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Zeno's Paradoxes

Updated on September 14, 2014
Parmenides | Source
Zeno | Source

The Eleatics: Parmenides and Zeno

Parmenides and his student Zeno, of Elea both, attempted to underwrite one of the most eccentric metaphysical views ever. From the point of view of studying philosophy, it is irrelevant whether they had any specific motives - which would be difficult if not impossible to discern anyway. These are the best known members of the so-called Eleatic school. The Eleatic philosophy is paradoxical and, appropriately, Zeno's surviving fragments are deliberately presented as paradoxes. It is obviously important to understand what a paradox is. The word is used in everyday language with meanings that are weaker than the one relevant in this discussion.

The student of philosophy has an abiding interest in the study of paradoxes because those unsolved puzzles test the limits of reasoning. If genuine paradoxes exist - in other words, if there are riddles and challenges in reasoning that cannot be possibly solved - this raises a fascinating issue about reasoning and the connection of reason to the what we take as the empirical world. Frankly, we have no proof that a paradox is genuine (that no solution is available) and it may well be that we couldn't have such a proof - or even that a proof to this effect would itself raise some paradox. Let us understand that, by definition, a paradox is a proof whose conclusion is unacceptable on some accepted grounds; and the proof appears on close scrutiny to be free of logical or other error. For instance, a valid and sound deductive argument that proves something we take on solid grounds as impossible is a paradox. Or an unassailably strong inductive argument whose conclusion is, again, impossible on some accepted grounds. Notice that, in a paradox, the conclusion does not hnave to be LOGICALLY impossible; it is deemed impossible on weaker grounds - for instance, it may be impossible on grounds provided by our ordinary experience or by our established convictions about how the laws of nature work. If we have a logically impossible conclusion and cannot find any flaw in the argument, we can deduce right away that at least one of the premises HAS to be false on logical grounds. If we have any reason, other than logical grounds, to insist on the truth of all the premises, we have, again, a paradox.

Studying paradoxes is not an idle pursuit or reasoning game - although it can also serve those purposes. As pointed out above, the stakes are as high as they can be with respect to reasoning. If there are unsolvable puzzles in the form of arguments concluding to an impossible conclusion - which means, if there are genuine paradoxes - we establish something about the inadequacy of reason. The issue is serious if paradoxes pop up in the foundations of theories. We may switch to pragmatic justifications about why we keep a theory that has paradoxes in its foundations (we may keep it insofar as it "works" and we may even push for the philosophic view that the truth of theoretical statements is itself a matter of relevant application and utility.) Historically, the discovery of paradoxes in the foundations of Set Theory spelled the end of an overly ambitious program that aimed to derive mathematics from a systematic study of formal logic. This is one example. Let us be clear that theories are also defeated by inconsistencies. An inconsistent collection of propositions (a theory) is one in which it is logically impossible for the theory's propositions to be true together in some logically possible setup (not necessarily the actual one, but SOME logically possible setup.) An inconsistent theory is not necessarily paradoxical. Suppose that we have theory [p1, p2, ..., pm], such that both pi and not-pi are in this theory. Unless one of pi or not-pi is eliminated (and it might be unacceptable to eliminate either one in the theory), then at least one of the remaining propositions in the theory must be false too. But even if we start removing propositions, the same applies: at least one of the remaining propositions must be false. In other words, we have a disaster in our hands. The theory entails, as we say, logical absurdity. In paradoxes, as we saw, it doesn't have to be logical absurdity that it entailed. For instance, in a typical Zenonian paradox, it is proven that nothing can ever complete any distance between points A and B. This does not contradict a logical truth; it contradicts what we take to be a truth about our physical universe. The proposition "I can move from A to B, where A and B are two different and connected points", is actually true but negating it does not yield logical absurdity. "I cannot move from A to B..." is not logically impossible; it is, we take it, physically impossible and also impossible in other relevant senses - but not logically impossible. Contrast negating the proposition "2+2=4" or "A triangle has three angles" or "if it is Monday, then it is Monday"; negating any one of those propositions yields a logical absurdity.

Since antiquity, it was noticed that concepts with which we are, as users of language, quite familiar and able to use, turn out to be breeding grounds for paradoxes when examined closely. This is the case with "truth" and also with the concepts Zeno assaults - "space" and "time." Zeno had a theoretical motivation in generating paradoxes. (What was said earlier about the irrelevance of motivations was referring to extra-philosophic or extra-theoretical considerations - for instance, trying to embarrass one's enemies because they cannot solve the puzzle.) The position Zeno was defending was the Parmenidean view.

Parmenides thought that he had a way of proving that, contrary to appearances, there isn't an ontological plurality of objects; there is only one thing. The earliest philosophic thinkers in the western tradition - the so-called Presocratics - were profoundly interested in figuring out what "really" happens behind the appearances we have by means of our senses. This is the rational interest that moves us from a mythical explanation - that lightning is a weapon used by an angry deity - to the proto-scientific account of an underlying and rationally accessible cause of lightning-events. This outlook is in evidence when, for instance, we say that "really, this table in front of you is a cloud of particles in constant motion." Of course, if you were discussing the table of our ordinary experience (round, brown, etc...), this shift to the "real" account would be fallacious (a fallacy of Irrelevance). Yet, we acquiesce to the notion that the "real" picture is the one provided by the causal account all the way down (to the metaphysics of moving particles.) This is the "real" (as distinguished from the apparent) account that the Presocratics were pursuing. Some of them are Monists (positing one underlying entity and accounting for the plurality of our experience by trying to figure out how the underlying entity yields other things - like atoms combining in Democritus' theory.) Others were Dualists - usually, this view has material and soul-like entities in its metaphysics. There must have been Pluralists (who defended the metaphysical integrity and irreducibility of the various things we access in experience) - especially among the so-called Sophists. A presocratic who has had modern disciples, Heraclitus, seems to have challenged the overall metaphysical project altogether - arguing that the imposition of stable entities, which we can study, is an unwarranted conventional trick and that the "real" picture is chaotic and always-in-flux. At least one of the presocratics, Anaximander, seems to have claimed that it SHOULD be the case that nothing exists : because to-exsist is an offense that is then properly redeemed by the ravishes inflicted on all changing things by time. (This view is sometimes called Nihilism - not to be cconfused with the Nietzschean concept of the same name. Although Anaximander does not deny that at least something exists, and so is not a Nihilist, he is something of a Normative Nihilist if his engmatic fragment B42 indeed makes the case that to-exist is to commit a moral offense, from which it can be reasonly inferred that nothing should exist perhaps.) Parmenides is a Radical Monist who not only posits that one and only one kind of thing exists but also denies that there are permutations that may yield multiple things (as in the Atomic Theory of Democritus).

We will check out Parmenides' views first, since Zenos's paradoxes belong to the same school of Presocratic philosophy. Parmenides is not a lunatic who denies the appearances. This is why his view is essentially paradoxical. Given that we do take it as true that multiple things exist, Parmenides' proof that this is not the case takes on a pradoxical character - we have a proof of something that cannot be true. Notice the use of the word "can" ion this last sentence. This "can" is not logical. "Triangles have two angles" is logically impossible to be true; so is "that shape over there both is and is not a triangle." The meanings of the words make it so that these propositions arre logical falsehoods or contradictions (sometimes also called invalidities.) But the proposition "nothing exists" is not a logical falsehood. The logical word in this sentence is "nothing." The logical form is "there is no such thing x so that, for any thing y, x and y are identical." (If you are an advanced student of Logic, you will detect that I am using Free Logic. But in standard Predicate Logic too, we can deny "there is an x such that x has the attribute F'" for any attribute, without getting a logical contradiction.) The non-logical word in the sentence is exists (at least on one view this is a non-logical word): again, nothing about the meaning of this word makes it so that "nothing exists" is logically impossible to be true. So, Parmenides' theory is not absurd; it is paradoxical.

Elea (Velia) today.
Elea (Velia) today.

How Eleatism Is Different from Rival Presocratic Schools

It is always a source of great philosophic excitement that the presocratic schools anticipate later, all the way to contemporary, trends in philosophy. (There are exceptions, of course.) The Eleatics are radically Anti-Empiricist and Rationalist. Unlike the Materialist early thinkers (Thales, for instance) the Eleatic motto is, to quote Parmenides himself, "to be is to-be-thought-of." This actually pushes in the direction of Berkeley Idealism ((better called Ideaism.) Methodologically, the type of inquiry the Eleatics enthrone is based on rational analysis. Whatever the apparent import of the senses may be, it is what the rational analysis shows that matters! What we have called "paradox" is inherent to the Eleatic school insofar as this school is viewed from the standpoint of common sense. The Elatics repudiate common sense from the beginning in that they require the highest available standards of certitude (affordable only through rigorous proof) and reject presumptively the testimony of the senses: as Descartes would complain a lot later, the senses deceive us and, insofar as we seek an approach that guarantees indubitable and incorrigible evidence, we ought to turn to pure rational analysis and swallow the paradoxical consequences. Descartes, famously, sought to discover principles that would permit a methodically certain study of empirical reality itself. The Eleatics do not appear interested in the study of the world of the senses - or perhaps they think they have already proven that the only reliable approach even to the world of the senses is through rational inquiry: what is left over (what is disproven) is rejected even if ordinary experience attests to it. Once again, this accounts for the inevitably paradoxical character of the philosophy.

The Eleatics agree with the Pythagoreans about the higher dignity of mathematical inquiry but the Pythagoreans move to a claim that mathematical objects are embodied in the material world. The Eleatics posit - and have proofs for the claim that - there is no divergence from the eternal One and, so, there can be no duality between the intelligible and the phenomenal. Plato was influenced by the Parmenideans (he pays tribute to them in the trilogy Parmenides, Sophist, Statesman) but he ends up being a Dualist in that he accepts two different kinds of beings - noumenal or intelligibles and phenomenal or sensible things.

The case of Heraclitus (or Heracleitus) is intriguing and we can only comment on it briefly. Heraclitean theory accepts that the very nature of things is radically unstable or in constant flux as the familiar metaphor has it. In extant fragments, Heraclitus makes certain radical claims about logic - eerily anticipating a school of our times called Dialeithism. According to this view, meaningful statements may have both truth-values - true AND false. In other words, contradictions are admissible. Heraclitus seems to embrace the extreme view that this is true of all meaningful statements. (A more moderate view, called today Paraconsistency, takes it that some contradictions may be rightly assertable.) "You both can and cannot step into the same river." A problem that is hiding behind this will allow us to go back to the Eleatics. The problem has to do with change. For Heraclitus, who has a philosophy of "becoming," it is only change that exists - so to speak. The problem is this. Suppose a green leaf becomes yellow when the Fall arrives. There has to be a sense in which both greenness and yellowness are predicated of the item to which we want to refer stably as "this leaf." It is the same leaf - or so we assert. It, then, must be both green and yellow. This seems wrong, but why? It seems that there is an easy solution to this riddle: the leaf, call it "a", is "green-at-time-t1" and "yellow-at-time-t2" with t1 and t2 being different time indeces. We take this as an obvious solution but we don't realize that the metaphysical status of TIME is by no means straightforward. Being intuitive Newtonians in our metaphysics, we don't puzzle over the nature of time. What kind of thing is this that we invoke to say that there different moments t1 and t2? Notice, by the way, how, try as you must, you are prone to conceive of t1 and t2 as two different points on a line (but this is a spatial arrangement!) Aristotle would solve the problem of change this way - by letting time do the job. Temporal modalities (like "IT WILL be true that p" or "IT HAS ALWAYS BEEN the case that p") have some affinity with POTENTIALITIES. The metaphysical status of the POTENTIAL is another notorious philosophic riddle. Aristotle accepted potentialities. (It is not a coincidence that the Catholic Church, whose Natural Law system has a central Aristotelian component, like to talk a lot about potential objects - for instance, the fetus as a potential person.) Without accepting potentialities, we are back to our initial predicament: the leaf, a, must be both green and yellow in some sense. But being green it cannot be yellow and being yellow it cannot be green. So, it must both be and not be what it is. This makes Heraclitus buy the bullet and accept contradictions as meaningful.

The Eleatics reject space and time as existing things. Parmenides takes the ONE (the only thing that exists) as being eternal and not-in-space. The Eleatic Melissus took the view that the ONE is omnitemporal - but this should be taken to mean that there is a separate thing that is the time-entity. Although change is a problem for the other presocratic schools, it is not for the Eleatics because their view is that nothing changes anyway. One of the most astounding developments in 20th-century science is the establishment of Einstein's theory of Special Relativity. The traditional metaphysical problems (cutting across all possible theories) cannot be expressed in the language of modern science. The scientific paradigm requires its own metaphysics but it cannot - it does not have the expressive resources - to ask or discuss broader metaphysical issues. (This does not prevent popularizing scientists from doing this but their authority does not carry over and they usually sound shallow when they try to tackle traditional metaphysics.) So, the question "what is the view of the universe according to the theory of special relativity?" is not itself a question for science to answer. Still, we can pose the question. Guess what happens. There is a good case to make that the metaphysical implications of Special Relativity are Eleatic!!!

Here is a Parmenidean type of proof that only one thing, the undifferentiated ONE, exists.

  1. Principle: "nothing" cannot exist; note that the verb "to be" serves a dual function - as connector of subject and predicate, and as a predicate that attributes existence to something; in the second sense, being cannot be predicated of nothing - by definition, nothing is what has the attribute "non-existent"; so it cannot possibly have the predicate "existent." So, nothing cannot possibly exist. [The modern school of philosophy known as the Analytic School has a critique of this. Something confused is allegedly going on. You can find this critique especially when Analytical philosophers attack the work of the 20th-century German philosopher Martin Heidegger who uses many Parmenidean twists in his own system. NOTICE something: does he take "nothing" to be naming some thing? Or does he take "nothing" to be an attribute - so we could try to say that "anything that is nothing has the attribute of nothingness predicated of it?" Can we do all this? Is it acceptable by the logical grammar of a language?]
  2. If anything exists, it cannot be nothing that exists. Assume that one thing exists which is something, not nothing. Call this the ONE.
  3. Now, for anything else, besides the ONE to exist, there must be a distinction we can draw between the ONE and that second thing - call it SECOND. But what is a distinction - what do we mean by difference? The attributes of difference are like the attributes of nothing. The difference between A and B is neither like A nor like B. Call the difference D. If D is like A or like B, then we are back to A, or to B. So, D is not like A and is not like B. But in some respects, D must be like A and like B since it allows us to compare A and B. So, D both is not and is like the things it compares. But this is absurd. So, there can be no such thing, no D. If there is no D, no difference, we never make it from the ONE to the SECOND.
  4. Metaphysically, the Eleatics do not need to take attributes themselves to be existents (that will be Plato's case.) For the Eleatic, what we are doing here is just a rational account - a noumenal or intelligible act (remember, for Parmenides, "it is the same thing to be and to be thought-of.") We just saw that, given the primordial ONE, it leads to logical absurdity to stipulate that a SECOND exists. So, a THIRD, and so on, cannot exist either - by the same reasoning.
  5. Since there can be no difference beteen the ONE and the SECOND, we never make it to the second! There is ONE and only that. This means that there can be no motion or change or differentiation -ever. There is no time - since time would have to be composed of differentiated units. There is no space, for the same reason. The ONE is nowhere and no-when.
  6. An ODD recent development: the universe of Special Relativity is Eleatic. The SPACETIME, which is the stuff of the universe in Relativistic Physics, is not itself in space or in time. Is Parmenides having the last laugh? (Spacetime is, however, populated by "frozen" events. The Eleatics have an ANTI-REALIST view of this: that all these items, besides the ONE, are mind-dependent; we think they are really out there but they are not. Physicists and other scientists tend to be REALISTS - although their metaphysics does not really matter for doing the science. A REALIST takes the things she is talking about (for instance, the events of spacetime) to be mind-independent or "really" out there even if no one is thinking about them.)
  7. Another corollary is that the ONE is eternal - not in time, not omnitemporal, but eternal. The ONE could not have been created because something else - what created the ONE - would have to be existent; but, as we saw in 3 above, this is impossible. The whole thing, the ONE, is frozen. What appears as an open-ended variety to our deluded empirical perceptions belies what is essentially a frozen, immovable UNITY of being.
  8. We can also try to prove, in this fashion, that the ONE could not possibly fail to exist.
  9. The kind of proof given above, in 3, can also be presented with different premises. All such Parmenidean proofs would be exploiting the point about the logical impossibility that "nothing" exists. This is the key.
  10. We will find out that Zeno's proofs follow the method known as Indirect Proof or Proof by Contradiction - whereas the proof above, in 3, is a direct deductive proof. Parmenides' proofs are deductive - the strong kind of reasoning that guarantees conclusions if the premises are true and the structure or form of the argument is valid. We have valid Parmenidean proofs. The question is: are they sound - do they have premises that are warranted to be accepted as true? Parmenides' proofs progress from premises to conclusion. An indirect proof - the kind we will find used by Zeno - reaches a conclusion X by positing not-X and deriving a contradiction from not-X and other warranted assumptions. By contradiction we mean any proposition that has the form "p and not-p" but also, more broadly, any proposition that is a logical falsehood - that cannot be true on logical grounds.


Zeno's Paradoxes 1

Zeno was credited in antiquity with over 50 paradoxes but only 4 are known to us - and we have sufficient information to attribute to him several others. He might have hit on a famous - still challenging - paradoxes known as the Sorites or the Heap. This is NOT a paradox related to space or time - like the paradoxes we will treat here. [Sorites paradoxes occur when you deal with small increments which are accepted to be insignificant: yet, at some point, some difference seems to have arisen. For instance, if you keep adding tiny morsels to a first morsel, you ought to have no heap. Yet, you look and eventually you have a heap. You can't tell at what point what was not a heap became a heap. It works in the other direction too: keep subtracting tiny pebbles from the heap; it does not make any difference each time you do; yet, you end up with something that is not a heap! The challenge here is actually logical, not about physical things. Think of the following as a solid rule of inference in logic: If X then Y; X/-> therefore, Y. Now try: "if we subtract tiny part from heap, still heap; we subtract tiny part from heap/.. therefore, still heap." But the conclusion is at some point turning out to be false - the heap is gone. And, yet, all we do is keep re-applying a valid rule of inference. It is not clear how Zeno might have used the HEAP paradox to lend support to Parmenidean philosophy.]

Before we move to the well-known paradoxes of Zeno, we can reflect on some other paradoxes that Zeno might have also pioneered. The target in all these is the claim that there are many atom-like things or there is space or there is time. The ATOMIC theory answers Zeno (we will have to figure this out later) but Zeno has a response in the form of a paradox. Here it is: suppose we have two atoms. This is enough, of course, to refute Parmenides who claims that only the ONE exists. Call them atoms A and B. What is there between them? There are two options: nothing or something. If nothing, then we run into absurdity as Parmenides showed. (Do we? Should we try independently? Well, we can think of a "traveling" problem. How do you get from A to B if there is "nothing" between them? The Atomic theorist has a problem here. The world of experience seems continuous and the Atomic theorist is proud to have a theory that can account for that. But the discontinuity of the atoms - if there is nothing between A and B - indicates that the "real" world is like frozen shots that have no option of transition between them. Then, the appearance of continuity remains illusionary - the deeper view of how things work does not support it. Moreover, we cannot explain on the basis of our physics or theory why we get an impression of continuity.)

The other option, recall, is that there is something between A and B. That would be atom C. So, we have satisfactory microcosmic or "real" view - but wait a minute: We will have to pose the same question: what is there between A and C (and between B and C for that matter)? The same options - nothing or something. We reject nothing again and we take something. Call this atom D. Again: what is there between C and D? Same all over: atom E. And so on, ad infinitum! As you will soon find out, Zeno now has us where he wants us. He has a proof that this situation is paradoxical. Hence, the premise that we have atoms deep down is refuted - since it leads to something that seems unacceptable.

A famous Zenonian paradox is the DICHOTOMY. It might be the best place to start.

  1. Assumption: you can get from point A to point B where A and B are the extremes of a line-segment AB. Our experience shows that this can be done. So, if we get absurdity out of this, we have a paradox insofar as we have no solution that shows what is wrong with the proof.
  2. Before you can get from A to B, you surely must first pass over the point that is exactly half-way between A and B. Let's indicate this point by the length it cuts from the beginning: AB/2.
  3. Before you get to the point AB/2, you must surely first pass over the point that is exactly half-way between A and AB/2: call this point AB/4.
  4. Before you get to AB/4 you must first pass over the point that is exactly half-way between AB/2 and AB/4: call this AB/8.
  5. Before you can get from AB/4 to AB/8, you definitely need first to pass over the point half-way between these two -- AB/16.
  6. And so on. This is repeated. We have no reason to abandon our principle that you first pass over the half-way point between those points. (Is this is a logical or a physical principle?)
  7. We notice that the points you have to keep traveling over, taken together, have denominators that form a mathematical series: you have to be traveling over AB/2, AB/4, AB/8, AB/16, AB/32, AB/64, AB/128, AB/256, .... This is an infinite series!
  8. You cannot cross an infinite series to completion. Suppose that a bell rings every time you cross one of those points. The last bell can never ring - it is logically impossible. So, you never make it form A to B.
  9. Put together 1 and 8: you can and you can't make it B. Logical absurdity - contradiction!
  10. Given 9 - we have reached a contradiction - we must reject our initial assumption 1: you CANNOT possibly make it to point B. Contrary to what experience suggests, we have a proof that it is not possible to cross any line segment. AB is a random line segment - nothing special about it. The proof applies to any conceivable line segment.
  11. Even if point B is "right next" to A you cannot get there. What are you moving over. If "nothing", then you are not moving. If something, call it C, then what is there between A and C? Same thing all over again: it must be a D. What is there between C and D? And so on - ad infinitum. We run into an infinite series of points to be crossed again.
  12. Motion is impossible - you have the proof based on reason and you should take this one instead of the unreliable evidence of your senses.

Another version of the DICHOTOMY paradox concludes that motion cannot even start at A when you try to cross from A to B.

A response you come across if you look this up on the internet, due to mathematicians who scoff at the paradox, is this: the theory of limits in mathematical analysis, which poor Zeno did not know, shows that we can approximate the length of AB: the sum SUM(1/2, 1/4, 1/8, 1/16, 1/32, ..., 1/x, 1/2x, ...) = 1. Hence, we have a finite length, not an infinite one; so, AB can be crossed.

This misses the point. Zeno lets you postulate any length you like for AB. Zeno's challenge is not about how to compute the length of AB but about how it is possible to reach the end of an infinite number of points. Understood as a geometrical segment, AB is indeed infinitely packed. Actually, the infinity of points between A and B is the size of the real numbers - it is called nondenumerable infinity and cannot be put into one-to-one correspondence with the natural numbers.

Another proposed solution draws on the distinction between geometrical and physical line segments. It is true that geometrical lines are infinitely packed but physical lines aren't. This thinking may lie behind Democritus's thinking about atoms. If you keep cutting a physical line in half, you will end up with material entities that can no longer be cut - the uncutables or indivisibles for which the Greek word is "atoms." As Aristotle pointed out, this goes against our ordinary experience. A Zenonian response to this, as we have seen, exploits the issue as to how atoms interact - what is between any two atoms that makes a "bridge" so to speak between them?

Could this be a genuine paradox?

In geometry, it is undeniable that we take the length of a segment, AB, as finite even though we cannot deny that the segment is packed with an infinity of points between A and B. Are we facing a LOGICAL absurdity here? There is a solution to this. Take a geometrical point to be like an empty set - no members. Let us symbolize with "{}". Now, take an infinite set with empty sets as members: {{}, {}, {}, ...}. Because there is only one empty set, we can write the set above as: {{}}. Notice that the set with the empty set as member is NOT itself empty. Think of the empty set as your dimensionless geometrical point of AB; you have an infinity of those points between A and B. Now, think of the length of AB as as {{}} - it is not empty. We can have a length - whatever it is said to be - even though the member (the point) is itself empty (without length.)

What you just saw has wider implications. Set theory came to the rescue to defuse a challenge. Contemporary solutions to Zeno's paradoxes also rely on using the right "tool" - like set theory in the above example. Insofar as the tool so employed is appropriate and effective in its theoretical applications, we can claim that our theories about motion are not susceptible to Zenonian paradoxes.

A related way of responding to a Zeno paradox is to deny at least one of its premises on the grounds that we don't have to be committed to some underlying theory that fits Zeno's purposes. Aristotle played this game well. We could, for instance, deny divisibility to elements of motion - like space and time. It is this divisibility postulate that falls into Zeno's hands and allows him to generate paradoxes.

There are, however, contemporary paradoxes that are Zenonian in their conception and recast Zeno-like challenges. Such paradoxes are about what are known as "supertasks." To see the connection consider that the Dichotomy paradox we considered above poses as its crucial thrust the impossibility of completing what is called a supertask: move over each and every one of an infinite or non-terminating series of points.

Zeno's Paradoxes II

"If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited." (Simplicius(a) On Aristotle's Physics, 140.29)

Achilles and theTortoise

"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." Aristotle,Physics VI:9, 239b15

"The [second] argument was called “Achilles,” accordingly, from the fact that Achilles was taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount" (Simplicius(b) On Aristotle's Physics, 1014.10)

The Arrow

"If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless." Aristotle, Physics VI:9, 239b5

The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b.30)

Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. (Diogenes Laertius Lives of Famous Philosophers, ix.72)

Notice the hinted solution in the commentary by Aristotle. Zeno assumes that time is composed of moments. Incidentally, this is the view that works for Newtonian physics. What would be an alternative view of time?

Bertrand Russell came up with another solution based on denying that the concept "motion," as we use it, agrees with what Zeno says.

The Stadium

The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite directions—the ones from the end of the stadium, the others from the middle—at equal speeds, in which he thinks it follows that half the time is equal to its double…. (Aristotle Physics, 239b33)

To see what this means check the drawing below. This puzzle seems to be based on a confusion about relative velocities.

The Stadium
The Stadium
Where is the error?
Where is the error?
The Dichotomy
The Dichotomy

© 2014 Odysseus Makridis


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