Chapter 1: Tokens, Tallies, and Tablets
What is Math?
· It began with numbers
· “Mathematics is universal and ubiquitous.”
What are numbers?
· “Numbers are some kind of mental construct, yet we feel that they would continue to have meaning even if humanity were wiped out by a global catastrophe and there were no minds left to contemplate them.”
· “Without numbers, civilization as we now know it could not exist. Numbers are everywhere.”
When do numbers begin? A brief history of numbers…
· Some 10,000 years ago in the Near East with little clay tokens. These were used instead of symbols. They were also used for keeping records, perhaps of taxes.
· Then, it evolved to clay envelopes with symbols on them. This method could be the beginning of inflation, because these Mesopotamian bureaucrats didn’t actually need the contents inside the envelopes.
· The oldest tally marks |||| are from some 37,000 years ago, found in a cave in the Lebombo mountains (border between Swaziland and South Africa). These evolved through the years to the first numerals.
· The first number system is said to be the Babylonian, with “base 60”.
· Another important system is that of the ancient Egyptians, they lived between 3150 BC and 31 BC.
o They used drawings for whole numbers (e.g. a slave for one million), and an eye for fractions.
Important thoughts
· “Math and culture co-evolve.”
· “Math seldom get credit for changing our world.”
· “The main mathematics that does lie on the surface is arithmetic.”
· “We are whole dependent on numbers.”
· “The inventions of number notation and arithmetic rank alongside those of language and writing as some of the innovations that differentiate us from trainable apes.”
· It began with numbers
· “Mathematics is universal and ubiquitous.”
What are numbers?
· “Numbers are some kind of mental construct, yet we feel that they would continue to have meaning even if humanity were wiped out by a global catastrophe and there were no minds left to contemplate them.”
· “Without numbers, civilization as we now know it could not exist. Numbers are everywhere.”
When do numbers begin? A brief history of numbers…
· Some 10,000 years ago in the Near East with little clay tokens. These were used instead of symbols. They were also used for keeping records, perhaps of taxes.
· Then, it evolved to clay envelopes with symbols on them. This method could be the beginning of inflation, because these Mesopotamian bureaucrats didn’t actually need the contents inside the envelopes.
· The oldest tally marks |||| are from some 37,000 years ago, found in a cave in the Lebombo mountains (border between Swaziland and South Africa). These evolved through the years to the first numerals.
· The first number system is said to be the Babylonian, with “base 60”.
· Another important system is that of the ancient Egyptians, they lived between 3150 BC and 31 BC.
o They used drawings for whole numbers (e.g. a slave for one million), and an eye for fractions.
Important thoughts
· “Math and culture co-evolve.”
· “Math seldom get credit for changing our world.”
· “The main mathematics that does lie on the surface is arithmetic.”
· “We are whole dependent on numbers.”
· “The inventions of number notation and arithmetic rank alongside those of language and writing as some of the innovations that differentiate us from trainable apes.”
Chapter 2: The Logic of Shape: First Steps in Geometry
2 main types of reasoning in math: symbolic (originated in numbers) and visual (pictures).
Euclid of Alexandria
· “The first systematic use of diagrams, together with a limited use of symbols and a heavy dose of logic.”
· He combined the use of pictures and the logical structure of proofs, both innovations.
· Euclid’s Elements provides a definitive treatment of the geometry of two dimensions (the plane) and three dimensions (space).
· The climax is the proof that there are precisely five regular solids: the tetrahedron (Earth), cube (Water), octahedron (Air), dodecahedron (Fire) and icosahedron – 20 equilateral triangles (Quintessence). He proved that there are no other regular solids, and these five actually exist.
· It’s important to mention that Euclid started by listing a number of definitions or axioms, also postulates (things we assume we can do) and common notions (derived by logic).
· “Euclid deduced each theorem from previous theorems and various axioms. He built a logical tower, which climbed higher and higher towards the sky, with the axioms as its foundations and logical deduction as the mortar that bound the bricks together.”
Pythagoras and the Pythagoreans
· They understood that mathematics is about abstract concepts, not reality.
· It was their belief that the universe is founded on numbers. They believed the number 10 had deep mystical significance.
· Hippasus of Metapontum proved them wrong in their attempt that everything could be proved using diagrams formed from lots and lots of copies of one basic shape. He proved that the diagonal of a unit square is irrational: not an exact fraction. He was drowned or expelled, and became one of the early victims of the irrationality, so to speak, of religious belief.
· The Greek theory of irrationals was invented by Eudoxus around 370 BC.
The golden mean
· (1 + √5) / 2
· It’s numerical value is roughly 1.618
· It is irrational.
· Extreme and mean ratio as Euclid called it. Regular pentagons are directly connected with it.
Archimedes
· His work is in circles, spheres, and cylinders, which we now associate with the number π (pi), which is roughly 3.14159
· To estimate π, he compared the circumference of a circle with the perimeters of two series of polygons: one series situated inside the circle, the other surrounding it.
Other Greeks
· With more instruments than a ruler and a compass (Euclid), the Greeks could do procedures called “neusis constructions”. Then, comes the conic sections: ellipse, hyperbola, and parabola. These were studied in detail by Apollonius of Perga.
· Two crucial ideas to human development: the first one was a systematic understanding of geometry. The second was the systematic use of logical deduction to make sure that what was being asserted could also be justified.
· Eratosthenes measured the size of the Earth. He knew that camel trains took 50 days to get from Alexandria to Syene, and they traveled a distance of 100 stadia each day. The circumference of the Earth was 250,000 stadia, approximately 39,250 km. Today the measure is 39,840 km.
“Ideas that survive stringent attempts to disprove them are more likely to be correct.”
Euclid of Alexandria
· “The first systematic use of diagrams, together with a limited use of symbols and a heavy dose of logic.”
· He combined the use of pictures and the logical structure of proofs, both innovations.
· Euclid’s Elements provides a definitive treatment of the geometry of two dimensions (the plane) and three dimensions (space).
· The climax is the proof that there are precisely five regular solids: the tetrahedron (Earth), cube (Water), octahedron (Air), dodecahedron (Fire) and icosahedron – 20 equilateral triangles (Quintessence). He proved that there are no other regular solids, and these five actually exist.
· It’s important to mention that Euclid started by listing a number of definitions or axioms, also postulates (things we assume we can do) and common notions (derived by logic).
· “Euclid deduced each theorem from previous theorems and various axioms. He built a logical tower, which climbed higher and higher towards the sky, with the axioms as its foundations and logical deduction as the mortar that bound the bricks together.”
Pythagoras and the Pythagoreans
· They understood that mathematics is about abstract concepts, not reality.
· It was their belief that the universe is founded on numbers. They believed the number 10 had deep mystical significance.
· Hippasus of Metapontum proved them wrong in their attempt that everything could be proved using diagrams formed from lots and lots of copies of one basic shape. He proved that the diagonal of a unit square is irrational: not an exact fraction. He was drowned or expelled, and became one of the early victims of the irrationality, so to speak, of religious belief.
· The Greek theory of irrationals was invented by Eudoxus around 370 BC.
The golden mean
· (1 + √5) / 2
· It’s numerical value is roughly 1.618
· It is irrational.
· Extreme and mean ratio as Euclid called it. Regular pentagons are directly connected with it.
Archimedes
· His work is in circles, spheres, and cylinders, which we now associate with the number π (pi), which is roughly 3.14159
· To estimate π, he compared the circumference of a circle with the perimeters of two series of polygons: one series situated inside the circle, the other surrounding it.
Other Greeks
· With more instruments than a ruler and a compass (Euclid), the Greeks could do procedures called “neusis constructions”. Then, comes the conic sections: ellipse, hyperbola, and parabola. These were studied in detail by Apollonius of Perga.
· Two crucial ideas to human development: the first one was a systematic understanding of geometry. The second was the systematic use of logical deduction to make sure that what was being asserted could also be justified.
· Eratosthenes measured the size of the Earth. He knew that camel trains took 50 days to get from Alexandria to Syene, and they traveled a distance of 100 stadia each day. The circumference of the Earth was 250,000 stadia, approximately 39,250 km. Today the measure is 39,840 km.
“Ideas that survive stringent attempts to disprove them are more likely to be correct.”