HIERAR

1. Elements

• About Mathematics (What is Math)

  • abstraction

  • history

• Mathematical method

  • rigor, Euclid

  • formal system

  • axiomatic system

  • primitive

  • definition

  • axiom

    • consistency

  • conjecture

  • inference rules

  • theory, theorem, lemma

  • expressiveness vs provability

• Foundations of Mathematics

  • Crisis in the Foundations of Mathematics

  • Hilbert's program

  • 24 Hilbert's problems

  • Axiomatization of Mathematics

  • Logicism

  • Tarski's undefinability theorem

  • Kurt Gödel's Incompleteness Theorems

• Philosophy of Mathematics

• Division of Mathematics

  • Subject matter

    • change, space, quantity, relation

  • Branches of Mathematics

    • discrete vs continuous

    • pure vs applied