X marks the spot: the lost inheritance of mathematics

1. Why this book?

Part I: The roots of mathematics

2. Sticks and stones

The unnumbered world

Stones: counting

Sticks: measuring

Multiplying the stick

Tricks with sticks

Rate of change

Units of multiply like rabbits

The volume?

Infinity

The numbered world

3. Abstraction, mistrust, and laziness

Abstraction

Mistrust

Premises

Methods of reasoning in proofs

Kinds of proof

Elegance

Laziness

Putting the three together

4. Algebra, geometry, analysis: the mathematical mindsets

Algebra

Here it is: value abstraction. Let's bring him out again, x the unknown, variable extraordinaire!

Geometry

Sticks are lines

When are shapes the same?

Shapes as tools

Analysis

Building machines

Function behavior and unknown functions

Logic

Logic in math

Logic and meaning

Digging up the roots

Part II: Theory in practice

5. Analytic geometry

Planes and space, numbers by numbers

Function is shape

Parametric graphing

The fault is not in our stars

Curves and surfaces

Coordinate systems

More dimensions than you con shake a stick at

6. Calculus: motion and size

The derivative

Limits

Derivative redux (reduced, that is)

When can we differentiate?

Function approximation and Taylor series

Vectors

Integral calculus

The fundamental theorem of calculus

Logarithm and exponential

The absence and presence of calculus

7. The language of Motion

Linear digression

8. Sound, notes, and harmonics

9. Probability and statistics

Cards and randomness

Combinatorics: counting without counting

Counting cards

Probability distributions: stick the chances

Stochastic processes

Statistics

10. Other geometries: not so straight, these sticks

The universe is bent

Shortest distance between two points

Metric spaces

Nature spiky in coast and leaf

Why are the called fractals?

Iteration and chaos

11. Algebra and the rise of abstraction

Construction and its limits

Imaginary and complex numbers

Algebraic structures: abstraction spreads wide its arms

Morphisms: preservatives added and multiplied

Algebraic view of geometry: slouching toward topology

Category theory

Part III: Toolkit of the theoretical universe

12. The smith and knight

How the user sees the tool

The maker's view of the tool

Beauty in the use of tools

Beauty in the making of tools: artists and artisans

Mathematics as toolkit

Using the toolkit

Expanding the toolkit

13. Building the theoretical universe

Fluids

Electricity and magnetism

14. Computers

Logic embodies

Binary arithmetic embodies

Computer programming

Modular programming, procedures, and functions

Speed, memory, bandwidth, price, size, and efficiency

Human thought and computer "thought"

Computer use in math and science

15. The theoretical universe of modern physics: toolkit included

Relativity

General relativity

Quantum mechanics

Quantum field theory

16. Math education and math in education

Welcome to math problem world

How to teach it

Math in education

Reclaiming the lost

A stone's throw

Index.