5 Most Fundamental Mathematical Ideas
Amavect ∧ Mycroftiv

1. Grammar, the formation of ideas.
2. Type, the invention of ideas.
3. Computer, the transformation of ideas.
4. Order, the relation of ideas.
5. Model, the interpretation of ideas.

Grammar forms the static language.
Ideas can be written and parsed, but don't otherwise do anything.
A sentence does not change.

Types are the objects and actions of a language, a description of the dynamic language.
A sentence is used to describe how sentences change.
A function describes how something transforms into another, including functions into other functions.
New types can be formed out of existing types, inventing new ideas.

Computing is the transformation of sentences.
Computation is the recognition of the grammar and carrying out the type instructions.
A sentence might be recognized, and a new sentence will be written.
A function might be carried out, transforming a value of a type into a new value of another type.

A grammar-type-computer system constitutes the entirety of a mathematical system.
For example:
  Natural Deduction
  ZFC set theory
  Peano arithmetic
  Martin-Löf Type Theory
  Reality itself.
But what do we do with these systems?

Order describes what to do within a grammar-type-computer system.
Find the relationships between things in those systems.
Prove that 1+1 computes to 2.
Show that TREE(3) is a massive number based on a well-quasi-ordering of rooted trees.

Models are about finding relationships between the systems themselves.
Find the relationship between ZFC and Peano arithmetic.
Find the relationship between calculus and reality.