2020 Transdisciplinary studies20 :Mathematics Learn from History

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Academic unit or major
Humanities and social science courses
Takuwa Yoshimi  Kato Fumiharu  Yamada Kotaro 
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Course description and aims

This course is designed and delivered to cultivate the following abilities, attributes, and perspectives which are appropriate and required for students who study at one of the top leading comprehensive universities of science and technology in Japan.
By the completion of this course, the students will have
1) the ability to recognize and explain the nature and broadening scope of certain fields and/or disciplines in science and engineering,
2) the ability to examine the ELSI (Ethical, Legal, and Social Implications/Influences) of those fields and/or disciplines and what role they should play in society and for society,
3) the attitude to seek the broad and transdisciplinary perspectives on science and engineering, and
4) the ability to develop an attitude to examine one's own field of study with a multi-dimensional framework.

This course is designed, developed, and offered jointly by the respective School and the Institute for Liberal Arts.

Traveling around the history of geometry in mathematics, from the classics of Euclidean geometry, via the 19th century discovery of non-Euclidean geometry, to the modern theories, we overview the historical footsteps of the human engagement across mathematical sciences and the philosophical and ideological developments that resulted therefrom. We also try to have a close look into the theories themselves, of the various kinds of geometries from classics to the modern.
The aim of this course lies in elucidating the historical background, processes, and the consequent significance, of the development of the geometrical theories, of which the important ideas can be found nowadays in diverse areas of mathematical sciences, and thereby fostering deeper insight into the modern mathematical sciences in general.

Student learning outcomes

Students will learn the contents of Euclidean geometry, as well as its axiomatic form and the logical outlook, by which they will have a good understanding of the history of the deductive methods by logical proofs, which has already been inherited from classical Greek. They will also be able to establish reasonable insight into the historical processes, from the struggles with the parallel postulate to the discovery of non-Euclidean geometry, and furthermore to the modern theories of geometry, whereby the fundamental sprit of mathematical sciences in general, not only geometrical theories, has been developed.


Ancient civilization, Greek philosophy, Euclidean geometry, parallel postulate, modern European philosophy, non-Euclidean geometry, Riemannian geometry, mathematical sciences

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

As with all the other courses in this category (400 Transdisciplinary Course), this course is offered in the "Active Learning" mode which requires students to take an active role in their own learning. Therefore, they are required to submit a summary report at the end of each session. (In case a student is not able to attend a class, he or she should inform the instructor of the reason for the absence in advance.) Class attendance is required and taken into account for grades.

Homework will be assigned at the end of a class session (a standard lecture), and the first half of the following class session will be devoted to the discussion on the homework.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Establishment of Euclidean geometry and its history (Kato) Details will be provided during each class session
Class 2 The history of the parallel postulate (Kato) Details will be provided during each class session
Class 3 Mathematics in the European "Aufklarung" and the discovery of non-Euclidean geometry (Kato) Details will be provided during each class session
Class 4 Mathematics and mathematical sciences in the system of knowledge Details will be provided during each class session
Class 5 A local "realization" of non-Euclidean geometry 1: Beltrami's pseudosphere (Yamada) Details will be provided during each class session
Class 6 A local "realization" of non-Euclidean geometry 2: lines on the pseudosphere (Yamada) Details will be provided during each class session
Class 7 A local "realization" of non-Euclidean geometry 3: triangles on the pseudosphere (Yamada) Details will be provided during each class session

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.


None required

Reference books, course materials, etc.

Jean Dieudonné "Abrégé d'histoire des mathématiques, 1700-1900", éditions Hermann, 1996 (ISBN-10: 2705660240)
John Stillwell "Mathematics and its history", Springer, 2010 (ISBN-10: 144196052X)

Assessment criteria and methods

For the credits of this course, as with all the other courses in this category (400 Transdisciplinary Course), students have to submit an original paper which addresses "the nature and scope" of the given field/discipline and its "social role." An important part of assessment is made on the quality of the paper. Details of the requirements of the paper will be explained in the first class meeting.

Related courses

  • LAH.S441 : Essence of Humanities and Social Sciences41:History of Mathematical Sciences
  • LAH.S433 : Essence of Humanities and Social Sciences37:History of Science

Prerequisites (i.e., required knowledge, skills, courses, etc.)

It is required to be familiar with calculus and linear algebra, at least at the level of university freshmen.

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