010 lessons from the history of mathematics

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,

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

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

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

• 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

– Universal Laws of motion

• Nearly every great name in math from Kepler (1600)

to Gauss (1850) worked on Celestial Mechanics

• Poincare and Chaos

2

r GMm

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”

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