Lecture 4, MATH 210G.01, Fall 2012 Greek Mathematics and Philosophy

1. Lecture 4, MATH 210G.01, Fall 2012Greek 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

2. Thales (624-547 BC): father of mathematical proof

3. 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

4. Pythagoras • (c. 580-500 BC)

5. 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 • True • B) False

6. 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

7. 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.

8. discovery that music was based on proportional intervals of numbers 1—4 • 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”

9. "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"

10. 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

11. 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 al • 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

12. Music of the Spheres

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

15. The Pythagorean Theorem

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

17. Plato Plato (428 BC – 348 BC),

18. Plato’s Cave Analogy

20. 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

21. Plato (left) and Aristotle (right)

22. 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.

23. 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

24. 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.

27. 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.””

28. 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.

29. 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.

31. 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.

32. Non-Euclidean geometries2 Non-Euclidean geometries 1 Non-Euclidean geometries 3

33. 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

34. Archimedes Possibly the greatest mathematician ever; Theoretical and practical

35. Other cultures • Avicenna (980-1037): propositional logic ~ risk analysis • Parallels in India, China, • Medieval (1200-1600) • Occam (1288-1347)