Day 2, MAT115

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Question of the Day: "Where exactly do numbers come from? Did humanity invent them? Or discover them?" (Joy of X, p. 4)
Discussion (5-10 minutes)
How might "primitive" people have counted? There are some good suggestions that I know of:
- First of all, all counting is based on the idea of a one-to-one correspondence:
  - What is "six"?
    We learn something of that in From fish to infinity. But
    
    "Six is more ethereal than six fish.... It applies to six of anything..., the ineffable thing they all have in common." (Joy of X, p. 4)
    So we use the counting numbers as our counting yardstick, and put them into one-to-one correspondence with a set of objects to discover its number.
    - "...numbers are shortcuts for counting by ones...." (p. 5)
    - Furthermore, "...Humphrey might realize he can keep counting forever." (p. 5) -- something that will be very important when we come to talk about infinity.
    Any two things that have the property of "six-ness" can be put into one-to-one correspondence, and that's just what Ernie did for Humphrey and his furry friend.
    Did you notice Humphrey using any other cues to help him keep track of the number of fish?
  - Rather than using counting numbers, as we do, this may have been done using body parts -- "one hand" of sheep, say -- meaning five. Folks doing this just can't keep "counting forever".
  - cairns: 'In Highland folklore it is believed that the Highland Clans, before they fought in a battle, each man would place a stone in a pile. Those who survived the battle returned and removed a stone from the pile. The stones that remained were built into a cairn to honour the dead.'
- Tallies:
  1. The history of the tally we use most (from Leslie, 1820):
  2. The Ishango bone (and other tally sticks):
- And then an unusual method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht described at the 2011 National Math Meetings. I have modified their algorithm slightly (to conform more to the original document, from 1820, from which they took the idea).
  - I'll illustrate by counting all students in our class:
    - Everyone stands up.
    - I'm going to count you off by "1/0" -- in that order. If you're a "1", sit down; if a "0", stay up.
    - I write down the last number said for the group.
    - Then I "do it again" with those left standing (writing the subsequent numbers down right to left).
    - I continue until there's only one student left (and so I write a 1 at the head of the string of numbers, and we're done).
    - In the end, I'll be able to tell you how many students are in the class, based on the numbers written.
  - Baggett and Ehrenfeucht proposed that primitive societies may have counted this way. Let's suppose you need to let the King know how many sheep you have:
    1. divide your sheep equally ("one for you, one for me") into two pens: either there is one left over, or not. You make a note of whether there is one left over or not.
    2. Send all the sheep in pen two (and any "left over") out to pasture, and then
    3. You divide the sheep in pen one into pens one and two: i.e., just do it again! And again, and again, and.... until you get down to a pen one with just sheep in it.
    4. Now let's see how we might record the results to send to the King (make sure to write the results from right to left):
      - 9 sheep
      - 22 sheep
      - 54 sheep
      Do you get 1,0,0,1 for 9 sheep, and 1,0,1,1,0 for 22?
    5. Can you go backwards? How many sheep is meant by 1,1,0,1,0?
  - The easiest way to illustrate the counting method is via a mathematical object called a tree -- which we will see over and over -- it's a very useful organizing concept.
    Let's see how we might use a tree to represent the solution to the 22 counting problem: in the linked example, we would get 10110 by writing the remainders from left-to-right starting from the bottom of the tree. (The result should always start with a 1 if done correctly, since we always end with one sheep!)
  - Can you go backwards? (reverse engineer the tree!):
    1. How many sheep is meant by 1,1,1,1,0?
    2. How many sheep is meant by 1,0,0,1,0,1,1,0,1?

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