Logical Methods

Logical Methods: The Art of Thinking Abstractly and Mathematically, The University Press, 2014 (in Norwegian)

The purpose of the book is to provide a solid foundation for a study in the sciences, and to introduce and explain the most important and essential concepts within mathematics and science.

Many think mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than performing calculations, manipulating symbols, and putting numbers into formulas. Mathematics is about discovering patterns, reasoning, finding counterexamples, and arguing logically. Mathematics is a way of thinking. It is an activity that is both extremely creative and challenging.

This book is intended as an introduction to scientific, in particular mathematical, thinking for entry level university or college students. The purpose of the book is to provide a solid foundation for a study in the sciences. It is designed to introduce and explain the most important, essential concepts within mathematics and science. Through 25 short chapters, in an original and visual presentation, the students will learn the basics within set theory, logic, proof methods, combinatorics, graph theory, and much more.

In the book you will find, among other things, answers to:

  • What does it mean to abstract and to understand?
  • What is a proof and a counterexample?
  • What does it mean that something follows logically from a set of premisses?
  • How can knowledge and information be represented and calculated with?
  • What is the connection between Morse code and Fibonacci numbers?
  • Why may it take billions of years to solve Hanoi's Tower?

"Logical methods" is especially appropriate for those who have not studied before, such that the transition to a university or college studies becomes easier. The book also has transfer value to other studies and is a useful and fascinating entry to scientific and logical thinking.

Roger Antonsen received The University Press Textbook Prize in 2013 for this book.

Table of Contents

0. The Art of Thinking Abstractly and Mathematically
1. Basic Set Theory
2. Propositional Logic
3. Semantics for Propositional Logic
4. Concepts in Propositional Logic
5. Proof, Conjectures, and Counterexamples
6. Relations
7. Functions
8. Some More Set Theory
9. Closures and Inductively Defined Sets
10. Recursive Functions
11. Mathematical Induction
12. Structural Induction
13. First-Order Languages
14. Representation of Quantified Propositions
15. Interpretation in Models
16. Reasoning about Models
17. Abstraction with Equivalences and Partitions
18. Combinatorics
19. Some More Combinatorics
20. Some Abstract Algebra
21. Graph Theory
22. Walks in Graphs
23. Formal Languages and Grammars
24. Natural Deduction

The University Press Textbook Prize

For the person that is to learn a subject, the subject arises as it is communicated. That is why the textbook is an important book for the learner, and that is precisely why it is difficult to write a textbook. The jury has considered a number of project sketches, and one project excelled. This year's recipient of the Textbook Prize has a doctorate in his field, he is also an award winning science communicator, but he is awarded the prize primarily because of his ability to create the subject in his textbook writing. He communicates with words a subject normally associated with numbers, he shows how mathematics is about understanding the world, and invites the reader to see mathematical activity as a creative and truth-seeking process.

Read more at the home pages of The University Press (in Norwegian).

Look inside

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The University Press

Cover and Illustrations

Cover, Marie Låte, 2014
Cover, Marie Låte, 2014

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Illustration from chapter 10 on recursive functions.
Illustration from chapter 10 on recursive functions.