EDWARD'S LECTURE NOTES:
More notes at http://tanguay.info/learntracker
C O U R S E 
Introduction to Mathematical Thinking
Keith Devlin, Stanford University
https://www.coursera.org/learn/mathematical-thinking
C O U R S E   L E C T U R E 
The Birth of Arithmetic
Notes taken on September 28, 2016 by Edward Tanguay
arithmetic
led to modern mathematics
addition, subtraction, multiplication, division
originally just addition and subtraction
counting
35,000 BCE or earlier
piles of pebbles
notches on bones and sticks
notched bones have been found
perhaps for
seasons
phases of the moon
monetary system
the first evidence we have for abstract numbers was money
Sumeria
5,000 BCE
the invention of numbers for monetary purposes
you didn't have numbers until you needed money
needed fro human transaction of goods and salaries
the essential reason why numbers came about was banking
bankers applied money to trade
today, bankers apply banking to figments of the imagination
but back then numbers were applied to actual physical goods
but not quite
the moment you introduce money, you immediately have something fictitious that depends on human agreement
1971 Denise Schmandt-Besserat
clay tokens from Sumeria, ca 5000 BCE
at burial sites and ruins of homes
tokens for things people would trade
represented goods, e.g. tokens that looked like tokens
Sumerian society was getting complex
you had tokens that represented the sheep
how to keep tokens secure
bankers kept tokens
took sheet of wet clay and sealed them
to show worth, they would impress the tokens on the wet clay
your worth is represented by the marks
layers of abstraction developing
then realized you didn't need the tokens inside, it's all about the indentations
then realized they just needed a flat tablet with abstract markings
measuring
building pyramids
seeing how much land you owned
counting
natural numbers
measuring
just thought of as units, not thought of as numbers, but were quantities, lengths, volumes
but area seems to be more complicated
the Greeks tried to combine them
discovered that the square root of two was irrational
pushed them apart again
because of the incommensurables
only at the end of the 19th century was this done in a system of numbers called the real numbers
the natural numbers were regarded as foundational
the human cognitive system captures both
natural numbers
counting
real numbers
lengths, areas
in the Principia, Newton
he's not talking about numbers, but quantities (real numbers)
calculus wasn't applied to numbers of distances and areas