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Triviality: Distinguishing what is trivial from what is nontrivial. What is assumed or understood. Prerequisites for duality.

• Integers
• Rationals. Proportionality.
• Fractions. Equivalence classes. Duality of numerator and denominator, vector and covector. Confusion of numerator as answer or as amount. Identification of denominator as unit. Relation to coordinate systems. Ambiguity inherent in expression. Use of explicit coordinate (denominator) but implicit meaning.
• Linear (algebra), linear functions, linearity (derivatives)
• Matrix, array
• Scalars
• Tensors
  • Tensors relate passive and active transformations, see Penrose. Tensors perhaps are related to breakdown in terms of positive and negative eigenvalues, see Orthogonal group and symmetry matrices.
  • Use of units (every answer is an amount and a unit) implicitly and explicitly. d/dx is vector (unit - denominator), dX is covector (amount - numerator). "Applying the unit" (such as "tenth") is a vector field, maps from a scalar field to a scalar field, from 3 to 0.3, not necessarily Cartesian. "Dropping the unit" is the covector field, maps us from the vector field to the scalar field, from 0.3 to 3, that is necessarily Cartesian. Units and scales can be most confusing because they are knowledge that is supposed, assumed, in the background. Consider conversion of units. Implicitness and explicitness of units. Vectors/units are lists "list different units". Covectors/amounts are distributions "combine like units". Implicitness is a sign of premathematics, explicitness of postmathematics.
  • Tensors are required for symmtery and invariants. And duality in general. And beauty? and the related topologies?
  • Partial derivatives are explicit, total derivatives implicit - this distinction between explicit and implicit.
  • Tensor symmetry: Wigner-Eckart.
• Note link to divisibility of numbers and prime decomposition.
• Rectangles, rectangular areas and volumes
• Rootedness in a world, our world. Partial world. Our relationsip with the world.

Nontrivial

• Square numbers and square roots and distances and metrics. Pythagorean theorem.
• Triangles and Geometry.
• Circles and spheres
• Real numbers
• Platonic solids
• Conic sections
• Power series
• Infinite sequences
• Worlds unto themselves. Wholeness. Total world with or without us.
• Rotations, reflections.
• Complex numbers
• Normality is a key tool for understanding a subworld unto itself.

In between

• Stitching: continuity, extension of domain, self superposition.