Mathematics: A Very Short Introduction

Timothy Gowers, Oxford University Press 2002
  1. Models
  2. Numbers and abstraction
  3. Proofs
  4. Limits and infinity
  5. Dimension
  6. Geometry
  7. Estimates and approximations
  8. Some frequently asked questions
Notable People: mathematician David Hilbert, Daniel Bernoulli, Maxwell, Boltzmann, Gauss, 

Models - simplified versions of the a part of the world being studied where exact calculations exist.
Circuit - an array of logic gates linked by edges, as a computational artefact.
Commutative law for addition - a+b=b+a
Associative law for addition - a+(b+c)=(a+b)+c
Commutative law for multiplication- ab=ba
Associative law for multiplication a(bc)=(ab)c
Multiplicative identity is 1 - 1a=a
Distributive law - (a+b)c=ac
Additive identity - 0 - 0+a=a
Additive inverse - a+b=0
Multiplicative inverses - a from 0 ther is c such that ac=1
Cancellation law for addition - a+b=a+c, then b=c
Cancellation law for multiplication - a !=0 and ab=ac, then b=c

A sensible policy for calculation and prediction is decide what level of accuracy you need and to achieve it as simply as possible, for which the simplification will only have a small effect on the answer.
Very useful models exist with almost no resemblance to the world.
Simpler or deliberately inaccurate models are useful in general understanding.
Models that have more parameters will yield better results, but that does not take into consideration social, economic, acts of god, etc. changes that can render a models obsolete.
Using models you can work out numerical relationships like between pressure temperature, volume.
A mathematical object is what is does.
A number systems is a collection of numbers with rules for how to do arithmetic.
Extending a number systems allows us to solve numbers we couldn't solve otherwise.

Chapter 3 proofs - reread chapter 1 and 2 and define each of the sub topics