Zeno՚S Paradoxes of Plurality: Zeno of Elea and Zeno՚S Paradoxes

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Zeno of Elea

  • Zeno of Elea was a student or philosophical associate of Parmenides. Almost everything we know about the life of Zeno comes from Plato՚s dialogue “Parmenides” .
  • His treatise is a defense against those who say or claim Parmenides՚ argument of “All is One” has many ridiculous consequences. He gets back to the advocates of plurality or motion by aiming to prove that “if there are many things” it suffers more ridiculous consequences.
  • “Parmenides argues for monism, Zeno argues for pluralism” .

Zeno՚S Paradoxes

  • Zeno wrote a book of Paradoxes defending Parmenides ′ philosophy. Unfortunately, we don ′ t have the original work and we know of his arguments through Aristotle and his commentators, particularly Simplices.
  • There were apparently 40 ‘paradoxes of plurality’ . of these only 2 most definitely survive, and a third one is partially attributed to him.
  • There are 4 arguments against motion (4 paradoxes of motion- The dichotomy paradox, Achilles and the tortoise paradox, the arrow paradox and the stadium paradox) .
  • Aristotle has also attributed two other paradoxes to Zeno- the paradox of place and the grain of millet paradox.

Paradoxes of Plurality

  • A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one՚s expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.
  • The paradoxes of plurality aim to give arguments against ontological pluralism. They start off with assuming plurality which further leads to contradiction. Therefore, it rejects plurality to avoid absurd conclusions.
  • The paradoxes of motion are closely related to the arguments against plurality. They are attacks on commonly held belief that motion is real, but motion is a kind of plurality- a process along plurality of places in a plurality of times. So, attacks on motion are also attacks on this kind of plurality.
  • By disapproving motion, Zeno disapproves the thesis of plurality in one of its major applications. However, he has offered more direct attacks on all kinds of plurality.

They are as follows:

  • Alike and Like
  • Limited and Unlimited
  • Large and Small
  • Infinite Divisibility

Alike and Like

  • “If things are many, they must be both like and unlike. But that is impossible; unlike things cannot be like, nor like things unlike.” (Parmenides 127 - 9)
  • Zeno argues that the assumption of plurality leads to a contradiction.
  • Consider a plurality of things- say, some people and some mountains. Both have the common property of being ‘heavy’ . But if they all have this property in common than they are really the same kind of things and there՚s no plurality.
  • By this reasoning Zeno establishes that plurality is one (many is not many) , which is a contradiction.
  • So, by reduction ad absurdum (a thesis must be accepted because its rejection would be untenable) , there is no plurality.

Plato՚S Response

  • He accuses Zeno of equivocating (use of ambiguous language) . A thing can be like some other thing in one respect, while not being alike in some other respects. Having one property in common doesn՚t make it identical with that thing.
  • Like in the above example, people and mountains are both heavy but they also have distinct properties like the people are intelligent, but the mountains are not.
  • Plato says that Zeno proves “something is many and one [in different aspects] , not that the unity is many or that plurality is one.”
  • So, there is no contradiction, and the paradox is resolved by Plato himself. This is one of Zeno՚s weakest paradoxes. It is rarely discussed now and is usually omitted from the list.

Limited and Unlimited (Argument from Denseness)

  • “If there are many, they must be as many as they are and neither more and nor less than that. But if they are as many as they are, they would be limited. If they are many, things that are unlimited. For there are always others between the things that are, and again others between those, so the things that are unlimited.” (Simplices (a) On Aristotle՚s Physics, 140.29 - 33)
  • Zeno argues that there cannot be more than one thing because they are both ‘limited’ and ‘unlimited’ which is a contradiction.
  • On the one hand, any collection must contain some definite or certain or fixed number of things, ‘neither more nor less’ . And if you have a fixed number of things, it must be ‘limited’ or ‘finite’ .
  • On the other hand, a collection of many things would always have something else between those, and something else between those, ad infinitum. Therefore, the collection is ‘unlimited’ or ‘infinite’ .
  • Therefore, as per Zeno՚s reasoning, there are no pluralities. Otherwise it leads to a contradiction.

Weakness of Zeno՚S Argument

  • According to the Standard Solution, the weakness lies in the assumption that “to keep them distinct, there must be a third thing separating them.” Are there always things between the things that are? Why must objects always be ‘densely’ ordered?
  • Imagine we have 10 apples in a line, there is 1 apple between 6th and 8th . But there is none between 6th and 7th . This view of Zeno presupposes that their being of different substances is not enough to render them distinct. But two objects can be distinct at a time simply by having a property the other does not have.
  • Though suppose one holds that some collection is dense, hence ‘unlimited’ or ‘infinite’ . Then also there remains problem with his first part of the argument. Zeno argues that because a collection contains a ‘definite’ no. of elements, it is also ‘limited’ . The main idea is- “If we admit plurality and divisibility, all the parts together will make up the whole of the universe. In order to be complete their no. must be finite.”
  • However, Cantor has showed us a way to understand infinite numbers in a way that makes them just as definite as finite numbers by giving the theory of ‘transfinite numbers.’ So mathematically, Zeno՚s reasoning is unsound when he says that because a collection has a definite number, it must be finite, and the first sub-argument is fallacious.

Large and Small (Argument from Finite Size)

  • “… if it should be added to something else that exists, it would not make it any bigger. For it were of no size and was added, it cannot increase in size. And so, it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing.” (Simplices (a) On Aristotle՚s Physics 139.9. -15)
  • ″ But if it exists, each thing must have some sizes and thickness, and part of it must be apart from rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and keep saying it forever. For no such part of it will be last, nor will there be one part not related to another.
  • Therefore, if there are many things, they must be both small and large; so small as to not have size, but so large as to be unlimited. ″ (Simplices (a) On Aristotle՚s Physics, 141.2 - 8)
  • First error is in his assumption that “If there is a plurality, then it must be composed of parts which are not themselves pluralities.”
  • This assumption is for the first part of the argument. Now we do not know why Zeno argued that if many things exist then they have no size at all, but we must assume this to make sense of the rest of the argument that follows. One interpretation from this is that it got be so because- things that are not pluralities cannot have a size or else they՚d be divisible into parts and thus be pluralities.
  • Example of university is given to counter this. We can say a university is a plurality of students, but students themselves can also be a plurality in a different context. Like, we might say that a student is a plurality of biological cells.
  • what՚s whole and what՚s part can vary. Zeno is confused about this notion of relativity and about part-whole reasoning.
  • Second error occurs in arguing that each part of plurality must have a non-zero size. By the contemporary notion of measure, we know that the measure function of a line segment can be defined properly to show that a line segment has a non-zero measure even though any point has a zero measure.

Infinite Divisibility (Argument from Complete Divisibility)

  • ″ whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible would have resulted if it has actually been divided … though perhaps nobody in fact could so divide it.
  • What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesis the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain … the body will either consist of points (and its constituents will be without magnitude) or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus, presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude. ″ (Aristotle On Generation and Corruption, 316 a 19)
  • This argument is not attributed to Zeno by Aristotle, but in On Aristotle՚s Physics, Simplices has said that it is originated by Zeno.
  • This is the most challenging of all paradoxes of plurality.

Aristotle has stated 3 steps or parts in this argument:

  • Assume an object is theoretically or hypothetically completely divided. By cutting the object into two non-overlapping parts, then similarly cutting these parts into parts and so on till the division is “exhaustive” .
  • “The elements” thus obtained will have no size or magnitude. If they have any magnitude, then the division was not complete after all.
  • Supposing that the body is divided into dimensionless parts, there are two possibilities- the elements are something, but they have zero size.
  • If the elements are nothing, the whole object would be a mere appearance or an illusion.
  • But if the elements are some things with zero size, the original object is composed of elements of zero size. They are referred to as point-parts. So, as the argument concludes, even if they are points since these are unexpended the body itself will be unexpended. Adding any sum of zeroes, even infinity, would yield a zero.
  • In summary, there is a problem of reassembly considering all the three possibilities. All three possibilities lead to absurdities, so the objects are not divisible into plurality of parts.

Standard Solution Response

  • We first need to ask: what is it that Zeno (or Aristotle) is dividing. Is it concrete or abstract?
  • If concrete, then we would reach ultimate constituents of matter such as quarks and electrons that cannot be further divided. According to quantum electrodynamics, they do have a zero size, but it would be incorrect to conclude that the whole object would have zero size.
  • If abstract, Zeno is again mistaken in saying that the total length, measure of all the zero-size elements is zero. It is wrong mathematically, as stated before.
  • Both of Zeno՚s paradoxes- ‘Argument from Finite Size’ and ‘Infinite Divisibility’ were generally considered to have shown that a continuous magnitude cannot be composed of points. But this was later on refuted by scientific, mathematical and philosophical discoveries.
  • So, there is not reassembly problem here and it breaks down Zeno՚s argument.

Conclusory Remarks

  • Zeno drew attention to the idea that reality as how it appears to be can be deceiving. He had a significant influence on his subsequent philosophers and scholars like Atomists and Aristotle.
  • His paradoxes caused mistrust in infinities and this influenced contemporary movements and discoveries. The paradoxes are often pointed to for a case study in how a philosophical problem can be dealt with. Before Zeno, philosophers majorly expressed their philosophy in poetry. He was the first philosopher to use prose arguments and this new method of presentation shaped all later philosophy, mathematics and science.
  • While studying Zeno՚s paradoxes, the time frame needs to be kept in mind when all these paradoxes were proposed. Also, these are not Zeno՚s original words which adds to the absurdity.
  • Zeno՚s paradoxes are now generally considered to be puzzles because of the wide agreement among experts that there is at least one acceptable resolution of the paradoxes. This resolution is the Standard Solution, which is not simply employing concepts that would undermine Zeno՚s reasoning but also the ones that are development of a coherent and fruitful system of mathematics and physical science. Aristotle՚s treatment does not seem to be adequate in front of that.
  • Even though a lot of time has passed, the paradoxes still seem to be fascinating and confusing at first glance. They make one think and question. Paradoxes of plurality are not as popular as paradoxes of motion and are usually missed while studying, but they are also quite interesting.

References

  • Cohen, S. Marc, P. Curd and C. D. C. Reeve, Readings in Ancient Greek Philosophy (Indianapolis, IN: Hackett Publishing Co. , 2011) , 4th edition.
  • Frankel, Hermann, Zeno of Elea՚s Attacks on Plurality, Vol. 63, No. 1 (1942) , pp 1 - 25, published by The John Hopkins University Press.
  • Internet Encyclopedia of Philosophy՚s entry of Pre-Socratics and Zeno՚s Paradoxes.
  • Stanford Encyclopedia of Philosophy՚s entry on Zeno՚s Paradoxes.

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