Mathematics as a
Human Endeavor
Ed Dickey
University of South Carolina
SCCTM Conference / 23 October 2009
1979 SCCTM Conference
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•
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20 October 1979
“Historical Anecdotes for the Math Class”
Teacher at Spring Valley High School
Taught from experience that injecting
history MOTIVATES and HELPS students
learn mathematics.
• I believe it then and I believe it now
Handouts
• Invite Letter from Patty Smith, SCCTM
President
• Handout (on blue mimeograph paper and I
STILL have 28!)
• Evaluation Form (32 attendees!)
Opening Quote
“ I have more than an impression- it amounts
to a certainty- that algebra is made repellent
by the unwillingness or inability of teachers
to explain why…
There is no sense of history behind the
teaching, so the feeling is given that the
whole system dropped down ready-made
from the skies, to be used by born jugglers.”
Jacques Barzun, Teacher in America
• Help students see mathematics as a
human endeavor that evolved through
men and women discovering and inventing
the many things we study in algebra and
school mathematics
• Help make connections to other cultures
Hypatia
• 5th Century AD
• Library of Alexandria
• Commentaries to works
of Diophantus and
Apollonius
• Edited Ptolemy and
Euclid
• Invented hydrometer
Hydrometer
To measure
specific gravity of
liquids
Hypatia
• Literary legend in Charles Kingsley 1953
novel: Hypatia – or New Foes with an Old
Face
• Portrayed the scholar as a “helpless,
pretentious, erotic heroine”
• Murdered by an angry mob of fanatical
monks who objected to her being a
woman who didn’t know her place
Gerolamo Cardano
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1501-1576
“eccentric” and “difficult”
Gyroscope gimbal
Auto Differential
Combination lock
Imaginary numbers
Two-axis gimbal set
Cardan Shaft
Combination Lock
Imaginary Numbers
• Contests to Solve Equations
• Degree 1 and 2 equations easy
• Degree 3 or Cubic
• Substitute
• Depressed cubic:
Solving Cubic Equation
• Where in (2), our depressed cubic
• Now introduce two new variables in (2)
• And get
Solving Cubic Equation
• Cardano let 3uv + p = 0 in (3), which
implies uv = -p/3 so substituting for uv and
multiplying by u3 he got
• Which is a quadratic in u3
• So using the quadratic formula
Solving Cubic Equation
• And
• Now work back from the substitutions to
get x in terms of a, b, c and d.
• When you do this, you must accept the
existence of imaginary numbers and in the
16th century only an eccentric like Cardano
would.
Solving the Cubic
• Along the backward substitution path
you reach:
t p
• A place where most would stop but
Cardano persisted and this gave him
the formula needed to win contests.
Cubic Formula
for
Example
x  2x  x  3  0
3
2
12  4
p  1  
3
3
2(1)3  9(1)( 1) 91
q  3

27
27
92
92
4
2
( ) 2 ( )3
3
3
 12 57  91  2 3
27
27
3
u 



2
4
27
6
.. And FINALLY
12 57  92  2
x

6
3
4
3
2
3
2
3
12 57  92  2
3(
)
6
3
• Generating 3 values of x
• Full explanation at
http://en.wikipedia.org/wiki/Cubic_function
#Cardano.27s_method
1

3
TI Nspire CAS
WolframAlpha
Derive
GeoGebra
Quartic anyone?
• Lodovico Ferrari (at 18!) discovered a
quartic formula in 1540.
• Cardano published it
in Ars Magna (1545)
• Many substitutions
and “nested
depressed cubics”
Quartic glimpse
Sigh…
SIGH…..
FINALLY:
Example
Solve
x  2 x  x  3x  4  0
4
3
Using Derive…..
2
Derive output (after .5 seconds)
So what about 5th degree?
• We are now in the mid 1500s
• The formula will be AWFUL!
• Isn’t there a formula to solve:
ax  bx  cx  dx  ex  f  0
5
4
3
2
Enter Niels Abel
• 1802-1829 (ouch!)
• “Abelian group”
Died young
• Contracted tuberculosis in Paris at
Christmas.
• Traveled by sled to visit his fiancee in
April but died after a short visit with her
on April 6
Abel’s Impossibility Theorem
• No general solution in radicals to
polynomial equations of degree five or
higher
• Fundamental Theorem of Algebra: every
polynomial with real or complex
coefficients can be solved with a complex
number.
• Proved using Galois Theory
Evariste Galois
• 1811-1832 (20 years old)
• Poisson denied a position
in the Academy
“incomprehensible work”
• Fought a duel on behalf off
a Mlle du Motel
• Stayed up all night writing
his papers
Example
Solve
x  x  2 x  x  3x  4  0
5
4
3
2
Derive output
Symbolism
• Cardano used no
algebraic symbols
• Variables like x were
written out as cosi
• Sample from
Tartaglia’s Nova
Scientia (1537)
Symbolisms
• François Viète initiated
the use of letters for
variables (end of 16th
Century)
• René Descartes then
Isaac Newton moved
algebraic symbolism
toward today’s
conventions
Hendrick van Heuraet (1634-1660)
Arc
length
using x
and y
for
y x
3
3
Giovanni Saccheri
(1667-1733)
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“the good little monk”
Actually a Jesuit
“Euclid Freed of Every Flaw”
Demonstrate that denying Euclid Parallel
Postulate leads to a contradiction
Saccheri Quadrilateral
Right Case
• Euclid’s 5th Postulate
• Given a line and a
point not on that line,
there is one and only
one line through the
given point parallel to
the given line
• Model of PLANE
geometry
Obtuse Case
• Assuming Euclid’s 5th Postulate is false
• Equivalent to NO parallel lines
• Leads to the conclusion that the sum of
the interior angles of a triangle are greater
than 180 degrees
• To Saccheri this was “absurd” but later it
was the basis for Elliptic Geometry
• Model of SPHERICAL Geometry
Acute Case
• Assuming Euclid’s 5th Postulate is false
• Equivalent to at least TWO parallel lines
• Leads to the conclusion that the sum of
the interior angles of a triangle are less
than 180 degrees
• To Saccheri this too was “absurd” but later
it was the basis for Hyperbolic Geometry
• Model for RELATIVITY (spacetime
Lorentz model)
Good Monk or Mathematician?
• Was Saccheri a “good monk” allowing the
prevalent view (Euclid’s) to define “absurd’
• Or was he a better mathematician allowing
the logic of his conclusions to win out
• He preserved his status and safety as a
monk and avoided conflict with the
prevailing Euclidean view point
• As time progressed, thinkers challenged
the prevailing views…
George Cantor
• 1845-1918
Set Theory and the Infinite
• Levels of infinity: countable and
uncountable
• Continuum Hypothesis (no set whose
cardinality is between the integers and the
reals)
• Gödel showed this cannot be proved or
disproved.
• Paradoxes… “nowhere dense”
Cantor Set
• Start with unit interval [0, 1]
• Delete open middle third (1/3, 2/3)
• Delete open middle third of each
remaining segment… infinitely
Sierpinski Carpet
Menger
Sponge
Koch Curve
• Generation 1:
• Generation 2:
• Generation 4:
“Pathological” Examples
A curve the is continuous but nowhere
differentiable:
Recursive Process
• Generation 1:
• Generation 2:
• Generation 4:
Recursive Process
• Generation 6:
• Generation 12:
Alan Turing
Turing Machine
• Thought experiment (not a real machine)
• Simulates what computer programs might
be able to do
• Church’s Thesis: Turing machine models
“effectively calcuable” functions or
mathematical propositions that are
provable
• Gödel's Theorem
Gödel's Theorem
• Incompleteness: there are
mathematical propositions
that cannot be proved
Turing Machine
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Tape
Head
Table
State Register
Unlimited tape and infinite memory
Similar to Gödel Numbering
The Wolfram Research Prize
• 2, 3 Turing Machine: $25,000
http://www.wolframscience.com/prizes/tm23/
• Won by Alex Smith
• 20-year-old undergraduate
engineering major at
Univ of Birmingham (UK)
Alan Turing, the man
• Was prosecuted in 1952
for the crime of being a
homosexual
• Required to undergo
chemical castration as
alternative to prison
• Committed suicide by
taking cyanide in 1954
British Gov’t Apology
• On 10 September 2009, Prime Minister
Gordon Brown apologizes on behalf of the
government
• http://www.telegraph.co.uk/news/newstopics/poli
tics/gordon-brown/6170112/Gordon-Brown-Improud-to-say-sorry-to-a-real-war-hero.html
“ Turing was a quite brilliant mathematician,
most famous for his work on breaking the
German Enigma codes.
It is no exaggeration to say that, without
his outstanding contribution, the history of
the Second World War could have been
very different. ”
“ Alan and the many thousands of other gay
men who were convicted, as he was
convicted, under homophobic laws, were
treated terribly. …
I am proud that those days are gone and
that in the past 12 years this Government
has done so much to make life fairer and
more equal for our LGBT community. This
recognition of Alan's status as one of
Britain's most famous victims of
homophobia is another step towards
equality, and long overdue.”
“ It is difficult to believe that in living
memory, people could become so
consumed by hate – by anti-Semitism, by
homophobia, by xenophobia and other
murderous prejudices – that the gas
chambers and crematoria became a piece
of the European landscape as surely as
the galleries and universities and concert
halls which had marked out the European
civilisation for hundreds of years. “
“ …. So on behalf of the British
government, and all those who live
freely thanks to Alan's work, I am very
proud to say: we're sorry.
You deserved so much better. “
Bring humanity into the teaching of
mathematics….
• Ideas for addressing Barzun’s challenge
• Highlighting the humanity and logic that
underlies what we study
Resources
• Web-based
• Books
• Other Print and Media Sources
• TV and Movies
Web-based
• MacTutor History of Mathematics Archive
http://www-history.mcs.st-andrews.ac.uk/history/
• Wikipedia
• Google Searches
• Mathematical Treasures:
http://mathdl.maa.org/jsp/search/searchResults.jsp?url=http://mathdl.maa.org/mathDL/46/?pa=content
&sa=viewDocument&nodeId=2591
• YouTube (2,650 videos “history of math”)
• Wolfram MathWorld
http://mathworld.wolfram.com/
Books
• Howard Eves
(1990), An
Introduction to the
History of
Mathematics (6th
Edition)
Books
By Mario Livio
– The Equation That Couldn’t
Be Solved: How
Mathematical Genius
Discovered the Language of
Symmetry
– The Golden Ratio
– Is God a Mathematician?
Books
• Victor Katz (2008), A History of
Mathematics (3rd Edition)
Books
• William Dunham (1991),
Journey through Genius:
The Great Theorems of
Mathematics
Books
• Berlinghoff and Gouvea
(2003), Math Through
the Ages: a Gentle
History for Teachers and
Others.
Books
• Swetz, Frank (1997), Learn from the
Masters. MAA.
Other Print and Media Sources
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American Mathematical Society books
Mathematics Association of America
Columns at www.maa.org
Keith Devlin “Devlin’s Angle” and “The
Math Guy” on PBS Morning Edition
http://www.stanford.edu/~kdevlin/MathGuy.html
• NCTM Journals
Oct 2009 Mathematics Teacher
• Euclid
• Sierpinski
TV and Movies
• BBC Documentary “Dangerous
Knowledge”
• Good Will Hunting (Cayley’s Formula,
Fourier Theory)
• A Beautiful Mind (story of John Forbes
Nash, Nobel Laureate Economics)
• Math in Movies
http://abel.math.harvard.edu/~knill/mathmovies/
Thank You
• Questions?
• Your own suggestions for helping connect
the teaching of mathematics to the human
beings who discovered or invented it?