The scissors incident
The prerequisites are included before the words that depend on them. Skip what you already know.
Start with step one → Return to the storyReading a Lean statement
High school
Separate the objects, their conditions, and the conclusion.
Functions
High school
A function gives one output for each permitted input.
Symmetry as a transformation
High school
A symmetry rearranges an object while preserving its rules.
Matrices as transformations
High school
A two-by-two table of numbers describes a transformation with its own arithmetic.
Vectors and subspaces
University
Vectors can be added and scaled; a subspace stays closed under those operations.
Powers and positive integers
Middle school
A power repeats multiplication; positive integers start at one.
A hypothetical counterexample
High school
Assume a forbidden solution exists, then keep its evidence together.
Curves defined by equations
High school
An equation describes pairs of coordinates satisfying it.
Prime numbers
Middle school
A prime is at least two and has no positive divisors except one and itself.
Comparing remainders
Middle school
Two integers can differ while having the same remainder.
The curve’s own addition
University
The curve’s points add: the straight line through two chooses their sum.
Torsion points E[ℓ]
University
The points killed by ℓ-fold addition keep the curve’s arithmetic in a finite cage.
Galois representations
University
Encode number-preserving symmetries as transformations of vectors.
Irreducibility
University
No nonzero proper subspace is preserved by every allowed symmetry.