Transcript X - York University

Greece
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The Origins of Scientific Thinking?
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
Greece is often cited as the place where
the first inklings of modern scientific
thinking took place.
Why there and not elsewhere?
Einstein’s 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

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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.
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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

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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.
Agamemnon’s Palace
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The Trojan War
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The Trojan War
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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
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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

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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

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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
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800 – 600 BCE
The Greek verbal culture could be written
down.
The heroic stories of the Trojan War
were written by Homer.

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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
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The first Olympic Games 776 BCE
The Polis (City-State)
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Independent governments arose all across the
Greek settlements.
Experimentation in forms of government:
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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.
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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

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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
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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.
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Socrates left no writings at all.
He is known to us primarily through the
works of Plato.

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It is hard to distinguish Socrates’ own thought
from Plato’s.
Socrates is an important figure in the
development of scientific reasoning, but…
He had no interest in the natural world.
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Back to the Pre-Socratics

Most PreSocratics 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
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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
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625-545 BCE
Phoenician parents?
Stories:
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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:
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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
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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
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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
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550-475 BCE
Student of
Anaximander?
Air – the fundamental
stuff
Cosmological view:
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Crystalline sphere of the
fixed stars
Earth in centre, planets
between
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Heraclitos of Ephesus
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Ephesus is 50 km N of
Miletos.
550?-475? BCE (i.e.,
about the same as
Anaximenes, but
uncertain)
Everything is Flux.
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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
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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
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Something either is or it is not.
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The law of the excluded middle
Therefore, nothing is that isn’t!
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It is impossible to be not being
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There is no such thing as empty space.
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Space is something and empty is nothing.
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Parmenides against Heraclitos
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If there is no space that is empty, the
universe is everywhere full and occupied.
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Therefore nothing actually changes.
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Therefore motion is impossible.
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The Fundamental Problem of
Viewpoint
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Focus on the whole – Parmenides
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Easier to grasp the unity of the world.
Difficult to explain processes, events,
changes.
Focus on the parts – Heraclitos
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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
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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
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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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Zeno’s Paradoxes

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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
stadium
—a running
track of
about 180
meters in
ancient
Greece.
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The Stadium
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Will the runner reach the other side of the
stadium?
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The Stadium Paradox
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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
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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
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Call the starting time t=0.
Before Achilles can pass the tortoise, he must
reach where the tortoise was at the start.
Call when Achilles reaches the tortoise’s starting
position t=1
By then, the tortoise has gone ahead.
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Achilles and the Tortoise, 3
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Now at time t=1, Achilles still must reach where the
tortoise is before he can pass it.
Every time Achilles reaches where the tortoise had been,
the tortoise is further ahead.
The tortoise must win the race.
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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
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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
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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

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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
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Pythagoras himself lived earlier than many
of the other Pre-Socratics and had some
influence on them:
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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
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The followers of Pythagoras were a closeknit group like a religious cult.
Vows of poverty.
Secrecy.
Special dress, went barefoot.
Strict diet:
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Vegetarian
Ate no beans.
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Everything is Number
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The Pythagoreans viewed number as the
underlying structure of everything in the
universe.
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Compare to Thales’ view of water,
Anaximander’s apeiron, Anaximenes’ air,
Heraclitos, change.
Pythagorean numbers take up space.

Like little hard spheres.
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Numbers and Music
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One of the discoveries attributed to
Pythagoras himself.
Musical scale:
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1:2 = octave
2:3 = perfect fifth
3:4 = perfect fourth
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Numbers and Music, contd.
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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
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The numbers 12, 8, 6 represent the
lengths of a ground note, the fifth above,
and the octave above the ground note.
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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
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Numbers that can be arranged to form a regular
figure (triangle, square, hexagon, etc.) are called
figurate numbers.
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The Tetractys
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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
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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
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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
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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
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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
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The 3-4-5 triangle: 32+42=52 or 9+16=25
The 5-12-13 triangle: 52+122=132 or
25+144=169
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Commensurability
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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
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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.

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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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.

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
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
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3. Therefore the opposite of the
assumption must be true.

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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
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The Pythagorean world view – that
everything that exists is ultimately a
numerical structure (and that numbers
mean just counting numbers—integers).
In their greatest triumph, the magical
Pythagorean theorem, lay a case that
cannot fit this world view.
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The Decline of the Pythagoreans
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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.
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