Greece

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Greece
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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?
• Einsteins answer
• The astonishing thing is that these discoveries
  the bases of science were made at all.

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The Origins of Ancient Greece

• What we call ancient Greece might better be
  called the ancient Aegean Civilizations.

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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 3000 1450 BCE.
• Had some kind of written language, never
  deciphered.
• Collapsed suddenly for unknown reasons.

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The Mycenaean Civilization

• On the Peloponnesus (the southern mainland)
  another civilization arose and flourished from
  about 1600-1200 BCE.
• The Mycenaeans adapted the Minoan writing system
  to their own language, Greek. But it was awkward
  to use.

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Mycenaea

• The peak of the Mycenaean civilization was the
  reign of Agamemnon, who took his people (the
  Greeks) to war against the Trojans.

Agamemnons Palace
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The Trojan War
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The Trojan War

• Approx. 1280 1180 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.

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The Dark Age of Greece

• 1200 800 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.

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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.

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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).

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The Phoenician Alphabet
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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.

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The Homeric Age

• 800 600 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

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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.

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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.

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The Pre-Socratics

• Thinkers living between about 600 450 BCE.
• So named because they (basically) predated
  Socrates.
• Known only through discussions of their thoughts
  in later works.
• Some fragments still exist.

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Socrates

• Lived in Athens, 470-399 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.

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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 Platos.
• Socrates is an important figure in the
  development of scientific reasoning, but
• He had no interest in the natural world.

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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.

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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.

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Thales of Miletos

• 625-545 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.

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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.

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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.
• The distance from A to B is the same as the
  distance from A to the ship.

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Anaximander of Miletos

• 611-547 BCE
• Student of Thales?
• Map of the known world
• Apeiron (the Boundless)
• The basic stuff of the world

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Anaximenes of Miletos

• 550-475 BCE
• Student of Anaximander?
• Air the fundamental stuff
• Cosmological view
• Crystalline sphere of the fixed stars
• Earth in centre, planets between

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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."

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Elea
Elea was a Greek colony in southern Italy.

• The minor Pre-Socratic, Xenophanes, fled from
  Colophon in Ionia to Elea to escape persecution.

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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.

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Parmenides and the Law of Contradiction

• Something either is or it is not.
• The law of the excluded middle
• Therefore, nothing is that isnt!
• It is impossible to be not being
• There is no such thing as empty space.
• Space is something and empty is nothing.

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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.

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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.

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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.

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Zeno of Elea

• 495-425 BCE
• Student of Parmenides
• Probably moved to Athens later and taught there,
  making his and Parmedies views better known.

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Zenos 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.

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The Stadium

• Consider a stadiuma running track of about 180
  meters in ancient Greece.

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The Stadium

• Will the runner reach the other side of the
  stadium?

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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.

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Achilles and the Tortoise

• Achilles, the mythical speedy warrior, is to have
  a footrace with a tortoise.
• Achilles gives the tortoise a head start.

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Achilles and the Tortoise, 2

• Call the starting time t0.
• Before Achilles can pass the tortoise, he must
  reach where the tortoise was at the start.
• Call when Achilles reaches the tortoises
  starting position t1
• By then, the tortoise has gone ahead.

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Achilles and the Tortoise, 3

• Now at time t1, 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.

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Achilles and the Tortoise, 4

• An animated demonstration of the paradox.

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Achilles and the Tortoise, 4

• An animated demonstration of the paradox.

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Achilles and the Tortoise, 4

• An animated demonstration of the paradox.

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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.

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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.

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Pythagoras of Samos

• Born between 580 and 569. Died about 500 BCE.
• Lived in Samos, an island off the coast of Ionia.

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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.

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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.

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Everything is Number

• The Pythagoreans viewed number as the underlying
  structure of everything in the universe.
• Compare to Thales view of water, Anaximanders
  apeiron, Anaximenes air, Heraclitos, change.
• Pythagorean numbers take up space.
• Like little hard spheres.

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Numbers and Music

• One of the discoveries attributed to Pythagoras
  himself.
• Musical scale
• 12 octave
• 23 perfect fifth
• 34 perfect fourth

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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 ¾, the fifth is 2/3.

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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.

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Figurate Numbers

• Numbers that can be arranged to form a regular
  figure (triangle, square, hexagon, etc.) are
  called figurate numbers.

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The Tetractys

• 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.

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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.

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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.

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The Pythagorean cosmos-- with 9 heavenly bodies
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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.

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The Pythagorean Theorem
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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.

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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 3-4-5 triangle 324252 or 91625
• The 5-12-13 triangle 52122132 or 25144169

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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.

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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.

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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 171.3 cm are
  commensurable because each is an exact multiple
  of the unit of measure 0.1 cm.
• 36.2 cm is exactly 362 units and 171.3 cm is
  exactly 1713 units.

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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.

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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.

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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.

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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.

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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.

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Proving Incommensurability, 3

• By the Pythagorean theorem, a2 a2 h2
• If 2a2 h2 then h2 must be even.
• If h2 is even, so is h.
• Therefore a must be odd. (Since they cannot both
  be even.)

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Proving Incommensurability, 4

• Since h is even, it is equal to 2 times some
  number, j. So h 2j. Substitute 2j for h in the
  formula given by the Pythagorean theorem
• 2a2 h2 (2j)2 4j2.
• If 2a2 4j2., then a2 2j2
• Therefore a2 is even, and so is a.
• But we have already shown that a is odd.

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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.

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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.

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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.

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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.

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The Flaw of Pythagoreanism

• The Pythagorean world view that everything that
  exists is ultimately a numerical structure (and
  that numbers mean just counting
  numbersintegers).
• In their greatest triumph, the magical
  Pythagorean theorem, lay a case that cannot fit
  this world view.

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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.