Lessons from the History of Mathematics Tom Osler Rowan University Is Mathematics a Humanity or a Science? • Humanities • The branch of learning regarded as having primarily a cultural character and use, including languages, literature, history and philosophy. • Sciences • A branch of study that is concerned with observation and classification of facts, and especially with the establishment or strictly the quantitative formulation of verifiable general laws by induction and by hypothesis, Introduction • Teaching math history for about 20 years • I am not a historian • Some of the opinions expressed are my personal view We discuss only Higher Mathematics • Every civilization needed a simple number system and primitive geometry • This is NOT what we discuss • Higher Mathematics • Algebra, Trigonometry, Geometry with proofs, Calculus etc The 5 minute history of math • There were only two principle explosions of mathematical creativity • Greek 600 BC to 200 AD • European 1500 to today The 5 minute history of math • Today math research is performed all over the world by both men and women • This was not true only 40 years ago, when most researchers were men with names of European descent • Note: Until 1800, mathematicians were known as “Geometers”, not mathematicians Lesson 1 The fragility of mathematics • Few cultures nourished math • Most people born with math gift never had a chance to develop it Lesson 2 Motivation for doing math • • • • • • • Play Challenge problems Exploring the mind of God Pythagorians 580-500 BC Everything is number Ptolemy ca 85-185 AD Kepler 1571-1630 Quote from Ptolemy • I know that I am mortal and the creature of a day; but when I search out the massed wheeling circles of the stars, my feet no longer touch the earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods. Quote from Kepler • “Why waste words? Geometry existed before the Creation, is co-eternal with the mind of God, is God himself (what exists in God that is not God himself?); geometry provided God with a model for the Creation and was implanted into man, together with God’s own likeness-and not merely conveyed to his mind through the eyes” Lesson 3 Math is unlike any other academic study • Does not change • Euclid’s Elements • No errors for 2300 years • God is a Mathematician William Blake 1757-1827 William Blake: The Ancient of Days William Blake: Newton The Tyger by William Blake • Tyger! Tyger! burning bright, In the forests of the night, What immortal hand or eye Could frame thy fearful symmetry? • In what distant deeps or skies Burnt the fire in thine eyes? On what wings dare he aspire? What the hand dare seize the fire? • And what shoulder, and what art? Could twist the sinews of thy heart? And when thy heart began to beat, What dread hand, and what dread feet? What the hammer? What the chain? In what furnace was thy brain? What the anvil? What dread grasp Dare its deadly terrors clasp? • When the stars threw down their spears, And watered heaven with their tears, Did he smile his work to see? Did he who made the Lamb, make thee? Tyger! Tyger! burning bright, In the forests of the night, What immortal hand or eye Dare frame thy fearful symmetry? Lesson 4 The mother of mathematics is astronomy/astrology • • • • • • Babylonian planet and star charts Greek Astronomy Eratosthenes 276-194 BC measures earth diameter (error 5%) Hipparchus 190-120 BC distance to moon (error 5%) and to sun (error 2000 %) Lesson 4 The mother of mathematics is astronomy/astrology • Motion of Sun, Moon, Planets and Stars – Develop Calendar of Seasons • Month – Lunar cycle • Year – Solar cycle – Predict future events – Astrology • Positions of Sun, Moon and Planets on the Ecliptic Zodiac Retrograde Motion of Mars • • • • • • • • • Apollonius 256-190 BC invents epicyclic motion http://www.lasalle.edu/~smithsc/Astronomy/retrograd.html Ptolemy – Almagest 150 AD Earliest trig tables Predicts planet positions with an accuracy one degree in 100 years Epicyclic Motion from Almagest Lesson 4 The mother of mathematics is astronomy/astrology • Navigation – Need to find longitude after 1500 • Newton (1643-1727) – Invents calculus – Law of gravity F GMm r 2 – Universal Laws of motion • Nearly every great name in math from Kepler (1600) to Gauss (1850) worked on Celestial Mechanics Lesson 5: The wheel in Mathematics is the decimal number system • If the Greeks had known it, they would have understood irrationals • Possibly they would have invented algebra and analysis Lesson 6: A second wheel is shorthand math symbolism • Before 1500, most problems in math were expressed by rhetorical language and not letters and symbols as we do in algebra. • “The square of the unknown added to twice the unknown plus 4 vanishes” x 2x 4 0 2 Lesson 7: Euclidean Geometry retarded Math development for almost 2000 years • • Today it is dead Even Newton wrote his principia in geometric language, and not with the calculus he invented

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# Lessons from the History of Mathematics