The schedule for the course can be found via the link to webTimeEdit top of the page.

28/3: Lecture notes for the sixth lecture were added.
13/2: Lecture notes for the eleventh lecture were added.
24/10: Lecture notes for the nineth lecture were added.
2/11: Schedule for the oral presentations and pairs for feedback published.
24/10: Lecture notes for the fifth lecture were added.
30/9: Lecture notes for the fourth lecture were added.
23/9: Lecture notes for the third lecture were added.
13/9: Lecture notes for the second lecture were updated.
12/9: Lecture notes for the first lecture were updated.

Lecturer: Ulf Persson
Course coordinator: Stefan Lemurell

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.

Lecture notes:

1. Thales and the beginnings
2. The books of Euclid
3. Alexander the Great and the Hellenistic period
4. The Decline of Greek Civilization and Mathematics
5. Renaissance Mathematics
6. Mathematics and Science
9. Gauss - the Prince of Mathematics
11. Foundations of Mathematics

Lectures:


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 Examination below).

Topics for presentation and essay:
Topics in red have already been chosen.

1. Higher plane curves in Greek mathematics
2. Different types of proofs of the Pythagorean theorem. Pros and cons.
3. Erathostenes computation of the circumference of the Earth. The mathematical and empirical aspects.
4. Herons formula for the area of a triangle given the length of the sides with a classical proof. Advantages and motivations of the theorem.
5. The relationship between music and mathematics according to Pythagoras.
6. The computation of Archimedes of a parabolic segment.
7. Archimedes work on hydrostatics.
8. Archimedes method of denoting high numbers.
9. The work of the Indian mathematician Brahmagupta.
10. The mathematics of the quadrivium in medieval Europe.
11. Relations between medieval astrology and mathematics.
12. The mathematical influence and contributions of Fibonacci.
13. Comparing the mathematics of England, France and Italy in the 14th century.
14. Why did mathematics advance so much in the 16th century, in particular the role of Italy. (Eliot)
15. The development of perspectives among Renaissance painters.
16. The origin of projective geometry
17. Leonardo da Vinci computed the center of gravity of a pyramid. How could he have proceeded?
18. The mathematical work of Copernicus.
19. The mathematical work of Kepler.
20. The mathematical work of Galileo.
21. Computation of the length of a day at a given time of the year and latitude using elementary geometry or spherical geometry.
22. Forerunners to Newton and Leibniz in the development of calculus.
23. A comparison between the notations of Leibniz and Newton
24. The history of the number e
25. The history of continued fractions and their basic mathematical features (Stepan)
26. The greatest mathematical advance in the 17th century and why?
27. Napier's presentation of logarithms and its connection with the work of Briggs.
28. How could you use trigonometric tables to simplify multiplication?
29. Solving the cubic equation trigonometrically according to Vieta.
30. Algebraic symbolic notation during the 17th century, in particular different ways of exhibiting equations.
31. Introduction to decimal expansions in Europe and various notations for it.
32. Bounds on the number of real roots to a polynomial according to Descartes.
33. The method of the false position.
34. Different algorithms for multiplication, in particular the so called Russian method
35. The unreasonable effectiveness of mathematics in the natural sciences, and its connections to mathematical platonism.
36. Mathematical paradoxes.
37. The rise of so called analytic geometry
38. The five most interesting mathematicians of either i) England, ii) Germany iii) Italy or iv) France during the 16th or 17th century
39. The origin of probability and the problem of points.
40. Counting and the pre-history of mathematics
41. Spherical geometry versus Euclidean geometry
42. Chinese mathematics during the 13th century
43. Contributions to mathematics made by the Persian Poet Omar Khayyam
44. The origin of algebra
45. The role of the universities in fostering mathematics in Europe during the medieval period.
46. The nature of commercial mathematics
47. The works of Apollonius
48. Types of problems in arithmetic, algebra and mensuration that interested ancient Egyptians.
49. Effect of Newton on the mathematics of Great Britain
50. Mathematics and the rise of the metric system
51. The development of matrices
52. The emergence of geometry in more than three dimensions.
53. Hamilton and the quaternions
54. A comparison between German and French mathematics during the 19th century (Susanne)
55. The rise of measure theory (Richard)
56. Celestial mechanics and its influence on mathematics
57. The hierarchy of infinities according to Cantor. (Jan)
58. The rise of hyperbolic geometry and its forerunners
59. Turing patterns; a historical person on the historical subject of explaining patterns in nature (Anna)
60. The history of the normal distribution
61. The history of infinity
62. Matematikens roll i ekonomisk forskning/teoribygge
63. The contributions of Évariste Galois
64. Newton vs Leibnitz in the invention of calculus (Luquene)
65. Topologins historia (Johan)
66. Fermat and his Last Theorem (Leonard)
67. The contributions of the Arabs to Mathematics. Example of Al Khawarizmi and Abu Al Wafa (Ababacar)
68. History of mathematical notation (Elisabeth)

Schedule for the student presentations:

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 peer review
9/12: Deadline for submission to teachers
16/12: Return from teachers with comments.
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 lectures. 
• 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 BibTex file.