Number Sense • Why do we need numbers? • Munduruku • Foraging Society • Numbers
Number Sense • Why do we need numbers? • Munduruku • Foraging Society • Numbers • What is culture? • What is nature?
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 Shepherds
English Shepherds
Base 1
Base 2 and 4
Base 10
Base 12 • Mathematical Advantages?
Base 12 • Mathematical Advantages?
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
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
Other Historical Bases • Base 20 • Word for 80 in French? • Vestiges of Base 8 • Cuneiform
Before Place Value 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
A FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with?
Other Factors -Trading -Urbanization 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 FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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? FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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 FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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 FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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 FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46) Get copies of Greek works through conquests, raids, or negotiations w/Byz’s
A 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 Medicine: •Advanced w/o microscope •Suturing w/animal gut •Mercury ointments for eye & skin diseases FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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 Medicine: •Advanced w/o microscope •Suturing w/animal gut •Mercury ointments for eye & skin diseases FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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 Medicine: •Advanced w/o microscope •Suturing w/animal gut •Mercury ointments for eye & skin diseases FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
A 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 Medicine: •Advanced w/o microscope •Suturing w/animal gut •Mercury ointments for eye & skin diseases FC.47 ARAB SCIENCE AND ITS LEGACY (c.750-1000 C.E.) Islamic belief there is no conflict between reason & faith Contact with Byz. & Persian civiliz’s (FC.46)
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: Music
