles God, it is comforting in much the way that faith in providence is a comfort. The cage supplies, imaginatively, a sense that one is supported by the universe: "I am standing on the floor of the cage. The laws of nature will hold me up. I will not slip into the void, because the Cartesian grid holds me in place." That is what it feels like to believe in the cage. There is a cosy feeling at the back of the mind; everything is fine, the world is solid, I am solid and I am in control. The cage sustains the universe and we control the cage. We need only understand how it works for it to work on us. This is nonsense scientifically, but I am talking about the imagination, not about science.

There is a comfort in the cage, but it is the comfort of an undemanding life, and that is the kind of comfort that wears thin with time. One pastes concepts over one's eyeballs and roams through the world looking for something to do, or something to believe in. Dissatisfaction sets in. Things fade, and the real world seems always somehow to be over the next hill. The bored and the angry rattle the bars of the cage, without knowing that there is much of anything beyond it.

Occasionally someone does see through the bars, perceives the cage for what it is, sees all creation on the further side, and falls back, terrified. Space can be frightening: the top of a cliff, the sky at night seen from a broad field. Gone is the false support of an imaginative construct. The cage is shattered, and in that moment of fear, the beginning of wisdom has a chance to do its work.

We can dismantle the cage in many ways. One of the simplest and the oldest is to shoot an arrow through it, Zeno's arrow. Zeno was a Greek philosopher who argued that the concept of motion, as used in the cage, leads to contradictions. He asked, "Where is the arrow during its flight?" If it is somewhere then it is not moving, and if it is moving it cannot be anywhere. To a mathematician, Zeno's arrow creates no difficulties. We set up a map of X and Y co-ordinates on a chart, and trace the trajectory of the arrow against it. For any point in time, we can mark the end points of the arrow and declare that the arrow lies between the two points. But this is not a picture of motion; this is a picture of a succession of parking places. How precisely, one moves from one parking place to the next is absent from the equation.

Zeno's arrow will not fly through the cage, and it would make no difference what sort of mathematics were used: a continuum like a pull of toffee, or a discrete model like a sequence of stepping stones. If the arrow is, it is not moving. Zeno's paradox is a direct consequence of substituting mathematics for nature. It makes no difference how intricate the mathematics may be or how accurate the science. The difficulty is over the nature of mathematics itself. To take an analogy, it does not matter how good the camera or how fine the photographer, a photo is not life. Nor does it become more lifelike by bringing the camera into sharper focus. And yet, the camera is true. It records everything in nature to the extent and in the manner that nature can be represented by a still image in two dimensions. Similarly, if we were smart enough, mathematics would describe everything in the physical world, to the extent that it is mathematical. It would be quite true, but not real.

So if space is not the space of the mathematicians, what is it? Ido not know, but, I know something of what it does, or rather, of what it permits. Space has to do with the relations between things. Space is not the relation, but, because of space, the relation can be expressed. Because of space, there can be more than one being in creation, and the beings that exist can also interact. Spatial pattern is then the expression of the relations which space allows.

Mathematics, as a sequence of mental constructs abstracted from the relations of physical space, physical beings, and human reasoning, gives rise to geometry, arithmetic, and logic. From this structure of relations in the mind we can manipulate and understand the structures of material things. But we understand them only to the extent that they resemble mathematics. In other words, we understand space to the extent that it has been brought to a halt.

We have forgotten time. If space permits two things to exist, it is time which allows each one to change itself and to change the other. Space is the accumulation of the changes wrought in time, and time itself moves forward to complete the patterns begun in space. Motion occurs, since the relations of space are incomplete and unstable, and each drives forward to a new state. The patterns of space exist and, if one could stop the clock, they might resemble the patterns of mathematics; but the patterns in space are not mathematics. They are more like the patterns of the dancers in a square dance. It is the life of the dance to move and for each figure to swing into the next. But when they stop dancing, it is not to produce a final pattern where they are left standing; rather, the dance dissolves and the dancers leave for the buffet. It was the dance that made the space, and space that allowed the dance.

The new cosmology seems to reflect this view of space by positing an expanding universe whose space is transformed along with the entities that exist within it. Objects, it would seem, do not fall into a pre-existing space. Rather, space is developed together with the things whose mutual relations space is to bear.

Space exists; but space as an immobile fixture, a backdrop to movement that leaves it unmoved, is an abstraction. And yet, there must be a backdrop or fixed point somewhere. The cage is right when it assumes some fixed structure as a matrix for the rest of creation. The mind cannot understand motion without understanding something which does not move, and the cage supplies such a thing. But the cage itself is not real; it is a mental construction of our own. It is not the cage that holds creation together, but God.

In creation itself there is no fixed, self-sustaining point. Whatever position I think I hold is held relative to other things, and is, in itself, not a position but a movement toward God. Ultimately, all movement is movement toward God, whether it is that of a stone or that of a human heart, and all space exists because of the relations established through creation. The stone never travels far, but the heart can indeed return to its end and its beginning. Recall that in the false world of the cage, prayer had a hollow and intrusive ring, but in the actual world it is as natural as birds flying north for the spring.

There is a converse to this statement that all movement is movement toward God: that which does not move, or does not appear to move, seems particularly close to heaven. This is true of the sky at night; it is true also of art. Every image seems to come from the eternal. By halting time, it presents an image of a possible final state of things, in which the truth of each one's nature will be apparent. This is particularly true of great art which, however natural it looks - perhaps the more natural it looks - seems to gather strength from beyond time, as if the light of glory were shining through it.

Leonardo da Vinci: The Virgin and Child with Saint Anne and Saint John the Baptist Courtesy The National Oallery,Trafalgar Square, London, UK. Used by permission.

I am looking at a reproduction of what some people have called the greatest drawing ever made. Leonardo da Vinci drew it as a preparatory sketch for an oil painting. It depicts the Virgin and child with Saint Anne and Saint John the Baptist. The artist made it in order to map out the figures and test their relative positions, the placing of the arms, and so on. One can see the traces of several attempts around the feet, and Saint Anne's left arm looks like an afterthought. The facial expressions must have been important to him, since here the drawing is most detailed.

It is the face of Mary, still and luminous, that pulls the picture together. At first glance one might say that it is the most natural expression portrayed, except that no woman has ever looked like that. More is conveyed here than the nature we experience. This is nature as it should be, and as it will be in glory. It is nature as the artist already discerns it to be in the midst of the approximations of daily life.

A drawing is an arrangement of figures in space, just like a geometric diagram. This tool, like mathematics, is a representation of some aspect of what space is and can be. But whereas mathematics knows only the objective and the non-personal, art takes the world from the opposite direction. Here, physical arrangement has been entirely transformed into gesture. The features of the face have risen from mere pattern to human expression. Space has been transformed into a medium of love.

Perhaps the da Vinci drawing is a vision of space fulfilled, of what space is tending to become through the transformations of time, in grace. And observe that this is a fully human vision. It is not the deterministic outcome of a non-rational process. The lines are determined not by the distance between material things, but by the ties of knowledge and love. The end is a human world, and not a diagram.

Science has looked for the fulfilment of knowledge in mathematics; in drawing we can see that the knowledge of nature is fulfilled in wisdom. But wisdom does not annihilate mathematics. There is mathematics in the picture, embedded in the patterns of the figures, in the curve of an aim. It is the mathematics you do not see that gives the composition its balance and elegance. But it does not remain lifelessly as pure mathematics. It is inscribed in the gestures of the figures, and it supports the expression of love that unites them.

The space in which we live is, like the drawing, open to what is human and personal. Space supports every kind of relation that the universe holds, from the distance between two posts to the shape of a smile. The physical world is open to what is more than physical, much more. No one need live in the cage, unless he truly prefers the incarcerated view.

And now that we are standing outside the cage, I am not even sure that it looks like a cage any more. The mathematical structures we define are more like a fish-net than a cage. We drag it behind ourselves, and gather within it the otherwise unformed material of the world. The cities, the machines, are instances of our transformation of the purely natural into the human, a transformation that is part of our vocation to name the creatures and bring them to the service of man and the glory of God. To be sure, the work is not always humane, but it remains ours to do. God instructed Adam to name the creatures, and it was to have been a work carried out in the light of grace. Without grace, and without prayer, the naming of names became a tissue of superstitions and a network of constructs without end: the cage. It is only by grace and prayer that the imagination can be renewed and reason set free.

It is fitting that the da Vinci drawing be incomplete. It is the incomplete sketch of a larger work. The greatest drawing is incomplete, just as the greatest theological work, Saint Thomas's Summa theologiae, was left unfinished. These attempts to presage a final order to the world, whether spatial or conceptual, must, to be perfect for this time, be incomplete in reference to that time where all will be completed by its Maker.

Furthermore, were you to visit the National Gallery in London where the drawing is displayed, you would not see it in its original condition. Somebody once put a bullet through it, but you would not notice the damage after the restoration that has been done. On earth, the pattern is incomplete, and often it seems to be spoiled beyond repair. But still, we can look up and see the stars.

Catherine Dalzell was born in Toronto. She holds degrees in mathematics and statistics from the Universities of Toronto, Oxford and Carnegie-Mellon. She converted to Catholicism in 1983. These articles were written in the 1980's and 1990's --- some while she was teaching at Memorial University in Newfoundland, and some while working as a statistical consultant for the Canadian Red Cross Society. She is now a full time homemaker and mother of three small children. She occasionally publishes under her married name of Collins. This is a companion piece to her previous article, "The Power to Act: Toward a Christian Experience of Time,"

Reprinted with permission from The Canadian Catholic Review (11[1993] 18-25). The Canadian Catholic Review has ceased publication.

Section Contents Copyright ©; Mark Alder and Catherine Collins 2000

This version: 1st April 2001

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