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Lecture 4, MATH 210G.03, Spring 2013
Greek Mathematics and
Philosophy
• Period 1: 650 BC-400 BC (pre-Plato)
• Period 2: 400 BC – 300 BC (Plato, Euclid)
• Period 3: 300 BC – 200 BC (Archimedes,
Appolonius, Eratosthenes
Thales (624-547 BC): father of mathematical
proof
In the diagram, the ratio of
the segments AD and DB
is the same as the ratio
of the segments AE and
EC
A) True
B) False
Pythagoras
• (c. 580-500 BC)
In the diagram, the
area of the square
with side a plus the
area of the square
with side b equals
the area of the
square with side c
A) True
B) False
Pythagorean philosophy
☺Transmigration of souls,
☺purification rites; developed rules of living
believed would enable their soul to achieve a
higher rank among the gods.
☺Theory that numbers constitute the true
nature of things, including music
C
D
E
F
G
A
B
C
1
9/8
5/4
4/3
3/2
5/3
15/8
2
• The diatonic: ratio of highest to lowest pitch is 2:1,
• produces the interval of an octave.
• Octave in turn divided into fifth and fourth, with ratios 3:2
and 4:3 …
• up a fifth + up a fourth = up an octave.
• fifth … divided into three whole tones, each corresponding
to the ratio of 9:8 and a remainder with a ratio of 256:243
• fourth into two whole tones with same remainder.
• harmony… combination… of … ratios of numbers
• … whole cosmos … and individual do not arise by a chance
combinations … must be fitted together in a "pleasing"
(harmonic) way in accordance with number for an order to
arise.
π discovery that music was based on
proportional intervals of numbers 1—4
π Believed the number system … and universe…
based on their sum (10)
π … swore by the “Tetractys” rather than by the
gods.
π Odd numbers were masculine and even were
feminine.
π Hippasos …discovered irrational
numbers…was executed.
π Hints of “heliocentric theory”
• "Bless us, divine number,
thou who generated gods
and men! O holy, holy
Tetractys, thou that
containest the root and
source of the eternally
flowing creation! For the
divine number begins with
the profound, pure unity
until it comes to the holy
four; then it begets the
mother of all, the allcomprising, all-bounding,
the first-born, the neverswerving, the never-tiring
holy ten, the keyholder of
all"
Clicker question
• The number 10 is a perfect number, that is, it
is equal to the sum of all of the smaller whole
numbers that divide into it.
• A) True
• B) False
۞Pythagoreans … believed… when someone was "in
doubt as to what he should say, he should always
remain silent”
۞…it was better to learn none of the truth about
mathematics, God, and the universe at all than to learn
a little without learning all
۞Pythagoreans’ inner circle,“mathematikoi”
("mathematicians”); outer circle, “akousmatikoi”
("listeners”)
۞… the akousmatikoi were the exoteric disciples who…
listened to lectures that Pythagoras gave out loud from
behind a veil.
۞Pythagorean theory of numbers still debated among
scholars.
۞Pythagoras believed in "harmony of the spheres”…
that the planets and stars moved according to
mathematical equations, which corresponded to
musical notes and thus produced a symphony
Music of the Spheres
• The square root of two is a rational number
(the ratio of two whole numbers)
A) True
B) False
The Pythagorean Theorem
Which of the two diagrams provide “visual
proof” of the Pythagoran theorem?
A) Left diagram only
B) Right diagram only
C) Both diagrams
Plato Plato (428 BC – 348 BC),
Plato’s Cave Analogy
• In Plato’s Divided Line, Mathematics falls
under the following category:
A)Highest form of true knowledge
B)Second highest form of true knowledge
C)A form of belief, but not true knowledge
D)A form of perception
Plato (left) and Aristotle (right)
Aristotle (384 BC – 322 BC)
• Aristotle’s logic: the
syllogism
• Major premise: All humans
are mortal.
• Minor premise: Socrates is
a human.
• Conclusion: Socrates is
mortal.
Epictetus and The Stoics (c 300 BC)
Stoics believed … knowledge attained through use of reason… Truth
distinguishable from fallacy; *even if, in practice, only an
approximation can be made.
• Modality (potentiality vs actuality).
• Conditional statements. (if…then)
• Meaning and truth
Euclid’s “Elements”
arranged in order many of Eudoxus's theorems, perfected
many of Theaetetus's, and brought to irrefutable
demonstration theorems only loosely proved by his
predecessors
Ptolemy once asked him if there were a shorter way to
study geometry than the Elements, …
In his aim he was a Platonist, being in sympathy with this
philosophy, whence he made the end of the whole
"Elements" the construction of the so-called Platonic
figures.
The axiomatic method
• The Elements begins with definitions and five
postulates.
• There are also axioms which Euclid calls
'common notions'. These are not specific
geometrical properties but rather general
assumptions which allow mathematics to
proceed as a deductive science. For example:
“Things which are equal to the same thing are
equal to each other.””
Euclid's Postulates
A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one
endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles
on one side is less than two right angles, then the two lines inevitably must intersect each other
on that side if extended far enough. This postulate is equivalent to what is known as the parallel
postulate.
Euclid's fifth postulate cannot be proven from others, though attempted by many people.
Euclid used only 1—4 for the first 28 propositions of the Elements, but was forced to invoke the
parallel postulate on the 29th.
In 1823,Bolyai and Lobachevsky independently realized that entirely self-consistent "nonEuclidean geometries" could be created in which the parallel postulate did not hold.
Euclid's Postulates
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a
straight line.
3. Given any straight line segment, a circle can be drawn having
the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough. This postulate is
equivalent to what is known as the parallel postulate.
• Euclid's fifth postulate cannot be proven from
others, though attempted by many people.
• Euclid used only 1—4 for the first 28
propositions of the Elements, but was forced
to invoke the parallel postulate on the 29th.
• In 1823,Bolyai and Lobachevsky
independently realized that entirely selfconsistent "non-Euclidean geometries" could
be created in which the parallel postulate did
not hold.
Non-Euclidean geometries 1
Non-Euclidean geometries2
Non-Euclidean geometries 3
Clicker question
• Euclid’s fifth postulate, the “parallel
postulate” can be proven to be a consequence
of the other four postulates
• A) True
• B) False
Archimedes
Possibly the greatest
mathematician ever;
Theoretical and
practical
Other cultures
• Avicenna (980-1037): propositional logic ~ risk
analysis
• Parallels in India, China,
• Medieval (1200-1600)
• Occam (1288-1347)
Some practice problems
• If a=3 and b=4, what is the length c of the
hypotenuse of the triangle?
c
3
4
• If a=5, b=4, c=3, d=3, and e=√5, find f.
e
d
c
f
3
b
4
a
• A ladder is 10 feet long. When the top of the
ladder just touches the top a wall, the bottom
of the ladder is 6 feet from the wall.
• How high is the wall?
• TV screen size is measured diagonally across
the screen. A widescreen TV has an aspect
ratio of 16:9, meaning the ratio of its width to
its height is 16/9. Suppose that a TV has a one
inch boundary one each side of the screen. If
Joe has a cabinet that is 34 inches wide, what
is the largest size wide screen TV that he can
fit in the cabinet?
Advanced
• The spherical law of cosines states
that, on a spherical triangle. Cos (c/R)
= (cos a/R) (cos b/R) + (sin a/R) (sin
b/R) cos γ where R is the radius of the
sphere. If the Earth’s radius is
6,371 km, find the distance from:
• from Seattle (48°N, 2°E) to Paris
(48°N, 122°W) if traveling due
east?
• from Lincoln, NE (40°N, 96°W) to
Sydney, Australia (34°S, 151°E).
Each of the following pictures
provide a scheme to prove the
Pythagorean theorem. In ach case,
explain how the proof follows from
the picture.
Explain the figure using the
Pythagorean theorem