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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4 311 3 4Mx= MQx out

midtermsoh

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oh = office hours oh

11 3 418 4 525

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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 ta4@paradise.caltech.edu

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!