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    Zenos Illustration of the Concept of Time Through Achilles and the Tortoise

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    By use of paradox, Zeno sets out specifically to demonstrate the flaws in our understanding of basic concepts such as time. It is thought that he does so in order to support the theories of Parmenides. By way of the ‘reductio ad absurdum’ method of following something’s consequences logically until it becomes absurd, Zeno presents the ‘Achilles and the tortoise’ paradox to show that the notion of time as infinitely divisible is inconsistent, and the paradox known as the Arrow to show that the notion of time as finitely divisible is inconsistent. Therefore he leaves us having to accept either that there is no such things as time, or that time is not divisible at all. Of course, this is only the case if his paradoxes themselves are consistent.

    Zeno attempts to illustrate something of the concept of time through ‘Achilles and the tortoise’. This proposes that there is a race between Achilles – the fastest runner in Greece – and a tortoise – a famously slow creature. The tortoise is given a head start. Zeno claims that Achilles would never overtake the tortoise if we were to adopt the opinion of time as infinitely divisible, because in order to do so, he first has to reach the point where the tortoise is. By this time the tortoise will have moved on, and so Achilles again will have to get to where the tortoise is. In such a way, Achilles will have to reach an infinite number of points representing where the tortoise has been before he is able to overtake him, and as it is impossible to complete an infinite series of tasks – reaching the points – Achilles can never catch the tortoise.

    This assumes that time is infinitely divisible as it is based on space being infinitely divisible, and due to the relationship between space and time that allows us to know that Achilles is moving, the two must be consistent with each other. So here Zeno demonstrates that such an understanding of time must be incorrect, or we would be forced to concede that Achilles can never overtake the tortoise, which would be as ridiculous as the idea of a man lactating or a spider ladder.

    This is similar to his Dichotomy, the conclusion of which Aristotle summed up by writing, ‘there is no motion, because it is necessary for something moving to arrive at the mid-point before the end-point’. Zeno argues that to reach any point, one must first reach the point half-way to that, and so on ad infinitum. Therefore, under this understanding of time as infinitely divisible for the same reasons as in the Achilles paradox, one would have to conclude that motion is impossible, because to get from A to B, one would have to cross an infinite series of points between A and B. Since it is impossible to complete something with is infinite by definition, one can never move. Hence in this way Zeno demonstrates that an understanding of time as infinitely divisible must be rejected or else one must accept that one cannot move, which most would be uncomfortable with doing.

    However, should one wish to defend this understanding of time, flaws can be pointed to in Zeno’s reasoning. As both of the paradoxes work on similar principles, they can be criticised together. For instance, Aristotle raised an issue with Zeno’s claim that a distance AB could be divided into an infinite amount of points. He claimed instead that a distance AB only potentially consists of an infinite amount of points. They only become actual when we make a division that distinguishes a point on the line. Hence Zeno is unjustified in presuming an infinite series of points on the line that represents distance to be travelled, as it would be impossible to actualise them all simultaneously. Therefore Zeno’s paradoxes cannot be used as valid methods of demonstrating that time cannot be infinitely divisible, as Achilles doesn’t have to cross an infinite series of points to reach the tortoise.

    Although Aristotle’s criticism seems insufficient, as it is hard to see how distinguishing between potential points and actual points has any relevance to the fact that an infinite number of points can exist, there is another way in which Zeno can be criticised perhaps more successfully. That is to distinguish between what it is to have crossed an infinite number of points on a line, and what it is to complete an infinite series of tasks. Zeno assumes that to cross one of the points is equivalent to completing a task, whereas there is no reason we cannot say that Bryony has completed the task of walking over a distance which can be divided into an infinite series of points. This is an action which Bryony can complete, as it does not involve infinite tasks, hence she can cross the line and can move, even though the line contains and infinite number of mini-distances within it. Therefore Zeno seems to fail in demonstrating that time is not infinitely divisible in this way.

    It could of course be argued that this criticism commits the definist fallacy, by phrasing the situation in a way favourable to its criticism by defining motion as completing a distance that is infinitely divisible, rather than completing an infinite series of actions. It seems that neither Zeno nor the critic can give compulsive enough grounds for asserting one definition of motion over the other, so we must concede at least that Zeno has indicated a possible flaw in an understanding of time as infinitely divisible. This is because, if his definition holds up as accurate, then his argument logically follows, yet clearly contradicts our view of the world (as most of us believe we can move), so would force a rejection of that particular notion of time. Having done this, Zeno moves on to attacking the understanding of time as finitely divisible.

    To deal with this, Zeno introduces the paradox known as the Arrow. As Aristotle stated; ‘it depends on the assumption that time is composed of instants’, which are finite divisions. At each single instant he claims that we can imagine a ‘snapshot’ of an arrow in flight. In this instant, the arrow is not moving, hence at each instant of its flight it is at rest. This is because for an arrow to move in an instant, it would have to be in one place at the beginning and a different place at the end. Yet this cannot happen, because by definition an instant is indivisible, so can have no beginning or end. Therefore throughout an arrow’s flight – a collection of instants – the arrow does not move. In this way Zeno seems to suggest that we ought to reject any understanding of time as finitely divisible, or else concede that an arrow is both moving and not moving.

    Aristotle had something to say about this as well. He argued that it is invalid for Zeno to move from saying the arrow does not move at an instant, to saying that he arrow does not move throughout the duration of its flight, as a flight is something more than a collection of instants. Therefore purely because it does not move at an instant, it does not follow that it does not move over a period of time. Therefore it is possible both to accept that time can be understood as finitely divisible, and that motion is possible.

    However, Aristotle seems ridiculously unjustified in asserting this. What he is essentially saying is that ‘the whole is more than the sum of the parts’, as he says if you add up all of the instants at which the arrow is not moving, you end up with more than just instants, and this is how motion is possible. Yet there seems no reason that this might be true; for instance if I have many drops of water then what I have is a puddle, and not a wombat. The sum of drops of water is a puddle with roughly the same properties as the drops of water, and nothing more – and all that a puddle is is drops of water. Hence it seems undefendable to claim that when you add instants together you somehow end up with time that is not a collection of instants.

    Disregarding Aristotle’s, there is another objection which can be raised against the validity of Zeno’s Arrow, and it is proposed by Warren. He argues that it is ambiguous for Zeno to state that the arrow is not moving at any instant, as if this can be said then it is also true that the arrow is not at rest. This is because ‘the properties of being at rest and being at motion are true only of things over periods of time’, meaning that without any timescale the notion of motion or non-motion ceases to have meaning. Hence Zeno cannot accurately state that an arrow is not moving at an instant, so as this is the base premise for his argument and it is not accurate, his argument cannot be accurate. Therefore, accepting this criticism, Zeno has failed to demonstrate that a finitely-divisible understanding of time poses a problem.

    However, Zeno has another paradox which also attempts to attack this understanding of time. Little of it survives other than Aristotle’s rather dismissive account, in which it is concluded that ‘half the time is equal to its double’, but an interpretation of it is as thus; imagine there to be three rows, each consisting of four units. The first row – A – is stationary. The second – B – is to the left of A and moves in the direction of A at a constant speed. The thirds row – C – is to the right of A and moves towards it at the same speed as B having started from the same distance away from A as B started. A, B and C represent atoms, and one B passes one A in one atomic fragment of time – hence making the assumption that time is finitely divisible. There is a point at which the situation will resemble the following; where the Bs and Cs have where all of the letters are aligned.

    At this point, the (B) will have passed 2 As and 4 Cs, hence it takes the same time for it to pass 2As as it does to pass 4Cs. Similarly, in the time it takes B to pass 1A it will have passed 2Cs. This raises the question of how many As it will pas in the time it takes to pass 1C. Logic tells that the answer would be half an A, but A is an atom so by definition indivisible. Therefore we are forced to say that when B passes 1C, it passes 1A since it clearly does not pass no As, so the only option is one. Therefore in the time it takes to pass 1A, B passes both one C and 2 Cs. This is contradictory, so shows that we must either accept that ‘half the time is equal to its double’, or reject that time is finitely divisible. The latter is what Zeno would have us do.

    This is a questionable paradox, because it could be argued that the solution would be to conclude that it is impossible for B to only pass one C in an atomic fragment of time due to the relative movement of both rows. So one would simply say that Zeno’s conclusion does not follow, as it relies on assuming that it is possible for B to pass only one C. Yet this raises the question of why it should not be possible: B must pass one C or else it would not have passed 2Cs it is necessary to pass the first before passing the second. It seems that the solution depends entirely upon the perspective from which the issue is approached, and due to its nature in this way, I feel it certainly highlights something of a problem, or at least an ungraspable element of our understanding of time.

    So, playing along with Zeno’s paradoxes it must be investigated as to what concept of time we are left with. If we are to eliminate that time is infinitely divisible and that it is finitely divisible, we must say either that time does not exist, or that it is not divisible. One possible understanding of this would be Parmenides’ concept of an eternal continuous present, which means that time is one, indivisible occurrence. To his credit, this seems to be as good an understanding of time as any, and does not seem to present any paradoxes. Even though Zeno’s paradoxes may not be entirely sound, the do present the limitations of any specific interpretation of time, as they cannot be conclusively proved wrong.

    Time is often referred to as a measure of change – a concept that arises relative to other things, rather than a thing in itself. So perhaps it can be used in any way we want it to be used, so long as it does not present a contradiction such as those in Zeno’s paradoxes. This would be in keeping with an anti-realist theory of truth, claiming that something is true if it coheres, rather than corresponds to the way things actually are. Hence an understanding of time would be a ‘true’ one so long as it coheres, or makes sense in the context which it is being used in. Therefore as long as it does not present an inconsistency it can be understood however we wish. As expressed by Lee Segall, the need to understand ‘time’ relative to other concepts such as motion means that ‘a man with one watch knows what time it is; a man with two watches is never quite sure’.

    In conclusion, perhaps Zeno can contribute that we are quick to accept time as a concept, but upon examination cannot justify that which we have been basing out understanding on. Time, it seems, cannot be restricted by defining it. So what Zeno has contributed to our understanding of time – quite to the annoyance of some, but to the amusement of myself – is that in fact we understand very little about it.

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    Zenos Illustration of the Concept of Time Through Achilles and the Tortoise. (2022, Nov 25). Retrieved from

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