Zeno's Paradox Resolved

This chapter argues that defined points in time and space, as expressed by mathematicians, prohibit progress. If we don't know exactly where we are, we can move. Temporal perception and our awareness of the present moment are the result of this uncertainty.

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Part 3. of the paper Time's Paradigm


"An abstract and summary of the chapters can be found on Home Page

"Or, click In a Nutshell, for a condensed overview of arguments.

"Universal Contraction is the core hypothesis of this paper: Read here...

"A short SF Thriller, based on concepts within this paper: Read here...

The illusion of progress is as a result of being unable to identify our true position in time and space, as infinity appears to stretch out before us in all directions. Mathematicians created defined points out of necessity, which in turn created paradoxical and unrealistic infinities, as in Zeno's Paradox of motion.

Cyclical Model for Time and Space

CHAPTER 3. Infinity

Infinity, they say: The unimaginably big and the unimaginably small; things too enormous to comprehend.

Quantum physics with its Uncertainty Principle and Relativity with its inertial Frames of Reference, both point towards a universe that does not allow motion if we know exactly where we are.  This hugely significant fact is the key stone to resolving Zeno's Paradox of motion, and will be explored in this chapter.  It brings the disciplines of psychology and physics together, to explain how we perceive the passage of time.

The Kalahari Effect:

As we wander across the Kalahari salt pans, we see only the horizon in front of us.  We look around and back and we are presented with the same vision.  In our 360 degree panorama, nothing exists, not a tree, not even a blade of grass.  We stand in the middle of this flat and bleached landscape and decide, rationally, that we are a certain size; in fact, we make an assumption that we are somewhere in the middle of time and space:  Some things are bigger than us and some are smaller; some things are in front of us and some behind; some things have happened and some things are going to happen.

Realistically, we cannot relate our position or size to anything out there in the Kalahari because there is no beginning or end in sight, so we can only use infinity to affirm that we are somewhere in the middle, of everything.  Somewhat similarly, we once believed that we and the Earth were the centre of the Universe.

We cannot accept that we might be closer to small than big, or vice-versa.  Over recent centuries the Universe has been explained to us as being huge, and not just a few miles up there emblazoned with little pin holes; so too, we have been obliged to understand that infinitesimally small objects are spinning around inside atoms.  Thus, we still remain in the middle of things and we are comfortable.

But where exactly in the grand scheme of things are we..?  Infinity on both ends comforts us, it means we are in the middle.  But is this not just an illusion?  The Kalahari Effect is all around us.  We stand in the middle of our perceived whereabouts; even if we walk a few miles this way or that, we are still in the middle because reality's horizon seems to stretch out before us in all directions into infinity.  Nothing has changed.

In this light, what would be the difference between us, say, living on a planet, or living on the nucleus of an atom?  None.  Except that, we humans are forced to exist in an environment composed of molecules of breathable gasses, and that influenced the size to which we have evolved.  But if conscious life forms did exist on the nuclei of atoms, they might likely have the same dilemma.  "Are we really in the middle?"

It is this ambiguity of 'not knowing' that allows us to pass effortlessly through our perceived existence, moving us consciously in both time and space, through a Block Universe where all of time already exists.  Only if we really don't know where we are or anything about our relative nature, are we capable of movement.  We take a step forwards and in reality nothing has changed, but we feel it has.

Being unable to determine the extent or finality of existence in any direction gives our consciousness the ability to experience movement within it.  It denies us the possibility of identifying our spacial or temporal location relative to it, if we have nowhere to measure from or to.  We may know where we are locally, and be able to pin point our position, say, between one tree and another, but that is all.  We can pass these trees and move on beyond them to a river, but we are no nearer the edge of existence having moved, nor are we further away.

Relative movement between frames of reference is a key theme in Special Relativity; if there is no stationary reference point in the Universe, then we can never knowingly be at rest, we are all in constant motion.  Likewise, a finite edge of existence, a wall beyond which nothing exists, would provide a relative point of reference to determine a finite point within.  It appears the Universe does not provide fixed points of any sort, otherwise we would all grind to a halt.

The Edge of the Universe

The standard model for any active medium or body that does not go on forever, is a cycle.  It then appears to go on forever, it satisfies the conundrum, while justifying the laws it exhibits.  Water flows down river, it reaches the ocean where it evaporates and clouds then return the water in the form of rain to the mountains.  Our planet Earth is not flat, but it used to be considered as such, and even today appears to be from the surface.  If we can accept that the Earth is round and so has no end to its surface dimension, then it is no great leap of faith to assume that existence, itself - the Universe, if that is the extent of it - has also a cyclical nature, thus giving the illusion of infinite progression.

Let's suppose that we are approaching a brick wall.  When we get to it, we are not in the middle of it, but on one side.  The wall has thickness, and if we make a hole in it we can reach the other side.  We could scale it and stand on top of the wall, and again we can see the other side, we can reach beyond it.  That fine, middle line in the wall is actually the separation between one side and the other. If this wall were the end - say, the edge of the universe and there was nothing beyond it - our side of the wall might exist, but the other side of the wall would not.  What we have is an odd circumstance: a wall with only thickness on one side.  Absurd!

We can whittle this down to atomic particles if we wish.   We can say that the end of a worm's nose is the end of the worm and the beginning of earth in front of it,  But we would be wrong to suggest there was a finite point at which one began and one ended.  The interaction between atoms of one and the other are indeterminate, complex and uncertain at any given moment; there is no finite line between them.  The worm is the earth and the earth is the worm.  If it were not so, then there would be some finite separation between them.

This paper subscribes to the notion that time is an alternate dimension of our physical state also constrained by the same principle: there are no finite points or moments, otherwise we would be denied motion.  If infinity stretches out in all dimensions we are free, we are not shackled, we can progress.  This is an illusion, and it is devilishly so!

The subtleties of perception through time are numerous, here is a similar illusory example:  When we are young we perceive time to crawl along from one year to the next; our teenage life seems to drag on forever.  But when we get into our 50s, suddenly five years goes by in a flash.  Five years, when we are ten years old, is fifty per cent of our entire existence; but only ten per cent of our life experience at age 50.  It's just a matter of time.

The illusion of size was mentioned before: there is no end to small; the smallest small cannot be defined.  Is the smallest small a tiny quantum particle whizzing in and out of existence in the blink of an eye?  Could there not be things much smaller than that?  Why not?  Infinity says there must be.  Is the biggest big the edge of our Universe?  How come there is nothing beyond?  Or, perhaps there is no edge of the Universe!  Fractal images show repeated structures scaling up and down, with no mathematical evidence to suggest that there is an end to big or small.

Solving Zeno's Paradox

They say, "You can never reach the door, because you have to go half way first, then half of what's left, then half of that, and so on.., ad infinitum;" which would make sense if distance were a divisible entity.  If a line from A to B were somehow segmented, then this argument could stand and we would never get anywhere.  Segments imply individuality, which equally suggests we could take some bits away or have nothing at all in places, as none are connected.

An intriguing analogy of unsegmented flow is the manufacture of flexible, ribbed hose, which in most instances appear to have individual, ringed segments along their length.  However, on closer examination it can be seen that the rib is one entire and continuous spiral.  Where it not so, then uninterrupted extrusion of the hose would be a great deal more difficult.

Like distance, time is also indivisible.  There are no moments, nor are there any points along the way.  That's what makes movement in time and space a reality.  Infinity has a purpose: it offers an illusion of flow; but it must remain as such, elusive.  Achilles cannot reach the tortoise because he cannot even depart from his start point, as all identified points have an infinity paradox leading towards and away from them.

Back to the door.  And infinity in this case is between you and it.  Behold, a paradox!  Under the constraints of perception, you consider yourself at a point in time and space...  This is a fallacy, as explained by the worm.  Moreover, you make the assumption that there are an infinite number of halves of halves between you and reaching the door, none of which actually exist as true identities.

Some mathematicians refute this.  To resolve Zeno's paradox of motion they have formulated a convergent trick with infinities, stating that an infinite number of reductions towards a whole number, actually adds up to that whole number.  So, a half plus a quarter plus an eighth and so on.., adds up to one - and so you reach the door.  This new theory of time presented here in these pages tells a different story: what if you could call those halves something else?

Consider a table on the other side of the door.  Then consider a point half way to your door, and we immediately discover that this point actually has more than one possible identity.  So, does it really exist?  I see an apple, but actually it could be an orange.  It may remind you somewhat of the argument quantum theorists had with that cat!  While they say particles may have all their states on offer in time, we might also say that points have all their identities on offer in space.

Quantum physics repeatedly offers solutions to the bizarre nature of time and progress, while Relativity is constrained and limiting.  Achilles catches up with and passes the tortoise because there is existence beyond the tortoise.

The door does not exist, not in itself; it is not a definable point, how could it be?  It has width, just as had that wall described above.  It is an area, not a point.  Crucially, an end is finite, and if the door were the end of everything then motion would cease. Space only exists between things, so neither would space continue.

If the table were not there beyond the door then we could not reach the door, because suddenly your halfway point would be definable.  Having a table on the other side of the door provides distance with a dilemma - it must go on.  You can now reach the door because all those halfway points have been re-classified.  Equally, if a point you wish to reach does not actually exist then it does not impede your progress.

By considering the elimination of definable points the 'key' to progress, a cyclical function for time, is the door we can now unlock.  No longer are we limited by ends, as in a linear model of progress, where every point between them can be precisely extrapolated and so, in order to facilitate motion, the ends must be dubbed non-relative.  A cyclical progression, by nature, has no points with certain identities, and infinity is fooled.

Infinity is not just an unfathomable distance, it is also the enigma of individuality.  Infinity does not give us the means to move, it denies it.  Continuity and the lack of individual points allows motion.  Infinity is created by mathematicians in their desire to utilize points to describe our life; points that do not in fact, exist; points which demand something as ridiculous as infinity exist, because those points are, themselves, unreal.

If points did exist then theoreticians would have every right to say you could never reach the door.  Reason dictates otherwise.

Mathematicians are not unaware of the horror they have created in infinities.  The likes of Leibniz, Newton and others realised many centuries ago that movement was impractical if individual points existed.  So they created another horror to vanquish the demon:  Calculus!  In part, this monster is defined as: the summation of infinitesimal differences.

In order to make sense of a body progressing through a distance it would clearly be impossible to accurately determine things like acceleration by dividing the distance up into tiny points and saying that from one point to the next it was going at 'this' velocity, and to the next point it was going at 'another' velocity.  There would be no continuous flow.  So, to eliminate this bumpy road, mathematicians devised an alternative strategy; they conceived: time divided by itself - or seconds per second - to dispense with all those points.

It worked.  Infinities were avoided and progress was possible.  This paper arrives at the same conclusion by the use of deduction rather than mathematics.

Setting Limits on Existence

Here's an example that has been puzzling scientists for a while:  Why can't we reach the temperature considered for the universe to be absolute zero (-273.15C), or zero Kelvin.  Physicists have got to within thousandths of a degree, but they just can't quite reach what could equally be described as Zero Energy or, for that matter, Zero Velocity.  However, considering these new circumstances it is hardly surprising; there must be something beyond absolute zero in order to reach it.  Unfortunately, they made a wall, they said there was nothing colder, and in so doing deprived themselves of their goal.

If we approached that invisible wall we were discussing earlier denoting the edge of our Universe, with the physical laws we assume today, it seems likely we would not make it.  No amount of energy would suffice to allow us to reach it, just as Special Relativity says of bodies approaching the speed of light.  Existence would begin to make no sense, as we tried to force our way towards it.  Scientific laws and principles would break down, logic would cease, and physics as we know it would fail in all aspects.  Somewhat like approaching a Black Hole, we could say, but a completely opposite experience.  So, could there be such a thing that begins or ends, absolutely?  Or could it mean that our known laws of physics are insufficient at this stage to see beyond these walls we have created for ourselves?

The theory that the Big Bang was the beginning of our Universe has, likewise, this same fundamental flaw.  There has to have been something before it, it was not the beginning; perhaps just a transition.  But if we do not search beyond it, we will never reach it.  The technology we have at present to detect such things is limited, unfortunately we seek only what we can understand at present.  Had we the technology, the formulas, the mindset to investigate a concept beyond these so called 'ends', we would certainly surpass them.  If we don't, we will never surpass them.  As you will later read, there is existence beyond both Zero Kelvin and the Big Bang.

There is no end to anything. Because everything is connected to the next thing, nothing, by itself, exists.  Thus, Zeno's Paradox is side-lined, eliminated, satisfying both the ancients and the contemporaries.

More crucially, neither can the moment we perceive and are aware of, the time we call the present.  Were this present moment to have a real, numerical, tangible identity then we would not be able to move - Either in time or in space.  Once again, we have to conclude: the present does not exist.

Now you may begin to understand how the past and the future can be explained as really existing at the same time as the present, even though we only think we are aware in the 'now'.  Everything exists, not just one thing.

This would not sit well with those of the Minkowski or Bergson faiths, who consider that time, like objects, are quantized and divided into ever smaller things.  Both described the passage of time as passing through layers, one moment after the next, almost as if the time line were made up of individual frames on a high speed camera.

Recent studies of various different terrestrial creatures, such as insects and birds, indicate that their perception of time is far faster than ours.  A movie runs at 24 frames per second and we watch it on TV as if it were real, life-like motion; to a fly or a pigeon this would be like looking at a family photo album, frame by frame and probably just as boring.  Their perception of motion is so fast that they can receive and process imagery at over 200 bits of information per second.

And then there are the stunning images of flocks of starlings, or schools of fishes in motion, all apparently synchronised to within thousandths of a second.  Beyond that, even modern standard cameras can take photographs at speeds of two or more thousandth of a second, but, as yet, a blank frame or one that has only half the information, has not been presented for scientific study.

However, later on we discuss the concept that while there may not be spatial layers we pass through, there could well be multiple wave functions that rationalize perception on our journey through time.  Minkowski might have visualized something more than a century ago with which we are only now catching up.

The Uncertainty Principle

Physics is also full of infinities.  In the 1960s, Wheeler and DeWitt finally came up with an equation which did away with the baffling infinities that kept on popping up when trying to unify Relativity with Quantum mechanics.  Infinity was tamed, a bit.  But, unfortunately this ugly beast has been perpetuated in myth for so long we can't seem to rid ourselves of it.

If we could do away with individual moments and points, accept that they are not allowed to exist, the present being one of them - where exactly are we?

We cannot pin-point where we are, as quantum physicists have explained perfectly: "Either the position of an object or its velocity can be determined, but not both at the same time."  The Uncertainty Principle.  So, if we are moving, our position is not certain.  And, as all things in the Universe are always moving, no position can be known.

We can experience motion but we don't know whose doing the moving, because there is no fixed point that does not move that we can point at and say: "That is not moving, so I must be!"  Moreover, as just explained, if there were such a stationary point in the Universe, then up would pop infinity and we would all grind to a halt.

There is the other end of motion to be considered.  Light!  A top speed, at which no mass can travel and beyond which is out of bounds.  Light travels at the same speed, regardless of any observers' motion or the platform from which the light was beamed, says Special Relativity.  Unlike us mortals, all floating about without a clue.  By keeping every point or reference frame moving, none can be identified as true points.  Thus, progress is deemed possible and we are aware of it.

However, this is only because motion is measured on a linear scale, and light speed is regarded as being at the end.  If time and distance are cyclical, then velocity (being a derivative) must also be.  In which case, there would be no ends of relative significance on a cyclical speedometer.  Added to which, no identifiable points within the cycle to impede progress.

The same thing would happen if we cut the circumference of a clock at midnight, and stretched it out like a ruler on a table.  We would then, in order to progress from one minute to the next, have to conclude that both ends of the day were not relative.

Some oddities are still to be understood about light: for one, that light does not accelerate.  It departs its launch platform at full speed, 300,000 km per second (c).  If we just measure the distance a pulse of light takes to pass between two mid points on its journey - which is the classical experiment - then, we don't know how much time it took to accelerate from its point of departure before reaching the first point.  If we try to simultaneously fire a pulse of light and start a clock ticking at the same time we will never succeed to a reasonable degree of accuracy.  This oddity does not seem to be of any great concern to physicists.

What if we said this: Because we cannot add vectors of speed (such as a moving platform) to the speed of a photon of light - otherwise it could be travelling faster than (c) - a photon's point of departure must be static.

From whose inertial reference frame can we assert this claim?  The observer on another platform watching this photon depart, or the reference frame of the person turning on the flash-light?  They both see the photon depart at (c) but it did not accelerate away from its point of departure, according to both observations.  So, its point of departure cannot be relative, just as the speed of the photon is not relative, either.

Such a concept is surely unacceptable in terms of reason and logic!  We'll see...

It may be acceptable to a tiny quantum particle, in as much as suggesting that it is quite likely to be at rest and moving at the same time (that cat again, and a superposition of states).  But this is a macro scenario we are discussing, concerning tangible items like flash-lights, and filaments in a bulb.  If we accept that we cannot have ends or beginnings, starts and stops, finalities and edges, otherwise things cease to move, both ends of a speeding photon of light are irrational.

Consider: light has no need for time; clocks stop at such speeds.  So, if we accept that there are two ends to motion, zero velocity and the speed of light, and infinity demands reconciliation, we can thus propose that these two points are one and the same thing - and a cycle is created.  Now time does not stop at the speed of light, it continues, it is simply that our current formulas cannot take such transitions into account because we are not looking beyond the speed of light or, for that matter, zero velocity.  In the wider sense, our cycle of motion can not have points verifying zero or light speed, anywhere in the cycle they are both equally valid, and a constant energy value of zero is established throughout.

The energy, required to create a photon of light, was in motion before impact.  The photon then impacted an object in its path and energy was again given off.  The fact that we only consider the passage of a beam of light, rather than what was going on before and after, is paramount to this paper's postulate.  Light is merely a change of state - as are we all.

So now, and from previous chapters we can assert that: our present awareness is as a result of sensory receptors; and our conscious consideration of progress is due to the illusion of an infinite horizon, there being no Zeno's Paradox due to there being no finite points in time or space.

Light and infinity, two curious components of our perceived existence, both seemingly made up to accommodate our necessary relative meanderings.  This article will be discussing such concepts in more depth in the following chapters.  First, in part 4., we consider the physical nature of time's flow and why it progresses in one direction only.

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Time's Paradigm is a philosophy of science paper advocating a Cyclical Time progression. It argues that linear scales like velocity create infinities at both ends, restricting motion, and are necessary mathematical parameters only.

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A synopsis of a work in progress. Copyright: A. Graham, 1988 - 2016

No unauthorised use of the material published or the concepts described herein is permitted.