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**Lessons from the History of Mathematics**

Tom Osler Rowan University

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**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,

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**Introduction Teaching math history for about 20 years**

I am not a historian Some of the opinions expressed are my personal view

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**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

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**The 5 minute history of math**

There were only two principle explosions of mathematical creativity Greek 600 BC to 200 AD European to today

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**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

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**Lesson 1 The fragility of mathematics**

Few cultures nourished math Most people born with math gift never had a chance to develop it

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**Lesson 2 Motivation for doing math**

Play Challenge problems Exploring the mind of God Pythagorians BC Everything is number Ptolemy ca AD Kepler

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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.

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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”

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**Lesson 3 Math is unlike any other academic study Does not change**

Euclid’s Elements No errors for years God is a Mathematician

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William Blake

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**William Blake: The Ancient of Days**

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William Blake: Newton

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**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?

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**And what shoulder, and what art. Could twist the sinews of thy heart**

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?

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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?

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**Lesson 4 The mother of mathematics is astronomy/astrology**

Babylonian planet and star charts Greek Astronomy Eratosthenes BC measures earth diameter (error 5%) Hipparchus BC distance to moon (error 5%) and to sun (error 2000 %)

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**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

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**Retrograde Motion of Mars**

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**invents epicyclic motion Ptolemy – Almagest 150 AD **

Apollonius BC invents epicyclic motion Ptolemy – Almagest 150 AD Earliest trig tables Predicts planet positions with an accuracy one degree in 100 years

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**Epicyclic Motion from Almagest**

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**Lesson 4 The mother of mathematics is astronomy/astrology**

Navigation Need to find longitude after 1500 Newton ( ) Invents calculus Law of gravity Universal Laws of motion Nearly every great name in math from Kepler (1600) to Gauss (1850) worked on Celestial Mechanics Poincare and Chaos

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**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

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**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”

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**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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