Technical Reasoning

A page from Oliver Byrne's version of Euclid's Elements, rendered in a color-coded style that resembles Bauhaus design.

Introduction

Welcome to technical reasoning!

This course discusses many aspects of logic and reasoning used in the technical sciences. It is meant to be of use to people who study

• mathematics
• natural and computational sciences
• statistics
• philosophy

Naturally mathematicians will care mostly about the sections on logic and mathematics, and may entirely skip the sections on science.

Physicists will likely find the math section contains some useful parts. However, to give one example, it is also likely that axiomatic set theory will be less useful.

Statisticians, computer scientists, and philosophers will all likewise each find some parts useful and other parts less useful.

Therefore I have tried to indicate a subsequence of study for several different disciplines.

• Guide for Mathematicians
• Guide for Physicists
• Guide for Computer Scientists
• Guide for Statisticians
• Guide for Philosophers

Prerequisites

This course assumes that the student understands basic algebra, geometry, and arithmetic.

It satisfies Wikiversity's prerequisite for discrete mathematics. Therefore if any course is listed on the portal as requiring discrete mathematics, then one should be well prepared for it by taking this course.

Lesson Schedule

Section 0: Introduction

This is an introduction to the course as a whole.

• Lesson 0: Introduction

Section 1: Examples and Motivations

We look at some ideas from mathematics, science, and philosophy, which motivate the desire for a system of logic.

• Lesson 0: Why Logic?

• Lesson 1: Mathematical Puzzles

• Lesson 2: Examples of Arguments

• Lesson 3: Examples of Mathematical Proofs

• Lesson 4: Example of Program Validation

Section 2: Formal Logic

We will inspect logic itself as an object of study, turning it into a symbolic and formal system.

• Lesson 0: Formalization of Logic
• Lesson 1: Structure of Arguments
• Lesson 2: Structure of Propositions
• Lesson 3: Proof Systems for Propositions
• Lesson 4: Axiom Systems for Propositions

• Lesson 5: Intuitive First-Order Logic
• Lesson 6: Predicates and Quantifiers: Syntax and Semantics
• Lesson 7: Proof Systems for First-Order Logic
• Lesson 8: Axiom Systems for First-Order Logic

Section 3: Mathematical Reasoning

We will apply lessons from formal logic to mathematical problems. Moreover, we will learn techniques of mathematical reasoning which are not easily understood by formal logic, such as the method of counting in two ways.

Principles

• Lesson 0: Introduction
• Lesson 1: Formal Proofs in Algebra and Geometry
• Lesson 2: Examples and Counter-examples
• Lesson 3: Contraposition and Contradiction
• Lesson 4: Proof by Induction
• Lesson 5: The Pigeonhole Principle
• Lesson 6: Counting in Two Ways
• Lesson 7: The Probabilistic Method in Combinatorics

Applications

• Lesson 8: Axiomatic Set Theory, Axioms and Numbers
• Lesson 9: Axiomatic Set Theory, Relations
• Lesson 10: Axiomatic Set Theory, Functions
• Lesson 11: Basic Number Theory
• Lesson 12: Basic Graph Theory

Section 4: Statistical and Scientific Reasoning

We will see that scientific reasoning is very different from mathematical reasoning, and yet mathematics can assist in scientific reasoning.

Section 5: Computational Reasoning

We will use mathematical methods to analyze computer algorithms, and use an extension of formal logic to analyze formal verification of computer programs.