IST 4Information and Logic
Lectures are at:paradise caltech edu/ist4/lectures htmlparadise.caltech.edu/ist4/lectures.html
Homeworks are at:paradise caltech edu/ist4/homeworks htmlparadise.caltech.edu/ist4/homeworks.html
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MQ1E h ift!Everyone has a gift!
Due Thursday 4/9/2015 by 10pmDue Thursday 4/9/2015 by 10pm
Please email PDFlastname-firstname.pdfpto [email protected]
Have your name inside the file as well...
languagelanguagepropertiesp p
Building Blocks
finite number of building blocks ‘infinitely’ many descriptionsy y p
Separation
Separation Apbetween syntax and semantics
S ti BSeparation Bbetween what is represented
d lit f ibilit ti Separation Cand reality, feasibility, time, space, ...
pbetween algorithms and implementation
Languages help in reasoning beyond our natural sensesbeyond our natural senses
Our number sense
Need a language for quantities!g g q
Why care about quantities Why care about quantities 40,000 years ago?
How will we remember quantities?
150 000 d h H i~150,000 years ago: modern humans - Homo sapiens
~60,000 years ago: verbal languages60,000 years ago verbal languages
~40,000 years ago: artificial memory for quantities
~5,000 years ago: written symbolic languages
~500 years ago: printing~500 years ago: printing
~50 years ago: electronic forms
What is this item?
Lebombo bone ~40 000 years agoLebombo bone 40,000 years ago
hint?
A very early device for di titi b m ki recording quantities by making
marks on a bone of a monkey
Early Counting Devices ~40,000 ago
lebombo bone
1970's excavations in Lebombo Mountains, a piece of the fibula of a baboon was found marked with 29 clearly fibula of a baboon was found marked with 29 clearly defined notches ??????
Lunar cycle (month): ~29.5 days Lu y (m ) 9. y
Source: Wikipedia
5 5
Day and night: 210 10
Moon – Earth (month): ~29 days
Earth – Sun (year): ~365 days
12 is a 12 is a ‘special’ numberspecial number..E l ?Examples?
B d Beyond natural numbers?
What was invented first?a. Written text
1. Numbers
a and 1 togethera and 1 together
Why do we need numbers?Why do we need numbers?
Tokens 8,000 ya, yPhysical Symbols:
shape = meaningshape = meaning
Source: DSB 2009 paper
one jar of oil one garment Idea: Represent goods with tokens
one sheep one jar of oil one garment
one ingot of metal
Token are physical symbols of real items – a language!
h b
Source: DSB 2009 paper
one honeycomb one garment
C th t k ith t k ?Can the system work with tokens?
Secure Transactions:E l (B ll ) Envelopes (Bulla) and Seals, 3,700BCand Seals, 3,700BC
cylinder seals
Does it work???Does it work???
New Idea: Mark on the outside to represent what is inside...
Writing: h i l b l t i i t d b l physical symbols to imprinted symbols
Envelope showing the imprint of three ovoid tokens with an incised line
3,300 BC
tokens with an incised line representing jars of oilSource: DSB 2009 paper
A crazy idea: No need for an envelope! A crazy idea: p
The tablet is born!
Tablet showing the impression of spheres and cones representing measures of grain3 100BC3,100BC
Source: DSB 2009 paper
One more crazy idea:Separate between the quantity and the item....
Too many jars of oils m y j ffor markings on the envelope / tablet
Source: DSB 2009 paper
Separate between the quantity and the itemSeparate between the quantity and the item...
The NUMBER is born!!!
10 1
The NUMBER is born!!!
110
Twelve jars
LLanguage for quantities!!!
Source: DSB 2009 paper
tablet featuring an account of 33 f il 3 100 BC il
oneten
33 measures of oil, 3,100 BC oil
Source: DSB 2009 paper
Language for quantities!!!
The item (jar of oil) is ‘left alone’It d t b lItems correspond to symbols
Symbols for oil, and other items – have a sound!
Sounds correspond to symbols
Collection of symbols correspond to words / meaning
b
General writing is born!!! Not covered in IST4....
rebus 22 dayL8
Denise Schmandt-Besserat1933 - Source: Her 2009 paper
(DSB 2009)
“Tokens and Writing: the C iti D l t ”Cognitive Development,”Will be posted on the class web page
Where did it start?Where did it start?
Babylonians and Egyptians5000 ~5000 years ago
The Egyptians Preferred 10 The Egyptians Preferred 0
How will you representHow will you representa trillion1,000,000,000,000?
2x100 + 7x10 + 6x1 = 276separation 2x100 + 7x10 + 6x1 = 276separat on
However, not a finite number of building blocks!
Babylonians
The first Positional number system
11
10
Separation: ‘meaningless’ symbols’ - modulationSeparat on mean ngless symbols modulat onThe smallest number (two) of building blocks!!
Babylonians
The first Positional number system
11
10
The Babylonian preferred 60 The Babylonian preferred 60
Babylonians
The first Positional number system
4x60 + 36x1 = 276
Babylonians were Masters of Abstractions
Why 60?
y
Why 60?Good for computation and representing fractions:
1/2 = 30/601/2 30/601/3 = 20/601/4 = 15/60 1/5 = 12/601/5 = 12/601/6 = 10/60
60 i th ll t b th t h 60 is the smallest base that has a simple representation of the first five fractions
Babylonian Number Systemsy y
What is the number?
31
What is the number?
31 60 31 189131x60 31+ = 1891
The Babyloniansknew everything!knew everything!
Deciphering the Deciphering the BabyloniansBabylonians
Otto NeugebauerAustria (BS) –> Germany (BS) –> Denmark (PHD) –> US (Brown U) Otto Neugebauer
1899-1990
Mathematics –> History of Exact Sciences
Participated in WWI, was POWEscaped Europe in 1939 WWIIEscaped Europe in 1939, WWII
His son:His son:Gerry Neugebauer1932-2014 (PhD ‘60)( )Millikan Professor of Physics, Emeritus
1945
Asger AaboeMathematics –> Teaching Math ->History of Exact Sciences
Denmark –> US (PhD 1957, Brown U) -> 1961,Yale U 1922–2007
Otto Neugebauer1899 19901899-1990
Otto Neugebauer1899-1990
Asger Aaboe1922–2007
Today you will get to be Otto and Asger!Today, you will get to be Otto and Asger!
1 159 2,15
Multiplication table for 9123
5161718
91827
, 52,242,332,422 514
567
1819?20
40
364554
63 1 3
2,513,04,306,0
27030789
10
4050??
631,121,211 30
1,3 6,07,3072 450
811011121
1,301,391,481 5713
14
1,572,6
Multiplication instead of division
????
GeometryGeometry
The Babyloniansknew everything!knew everything!
even Geometryeven Geometry...
3
945
The Babyloniansknew everything!knew everything!
even Geometryhint even Geometry...4
hint
161
The Babyloniansknew everything!knew everything!
even Geometryeven Geometry...
3
945
The Babyloniansknew everything!knew everything!
even Geometryeven Geometry...
3
945
The Babyloniansknew everything!knew everything!
even Geometryeven Geometry...
12 again!!12
12 again!!
2,24?12
Which Number Does not Belong?g
is a space between two digits
Which Number Does not Belong?g
139 25512763139 25512763
64-1 128-1 256-1
Take-homes from today:
The language of numbers andwritten languages evolved from the need for secure from the need for secure transactions
The Babylonians knew EVERYTHING!The Babylonians knew EVERYTHING!