Snow was sifting down in the afternoon twilight as I made my way home from the university library. So far I haven’t been able to find a single serious and extensive inquiry into the possible discreteness of time. Writers who bring up the idea at all barely mention it.
As I was passing the building that houses the university’s Department of Astronomy and Geophysics, I remembered that Michael Coventree’s office is in there. The lights were on in the lobby, so I walked over to the door and looked in. Interesting looking machines and displays lined both walls. The door was not locked, so I went inside for a look-see.
Right away a sign on a bulky green cabinet confronted me with a bold inquiry: “Do you have the right time?” In smaller print it informed me that the clockwork inside the cabinet received radioed time signals from Fort Collins, Colorado, making a precise correction for the six milliseconds required for the signals to travel that far. The time shown was reasonably close to that shown on my watch.
To the left stood a seismograph, an instrument which registers earthquake tremors. An ultrasensitive pen records on moving graph paper the precise moment at which a quake wave arrives. When this information is correlated with quake times recorded on other instruments at known distances from each other, geophysicists can calculate the location of a quake’s epicenter.
Embedded in the opposite wall was a great circular star-map of the night sky over our city. The viewer was invited to manipulate controls which would display the appearance of the heavens at any desired time of the year. I was trying to reproduce the constellations currently overhead when I heard a door open to my left.
As I turned, a familiar British voice exclaimed, “Oh, it’s you, John!”
It was Michael himself in topcoat and ear-lug cap, with a long tawny scarf around his neck. As we shook hands I explained that on my way home, I was lured inside by the sight of all the interesting equipment around us.
I pointed to a clock which prominently displayed the time in Greenwich, England, saying, “I see that you like to know what time it is back home in the UK.”
He smiled. “The whole world relies on that time.”
“We can talk about that as we head home,” I replied.
Michael hesitated rather obviously, so I asked, “You are going home now, aren’t you?”
He showed me that his office had a pebbled glass window through which he could make out blurry forms in the entry lobby. Then he explained that he had supposed my figure, indistinctly seen through the glass, was that of a Dr. Barry Bard who was to join him for dinner at the Faculty Club.
“Who is Dr. Bard?” I asked.
“Barry is a brilliant psychophysicist with an elephant-sized memory for publications and their contents. He’s a veritable walking encyclopedia of information on all sorts of subjects. He has been working out over at the gym. When he finishes, he’ll come here. Should be arriving shortly. Would you like to come and eat with us?”
I explained that I was expected home for dinner, so Michael invited me into his office to chat until Barry would show up.
As we were loosening our coats, my eye was caught by what was on a shelf at the end of his long, narrow office: a magnificent scale model of a railroad locomotive. As I moved to look at it more closely, Michael said, “That’s the engine of the Royal Scot – a British train, don’t you know. Model railroading is my hobby. I have a big layout at home. The trains can be scheduled by computer so they pass or meet each other without collisions. You’d have fun coordinating the timing of all the switches.”
Michael mentioned that Jack and Marie had told him about my reasons for supporting the hypothesis of the discreteness of time. He already knew that I was skeptical about time flowing like a stream. Before we could get into a discussion of such things, the door opened and Michael’s younger colleague came in – a bright-eyed, good-looking man in a black leather jacket and peaked leather cap.
Michael made the introductions. As we shook hands, Michael found a third chair and suggested we should all sit and chat for a while.
“How’s your work going?” asked the newcomer.
“Well,” I replied, “I think I really need a psychologist.”
“How’s that?” he asked.
“I’m trying to understand the origin and nature of time. But no matter which aspect of time I focus on, my thoughts inevitably take off in two opposite directions. I’d like to find a neat and tidy, consistent, conflict-free way to a harmonious point of view.”
“Say more,” the psychologist said.
I quickly recollected a little list of problems. “If I think of time as moving from future possibilities through the actual present and on into the powerless past, I face the fact that what happened in the past nevertheless affects what is happening now and modifies what can happen in the future. Surely time can’t be moving in both directions at once. Both vectors seem perfectly reasonable, but they clash.
“Again, if I think of time as the objective, actual process of change that keeps altering everything, I have to admit that many things won’t change much for ages. Year after year numbers and other general concepts remain constant. My own memories and values tend to persist and my personal identity is stable.
“I believe that time always advances at a uniform rate, but the followers of Einstein claim that time as clocked can be speeded up or slowed down by changing the velocity and location of clocks.
“It’s exciting to realize that time is always bringing on new things but, on the other hand, it’s painful to watch important established things deteriorating and people I love passing away. Life moves on by doing away with existing things, people and organizations.
“You see? No matter how I try to grab hold of time, it slips out of my grasp in two opposite directions. Time seems to be inherently contrary. It’s my belief that we need a straightforward, all-inclusive understanding of time. If we can’t find one, we won’t ever be able to say that we have fully explained anything that happens. So time is giving me – if you’ll pardon the expression – a ‘chronic, splitting’ headache. This provoking, uneasy conflict I call my ‘Time Syndrome.’”
Michael broke in. “I hear that these days you are also trying to decide whether time proceeds by quantum blips or by a continuous flow. Let me try to shed a little light on your problem with these opposites. In traffic at rush hour or holiday weekends on the freeway, the vehicles move along bumper to bumper much like the cars of a coupler-linked train. Yet they are actually separated from each other by unoccupied spaces. If you pay special attention to the spaces, the traffic can be seen as a lineup of discrete units. But if you think of the long line of traffic as a uniformly moving whole, like a train, you will naturally think of it as continuous.
“Very often what we see depends on how we look at it. Anything whatever can be looked at either as a whole or analyzed into constituent parts or regions. A line of music can be considered as a succession of single notes or as a continuous melody. If we happen to tune in on someone reciting a familiar poem, we may simply identify the poem by name, or tag which stanza we’re hearing, or anticipate the line that’s coming next, or recall what an indistinct word must really have been. What we get out of the experience depends on the focus we adopt.”
“Michael, you would have made a good psychologist,” chuckled Barry. “Attention is of the very essence of consciousness. Attention can divide something at will into imaginary sections, even if that something is as homogeneous as the clear blue sky. By directing our focused attention we manage to discriminate between ‘this’ and ‘that,’ between ‘here’ and ‘there’ or ‘now’ and ‘then.’ Once our attention has divided a field of perception, we can select any section or part of it for emphasis, for further examination, or whatever. We ignore everything else.
“It takes real effort, however, to focus for very long on a single object, one that doesn’t appear to be changing. Attention doesn’t like having to keep renewing its concentration again and again upon the same old thing. It will break loose and begin to search for something new in that field. Time is definitely and deeply involved in the effort of paying attention.
“I’m sure both of you have seen that line drawing called the ‘reversing staircase illusion’, haven’t you?” Michael and I both nodded.
“After looking down on the top surfaces of the ‘steps,’ you suddenly find yourself looking up at the undersides of those same steps. It’s the same with a line drawing of a pile of cubes. At first they may seem to be projecting toward you, but then they will quickly reverse and seem to be facing away from you. These surprising transformations depend upon which line or surface your attention happens to focus upon when it wanders. And, with time, it will inevitably wander.
“Our brains and nervous systems will not handle rival inputs simultaneously. They refuse to accept two opposite perceptions or two opposite abstract conceptions at the same time. We can’t pay attention to both of two contradictory thoughts at once. Time maintains a rigid control over both what we can perceive and what we can think.
“There are always at least two incompatible ways of understanding anything. The coffee cup which is half empty can also be considered to be half full. Even scientists may interpret their data and phenomena in quite opposite ways. For example, one view is that light is composed of photon particles. But from another point of view light seems to be characterized by waves. Who can think of it in both modes at once? As you seem to have discovered, all aspects of time, like everything else, can be approached from two opposite directions. The human race has never been able to agree unanimously about anything. No matter how any given situation may be assessed, it can always be countered by an opposing judgment.
“So, John, don’t be discouraged if you can’t come up with a completely consistent and unchallengeable conception of time. And don’t expect that everybody will understand time the way you do.”
Michael broke in. “For astronomers in ancient times Earth was the center of the cosmos. Eventually Copernicus, Galileo and Newton took a Sun-centered approach and overturned traditional Ptolemaic astronomy. Then Einstein’s view of gravitation challenged Newton’s conception. Einstein’s understanding of physics has been undermined in certain respects by quantum physics, and keen thinkers are at work trying to reconcile the two points of view.
“From within this quantum vision of reality you, John, are now challenging our inherited commonplace belief that time flows uniformly and continuously – an assumption which physical science has never actually been able to justify. Since no two intervals of time ever occur at once, their lengths cannot be directly compared by placing them side by side.”
These mini-lectures from the two professors quite astonished me. I had walked into the building never suspecting that I would be walking into sympathetic understanding.
I wanted to hear more from the two men, but realized that I was keeping them from their dinner and also delaying my own dinner at home. So I stood up, thanked them heartily for their time and counsel, shook hands in friendly farewell, and we all stepped out into the silent, snowy evening. They went their way, and I walked homeward.
The discrete and the continuous
What the two men said has set me thinking about the fact that intelligent, responsible individuals can look at the same situation and yet come up with entirely different accounts of what they saw. Different aspects or levels of a given situation may catch the attention of different people.
When Kay and I visited the Tower of London a few years ago, several suits of armor were on display. Most of the other visitors were exclaiming about the size and shape of a bulky chain-mail “shirt” which had belonged to portly King Henry VIII. What drew my attention, however, were the little interlocking metal rings which gave flexibility to the armor.
I edged as close as I could to the display in order to see how the rings had been assembled into that ingenious metal fabric. Their linkage created intersecting hexagons wherever I looked throughout the whole metal garment. Some particular ring and its linkage … a hexagonal pattern … patterns of patterns … a whole garment – what a visitor might see in that armor would certainly depend on his or her particular interest. Both discreteness and continuity were there for the looking.
A knitted sweater invites the same kind of selective approach. The knitting process begins with continuous yarn. That yarn actually consists of discrete fibers which have been twisted together. The twist would not be maintained in those fibers were it not for rough projections on the individual cells which make up the fibers. Each of those cells is itself a unified entity composed of discrete molecules, each of which is composed of its own set of discrete atoms. To construct the knitted fabric, loops in the yarn are pulled through loops which were formed earlier. Each loop-through-a-loop makes a stitch. Stitches may be formed in several different ways. Some are simple loops through loops. Others double back, cross, spread or reverse. Each stitch is quite distinguishable, but when some of the various kinds have been formed in a certain order and position, interesting textural patterns emerge in the resulting fabric. Both continuity and discreteness thus appear together in the yarn, the stitches, the patterns, the fabric and the sweater – not to mention the living person who happens to be wearing it.
Time and points of view
Time, like that coat of mail and sweater, could possibly be continuous in one respect and discrete in another. One’s view of time may simply depend on which of the two possible aspects one has chosen to focus upon.
Although they are opposites, continuity and discreteness can obviously coexist quite comfortably in the same phenomenon. But that is not to say that moments of time really do bear much resemblance to a series of knots tied in a string.
Now that I think of it, just having a particular personal interest is not a completely adequate explanation of why one person sees continuity in a display while another pays attention to its discrete elements. My interest in the construction of chain mail moved me to get closer to King Henry’s shirt all right, but if I hadn’t been able to get close enough, I would not have been able to discern the little rings, how they were interlocked and how hexagonal patterns emerged from that simple linkage. One has to look closely too at a sweater in order to discern the structure of the stitches and of the yarn which connects them. What we perceive often thus depends on our distance from the objects. But the objects are what they are whether we perceive them or not.
A rainbow arching over the landscape has a certain wholeness, a perfectly semicircular arc, a spectrum of colors, an apparent location, and several minutes of duration. Yet that rainbow is actually the visible product of sunlight shining on small, discrete droplets of rain which remain invisible from a distance.
When I turn over my hourglass, the fine sand in its upper bulb starts flowing down through a narrow neck into the bulb below. From an ordinary viewing distance the flow seems to be a continuously steady stream. A closer look, however, will reveal that it is actually a cascade of tiny, discrete grains of sand.
What we see is thus often a matter of scale – the relation between apparent size and visual distance. Raindrops and fine grains of sand are simply too small and too many for our minds to handle easily, so we blur them together. When viewed with the unaided eye, immense galaxies, enormous stars and giant planets appear to be mere dots of light in the night sky. However today’s telescopes and instrumentation can detect radiation other than visible light (X-rays, radio waves), and reveal all sorts of intriguing features in distant celestial phenomena.
The invention of powerful electron microscopes has raised to a visible scale minuscule realms of reality which used to be quite hidden. Once we were able to see only an organism, but now we can see its tiny cells. Even within one of its cells we can now view an amazing number of functioning components whose elements are tinier still. The development of incredibly sensitive instrumentation has helped us to realize that substances which seem perfectly solid are actually composed of discrete molecular, atomic, and subatomic units.
So far however no device has been invented which could measure the duration of a Now-state – a blip of time.
Some small things may be undetectable because of their size, but others cannot be seen to be what they are because they move too swiftly for our eyes and brains to follow them. An individual item can be lost in a seemingly continuous blur. When my circular saw is running, I can’t see its individual teeth. Could this be the case with time?
Although time’s discrete pulses are not detectable, we may be living on and on because of them. Our life appears to have a “continuity” which is not easily resolvable into distinctly discrete parts or ultrabrief separate events. It does involve successive heartbeats, breaths, eye-blinks and nerve firings, as well as periodic routine episodes such as feeding, sleeping, urination and defecation. But these repetitive, systemic, vital activities are so complexly interwoven, interdependent and overlapping that “life” does not appear to be something that happens moment by moment. Life seems to be continuous.
Somehow both conceptions of time – discreteness and continuity – may be valid. The controversy around them would never have arisen if either notion had made no sense at all.
The status of continuity
We can not only see and comprehend continuity, we can easily create it. Spread a handful of beans randomly over a table and look for a minute at what you have done. Soon you will be noticing, not just individual beans, but patterns. Several beans will seem grouped together, joined by imaginary lines which form the group into a triangle, a tree, a car, an animal or some other likeness. Obviously our imagination is eminently capable of creating continuity.
Starting from the top of this page, let your eye move from line to line on down to the bottom of the page. From the top line all the way down the page you will be able to follow an unintended, wobbly but continuous line of blank spaces between printed words.
Geometry would be impoverished if our minds were unable to “project” a certain segment of a line beyond where it presently ends. Projected straight lines may be imagined to stretch to infinity, even though our optical sight cannot follow them so far. The projected portion of a line is always considered to be a continuity – never a series of closely juxtaposed but discrete dots. Although projected continuity appears to be the product of imagination, it is nevertheless amazingly useful in surveying and astronomy.
Imagined continuity enables us to create useful intellectual abstractions. Rather than deal with individual items one at a time, our minds find it more convenient to embrace a lot of them as the same kind under one name and concept. “The robin” refers to all robins in general, not to any particular robin’s distinctive features. Reducing a set of different things to a generalized abstraction ignores all the individual differences between them.
Small things can easily be lost in the generalizing concepts that our minds are so prone to develop. Few people who pick up a brick ever look for the grains of sand that went into its constitution. The brick in turn will easily get lost in a wall, a wall in a house, a house in a town, a town in a province or state. When studying a painting, who gives even the slightest consideration to individual particles of pigment without which there would be no painting at all? The particular needs and peculiarities of individual persons are easily overlooked in the lumped-together, abstract statistics which may be published concerning the population of a certain area. Whether our minds are congenitally lazy or whether abstract thinking is more efficient, sometimes we much prefer to think in terms of homogeneous rock and continuous ropes rather than deal with the discrete crystals in a rock sample or the separate strands in a cable.
This preference for abstract continuity has long dominated theoretical thinking about time. Having treated time abstractly as an extended, homogeneous continuity, intellectuals instinctively recoil from the possibility that time may actually consist of a succession of discrete moments. This reaction is understandable. After all, it is hard to see that a split second on a clock could contribute anything much by itself toward far-ranging concepts such as derivation, development, growth, tendency, purposefulness or destiny – all of which involve an abstract continuity of time.
A clock simply counts up the number of distinct, regularly repeated movements or vibrations which have occurred since that day’s midnight or high noon. It thus assigns a numerical name – a number – to the present moment, which is then called “the time.” That single figure is an abstract, intellectual, “shorthand” way of skipping from the beginning of a succession of discrete, countable events to the latest of those events without having to pay any particular attention to anything which occurred in between.
“The time” as shown on a clock is not the actual reality of cosmic time; it is simply an arithmetical total of elapsed units, a kind of measurement. It is important toremember that neither a measuring instrument nor any of its measurements is ever identical with what has been measured.Description is one thing; existential reality is another.
Numbers are the most abstract of all abstractions. The same number may refer to a constellation of stars, a highway, a year, a building, a person or any other kind of object. For abstract arithmetical purposes, all of these various things, despite their obvious distinctive differences, are reduced to the omnibus status of “units.” Numbers are an invention of human intellectual imagination. They do not occur in the world as concrete actualities. Numbers by themselves alone do not produce anything actual. They do not change anything nor do they prevent anything from happening.
Mathematicians consider the series of rational numbers to be the intellectual model which is closest to a perfect continuum. A true continuum consists of purely conceptual possibilities. The color continuum, for example, is made up of all the possible hues, tints and shades of colors which could result from all possible combinations of the frequencies of visible light. The ordered string of rational numbers is an ordered listing of all the possible factors and results of possible arithmetical operations. Mathematicians are agreed that the rational numbers form a densely continuous series. That is, between any two numbers there is always another number – perhaps fractional, but a number nonetheless. The number continuum extends from negative infinity to positive infinity. There is no first number or last number – no beginning and no ending.
The word “continuous” means “holding together,” that is, uninterrupted. A continuum therefore must not have any breaks, gaps, edges, starts, stops or any other signs of partitioning. Any selection or actualization of a specific portion of a continuum must come from a source foreign to it and completely outside of it. It takes a painter to choose colors and actually mix colored pigments. Mathematical operations are performed by mathematicians who select and manipulate specific numbers. Their operations however do not do anything to affect the integrity of the number continuum. It just keeps on being the same abstract intellectually-ordered set of imaginary numerical possibilities, any one of which might provide a referential meaning for the xs or ys of algebra.
Most contemporary intellectuals are convinced that time must be a continuum – an uninterrupted, homogeneous, one-dimensional whole. If time is truly another continuum among others, however, it must be only another abstract product of intellectual imagination which has no physical power to make any difference in the real world. As a continuum, time by itself must be unable to divide the story of the world and the universe into actual periods large or small. As a continuum, time cannot have been responsible for the occurrence of the distinct, actual events which have punctuated the world’s actual story. Stars were born and died. Geological ages rose and fell. Biological species appeared and later became extinct. Volcanoes erupted and subsided. These episodes began and ended aeons before humans appeared with their penchant for dividing, selecting and taking things apart. Nothing in time, when considered as a continuum, can explain the appearance and disappearance of well-demarcated actual events such as a wave breaking or an explosion. Something outside of or beyond a continuum has to be invoked to account for the variegated staccato of events that have stammered out the world’s historical existence ever since the Big Bang. It is reasonable to ask, Who or what then has been giving actual existence to the distinctly different particular contents of the world during time’s unbroken and serene subsistence?
A common assumption has been that time was included in the crop of diverse phenomena which arose out of the Big Bang. Time’s unique difference from all the other aboriginal products is seldom given due attention. Primeval energy is believed to have condensed into scattered, discrete, material particles which eventually congregated into galaxies of stars and planets. Time however did not condense. Space began to expand in all directions at once. Time was already wherever anything changed. The immense heat of the primal explosion began cooling down toward the chill of deep space. Time has never lost its original powerful efficacy. Without time nothing else could have happened, then or now.
Gaps?
If time is not a continuum but rather a succession of discrete moments separated by “gaps,” it must be admitted that the notion of a gap in time is hard to comprehend. Is a gap an utter absence of contents? Sheer emptiness? A complete void? My mind is completely helpless when I try to think of nothing at all.
The question “How long is one of those intervals between moments?” assumes that time would be continuing during the hiatus. A gap between two moments, however, need not be either a continuation or a mutation of ordinary time, nor need it be absolutely nothing at all. Then what kind of reality could it be?
Perhaps there is some uninterrupted “beyond,” some “out-of-this-world,” time-like, coherent background reality which provides support for all moments of cosmic time as well as for the gaps between them. If that transcendent, continuous Background Reality were known or believed to engage in periodic creative activity, that could settle the “continuum or discreteness” question concerning the nature of time. Both positions could be right in their own way, and yet incomplete.
Perhaps the true situation is somewhat like being in a dilapidated, windowless, old barn whose siding boards have cracks between them which let the sunlight shine through. Like those siding boards, discrete moments of time – Now-states – may be taken as actualities which stand in front of a continuous background. That background is what occupies the gaps between Now-states – like the light which shines through each and every crack between siding boards. This background can easily be ignored as something which is merely there. But what if that continuous background should not only have its own duration, but really be the energetically effective, transcendent source of time’s discrete moments or Now-states?
A gap between two successive moments could then be understood as a pause between creative pulses – between the last and the next – during which a new menu of possible choices is being readied to be offered to each and every entity by the Source of time.
If time is not only discrete but orderly, there must be some kind of ongoing continuous Being in control of the process. The transcendent continuity which sustains the moments of ordinary time in an orderly sequence is obviously not “out front” like the possibilities and actualized events which it generates. Time as we know it could be, as it were, a “rear-screen projection,” not of images but of existents. The Source of the time-process is behind the scenes. During each pause in the time process, the Creator could conceivably be storing information which has just arrived from the creatures’ choices and also revising accordingly the options to be offered next to them.
This afternoon Michael and Dr. Bard proposed a “psychological” approach which, they thought, would resolve the two opposed views: time’s continuity versus its discreteness. Just as a continuous line of traffic can be seen as a succession of discrete cars, a complex situation can be perceived as a whole or, alternatively, divided into parts, regions and levels. I see now that that psychological approach doesn’t really go to the heart of my problem.
I really want to know for sure whether time out there everywhere actually comes in discrete moments, irrespective of whether or not humans divide up Earth’s motion into units such as seconds. What really is, is what it is, and what it is does not depend upon what anyone thinks it is. If time actually comes into existence unit by unit, it is not the same as an imagined abstract, continuous whole which people divide into segments by clocks. An outside job is quite different from the latter kind of “inside job.” I see that I have come to believe that cosmic time with its gaps must be an “outside job.”
