Day 9, Mat305/594, History of Mathematics

Last time: Section 3.1 Next time: Section 3.3

Today:

Announcements
- Your homework sections 2.3 and 2.4 is due today.
- Worksheet for section 3.3 (for next time).
Today:
- Wrapping up section 3.1 with an experiment (can I have your Thales sheets, just to inspect?)
- Commercial (Adam); Tom for Monday. Any volunteers for Wednesday, next week?
- Lessons from section 3.2
Mathematical Elements, lessons and problems from section 3.1: the Beginnings of Greek Mathematics
- Into Greece.... (source)
- The Greeks:
  - travellers/migrants (p. 86)
  - colonizers (colonies were "conduits through which Greek culture flowed to the world of the "Barbarians"....) (p. 86)
  - Mathematics came from the colonies, not from Mainland Greece (p. 86)
  - The Greeks took the Phoenician alphabet, and swapped some consonents for vowels. The Ionian variant was adopted "universally" in 403 BCE. (p. 86)
  - Greeks of Asia Minor introduced coinage in 700 BCE.
  - Their education helped create "gentleman amateurs" (rather than arcane priesthoods): education was for the masses. So was their language.
  - "uninhibited": they had "no sacred writings or rigid dogmas that required the mind's subservience." (p. 87)
  - Greeks cities worked as individuals. The city-states fought together, but could not easily be unified under a single ruler.
  - Philip II of Macedonia finally unified Greece by defeating its combined forces at Chaeronea in 338BCE.
  - Then Philip's son Alexander the Great (356-323 BCE) finished the job, and carried Greek civilization to the ends of the known world.
  - The time from "...Alexander the Great into the first century B.C. ... formed a brilliant period of history known ... as the Hellenistic Age." (p. 88)
  - Greek mathematics reached its zenith with Euclid, Archimedes, and Appolonius.
- Who was Thales of Miletus?
  - Circa 625-547 B.C.
  - First of the Seven Sages of Greece
  - Credited with the maxim "Know thyself."
  - Strangest thing he had ever seen: "An aged tyrant."
  - Aesop's Mule (originally told by Plutarch -- search for Thales)
  - Olive oil, tripping and falling, yada yada....
  - Credited with heightening deductive reasoning in mathematics.
  - Mathematics:
    - See page 89. Do page 89!
    - May have travelled to Egypt (knew Egyptian mathematics, at any rate), and credited with calculating the height of the Great Pyramid of Cheops.
    - Credited with the method for calculating the distance of a ship from shore. Let's see if we can use this method to calculate the distance of an object on campus!
A few tidbits from Kline's Mathematical Thought from Ancient to Modern Times:
- "...in the history of mathematics the Greeks are the surpreme event."
- "One of the great problems of the history of civilization is how to account for the brilliance and creativity of the ancient Greeks." (p. 24)
- Greeks had much commerce with the Egyptians and Babylonians.
- In the time of Thales "Miletus was a great and wealthy trading city on the Mediterranean."
- Gnomon -- an upright stick whose shadow was used to tell time. In Pythagoras's time, it meant a carpenter's square.
- Greek sources are very spotty: much of our knowledge is very suspect. "Our chief sources for the Greek mathematical works are Byzantine Greek codices (manuscript books) written 500 to 1500 years after the Greek works were originally composed."
- "The reconstruction of the history of Greek mathematics ... has been an enormous and complicated task. Despite the extensive efforts of scholars, there are gaps in our knowledge and some conclusions are arguable. Nevertheless the basic facts are clear." (p. 27)
Mathematical Elements, lessons and problems from section 3.2
- History and Geography
  - Pythagoras (approx 580-500 BCE): "little can be said with any certainty" (p. 92)
  - Pythagoras's trajectory: Samos ${\longrightarrow}$ Phoenicia ${\longrightarrow}$ Egypt ${\longrightarrow}$ Crotona
  - Education
    - The seven liberal arts
      - Quadrivium:
        
        arithmetica
        harmonia (music)
        geometria
        astrologia
      - Trivium (added in the Middle Ages):
        
        logic
        grammar
        rhetoric
    - School of Crotona
      - Students divided into
        
        acoustici -- "listeners" -- novices (three years)
        mathematici -- inner circle
      - Women were allowed (including Theano -- wife, child, student?)
      - Political, philosophical, religious
      - Secrecy, honor, virtue valued highly
      - Strict taboos, dietary restrictions, etc.
    - Pythagoreans persisted for centuries even after Pythagoras's violent end.
  - Nicomachus of Gerasa
  - Zeno of Elea
- Mathematics
  - Dodecahedron -- Platonic solids
  - Theano
  - tetractys -- "holy fourfoldness" (fire, water, air, earth)
  - pentagram, sign of the secret society
  - mathematics is the greatest purification
  - everything is number ( ${m\in{N}$ , the natural numbers)
  - "music of the spheres" -- the "seven known planets" (including the moon and sun, but not including the earth, of course!;) travel through space along spheres, generating tones that only Pythagoras can hear.
  - number mysticism:
    1. reason
    2. man
    3. women
    4. justice (2*2, the first product of equals -- ignoring reason, which is itself just, evidently!;)
    5. marriage (2+3)
    6. even (feminine and earthy -- after 2 of course!)
    7. odd (masculine and divine -- after 3 of course!)
  - Nicomachus's Introductio Arithmeticae
    - "Despite a lack of originality and a mathematical poverty..."
    - it "... became a leading textbook in the Latin West from the time it was written until the 1500s." (p. 96)
    - Figurative numbers
      - triangular
      - square
      - (but also pentagonal, hexagonal, etc.)
      - Some results:
        
        ${s_n=t_{n}+t_{n-1}}$
        
        ${t_{n}=\frac{n(n+1)}{2}}$
        
        ${\sum_{i=1}^n(2i-1)=s_n}$
        
        ${\sum_{i=1}^ni^3=t_n^2}$ (proof by induction)
  - Zeno of Elea, and the Eleatic school (circa 450 BCE)
    - "Zeno pointed out the logical absurdities arising from the concept of 'infinite divisibility' of time and space." (p. 104)
    - Four paradoxes
      - Achilles and the tortoise:
        A fleet-of-foot Achilles is unable to catch a plodding tortoise which has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward some distance.
      - The dichotomy:
        Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled.
      - The arrow:
        An arrow in flight has an instantaneous position at a given instant of time. At that instant, however, it is indistinguishable from a motionless arrow in the same position, so how is the motion of the arrow perceived?
      - Stade paradox:
        Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time. (source)
    - We love the famous stories (Zeno and Diogenes), even when they're false....
    - Led to the Greeks' "horror of the infinite"
Mathematical Elements, lessons and problems from section 3.3

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