Blog articles/Mathematics

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Categories: Tutorials | Mathematics

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

For further references see the:

• Haskell math community.
• Mathematics of monads

Contents 1 Haskell for mathematics 1.1 General 1.2 Calculus and Differential Geometry 1.3 Algebraic Topology and Geometry 1.4 Geometry 1.5 Group theory 1.6 Set theory 1.7 Ring theory 1.8 Number theory 1.9 Cryptography and coding theory 1.10 Logic 1.11 Numerics 2 Theorem proving 3 Quantum computing 4 Mathematics of Haskell 4.1 Category theoretic 4.2 Lambda and combinator calculi 4.3 Other formal logics 4.4 Miscellaneous

1 Haskell for mathematics

1.1 General

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

1.2 Calculus and Differential Geometry

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

1.3 Algebraic Topology and Geometry

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

1.4 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

1.5 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

1.6 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

1.7 Ring theory

• Polynomials (update)

1.8 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

1.9 Cryptography and coding theory

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

1.10 Logic

• Laws of Form: An Opinion

1.11 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

2 Theorem proving

• Quick and Dirty Theorem Prover

3 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)
• The Frame Of Reference Monad

4 Mathematics of Haskell

4.1 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.

4.2 Lambda and combinator calculi

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

4.3 Other formal logics

• From Löb's Theorem to Spreadsheet Evaluation

4.4 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

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