Towards Modern Mathematics

LecturerTomoki NAKANISHI, Professor
DepartmentInstitute of Liberal Arts & Sciences, 2008 Spring
Recommended for:Humanities Students (2)

Course Aims

When we mention mathematics, we consider something that has been recognized, abstracted and formularized by both humans and mathematical logic concerned with that naturally occurring phenomenon known as "numbers". That such a thing is possible is already astonishing, and because mathematics is so intrinsically linked to the natural world in which we live, even if you study the seemingly unrelated field of humanities you will still find that mathematics is closely related in various ways, naturally. Through these lectures, by learning the ways of thinking of the fundamentals of modern mathematics "Sets and Maps", and "Structures and Homomorphisms", we aspire that students will either discover a target for study, or will have their abilities for seeing new hidden mathematical phenomena strengthened.

Key Features

These lectures aim to promote the learning of the fundamentals of modern mathematics, bearing in mind students who have little occasion for studying mathematics in a typically humanities department.

In the early half of the 20th century, mathematics was reorganized around the fundamental ideas of "Sets and Maps", and "Structures and Homomorphisms". Consequently, when anyone had some sort of motive or reason for studying mathematics, a firm background in "Sets and Maps" was required, and understanding of the idea of "Structures and Homomorphisms" would give us new insight into its theory.

The majority of textbooks concerning "Structures and Homomorphisms" are students with mathematics majors, and studying these normally requires professional study and appropriate background knowledge. These lectures will start with "Sets and Maps" from first basis, with the final goal of understanding the significance of the idea of "Structure and Homomorphisms". In this way, reaching the idea of "Structures and Homomorphisms" by the shortest possible route, the contents of this course are quite unique.

In these lectures, through learning the basics and fundamental ways of thinking behind modern mathematics, hopefully everybody will learn to be able to study advanced mathematics in future for their own goals and needs. The tip to the lessons is not to just flick over the lecture notes, but to pretend to attend the lectures themselves and take your own for personal study.


We don't use any reference books or textbooks. All necessary materials will be dealt with in class.

Course Schedule

Part1:Sets and Mapping

Session Contents
1 Mathematical Logic
2 Construction of Sets
3 Sets and Maps
4 Equivalence Relations and Quotient Sets
5 Countable Sets

Part2:Vector Spaces and Linearity

Session Contents
6 Vector Spaces
7 Subspaces / Linear Independence
8 Basis and Dimensions
9 Linear Maps
10 Isomorphisms
11 Dimension Theorem
12 Linear Equations
13 What are real numbers?

Lecture Handouts


These lecture notes are part-corrected manuscripts of lecture notes created for the 2008 Spring Semester class "Towards Modern Mathematics".

At the time of these being drawn up there were two fundamental ideas behind them, as written below.

The first was so that students taking the class might be able to reread over the notes and lesson content at a later date and study it again by themselves. This is because realizing the significance of mathematics or that of the lesson completely in the given time is extremely difficult. Thus, these lecture notes are not just simplified abbreviations of the lesson contents, but necessary and fully written, and consideration has been taken so that their information can be as self-contained as possible.

The second is that it is difficult to practice and thus master mathematics' concepts and symbols. Mathematics makes use of a great deal of otherwise unseen notions and signs. To learn these, in the same way as learning grammar how to spell words in a foreign language, actual repetition and use of them is required. When these notes were being transcribed, consideration was taken to mastering any new notions or symbols and their correct usage.

Therefore, in actual lessons while summing up the explanations of key points the majority of the lecture notes are written up on the blackboard, and students learn each subject by transcribing these into their own notes.

OCW Support Staff's Yasutaka Nakashima (Nagoya University Graduate School of Mathematics) has kindly provided us with clean copy, error-corrected copies of the lecture notes. We give him our gratitude for his contribution.

Lecture Notes

Note:All files are in Japanese

Session 1
Lectuer Notes
Distributed Materials

Session 2 Lectuer Notes
Practice Problems

Session 3 Lectuer Notes
Practice Problems

Sessoin 4 Lectuer Notes
Practice Problems
Distributed Materials

Session 5 Lectuer Notes

Session 6 Lectuer Notes
Practice Problems

Session 7 Lectuer Notes
Practice Problems

Session 8

Lectuer Notes
Practice Problems

Session 9 Lectuer Notes
Practice Problems

session 10 Lectuer Notes
Practice Problems

Session 11 Lectuer Notes

Session 12 Lectuer Notes

Session 13 Lectuer Notes


  1. In order to sit the end of term exam it is necessary to pass both the Part 1 Test and also the Part 2 Report. (For those who do not succeed at the Part 1 test, we allow an extra chance to resit. If the Part 2 Report is not sufficient students will be asked to resubmit).
  2. The final assessment (Excellent/Good/Pass/Fail) is decided on the marks from the final exam.

Last updated

May 08, 2020