Overview

History and motivation

• Introduction
• The History of Mathematics
        Stone age, Babylonians, Egypts
        Greeks: Thales, Pythagoras, Euclid, Aristoteles
        What is mathematics about?
• Some early proofs
        Thales' theorem
        Pythagoras' theorem
        The square root of 2 is irrational
        There are infinitely many prime numbers

Kinds of reasoning/arguments

•       inductive/deductive
        direct/indirect
        formal/contentful
        fallacious

The structure of mathematical theories

• Definitions
• Aspects of axiomatizations
        completeness
        independence
  
• Axiomatic theories
        Geometry: Euclid, Hilbert
        Natural numbers: Peano
  

Case study I

• Probability theory
        Let's Make a Deal
        HIV Testing

The nature of mathematical proof

• Laws of reasoning: logic
        Aristotle's syllogisms
        Propositional logic
        Predicate logic
• Syntax vs. semantics
• Methods of proof
        direct/indirect
        mathematical induction

Case study II

• Cantor's theory of the infinite
        sets
        sets of numbers: N, Z, Q, R
        equinumerosity
        diagonal argument

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