The schedule for the course can be found via the link to
top of the page.
: Lecture notes for the sixth lecture
: Lecture notes for the eleventh lecture
: Lecture notes for the nineth lecture
Schedule for the oral presentations and pairs for feedback
24/10: Lecture notes for the fifth lecture
30/9: Lecture notes for the fourth lecture
23/9: Lecture notes for the third lecture
13/9: Lecture notes for the second lecture
12/9: Lecture notes for the first lecture
The course will present a historical and a philosophical perspective
on mathematics. A useful background reference would be the three volumes
of Morris Kline - Mathematical Thought from Ancient to Modern Times.
However, the course will not be a straightforward historical
course,although history will be a useful way of organizing the material.
A general theme of the course will be mathematics and space.Thus we
will start with the axiomatic method of geometry initiated by the
Greeks. We will discuss Euclid and the strengths and weaknesses of its
Geometry. We will put special emphasis on the axiomatic approach that
has served as a basis of presentation as well as justification for
mathematics ever since.
We will then discuss the later contributions to mathematics, by
Apollonius and Archimedes. Its applications, especially to Astronomy.
After the dark Ages we will discuss the mathematical basis for spatial
representation on a plane - the theory of perspective and its
mathematical development of projective geometry. We will touch upon the
influx of algebra due to Descartes and how it opened up a whole slew of
new geometrical objects. We will discuss Hyperbolic geometry in some
detail, especially its philosophical consequences. Then we will discuss
the expansion of geometry in the 19th century, the birth of modern
Algebraic Geometry, Differential Geometry and its implications on
physics, via Relativity Theory.
During the 19th century there was a further influx of
algebra,especially linear algebra. In this connection we will digress on
the birth of Group theory and its influence on Geometry, especially as
seen by Kleins Erlangen program.
The 20th century brought other developments such as topology and the
important influence of algebra, in so called algebraic topology,
especially euler characteristics and cohomology theories. In the field
of analysis it became important to measure complicated sets leading to
modern measure theory and modern mathematical probability.
We will then connect with the axiomatic perspective and discuss the
crisis of the foundations of mathematics. We will discuss Hilberts
axioms for Geometry and his formal instrumental approach to axioms and
his wish of being able to prove the consistency of mathematics, and how
those efforts were foiled by people such as Gödel and Turing during the
Golden decade of mathematical logic.
The course will consist in lectures during which discussions are
encouraged. Students will also be given assignments and work out more
detailed reports and give presentations in front of the class.
Student presentation and essay:
Students are divided into pairs (or a group of three). This
division is marked by colors in the schedule for the oral presentations
below. The purpose of this is for the pair to give each other initial
feedback on both oral presentation, and the text and contents of the
essay. For the oral presentation the pair should meet before the
"official"presentation and give a trial presentation in order to get
feedback. For the essay, the friend should make a peer review on a first
draft,before the essay is submitted to the teachers.
The oral presentation should be in English and 15 minutes long and is
followed by questions and discussions including constructive feedback on
the presentation. The technique used for the presentation is up to the
speaker to decide.
The essay should be not shorter than 4 pages and not longer than 10. It
should be written in English using LaTex (a template provided under
Topics for presentation and essay:
Topics in red have already been chosen.
- Higher plane curves in Greek mathematics
- Different types of proofs of the Pythagorean theorem. Pros and cons.
- Erathostenes computation of the circumference of the Earth. The
mathematical and empirical aspects.
- Herons formula for the area of a triangle given the length of the
sides with a classical proof. Advantages and motivations of the
- The relationship between music and mathematics according to
- The computation of Archimedes of a parabolic segment.
- Archimedes work on hydrostatics.
- Archimedes method of denoting high numbers.
- The work of the Indian mathematician Brahmagupta.
- The mathematics of the quadrivium in medieval Europe.
- Relations between medieval astrology and mathematics.
- The mathematical influence and contributions of Fibonacci.
- Comparing the mathematics of England, France and Italy in the 14th
- Why did mathematics advance so much in the
16th century, in particular the role of Italy. (Eliot)
- The development of perspectives among Renaissance painters.
- The origin of projective geometry
- Leonardo da Vinci computed the center of gravity of a pyramid. How
could he have proceeded?
- The mathematical work of Copernicus.
- The mathematical work of Kepler.
- The mathematical work of Galileo.
- Computation of the length of a day at a given time of the year and
latitude using elementary geometry or spherical geometry.
- Forerunners to Newton and Leibniz in the development of calculus.
- A comparison between the notations of Leibniz and Newton
- The history of the number e
- The history of continued fractions and
their basic mathematical features (Stepan)
- The greatest mathematical advance in the 17th century and why?
- Napier's presentation of logarithms and its connection with the work
- How could you use trigonometric tables to simplify multiplication?
- Solving the cubic equation trigonometrically according to Vieta.
- Algebraic symbolic notation during the 17th century, in particular
different ways of exhibiting equations.
- Introduction to decimal expansions in Europe and various notations
- Bounds on the number of real roots to a polynomial according to
- The method of the false position.
- Different algorithms for multiplication, in particular the so called
- The unreasonable effectiveness of mathematics in the natural
sciences, and its connections to mathematical platonism.
- Mathematical paradoxes.
- The rise of so called analytic geometry
- The five most interesting mathematicians of either i) England, ii)
Germany iii) Italy or iv) France during the 16th or 17th century
- The origin of probability and the problem of points.
- Counting and the pre-history of mathematics
- Spherical geometry versus Euclidean geometry
- Chinese mathematics during the 13th century
- Contributions to mathematics made by the Persian Poet Omar Khayyam
- The origin of algebra
- The role of the universities in fostering mathematics in Europe
during the medieval period.
- The nature of commercial mathematics
- The works of Apollonius
- Types of problems in arithmetic, algebra and mensuration that
interested ancient Egyptians.
- Effect of Newton on the mathematics of Great Britain
- Mathematics and the rise of the metric system
- The development of matrices
- The emergence of geometry in more than three dimensions.
- Hamilton and the quaternions
- A comparison between German and French
mathematics during the 19th century (Susanne)
- The rise of measure theory
- Celestial mechanics and its influence on mathematics
- The hierarchy of infinities according to
- The rise of hyperbolic geometry and its forerunners
- Turing patterns; a historical person on
the historical subject of explaining patterns in nature
- The history of the normal distribution
- The history of infinity
- Matematikens roll i ekonomisk forskning/teoribygge
- The contributions of Évariste Galois
- Newton vs Leibnitz in the invention of
- Topologins historia (Johan)
- Fermat and his Last Theorem
- The contributions of the Arabs to
Mathematics. Example of Al Khawarizmi and Abu Al Wafa
- History of mathematical notation
Schedule for the student
9/11: Susanne Jan Johan
Stepan Luquene Anna
16/11:Ababacar Richard Leonard
Eliot Kira Elisabeth
Schedule for the written essay:
2/12: Deadline for submission
to fellow student for peer review
7/12: Deadline for return from
9/12: Deadline for submission
16/12: Return from teachers
22/12: Final submission
Course requirements are given in the syllabus.
The examination will consist of three parts:
- Compulsory attendance and participation at the lectures
- Give a short oral presentation in English on a topic related to the
- Write an essay that summarize and discuss a selection of the
topics covered inthe lectures. Apart from writing the essay it will
also include giving written feedback to a fellow student's
essay. A template for the essay that should bu used and a sample