2020 Essence of Humanities and Social Sciences48:History of Mathematical Sciences

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Humanities and social science courses
Takuma Naoki 
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Course description and aims

Predictions of complex system like weather and economy are widely conducted today. Among them, prediction based on mathematical modeling takes an important role.
By reading some chapters of the textbook on the prediction science written by David Orrell: Apollo’s Arrow: The Science of Prediction and the Future of Everything (2007), this lecture examines the history of science of prediction with mathematical modelling, especially on weather and climate.
By tracing the concrete history of prediction science, the lecture aims at deepening the students' understanding of prediction science, especially its limitation and effectiveness from mathematical point of view.

Student learning outcomes

1) To grasp roughly the historical development of prediction science of complex system, especially of weather and climate, along with related mathematical concepts like "sensitivity to initial condition," "modeling error," and "computational irreducibility."
2) To learn that a prediction of phenomena described by deterministic equation was considered easy in the early years, but later it turned out to be difficult than expected.
3) To understand that it is difficult for a new proposition which challenges the existing paradigm to be accepted.


complex system, chaos, science of prediction, sensitivie to initial condition, sensitivity to parameterization, model errror, coomputational irreducibility, paradigm.

Competencies that will be developed

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

Class flow

The lecturer summarizes important discourses of the Chapter 3 and 4 and of some sections of chapter 7 of the textbook, using slides.
If question arises, feel free to ask question during the class.
At the end of the class, students fill in comment sheets. If there are important comments among them, the lecturer make answer to it in the next class.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Orientation and Introduction. After explaining the lecture plan and the evaluation of grades, the lecturer makes elementary explanation of chaos and complex system. To understand roughly the concept of chaos and complex system. As preparation for the next class, students are required to read Appendix I to III, in addition fo Chapter 3.
Class 2 Reading of Chapter 3 ("Divide and Conquer: The Gospel of Deterministic Science") (1st half) We trace the successful history of deterministic science and its straightforward progress before the recognition of “chaos” and “complex system.”
Class 3 Reading Chapter 3 (2nd half). We tarce the anguished history of science of prediction after the recognition of “chaos” and “complex system (with emergent properties).” Students understand what is “chaos” where prediction is difficult due to sensitivity to initial conditions and what is “complex system (with emergent properties)” whose exact prediction is “essentially impossible.”
Class 4 Reading of Chapter 4 ("Red Sky at Night: Predicting the Weather") (1) (Early sections) We trace briefly the development of mathematical modeling in the early history of weather forecast.
Class 5 Reading of Chapter 4 (2) (Middle Sections). We trace the history of weather forcast in the age computer. We look back the beginning of GCM (General Circulation Model or Global Climate Model), accentuation of chaos by Edward Lorenz, etc.
Class 6 Reading of Chapter 4 (3) (Late sections). We trace the hisotry of identifying the cause of the prediction error. We learn that the error of the weather forecast is less due to the sensitivity to initial condition than the error of the modeling itself (“model error”). We also understand why meteorologist want to attribute the cause of prediction error to the sensitivity of chaos.
Class 7 Reading of some sections of Chapter 7 which are related to Global Warming. We deepen our understanding of what and to which extent we can tell about the future Climate Change

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.


David Orrell (2007), Apollo’s Arrow: The Science of Prediction and the Future of Everything, Harper Perennial.

Reference books, course materials, etc.

Students are recommended to read the relevant literature referred in the textbook. Precise information of the literature is listed in the bibliography of the textbook.

Assessment criteria and methods

Class participation 30%, Mid-term report 20%, Term essay 50%.
Class participation score is evaluated mainly based on comment sheet submitted at the end of each class.
Outline of the Mid-term report and the Term essay will be explained in the 1st class. Detailed instruction will be given in later classes.

Related courses

  • LAH.S433 : Essence of Humanities and Social Sciences37:History of Science
  • LAH.T102 : History of Science A
  • LAH.T202 : History of Science B
  • LAH.T302 : History of Science C

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

General knowledge of mathematics and physical sciences at undergraduate level.


No classes will be given on April 8 (Wed).
Seven total classes will be held for this course: April 15 (Wed), April 22 (Wed), May 13 (Wed), May 20 (Wed), May 27 (Wed), June 3 (Wed), ※ June 3 (Wed) 7-8 period.
※Please take note that on June 3 (Wed), after the regular class, the seventh class will be held in the 7-8 period.

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