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5.1: Pre-Socratics

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    27Pre-Socratics

    Heraclitus51

    Heraclitus of Ephesus (/ˌhɛrəˈklaɪtəs/; Greek: Ἡράκλειτος ὁ Ἐφέσιος, Hērákleitos ho Ephésios; c. 535 – c. 475 BC) was a pre-Socratic Greek philosopher, and a native of the city of Ephesus,[2] then part of the Persian Empire. He was of distinguished parentage. Little is known about his early life and education, but he regarded himself as self-taught and a pioneer of wisdom. From the lonely life he led, and still more from the apparently riddled and allegedly paradoxical nature of his philosophy and his stress upon the needless unconsciousness of humankind, he was called "The Obscure" and the "Weeping Philosopher".

    Heraclitus was famous for his insistence on ever-present change as being the fundamental essence of the universe, as stated in the famous saying, "No man ever steps in the same river twice" (see panta rhei, below). This position was complemented by his stark commitment to a unity of opposites in the world, stating that "the path up and down are one and the same". Through these doctrines Heraclitus characterized all existing entities by pairs of contrary properties, whereby no entity may ever occupy a single state at a single time. This, along with his cryptic utterance that "all entities come to be in accordance with this Logos" (literally, "word", "reason", or "account") has been the subject of numerous interpretations.

    Life

    The main source for the life of Heraclitus is Diogenes Laërtius, although some have questioned the validity of his account as "a tissue of Hellenistic anecdotes, most of them obviously fabricated on the basis of statements in the preserved fragments." Diogenes said that Heraclitus flourished in the 69th Olympiad, 504–501 BC. All the rest of the evidence — the people Heraclitus is said to have known, or the people who were familiar with his work — confirms the floruit. His dates of birth and death are based on a life span of 60 years, the age at which Diogenes says he died, with the floruit in the middle.

    Diogenes says that he abdicated the kingship ( which can mean either that it was fundamentally wrong or that he considered it toilsome. Two extant letters between Heraclitus and Darius I, quoted by Diogenes, are undoubtedly later forgeries.

    With regard to education, Diogenes says that Heraclitus was "wondrous" (thaumasios, which, as Socrates explains in Plato's Theaetetus and Gorgias, is the beginning of philosophy) from childhood. Diogenes relates that Sotion said he was a "hearer" of Xenophanes, which contradicts Heraclitus' statement (so says Diogenes) that he had taught himself by questioning himself. Burnet states in any case that "... Xenophanes left Ionia before Herakleitos was born." Diogenes relates that as a boy Heraclitus had said he "knew nothing" but later claimed to "know everything." His statement that he "heard no one" but "questioned himself," can be placed alongside his statement that "the things that can be seen, heard and learned are what I prize the most."

    Diogenes relates that Heraclitus had a poor opinion of human affairs. He believed that Hesiod and Pythagoras lacked understanding though learned and that Homer and Archilochus deserved to be beaten. Laws needed to be defended as though they were city walls. Timon is said to have called him a "mob-reviler." Heraclitus hated the Athenians and his fellow Ephesians, wishing the latter wealth in punishment for their wicked ways. Says Diogenes: "Finally, he became a hater of his kind (misanthrope) and wandered the mountains ... making his diet of grass and herbs."

    Heraclitus' life as a philosopher was interrupted by dropsy. The physicians he consulted were unable to prescribe a cure. Diogenes lists various stories about Heraclitus' death: In two versions, Heraclitus was cured of the dropsy and died of another disease. In one account, however, the philosopher "buried himself in a cowshed, expecting that the noxious damp humour would be drawn out of him by the warmth of the manure", while another says he treated himself with a liniment of cow manure and, after a day prone in the sun, died and was interred in the marketplace. According to Neathes of Cyzicus, after smearing himself with dung, Heraclitus was devoured by dogs.

    Works

    Diogenes states that Heraclitus' work was "a continuous treatise On Nature, but was divided into three discourses, one on the universe, another on politics, and a third on theology." Theophrastus says (in Diogenes) "...some parts of his work [are] half-finished, while other parts [made] a strange medley."

    Diogenes also tells us that Heraclitus deposited his book as a dedication in the great temple of Artemis, the Artemisium, one of the largest temples of the 6th century BC and one of the Seven Wonders of the Ancient World. Ancient temples were regularly used for storing treasures, and were open to private individuals under exceptional circumstances; furthermore, many subsequent philosophers in this period refer to the work. Says Kahn: "Down to the time of Plutarch and Clement, if not later, the little book of Heraclitus was available in its original form to any reader who chose to seek it out." Diogenes says: "the book acquired such fame that it produced partisans of his philosophy who were called Heracliteans."

    As with other pre-Socratics, his writings survive now only in fragments quoted by other authors.

    Ancient characterizations

    "The Obscure"

    At some time in antiquity he acquired this epithet denoting that his major sayings were difficult to understand. According to Diogenes Laërtius, Timon of Phlius called him "the riddler" (αἰνικτής ainiktēs), and explained that Heraclitus wrote his book "rather unclearly" (asaphesteron) so that only the "capable" should attempt it. By the time of Cicero he had become "the dark" (ὁ Σκοτεινός — ho Skoteinós) because he had spoken nimis obscurē, "too obscurely", concerning nature and had done so deliberately in order to be misunderstood. The customary English translation of ὁ Σκοτεινός follows the Latin, "the Obscure."

    The "weeping philosopher"

    Diogenes Laërtius ascribes the theory that Heraclitus did not complete some of his works because of melancholia to Theophrastus. Later he was referred to as the "weeping philosopher," as opposed to Democritus, who is known as the "laughing philosopher." If Stobaeus writes correctly, Sotion in the early 1st century CE was already combining the two in the imaginative duo of weeping and laughing philosophers: "Among the wise, instead of anger, Heraclitus was overtaken by tears, Democritus by laughter." The view is expressed by the satirist Juvenal:

    The first of prayers, best known at all the temples, is mostly for riches... Seeing this then do you not commend the one sage Democritus for laughing... and the master of the other school Heraclitus for his tears?

    The motif was also adopted by Lucian of Samosata in his "Sale of Creeds," in which the duo is sold together as a complementary product in the satirical auction of philosophers. Subsequently, they were considered an indispensable feature of philosophic landscapes. Montaigne proposed two archetypical views of human affairs based on them, selecting Democritus' for himself. The weeping philosopher may have been mentioned in William Shakespeare's The Merchant of Venice. Donato Bramante painted a fresco, "Democritus and Heraclitus," in Casa Panigarola in Milan.

    Fragments of Heraclitus52

    1 It is wise for those who hear, not me, but the universal Reason, to confess that all things are one.

    2 To this universal Reason which I unfold, although it always exists, men make themselves insensible, both before they have heard it and when they have heard it for the first time. For notwithstanding that all things happen according to this Reason, men act as though they had never had any experience in regard to it when they attempt such words and works as I am now relating, describing each thing according to its nature and explaining how it is ordered. And some men are as ignorant of what they do when awake as they are forgetful of what they do when asleep.

    3 Those who hear and do not understand are like the deaf. Of them the proverb says: "Present, they are absent."

    4 Eyes and ears are bad witnesses to men having rude souls.

    5 The majority of people have no understanding of the things with which they daily meet, nor, when instructed, do they have any right knowledge of them, although to themselves they seem to have.

    6 They understand neither how to hear nor how to speak.

    7 If you do not hope, you will not win that which is not hoped for, since it is unattainable and inaccessible.

    8 Gold-seekers dig over much earth and find little gold.

    9 Debate.

    10 Nature loves to conceal herself.

    11 The God whose oracle is at Delphi neither speaks plainly nor conceals, but indicates by signs.

    12 But the Sibyl with raging mouth uttering things solemn, rude and unadorned, reaches with her voice over a thousand years, because of the God.

    13 Whatever concerns seeing, hearing, and learning, I particularly honor.

    14 Polybius iv. 40. Especially at the present time, when all places are accessible either by land or by water, we should not accept poets and mythologists as witnesses of things that are unknown, since for the most part they furnish us with unreliable testimony about disputed things, according to Heraclitus.

    15 The eyes are more exact witnesses than the ears.

    16 Much learning does not teach one to have understanding, else it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus.

    17 Pythagoras, son of Mnesarchus, practised investigation most of all men, and having chosen out these treatises, he made a wisdom of his own--much learning and bad art.

    18 Of all whose words I have heard, no one attains to this, to know that wisdom is apart from all.

    19 There is one wisdom, to understand the intelligent will by which all things are governed through all.

    20 This world, the same for all, neither any of the gods nor any man has made, but it always was, and is, and shall be, an ever living fire, kindled in due measure, and in due measure extinguished.

    21 The transmutations of fire are, first, the sea; and of the sea, half is earth, and half the lightning flash.

    22 All things are exchanged for fire and fire for all things, just as wares for gold and gold for wares.

    23 The sea is poured out and measured to the same proportion as existed before it became earth.

    24 Craving and Satiety.

    25 Fire lives in the death of earth, air lives in the death of fire, water lives in the death of air, and earth in the death of water.

    26 Fire coming upon all things, will sift and seize them.

    27 How can one escape that which never sets?

    28 Lightning rules all.

    29 The sun will not overstep his bounds, for if he does, the Erinyes, helpers of justice, will find him out.

    30 The limits of the evening and morning are the Bear, and opposite the Bear, the bounds of bright Zeus.

    31 If there were no sun, it would be night.

    32 The sun is new every day.

    33 Diogenes Laertius i. 23. He (scil. Thales) seems, according to some, to have been the first to study astronomy and to foretell the eclipses and motions of the sun, as Eudemus relates in his account of astronomical works. And for this reason he is honored by Xenophanes and Herodotus, and both Heraclitus and Democritus bear witness to him.

    34 Plutarch, Qu. Plat. viii. 4, p. 1007. Thus Time, having a necessary union and connection with heaven, is not simple motion, but, so to speak, motion in an order, having measured limits and periods. Of which the sun, being overseer and guardian to limit, direct, appoint and proclaim the changes and seasons which, according to Heraclitus, produce all things, is the helper of the leader and first God, not in small or trivial things, but in the greatest and most important.

    35 Hesiod is a teacher of the masses. They suppose him to have possessed the greatest knowledge, who indeed did not know day and night. For they are one.

    36 God is day and night, winter and summer, war and peace, plenty and want. But he is changed, just as when incense is mingled with incense, but named according to the pleasure of each.

    37 Aristotle, de Sensu 5, p. 443 a 21. Some think that odor consists in smoky exhalation, common to earth and air, and that for smell all things are converted into this. And it was for this reason that Heraclitus thus said that if all existing things should become smoke, perception would be by the nostrils.

    38 Souls smell in Hades.

    39 Cold becomes warm, and warm, cold; wet becomes dry, and dry, wet.

    40 It disperses and gathers, it comes and goes.

    41 Into the same river you could not step twice, for other <and still other> waters are flowing.

    42 †To those entering the same river, other and still other waters flow.†

    43 Aristotle, Eth. Eud. vii. 1, p. 1235 a 26. And Heraclitus blamed the poet who said, "Would that strife were destroyed from among gods and men." For there could be no harmony without sharps and flats, nor living beings without male and female which are contraries.

    44 War is the father and king of all, and has produced some as gods and some as men, and has made some slaves and some free.

    45 They do not understand: how that which separates unites with itself. It is a harmony of oppositions, as in the case of the bow and of the lyre.

    46 Aristotle, Eth. Nic. viii. 2, p. 1155 b 1. In reference to these things, some seek for deeper principles and more in accordance with nature. Euripides says, "The parched earth loves the rain, and the high heaven, with moisture laden, loves earthward to fall." And Heraclitus says, "The unlike is joined together, and from differences results the most beautiful harmony, and all things take place by strife."

    47 The hidden harmony is better than the visible.

    48 Let us not draw conclusions rashly about the greatest things.

    49 Philosophers must be learned in very many things.

    50 The straight and crooked way of the woolcarders is one and the same.

    51 Asses would choose stubble rather than gold.

    52 Sea water is very pure and very foul, for, while to fishes it is drinkable and healthful, to men it is hurtful and unfit to drink.

    53 Columella, de Re Rustica viii. 4. Dry dust and ashes must be placed near the wall where the roof or eaves shelter the court, in order that there may be a place where the birds may sprinkle themselves, for with these things they improve their wings and feathers, if we may believe Heraclitus, the Ephesian, who says, "Hogs wash themselves in mud and doves in dust."

    54 They revel in dirt.

    55 Every animal is driven by blows.

    56 The harmony of the world is a harmony of oppositions, as in the case of the bow and of the lyre.

    57 Good and evil are the same.

    58 Hippolytus, Ref. haer. ix. 10. And good and evil (scil. are one). The physicians, therefore, says Heraclitus, cutting, cauterizing, and in every way torturing the sick, complain that the patients do not pay them fitting reward for thus effecting these benefits-- †and sufferings†.

    59 Unite whole and part, agreement and disagreement, accordant and discordant; from all comes one, and from one all.

    60 They would not know the name of justice, were it not for these things.

    61 Schol. B. in Iliad iv. 4, p. 120 :Bekk. They say that it is unfitting that the sight of wars should please the gods. But it is not so. For noble works delight them, and while wars and battles seem to us terrible, to God they do not seem so. For God in his dispensation of all events, perfects them into a harmony of the whole, just as, indeed, Heraclitus says that to God all things are beautiful and good and right, though men suppose that some are right and others wrong.

    62 We must know that war is universal and strife right, and that by strife all things arise and † are used †

    63 For it is wholly destined ...

    64 Death is what we see waking. What we see in sleep is a dream.

    65 There is only one supreme Wisdom. It wills and wills not to be called by the name of Zeus.

    66 The name of the bow is life, but its work is death.

    67 Immortals are mortal, mortals immortal, living in their death and dying in their life.

    68 To souls it is death to become water, and to water it is death to become earth, but from earth comes water, and from water, soul.

    69 The way upward and downward are one and the same.

    70 The beginning and end are common.

    71 The limits of the soul you would not find out, though you should traverse every way.

    72 To souls it is joy to become wet.

    73 A man when he is drunken is led by a beardless youth, stumbling, ignorant where he is going, having a wet soul.

    74 The dry soul is the wisest and best.

    75 †The dry beam is the wisest and best soul.†

    76 †Where the land is dry, the soul is wisest and best.†

    77 Man, as a light at night, is lighted and extinguished.

    78 Plutarch, Consol. ad Apoll. 10, p. 106. For when is death not present with us? As indeed Heraclitus says: Living and dead, awake and asleep, young and old, are the same. For these several states are transmutations of each other.

    79 Time is a child playing at draughts, a child's kingdom.

    80 I have inquired of myself.

    81 Into the same river we both step and do not step. We both are and are not.

    82 It is weariness upon the same things to labor and by them to be controlled.

    83 In change is rest.

    84 A mixture separates when not kept in motion.

    85 Corpses are more worthless than excrement.

    86 Being born, they will only to live and die, or rather to find rest, and they leave children who likewise are to die.

    87 Plutarch, de Orac. def. 11, p. 415: Those who adopt the reading hêbôntos (i. e. at man's estate, see Hesiod, fr. 163, ed. Goettling) reckon a generation at thirty years, according to Heraclitus, in which time a father may have a son who is himself at the age of puberty.

    88 Io. Lydus de Mensibus iii. 10, p. 37, ed. Bonn.: Thirty is the most natural number, for it bears the same relation to tens as three to units. Then again it is the monthly cycle, and is composed of the four numbers 1, 4, 9,16, which are the squares of the units in order. Not without reason, therefore, does Heraclitus call the month a generation.

    89 In thirty years a man may become a grandfather.

    90 M. Antoninus vi. 42. We all work together to one end, some consciously and with purpose, others unconsciously. Just as indeed Heraclitus, I think, says that the sleeping are co-workers and fabricators of the things that happen in the world.

    91 The Law of Understanding is common to all. Those who speak with intelligence must hold fast to that which is common to all, even more strongly than a city holds fast to its law. For all human laws are dependent upon one divine Law, for this rules as far as it wills, and suffices for all, and overabounds.

    92 Although the Law of Reason is common, the majority of people live as though they had an understanding of their own.

    93 They are at variance with that with which they are in most continual association.

    94 We ought not to act and speak as though we were asleep.

    95 Plutarch, de Superst. 3, p. 166: Heraclitus says: To those who are awake, there is one world in common, but of those who are asleep, each is withdrawn to a private world of his own.

    96 For human nature does not possess understanding, but the divine does.

    97 The thoughtless man understands the voice of the Deity as little as the child understands the man.

    98 Plato, Hipp. maj. 289 B. And does not Heraclitus, whom you bring forward, say the same, that the wisest of men compared with God appears an ape in wisdom and in beauty and in all other things?

    99 Plato, Hipp. maj. 289 A. You are ignorant, my man, that there is a good saying of Heraclitus, to the effect that the most beautiful of apes is ugly when compared with another kind, and the most beautiful of earthen pots is ugly when compared with maidenkind, as says Hippias the wise.

    100 The people must fight for their law as for their walls.

    101 Greater fates gain greater rewards.

    102 Gods and men honor those slain in war.

    103 Presumption must be quenched even more than a fire.

    104 For men to have whatever they wish, would not be well. Sickness makes health pleasant and good; hunger, satiety; weariness, rest.

    105 It is hard to contend against passion, for whatever it craves it buys with its life.

    106 †It pertains to all men to know themselves and to learn self-control.†

    107 †Self-control is the highest virtue, and wisdom is to speak truth and consciously to act according to nature.†

    108 It is better to conceal ignorance, but it is hard to do so in relaxation and over wine.

    109 † It is better to conceal ignorance than to expose it. †

    110 It is law, also, to obey the will of one.

    111 For what sense or understanding have they? They follow minstrels and take the multitude for a teacher, not knowing that many are bad and few good. For the best men choose one thing above all--immortal glory among mortals; but the masses stuff themselves like cattle.

    112 In Priene there lived Bias, son of Teutamus, whose word was worth more than that of others.

    113 To me, one is ten thousand if he be the best.

    114 The Ephesians deserve, man for man, to be hung, and the youth to leave the city, inasmuch as they have banished Hermodorus, the worthiest man among them, saying: "Let no one of us excel, and if there be any such, let him go elsewhere and among other people."

    115 Dogs, also, bark at what they do not know.

    116 By its incredibility, it escapes their knowledge.

    117 A stupid man loves to be puzzled by every discourse.

    118 The most approved of those who are of repute knows how to cheat. Nevertheless, justice will catch the makers and witnesses of lies.

    119 Diogenes Laert. ix. 1. And he (Heraclitus) used to say that Homer deserved to be driven out of the lists and flogged, and Archilochus likewise.

    120 One day is like all.

    121 A man's character is his daemon.

    122 There awaits men after death what they neither hope nor think.

    123 And those that are there shall arise and become guardians of the living and the dead.

    124 Night-roamers, Magians, bacchanals, revelers in wine, the initiated.

    125 For the things which are considered mysteries among men, they celebrate sacrilegiously.

    126 And to these images they pray, as if one should prattle with the houses knowing nothing of gods or heroes, who they are.

    127 For were it not Dionysus to whom they institute a procession and sing songs in honor of the pudenda, it would be the most shameful action. But Dionysus, in whose honor they rave in bacchic frenzy, and Hades are the same.

    128 Iamblichus, de Mysteriis v. 16. I distinguish two kinds of sacrifices. First, those of men wholly purified, such as would rarely happen in the case of a single individual, as Heraclitus says, or of a certain very few men. Second, material and corporeal sacrifices and those arising from change, such as are fit for those still fettered by the body.

    129 Atonements.

    130 When defiled, they purify themselves with blood, just as if any one who had fallen into the mud should wash himself with mud!

    ZENO’S PARADOXES53

    Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (ca. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's Parmenides (128a–d), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." (Parmenides 128d). Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point (Parmenides 128a–b).

    Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them. Three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below.

    Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates.

    Some mathematicians and historians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution. Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems.

    The origins of the paradoxes are somewhat unclear. Diogenes Laertius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the Achilles and the tortoise paradox. But in a later passage, Laertius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.

    Paradoxes of motion

    Achilles and the tortoise

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    Achilles and the tortoise

    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. – as recounted by Aristotle, Physics VI:9, 239b15

    In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

    Dichotomy paradox

    That which is in locomotion must arrive at the half-way stage before it arrives at the goal.– as recounted by Aristotle, Physics VI:9, 239b10

    Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

    This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

    This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. An alternative conclusion, proposed by Henri Bergson, is that motion (time and distance) is not actually divisible.

    This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox. Some, like Aristotle, regard the Dichotomy as really just another version of Achilles and the Tortoise.

    There are two versions of the dichotomy paradox. In the other version, before Homer could reach the end of the path, he must reach half of the distance to it. Before reaching the last half, he must complete the next quarter of the distance. Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be accomplished before he could reach the end of the path. Expressed this way, the dichotomy paradox is very much analogous to that of Achilles and the tortoise.

    Arrow paradox

    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.

    – as recounted by Aristotle, Physics VI:9, 239b5

    In the arrow paradox (also known as the fletcher's paradox), Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

    Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.

    Three other paradoxes as given by Aristotle

    Paradox of Place

    From Aristotle:

    if everything that exists has a place, place too will have a place, and so on ad infinitum.

    Paradox of the Grain of Millet

    Description of the paradox from the Routledge Dictionary of Philosophy:

    The argument is that a single grain of millet makes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, an absurd conclusion.

    Aristotle's refutation:

    Zeno is wrong in saying that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.

    Description from Nick Huggett:

    This is a Parmenidean argument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound.

    The Moving Rows (or Stadium)

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    The moving rows

    From Aristotle:

    concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time.

    For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius' commentary On Aristotle's Physics.

    Proposed solutions

    Simplicius of Cilicia

    According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

    Aristotle

    Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities"). Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."

    Thomas Aquinas

    Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."

    Archimedes

    Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: Geometric series, 1/4 + 1/16 + 1/64 + 1/256 + · · ·, The Quadrature of the Parabola.) Modern calculus achieves the same result, using more rigorous methods (see convergent series, where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent). These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.

    Bertrand Russell

    Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.

    Nick Huggett

    Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest.

    Peter Lynds

    Peter Lynds has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist. Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. For more about the inability to know both speed and location, see Heisenberg uncertainty principle.

    Hermann Weyl

    Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space, if not on time and space themselves. According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem". According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.

    Hans Reichenbach

    Hans Reichenbach has proposed that the paradox may arise from considering space and time as separate entities. In a theory like general relativity, which presumes a single space-time continuum, the paradox may be blocked.

    The paradoxes in modern times

    Infinite processes remained theoretically troublesome in mathematics until the late 19th century. The epsilon-delta version of Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.

    While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise.

    Popular literature often misrepresents Zeno's arguments. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?

    Debate continues on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics: An Introduction (2010) Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'".

    Bertrand Russell offered a "solution" to the paradoxes based on the work of Georg Cantor, but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns (if they have any)."

    Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it. She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows:

    • P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise
    • P2: it is impossible for Achilles to traverse an infinite number of divisions
    • C: therefore, Achilles can never surpass the tortoise

    Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution.

    An ancient Chinese philosophic equivalent

    Ancient Han Chinese philosophers from the Mohist School of Names during the Warring States period of China (479-221 BCE) independently developed equivalents to some of Zeno's paradoxes. The scientist and historian Sir Joseph Needham, in his well regarded academic work Science and Civilisation in China, describes an ancient Chinese paradox from the surviving Mohist School of Names book of logic which states, in the archaic ancient Chinese script, "a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted." Several other paradoxes from this philosophical school (more precisely, movement) are known, but their modern interpretation is more speculative.

    Quantum Zeno effect

    In 1977, physicists E. C. G. Sudarshan and B. Misra studying quantum mechanics discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system. This effect is usually called the "quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.

    Zeno behaviour

    In the field of verification and design of timed and hybrid systems, the system behaviour is called Zeno if it includes an infinite number of discrete steps in a finite amount of time. Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.

    In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.


    This page titled 5.1: Pre-Socratics is shared under a CC BY license and was authored, remixed, and/or curated by Noah Levin (NGE Far Press) .

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