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Topic #A7



24 September 2001

Scribes: Rose Healy and Jacqueline Tsai


These minutes were spoken on 26 September; for another version,

go to the unspoken minutes




There is little writing preserved of Zeno of Elea, although during his lifetime he wrote a great amount.  Most of our sources are reports given by philosophers:  Plato documented both Parmenides’ and Zeno’s philosophies; Aristotle found Zeno’s arguments challenging, and so worth discussing; and Simplicius added information about scholarship concerning Zeno during a later period.  It is only from these sources that we can have an accurate idea of where Zeno was going with his arguments.  Yet it still remains unclear how many other arguments existed and how exactly the arguments are meant to be understood.


First, it is necessary to reflect back on Parmenides’ philosophies and remind ourselves of what he thought was achieved by his poem.  Parmenides states:


         For you will not find thinking apart from what-is, on which it


         For its expression.  For apart from what-is nothing else

         Either is or will be, since what-is is what Fate bound

         To be entire and changeless.  Therefore all those things which mortal


         Trusting in their true reality, have proposed, are no more than

                  names –

         Both birth and perishing, both being and not being,

         Change of place, and alteration of bright colouring.

         Now, since there is a last limit, what-is is complete,

         From every side like the body of a well-rounded sphere,

         Everywhere of equal intensity from the centre.  For it must not be

         Somewhat greater in one part and somewhat smaller in another.

(F8, lines 34-45, p.60)


Parmenides’ crucial idea is that the world is one thing and changeless (line 38)  Non-existence and existence are illusory names for one reality (lines 38-41).  For Parmenides, what-is cannot stand for other things because there is no variation in the density, quantity, or intensity of being.  Everything remains the same from the inside out.  Parmenides provides more detail:


         Thus birth has been extinguished and perishing made inconceivable.

         Nor can it be divided, since all alike it is.  Nor is there

         More of it here and an inferior amount of it elsewhere,

         Which would restrain it from cohering, but it is all full of what-is.

         And so it is all coherent, for what-is is in contact with what-is.

         Now, changeless within the limits of great bonds,

         It is without beginning and without end, since birth and perishing

         Have been driven far off, and true trust has cast them away.

         It stays in the same state and in the same place, lying by itself,

         And so it stays firmly as it is, for mighty Necessity

         Holds it in the bonds of a limit which restrains it all about,

                                                               (F8, lines 21-31, p.60)


If something exists, then it is logically required to have all these features.  This is the only existence that we have knowledge of, and there is an awareness of one, undifferentiated reality.  


A student then commented on the idea of sphere being the boundary of the universe.  She stated that if there is a sphere, then there must be something outside the sphere.  Prof. Hutchinson pointed out that lines 30-31 and 42-43 explicitly state that the sphere is the limit.   This was an important comment because the idea of limit becomes a crucial concept to all natural philosophy.  Fate and necessity is one way of explaining limit: the idea that it is not literally spacial, and logic is what imposes the constraint on certain limits.  However, the idea that the sphere has a spacial limit creates problems.  Melissus disagrees with the conclusion that a limit exists, and argues that there should be no limit (p.82-83).  Parmenides thought the universe was smooth, solid, and stable.  The sphere has the most uniformity in all dimensions, which is why Parmenides uses it (F8, lines 43-45, p.60).  Unfortunately, this creates edge problems.


Epicurus, on the other hand, argued that the universe is not finite by the use of the javelin experiment.  If one threw a javelin, it would either bounce back or fly away.  If the javelin bounced back, this would be an indication that there exists a limit to our universe.  The fact that the javelin flies away shows that the universe must be infinite.  So, for Epicurus, the universe is very gappy with a finite number of atoms, shapes, and sizes, and an infinite amount of space (imagine a scattered rice pudding with the raisins representing the cosmos).  This means that there are larger and smaller infinities.


Infinity was a very interesting concept for these philosophers to be dealing with as it wasn’t until the 16th and 17th centuries that Leibniz and Descartes found new ways of expressing infinity.  It is very difficult to think clearly about infinity, and it wasn’t until recently that we developed techniques to do so.


Upon first glance, it seems that Zeno defended the view of Parmenides with the use of a vast basket of techniques.  This is displayed in the discussion between Socrates, Parmenides, and Zeno (T1, p.74).  It is actually a piece of historical fiction that Plato created based on his conception of Zeno’s task for philosophy, and shows great originality in getting more clear about infinity.  Zeno was a great constructive thinker, which is evident by the organization of his arguments.  His display of parallel structures of reasoning was the first step towards logic, laws of thought, and inference.  Logic was taken further by Plato through exercises, then completed by Aristotle in his development of a full-fledged logic system.


Plato believed that Zeno’s intentions were to defend Parmenides from attack.  Zeno did so by showing that the idea of plurality has equally ludicrous conclusions as Parmenides’ idea of ‘one’ (F1, p.78).  We can be satisfied with Plato’s opinion of this matter since Plato had Zeno’s book of arguments (which we do not), and since Plato was a clever person.  However, there are those who believe that instead of defending Parmenides, Zeno’s intentions were actually to attack him.  Unfortunately, there is not enough evidence to completely support either of these opinions.  Nevertheless, one is inclined to think that Zeno’s intentions were indeed to defend Parmenides.


Here are now a few examples of Zeno’s arguments:


1.    The Achilles

This involves a race where Achilles is competing against a slower runner (e.g. a tortoise).  Achilles has a handicap and starts behind the tortoise.  Zeno argues that once the race has started, Achilles will never be able to overtake the tortoise.  This is because Achilles must first reach the place the tortoise started, but by then the tortoise has moved on.  Achilles must then reach the place the tortoise has reached now, but the tortoise will have moved again, and this continues ad infinitum.  (T3, p.76)


2.    The Arrow

Zeno argues that a moving arrow is not actually in motion.  This is because at any given moment the arrow is occupying space of its own size.  By definition, for something to occupy space of its own size, it is then still.  And, since an arrow is still at any given moment, it is not in motion.  (T3, p.76)


3.    The Existence of Place

Zeno argued that, “if there is a place, it will be in something, because everything that exists is in something.  But what is in something is in a place.  Therefore the place will be in a place, and so on ad infinitum.  Therefore, there is no such thing as place.”  (T4, p.79)


It is difficult to know what to make of Zeno’s arguments.  Are they a challenge?  Were they meant to be a joke?  Indeed, many people who have read these arguments attempt to refute them.  Not many have ever been inclined to believe them.  Zeno created complex arguments and arranged them in a complex way with conclusions that no one believed.  This is a rather curious fact. 


What these arguments actually show is that even though Parmenides’ poem did not presuppose the concept of infinity, Zeno was the first person to see that this concept has unimagined complexities.  When we think of the modern concept of infinity, we use mathematical analysis to solve these problems.  The recent achievement of calculus enables us to understand our infinite reality better.  But, during ancient times, philosophers did not have the mathematical knowledge to guide them to their conclusions.  Rather, they had to use their common sense to understand infinity.


Simplicius quoted Zeno on the matter of the finite and infinite:


If there are many things, they are bound to be as many as they are, neither more nor less; but if they are as many as they are, they are finite in number.  If there are many things, there are infinitely many things, since there are always other things between any two given things, and others again between any two of those, and so things are infinite in number.  (F1, p.78)


It is reasonable to suggest that “if there are many things, they are bound to be as many as they are, neither more nor less.”  So it is really the second part of this argument that is the problem.  This section was made more explicit in part of the argument that Simplicius did not quote.  If there are many things, then the world is carved into particular things.  But, if there are 70 people in this room, there are no people in between those people, and so on, and so on.  There are separate clumps of reality, and we would be able to count the different ways of looking at reality e.g. the number of arms or cells that exist.  Furthermore, any way of marking chunks of reality can be replaced by another way.


There is a definite format to Zeno’s argument:


              A                             (assumption)

         1.  If A then P                         (argument supplied)

         2.  If A then not P  


Note that premises 1 and 2 are exactly contrary.  The only conclusion that follows is that the assumption must be abandoned.


One student had trouble accepting the first part of this argument.  Prof. Hutchinson explained that if there are many things, then there must be a number.  Once there is a number to something, it is finite.  He then added that to struggle with Zeno’s arguments is the right condition to be in.  They require you to exert extra sweat, and are ultimately a call to simple exercise and logical clarity.