1.River Nile→Egypt:flooding of the Nile, calendar
measurement: used their bodies to measure the world: A palm was the width of a hand, a cubit an arm length from elbow to fingertips, Land cubits, strips of land measuring a cubit by 100,
fingers: 10 base, no place value: The sign for 1 was a stroke; 10, a heel bone; 100, a coil of rope;1,000, a lotus plant.
Rhind Papyrus(recorded by a scribe called Ahmes 1650BC): how to multiply two large numbers together: binary system(Lebniz 3000 later);
Mancala: divide 10 loaf between 9 people; 非洲棋棋盘: area of circle(diameter 9) = square(sides 8), pi,
the Eye of Horus: golden ratio
Pythagoras' Theorem: 345(concrete number not general proof)
Moscow Papyrus: volume of pyramid
Damascus:
The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.
12 knucle and 5 finger: 60 base, place value
The Babylonians' calendar was based on the cycles of the moon, cycles were recorded: angular measurement: 360 degrees in a full circle
zero 0
quadratic equation: geometric trick
Plimpton 322: Pythagoras' Theorem:
Greek, Palmyra in central Syria
Pythagoras' Theorem
music and the discovery of the harmonic series.
Hippasus: irrational number
Timaeus: Platonic solids: The tetrahedron(四面体) represented fire.
The icosahedron(二十面体), made from 20 triangles, represented water.
The stable cube was Earth.
The eight-faced octahedron was air.
the dodecahedron,made out of 12 pentagons, was reserved for the shape
which captured Plato's view of the universe.
Hypatia
2.
China
decimal place-value system
According to legend, the first sovereign of China, the Yellow Emperor, had one of his deities create mathematics in 2800BC, believing that number held cosmic significance. And to this day, Odd numbers are seen as male, even numbers, female.
Legend has it that thousands of years ago, Emperor Yu was visited by a sacred turtle that came out of the depths of the Yellow River. On its back were numbers arranged into a magic square, a little like this.
Chinese remainder theorem
Brahmagupta
pi=4(1-1/3+1/5-1/7+...)
3.The Frontiers of Space
Mountains of the Moon尼罗河之旅→Piero della Francesca(1415~1492), Urbino, northern Italy: perspective透視法, The Flagellation of Christ被鞭挞的耶稣→In France, Germany, Holland and Britain: mathematics of objects in motion
village of Descartes, Loire Valley→Descartes (1596~1650 France): lied in bed, soldier(mercenary),1628 in the Bavarian Army: the key was to build philosophy on the indisputable facts of mathematics.Numbers could brush away the cobwebs of uncertainty.
(left army)→Leiden, Holland: 1637 link algebra and geometry
→Church, Marin Mersenne: Parisian monk, 17th century Internet hub, publicised some new findings on the properties of numbers by →
Pierre de Fermat (1601～1665 France) → Beaumont-de-Lomagne near Toulouse→magistrate, hobby; invent modern number theory: Last Theorem, Little Theorem:密码的基础; 除四余一的指數=a^2+b^2
→Isaac Newton (1643 ~1727 England)→Grantham→village of Woolsthorpe→stepfather, Great Plague of 1665, came back to Lincolnshire from Cambridge: new theory of light, discovered gravitation, scribbled out Calculus(vs Greece: static geometry): circulate his thoughts among friends; professor, an MP, Warden of the Royal Mint in the City of London
→Gottfried Leibniz (1646~1716 Germany)→Hanover→ invent practical calculating machines that worked on the binary system→gardens of Herrenhausen: maze→Within five years, he'd worked out the details of the calculus, happy to make his work known, (Quite often revolutions in mathematics are about producing the right language to capture a new vision) Leibniz's notation
→Basel, Switzerlandl:commercial hub of the entire Western world→Bernoullis*6: Johann I, Jakob: worshipped Leibniz, distribute calculus in the scientific community; get the ball from the top to the bottom in the fastest time possible: cycloid, calculus of variation
→Leonhard Euler (1707~1783 Swiss)→boat across the Rhine→1728, 1766~83 by the help of Daniel Bernoullis, St Petersburg, Russia→calculus of variation, Fermat's theory of numbers: crystallised in Euler's hands; created topology and analysis, notation e and i, popularised the use of the symbol π; prime numbers, optics, astronomy,devised a new system of weights and measures, wrote a textbook on mechanics, and even found time to develop a new theory of music; 13 children, 5 survived to adulthood, lost most of his eyesight→1735 1+1/4+1/9+1/16=π^2/6(Basel problem, the Bernoullis tried and failed to solve it)
Both France and Germany were caught up in the age of revolution that was sweeping Europe in the late 18th century.
France: Napoleon(1769~1821), usefulness of mathematics: Joseph Fourier(1768~1830)
German: Wilhelm von Humboldt (1769～1859),valued mathematics for its own sake
→Gottingen→Carl Friedrich Gauss (1777~1855 Germany): Prince of Mathematics→father was a stonemason, criticized Euclid's geometry at 12; at 15, discovered a new pattern in prime numbers, which had eluded mathematicians for 2,000 years; at 19, discovered the construction of a 17-sided figure→keep a diary in Latin: first intimations of the theory of elliptic functions, Riemann ζ function→imaginary numbers→distrustful and grumpy, dismissal or lack of interest in the work of lesser mortals→Tower of Gauss
Transylvania, Romania →Jámos Bolyai(1802~1860 Romania): hyperbolic geometry, army → Nikolai Lobachevsky(1792~1856 Russian)
→Bernhard Riemann(1826-1866 German): the only one Gauss supported, very shy→Luneburg, northern Germany→1852 lecture on the foundations of geometry1852, multi-dimensional space →(read of) Legendre(1752~1833) → La Defense: hypercube architecture
4.
1900, Sorbonne, Paris, International Congress of Mathematicians→
David Hilbert (1862~1943 Germany): 23 most important problems, set the agenda for 20th-century maths and he succeeded
→Halle, East Germany→Georg Cantor (1845~1918 German): the first person to really understand the meaning of infinity and give it mathematical precision; →George Handel (1685~1759 Germany)→ different infinities, manic depression, continuum hypothesis: Is there an infinity sitting between the smaller infinity of all the whole numbers and the larger infinity of the decimals?
→Henri Poincaré (1854~1912 France): (Bertrand Russell (1872~1970 England) regards him as the greatest man France had produced)→Paris→very strict about his working da, two hours of work in the morning and two hours in the early evening; In 1885, King Oscar II
of Sweden and Norway offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all whether the solar system would continue turning like clockwork, or might suddenly fly apart: 3 body problem
→Kaliningrad(Konigsberg) 7 bridges→solved by 1735 Euler →topology→St Petersburg→1904 Poincaré conjecture→2002 solved by Grisha Perelman(1966~ Russia)
→Gottingen→although there are infinitely many equations, there are ways to divide them up so that they are built out of just a finite set,
like a set of building blocks →1930 'Wir mussen wissen, wir werden wissen.'→Vienna→Kurt Godel (1906~1978 Austria)→Incompleteness Theorem: within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove. "This statement cannot be proved."~(∃r:∃s:(P(r,s)∧(s=g(sub(f2(y))))))
→The Institute for Advanced Study, Princeton, New Jersey→Hermann Weyl(1885~1955 Germany)→John Von Neumann(1903~1957 Hungary)→Albert Einstein (1879~1955 Germany) & Kurt Godel
→Paul Cohen→proved continuum hypothesis: 2 system→Peter Sarnak
Sofia Kovalevskaya(1850～1891 Russia), Emmy Noether(1882~1935 Germany)→Julia Robinson(1919 America)→Phoenix, Arizona→UC Berkeley→marry Raphael Robinson→Hilbert's tenth problem: if there was some universal method that could tell whether any equation had whole number solutions or not→Robinson hypothesis→St Petersburg→ Yuri Matiyasevich→29th May 1832, Evariste Galois, Paris→Andre Weil: algebraic geometry→Nicolas Bourbaki→Alexandre Grothendieck
