1. 1 Pythagorean creed-everything is counted.
1.2 the first irrational number
1.3 the riddle of irrational numbers
1.4 the mystery of continuity
1.5 Deking division
1.6 principle of continuous induction
1.7 Regeneration of "Everything is Count"
Various proofs of 1.8 Pythagorean theorem
1.9 Irrational Numbers and the First Mathematical Crisis
1. 10 "Everything counts" in ancient China culture.
1. 1 1 split in two and split in three.
Chapter II What Geometry Is True —— Axiom of Non-Euclidean Geometry and Modern Mathematics/19
2. 1 Euclid axiomatic method
2.2 Is Euclid's geometric theorem true?
2.3 Discovery of Non-Euclidean Geometry
2.4 Which one is correct?
2.5 What is axiom?
2.6 The difference between the tangents from outside the circle to the circle in ancient and modern times.
2.7 Diversity and limitations of definitions
Chapter III Ghosts of Variables, Infinitesimals and Quantities-The Second Mathematical Crisis and the Concept of Limit /32
3. 1 How does mathematics describe movement and change?
3.2 Instantaneous velocity
3.3 Is differential the ghost of quantity?
3.4 Infinitely small regeneration
3.5 Calculus has no limit
Chapter IV How many natural numbers are there-the concept of "real infinity" in mathematics /50
4. 1 Galileo's riddle
4.2 Cantor, the pioneer who broke into the infinite kingdom.
4.3 Hilbert's Infinite Hotel
Is 4.4 infinity the same?
4.5 How many natural numbers are there
4.6 Confessions of Rational Numbers
4.7 Different expressions of infinite prime numbers
4.8 Rigidity of Mathematics
Chapter V The Great Wave Caused by Russell Paradox-The Third Mathematical Crisis /67
5. 1 logic-Set a number
5.2 Russell Paradox
5.3 Hierarchy Theory of Sets
5.4 Axiomatization of Set Theory
5.5 continuum hypothesis
5.6 the horizon is still ahead
5.7 Paradox and Crisis
Chapter VI What is Number-Some Views on the Essence of Mathematical Objects /79
6. What is11?
6.2 Platonism-Numbers exist in the world of ideas.
6.3 Nominalism-Numbers are symbols on paper or specific concepts in the mind.
6.4 Kant: Numbers are abstract entities created by thinking.
6.5 Traditional view-mathematical rules are just human practices.
6.6 logicism-arithmetic is a part of logic.
6.7 Intuitionism-Mathematical concept is an independent intellectual activity.
6.8 formalism-turning mathematics into an operation about the arrangement of finite symbols.
6.9 Debate and unification
6. Existence and structure of10
6. 1 1 0.9= 1?
Chapter 7 is true, but it can't be proved-Godel Theorem /98
7. 1 godel theorem
7.2 liar paradox and Richard paradox
7.3 How many kinds of arithmetic are there?
7.4 Advantages and limitations of mathematics
7.5 the limitations of mathematics and encryption
7.6 Limitations of Mathematics and Games
Chapter 8 Mathematics and Structure —— The View of Bourbaki School/109
8. 1 is behind the logical long chain.
8.2 Various Appendices
8.3 Basic structure
8.4 The Art of Analysis and Synthesis
8.5 Bourbaki School and New Digital Movement
Chapter 9 Fate or Freedom of Will —— Mathematical Thinking on Necessity and Chance/125
9. 1 Two opposing philosophical views
9.2 Necessity comes from accidents
9.3 From Necessity to Chance
9.4 Can a storm or phlegm affect the fate of a nation?
9.5 What is necessity? What is chance?
9.6 filing principle
9.7 In five hundred years, there will be a king.
Chapter 10 can the example prove the geometric theorem-the unity of opposites of deduction and induction/143
10. 1 example method-deductive support induction method
The geometric theorem of 10.2 can also be proved by examples.
10.3 further thinking
10.4 verify the theorem of triangle interior angle sum.
10.5 exact mathematics and approximate mathematics
10.6 example method and dynamic geometry
Chapter 1 1 The computer is changing mathematics/155
Machine proof of 1 1. 1 four-color theorem
1 1.2 is the theorem proved by computer reliable?
1 1.3 Mathematics and computers develop together.
1 1.4 Chapter 9 Arithmetic Thought
1 1.5 Brief Introduction of Geometric Information Search System
1 1.6 Brief introduction of machine authentication software
Chapter 12 Random Thoughts on Mathematics and Philosophy/174
12. 1 The field of mathematics is expanding and the field of philosophy is shrinking.
12.2 Mathematics has always influenced philosophy.
12.3 abstraction and concreteness
12.4 When it comes to specific issues, the language must be precise and rigorous.
12.5 personal and general
12.6 things and concepts
12.7 "I don't need this assumption.
12.8 verification and forgery
12.9 The mathematical world is created by human beings, but it exists objectively.
The totality of things
12. 1 1 constant is changing.
12. 12 prediction
Nothing is exactly the same
12. 14 poles meet.
12. 15 is in doubt.
12. 16 quantitative and qualitative changes
12. 17 Russell and "matter element"
Reference /20 1