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
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Teaching math history for about 20
years
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I am not a historian
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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
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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
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“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
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Does not change
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Euclid’s Elements
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No errors for 2300
years
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God is a Mathematician
William Blake 1757-1827
William Blake: The Ancient of Days
William Blake: Newton
The Tyger
by William Blake
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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
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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
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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
GMm
F
2
r
– 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
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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
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Today it is dead
Even Newton wrote his principia in
geometric language, and not with the
calculus he invented