Blog articles/Mathematics

Articles using Haskell for mathematics, and the mathematics of Haskell.

For further references see the:

• Haskell math community.
• Mathematics of monads

Haskell for mathematics

General

• Eleven Reasons to use Haskell as a Mathematician
• Haskell for Maths: commutative algebra, combinatorics, number theory, and group theory libraries (blog, hackage)
• Learn Maths with Haskell
• Prototyping thought
• Why Haskell?

Calculus and Differential Geometry

• Practical Synthetic Differential Geometry
• Non-standard analysis, automatic differentiation, Haskell

Algebraic Topology and Geometry

• Algebraic Topology in Haskell
• More Low Cost Geometric Algebra
• Geometric Algebra for Free!

Geometry

• Haskell, PDF and Penrose Tilings
• Visualizing 2D convex hull using Gtk and OpenGL in Haskell
• Calculating the reflect-rotate-translate normal form for an isometry of the plane in Haskell, and verifying it with QuickCheck.
• Drawing fractals in Haskell with a cursor graphics DSEL and a cute list representation

Group theory

• Computational Group Theory in Haskell
• Carry bits and group cohomology
• Monads from Algebra and the the Gray Code from Groups
• Infinite lazy Knuth-Bendix completion for monoids in Haskell

Set theory

• Ordinals in Haskell
  • Countable ordinals
  • Uncountable ordinals and better representations for countable ordinals
  • Ord, Countable Ordinals, and an Idea of sigfpe
• Constructability, Uncountability, and ω-Haskell
• Defining a power set in one line
  • Commentary on this powerset implementation

Ring theory

• Polynomials (update)

Number theory

• Number theory and Haskell:
  • The Division Algorithm
  • QuickCheck and Divisors
  • Greatest Common Divisor
  • GCD and Higher Order Functions
  • The End of GCD

• A foray into number theory with Haskell

Cryptography and coding theory

• Feistel Ciphers and DES in Haskell
• Arithmetic coding in Haskell
• Two-dimensional spatial hashing with space-filling curves

Logic

• Laws of Form: An Opinion
• Incomplete and Utter Introduction to Modal Logic, Pt. 1
• Incomplete and Utter Introduction to Modal Logic, Pt. 2

Numerics

• The Division Bell
  • Part One
  • Part Two
  • Part Three
  • Part Four (Conclusion)
  • The Divisive Aftermath

• Overloading Haskell numbers
  • Part 1: symbolic expressions
  • Part 2: Forward Automatic Differentiation
  • Part 3: Fixed Precision
  • Dynamic epsilons in Haskell with a bit of type hackery

Theorem proving

• Quick and Dirty Theorem Prover

Quantum computing

• The Essence of Quantum Computing
• Monads for vector spaces, probability and quantum mechanics pt. I
• Monads, Vector Spaces and Quantum Mechanics pt. II
• Independence, entanglement and decoherence with the quantum monad
• The Shor Quantum Error Correcting Code (and a Monad for Heat)

Mathematics of Haskell

Category theoretic

• Category Theory and the Category of Haskell programs:
  • Part 1
  • Part 2
  • Part 3

• Category theory and Haskell

• Why isn't ListT list a monad?
• Reverse Engineering Machines with the Yoneda Lemma
• Variable substitution gives a...
• Games, Strategies and the Self-Composition of the List Monad.

Lambda and combinator calculi

• Programs are Proofs: Models and Types in Lambda Calculus
• Don't need the Y combinator

Other formal logics

• From Löb's Theorem to Spreadsheet Evaluation

Miscellaneous

• ML, Haskell and Coq
• Infinitesimal Types
• Djinn, Coq, Monad and a bit of Haskell
• Peano's Axioms IV: Advanced Functions and Integers
• Formal Power Series and Haskell