Credits
This year History of Mathematics (HoM) is
offered as a 5credit module (MATH281) to BA2 students and a 7.5credit module
(MATH304) to BA3 students. MATH281 (with 33 contact hours) is a subset of
MATH304 (with 47 contact hours). Detailed announcements are given in Moodle
and on departmental notice boards (outside D204
& outside D115).
The
purpose of this course is to:
 Provide
students with a framework for appreciating the historical development of
mathematics. 
At
the end of the course students should be able to:
 Outline
in very general terms the timeline of the development of mathematics
 Describe
significant historical periods when key changes in mathematical thought
occurred and new areas emerged
 Summarise
some important contributions of prominent mathematicians
 Explain
how topics arising in school mathematics developed historically
 Discuss
important examples of cultural factors influencing the development of
mathematics
 Discuss
the technical details of specific mathematical problems pertinent to 25,
above
 Situate
points 25, above, in a broader historical context
Teaching
methods (with approximate hours allocated to each, MATH281/MATH304):

Lectures
(23/31, including possible lab sessions)


Seminars
(5/8)


Tutorials
(5/8) 
Assessment
instruments:

Journal 

Handin
homework (with possible use of GeoGebra
& Maple worksheets)


Essay 

Final
examination (70%) 
Typical
questions motivating the course:
 How
was the problem of the solution of polynomial equations solved?
 How
have geometry and algebra interacted over time?
 What
lead up to the discovery of the calculus?
 Did
we always have the mathematical notation we use today?
 What,
if anything, is particularly important about ‘Irish’ mathematics?
 Which
cultures contributed importantly to mathematics?
 How
did it all start?
 What
makes a mathematical result ‘important’?
 How
do application and abstraction interact in the development of mathematics?
 Who
are the ‘big’ names in the discipline?
 How
have rigour and intuition interacted in mathematical history?
 How
has the notion of certainty evolved in mathematics?
Key resources (others will appear on the
course's Moodle site):
 Fundamental
 Mac Tutor
History of Mathematics
 Books/extracts
 Derbyshire, J: Unknown
Quantity: A real and imaginary history of algebra with Errata,
Joseph Henry Press, Washington DC 2006
 Hodgkin, L:
A History of Mathematics: From Mesopotamia to Modernity, OUP
2005; introduction
(U Laval,
Québec);
entire
book (FMCS, Tehran); MAA review
 Grabiner,
JV: Why
Proof? A Historian's Perspective [missing
pages 148,149, 155,156, 162 & 163] in Gila Hanna &
Michael de Villiers (eds) Proof and Proving in Mathematics Education
(19^{th} ICMI
Study, 2012)
 Web
 MAA's Convergence
 Rosen's
translation (1831) of alKhwarizmi's aljabr w'al muqabala
 Witmer's
translation (1963; Dover 1993) of Cardano's Ars
Magna (1545)
 Other resources
 Math history books in
library
 GeoGebra
Tube
Past exam papers: 2008,
2010, 2012
Tentative timeline:
Century 
History 
Mathematics 
BC 
Mesopotamia (incl. Babylon) & Egypt ... 
Babylon:
Cuneiform
tablet (c. 1700 BCE)
Egypt:
Rhind
papyrus (c. 1550 BCE) 
BCAD 
Greece & China ... 
Greece
& China 
AD 
India & Japan... 
India
& Japan 
AD 
Irish 'Golden Age' 600850 
Geometry
& Computus 



13 
Frederick
II (b. 1194) was Holy Roman Emperor from 1220 until 1250. 
Fibonacci,
famous for his sequence (published in his Liber Abaci of 1202),
tackled the solution of a cubic equation posed by Johannes of Palermo, in
his Flos of 1225. 
14 


15 


16 

Cardano
and Tartaglia
were involved in intrigue surrounding the general solution of a cubic
polynomial, published in Cardano's Ars Magna of 1545. 
17 
England: Charles I &
II with Oliver Cromwell in between
...
Europe: the glories of the Hapsburgs and of
Louis XIV; securing
Vienna
against the Ottoman empire ... 
Newton
and Leibniz
were rivals in what must be the greatest battle for priority in the
history of mathematics: who invented the calculus? Jacob
and Johann
Bernoulli belonged to the Leibnizian tradition, as did L'Hopital
who tried to 'buy' all the latter's work. L'Hopital wrote the first
textbook on the calculus without giving due credit to Johann B  more
skullduggery! 
18 
Prussia's Frederick the Great ...
American & French revolutions 
Following Johann's work, Euler
made huge advances in the establishment of analysis with outstanding
original contributions to the calculus as well as pioneering work on
notation and textbooks. 
19 
Just about everything in France ...
The British Empire dominates the world ...
Ireland's famine ...
Both Germany &
Italy unite ... 
It wasn't until this century that the concept of limit was
well established with the work of Cauchy,
Weierstrass
and others. The completeness of the real line (Dedekind,
1872) provided a rigorous understanding of how limits extend the rational
numbers to the real numbers.
Abstract algebra
emerged from determinants
to Grassmann
algebras, via matrices,
quaternions,
vector spaces, fields
and much more. The dramatis personae include: Cauchy,
Cayley,
Galois, Hamilton,
Klein,
Sylvester,
...

20 
Two World Wars & many others ...
Global use of millions of gadgets depending on mathematics!
Most of us were born! 
The theory of integration was developed into the 20th
century with the work of Lebesgue
and others. 



Personal
mathematics genealogy
