Number Sense

• Why do we need numbers?• Munduruku– Foraging Society– Numbers

English Number Munduruku Number

One Pug

Two Xep Xep

Three Ebapug

Four Ebadipdip

Five/Handful Pug Pogdi

Some/ Not Many Adesu/ Adegu

Many/ Really Many Ade/ Ade Ma

Two Handfuls Xep Xep Pogdi

Number Sense

• Why do we need numbers?• Munduruku– Foraging Society– Numbers – What is culture? – What is nature?

English Number Munduruku Number

One Pug

Two Xep Xep

Three Ebapug

Four Ebadipdip

Five/Handful Pug Pogdi

Some/ Not Many Adesu/ Adegu

Many/ Really Many Ade/ Ade Ma

Two Handfuls Xep Xep Pogdi

Munduruku Dot Placement Experiment

• =

Munduruku Dot Placement Experiment

Little Kids, Dot Placement

Little Kids, Dot Placement

Dealing with Big Numbers

• Exact Sense for Small Numbers

Dealing with Big Numbers

• Exact Sense for Small Numbers• Rough Sense of Big Numbers

Dealing with Big Numbers

• Exact Sense for Small Numbers• Rough Sense of Big Numbers

Dealing with Big Numbers

• Exact Sense for Small Numbers• Rough Sense of Big Numbers

Dealing with Big Numbers

• Exact Sense for Small Numbers• Rough Sense of Big Numbers– 10 vs. 13: 90 % Accurate– 20 vs. 26: 90 % Accurate

Dealing with Big Numbers

• Munduruku Number Sense

English ShepherdsNumber Derbyshire Weardale Tong Kirkby Lonsdal

eWensleydale Lincolnshire

1 Yain Yan Yan Yaan Yain Yan2 Tain Teyan Tan Tyaan Tain Tan3 Eddero Tethera Tether Taed'ere Eddero Tethera4 Pederro Methera Mether Mead'ere Peddero Pethera5 Pitts Tic Pick Mimp Pitts Pimp6 Tayter Yan-a-tic Sesan Haites Tayter Sethera7 Later Teyan-a-tic Asel Saites Later Lethera8 Overro Tethera-tic Catel Haoves Overro Hovera9 Coverro Methera-tic Oiner Daoves Coverro Covera

10 Dix Bub Dick Dik Disc Dik11 Yain-dix Yan-a-bub Yanadick Yaan'edik Yain disc Yan-a-dik12 Tain-dix Teyan-a-bub Tanadick Tyaan'edik Tain disc Tan-a-dik13 Eddero-dix Tethera-bub Tetheradick Tead'eredik Ederro disc Tethera-dik14 Peddero-dix Methera-bub Metheradick Mead'eredik Peddero disc Pethera-dik

15 Bumfitt Tic-a-bub Bumfit Boon, buom, buum Bumfitt Bumfit

16 Yain-o-bumfitt Yan-tic-a-bub Yanabum Yaan'eboon Bumfitt yain Yan-a-bumfit17 Tain-o-bumfitt Teyan-tic-a-bub Tanabum Tyaan'eboon Bumfitt tain Tan-a-bumfit

18 Eddero-o-bumfitt

Tethea-tic-a-bub Tetherabum Tead'ereboon Bumfitt ederro Tethera-bumfit

19 Peddero-o-bumfitt

Methera-tic-a-bub Metherabum Mead'ereboon Bumfitt

peddero Pethera-bumfit

20 Jiggit Gigget Jigget Buom'fit, buum'fit Jiggit Figgot

English ShepherdsNumber Derbyshire Weardale

1 Yain Yan2 Tain Teyan3 Eddero Tethera4 Pederro Methera5 Pitts Tic6 Tayter Yan-a-tic7 Later Teyan-a-tic8 Overro Tethera-tic9 Coverro Methera-tic

10 Dix Bub11 Yain-dix Yan-a-bub12 Tain-dix Teyan-a-bub13 Eddero-dix Tethera-bub14 Peddero-dix Methera-bub

15 Bumfitt Tic-a-bub

16 Yain-o-bumfitt Yan-tic-a-bub17 Tain-o-bumfitt Teyan-tic-a-bub

18 Eddero-o-bumfitt Tethea-tic-a-bub

19 Peddero-o-bumfitt Methera-tic-a-bub

20 Jiggit Gigget

Base 1Number Numerical Linguistic

1 1 Yan2 11 Yan Yan3 111 Yan Yan Yan4 1111 Yan Yan Yan Yan5 11111 Yan Yan Yan Yan Yan 6 111111 Yan Yan Yan Yan Yan Yan7 1111111 Yan Yan Yan Yan Yan Yan Yan8 11111111 Yan Yan Yan Yan Yan Yan Yan Yan9 111111111 Yan Yan Yan Yan Yan Yan Yan Yan Yan

10 1111111111 Yan Yan Yan Yan Yan Yan Yan Yan Yan Yan

Base 2 and 4Number Base 2 Base 4

1 1 1

2 10 2

3 11 3

4 100 10

5 101 11

6 110 12

7 111 13

8 1000 20

9 1001 21

10 1010 22

Base 10Number Derbyshire Weardale

1 Yain Yan2 Tain Teyan3 Eddero Tethera4 Pederro Methera5 Pitts Tic6 Tayter Yan-a-tic7 Later Teyan-a-tic8 Overro Tethera-tic9 Coverro Methera-tic

10 Dix Bub11 Yain-dix Yan-a-bub12 Tain-dix Teyan-a-bub13 Eddero-dix Tethera-bub14 Peddero-dix Methera-bub15 Bumfitt Tic-a-bub16 Yain-o-bumfitt Yan-tic-a-bub17 Tain-o-bumfitt Teyan-tic-a-bub18 Eddero-o-bumfitt Tethea-tic-a-bub19 Peddero-o-bumfitt Methera-tic-a-bub20 Jiggit Gigget

Base 12

• Mathematical Advantages?

Base 12

• Mathematical Advantages?x 1 2 3 4 5 6 7 8 9 A B

1 1 2 3 4 5 6 7 8 9 A B

2 2 4 6 8 A 10 12 14 16 18 1A

3 3 6 9 10 13 16 19 20 23 26 29

4 4 8 10 14 18 20 24 28 30 34 38

5 5 A 13 18 21 26 2B 34 39 42 47

6 6 10 16 20 26 30 36 40 46 50 56

7 7 12 19 24 2B 36 41 48 53 5A 65

8 8 14 20 28 34 40 48 54 60 68 74

9 9 16 23 30 39 46 53 60 69 76 83

A A 18 26 34 42 50 5A 68 76 84 92

B B 1A 29 38 47 56 65 74 83 92 A1

Base 12

• Mathematical Advantages?• Vestiges of Base 12

Base 12

• Mathematical Advantages?• Vestiges of Base 12– Feet– Time– Dozen/Gross/Grocer– Counting

Eleven And Twelve

German French SpanishOne ein(s) un(e) un(e)Two zwei deux deuxThree drei trois troisFour vier quatre quatreFive fünf cinq cinqSix sechs six sixSeven sieben sept septEight acht huit huitNine neun neuf neufTen zehn dix dixEleven elf onze onzeTwelve zwölf douze douzeThirteen dreizehn treize treizeFourteen vierzehn quatorze quatorzeFifteen fünfzehn quinze quinze

Other Historical Bases

• Base 20– Word for 80 in French?

• Vestiges of Base 8

Other Historical Bases

• Base 20– Word for 80 in French?

• Vestiges of Base 8

Language Nine New

French Neuf Neuf

Spanish Nueve Nuevo

German Neun Neu

Norwegian Ni Ny

Other Historical Bases

• Base 20– Word for 80 in French?

• Vestiges of Base 8• Cuneiform

Before Place ValueNumber Roman

Numeral1 I

5 V

10 X

50 L

100 C

500 D

1000 M

1861: MDCCCLXI

Place Value

• Our Base 20 Shepherd

One Number System in the Vedas

• 422,396– 6 – 9 dasa– 3 Sata– 2 Sahasra– 2 Ayuta– 4 Niyuta

One Number System in the Vedas

• 422,396– 6 – 9 dasa– 3 Sata– 2 Sahasra– 2 Ayuta– 4 Niyuta

• What about 403,396?– How many Ayuta?– Shunya 0

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Contact with?

Islamic belief there is no conflict between reason & faith

Other Factors-Trading-Urbanization

FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Get copies of Greek works through?

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Math:

• Fused Indian place value digits w/Grk math Algebra & trigonometry w/sine, cosine, &

cotangent functions

Arabs take Greek math & science & do their own original work

Physics & optics:

• Calculated earth’s size

•Proved light goes fr. object to eye

•Formulae for specific & absolute weights

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

Medicine:

•Advanced w/o microscope

•Suturing w/animal gut•Mercury ointments for eye &

skin diseases

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Math:

• Fused Indian place value digits w/Grk math Algebra & trigonometry w/sine, cosine, &

cotangent functions

Arabs take Greek math & science & do their own original work

Physics & optics:

• Calculated earth’s size

•Proved light goes fr. object to eye

•Formulae for specific & absolute weights

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

Medicine:

•Advanced w/o microscope

•Suturing w/animal gut•Mercury ointments for eye &

skin diseases

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Math:

• Fused Indian place value digits w/Grk math Algebra & trigonometry w/sine, cosine, &

cotangent functions

Arabs take Greek math & science & do their own original workPhysics & optics:

• Calculated earth’s size

•Proved light goes fr. object to eye

•Formulae for specific & absolute weights

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

Medicine:

•Advanced w/o microscope

•Suturing w/animal gut•Mercury ointments for eye &

skin diseases

AFC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.)

Extensive funding through religious foundations (waqfs) & caliph’s “House of Wisdom”

Arabs willing & able to provide accurate versions of ancient texts thanks to:

Excellent translators since new converts must read Quran in the orig. Arabic

Get copies of Greek works through conquests, raids,

or negotiations w/Byz’s

Math:

• Fused Indian place value digits w/Grk math Algebra & trigonometry w/sine, cosine, &

cotangent functions

Arabs take Greek math & science & do their own original work

Physics & optics:

• Calculated earth’s size

•Proved light goes fr. object to eye

•Formulae for specific & absolute weights

Contact with Byz. & Persian civiliz’s (FC.46)

Islamic belief there is no conflict between reason & faith

Medicine:

•Advanced w/o microscope

•Suturing w/animal gut•Mercury ointments for eye &

skin diseases

Decline of Intellectual Life

• Outside Invaders• Ulama• Sufis

Spread to the West

• Means of Spread– Hajj– Trade

• Influence– Islamic Spain• Greek Learning

– Crusades• Arabic Number by 1200s• ShopkeepersScientists

Conclusion

• Cultural– Number Lines– Bases– Cultural Breakthroughs• Place Value• Zero• Multiplication, Division, Logs, not to mention Calculus

Conclusion

• Cultural– Number Lines– Bases– Cultural Breakthroughs

• Place Value• Zero• Multiplication, Division, Logs, not to mention Calculus

• Natural– Cultural Breakthroughs describe something real: Logs, Calculus,

Patterns in Dot Discernment– Math works in different bases– Music…

Conclusion: MusicCommon name

Examplename

Hz

FundamentalA2,

110

OctaveA3

220

Perfect FifthE4

330

OctaveA4

440

Major ThirdC♯5

550

Perfect FifthE5

660

Harmonic seventhG5

770

OctaveA5

880