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Mathematics as a Human Endeavor
Ed Dickey
University of South Carolina
SCCTM Conference / 23 October 2009
1979 SCCTM Conference
• 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
• 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:
• A place where most would stop but Cardano persisted and this gave him the formula needed to win contests.
pt
Cubic Formula
for
Example
032 23 xxx
3
4
3
11
2 p
27
91
27
)1)(1(9)1(23
3
q
6
2915712
27
)34
(
4
)2792(
22792
3
23
3
32
u
.. And FINALLY
• Generating 3 values of x
• Full explanation at http://en.wikipedia.org/wiki/Cubic_function#Cardano.27s_method
3
1
)6
2925712(3
34
6
2925712
3
23
3
23
x
TI Nspire CAS
WolframAlpha
Derive
GeoGebra
Quartic anyone?
• Cardano published it in Ars Magna (1545)
• Many substitutions and “nested depressed cubics”
• Lodovico Ferrari (at 18!) discovered a quartic formula in 1540.
Quartic glimpse
Sigh…
SIGH…..
FINALLY:
Example
Solve
Using Derive…..
0432 234 xxxx
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:
02345 fexdxcxbxax
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
0432 2345 xxxxx
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
33 xy
Giovanni Saccheri (1667-1733)
• “the good little monk”
• Actually a Jesuit
• “Euclid Freed of Every Flaw”
• Demonstrate that denying Euclid Parallel Postulate leads to a contradiction
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…
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
• 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/politics/gordon-brown/6170112/Gordon-Brown-Im-proud-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 Archivehttp://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 MathWorldhttp://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.
• 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
Other Print and Media Sources
Oct 2009 Mathematics Teacher
• Euclid
• Sierpinski
• 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/
TV and Movies
Thank You
• Questions?
• Your own suggestions for helping connect the teaching of mathematics to the human beings who discovered or invented it?
Recommended