Why Should Historical Truth
Matter to Teachers of
Mathematics?
Dispelling Myths while Promoting
Maths
Judith V. Grabiner
Pitzer College, Claremont, California
jgrabiner@pitzer.edu
1. First Myth: The social history
of mathematics is easy; just
determine what nation or group
your mathematician comes from
and generalize
2. All modern mathematics
comes from Christian men in
the Graeco-European
tradition.
Notable mathematicians of the past who are
included on the 2009 MAA Poster,
“Women of Mathematics”
Hypatia of Alexandria (ca. 355-415)
Gabrielle du Châtelet (1706-1749)
Maria Gaetana Agnesi (1718-1799)
Caroline Herschel (1750-1848)
Marie-Sophie Germain (1776-1831)
Ada Lovelace (1815-1852)
Florence Nightingale (1820-1910)
Christine Ladd-Franklin (1847-1930)
Sofia Kovalevskaia (1850-1890)
Charlotte Angas Scott (1858-1931)
Grace Chisolm Young (1868-1944)
Emmy Noether (1882-1935)
Ann Johnson Pell Wheeler (1883-1966)
Dame Mary Cartwright (1900-1998)
Mina Rees (1902-1997)
Ruth Moufang (1905-1977)
Olga Taussky-Todd (1906-1995)
Grace Hopper (1906-1992)
Emma Lehmer (1906-2007)
Cora Ratto de Sadosky (1912-1981)
Hanna Neumann (1914-1971)
Julia Bowman Robinson (1919-1985)
Olga Ladyzhenskaya (1922-2004)
Olga Arsen’enva Oleinik (1925-2001)
Etta Zubner Falconer (1933-2002)
Over 30% of all U. S. Ph. Ds. in math now are
women. Biological change? Oh, sure.
You can still buy the colorful
MAA “Women of Math” poster
• http://www.maa.org/pubs/posterW.p
df
Muhammad ibn Musa al-Khwarizmi
(c. 780 – 850)
Name Latinized as
Algorismus
Name then confused with Arithmos
Name & method became “Algorithm”
_________________________________
His book:
Al-kitab al-muhtasar fi hisab al-jabr wa’l’muqabala
became “Algebra”
The picture illustrates a bow
and bowstring, or cord, or
(Greek) “chord”
The sine is the half-chord
Since the sine is the half-chord:
Sanskrit: jya - ardha (chord –half)
Shortened into jya or jiva
Transliterated into Arabic as: jiba
RD WTHT VWLS:
jaib = bay, inlet, cavity
Translated into Latin as: Sinus
3. There wasn’t any real
mathematics in the European
Middle Ages. After the decline
of Greek mathematics,
nothing important happened
mathematically in Europe until
the Renaissance.
Merton mean-speed theorem
If the velocity is changing uniformly,
Distance covered in time t
= distance covered by average speed
(Vmax+ Vmin)/2
in the same time
Stated by
William of Heytesbury, 1335,
at Merton College, Oxford
Diagram is first drawn by Nicole Oresme, 1350
Area under velocity graph
= distance covered in time t
= distance covered by average speed
(Vmax+ Vmin)/2
in the same time
Oresme’s diagram
Galileo’s diagram, from
his epoch-making book
Two New Sciences,
1633
1/2 + 2/4 + 3/8 + 4/16 + 5/32 + … k/2k + … Sum?
1/2 + 1/4 + 1/8 + 1/16 + 1/32 +…
+ 1/4 + 1/8 + 1/16 + 1/32 +…
= 1
= 1/2
+ 1/8 + 1/16 + 1/32 +… = 1/4
+ 1/16 + 1/32 +… = 1/8
+ 1/32 +… = 1/16, etc.
Right column adds up to
2
So the sum is 2. This solution is due to Richard Swineshead
(Suiseth), 14th century, nicknamed “Calculator.”
Proof that the harmonic series diverges:
Nicole Oresme, 14th Century
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + …
= 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + …
≥
1/2 + (1/ 4 + 1/ 4) + (1/8 + 1/8 + 1/8 + 1/8) + …
=
1/2 +
1/2
+
1/ 2
which exceeds any given quantity.
+
. ..
4. Newton invented the
calculus just so he could do
his physics.
D. T. Whiteside, ed., The Mathematical
Papers of Isaac Newton, 8 volumes,
Cambridge University Press, 1967 – 1981
The papers on the discovery of the calculus
are in Volume I, covering 1664 – 1666
(in Latin – sorry!)
5. Serious statistical thinking
in the sciences begins in the
natural sciences; the social
sciences learned this from
natural science and copied it
so they’d look scientific.
Adolphe Quetelet, 1796-1874
Quetelet: “Curve of possibilities”: 1830s
Quetelet, heights of French conscripts, early 19th century
(dip in graph greatly exaggerated)
1.57 meters
N = 100,000.
< 1.57 m, 2275 more than predicted
Between 1.570 and 1.624 m, 2114 fewer than predicted
6. The mathematical
approach can solve any
problem.
• Augustin-Louis Cauchy (1789 - 1857)
• Auguste Comte (1798 - 1857)
Check it out!
Victor J. Katz, A History of
Mathematics: An Introduction
(a 950-page Introduction)
Best is the Third Edition,
Addison-Wesley, 2009
“Convergence”: Using historical materials to teach
mathematics
http://mathdl.maa.org/mathDL/46/
History and Pedagogy of Mathematics Newsletter:
http://www.clab.edc.uoc.gr/hpm/NewsLetters.htm
Let’s ask again: Why should historical truth
matter to teachers of mathematics?
1.
2.
3.
4.
Examples
Mathematical practice
In general, truth matters.
Mathematics evolves.