Shall we continue for the sake of completeness?

PYTHAGORAS OF SAMOS

Biography – Who was Pythagoras

It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician. But, although his contribution was clearly important, he nevertheless remains a controversial figure.

He left no mathematical writings himself, and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. Indeed, it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers.

The school he established at Croton in southern Italy around 530 BCE was the nucleus of a rather bizarre Pythagorean sect. Although Pythagorean thought was largely dominated by mathematics, it was also profoundly mystical, and Pythagoras imposed his quasi-religious philosophies, strict vegetarianism, communal living, secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewellery, never passing an ass lying in the street, never eating or even touching black fava beans, etc) .

The members were divided into the “mathematikoi” (or “learners“), who extended and developed the more mathematical and scientific work that Pythagoras himself began, and the “akousmatikoi” (or “listeners“), who focused on the more religious and ritualistic aspects of his teachings. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and, in 460 BCE, all their meeting places were burned and destroyed, with at least 50 members killed in Croton alone.

The over-riding dictum of Pythagoras’s school was “All is number” or “God is number”, and the Pythagoreans effectively practised a kind of numerology or number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or “wandering stars”; etc. Odd numbers were thought of as female and even numbers as male.

The holiest number of all was “Tetractys” or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans’ intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands.

However, Pythagoras and his school – as well as a handful of other mathematicians of ancient Greece – was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement.

Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth.

The Pythagorean Theorem

He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”).

Written as an equation: a² + b² = c².

What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular( shape.

The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (3² + 4² = 5², as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer “Pythagorean triples”, starting with (5, 12 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), etc. It should be noted, however that (6, 8, 10) is not what is known as a “primitive” Pythagorean triple, because it is just a multiple of (3, 4, 5).

Pythagoras’ Theorem and the properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from Babylon and Egypt, dating from over a thousand years earlier. One of the simplest proofs comes from ancient China, and probably dates from well before Pythagoras’ birth. It was Pythagoras, though, who gave the theorem its definitive form, although it is not clear whether Pythagoras himself definitively proved it or merely described it. Either way, it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.

It soon became apparent, though, that non-integer solutions were also possible, so that an isosceles triangle with sides 1, 1 and √2, for example, also has a right angle, as the Babylonians had discovered centuries earlier. However, when Pythagoras’s student Hippasus tried to calculate the value of √2, he found that it was not possible to express it as a fraction, thereby indicating the potential existence of a whole new world of numbers, the irrational numbers (numbers that can not be expressed as simple fractions of integers). This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers, and the existence of a number that could not be expressed as the ratio of two of God’s creations (which is how they thought of the integers) jeopardized the cult’s entire belief system.

Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world. But the replacement of the idea of the divinity of the integers by the richer concept of the continuum, was an essential development in mathematics. It marked the real birth of Greek geometry, which deals with lines and planes and angles, all of which are continuous and not discrete.

Among his other achievements in geometry, Pythagoras (or at least his followers, the Pythagoreans) also realized that the sum of the angles of a triangle is equal to two right angles (180°), and probably also the generalization which states that the sum of the interior angles of a polygon with n sides is equal to (2n – 4) right angles, and that the sum of its exterior angles equals 4 right angles. They were able to construct figures of a given area, and to use simple geometrical algebra, for example to solve equations such as a(a – x) = x² by geometrical means.

The Pythagoreans also established the foundations of number theory, with their investigations of triangular, square and also perfect numbers (numbers that are the sum of their divisors). They discovered several new properties of square numbers, such as that the square of a number n is equal to the sum of the first n odd numbers (e.g. 4² = 16 = 1 + 3 + 5 + 7). They also discovered at least the first pair of amicable numbers, 220 and 284 (amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220).

Music Theory

Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. For instance, playing half a length of a guitar string gives the same note as the open string, but an octave higher; a third of a length gives a different but harmonious note; etc.

Non-whole number ratios, on the other hand, tend to give dissonant sounds. In this way, Pythagoras described the first four overtones which create the common intervals which have become the primary building blocks of musical harmony: the octave (1:1), the perfect fifth (3:2), the perfect fourth (4:3) and the major third (5:4). The oldest way of tuning the 12-note chromatic scale is known as Pythagorean tuning, and it is based on a stack of perfect fifths, each tuned in the ratio 3:2.

The mystical Pythagoras was so excited by this discovery that he became convinced that the whole universe was based on numbers, and that the planets and stars moved according to mathematical equations, which corresponded to musical notes, and thus produced a kind of symphony, the “Musical Universalis” or “Music of the Spheres”.

Accurate history or not, here is something also relevant, given that it is so widely accepted as true about Plato:

Plato FAQ: "Let no one ignorant of geometry enter"

"Let no one ignorant of geometry enter"

Tradition has it that this phrasewas engraved at the door of Plato's Academy, the school he had founded in Athens.

More detail: https://www.storyofmathematics.com/greek_plato.html

Biography: What was Plato Known for

Although usually remembered today as a philosopher, Plato was also one of ancient Greece’s most important patrons of mathematics.

Inspired by Pythagoras, he founded his Academy in Athens in 387 BCE, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: “Let no-one ignorant of geometry enter here”.

Plato played an important role in encouraging and inspiring Greek intellectuals to study mathematics as well as philosophy. His Academy taught mathematics as a branch of philosophy, as Pythagoras had done, and the first 10 years of the 15 year course at the Academy involved the study of science and mathematics, including plane and solid geometry, astronomy and harmonics. Plato became known as the “maker of mathematicians”, and his Academy boasted some of the most prominent mathematicians of the ancient world, including Eudoxus, Theaetetus and Archytas.

He demanded of his students accurate definitions, clearly stated assumptions, and logical deductive proof, and he insisted that geometric proofs be demonstrated with no aids other than a straight edge and a compass. Among the many mathematical problems Plato posed for his students’ investigation were the so-called Three Classical Problems (“squaring the circle”, “doubling the cube” and “trisecting the angle”) and to some extent these problems have become identified with Plato, although he was not the first to pose them.

Platonic Solids

Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes, which he maintained were the basis for the whole universe, and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles, and which for Plato represented fire), the octahedron (composed of 8 triangles, representing air), the icosahedron (composed of 20 triangles, and representing water), the cube (composed of 6 squares, and representing earth), and the dodecahedron (made up of 12 pentagons, which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”).

The tetrahedron, cube and dodecahedron were probably familiar to Pythagoras, and the octahedron and icosahedron were probably discovered by Theaetetus, a contemporary of Plato. Furthermore, it fell to Euclid, half a century later, to prove that these were the only possible convex regular polyhedra. But they nevertheless became popularly known as the Platonic Solids, and inspired mathematicians and geometers for many centuries to come. For example, around 1600, the German astronomer Johannes Kepler devised an ingenious system of nested Platonic solids and spheres to approximate quite well the distances of the known planets from the Sun (although he was enough of a scientist to abandon his elegant model when it proved to be not accurate enough).

One more from Lucian, similar to Hermotimus, this time from Incaromenippus, An Aerial Expedition. The main relevant part is in bold below but I left text before and after since it is so colorful:

Menippus. Well, a very short survey of life had convinced me of the absurdity and meanness and insecurity that pervade all human objects, such as wealth, office, power. I was filled with contempt for them, realized that to care for them was to lose all chance of what deserved care, and determined to grovel no more, but fix my gaze upon the great All. Here I found my first problem in what wise men call the universal order. I could not tell how it came into being, who made it, what was its beginning, or what its end. But my next step, which was the examination of details, landed me in yet worse perplexity. I found the stars dotted quite casually about the sky, and I wanted to know what the sun was. Especially the phenomena of the moon struck me as extraordinary, and quite passed my comprehension; there must be some mystery to account for those many phases, I conjectured. Nor could I feel any greater certainty about such things as the passage of lightning, the roll of thunder, the descent of rain and snow and hail.

In this state of mind, the best I could think of was to get at the truth of it all from the people called philosophers; they of course would be able to give it me. So I selected the best of them, if solemnity of visage, pallor of complexion and length of beard are any criterion—for there could not be a moment's doubt of their soaring words and heaven-high thoughts—and in their hands I placed myself. For a considerable sum down, and more to be paid when they should have perfected me in wisdom, I was to be made an airy metaphysician and instructed in the order of the universe. Unfortunately, so far from dispelling my previous ignorance, they perplexed me more and more, with their daily drenches of beginnings and ends, atoms and voids, matters and forms. My greatest difficulty was that, though they differed among themselves, and all they said was full of inconsistency and contradiction, they expected me to believe them, each pulling me in his own direction.

Friend. How absurd that wise men should quarrel about facts, and hold different opinions on the same things!

Menippus. Ah, but keep your laughter till you have heard something of their pretentious mystifications. To begin with, their feet are on the ground; they are no taller than the rest of us 'men that walk the earth'; they are no sharper-sighted than their neighbors, some of them purblind, indeed, with age or indolence. And yet they say they can distinguish the limits of the sky, they measure the sun's circumference, take their walks in the supra-lunar regions, and specify the sizes and shapes of the stars as though they had fallen from them. Often one of them could not tell you correctly the number of miles from Megara to Athens, but has no hesitation about the distance in feet from the sun to the moon. How high the atmosphere is, how deep the sea, how far it is round the earth— they have the figures for all that. Moreover, they have only to draw some circles, arrange a few triangles and squares, add certain complicated spheres, and lo, they have the cubic contents of Heaven.

Then, how reasonable and modest of them, dealing with subjects so debatable, to issue their views without a hint of uncertainty; thus it must be and it shall be; contra gentes they will have it so. They will tell you on oath the sun is a molten mass, the moon inhabited, and the stars water-drinkers, moisture being drawn up by the sun's rope and bucket and equitably distributed among them.

How their theories conflict is soon apparent; next-door neighbors? No, they are miles apart. In the first place, their views of the world differ. Some say it had no beginning, and cannot end; others boldly talk of its creator and his procedure. What particularly entertained me was that these latter set up a contriver of the universe, but fail to mention where he came from, or what he stood on while about his elaborate task, though it is by no means obvious how there could be place or time before the universe came into being.

Friend. You really do make them out very audacious conjurers.

Menippus. My dear fellow, I wish I could give you their lucubrations on ideas and incorporeals, on finite and infinite. Over that point, now, there is fierce battle; some circumscribe the All, others will have it unlimited. At the same time they declare for a plurality of worlds, and speak scornfully of others who make only one. And there is a bellicose person who maintains that war is the father of the universe.

As to Gods, I need hardly deal with that question. For some of them God is a number; some swear by dogs and geese and plane-trees. [note: Socrates made a practice of substituting these for the names of Gods in his oaths.] Some again banish all other Gods, and attribute the control of the universe to a single one; I got rather depressed on learning how small the supply of divinity was. But I was comforted by the lavish souls who not only make many, but classify; there was a First God, and second and third classes of divinity. Yet again, some regard the divine nature as unsubstantial and without form, while others conceive it as a substance. Then they were not all disposed to recognize a Providence; some relieve the Gods of all care, as we relieve the superannuated of their civic duties; in fact, they treat them exactly like supernumeraries on the stage. The last step is also taken, of saying that Gods do not exist at all, and leaving the world to drift along without a master or a guiding hand.

Well, when I heard all this, I dared not disbelieve people whose voices and beards were equally suggestive of Zeus. But I knew not where to turn for a theory that was not open to exception, nor combated by one as soon as propounded by another. I found myself in the state Homer has described; many a time I would vigorously start believing one of these gentlemen; “But then came second thoughts.”

So in my distress I began to despair of ever getting any knowledge about these things on earth. The only possible escape from perplexity would be to take to myself wings and go up to Heaven.

Also directly relevant to our topic tonight is Usener 376:

U376

Cicero Academica II.30.97 (Lucullus): They will not get Epicurus, who despises and laughs at the whole of dialectic, to admit the validity of a proposition of the form “Hermarchus will either be alive tomorrow or not alive,” while dialecticians demand that every disjunctive proposition of the form “either x or not-x” is not only valid but even necessary. See how on his guard the man is whom your friends think slow; for “If,” he says, “I admit either of the two to be necessary, it will follow that Hermarchus must either be alive tomorrow or not alive; but as a matter of fact in the nature of things no such necessity exists.” Therefore let the dialecticians, that is, Antiochus and the Stoics, do battle with this philosopher, for he overthrows the whole of dialectic.

Cicero, On The Nature of The Gods, I.25.70 (Cotta speaking): Epicurus did the same sort of thing in his argument with the logicians. It is an axiom of the traditional logic that in every disjunctive proposition of the form “X either is … or is not …” one of the alternatives must be true. He was afraid that if he admitted anything of this sort, then in a proposition such as “Tomorrow Epicurus will either be alive or he will not be alive,” one or the other of the statements would be a necessary truth: so to avoid this he denied that there was any logical necessity at all in a disjunction proposition, which is too stupid for words!

Cicero, On Fate, 10.21: Now here, first of all, if it were my desire to agree with Epicurus and deny that every proposition is either true or false, I would rather accept that blow than agree that all things come about through fate; for the former opinion gives some scope for discussion, but the latter is intolerable. So Chrysippus strains every sinew in order to convince us that every proposition is either true or false. Epicurus is afraid that, if he concedes this, he will have to concede that whatever comes about does so through fate; for if either the assertion or the denial is true from eternity, it will also be certain – and if certain, also necessary. [cf. Ibid., 9.19]

Potentially Relevant in Addition:

U380 (this may be a good clue to those categories in which dialectical formal logic is especially to be distrusted:

Aetius, Doxography, I.29.6 [p. 326 Diels] (Plutarch, I.29.2; Stobaeus Anthology, Physics 7.9): Epicurus says that chance is a cause which is uncertain with respect to persons, times, and places.

I would suggest that:

1 - Epicurus anticipated the need to say this by about 2000 years,

2 - That the need to say it was planted firmly at least as far back as Plato, and by every Greek philosopher before and after who rejected the senses as the ultimate source of all reliable source of information, and

3 - Orwell was right and in league with Epicurus in identifying lack of confidence in the senses as the ultimate weapon of those who consider themselves "the golden" elite:

Quote

The Party told you to reject the evidence of your eyes and ears. It was their final, most essential command. His heart sank as he thought of the enormous power arrayed against him, the ease with which any Party intellectual would overthrow him in debate, the subtle arguments which he would not be able to understand, much less answer. And yet he was in the right! They were wrong and he was right. The obvious, the silly, and the true had got to be defended. Truisms are true, hold on to that! The solid world exists, its laws do not change. Stones are hard, water is wet, objects unsupported fall toward the earth's center.

With the feeling that he was speaking to O'Brien, and also that he was setting forth an important axiom, he wrote:

Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.

George Orwell Nineteen Eight-Four - Book 1, Chapter 7.

(I know nothing about that page except that it came up first in my googling for a well-formatted copy.)

Episode Eighty-Eight of Lucretius Today is now available. In this episode we continue into Book Six to discuss the mysteries of the waters of the Nile and sulfur pits which are fatal to birds. Thanks to Don for reading this week:

In editing this episode I wanted to remember to include this line from Virgil which we discussed briefly but probably not nearly enough.

It's one of the great Latin lines with an Epicurean pedigree:

Welcome to Episode Eighty-Nine of Lucretius Today.

I am your host Cassius, and together with my panelists from the EpicureanFriends.com forum, we'll walk you through the six books of Lucretius' poem, and discuss how Epicurean philosophy can apply to you today. We encourage you to study Epicurus for yourself, and we suggest the best place to start is the book, "Epicurus and His Philosophy" by Canadian professor Norman DeWitt.

For anyone who is not familiar with our podcast, please visit EpicureanFriends.com where you will find our goals and our ground rules. If you have any questions about those, please be sure to contact us at the forum for more information.

In this Episode 89 we will read approximately Latin lines 830-917 as we discuss the hot and cold springs and similar phenomena.

Now let's join Martin reading today's text.

Latin Lines 830-917

Munro Notes-

830-839 : sometimes this exhalation causes a partial void, so that the bird cannot support itself on the wing, but falls down and perishes.

840-847: the water of wells is colder in summer, because they let out their seeds of heat through the earth which is then rarefied by heat: the contrary is the case in winter for the contrary reason.

879-905: there is also a cold fountain which ignites tow or pine-wood put over it: it contains many seeds of latent fire, which rise up and set on fire this tow or wood, as flame will light a freshly extinguished wick, before actual contact.

906-916: to discuss now the magnet, a stone which has the power of attracting iron, and communicating this power to a series of pieces of iron.

917-920: but many points have to be cleared up, before we come to the actual question.

Browne 1743

[830] Or else, sometimes, the force and rising blasts of these Averni dispel the air that lies between the birds and the earth, and the intermediate space becomes a void. Here, when birds are carried by their flight, they immediately flutter in the air, they clap their wings in vain, their pinions flag, and when they can no longer bear them up, nature must drive them down upon the earth with all their weight; and as they, helpless, in the vacuum lie, they breathe their soul abroad through every pore.

[840] The water in some wells, we find, is cold in summer, because the earth is rarefied by the sun's heat, and by that means the seeds of fire it contains within break freely out into the Air: And therefore the more the earth is affected by the heat, the colder the water will be that is enclosed within. But when the earth is contracted with the cold, when its surface grows close, and its pores are stopped, this restraint hinders the heat from flying out; it is then squeezed together into the wells, and the water becomes hot.

[848] There is a fountain, near the temple of Jupiter Ammon, that is cold in the day, and hot by night. Men strangely wonder at the quality of this spring, and imagine that when the night has spread her dreadful darkness over the world the water is warmed by the violent heat of the sun through the body of earth. But this reason is far from being true; for if the sun, striking upon the open body of water, is not able to warm even the surface of it, when it receives the force of his descending rays with all their heat, how can he warm the water, and infuse his heat through so thick a body as the Earth; especially since he is scarce able, with his scorching beams, to pierce through the walls of our houses?

[861] What then is the reason? Doubtless this: because the Earth, near this fountain, is more rare and spongy then it is in other places, and contains within it many seeds of fire near the body of the water itself. Here, when the night has spread the world with dewy shades, the earth below grows initially cold, and is contracted; by this means it is compressed, as with your hand, and squeezes out those seeds of fire into the spring, which make the water warm to feel and taste. But when the sun has driven away the night with his bright rays, and with his heat has rarefied the Earth, and made it loose, these seeds of fire return into their former place, and all the heat that warmed the spring retires within the earth again, and so the fountain in the day is cold.

[874] Besides, the water in the day is strongly moved by the sun's rays, and by his trembling streams of heat grows rare, and so lets out the seeds of fire it held by night; just as by the heat it shakes off seeds of cold, and melts the ice, and loosens all its bonds.

[879] There is likewise a cold spring over which if you place tow or flax it immediately takes fire and is all in a blaze. A torch, newly extinguished, in the same manner, gently drawn over the surface, is lighted by this water, and flames out at every breath of air. And no wonder, for there are many seeds of fire in the water itself, and many must needs rise out of the earth, and ascend through all the fountain, and flow abroad, and make their way into the air, but yet they are not so hot as to set the spring on fire.

[890] Besides, the innate force of these seeds, dispersed through the water, compels them to move upwards, and to unite upon the surface; as we see sometimes a fountain of sweet water bubble up in the middle of the sea, and beat off the salt waves that are about it. The sea affords many of these springs that bring a seasonable relief to the thirsty mariners by throwing out streams of fresh water among the salt. The seeds of fire may in the same manner break through the water of this fountain, and flow out into the tow. Here, when they unite and stick to the body of the torch, they immediately fall into a flame; for flax and tow contain many seeds of fire within which make them easily disposed to burn.

[900] Have you not observed, when you hold a candle newly extinguished to another that is lighted it catches fire before it touches the flame? A torch likewise, by that same rule, will do the same; and many other things will take fire at a distance, before the flame reaches them. And this you may imagine is the case of the fountain above-mentioned.

[906] And now I shall begin to show by what power of nature it is that the stone (which the Greeks call a magnet, from the country that produces it, for it is found in the region of the Magnetes) has the virtue to attract iron. Men are amazed at the qualities of this stone, for it will make a chain of several little rings of iron, without a link between, to hang together entirely from itself. You may sometimes see five or more hanging straight down, and play in the gentle air, as they stick close and depend at the bottom one upon another; the ring that follows feels the attraction and power of the stone from that above it. So strongly is the virtue of the magnet communicated to the several rings; it acts with so great a force.

[917] In inquiries of this nature many things are to be first proved before we can fix upon the true cause; we must trace the subject through many long and intricate difficulties; and therefore I beg you will hear me with a willing mind, and with the closest attention.

Munro 1886

[830] Sometimes too this power and exhalation of Avernus dispels whatever air lies between the birds and earth, so that almost a void is left there. And when the birds have arrived in their flight just opposite this spot, at once the buoyant force of their pinions is crippled and rendered vain and all the sustaining efforts of their wings are lost on both sides. So when they are unable to buoy themselves up and lean upon their wings, nature, you know, compels them by their weight to tumble down to earth, and lying stark through what is now almost a void they disperse their soul through all the openings of their body.

[840] Again during summer the water in wells becomes colder, because the earth is rarefied by heat and rapidly sends out into the air whatever seeds of heat it happens to have. The more then the earth is drained of heat, the colder becomes the water which is hidden in the earth. Again when all the earth is compressed by cold and contracts and so to say congeals, then, you are to know, while it contracts, it presses out into the wells whatever heat it contains itself.

[848] At the fane of Hammon there is said to be a fountain which is cold in the daylight and hot in the night-time. This fountain men marvel at exceedingly and suppose that it suddenly becomes hot by the influence of the fierce sun below the earth, when night has covered the earth with awful darkness. But this is far far removed from true reason. Why when the sun though in contact with the uncovered body of the water has not been able to make it hot on its upper side, though his light above possesses such great heat, how can he below the earth which is of so dense a body boil the water and glut it with heat? Above all, when he can scarcely with his burning rays force his heat through the walls of houses.

[861] What then is the cause? This sure enough: the earth is more porous and warmer round the fountain than the rest of the earth, and there are many seeds of fire near the body of water. For this reason when night has buried the earth in its dewy shadows, the earth at once becomes quite cold and contracts: in this way just as if it were squeezed by the hand it forces out into the fountain whatever seeds of fire it has; and these make the water hot to the touch and taste. Next when the sun has risen and with his rays has loosened the earth and has rarefied it as his heat waxes stronger, the first-beginnings of fire return back to their ancient seats and all the heat of the water withdraws into the earth: for this reason the fountain becomes cold in the daylight.

[874] Again the liquid of water is played upon by the sun’s rays and in the daytime is rarefied by his throbbing heat; and therefore it gives up whatever seeds of fire it has; just as it often parts with the frost which it holds in itself, and thaws the ice and loosens its bonds.

[879] There is also a cold fountain of such a nature that tow, often when held over it, imbibes fire forthwith and emits flame; a pine torch in like manner is lighted and shines among the waters, in whatever direction it swims under the impulse of the winds. Because sure enough there are in the water very many seeds of heat, and from the earth itself at the bottom must rise up bodies of fire throughout the whole fountain and at the same time pass abroad in exhalations and go forth into the air, not in such numbers however that the fountain can become hot, for these reasons a force compels those seeds to burst out through the water and disperse abroad and to unite when they have mounted up.

[890] In the sea at Aradus is a fountain of this kind, which wells up with fresh water and keeps off the salt waters all round it; and in many other quarters the sea affords a seasonable help in need to thirsting sailors, vomiting forth fresh waters amid the salt. In this way then those seeds may burst forth through that fountain and well out; and when they are met together in the tow or cohere in the body of the pine-torch, they at once readily take fire, because the tow and pinewood contain in them likewise many seeds of latent fire.

[900] See you not too that, when you bring a newly extinguished wick near night-lamps it catches light before it has touched the flame; and the same with the pinewood? And many things beside catch fire at some distance touched merely by the heat, before the fire in actual contact infects them. This therefore you must suppose to take place in that fountain as well.

[906] Next in order I will proceed to discuss by what law of nature it comes to pass that iron can be attracted by that stone which the Greeks call the Magnet from the name of its native place, because it has its origin within the bounds of the country of the Magnesians. This stone men wonder at; as it often produces a chain of rings hanging down from it. Thus you may see sometimes five and more suspended in succession and tossing about in the light airs, one always hanging down from one and attached to its lower side, and each in turn one from the other experiencing the binding power of the stone: with such a continued current its force flies through all.

[917] In things of this kind many points must be established before you can assign the true law of the thing in question, and it must be approached by a very circuitous road; wherefore all the more I call for an attentive ear and mind.

Bailey 1921

[830] It may happen, too, sometimes that this force and effluence of Avernus dispels all the air that is situate between the birds and the ground, so that there is left here an almost empty space. And when the birds in their flight have come straight over this place, on a sudden the lifting force of their pinions is crippled and useless, and all the effort of their wings fails on either side. And then, when they cannot support themselves or rest upon their wings, of course nature constrains them to sink by their weight to the ground, and lying in death in what is now almost empty void, they scatter abroad their soul through all the pores of their body

[840] Moreover, the water in wells becomes colder in summer, because the earth grows porous with the heat, and if by chance it has any seeds of heat of its own, it sends them abroad into the air. The more then earth is exhausted of its heat, the colder too becomes the moisture which is hidden in the earth. Moreover, when all the earth is hard pressed with cold, and contracts and, as it were, congeals, of course it comes to pass that, as it contracts, it squeezes out into the wells any heat it bears in itself.

[848] There is said to be near the shrine of Ammon a fountain, cold in the daylight and warm in the night time. At this fountain men marvel overmuch, and think that it is made to boil in haste by the fierceness of the sun beneath the earth, when night has shrouded earth in dreadful darkness. But this is exceeding far removed from true reasoning. For verily, when the sun, touching the uncovered body of the water, could not make it warm on the upper side, though its light in the upper air enjoys heat so great, how could it beneath the earth with its body so dense boil the water and fill it with warm heat? and that when it can scarcely with its blazing rays make its hot effluence pierce through the walls of houses.

[861] What then is the reason? We may be sure, because the ground is rarer and warmer around the fountain than the rest of the earth, and there are many seeds of fire near the body of the water. Therefore, when night covers the earth with the shadows that bring the dew, straightway the earth grows cold deep within and contracts. By this means it comes to pass that, as though it were pressed by the hand, it squeezes out into the fountain all the seeds of fire it has, which make warm the touch and vapour of the water. Then when the rising sun has parted asunder the ground with his rays, and has made it rarer, as his warm heat grows stronger, the first-beginnings of fire pass back again into their old abode, and all the heat of the water retires into the earth. For this cause the fountain becomes cold in the light of day.

[874] Moreover, the moisture of the water is buffeted by the sun’s rays, and in the light grows rarer through the throbbing heat; therefore it comes to pass that it loses all the seeds of fire that it has; just as often it gives out the frost that it contains in itself, and melts the ice and loosens its bindings.

[879] There is also a cold spring, over which if tow be held, it often straightway catches fire and casts out a flame, and a torch in like manner is kindled and shines over the waters, wherever, as it floats, it is driven by the breezes. Because, we may be sure, there are in the water very many seeds of heat, and it must needs be that from the very earth at the bottom bodies of fire rise up through the whole spring, and at the same time are breathed forth and issue into the air, yet not so many of them that the spring can be made hot. Moreover, a force constrains them suddenly to burst forth through the water scattered singly, and then to enter into union up above.

[890] Even as there is a spring within the sea at Aradus, which bubbles up with fresh water and parts the salt waters asunder all around it; and in many other spots too the level sea affords a welcome help to thirsty sailors, because amid the salt it vomits forth fresh water. So then those seeds are able to burst out through that spring, and to bubble out into the tow; and when they gather together or cling to the body of the torch, readily they blaze out all at once, because the tow and torches too have many seeds of hidden fire in themselves.

[900] Do you not see too, when you move a wick just extinguished near a night-lamp, that it is kindled before it has touched the flame, and a torch in like manner? And many other things as well are touched first by the mere heat and blaze out at a distance, before the fire soaks them close at hand. This then we must suppose comes to pass in that spring too.

[906] For what follows, I will essay to tell by what law of nature it comes to pass that iron can be attracted by the stone which the Greeks call the magnet, from the name of its native place, because it has its origin within the boundaries of its native country, the land of the Magnetes. At this stone men marvel; indeed, it often makes a chain of rings all hanging to itself. For sometimes you may see five or more in a hanging chain, and swaying in the light breezes, when one hangs on to the other, clinging to it beneath, and each from the next comes to feel the binding force of the stone: in such penetrating fashion does its force prevail.

[917] In things of this kind much must be made certain before you can give account of the thing itself, and you must approach by a circuit exceeding long: therefore all the more I ask for attentive ears and mind.

LOL. I was afraid I missed it so good!

Did you come to a conclusion there Don?

No problem! Also if you're on a desktop, there is now a "Conversations" button at bottom right. You can click that and a list of users should pop up and you can either click on a user you see there, or hit the "plus" sign.

Since this is directly on point in the Physics forum, I made a crosslink to that location: Materiality - Atoms and Void - Properties And Qualities and Emergence

Thanks to Don for this one: Emergence (Epicurean "accidents")

Quote from Don

Epicurus/Lucretius talked about as "accidents":

Or, if one looks back at the Latin - EVENTUM. I suppose I've had too many car accident cases to like that word "accident."

"Eventum" (or "event") seems to me to convey less connotation of "chance" or "luck" than does "accident."

And it seems to me to be clear that these emergent events are in fact not unlimited randomness or "luck" or in any sense "anything can happen."

These emergent events arise within the limits of the combinations of the specific elements and void which are involved.

The ant example it starts with is a good example of that.

"More than the sum of its parts" is a good line.

So is there a separate video on consciousness as an emergent property?

This exercise is helping me see the connection between this "formal logic" problem and the problem of "necessity."

Since Epicurus was rejecting "necessity" in human life, in favor of "free will," then it's logical he would be suspicious of too-broad claims of "necessity" in anything involving human life.

So when DeLacy says:

combine Epicurus' rejection of dialectic with his rejection of "necessity" and it seems to me that you have a pretty sweeping rejection of the reliability of syllogistic logic in virtually every aspect of human affairs. That doesn't mean syllogistic logic isn't reliable in regard to "material" issues, because the letter to Herodotus points out that most things in the universe are as they have been set in motion from the "formation of the world."

So it looks like you end up with both necessity (and formal logic) being useful in most purely non-living affairs, but "free will," and therefore freedom from formal logic, in the affairs of living things with freedom of action.

These excerpts from a nearby post are also relevant:

Especially this part from Philip DeLacy as to Philodemus' "On Methods of Inference" -->

The last reference I would throw into this pot is a comment by Richard Dawkins in which he seems to also place Aristotle in Plato's camp:

Last excerpt illustrating someone who fell victim to this issue, from Heller's biography "Ayn Rand and the World She Made":

One more point Macario: the messenger system in this software is called "conversations" so be sure to look for that word instead of "messages"

Examples:

Reference in DeWitt as to the Hermarchus example (however DeWitt is probably wrong in this first one to say Epicurus "ignored" the issue - the reason we have the example is that he was giving the proper response to the problem):

Seaching for "dialectic" in EAHP produces a huge number of hits. Here are some of the most on point:

Another:

Another, as to education:

Quote from Martin

Epicurus knew and even Aristotle was aware of that binary logic might be applicable in full only to timeless sentences and those which refer to past events but not to events in the future.

YES - that is the issue, and I think it's absolutely critical that people understand that before, during, and after they try to absorb the presentation, because otherwise the problem diagnosed in the Hermotimus excerpt is undetectable.

Do you have any specific references on those two categories (1) timeless sentences (2) future events?

I believe part of what you are referring to must include the Epicurus reference to not being willing to state whether Hermarchus will be alive or dead tomorrow.

And this comment is not directed just to Martin - everyone who is at all interested in this issue needs an understanding of this, so we need to develop means of explaining it that are as memorable as possible.

Martin I have glanced at both the the lit.edu material and your additional notes.

This looks to be a very interesting presentation!

Thinking forward to how you begin the presentation, I do not see anything in the Lit.edu material as to the meaning of the variables that will be used in the tables, and I presume that is one of the central issues that you will be describing in terms of how these operations are not necessarily tied to reality.

Is there a way to summarize or add to the handout picture the nature of this issue? I think you will be very thoroughly explaining how, given the premises of the exercise, the results of formal logic are reached.

So is the issue in the "premises of the exercise" themselves? And how do we start off the presentation emphasizing that aspect, so that we do not get lost in the weeds?

I am reminded of this from Hermotimus:

Quote

Perhaps an illustration will make my meaning clearer: when one of those audacious poets affirms that there was once a three-headed and six-handed man, if you accept that quietly without questioning its possibility, he will proceed to fill in the picture consistently—six eyes and ears, three voices talking at once, three mouths eating, and thirty fingers instead of our poor ten all told; if he has to fight, three of his hands will have a buckler, wicker targe, or shield apiece, while of the other three one swings an axe, another hurls a spear, and the third wields a sword. It is too late to carp at these details, when they come; they are consistent with the beginning; it was about that that the question ought to have been raised whether it was to be accepted and passed as true. Once grant that, and the rest comes flooding in, irresistible, hardly now susceptible of doubt, because it is consistent and accordant with your initial admissions. That is just your case; your love-yearning would not allow you to look into the facts at each entrance, and so you are dragged on by consistency; it never occurs to you that a thing may be self- consistent and yet false; if a man says twice five is seven, and you take his word for it without checking the sum, he will naturally deduce that four times five is fourteen, and so on ad libitum. This is the way that weird geometry proceeds: it sets before beginners certain strange assumptions, and insists on their granting the existence of inconceivable things, such as points having no parts, lines without breadth, and so on, builds on these rotten foundations a superstructure equally rotten, and pretends to go on to a demonstration which is true, though it starts from premises which are false.

Just so you, when you have granted the principles of any school, believe in the deductions from them, and take their consistency, false as it is, for a guarantee of truth. Then with some of you, hope travels through, and you die before you have seen the truth and detected your deceivers, while the rest, disillusioned too late, will not turn back for shame: what, confess at their years that they have been abused with toys all this time? so they hold on desperately, putting the best face upon it and making all the converts they can, to have the consolation of good company in their deception; they are well aware that to speak out is to sacrifice the respect and superiority and honor they are accustomed to; so they will not do it if it may be helped, knowing the height from which they will fall to the common level. Just a few are found with the courage to say they were deluded, and warn other aspirants. Meeting such a one, call him a good man, a true and an honest; nay, call him philosopher, if you will; to my mind, the name is his or no one's; the rest either have no knowledge of the truth, though they think they have, or else have knowledge and hide it, shamefaced cowards clinging to reputation.

How do we make that point at the very introduction of the topic?