Math Foundations A (1-79) by Norman J Wildberger

source:  2017年2月1日
Does modern pure mathematics make logical sense? No, unfortunately there are serious problems! Foundational issues have been finessed by modern mathematics, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. Aimed for a general audience, interested in mathematics, or willing to learn.

9:55 1: What is a number?
10:07 2: Arithmetic with numbers
9:36 3: Laws of Arithmetic
10:07 4: Subtraction and Division
9:23 5: Arithmetic and maths education
8:16 6: The Hindu-Arabic number system
10:03 7: Arithmetic with Hindu-Arabic numbers
9:56 8: Division
6:29 9: Fractions
9:42 10: Arithmetic with fractions
6:30 11: Laws of arithmetic for fractions
9:22 12: Introducing the integers
9:15 13: Rational numbers
9:43 14: Rational numbers and Ford Circles
10:01 15: Primary school maths education
7:38 16: Why infinite sets don't exist
9:58 17: Extremely big numbers
8:14 18: Geometry
9:05 19: Euclid's Elements
9:39 20: Euclid and proportions
7:33 21: Euclid's Books VI--XIII
8:01 22: Difficulties with Euclid
8:49 23: The basic framework for geometry (I)
9:39 24: The basic framework for geometry (II)
9:41 25: The basic framework for geometry (III)
6:52 26: The basic framework for geometry (IV)
9:34 27: Trigonometry with rational numbers
9:14 28: What exactly is a circle?
8:39 29: Parametrizing circles
9:53 30: What exactly is a vector?
9:23 31: Parallelograms and affine combinations
9:25 32: Geometry in primary school
7:55 33: What exactly is an area?
9:45 34: Areas of polygons
10:09 35: Translations, rotations and reflections (I)
9:51 36: Translations, rotations and reflections (II)
9:58 37: Translations, rotations and reflections (III)
9:20 38: Why angles don't really work (I)
9:52 39: Why angles don't really work (II)
9:50 40: Correctness in geometrical problem solving
8:11 41: Why angles don't really work (III)
9:43 42a: Deflating modern mathematics: the problem with `functions'
6:27 42b: Deflating modern mathematics: the problem with `functions'
9:52 43: Reconsidering `functions' in modern mathematics
9:54 44: Definitions, specification and interpretation
9:59 45: Quadrilaterals, quadrangles and n-gons
9:46 46: Introduction to Algebra
9:16 47: Baby Algebra
8:11 48a: Solving a quadratic equation
7:08 48b: Solving a quadratic equation
10:06 49: How to find a square root
9:32 50: Algebra and number patterns
9:54 51: More patterns with algebra
10:05 52: Leonhard Euler and Pentagonal numbers
8:48 53: Algebraic identities
9:58 54: The Binomial theorem
10:04 55: Binomial coefficients and related functions
10:08 56: The Trinomial theorem
9:47 57: Polynomials and polynumbers
9:52 58: Arithmetic with positive polynumbers
9:20 59: More arithmetic with polynumbers
9:39 60: What exactly is a polynomial?
9:53 61: Factoring polynomials and polynumbers
7:58 62: Arithmetic with integral polynumbers
10:10 63: The Factor theorem and polynumber evaluation
45:17 64: The Division algorithm for polynumbers
49:53 65: Row and column polynumbers
28:20 66: Decimal numbers
44:23 67: Visualizing decimal numbers and their arithmetic
39:52 68: Laurent polynumbers (the New Years Day lecture)
37:15 69: Translating polynumbers and the Derivative
36:00 70: Calculus with integral polynumbers
36:29 71: Tangent lines and conics of polynumbers
37:50 72: Graphing polynomials 
39:52 73: Lines and parabolas I
38:19 74: Lines and parabolas II
28:09 75: Cubics and the prettiest theorem in calculus
34:33 76: An introduction to algebraic curves
45:50 77: Object-oriented versus expression-oriented mathematics
35:20 78: Calculus on the unit circles
31:56 79: Calculus on a cubic: the Folium of Descartes