SC/STS 3760, II 1 Greece. 2 The Origins of Scientific Thinking? Greece is often cited as the place where the first inklings of modern scientific thinking.

SC/STS 3760, IISC/STS 3760, II11

GreeceGreece


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

Greece is often cited as the place Greece is often cited as the place where the first inklings of modern where the first inklings of modern scientific thinking took place.scientific thinking took place.

Why there and not elsewhere?Why there and not elsewhere? Einstein’s answer:Einstein’s answer:

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


The Origins of Ancient GreeceThe Origins of Ancient Greece

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


The Aegean CivilizationsThe Aegean Civilizations

There have been civilizations in the There have been civilizations in the Aegean area almost as long as there Aegean area almost as long as there have been in Mesopotamia and Egypt.have 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 – 1450 BCE.Existed from about 3000 – 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.


The Mycenaean CivilizationThe Mycenaean Civilization

On the Peloponnesus (the southern On the Peloponnesus (the southern mainland) another civilization arose mainland) another civilization arose and flourished from about 1600-1200 and flourished from about 1600-1200 BCE.BCE.

The Mycenaeans adapted the Minoan The Mycenaeans adapted the Minoan writing system to their own writing system to their own language, Greek. But it was awkward language, Greek. But it was awkward to use.to use.


MycenaeaMycenaea

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

Agamemnon’s Palace


The Trojan WarThe Trojan War


The Trojan WarThe Trojan War

Approx. 1280 – 1180 BCE.Approx. 1280 – 1180 BCE. Mycenaea versus Troy.Mycenaea versus Troy. Won by the Greeks, but the war Won by the Greeks, but the war

depleted their fighting forces.depleted their fighting forces. Mycenaea was invaded by Dorians Mycenaea was invaded by Dorians

about 1200 BCE, and its culture about 1200 BCE, and its culture destroyed.destroyed.


The Dark Age of GreeceThe Dark Age of Greece

1200 – 800 BCE1200 – 800 BCE The organized Greek civilization was The organized Greek civilization was

destroyed by the invading Dorians.destroyed by the invading Dorians. 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 What they had in common was

spoken Greek and memories of past spoken Greek and memories of past greatness.greatness.


PhoeniciaPhoenicia

Around 1700 BCE, in the Near East, Around 1700 BCE, in the Near East, what is now Lebanon, a civilization what is now Lebanon, a civilization developed with both Mesopotamian developed with both Mesopotamian and Egyptian influences.and Egyptian influences.

The Greeks later called the people The Greeks later called the people from there “Phonecians” – meaning from there “Phonecians” – meaning traders in purple.traders in purple.


Phoenician WritingPhoenician Writing

Phoenicians developed a style of Phoenicians developed a style of writing that combined Mesopotamian writing that combined Mesopotamian cuneiform and Egyptian heiratic.cuneiform and Egyptian heiratic.

It had 22 distinct characters, each It had 22 distinct characters, each representing a particular sound (a representing a particular sound (a consonant).consonant).


The Phoenician AlphabetThe Phoenician Alphabet


The Phoenician Alphabetic was The Phoenician Alphabetic was PhoneticPhonetic

Since each character represented a Since each character represented a sound, rather than a meaning, the sound, rather than a meaning, the characters could be used to characters could be used to represent words in an entirely represent words in an entirely different language.different language.

The Greeks adapted the Phoenician The Greeks adapted the Phoenician script to their own language and script to their own language and produced an produced an alphabetalphabet..


The Homeric AgeThe Homeric Age 800 – 600 BCE800 – 600 BCE The Greek verbal culture could be The Greek verbal culture could be

written down.written down. The heroic stories of the Trojan WarThe heroic stories of the Trojan War

were written by Homer. were written by Homer. The Iliad, The OdysseyThe Iliad, The Odyssey

Greek mythology and folk knowledge Greek mythology and folk knowledge were recorded by Hesiod.were recorded by Hesiod. Theogony, Works and DaysTheogony, Works and Days


The Greek Civilization Takes OffThe Greek Civilization Takes Off

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

Independent governments arose all Independent governments arose all across the Greek settlements.across the 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 common language, religion, and literature.literature.


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 human-like. – The Greek gods were too human-like. LanguageLanguage – Phonetic alphabet encouraged – Phonetic alphabet encouraged

literacy.literacy. TradeTrade – The Greeks became traders and – The Greeks became traders and

travellers, bringing home new ideas.travellers, bringing home new ideas. DemocracyDemocracy – Democratic governments, where – Democratic governments, where

they existed, encouraged independent thought.they existed, encouraged independent thought. SlaverySlavery – Greeks (like many other cultures) had – Greeks (like many other cultures) had

slaves who did the menial work.slaves who did the menial work.


The Pre-SocraticsThe Pre-Socratics

Thinkers living between about 600 – Thinkers living between about 600 – 450 BCE.450 BCE.

So named because they (basically) So named because they (basically) predated Socrates.predated Socrates.

Known only through discussions of Known only through discussions of their thoughts in later works.their thoughts in later works.

Some fragments still exist.Some fragments still exist.


SocratesSocrates

Lived in Athens, 470-399 BCE.Lived in Athens, 470-399 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

“Socratic” method.“Socratic” method.


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 Socrates’ own It is hard to distinguish Socrates’ own

thought from Plato’s.thought from Plato’s. Socrates is an important figure in the Socrates is an important figure in the

development of scientific reasoning, development of scientific reasoning, but…but…

He had no interest in the natural world.He had no interest in the natural world.


Back to the Pre-SocraticsBack to the Pre-Socratics

Most Pre-Most 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 the Aegean Sea known as Sea known as Ionia.Ionia. This is now This is now

part of Turkey.part of Turkey.


Wondering about NatureWondering about Nature

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

justify their answers.justify their answers.


Thales of MiletosThales of Miletos 625-545 BCE625-545 BCE Phoenician parents?Phoenician parents? Stories:Stories:

Predicted solar eclipse Predicted solar eclipse of May 28, 585 BCEof May 28, 585 BCE

Falling into a wellFalling into a well Olive pressOlive press

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


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 If two straight lines intersect, opposite angles intersect, opposite angles are equal.are equal.


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 angle from the ship and place a marker there.a 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 At B, turn at a right angle away from the shore and away from the shore and walk until the marker at C walk until the marker at C and the ship are in a straight and the ship are in a straight line. Call that A’.line. Call that A’.

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


Anaximander of Anaximander of MiletosMiletos

611-547 BCE611-547 BCE Student of Thales?Student of Thales? Map of the known Map of the known

worldworld ApeironApeiron (the (the

Boundless)Boundless) The basic stuff of The basic stuff of

the worldthe world


Anaximenes of MiletosAnaximenes of Miletos 550-475 BCE550-475 BCE Student of Student of

Anaximander?Anaximander? AirAir – the – the

fundamental stufffundamental stuff Cosmological view:Cosmological view:

Crystalline sphere of Crystalline sphere of the fixed starsthe fixed stars

Earth in centre, Earth in centre, planets betweenplanets between


Heraclitos of EphesusHeraclitos of Ephesus

Ephesus is 50 km N Ephesus is 50 km N of Miletos.of Miletos.

550?-475? BCE (i.e., 550?-475? BCE (i.e., about the same as about the same as Anaximenes, but Anaximenes, but uncertain)uncertain)

Everything is Flux.Everything is Flux. Fire fundamentalFire fundamental "You can't step in "You can't step in

the same river the same river twice."twice."


EleaElea

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

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


Parmenides of EleaParmenides of Elea 510-??510-?? Student of the exiled Student of the exiled

XenophanesXenophanes The goal of philosophy The goal of philosophy

is to attain the truth.is 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 appearance from reality.reality. Nature is comprehensible Nature is comprehensible

and logical.and logical.


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 isn’t!isn’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 is

nothing.nothing.


Parmenides against HeraclitosParmenides against Heraclitos

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

universe is everywhere full and universe is everywhere full and

occupied.occupied.

Therefore nothing actually changes.Therefore nothing actually changes.

Therefore motion is impossible.Therefore motion is impossible.


The Fundamental Problem of The Fundamental Problem of ViewpointViewpoint

Focus on the whole – ParmenidesFocus on the whole – Parmenides Easier 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 – HeraclitosFocus on the parts – Heraclitos

Easier to explain changes as Easier to explain changes as rearrangements of the parts.rearrangements of the parts.

Difficult to make sense of all that is.Difficult to make sense of all that is.


The Perils of LogicThe Perils of Logic

Reasoning with logic inevitably Reasoning with logic inevitably begins with assumed premises, begins with assumed premises, which may or may not be true.which may or may not be true.

The reasoning itself may or may not The reasoning itself may or may not be valid – though this can be be valid – though this can be checked.checked.

The truth of conclusions depends on The truth of conclusions depends on the truth of the premises and the the truth of the premises and the validity of the argument.validity of the argument.


Zeno of EleaZeno of Elea

495-425 BCE495-425 BCE Student of Student of

ParmenidesParmenides Probably moved to Probably moved to

Athens later and Athens later and taught there, making taught there, making his and Parmedies’ his and Parmedies’ views better known.views better known.


Zeno’s ParadoxesZeno’s Paradoxes

Paradox, from the Greek meaning Paradox, from the Greek meaning “contrary to opinion.”“contrary to opinion.”

Showed that logic can lead to Showed that logic can lead to conclusions which defy common conclusions which defy common sense.sense. Hard to say whether he was attacking Hard to say whether he was attacking

common sense beliefs (as seems common sense beliefs (as seems probable), or demonstrating the dangers probable), or demonstrating the dangers of reasoning by logical deduction.of reasoning by logical deduction.


The StadiumThe Stadium Consider a Consider a

stadiumstadium—a —a running running track of track of about 180 about 180 meters in meters in ancient ancient Greece.Greece.


The StadiumThe Stadium

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


The Stadium ParadoxThe Stadium Paradox

Before the runner can reach the finish line, the mid-Before the runner can reach the finish line, the mid-point must be reached.point must be reached.

Before that, the ¼ point. Before that 1/8, 1/16, 1/32, Before that, the ¼ point. Before that 1/8, 1/16, 1/32, 1/64,… and an infinite number of prior events.1/64,… and an infinite number of prior events.

The runner never can leave the starting block.The runner never can leave the starting block.


Achilles and the TortoiseAchilles and the Tortoise

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

Achilles gives the tortoise a head start.Achilles gives the tortoise a head start.


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 tortoise’s Call when Achilles reaches the tortoise’s

starting position t=1starting position t=1 By then, the tortoise has gone ahead.By then, the tortoise has gone ahead.



Now at time t=1, Achilles still must reach where Now at time t=1, Achilles still must reach where the tortoise is before he can pass it.the tortoise is before he can pass it.

Every time Achilles reaches where the tortoise Every time Achilles reaches where the tortoise had been, the tortoise is further ahead.had been, the tortoise is further ahead.

The tortoise must win the race.The tortoise must win the race.



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








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 At any single moment, the arrow is in a

single place, therefore, not moving.single place, therefore, not moving.


The Flying Arrow, 2The Flying Arrow, 2

At every moment of its flight, the arrow is At every moment of its flight, the arrow is not moving. If it were, it would occupy more not moving. If it were, it would occupy more space that it does, which is impossible.space that it does, which is impossible.

There is no such thing as motion.There is no such thing as motion.


Pythagoras of SamosPythagoras of Samos

Born between 580 Born between 580 and 569. Died and 569. Died about 500 BCE.about 500 BCE.

Lived in Samos, an Lived in Samos, an island off the coast island off the coast of Ionia.of Ionia.


Pythagoras and the PythagoreansPythagoras and the Pythagoreans

Pythagoras himself lived earlier than Pythagoras himself lived earlier than many of the other Pre-Socratics and many of the other Pre-Socratics and had some influence on them:had some influence on them: E.g., Heraclitos, Parmenides, and ZenoE.g., Heraclitos, Parmenides, and Zeno

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


The Pythagorean CultThe Pythagorean Cult

The followers of Pythagoras were a The followers of Pythagoras were a close-knit group like a religious cult.close-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:

VegetarianVegetarian Ate no beans.Ate no beans.


Everything is NumberEverything is Number

The Pythagoreans viewed number as The Pythagoreans viewed number as the underlying structure of the underlying structure of everything in the universe.everything in the universe. Compare to Thales’ view of water, Compare to Thales’ view of water,

Anaximander’s Anaximander’s apeironapeiron, Anaximenes’ , Anaximenes’ air, Heraclitos, change.air, Heraclitos, change.

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


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 = octave 2:3 = perfect fifth2:3 = perfect fifth 3:4 = perfect fourth3:4 = perfect fourth


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

Relative string lengths for notes of the scale from lowest Relative string lengths for notes of the scale from lowest note (bottom) to highest. note (bottom) to highest.

The octave higher is half the length of the former. The The octave higher is half the length of the former. The fourth is ¾, the fifth is 2/3.fourth is ¾, the fifth is 2/3.


Geometric HarmonyGeometric Harmony

The numbers 12, 8, 6 represent the The numbers 12, 8, 6 represent the lengths of a ground note, the fifth lengths of a ground note, the fifth above, and the octave above the ground above, and the octave above the ground note.note. Hence these numbers form a “harmonic Hence these numbers form a “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 “geometric Fantastic! A cube is in “geometric

harmony.”harmony.”


Figurate NumbersFigurate Numbers

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


The TetractysThe Tetractys

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

Called the tetrad or Called the tetrad or tetractys.tetractys.


The significance of the TetractysThe significance of the Tetractys

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

Tetractys, which contains the fount and Tetractys, which contains the fount and root of eternal nature.”root of eternal nature.”


Pythagorean CosmologyPythagorean Cosmology

Unlike almost every other ancient Unlike almost every other ancient thinker, the Pythagoreans did not thinker, the Pythagoreans did not place the Earth at the centre of the place the Earth at the centre of the universe.universe.

The Earth was too imperfect for such The Earth was too imperfect for such a noble position.a noble position.

Instead the centre was the “Central Instead the centre was the “Central Fire” or, the watchtower of Zeus.Fire” or, the watchtower of Zeus.


The The Pythagorean Pythagorean

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


The Pythagorean Cosmos and the TetractysThe Pythagorean Cosmos and the Tetractys

To match the To match the tetractys, tetractys, 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 the central fire, and fire, and invisible to invisible to human eyes.human eyes.


The Pythagorean TheoremThe Pythagorean Theorem


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

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

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


Well-known Special CasesWell-known Special Cases

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

The best known special cases areThe best known special cases are The 3-4-5 triangle: 3The 3-4-5 triangle: 322+4+422=5=522 or 9+16=25 or 9+16=25 The 5-12-13 triangle: 5The 5-12-13 triangle: 522+12+1222=13=1322 or or

25+144=16925+144=169


CommensurabilityCommensurability

Essential to the Pythagorean view Essential to the Pythagorean view that everything is ultimately number that everything is ultimately number is the notion that the same scale of is the notion that the same scale of measurement can be used for measurement can be used for everything.everything.

E.g., for length, the same ruler, E.g., for length, the same ruler, perhaps divided into smaller and perhaps divided into smaller and smaller units, will ultimately measure smaller units, will ultimately measure every possible length exactly.every possible length exactly.

This is called This is called commensurabilitycommensurability..


Commensurable NumbersCommensurable Numbers

Numbers, for the Pythagoreans, mean Numbers, for the Pythagoreans, mean the natural, counting numbers.the natural, counting numbers.

All natural numbers are commensurable All natural numbers are commensurable because the can all be “measured” by because the can all be “measured” by the same unit, namely 1. the 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.


Commensurable MagnitudesCommensurable Magnitudes

A magnitude is a measurable A magnitude is a measurable quantity, for example, length.quantity, for example, length.

Two magnitudes are commensurable Two magnitudes are commensurable if a common unit can be laid off to if a common unit can be laid off to measure each one exactly.measure each one exactly. E.g., two lengths of 36.2 cm and 171.3 E.g., two lengths of 36.2 cm and 171.3

cm are commensurable because each is cm are commensurable because each is an exact multiple of the unit of measure an exact multiple of the unit of measure 0.1 cm.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.


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, ultimately has a numerical structure, and numbers mean some tiny spherical and numbers mean some tiny spherical balls that occupy space, then balls that occupy space, then everything in the world is ultimately everything in the world is ultimately commensurable with everything else.commensurable with 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.


IncommensurabilityIncommensurability

The (inconceivable) opposite to The (inconceivable) opposite to commensurability is commensurability is incommensurability,incommensurability, the situation where no common measure the situation where no common measure between two quantities exists.between two quantities exists.

To prove that two quantities are To prove that two quantities are commensurable, one need only find a commensurable, one need only find a single common measure.single common measure.

To prove that quantities are To prove that quantities are inincommensurable, it would be necessary commensurable, it would be necessary to prove that no common measures could to prove that no common measures could possibly exist. possibly exist.


The Diagonal of the SquareThe Diagonal of the Square

The downfall of the The downfall of the Pythagorean world Pythagorean world view came out of view came out of their greatest their greatest triumph the triumph the Pythagorean Pythagorean theorem.theorem.

Consider the Consider the simplest case, the simplest case, the right triangles right triangles formed by the formed by the diagonal of a square. diagonal of a square.


Proving IncommensurabilityProving Incommensurability

If the diagonal and If the diagonal and the side of the the side of the square are square are commensurable, commensurable, then they can each then they can each be measured by be measured by some common unit.some common unit.

Suppose we choose Suppose we choose the the largest largest common unit of common unit of lengthlength that goes that goes exactly into both.exactly into both.


Proving Incommensurability, 2Proving Incommensurability, 2 Call the number of Call the number of

times the measuring times the measuring unit fits on the diagonal unit fits on the diagonal hh and the number of and the number of time it fits on the side time it fits on the side of the square of the square aa..

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


Proving Incommensurability, 3Proving Incommensurability, 3

By the Pythagorean By the Pythagorean theorem, theorem, aa22 + a + a22 = = hh22

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 must

be odd. (Since they be odd. (Since they cannot both be cannot both be even.)even.)


Proving Incommensurability, 4Proving Incommensurability, 4 Since Since hh is even, it is is even, it is

equal to 2 times some equal to 2 times some number, number, j. j. So So hh = 2 = 2jj. . Substitute 2Substitute 2jj for for hh in the in the formula given by the formula given by the Pythagorean theorem:Pythagorean theorem:

22aa22 = h = h22 = (2 = (2jj))22 = 4 = 4jj22.. If 2If 2aa22 = 4= 4jj22., then ., then aa22 = =

22jj22 Therefore Therefore aa22 is even, is even,

and so is and so is aa.. But we have already But we have already

shown that shown that aa is odd. is odd.


Proof by ContradictionProof by Contradiction

This proof is typical of the use of This proof is typical of the use of logic, as championed by Parmenides, logic, as championed by Parmenides, to sort what is true and what is false to sort what is true and what is false into separate categories.into separate categories.

It is the cornerstone of Greek It is the cornerstone of Greek mathematical reasoning, and also is mathematical reasoning, and also is used throughout ancient reasoning used throughout ancient reasoning about nature.about nature.


The Method of Proof by The Method of Proof by ContradictionContradiction

1. Assume the opposite of what you 1. Assume the opposite of what you wish to prove:wish to prove: Assume that the diagonal and the side Assume that the diagonal and the side

are commensurable, meaning that at are commensurable, meaning that at least one unit of length exists that least one unit of length exists that exactly measures each.exactly measures each.



2. Show that valid reasoning from that 2. Show that valid reasoning from that premise leads to a logical contradiction.premise leads to a logical contradiction. That the length of the side of the square must That the length of the side of the square must

be both an odd number of units and an even be both an odd number of units and an even

number of units.number of 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 The only thing that could be wrong is the

assumption that the lengths are assumption that the lengths are

commensurable.commensurable.



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 lengths are commensurable and that is false, then it must be the case that the false, then it must be the case that the lengths are incommensurable.lengths are incommensurable.

Note that the conclusion logically follows Note that the conclusion logically follows even though at no point were any of the even though at no point were any of the possible units of measure specified.possible units of measure specified.


The Flaw of PythagoreanismThe Flaw of Pythagoreanism

The Pythagorean world view – that The Pythagorean world view – that everything that exists is ultimately a everything that exists is ultimately a numerical structure (and that numerical structure (and that numbers mean just counting numbers mean just counting numbers—integers).numbers—integers).

In their greatest triumph, the magical In their greatest triumph, the magical Pythagorean theorem, lay a case Pythagorean theorem, lay a case that cannot fit this world view.that cannot fit this world view.


The Decline of the PythagoreansThe Decline of the Pythagoreans

The incommensurability of the The incommensurability of the diagonal and side of a square sowed diagonal and side of a square sowed a seed of doubt in the minds of a seed of doubt in the minds of Pythagoreans.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.

SC/STS 3760, II 1 Greece. 2 The Origins of Scientific Thinking? Greece is often cited as the place where the first inklings of modern scientific thinking.

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