Math Foundations/A

Available as a playlist https://www.youtube.com/playlist?list=PL5A714C94D40392AB (1-79)

What is a number?
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Objectives

 * Point out the logical weaknesses in Set Theory, Analysis
 * Create a new framework for mathematics foundations
 * Provide a resource for teachers

Main Principles

 * Start from the beginning
 * Keep things simple and natural
 * Keep an open mind

Natural Numbers

 * Indicated by tallymarks
 * Concepts of the successor, and comparison
 * Variable names can be assigned to numbers, such as n or m

Arithmetic with numbers
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Introduction

 * The definition of Natural number
 * The operation of summing

Properties
Addition is Multiplication (i.e. the product of two numbers)
 * Commutative
 * Associative
 * and each number has a successor
 * Exercise: Prove these laws

Laws of Arithmetic
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Review

 * The commutative law of multiplication - rotation

Further Proofs

 * The associative law for multiplication
 * Distributive laws (left and right)

The 'Standard' Treatment

 * This is not Set Theory
 * Overly complicated
 * Unintuitive
 * Divorced from reality

Subtraction and Division
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Number Theory

 * Factors
 * Squares
 * Primes (and Composites)