Greece. Greece. The Origins of Ancient Greece. The Origins of Scientific Thinking?

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1 Greece 1 The Origins of Scientific Thinking? Greece is often cited as the place where the first inklings of modern scientific thinking took place. Why there and not elsewhere? Einstein s s answer: The astonishing thing is that these discoveries [the bases of science] were made at all. 2 The Origins of Ancient Greece What we call ancient Greece might better be called the ancient Aegean Civilizations. 3

2 The Aegean Civilizations There have been civilizations in the Aegean area almost as long as there have been in Mesopotamia and Egypt. The earliest known in the area was the Minoan Civilization on the island of Crete. Existed from about BCE. Had some kind of written language, never deciphered. Collapsed suddenly for unknown reasons. 4 The Mycenaean Civilization On the Peloponnesus (the southern mainland) another civilization arose and flourished from about BCE. The Mycenaeans adapted the Minoan writing system to their own language, Greek. But it was awkward to use. 5 Mycenaea The peak of the Mycenaean civilization was the reign of Agamemnon, who took his people (the Greeks )) to war against the Trojans. Agamemnon s Palace 6

3 The Trojan War 7 The Trojan War Approx BCE. Mycenaea versus Troy. Won by the Greeks, but the war depleted their fighting forces. Mycenaea was invaded by Dorians about 1200 BCE, and its culture destroyed. 8 The Dark Age of Greece BCE The organized Greek civilization was destroyed by the invading Dorians. Knowledge of writing was lost. People lived in isolated villages. What they had in common was spoken Greek and memories of past greatness. 9

4 Phoenicia Around 1700 BCE, in the Near East, what is now Lebanon, a civilization developed with both Mesopotamian and Egyptian influences. The Greeks later called the people from there Phonecians meaning traders in purple. 10 Phoenician Writing Phoenicians developed a style of writing that combined Mesopotamian cuneiform and Egyptian heiratic. It had 22 distinct characters, each representing a particular sound (a consonant). 11 The Phoenician Alphabet 12

5 The Phoenician Alphabetic was Phonetic Since each character represented a sound, rather than a meaning, the characters could be used to represent words in an entirely different language. The Greeks adapted the Phoenician script to their own language and produced an alphabet. 13 The Homeric Age BCE The Greek verbal culture could be written down. The heroic stories of the Trojan War were written by Homer. The Iliad, The Odyssey Greek mythology and folk knowledge were recorded by Hesiod. Theogony, Works and Days 14 The Greek Civilization Takes Off The first Olympic Games 776 BCE The Polis (City-State) Independent governments arose all across the Greek settlements. Experimentation in forms of government: Monarchies, Aristocracies, Dictatorships, Oligarchies, Democracies Independent units, but tied together by a common language, religion, and literature. 15

6 Assertion: Scientific Thinking Began in Ancient Greece Possible explanations given: Religion The Greek gods were too human-like. Language Phonetic alphabet encouraged literacy. Trade The Greeks became traders and travellers, bringing home new ideas. Democracy Democratic governments, where they existed, encouraged independent thought. Slavery Greeks (like many other cultures) had slaves who did the menial work. 16 The Pre-Socratics Thinkers living between about BCE. So named because they (basically) predated Socrates. Known only through discussions of their thoughts in later works. Some fragments still exist. 17 Socrates Lived in Athens, BCE. Set the direction of Western philosophical thinking. The goal of philosophy to discover the truth. Reasoning, the supreme method. Pursued by asking questions, the dialectical, or Socratic method. 18

7 Socrates, contd. Socrates left no writings at all. He is known to us primarily through the works of Plato. It is hard to distinguish Socrates own thought from Plato s. Socrates is an important figure in the development of scientific reasoning, but He had no interest in the natural world. 19 Back to the Pre-Socratics Most Pre- Socratics came from the Greek colonies on the eastern side of the Aegean Sea known as Ionia. This is now part of Turkey. 20 Wondering about Nature The importance of the Pre-Socratics is that they appear to be the first people we know of who asked fundamental questions about nature, such as What is the world made of? And then they provided reasons to justify their answers. 21

8 Thales of Miletos BCE Phoenician parents? Stories: Predicted solar eclipse of May 28, 585 BCE Falling into a well Olive press Water is the basic stuff of the world. 22 Thales and Mathematics Thales is said to have brought Egyptian mathematics to Greeks. Examples: All triangles constructed on the diameter of a circle are right triangles. The base angles of isosceles triangles are equal. If two straight lines intersect, opposite angles are equal. 23 Measuring the distance of a ship from shore From the desired point on the shore, A, walk off a known distance to point C, at a right angle from the ship and place a marker there. Continue walking the same distance again to B. At B, turn at a right angle away from the shore and walk until the marker at C and the ship are in a straight line. Call that A. A The distance from A A to B is the same as the distance from A to the ship. 24

9 Anaximander of Miletos BCE Student of Thales? Map of the known world Apeiron (the Boundless) The basic stuff of the world 25 Anaximenes of Miletos BCE Student of Anaximander? Air the fundamental stuff Cosmological view: Crystalline sphere of the fixed stars Earth in centre, planets between 26 Heraclitos of Ephesus Ephesus is 50 km N of Miletos. 550?-475? BCE (i.e., about the same as Anaximenes,, but uncertain) Everything is Flux. Fire fundamental "You can't step in the same river twice." 27

10 Elea Elea was a Greek colony in southern Italy. The minor Pre-Socratic, Xenophanes, fled from Colophon in Ionia to Elea to escape persecution. 28 Parmenides of Elea 510-?? Student of the exiled Xenophanes The goal of philosophy is to attain the truth. The path to truth is via reason and logic. Reason will distinguish appearance from reality. Nature is comprehensible and logical. 29 Parmenides and the Law of Contradiction Something either is or it is not. The law of the excluded middle Therefore, nothing is that isn t! It is impossible to be not being There is no such thing as empty space. Space is something and empty is nothing. 30

11 Parmenides against Heraclitos If there is no space that is empty, the universe is everywhere full and occupied. Therefore nothing actually changes. Therefore motion is impossible. 31 The Fundamental Problem of Viewpoint Focus on the whole Parmenides Easier to grasp the unity of the world. Difficult to explain processes, events, changes. Focus on the parts Heraclitos Easier to explain changes as rearrangements of the parts. Difficult to make sense of all that is. 32 The Perils of Logic Reasoning with logic inevitably begins with assumed premises, which may or may not be true. The reasoning itself may or may not be valid though this can be checked. The truth of conclusions depends on the truth of the premises and the validity of the argument. 33

12 Zeno of Elea BCE Student of Parmenides Probably moved to Athens later and taught there, making his and Parmedies views better known. 34 Zeno s s Paradoxes Paradox, from the Greek meaning contrary to opinion. Showed that logic can lead to conclusions which defy common sense. Hard to say whether he was attacking common sense beliefs (as seems probable), or demonstrating the dangers of reasoning by logical deduction. 35 Consider a stadium a a running track of about 180 meters in ancient Greece. The Stadium 36

13 The Stadium Will the runner reach the other side of the stadium? 37 The Stadium Paradox Before the runner can reach the finish line, the mid-point must be reached. Before that, the ¼ point. Before that 1/8, 1/16, 1/32, 1/64, and an infinite number of prior events. The runner never can leave the starting block. 38 Achilles and the Tortoise Achilles, the mythical speedy warrior, is to have a footrace with a tortoise. Achilles gives the tortoise a head start. 39

14 Achilles and the Tortoise, 2 Call the starting time t=0. Before Achilles can pass the tortoise, he must reach where the tortoise was at the start. Call when Achilles reaches the tortoise s s starting position t=1 By then, the tortoise has gone ahead. 40 Achilles and the Tortoise, 3 Now at time t=1, Achilles still must reach where the tortoise is before he can pass it. Every time Achilles reaches where the tortoise had been, the tortoise is further ahead. The tortoise must win the race. 41 Achilles and the Tortoise, 4 An animated demonstration of the paradox. 42

15 Achilles and the Tortoise, 4 An animated demonstration of the paradox. 43 Achilles and the Tortoise, 4 An animated demonstration of the paradox. 44 The Flying Arrow Imagine an arrow in flight. Is it moving? Motion means moving from place to place. At any single moment, the arrow is in a single place, therefore, not moving. 45

16 The Flying Arrow, 2 At every moment of its flight, the arrow is not moving. If it were, it would occupy more space that it does, which is impossible. There is no such thing as motion. 46 Pythagoras of Samos Born between 580 and 569. Died about 500 BCE. Lived in Samos, an island off the coast of Ionia. 47 Pythagoras and the Pythagoreans Pythagoras himself lived earlier than many of the other Pre-Socratics and had some influence on them: E.g., Heraclitos,, Parmenides, and Zeno Very little is known about what Pythagoras himself taught, but he founded a cult that promoted and extended his views. Most of what we know is from his followers. 48

17 The Pythagorean Cult The followers of Pythagoras were a close- knit group like a religious cult. Vows of poverty. Secrecy. Special dress, went barefoot. Strict diet: Vegetarian Ate no beans. 49 Everything is Number The Pythagoreans viewed number as the underlying structure of everything in the universe. Compare to Thales view of water, Anaximander s apeiron, Anaximenes air, Heraclitos,, change. Pythagorean numbers take up space. Like little hard spheres. 50 Numbers and Music One of the discoveries attributed to Pythagoras himself. Musical scale: 1:2 = octave 2:3 = perfect fifth 3:4 = perfect fourth 51

18 Numbers and Music, contd. Relative string lengths for notes of the scale from lowest note (bottom) to highest. The octave higher is half the length of the former. The fourth is i ¾, the fifth is 2/3. 52 Geometric Harmony The numbers 12, 8, 6 represent the lengths of a ground note, the fifth above, and the octave above the ground note. Hence these numbers form a harmonic progression. A cube has 12 edges, 8 corners, and 6 faces. Fantastic! A cube is in geometric harmony. 53 Figurate Numbers Numbers that can be arranged to form a regular figure (triangle, square, hexagon, etc.) are called figurate numbers. 54

19 Special significance was given to the number 10, which can be arranged as a triangle with 4 on each side. Called the tetrad or tetractys. The Tetractys 55 The significance of the Tetractys The number 10, the tetractys,, was considered sacred. It was more than just the base of the number system and the number of fingers. The Pythagorean oath: By him that gave to our generation the Tetractys,, which contains the fount and root of eternal nature. 56 Pythagorean Cosmology Unlike almost every other ancient thinker, the Pythagoreans did not place the Earth at the centre of the universe. The Earth was too imperfect for such a noble position. Instead the centre was the Central Fire or, the watchtower of Zeus. 57

20 The Pythagorean cosmos -- with 9 heavenly bodies 58 The Pythagorean Cosmos and the Tetractys To match the tetractys, another heavenly body was needed. Hence, the counter earth, or antichthon, always on the other side of the central fire, and invisible to human eyes. 59 The Pythagorean Theorem 60

21 The Pythagorean Theorem, contd. Legend has it that Pythagoras himself discovered the truth of the theorem that bears his name: That if squares are built upon the sides of any right triangle, the sum of the areas of the two smaller squares is equal to the area of the largest square. 61 Well-known Special Cases Records from both Egypt and Babylonia as well as oriental civilizations show that special cases of the theorem were well known and used in surveying and building. The best known special cases are The triangle: =5 2 or 9+16=25 The triangle: =13 2 or = Commensurability Essential to the Pythagorean view that everything is ultimately number is the notion that the same scale of measurement can be used for everything. E.g., for length, the same ruler, perhaps divided into smaller and smaller units, will ultimately measure every possible length exactly. This is called commensurability. 63

22 Commensurable Numbers Numbers, for the Pythagoreans, mean the natural, counting numbers. All natural numbers are commensurable because the can all be measured by the same unit, namely 1. The number 25 is measured by 1 laid off 25 times. The number 36 is measured by 1 laid off 36 times. 64 Commensurable Magnitudes A magnitude is a measurable quantity, for example, length. Two magnitudes are commensurable if a common unit can be laid off to measure each one exactly. E.g., two lengths of 36.2 cm and cm are commensurable because each is an exact multiple of the unit of measure 0.1 cm cm is exactly 362 units and cm is exactly 1713 units. 65 Commensurability is essential for the Pythagorean view. If everything that exists in the world ultimately has a numerical structure, and numbers mean some tiny spherical balls that occupy space, then everything in the world is ultimately commensurable with everything else. It may be difficult to find the common measure, but it just must exist. 66

23 Incommensurability The (inconceivable) opposite to commensurability is incommensurability, the situation where no common measure between two quantities exists. To prove that two quantities are commensurable, one need only find a single common measure. To prove that quantities are incommensurable, it would be necessary to prove that no common measures could possibly exist. 67 The Diagonal of the Square The downfall of the Pythagorean world view came out of their greatest triumph the Pythagorean theorem. Consider the simplest case, the right triangles formed by the diagonal of a square. 68 Proving Incommensurability If the diagonal and the side of the square are commensurable, then they can each be measured by some common unit. Suppose we choose the largest common unit of length that goes exactly into both. 69

24 Proving Incommensurability, 2 Call the number of times the measuring unit fits on the diagonal h and the number of time it fits on the side of the square a. It cannot be that a and h are both even numbers, because if they were, a larger unit (twice the size) would have fit exactly into both the diagonal and the side. 70 Proving Incommensurability, 3 By the Pythagorean theorem, a 2 + a 2 = h 2 If 2 2a 2 = h 2 then h 2 must be even. If h 2 is even, so is h. Therefore a must be odd. (Since they cannot both be even.) 71 Proving Incommensurability, 4 Since h is even, it is equal to 2 times some number, j. So h = 2j. 2. Substitute 2j for h in the formula given by the Pythagorean theorem: 2a 2 = h 2 = (2j) 2 = 4 4j 2. If 2 2a 2 = 4 4j 2., then a 2 = 2j 2 Therefore a 2 is even, and so is a. But we have already shown that a is odd. 72

25 Proof by Contradiction This proof is typical of the use of logic, as championed by Parmenides, to sort what is true and what is false into separate categories. It is the cornerstone of Greek mathematical reasoning, and also is used throughout ancient reasoning about nature. 73 The Method of Proof by Contradiction 1. Assume the opposite of what you wish to prove: Assume that the diagonal and the side are commensurable, meaning that at least one unit of length exists that exactly measures each. 74 The Method of Proof by Contradiction 2. Show that valid reasoning from that premise leads to a logical contradiction. That the length of the side of the square must be both an odd number of units and an even number of units. Since a number cannot be both odd and even, something must be wrong in the argument. The only thing that could be wrong is the assumption that the lengths are commensurable. 75

26 The Method of Proof by Contradiction 3. Therefore the opposite of the assumption must be true. If the only assumption was that the two lengths are commensurable and that is false, then it must be the case that the lengths are incommensurable. Note that the conclusion logically follows even though at no point were any of the possible units of measure specified. 76 The Flaw of Pythagoreanism The Pythagorean world view that everything that exists is ultimately a numerical structure (and that numbers mean just counting numbers integers). In their greatest triumph, the magical Pythagorean theorem, lay a case that cannot fit this world view. 77 The Decline of the Pythagoreans The incommensurability of the diagonal and side of a square sowed a seed of doubt in the minds of Pythagoreans. They became more defensive, more secretive, and less influential. But they never quite died out. 78