Transcript Teaching Mathematics, History, and the History of Mathematics Dedicated to the memory of Louise Karlquist (who knew all the QA numbers by heart)

Teaching Mathematics, History,
and the History of Mathematics
Dedicated to the memory of
Louise Karlquist
(who knew all the QA numbers by heart)
And thanks to
• The Ohio State University Libraries
particularly
• Danny Dotson
• Mary Scott
Fermat’s Last Theorem
1993
• Andrew Wiles lectures in Cambridge
• “Modular forms, elliptic curves, and Galois
representations”
• Concluded with Fermat’s last theorem
• Xn + Yn = Zn is impossible in positive whole
numbers if n > 2
• Flurry of email
• Front page New York Times
Found a gap in the proof
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Proof withdrawn in December 1993
Wiles student Richard Taylor contributed
Proof complete by October 1994
Published April 1995, Annals of Mathematics
Greek Texts of Euclid
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J.L. Heiberg – 1883 – 1916
Sir Thomas L. Heath, Dover
Stamatis, 1974 (in library)
New (cheap) reprint with new translation by
Richard Fitzpatrick
OSU’s Euclid, 1570
• The elements of Geometrie of the most auncient
Philosopher Euclid of Megara. Faithfully (now first)
translated into the Englishe toung by H. Billingsley,
Citizen of London. Whereunto are annexed certaine
Scholies, Annotations, and Inuentions, of the best
Mathematicians both of time past, and in this our age.
With a very fruitful Paeface made by M.I. Dee.
Al-Tusi
Plimpton 322
Pythagorean Triples in P-322
• 32 + 42 = 52
• 52 + 122 = 132
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1192 + 1202 = 1692
652 + 722 = 972
5412 + 5462 = 7692
127092 + 135002 = 185412
Plimpton 322
• Otto Neugebauer: “The Exact Sciences in
Antiquity”
• R.C. Buck: “Sherlock Holmes in Babylon”
• Eleanor Robson: Words and Pictures,
New Light on Plimpton 322
Plimpton 322
Bill Casselman on Plimpton 322
Pythagorean Triples
• X2 + Y2 = Z2
• X = m2 – n 2
• Y = 2mn
• Z = m2 + n 2
Diophantus of Alexandria 200-284
• Links to the MacTutor history of math site at St.
Andrews, Scotland
Bachet’s Diophantus Cover Page
Pierre de Fermat 1601 -1665
Fermat’s Marginal Note
Fermat in the margin
Cubem autem in duos cubos,
aut quadratoquadratum in duos quadratoquadratos,
et generaliter nullum in infinitum ultra quadratum potestatem
in duos eiusdem nominis fas est dividere:
cuius rei demonstrationem mirabilem sane detexi.
Hanc marginis exiguitas no caparet.
Fermat’s Last Theorem
• NOVA page from PBS
• MacTutor Page
• Sophie Germain
• Kummer and algebraic number theory
• Andrew Wiles, 1993
X4 + Y4 ≠ Z4
• Fermat really proved this case
• The “method of infinite descent”
Leonhard Euler 1707-1783
X3 + Y3 ≠ Z3
• Euler’s contribution 1770
• Small gap fixed by Gauss
Sophie Germain 1776-1831
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Pen-name letters to Gauss
Sophie Germain primes
p and 2p+ 1 both prime
3 (and 7); 5 (and 11); 11(and 23)
Case I
Xp + Yp ≠ Zp if p does not go into X,Y,Z
Gabriel Lame 1795-1870
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Cyclotomic integers
ζ = cos (2π/p) + I sin(2π/p)
Xp + Yp = (X+Y)(X + ζ Y)…(X+ ζp-1Y)
Arithmetic in the ring Z[ζ]
Unique factorization into primes??
Nope, too bad.
Ernst Eduard Kummer 1810-1893
• Ideals and ideal numbers
• Unique factorization into ideal factors
• “class number” measures failure of prime
factorization of numbers
• Regular Primes
• Kummer: criterion for regular primes, FLT for
regular primes
• 3,5,7,11,13,17,19,23,29,31, (not 37), 41, 43,53,
59, 61, (not 67), 71, 73, 79, . . .
So – FLT motivated a lot of algebraic
number theory
A Century of Computation
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Wolfskehl prize
Flurry of wrong proofs
Regular primes not so hard
Irregular primes tough but possible
Exponent by exponent
Try out new computers!
Dead end?
What next?
• “Elliptic Curves”
• Arose from integrals trying to measure the length
of an ellipse
• Not an ellipse! Cubic
• Group Structure
• Really hot stuff starting in the 50s
• Main line algebraic number theory
• Andrew Wiles – dissertation at Cambridge
Taniyama – Shimura – Weil
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Technical conjecture describing elliptic curves
Frey curve 1984
TSW implies Fermat
Once again, FLT inspired main line math
Andrew Wiles started working (secretly) on TSW
Seven years in the attic
Andrew Wiles in Cambridge
Back to 1993
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Proof withdrawn in December 1993
Wiles student Richard Taylor contributed
Proof complete by October 1994
Published April 1995, Annals of Mathematics
Full force of Taniyama-Shimura-Weil now proved
Maybe a Moral?
• Fermat’s Last Theorem easily understood and
looks like just a puzzle
• Motivated a great deal of mathematics
– Rings of Algebraic Integers
– Applications of Elliptic Curves
– Even more Galois Theory
• Mathematics swings from very concrete to the
very abstract and back again.
What about Math 504?
• Required by State of Ohio for a secondary
teaching license
• Strongly recommended by the College of
Education for admission to the MSAT program
• Audience is mostly math majors who aspire to
high school teaching
• Varying math skills, writing skills, history skills,
geography skills, . . . . . . . .
What to emphasize??
• History vs. Heritage?
– Grattan-Guinness
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Mathematics as a human endeavor?
Historical Approach?
Capstone for a Math major?
Using History to teach Math?
Third Writing Course
• Babcock Committee 1988/McHale Report 2006
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Book Review
Biography
Long Paper
Oral Presentation
But only ten weeks . . .