History of Mathematics for PS421 Maurice OReilly CASTeL, Dept of Mathematics, St Patrick’s College Drumcondra History of Mathematics for PS421 20/4/2012

History of Mathematics for PS421

Maurice OReilly CASTeL, Dept of Mathematics,

St Patrick’s College Drumcondra

History of Mathematics for PS421 20/4/2012


1. HoM & me2. History in Maths Education3. HoM Timeline & Themes4. HoM Resources5. HoM & Ireland6. HoM Issues7. Conclusion

The plan


• Gerdes on ethnomathematics, c. 1992• Baghdad & Monasterboice, 1997• IMTA in Dundalk, 1996-7• Computus (Ó Cróinín, McCarthy)• Stevick on bookarts• ICMI & ICME, Monterrey, Mexico, 2008• van Maanen at MEI 3, Dublin, 2009• HPM ESU 6, Vienna, 2010• WG12, CERME7, Rzeszów, 2011• PS421, 2012

1. HoM & me


Gerdes on ethnomathematics, c. 1992


Baghdad & Monasterboice, 1997


Where do the history of maths and the curriculum meet? NUMBERS ALGEBRA PROBABILITY GEOMETRY CALCULUS FUNCTIONS

IMTA in Dundalk, 1996-7


Computus (Ó Cróinín, McCarthy)


Stevick on Early Irish & English Bookarts


ICMI & ICME, Monterrey, Mexico, 2008


van Maanen at MEI 3, Dublin, 2009

x² + 10x = 39


HPM ESU 6, Vienna, 2010


WG12, CERME7, Rzeszów, 2011


PS421, DCU 2012


1. HoM & me2. History in Maths Education3. HoM Timeline & Themes 4. HoM Resources5. HoM & Ireland6. HoM Issues7. Conclusion

The plan


• International setting• International Commission on Mathematical

Instruction (ICMI)• History and Pedagogy of Mathematics (HPM)• ICMI Study: History in Mathematics Education• European Summer School (ESU) on History and

Epistemology in Mathematics Education• Congress of the European Society for Research in

Mathematics Education (CERME)• Why history?

2. History in Maths Education


A view of the international setting


International Commission on Mathematical Instruction

at ICM, Rome 1908by Felix Kleinreconstituted/reborn 1952first ICME, Lyon 1969

(ICMI)


International Study Group on the Relations betweenthe History and Pedagogy of Mathematics (HPM)

Not:

but …


•formally affiliated to ICMI at ICME 3, Karlsruhe 1976

•conferences: satellites at ICME (starting ICME 5, Adelaide 1984)European Summer Schools (ESU, since Montpellier 1993)working groups at ICME & CERME

•10th ICMI Study devoted to H in ME

•HPM Newsletter – online


• Napier & Briggs meeting, Edinburgh 1616• Does HoM have a role in ME?• Relation between how students achieve

understanding and the historical construction of mathematical thinking

• Integration of HoM into teaching – including use of original sources

• Use of technology, e.g. GeoGebra• HoM on the Web, & its use for ME

10th ICMI Study


• Letter from Leibniz to Riccati, c. 1715 on 1-1+1-1+ … = ½

• Cayley’s 1878 paper on groups of order 6: a,b: a² = b³ = 1, ab = ba

• Hamilton’s perception of multiplication of complex numbers: z1z2

Examples:


• Uffe Janqvist on history as tool or history as goal

• Michael Fried on how can ME accommodate HoM

• Michael Glaubitz & James Kiernan on using original sources

• Olivier Bruneau on ICT & pedal curves, C 17th- 19th

• Man Keung Siu on Sino-European dialogue, 1587 …

ESU 6, Vienna 2010


WG12, CERME7, Rzeszów, 2011

Why history?

“… mathematics, as persented to students, fails on two major grounds which you’ve pointed out and that it’s taught in a very dry formal way without any explanation of the origins and the motivations of where things are coming from. That is a terrible mistake and shouldn’t happen, but people are human.”

Michael Atiyah @ RIAIMS Bulletin #58 (2006), p49

Consider student activities:

National Committee for Commemorative Plaques in Science and Technology

Women in Mathematics Day: IrelandUL 27th April 2010



The plan


• Beginnings• Greece, India, China• Arabic & Persian mathematics• Europe to early 16th century• Solving polynomials• Calculus & physics• Rigour & Foundations• Abstraction, specialisation & new synergies

3. HoM Timeline & Themes


Plimpton 322c. 1800 BCEMesopotamiaCuneiformSexagesimalPythagorean triples

Beginnings


Rhind papyrusc. 1650 BCE by the scribe AhmesHieratic Egyptian scriptCalculations with fractions; areas & volumes

710 = 23+130


“Pythagoras’ theorem”from Euclid

Archimedes and using inscribed &

circumscribed polygons

from length, area or volume?

GREECE


Brahmagupta’s theorem

(7th cent CE) Number systemwith place values

and zero

Negative numbers

Trigonometry

INDIA

|AF|=|FD|


Jiuzhang Suanshu or

The Nine Chapters onthe Mathematical Art

Variety of practicalproblems with solutions

c. 100 BCE

CHINA

“Pythagoras” 500-200 BCE

Textual (rather than symbolic), but ‘justified’ by geometry.Specific examples, yet general approach: algorithm

Al-Khwarizmi’s al-jabr wa-l-muqabala

x² + 10x = 39

Omar Khayyam’s solution of x³+cx=d

The parabola x²=-√c yintersects the semi-circle

(having radius dc) at B & D.

x=|BE| is the solution of the cubic.

1048-1131 CE

Europe in the Middle Ages

Trivium (logic, rhetoric & grammar)Quadrivium (arithmetic, geometry, music & astronomy)Boethius (480-524)Isidore of Seville (560-636) – EtymologiaeGathering universal knowledgeCalendar studies important Translations: geometry, number systemsGreek knowledge through Arab world, extended by themLeonardo di Pisa (Fibonacci, c. 1170-1240)Nicole Oresme (1320-1382) – kinematics, exponentials

ScotiiCharlemangeBede

c. 500-1500

Solution of cubic & quartic

Tartaglia (1499-1557) & Cardano (1501-1576)

La Géometrie, 1637

Ars Magna, 1545

Algebrization of geometry

1540-1640

what about Fermat, Wallis, Pascal, …and logarithms – who thought of them?

Calculus …

Leibniz (1646-1716) & Newton (1642-1727)

Differential equations; books

The priority dispute!!

… becomes analysis

1660-1780

Johann Bernoulli (1667-1748) & l’Hôpital (1661-1704)

what about other Bernoullis, Taylor, …doubting Berkeley …also d’Alembert & Agnesi?

Leibniz’ characteristic triangle

transmutation ruleand Gregory-Madhava-Leibniz series:

Political turmoil in FranceLagrange (1736-1813)

Theory of equationsMonge (1746-1818)

Analytic & differential geometry)Laplace (1749-1827)

Statistical inferenceLegendre (1752-1833)

Number theory

… but maths is healthy

1780-1820

number theory, congruences, fundamental theorem of algebra, non-Euclidean and differential geometry, etc., etc.Is England still in the races?

FTA

w=z²+z+1replace 2 by 3, 4, …

As r=|z| increases, thelocus of w must pass through 0, and so …

Complex numbers?



The plan


• Original sources• HoM books• The Web

4. HoM Resources



The plan


• Early medieval Ireland• Irish mathematicians• Foreign mathematicians in Ireland

5. HoM in Ireland

Mathematics in Early Medieval Ireland

Two areas: the Computus & Geometry

Imago LeonisEchternach Gospels f.75v

Fr Ingram lecture, IMTA, Carlow, 2000

St MatthewBook of Moling , f.12v

DIT, February 2008

Insular manuscripts

Carpet page (Luke)Lindisfarne f.138v

Galway Computus conference, 2008

Hamilton in Cabra

Enormous insights:Conical refractionGeneral theory of dynamicsQuaternions

graffiti

tea @ Dunsink

Seán Keating’s version

plaque

Casey in Reykjavík!? New triangle geometry of:•Lemoine•Brocard•Casey

1820-1891

Alastair Wood on Stokes

George Gabriel Stokes 1819 - 1903 An Irish Mathematical Physicist, IMS Bulletin #35

Boole in CorkAn investigation into the Laws

of Thought, on Which are founded the Mathematical

Theories of Logic and Probabilities (1854)

Alicia Boole Stottworked on polytopes:there are six regular polytopes on four dimensions and that they are bounded by 5, 16 or 600 tetrahedra, 8 cubes, 24 octahedra or 120 dodecahedra.

Philip Boland on Gosset

A biographical glimpse of WS Gosset, IMS Newsletter #8



The plan


• Proof & mathematical truth• Notation• Eurocentrism• War & maths• Women in maths• Maths & applications

6. HoM Issues


The History of Europe 2Aidan Hickey

Pythagoras’ theorem

Pascal’s triangle

Gaussian elimination

Eurocentrism

• Picture removed(awaiting artist’spermission)

“Since researching my essay on women mathematicians my attitude towards mathematics has been altered. Although of course these male mathematicians achieved great things during the 16th & 17th century, I can’t help but think about women such as Sophie Germain and Maria Agnesi. These women had to fight so much harder than their male counterparts to achieve or succeed in this area. They did not receive the same respect or education that they should have been entitled to.”

from (female) student’s learning journalHoM module, spring 2010



The plan

“Some treatment of the history of mathematics is very important I think and part of that history is, of course, talking about the people and where they came from with their contribution and it also gives you a chance to explain the motivational origins – the roots if you want – and to follow these things back into the past”

Michael Atiyah @ RIAIMS Bulletin #58 (2006), p49


See the big picture with HoM!


“As from his small windowthe astronaut sees all that he has sprung from”

Alphabets by Seamus Heaney

Documents

History of Mathematics for PS421 Maurice OReilly CASTeL, Dept of Mathematics, St Patrick’s College Drumcondra History of Mathematics for PS421 20/4/2012