# What Makes Functional and Object-oriented Programming Equal

http://codinghelmet.com/articles/what-makes-functional-and-object-oriented-programming-equal

Comparison of 4 Functional Programming (FP : F#, Lisp/Clojure Haskell, Scala,… ) concepts in Object-Oriented (OO: C#, C++, Java,… ) :

[Not covered here] : There are other FP techniques lacking in OO: Functor (FoldMap), Monad, etc.

1. Function:

2. Closure (variable binding)

3. Currying

4. Function Composition

Conclusion:

Video Game Animation: Verlet Integration

AI: Stats, Probability, Calculus, Linear Algebra

Search Engine : PageRank: Linear Algebra

Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad

Program “Proof” : Propositions as Types, HoTT

https://awalterschulze.github.io/blog/post/neglecting-math-at-university/

Abstraction: Monoid, Category

Category

# Python in Functional Programming Style

Functional Programming has the following key styles:

1) Lambda function:

2) Map, Filter, Reduce

# A Functional Programmer’s Guide to Homotopy Type Theory (HoTT)

Since April 2019 until I re-visit this Youtube video on 12 August 2019, I can now totally understand his speech after a pause of 4 months by viewing other related Youtube (below prerequisite) videos on Category Theory, Type Theory, Homotopy Type Theory.

That is the technique of self-study:

1. First go through the whole video,
2. Don’t understand? view other related simpler videos.
3. Repeat 1.

Prerequisite knowledge:

1. Homotopy
2. Type Theory
3. Homotopy Type Theory
4. Bijection = Isomorphism
5. Functional Programming in Category Theory Concept: Monad & Applicative

Two Key Takeaway Points:

1. In the Homotopy “Space” : Programs are points in the space, Paths are Types.
2. “Univalence Axiom” : Paths Induce Bijection, vice versa.

# Knowing Monads Through The Category Theory

While Mathematicians like to talk non-sensical abstract idea, Informaticians want to know how to apply the idea concretely:

Mathematical Parlance:

Monoid = Identity + Associative

Endo-functor = functor between 2 same categories

IT Parlance:

Monad is a ‘function’ to wrap the ‘side effects’ (exception errors, I/O,… ) so that function composition in ‘pipeline‘ chained operation sequence is still possible in pure FP (Functional Programming, which forbids side-effects).

This allows monads to simplify a wide range of problems, like handling potential undefined values (with the Maybe monad), or keeping values within a flexible, well-formed list (using the List monad). With a monad, a programmer can turn a complicated sequence of functions into a succinct pipeline that abstracts away additional data management, control flow, or side-effects.[2][3]

# The Evolution of Software

Introduction:

• The curse of Immutability in Functional Programming – no “Looping” (recursion ok), no Date, no Random, …no I/O …

# Alejandro Serrano: Category Theory Through Functional Programming

(Part 1/3) – λC 2017

What is Category ?

Objects

Morphism (Arrows )

Rule1: Associative

Rule 2: Identity

A <– C –> B

Product of Categories : A x B

Unique

Sum of Categories: A + B

Unique

(Either a b)

Co-Product

Reverse all arrows.

Unique

Functor F: C-> D

Mapping of all objects (A, B) in categories C,D

Mapping of arrows f
f : A -> B
Ff : FA -> FB (preservation)
F Id = Id
F (f。g) = Ff。Fg

Example:

Constant C -> F
FC = k
Ff = Id

Arrow Functor F: C -> D

For any object A in C,

F A = D -> A
(Functional Type is also Type)

Functors compose !

Category of categories:

Objects: categories

Arrows : Functors

Mapping of arrows.

Mapping of Objects = predefined

(Part 2/3) – λC 2017

(part 3/3) – λC 2017

# Mathematical Functions vs Programming Functions

Key Points:

• Pure Function
• Partial Function
• Total Function

View at Medium.com

# Functional Programming with Kotlin

3) Higher–Order Function, Closure

View at Medium.com

# Higher Order Function

As Tikhon Jelvis explained in his response, functions map sets to sets, and functions themselves form sets. This is the essence of the untyped lambda calculus. Unfortunately, untyped lambda calculus suffers from the Kleene–Rosser paradox (later simplified to Curry’s paradox).

This paradox can be removed by introducing types, as in the typed lambda calculus. Simple types are equivalent to sets, but  in order to pass a function as an argument to another function (or return one), we have to give this function a type. To really understand what a function type is, you need to look into category theory.

The categorical model for the typed lambda calculus is a category in which objects are types and morphism are functions. So if you want to have higher order functions, you have to be able to represent morphisms as objects — in other words, create a type for functions. This is possible only if the category is cartesian closed. In such a category you can define product types and exponential types. The latter correspond to function types.

So that’s a mathematical explanation for higher order.

# Functional Programming for the Object Oriented – Øystein Kolsrud

• Imperative
• Object- Oriented
• Functional Programming

Part 2: Example – The 8 Queens Problem

Note: A simpler Haskell coding here.

# Programming and Math

Category Theory (CT) is like Design Pattern, only difference is CT is a better mathematical pattern which you can prove, also it has no “SIDE-EFFECT” and with strong Typing.

The examples use Haskell to explain the basic category theory : product, sum, isomorphism, fusion, cancellation, functor…

# BM Category Theory : Motivation and Philosophy

Object-Oriented  has 2 weaknesses for Concurrency and Parallel programming :

1. Hidden Mutating States;
2. Data Sharing.

Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra… =>

$\boxed {\text {CT reveals the way how our brain works by analysing, reasoning about structures !}}$

Our brain works by:  1) Abstraction 2) Composition 3) Identity (to identify)

What is a Category ?
1) Abstraction:

•  Objects
• Morphism (Arrow)

2) Composition: Associative
3) Identity

Notes:

• Small  Category with “Set” as object.
• Large Category without Set as object.
• Morphism is a Set : “Hom” Set.

Example in Programming
:

• Object : Types Set
• Morphism : Function “Sin” converts degree to R: $\sin \frac {\pi}{2} = 1$

Note: We just look at the Category “Types Set” from external Macroview, “forget ” what it contains, we only know the “composition” (Arrows) between the Category “Type Set”, also “forget” what these Arrows (sin,cosin, tgt, etc) actually are, we only study these arrows’ behavior (Associativity).

2.1 : Function of Set, Morphism of Category

Set: A function is

• Surjective (greek: epic / epimorphism 满射),
• Injective (greek : monic / monomorphism 单射)

Category:  [Surjective]

g 。f = h 。f
=> g = h (Right Cancellation )

2.2 Monomorphism

f 。g = f 。h
=> g = h
(Left cancellation)

$\boxed { \text {Epimorphism + Monomorphism =? Isomorphism }}$