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Zeno of Elea on Plurality March 7, 2007

Posted by Ninja Clement in Philosophy.
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Question: What were Zeno of Elea’s reasons for claiming that “if things are many, they must be both small and large – so small as to have no size and so large as to be infinite”? (Diels and Kranz, Zeno of Elea fragments 1 – 2)

Answer: Zeno’s argument has two limbs, the first of which is not preserved in its entirety. If things are many (if there is plurality), these units would either have no magnitude or magnitude.

  1. If there are many things, then there must be ultimate parts that are not themselves divisible into parts. If they were divisible, then they would be composites, in which case the parts that make up composites would be more ultimate than the things they make up. So the most ultimate parts are not divisible. If they are indivisible, then they have no size, for size implies divisibility. Everything is therefore is made of parts with no magnitude. Anything without magnitude is infinitely small. A summation of infinitely small parts is also an infinitely small thing. So even a composite made up of ultimate parts is an infinitely small thing. In fact, since the result of adding or subtracting a sizeless object from anything is no change at all, sizeless object are, literally, nothing. Therefore all things are infinitely small (they have no magnitude at all) or consist of things that are infinitely small.
  2. What exists must have size. Something that has size can change the size of anything it is added or subtracted to (or else the first thing would be, literally, nothing). Whatever has size is divisible into parts. Those parts, no matter how small, have size and so they are divisible. The parts consist of parts that are themselves divisible, and so on, ad infinitum. Everything therefore is made up of parts with unlimited magnitude. Anything with unlimited magnitude is infinitely large. Therefore all things are infinitely large (they have unlimited magnitude) or consist of things that are infinitely large.

Since nothing can be both infinitely large and infinitely small at the same time, the claim that there are many things is false.  There is only one thing. That thing is not divisible (it is not even in space and time). Each unit has no magnitude or magnitude. 

  1. If no magnitude, then each unit is infinitely small.
  2. If magnitude, then each unit has size and thickness. It can be divided into parts, each of which is at a certain distance from other parts (if there is a plurality of things, they must be separable in space). This goes on ad infinitum, since there is no subdivision of things so small it cannot be divided – that is, so small it does not have one part and another part at a certain distance from each other.

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