Abstract vs instances

This concept of abstract vs instances comes up everywhere and should probably use the same language.

The core idea is that there are things are and there properties of things

abstract / instance universal / existential theorem / proof universal / particular

All of these concepts are from different disciplines but are equivalent. I'm trying to figure out what the fundamental idea is.

The Dance of Abstraction and Instantiation: From Plato to Programming

In the world of philosophy, mathematics, computer science, and logic, there exists a pervasive underlying concept that is central to our understanding of various systems: the interplay of abstraction and instantiation. This principle allows us to discern the common patterns and structures behind diverse phenomena and to operate in the realm of both the concrete and the abstract.

The roots of this fundamental concept can be traced back to ancient Greek philosophy, specifically to Plato's Theory of Forms. Plato posited that our world is filled with imperfect copies of perfect, unchanging archetypes, or Forms, which exist in an abstract realm.

Let's delve into how this concept permeates different fields and phenomena.

1. Classes and Instances in Object-Oriented Programming

In object-oriented programming, a class is the abstract blueprint or archetype (akin to Plato's Form) that defines a type of object. It delineates the properties (data) and behaviors (methods) that a specific instance of the class should have.

For example, consider a class Bird:

class Bird:
    def __init__(self, name, can_fly=True):
        self.name = name
        self.can_fly = can_fly

    def fly(self):
        if self.can_fly:
            return f"{self.name} is flying!"
            return f"{self.name} can't fly!"

An instance of this class is a concrete realization or a specific embodiment of the abstract blueprint:

penguin = Bird("Penguin", can_fly=False)
print(penguin.fly())  # Output: Penguin can't fly!

2. Theorems and Proofs in Mathematics

Mathematical reasoning also manifests this principle. A theorem is an abstract, general statement that has been proven to be true within the system of mathematical axioms. It is the ideal, abstract truth (akin to a Form).

For instance, the Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can write this as: a² + b² = c².

The proof of a theorem is the concrete sequence of logical deductions and mathematical operations that demonstrates the truth of the theorem. Each proof is an instantiation of the theorem:

Proof (for the Pythagorean theorem):

Consider a right-angled triangle. Construct a square on each side. 

The square on the hypotenuse (c²) is made up of two types of pieces: 
- 4 right-angled triangles (each with area ½ab).
- 1 small square in the center (with area (a - b)²).

Therefore, c² = 4*(½ab) + (a - b)²
Or, c² = 2ab + a² + b² - 2ab
Simplifying, we get: c² = a² + b²

3. Quantifiers in Logic

In the realm of logic, quantifiers provide the means to talk about abstract, general properties of sets or classes of objects.

The universal quantifier (∀) is used to make an abstract claim about all elements in a certain domain. For example, ∀x (Bird(x) → CanFly(x)) symbolizes the statement "For all x, if x is a bird, then x can fly".

The existential quantifier (∃) asserts the existence

of at least one instance within a certain domain that satisfies a particular property. For instance, ∃x (Bird(x) ∧ ¬CanFly(x)) symbolizes "There exists an x such that x is a bird and x cannot fly".

Both quantifiers play a role in defining abstract properties and then deriving specific cases or instances.

Abstraction and instantiation form the backbone of logical, mathematical, and computational thinking. This concept traverses the path from the purest realms of philosophy (as in Plato's Forms) to the very concrete domains of programming and engineering, proving its worth as a fundamental tool in comprehending and modeling the world around us. This dance between the abstract and the concrete, the ideal and the actual, the general and the specific, allows us to bridge the gap between the world as we imagine it and the world as we experience it.