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SC/STS 3760, II SC/STS 3760, II 1 Greece Greece SC/STS 3760, II SC/STS 3760, II 2 The Origins of Scientific Thinking? The Origins of Scientific Thinking? Greece is often cited as the place where Greece is often cited as the place where the first inklings of modern scientific the first inklings of modern scientific thinking took place. thinking took place. Why there and not elsewhere? Why there and not elsewhere? Einstein Einstein’ s answer: s answer: The astonishing thing is that these The astonishing thing is that these discoveries [the bases of science] were made discoveries [the bases of science] were made at all. at all.” SC/STS 3760, II SC/STS 3760, II 3 The Origins of Ancient Greece The Origins of Ancient Greece What we What we call ancient call ancient Greece Greece might better might better be called be called the ancient the ancient Aegean Aegean Civilizations. Civilizations.
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Page 1: Greece - York University · Greece might better be called the ancient Aegean Civilizations. 2 SC/STS 3760, II 4 The Aegean Civilizations There have been civilizations in the Aegean

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GreeceGreece

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The Origins of Scientific Thinking?The Origins of Scientific Thinking?

Greece is often cited as the place where Greece is often cited as the place where the first inklings of modern scientific the first inklings of modern scientific thinking took place.thinking took place.Why there and not elsewhere?Why there and not elsewhere?EinsteinEinstein’’s answer:s answer:

““The astonishing thing is that these The astonishing thing is that these discoveries [the bases of science] were made discoveries [the bases of science] were made at all.at all.””

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The Origins of Ancient GreeceThe Origins of Ancient Greece

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

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The Aegean CivilizationsThe Aegean Civilizations

There have been civilizations in the There have been civilizations in the Aegean area almost as long as there have Aegean area almost as long as there have been in Mesopotamia and Egypt.been in Mesopotamia and Egypt.The earliest known in the area was the The earliest known in the area was the Minoan Civilization on the island of Crete.Minoan Civilization on the island of Crete.

Existed from about 3000 Existed from about 3000 –– 1450 BCE.1450 BCE.Had some kind of written language, never Had some kind of written language, never deciphered.deciphered.Collapsed suddenly for unknown reasons.Collapsed suddenly for unknown reasons.

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The Mycenaean CivilizationThe Mycenaean Civilization

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

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MycenaeaMycenaeaThe peak of The peak of the Mycenaean the Mycenaean civilization was civilization was the reign of the reign of Agamemnon, Agamemnon, who took his who took his people (the people (the ““GreeksGreeks””) to ) to war against war against the Trojans.the Trojans.

Agamemnon’s Palace

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The Trojan WarThe Trojan War

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The Trojan WarThe Trojan War

Approx. 1280 Approx. 1280 –– 1180 BCE.1180 BCE.MycenaeaMycenaea versus Troy.versus Troy.Won by the Greeks, but the war depleted Won by the Greeks, but the war depleted their fighting forces.their fighting forces.MycenaeaMycenaea was invaded by was invaded by DoriansDorians about about 1200 BCE, and its culture destroyed.1200 BCE, and its culture destroyed.

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The Dark Age of GreeceThe Dark Age of Greece

1200 1200 –– 800 BCE800 BCEThe organized Greek civilization was The organized Greek civilization was destroyed by the invading destroyed by the invading DoriansDorians..Knowledge of writing was lost.Knowledge of writing was lost.People lived in isolated villages.People lived in isolated villages.What they had in common was spoken What they had in common was spoken Greek and memories of past greatness.Greek and memories of past greatness.

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PhoeniciaPhoenicia

Around 1700 BCE, in the Near East, what Around 1700 BCE, in the Near East, what is now Lebanon, a civilization developed is now Lebanon, a civilization developed with both Mesopotamian and Egyptian with both Mesopotamian and Egyptian influences.influences.The Greeks later called the people from The Greeks later called the people from there there ““PhoneciansPhonecians”” –– meaning traders in meaning traders in purple.purple.

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Phoenician WritingPhoenician Writing

Phoenicians developed a style of writing Phoenicians developed a style of writing that combined Mesopotamian cuneiform that combined Mesopotamian cuneiform and Egyptian and Egyptian heiraticheiratic..It had 22 distinct characters, each It had 22 distinct characters, each representing a particular sound (a representing a particular sound (a consonant).consonant).

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The Phoenician AlphabetThe Phoenician Alphabet

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The Phoenician Alphabetic was The Phoenician Alphabetic was PhoneticPhonetic

Since each character represented a sound, Since each character represented a sound, rather than a meaning, the characters rather than a meaning, the characters could be used to represent words in an could be used to represent words in an entirely different language.entirely different language.The Greeks adapted the Phoenician script The Greeks adapted the Phoenician script to their own language and produced an to their own language and produced an alphabetalphabet..

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The Homeric AgeThe Homeric Age800 800 –– 600 BCE600 BCEThe Greek verbal culture could be written The Greek verbal culture could be written down.down.The heroic stories of the Trojan WarThe heroic stories of the Trojan Warwere written by Homer.were written by Homer.

The Iliad, The OdysseyThe Iliad, The OdysseyGreek mythology and folk knowledge Greek mythology and folk knowledge were recorded by Hesiod.were recorded by Hesiod.

TheogonyTheogony, Works and Days, Works and Days

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The Greek Civilization Takes OffThe Greek Civilization Takes Off

The first Olympic Games 776 BCEThe first Olympic Games 776 BCEThe Polis (CityThe Polis (City--State)State)

Independent governments arose all across the Independent governments arose all across the Greek settlements.Greek settlements.Experimentation in forms of government:Experimentation in forms of government:

Monarchies, Aristocracies, Dictatorships, Monarchies, Aristocracies, Dictatorships, Oligarchies, DemocraciesOligarchies, Democracies

Independent units, but tied together by a Independent units, but tied together by a common language, religion, and literature.common language, religion, and literature.

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Assertion: Scientific Thinking Assertion: Scientific Thinking Began in Ancient GreeceBegan in Ancient Greece

Possible explanations given:Possible explanations given:

ReligionReligion –– The Greek gods were too humanThe Greek gods were too human--like. like.

LanguageLanguage –– Phonetic alphabet encouraged literacy.Phonetic alphabet encouraged literacy.

TradeTrade –– The Greeks became traders and The Greeks became traders and travellerstravellers, , bringing home new ideas.bringing home new ideas.

DemocracyDemocracy –– Democratic governments, where they Democratic governments, where they existed, encouraged independent thought.existed, encouraged independent thought.

SlaverySlavery –– Greeks (like many other cultures) had slaves Greeks (like many other cultures) had slaves who did the menial work.who did the menial work.

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The PreThe Pre--SocraticsSocratics

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

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SocratesSocrates

Lived in Athens, 470Lived in Athens, 470--399 BCE.399 BCE.Set the direction of Western Set the direction of Western philosophical thinking.philosophical thinking.The goal of philosophy The goal of philosophy –– to discover the truth.to discover the truth.Reasoning, the supreme Reasoning, the supreme method.method.

Pursued by asking questions, the dialectical, or Pursued by asking questions, the dialectical, or ““SocraticSocratic”” method.method.

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Socrates, contd.Socrates, contd.

Socrates left no writings at all.Socrates left no writings at all.He is known to us primarily through the He is known to us primarily through the works of Plato.works of Plato.

It is hard to distinguish SocratesIt is hard to distinguish Socrates’’ own thought own thought from Platofrom Plato’’s.s.

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

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Back to the PreBack to the Pre--SocraticsSocraticsMost PreMost Pre--Socratics came Socratics came from the Greek from the Greek colonies on the colonies on the eastern side of eastern side of the Aegean Sea the Aegean Sea known as Ionia.known as Ionia.

This is now part This is now part of Turkey.of Turkey.

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Wondering about NatureWondering about Nature

The importance of the PreThe importance of the Pre--Socratics is that Socratics is that they appear to be the first people we they appear to be the first people we know of who asked fundamental questions know of who asked fundamental questions about nature, such as about nature, such as ““What is the world What is the world made of?made of?””

And then they provided reasons to justify And then they provided reasons to justify their answers.their answers.

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Thales of Thales of MiletosMiletos625625--545 BCE545 BCEPhoenician parents?Phoenician parents?Stories:Stories:

Predicted solar eclipse of Predicted solar eclipse of May 28, 585 BCEMay 28, 585 BCEFalling into a wellFalling into a wellOlive pressOlive press

WaterWater is the basic stuff of is the basic stuff of the world.the world.

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Thales and MathematicsThales and Mathematics

Thales is said to have Thales is said to have brought Egyptian brought Egyptian mathematics to Greeks. mathematics to Greeks. Examples:Examples:

All triangles constructed on All triangles constructed on the diameter of a circle are the diameter of a circle are right triangles.right triangles.The base angles of isosceles The base angles of isosceles triangles are equal.triangles are equal.If two straight lines intersect, If two straight lines intersect, opposite angles are equal.opposite angles are equal.

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Measuring the distance of a ship Measuring the distance of a ship from shorefrom shore

From the desired point on the From the desired point on the shore, A, walk off a known shore, A, walk off a known distance to point C, at a right distance to point C, at a right angle from the ship and place a angle from the ship and place a marker there.marker there.Continue walking the same Continue walking the same distance again to B.distance again to B.At B, turn at a right angle away At B, turn at a right angle away from the shore and walk until from the shore and walk until the marker at C and the ship are the marker at C and the ship are in a straight line. Call that Ain a straight line. Call that A’’..The distance from AThe distance from A’’ to B is the to B is the same as the distance from A to same as the distance from A to the ship.the ship.

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Anaximander of Anaximander of MiletosMiletos

611611--547 BCE547 BCEStudent of Thales?Student of Thales?Map of the known Map of the known worldworldApeironApeiron (the (the Boundless)Boundless)

The basic stuff of the The basic stuff of the worldworld

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AnaximenesAnaximenes of of MiletosMiletos550550--475 BCE475 BCEStudent of Student of Anaximander?Anaximander?AirAir –– the fundamental the fundamental stuffstuffCosmological view:Cosmological view:

Crystalline sphere of the Crystalline sphere of the fixed starsfixed starsEarth in centre, planets Earth in centre, planets betweenbetween

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HeraclitosHeraclitos of Ephesusof Ephesus

Ephesus is 50 km N of Ephesus is 50 km N of MiletosMiletos..550?550?--475? BCE (i.e., 475? BCE (i.e., about the same as about the same as AnaximenesAnaximenes, but , but uncertain)uncertain)Everything is Flux.Everything is Flux.

Fire fundamentalFire fundamental"You can't step in the "You can't step in the same river twice."same river twice."

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EleaElea

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

Elea was a Elea was a Greek Greek colony in colony in southern southern Italy.Italy.

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Parmenides of EleaParmenides of Elea510510--????Student of the exiled Student of the exiled XenophanesXenophanesThe goal of philosophy is The goal of philosophy is to attain the truth.to attain the truth.The path to truth is via The path to truth is via reason and logic.reason and logic.Reason will distinguish Reason will distinguish appearance from reality.appearance from reality.

Nature is comprehensible Nature is comprehensible and logical.and logical.

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Parmenides and the Law of Parmenides and the Law of ContradictionContradiction

Something either Something either isis or it or it is not.is not.

The law of the excluded middleThe law of the excluded middle

Therefore, nothing Therefore, nothing is is that that isnisn’’t!t!

It is impossible to be It is impossible to be not beingnot being

There is no such thing as There is no such thing as emptyempty space.space.

SpaceSpace is something and is something and emptyempty is nothing.is nothing.

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Parmenides against Parmenides against HeraclitosHeraclitos

If there is no space that is empty, the If there is no space that is empty, the

universe is everywhere full and occupied.universe is everywhere full and occupied.

Therefore nothing actually changes.Therefore nothing actually changes.

Therefore motion is impossible.Therefore motion is impossible.

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The Fundamental Problem of The Fundamental Problem of ViewpointViewpoint

Focus on the whole Focus on the whole –– ParmenidesParmenidesEasier to grasp the unity of the world.Easier to grasp the unity of the world.Difficult to explain processes, events, Difficult to explain processes, events, changes.changes.

Focus on the parts Focus on the parts –– HeraclitosHeraclitosEasier to explain changes as rearrangements Easier to explain changes as rearrangements of the parts.of the parts.Difficult to make sense of all that is.Difficult to make sense of all that is.

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The Perils of LogicThe Perils of Logic

Reasoning with logic inevitably begins with Reasoning with logic inevitably begins with assumed premises, which may or may not assumed premises, which may or may not be true.be true.The reasoning itself may or may not be The reasoning itself may or may not be valid valid –– though this can be checked.though this can be checked.The truth of conclusions depends on the The truth of conclusions depends on the truth of the premises and the validity of truth of the premises and the validity of the argument.the argument.

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Zeno of EleaZeno of Elea

495495--425 BCE425 BCEStudent of ParmenidesStudent of ParmenidesProbably moved to Probably moved to Athens later and taught Athens later and taught there, making his and there, making his and ParmediesParmedies’’ views better views better known.known.

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ZenoZeno’’s Paradoxess Paradoxes

Paradox, from the Greek meaning Paradox, from the Greek meaning ““contrary to opinion.contrary to opinion.””Showed that logic can lead to conclusions Showed that logic can lead to conclusions which defy common sense.which defy common sense.

Hard to say whether he was attacking Hard to say whether he was attacking common sense beliefs (as seems probable), common sense beliefs (as seems probable), or demonstrating the dangers of reasoning by or demonstrating the dangers of reasoning by logical deduction.logical deduction.

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The StadiumThe StadiumConsider a Consider a stadiumstadium——a running a running track of track of about 180 about 180 meters in meters in ancient ancient Greece.Greece.

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The StadiumThe Stadium

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

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The Stadium ParadoxThe Stadium Paradox

Before the runner can reach the finish line, the midBefore the runner can reach the finish line, the mid--point point must be reached.must be reached.Before that, the Before that, the ¼¼ point. Before that 1/8, 1/16, 1/32, point. Before that 1/8, 1/16, 1/32, 1/64,1/64,…… and an infinite number of prior events.and an infinite number of prior events.The runner never can leave the starting block.The runner never can leave the starting block.

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Achilles and the TortoiseAchilles and the Tortoise

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

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Achilles and the Tortoise, 2Achilles and the Tortoise, 2

Call the starting time t=0.Call the starting time t=0.Before Achilles can pass the tortoise, he must Before Achilles can pass the tortoise, he must reach where the tortoise was at the start.reach where the tortoise was at the start.Call when Achilles reaches the tortoiseCall when Achilles reaches the tortoise’’s starting s starting position t=1position t=1By then, the tortoise has gone ahead.By then, the tortoise has gone ahead.

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Achilles and the Tortoise, 3Achilles and the Tortoise, 3

Now at time t=1, Achilles still must reach where the Now at time t=1, Achilles still must reach where the tortoise is before he can pass it.tortoise is before he can pass it.Every time Achilles reaches where the tortoise had been, Every time Achilles reaches where the tortoise had been, the tortoise is further ahead.the tortoise is further ahead.The tortoise must win the race.The tortoise must win the race.

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Achilles and the Tortoise, 4Achilles and the Tortoise, 4

An animated demonstration of the paradox.An animated demonstration of the paradox.

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Achilles and the Tortoise, 4Achilles and the Tortoise, 4

An animated demonstration of the paradox.An animated demonstration of the paradox.

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Achilles and the Tortoise, 4Achilles and the Tortoise, 4

An animated demonstration of the paradox.An animated demonstration of the paradox.

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The Flying ArrowThe Flying Arrow

Imagine an arrow in flight. Is it moving?Imagine an arrow in flight. Is it moving?Motion means moving from place to place. Motion means moving from place to place. At any single moment, the arrow is in a single At any single moment, the arrow is in a single place, therefore, not moving.place, therefore, not moving.

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The Flying Arrow, 2The Flying Arrow, 2

At every moment of its flight, the arrow is not At every moment of its flight, the arrow is not moving. If it were, it would occupy more space moving. If it were, it would occupy more space that it does, which is impossible.that it does, which is impossible.There is no such thing as motion.There is no such thing as motion.

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Pythagoras of SamosPythagoras of Samos

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

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Pythagoras and the PythagoreansPythagoras and the Pythagoreans

Pythagoras himself lived earlier than many Pythagoras himself lived earlier than many of the other Preof the other Pre--Socratics and had some Socratics and had some influence on them:influence on them:

E.g., E.g., HeraclitosHeraclitos, Parmenides, and Zeno, Parmenides, and Zeno

Very little is known about what Pythagoras Very little is known about what Pythagoras himself taught, but he founded a cult that himself taught, but he founded a cult that promoted and extended his views. Most of promoted and extended his views. Most of what we know is from his followers.what we know is from his followers.

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The Pythagorean CultThe Pythagorean Cult

The followers of Pythagoras were a closeThe followers of Pythagoras were a close--knit group like a religious cult.knit group like a religious cult.Vows of poverty.Vows of poverty.Secrecy.Secrecy.Special dress, went barefoot.Special dress, went barefoot.Strict diet:Strict diet:

VegetarianVegetarianAte no beans.Ate no beans.

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Everything is NumberEverything is Number

The Pythagoreans viewed number as the The Pythagoreans viewed number as the underlying structure of everything in the underlying structure of everything in the universe.universe.

Compare to ThalesCompare to Thales’’ view of water, view of water, AnaximanderAnaximander’’s s apeironapeiron, , AnaximenesAnaximenes’’ air, air, HeraclitosHeraclitos, change., change.

Pythagorean numbers take up space.Pythagorean numbers take up space.Like little hard spheres.Like little hard spheres.

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Numbers and MusicNumbers and Music

One of the discoveries attributed to One of the discoveries attributed to Pythagoras himself.Pythagoras himself.Musical scale:Musical scale:

1:2 = octave1:2 = octave2:3 = perfect fifth2:3 = perfect fifth3:4 = perfect fourth3:4 = perfect fourth

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Numbers and Music, contd.Numbers and Music, contd.

Relative string lengths for notes of the scale from lowest note Relative string lengths for notes of the scale from lowest note (bottom) to highest. (bottom) to highest. The octave higher is half the length of the former. The fourth iThe octave higher is half the length of the former. The fourth is s ¾¾, , the fifth is 2/3.the fifth is 2/3.

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Geometric HarmonyGeometric Harmony

The numbers 12, 8, 6 represent the The numbers 12, 8, 6 represent the lengths of a ground note, the fifth above, lengths of a ground note, the fifth above, and the octave above the ground note.and the octave above the ground note.

Hence these numbers form a Hence these numbers form a ““harmonic harmonic progression.progression.””

A cube has 12 edges, 8 corners, and 6 A cube has 12 edges, 8 corners, and 6 faces.faces.

Fantastic! A cube is in Fantastic! A cube is in ““geometric harmony.geometric harmony.””

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Figurate NumbersFigurate Numbers

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

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The The TetractysTetractys

Special significance Special significance was given to the was given to the number 10, which can number 10, which can be arranged as a be arranged as a triangle with 4 on triangle with 4 on each side.each side.Called the tetrad or Called the tetrad or tetractystetractys..

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The significance of the The significance of the TetractysTetractys

The number 10, the The number 10, the tetractystetractys, was , was considered sacred. considered sacred. It was more than just the base of the It was more than just the base of the number system and the number of number system and the number of fingers.fingers.The Pythagorean oath:The Pythagorean oath:

““By him that gave to our generation the By him that gave to our generation the TetractysTetractys, which contains the fount and root , which contains the fount and root of eternal nature.of eternal nature.””

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Pythagorean CosmologyPythagorean Cosmology

Unlike almost every other ancient thinker, Unlike almost every other ancient thinker, the Pythagoreans did not place the Earth the Pythagoreans did not place the Earth at the centre of the universe.at the centre of the universe.The Earth was too imperfect for such a The Earth was too imperfect for such a noble position.noble position.Instead the centre was the Instead the centre was the ““Central FireCentral Fire””or, the watchtower of Zeus.or, the watchtower of Zeus.

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The The Pythagorean Pythagorean

cosmoscosmos---- with 9 with 9 heavenly heavenly bodiesbodies

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The Pythagorean Cosmos and the The Pythagorean Cosmos and the TetractysTetractys

To match the To match the tetractystetractys, , another another heavenly body heavenly body was needed.was needed.Hence, the Hence, the counter earth, counter earth, or or antichthonantichthon,,always on the always on the other side of other side of the central fire, the central fire, and invisible to and invisible to human eyes.human eyes.

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The Pythagorean TheoremThe Pythagorean Theorem

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The Pythagorean Theorem, contd.The Pythagorean Theorem, contd.

Legend has it that Pythagoras Legend has it that Pythagoras himself discovered the truth of himself discovered the truth of the theorem that bears his name:the theorem that bears his name:

That if squares are built upon the That if squares are built upon the sides of sides of anyany right triangle, the sum right triangle, the sum of the areas of the two smaller of the areas of the two smaller squares is equal to the area of the squares is equal to the area of the largest square.largest square.

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WellWell--known Special Casesknown Special Cases

Records from both Egypt and Babylonia as Records from both Egypt and Babylonia as well as oriental civilizations show that well as oriental civilizations show that special cases of the theorem were well special cases of the theorem were well known and used in surveying and building.known and used in surveying and building.The best known special cases areThe best known special cases are

The 3The 3--44--5 triangle: 35 triangle: 322+4+422=5=522 or 9+16=25or 9+16=25The 5The 5--1212--13 triangle: 513 triangle: 522+12+1222=13=1322 or or 25+144=16925+144=169

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CommensurabilityCommensurability

Essential to the Pythagorean view that Essential to the Pythagorean view that everything is ultimately number is the everything is ultimately number is the notion that the same scale of notion that the same scale of measurement can be used for everything.measurement can be used for everything.E.g., for length, the same ruler, perhaps E.g., for length, the same ruler, perhaps divided into smaller and smaller units, will divided into smaller and smaller units, will ultimately measure every possible length ultimately measure every possible length exactly.exactly.This is called This is called commensurabilitycommensurability..

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Commensurable NumbersCommensurable Numbers

Numbers, for the Pythagoreans, mean the Numbers, for the Pythagoreans, mean the natural, counting numbers.natural, counting numbers.All natural numbers are commensurable All natural numbers are commensurable because the can all be because the can all be ““measuredmeasured”” by the by the same unit, namely 1. same unit, namely 1.

The number 25 is measured by 1 laid off 25 The number 25 is measured by 1 laid off 25 times.times.The number 36 is measured by 1 laid off 36 The number 36 is measured by 1 laid off 36 times.times.

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Commensurable MagnitudesCommensurable Magnitudes

A magnitude is a measurable quantity, for A magnitude is a measurable quantity, for example, length.example, length.Two magnitudes are commensurable if a Two magnitudes are commensurable if a common unit can be laid off to measure common unit can be laid off to measure each one exactly.each one exactly.

E.g., two lengths of 36.2 cm and 171.3 cm E.g., two lengths of 36.2 cm and 171.3 cm are commensurable because each is an exact are commensurable because each is an exact multiple of the unit of measure 0.1 cm.multiple of the unit of measure 0.1 cm.

36.2 cm is exactly 362 units and 171.3 cm is 36.2 cm is exactly 362 units and 171.3 cm is exactly 1713 units.exactly 1713 units.

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Commensurability is essential for Commensurability is essential for the Pythagorean view.the Pythagorean view.

If everything that exists in the world If everything that exists in the world ultimately has a numerical structure, and ultimately has a numerical structure, and numbers mean some tiny spherical balls numbers mean some tiny spherical balls that occupy space, then everything in the that occupy space, then everything in the world is ultimately commensurable with world is ultimately commensurable with everything else.everything else.It may be difficult to find the common It may be difficult to find the common measure, but it just measure, but it just mustmust exist.exist.

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IncommensurabilityIncommensurabilityThe (inconceivable) opposite to The (inconceivable) opposite to commensurability is commensurability is incommensurability,incommensurability, the the situation where no common measure between situation where no common measure between two quantities exists.two quantities exists.To prove that two quantities are To prove that two quantities are commensurable, one need only find a single commensurable, one need only find a single common measure.common measure.To prove that quantities are To prove that quantities are inincommensurable, it commensurable, it would be necessary to prove that no common would be necessary to prove that no common measures could possibly exist. measures could possibly exist.

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The Diagonal of the SquareThe Diagonal of the SquareThe downfall of the The downfall of the Pythagorean world Pythagorean world view came out of view came out of their greatest triumph their greatest triumph the Pythagorean the Pythagorean theorem.theorem.Consider the simplest Consider the simplest case, the right case, the right triangles formed by triangles formed by the diagonal of a the diagonal of a square. square.

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Proving IncommensurabilityProving IncommensurabilityIf the diagonal and If the diagonal and the side of the square the side of the square are commensurable, are commensurable, then they can each be then they can each be measured by some measured by some common unit.common unit.Suppose we choose Suppose we choose the the largest common largest common unit of lengthunit of length that that goes exactly into goes exactly into both.both.

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Proving Incommensurability, 2Proving Incommensurability, 2Call the number of times Call the number of times the measuring unit fits on the measuring unit fits on the diagonal the diagonal hh and the and the number of time it fits on number of time it fits on the side of the square the side of the square aa..

It cannot be that It cannot be that aa and and hhare both even numbers, are both even numbers, because if they were, a because if they were, a larger unit (twice the larger unit (twice the size) would have fit size) would have fit exactly into both the exactly into both the diagonal and the side.diagonal and the side.

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Proving Incommensurability, 3Proving Incommensurability, 3

By the Pythagorean By the Pythagorean theorem, theorem, aa22 + a+ a22 = h= h22

If 2If 2aa22 = h= h22 then then hh22

must be even.must be even.If If hh22 is even, so is is even, so is hh..Therefore Therefore aa must be must be odd. (Since they odd. (Since they cannot both be even.)cannot both be even.)

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Proving Incommensurability, 4Proving Incommensurability, 4Since Since hh is even, it is equal is even, it is equal to 2 times some number, to 2 times some number, j. j. So So hh = 2= 2jj. Substitute . Substitute 22jj for for hh in the formula in the formula given by the Pythagorean given by the Pythagorean theorem:theorem:22aa22 = h= h22 = (2= (2jj))22 = 4= 4jj22..If 2If 2aa22 = 4= 4jj22., then ., then aa22 = = 22jj22

Therefore Therefore aa22 is even, and is even, and so is so is aa..But we have already But we have already shown that shown that aa is odd.is odd.

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Proof by ContradictionProof by Contradiction

This proof is typical of the use of logic, as This proof is typical of the use of logic, as championed by Parmenides, to sort what championed by Parmenides, to sort what is true and what is false into separate is true and what is false into separate categories.categories.It is the cornerstone of Greek It is the cornerstone of Greek mathematical reasoning, and also is used mathematical reasoning, and also is used throughout ancient reasoning about throughout ancient reasoning about nature.nature.

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The Method of Proof by The Method of Proof by ContradictionContradiction

1. Assume the opposite of what you wish 1. Assume the opposite of what you wish to prove:to prove:

Assume that the diagonal and the side are Assume that the diagonal and the side are commensurable, meaning that at least one commensurable, meaning that at least one unit of length exists that exactly measures unit of length exists that exactly measures each.each.

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The Method of Proof by The Method of Proof by ContradictionContradiction

2. Show that valid reasoning from that premise 2. Show that valid reasoning from that premise leads to a logical contradiction.leads to a logical contradiction.

That the length of the side of the square must be That the length of the side of the square must be both an odd number of units and an even number of both an odd number of units and an even number of units.units.

Since a number cannot be both odd and even, Since a number cannot be both odd and even, something must be wrong in the argument. something must be wrong in the argument.

The only thing that could be wrong is the assumption The only thing that could be wrong is the assumption that the lengths are commensurable.that the lengths are commensurable.

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The Method of Proof by The Method of Proof by ContradictionContradiction

3. Therefore the opposite of the 3. Therefore the opposite of the assumption must be true. assumption must be true.

If the only assumption was that the two If the only assumption was that the two lengths are commensurable and that is false, lengths are commensurable and that is false, then it must be the case that the lengths are then it must be the case that the lengths are incommensurable.incommensurable.Note that the conclusion logically follows even Note that the conclusion logically follows even though at no point were any of the possible though at no point were any of the possible units of measure specified.units of measure specified.

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The Flaw of The Flaw of PythagoreanismPythagoreanism

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

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The Decline of the PythagoreansThe Decline of the Pythagoreans

The incommensurability of the diagonal The incommensurability of the diagonal and side of a square sowed a seed of and side of a square sowed a seed of doubt in the minds of Pythagoreans.doubt in the minds of Pythagoreans.They became more defensive, more They became more defensive, more secretive, and less influential.secretive, and less influential.But they never quite died out.But they never quite died out.