1. Lessons from the History of Mathematics
Tom Osler
Rowan University

2. 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,

3. Introduction Teaching math history for about 20 years
I am not a historian
Some of the opinions expressed are my personal view

4. 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

5. The 5 minute history of math
There were only two principle explosions of mathematical creativity
Greek 600 BC to 200 AD
European to today

6. 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

7. Lesson 1 The fragility of mathematics
Few cultures nourished math
Most people born with math gift never had a chance to develop it

8. Lesson 2 Motivation for doing math
Play
Challenge problems
Exploring the mind of God
Pythagorians BC
Everything is number
Ptolemy ca AD
Kepler

9. 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.

10. 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”

11. Lesson 3 Math is unlike any other academic study Does not change
Euclid’s Elements
No errors for years
God is a Mathematician

13. William Blake

14. William Blake: The Ancient of Days

15. William Blake: Newton

17. 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? 

18. 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? 

19. 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? 

20. 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 %)

21. 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

24. Retrograde Motion of Mars

25. 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

26. Epicyclic Motion from Almagest

27. 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

29. 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

30. 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”

31. 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