# NMAK15016U History of Mathematics 2 (Hist2)

MSc Programme in Mathematics

**History of geometry 1770-1920.**

During the indicated period geometry underwent a development that had wide ranging influence on our understanding of

1. what mathematics is and

2. the nature of space.

It is often mentioned as one of the few examples of a revolution
in mathematics.' Arround 1830 the discussion of the parallel
postulate led to the creation of non-Euclidean geometry. Whether
this geometry was consistent remained an open question until
Gauss' and Riemann's works on differential geometry made it
possible to create a model of the non-Euclidean plane as a surface
of constant negative Gauss curvature. This model was also
interpreted in projective geometry that was also devellopped in the
19th century. The century ended with different new attempts to give
axiomatic descriptions of geometry, among which Hilbert's is
the most famous. The considerations concerning non-Euclidean
geometry was not only an exercise in axiomatics. For all the actors
it was also a question of understanding the nature of (physical)
space. The discovery of non-Euclidean geometry led to a rejection
of Kant's opinion that geometry (for Kant this meant Euclidean
geometry) was an a priori but synthetic intuition. Instead various
empirical, conventional or formalistic epistemologies were put
forward. The mathematical and philosophical considerations of the
19th century created a background for the revolutionary ideas that
Einstein put forward in his special and in particular general
theory of relativity. In the course all these interacting subjects
will be discussed.

Students are required to take an active part and give seminars.

During the course the student will learn to investigate the history of a piece of mathematics, to analyze a mathematical text from the past, and to use the history of mathematics as a background for reflections on philosophical and sociological questions regarding mathematics. Moreover the course will give the students a more mature view on the mathematical subject in question. The course will be particularly relevant for students who aim for a career in the gymnasium (high school) but all mathematics students can benefit from it.

Knowledge:

After having completed the course, the student will have a rather
deep knowledge of the history of geometry in the period 1770 to
1920 and about the historiographical questions related to this
history

Skills:

After having completed the course the student will be able to

1. Read a mathematical text on foundational issues concerning
geometry from the period 1770 to 1920 (in translation if
necessary).

2. Find primary and secondary literature on the subject of the
course.

Competences:

After having completed the course the student will be able to

1. Communicate orally as well as in written form about the selected
topic from the history of mathematics (history of geometry).

2. Analyse a primary historical text (if necessary in
translation) within the subject of the course.

3. Analyse, evaluate and discuss a secondary historical text on the
subject of the course.

4. Use the historical topic of the course in connection
with mathematics teaching and more generally reflect on the
development of the selected topic.

5. Place a concrete piece of mathematics from the selected topic in
its historical context.

6. Independently formulate and analyze historical questions within
a wide field of the history of mathematics.

7. Use the history of mathematics as a background for reflections
about the philosophical and social status of mathematics.

8. Use modern historiographical methods to analyze problems in the
history of mathematics.

Primary sources (mostly in English translations) and secondary papers.

- Category
- Hours
- Exam
- 1
- Lectures
- 35
- Preparation
- 149
- Theory exercises
- 21
- Total
- 206

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- Credit
- 7,5 ECTS
- Type of assessment
- Oral examination, 30 minutes30 minutes oral exam with 30 minutes preparation time. The student will start the exam by giving a 10 minutes version of the seminar presentation.
- Exam registration requirements
In order to qualify for the exam the student must give a 1½ hour seminar presentation during the course and prepare written materials about the subject of the seminar for the use of the other students.

- Aid
- Only certain aids allowed
During the 30 minutes preparation time all aids are permitted. During the exam itself the student is allowed to consult a note with at most 20 words. Other aids are not permitted.

- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
- Re-exam
Same as ordinary exam. If the student has not presented the required seminar, he or she must hand in a 20 page written presentation of one of the seminar questions no later than two weeks before the beginning of the re-exam week.

##### Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

### Course information

- Language
- English
- Course code
- NMAK15016U
- Credit
- 7,5 ECTS
- Level
- Full Degree Master
- Duration
- 1 block
- Placement
- Block 2
- Schedule
- C
- Course capacity
- No limit
- Continuing and further education
- Study board
- Study Board of Mathematics and Computer Science

##### Contracting department

- Department of Mathematical Sciences

##### Course responsibles

- Jesper Lützen (6-727b7a806b744673677a6e34717b346a71)