Day 10, Mat305/594, History of Mathematics

Last time: Section 3.2

Next time: Section 3.4/3.5

Today:

• Announcements
  • Your homework section 2.3 is returned today. Graded:
    • 1.a, d; 2.a, c; 4.a; 6.a; 10; 19 (6 and 7)
      • #4: think like an Egyptian!
      • #6:
        • you need to show this in general, not just for a specific value of n.
        • Some of you lost track of the point about the unit fractions.
      • #10: show work (or at least verify in general)
        • One found: (works for all n, whatever their divisibility!) -- see p. 44.
        • Two found:
        • Several found:
      • #19: punt level was high!
        • Since false position was called for, you should try to get it in there somehow!

  • I will have graded Section 2.4 by next time (1a, 3, 6, 9)

  • Worksheet for selections from sections 3.4/3.5 (for next time).
  • Thales pictures are on-line:
    • Check out your own, and
    • Check out your peers' versions!
    See if your results are consistent with others (and if your results are sufficiently general)

• Today:
  • Wrapping up section 3.2: Zeno
  • Commercial (Tom); who's up for Wednesday?
  • Lessons from section 3.3

• 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 Phoenicia Egypt Crotona
    • Education
      • The seven liberal arts
        • Quadrivium:
          1. arithmetica
          2. harmonia (music)
          3. geometria
          4. astrologia
        • Trivium (added in the Middle Ages):
          1. logic
          2. grammar
          3. 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 (, 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:
          • 

          • 

          • 

          • (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
  • The Pythagorean theorem
    • "This is a theorem that may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs." (source)
    • Earliest proof perhaps from China;
    • Suggestion of the general solution from Babylonia, as we've seen previously, from Plimpton 322 (1900-1600 BCE): a tablet containing "Pythagorean triples" (many years before Pythagoras, 6th century BCE)
    • Geometric proof (p. 107)
    • Algebraic solutions for special cases
      • Pythagoras: using successive squares
      • Plato: using consecutive odd squares
      • We already derived the general form of a solution: in class, during our discussion of the Babylonians and Plimpton.
  • The "Crisis of Incommensurable Quantities"
    • That is irrational. (Proof by contradiction -- famous and easy!) Careful: this one can get you killed....

    • Theon of Smyrna concocts an approximation scheme
      • Let's look at it from another perspective....
      • Newton improved on it, with his "method".

    • Eudoxus of Cnidus concocts a theoretical paradigm for dealing with "the unutterable" at his school in Cyzicus (p. 117).