In my opinion, the following top-down hierarchy should be instilled into students’ hearts. The importance of them degrades by level.

  1. The Prime Mover

    • To know.

      Wir müssen wissen. Wir werden wissen - David Hilbert

    • To predict and control via mechanized computation.

  2. Wisdom

    Ideas and philosophy which generate principles and techniques. It is about thought, motivation and inspiration.

    Examples

    • Axiomatic method in mathematics.
    • From continuous case to discrete case.
    • Convert ODE and PDE to algebraic equation.
    • Accurate analytical computation to approximation numerical computation, but with rigorous proof about existence, uniqueness and stability of the solution, error estimate.
    • Infinite dimension to finite dimension.
    • Infinite set to basis with finite or countable elements.
    • Extract governing parameters of a complex object.
    • Elevation: low level problem is perfectly solved by using a high level tool.

  3. Tao

    Generalized principles and abstract framework.

    Examples

    • Least action principle.
    • Variational principle.
    • Spectrum of linear operators.
    • Principle of linear superposition.

  4. Techniques

    Examples

    • Integration rules.
    • Complex number arithmetics.
    • Tensor manipulations.

  5. Representations

    User interface with different styles and flavors.

    Examples

    • Rigorous and logical.
    • Humorous and relaxed.
    • Static slides and dynamic demonstration.