Lecture 4, MATH 210G.02, Fall 2017 Greek Mathematics and Philosophy Goals: Learn a few of the theorems proved by Greek mathematicians Understand some of the Greek philosophy in which mathematics was developed

Lecture 4, MATH 210G.02, Fall 2015 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 windmill 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 True 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

The diatonic: ratio of highest to lowest pitch is 2:1, B 1 9/8 81/64 4/3 3/2 27/16 243/128 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. https://en.wikipedia.org/wiki/Pythagorean_tuning

Believed the number system … and universe… based on their sum (10) … swore by the “Tetractys” rather than by the gods. Odd numbers were masculine and even were feminine. Hippasos …discovered irrational numbers…was executed. Hints of “heliocentric theory” discovery that music was based on proportional intervals of numbers 1—4

"Bless us, divine number, thou who generated gods and men "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 all-comprising, all-bounding, the first-born, the never-swerving, 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

Music of the Spheres

…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 … believed… when someone was "in doubt as to what he should say, he should always remain silent” 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

The square root of two is a rational number (the ratio of two whole numbers) True False

The Pythagorean Theorem

The Pythagorean Theorem

Which of the two diagrams provide “visual proof” of the Pythagoran theorem? Left diagram only B) Right diagram only C) Both diagrams together

Plato (428 BC – 348 BC),

Plato’s Cave Analogy

Note ratios: AB:BC :: CD:DE:: AC:CE

In Plato’s Divided Line, Mathematics falls under the following category: Highest form of true knowledge Second highest form of true knowledge A form of belief, but not true knowledge 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 "non-Euclidean 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.

A visual paradox…

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 "non-Euclidean geometries" could be created in which the parallel postulate did not hold.

Non-Euclidean geometries2

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)

Exercises

Explain how the Pythagorean theorem follows from the picture using the formula for the area of a trapezoid

Explain how the Pythagorean theorem follows from the picture

Explain how the Pythagorean theorem follows from the picture

Advanced: Explain how the Pythagorean theorem follows from the picture Solution can be found here: Euclid's proof

Prove that the area of the big hexagon is the sum of the areas of the smaller ones

Solution: the trick is that the area of a (regular) hexagon is a fixed multiple of that of the square of its side. This area is 3sqrt(3)/2 s^2 since a hexagon is the union of six equilateral triangles which have area sqrt(3)/4 s^2

Assuming: the area of a semicircle of diameter d is Prove that the area of the big semicircle is the sum of the areas of the smaller ones

Solution: Given the hint, if the triangle has sides a,b,c then the areas of the semicircles are pi a^2/8, pi b^2/8 and pi c^2/8. Multiplying all terms by 8/pi and applying the Pythagorean theorem gives the result.

Some practice problems If a=3 and b=4, what is the length c of the hypotenuse of the triangle? c 3 4

Some practice problems Solution: by the Pythagorean theorem, c^2=9+16=25 so c=5 c 3 4

e If a=5, b=4, c=3, d=3, and e=√5, find f. d c f 3 b 4 a

e Solution: If a=5, b=4, c=3, d=3, and e=√5, find f. d c f 3 b 4 a

Explain the lengths of the sides of the Pythagorean spiral

Solution: Just apply the Pythagorean theorem successively to each successive triangle. The square of side of next hypotenuse is 1+square of side of previous hypotenuse.

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?

Solution: The ladder’s length of 10 feet is the length of the hypotenuse of a right triangle whose height is the height along the wall and whose base is the base along the ground. So 10^2 = 6^2+h^2 where h is the height of the wall, or h^2=64, h=9 How high is the wall?

TV screen size is measured diagonally across the screen 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?

Solution: W=(16/9)H where W is the width and H is the height of the TV Solution: W=(16/9)H where W is the width and H is the height of the TV. The square of the diagonal is W^2+H^2=W^2(1+(9/16)^2). If W^2\leq 34^2 then D\leq 34x sqrt(1+(9/16)^2)\approx 39

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).