Time, Space and Philosophy

Christopher Ray

• Publisher: Routledge
• Year: 2002
• eBook ISBN: 9781134968480
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

This book provides a comprehensive, up-to-date and accessible introduction to the philosophy of space and time.
Ray considers in detail the central questions of space and time which arizse from the ideas of Zeno, Newton, Mach, Leibniz and Einstein. Time, Space and Philosophy extends the debate in many areas:absolute simultaneity is examined as well as black holes, the big bang and even time travel.
Time, Space and Philosophy will be invaluable to the student of philosophy and science and will be of considerable interest to mathematics students. The clear, non-technical approach should also make it suitable to for the general reader.

1
ZENO AND THE LIMITS
OF SPACE AND TIME

INTRODUCTION

We typically think of space and time as three dimensions plus one. Mathematicians tell us that each dimension may be continuously subdivided. But they also tell us that we may construct model universes with rather different properties. We may have other structures which may not be continuously sub-divided. And, to complicate matters, we may construct worlds with whatever dimensionality we please. So, can we really chop ‘real’ space and time up as small as we like?

The pre-Socratic philosopher Zeno of Elea—a Greek settlement in Southern Italy—is said to be responsible for five ‘paradoxes’ which wrestle with the properties of space, time, and motion. The main focus of Zeno’s paradoxes is the ‘small-scale’ character of space and time. Is this smallscale structure really continuous, or is it ‘indivisibly atomistic’ or ‘discrete’ in some sense? If threedimensional space is a continuum, then we may continuously and indefinitely sub-divide its parts. But if space or time are discrete in some way, then any process of sub-division will have a definite limit. Aristotle gives a brief and perhaps incomplete account of the first four paradoxes in his Physics, and Simplicius discusses the fifth in his commentary on Aristotle.¹ Zeno is thought to have produced his ideas around 460 BC. We shall review Zeno’s discussion, and we shall find that these paradoxes do identify some real difficulties for our ‘continuum’ view of space and time.

Many mathematicians and philosophers believe that a thorough acquaintance with the mathematics of the continuum should be sufficient to dispel any worries that might arise from Zeno’s paradoxes. But the problems raised by Zeno live on and somewriters, including the philosopher Wesley Salmon and the theoretical physicist Roger Penrose, advise against any uncritical and complete acceptance of the role of the continuum in our physical theories.² A related problem, suggested by James Thomson in 1954, concerns the paradoxical nature of any supertask consisting of an infinite number of tasks. I shall argue that this problem is genuinely paradoxical, on the mathematicians’ own terms. But I shall not join Zeno in rejecting the reality of a complex, diverse world. I shall merely question the extent to which mathematics and geometry may serve as an adequate model for the physical world.

Imagine that we have two theories about the way objects move in the world. One theory assumes that space and time may be continuously sub-divided. The other denies this. But also imagine that both theories are perfectly consistent with every measurement and observation we can possibly make. If we can actually construct such an empirically impeccable rival to the ‘continuum’ theory, then we might begin to wonder about the status of the continuum. We may be willing to admit that it gives us an extremely useful way of organising our experience. But should we believe that the world is really like that? The advantage of mathematics is that it helps us to think clearly about those structures which we believe to be the actual structures of the world; but the problem with mathematics is that it allows us to generate all sorts of weird and wonderful possible structures for the world. The job of sorting out which, if any, we should accept as the ‘real’ picture is left to the physicist. And sometimes the choice is far from straightforward.

DIVISIBILITY VERSUS INDIVISIBILITY

Zeno’s paradoxes of space, time, and motion attack the very idea of the divisibility of space and time. We begin by imagining a distance or a temporal duration which is divided by two; and we imagine that the process of division is continued. Why may we not imagine that the process could continue indefinitely? Zeno tells us that any assumption that the process could go on indefinitely will lead us into logical contradictions. But he also argues that any assumption that the process has some definite limit also leads us into just as much trouble. The first four paradoxes reveal the dilemma:

1. Achilles and the tortoise
   Zeno asks us to imagine a race between Achilles and a tortoise in which the tortoise is allowed to start first. After an agreed time, Achilles sets off in pursuit. Although it seems entirely obvious that the race is a mis-match and that Achilles will all too soon overtake the tortoise, Zeno raises a doubt in our minds. For in order to overtake the tortoise Achilles must first reach the point where the tortoise was when Achilles was given the signal to start in pursuit. Let us call this first point P. But when he reaches point P, the tortoise will now be a little further on at point Q. Achilles now must reach Q if he is to catch the tortoise. Yet when he arrives at Q the tortoise is still ahead at R. When Achilles gets to R, the tortoise has reached S. The race continues just like this: every time Achilles reaches the tortoise’s last ‘staging-post’ the tortoise has moved further on to a new post. Of course, the distance between the two gets shorter and shorter all the time. But Achilles is always behind! So despite first appearances Achilles cannot even catch let alone overtake the tortoise.
2. The racecourse (or dichotomy paradox)
   Here Zeno not only argues that an athlete would never finish, say, a 100-metre race, it also seems that the athlete could not even get started! To reach the end of the track, the athlete would first have to reach the 50-metre point. Having run 50 metres, the athlete would now have to reach the half-way point between the 50-metre point and the finish line. That would take the athlete to the 75 metre mark. But now the athlete would have to reach the half-way point between this mark and the finish. No matter how far the athlete gets down the track, there would always be yet another ‘half-way’ point to reach between the point where the athlete is and the finishing line. So the athlete would get closer and closer to the end of the track, but would never actually reach the finish. For there would be an infinite number of half-way points ahead of the athlete. This might seem bad, but an associated argument implies that the race would not even begin. For to reach the finishing line demands that the athlete would first need to reach the 50-metre mark; and to reach the 50-metre mark demands that the athlete would already have reached the 25-metre point; and to reach that point would require that athlete to have got to the 12.5 metre mark; and so on. As we keep dividing the distance by two, we get closer to the startingline, but we never actually reach it. And we may divide these distances an infinite number of times. So to reach the end of the track there would be an infinite number of distances to run through. Indeed, no matter how short the track, there would always be an infinite number of distances ahead. The athlete would be stuck at the start. To go any distance at all, the athlete would have to run through an infinite number of distances— and how could that be possible?
3. The arrow
   Take a high-speed photograph of an arrow in flight and you may find it hard to disagree with Zeno’s assertion that such an arrow occupies exactly that space which is equal to its own shape and size. We seem to have captured the arrow at an instant of time. At such an instant the arrow is motionless. If it were not motionless, the instant of time could be sub-divided: now the arrow is here, now there. Yet the entire flight of the arrow could be captured in a series of instantaneous photographs. At every instant, the arrow is motionless. There is no time between the instants for the arrow to move on to the next instant. For such a time would be composed of instants itself. So how can an always motionless object move?
4. The moving rows (or the stadium)
   Imagine a stadium in which a column of soldiers passes a column of soldiers at attention so that each step brings every soldier in the moving column into line with the next comrade in the stationary column; a third column of soldiers is also moving, but in the opposite direction, so that with each step the soldiers here also are brought into line with the next comrade along in the stationary column; see Figure 1 (p. 9). With each step, each soldier in each moving column encounters one comrade in the stationary column but two comrades in the oppositely moving column. Now imagine that each soldier represents an indivisible minimum unit of length and that each step represents an indivisible minimum unit of time. Surely we can ask the question: at what instant and in what position did the two moving columns align so that each soldier was alongside the next (rather than the next-but-one) soldier in the adjacent moving column? If we can sub-divide the time for the step and the space between steps there is no problem at all. For they will meet after half a step. But we have supposed that there is no such thing as half of one of our units of length or time—since they are indivisible minima. So either the question is unreasonable (and why should this be?) or we are wrong to suppose that space and time consist in indivisible minima.

Moving rows paradox. Two rows (X and Z) move by a stationary row (Y) as shown. In the top diagram, X¹ and ^Zl are in adjacent columns, X¹ to the left and Z¹ to the right. An instant later, X¹ and Z¹ have shifted their positions, so that they are still in adjacent columns but with X¹ now to the right of Z¹ as shown in the lower diagram. Zeno’s problem is this: when and where were X¹and Z¹ in alignment vertically? Given that the change of position took place in the shortest possible time, we cannot say that they were in line in half this time. And, because the change of position involves the shortest possible distance, we cannot say that they were in line when they had moved through half this distance.

Figure 1 Zeno’s moving rows or stadium paradox

In the first two paradoxes, Zeno tries to illustrate the absurdity of believing that a line may be divided up into progressively smaller chunks ad infinitum. And there is something seductive in his argument. For how can I move from A to B when I first must move to some point in between? And whatever point I choose and no matter how many times I do this, there is always going to be yet another point in between. Zeno warns us against saying that sooner or later I must reach the smallest possible ‘indivisible’distance. For this discrete view of space too will generate problems, as demonstrated by the fourth paradox. Some writers approach Zeno’s paradoxes with confidence, saying that just a little modern calculus will be sufficient to dispel any worries which the paradoxes may produce.³ Ian Stewart identifies the central issue in Zeno as the way we think of infinitesimal quantities; and says that only in the last hundred and fifty years or so have we begun to see the problem in a way that helps us to resolve the paradoxes without too many qualms. Stewart asks:

Can a line be thought of as a sequence of points? Can a plane be sliced up into parallel lines? The modern view is ‘yes’, the verdict of history an overwhelming ‘no’; the main reason being that the interpretation of the question has changed.

(Stewart 1987:66)⁴

Mathematicians now seem to have few worries about continuous subdivisions. What has changed is their attitude towards infinitesimal quantities. Such quantities are not regarded as extensionless points in space or in time. If we regard points as having no extension, then we fall victim to Zeno’s fifth paradox: that of plurality—said by G.E.L. Owen and others to be Zeno’s primary concern and to underlie the other four paradoxes.⁵ Indeed Owen argues that we should regard the paradoxes as providing a coordinated attack on the reality of space, time, and motion. The first two paradoxes challenge the idea that space and time can be continuously sub-divided and the second two attack the notion that there are indivisible minima of space and time; so that Zeno’s overall judgement may be summarised thus: ‘no method of dividing anything into spatial or temporal parts can be described without absurdity.’⁶ The fifth paradox discourages us from regarding the end result of some continuous subdivision as either an extensionless quantity like a point or a quantity with some definite if minute extension:

5 The paradox of plurality
Zeno, according to Simplicius, asks how even an infinite number of extensionless distances could add up to a finite distance and how an extended body can consist of an infinite number of parts (geometrical points?) which themselves have no extension; such a distance or such a body must be infinitely small—i.e. it must be just like its constituent parts: extensionless.⁷ Yet if we allow these constituent parts to have some finite size—however small—then the body must be infinite in size.⁸

Owen points out that this paradox, taken together with the first four, may be seen as providing reasons for Zeno’s view of the world as a single global entity rather than as made up of parts, whether these are indivisibly small or continuously divisible. As soon as we start to sub-divide we run into difficulties. So the sensible thing to do is to resist the temptation to divide the world up at all! Zeno’s world is a single body which may not be sub-divided in any way without absurdity.

INFINITESIMALS AND LIMITS

Must we accept Zeno’s conclusions? The answer seems to lie in our attitude towards the ‘end’ result of an unending process of sub-division, to the idea of infinitesimals. It is a mistake to regard them as having some ‘constant’ value whether this be the ‘zero’ of extensionless objects or points, or whether it is the non-zero value of the shortest possible distance or time. In both cases we would fall straight into one or other of Zeno’s traps. We need a different approach if we are to avoid the traps altogether. The way out was first suggested by the French mathematician Cauchy in 1821: he introduced the idea of a limit; and the notion of the infinitesimal was absorbed into this more coherent concept.⁹ And, some thirty years later, Weierstrass showed that we could move the debate from the realm of geometry to that of arithmetic, from ideas of spatial and temporal distances to those of functions. Instead of talking about ever-decreasing distances along a straight line, we could talk with a little more rigour about infinite series converging on limiting values in terms of functions and real numbers.

The problem may be highlighted by considering how we should answer...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• DEDICATION
• PREFACE
• INTRODUCTION
• 1: ZENO AND THE LIMITS OF SPACE AND TIME
• 2: CLOCKS, GEOMETRY AND RELATIVITY
• 3: TRAVELLING LIGHT
• 4: A CONVENTIONAL WORLD?
• 5: NEWTON AND THE REALITY OF SPACE AND TIME
• 6: MACH AND THE MATERIAL WORLD
• 7: EINSTEIN AND ABSOLUTE SPACETIME
• 8: TIME TRAVEL
• 9: EINSTEIN’S GREATEST MISTAKE?
• 10: COSMOLOGICAL CONUNDRUMS
• CONCLUSION: RELATIVITY—JUST ANOTHER BRICK IN THE WALL?
• NOTES
• SELECT BIBLIOGRAPHY