[FOM] Object-Oriented Programming (OOP)

[Victor Makarov]

Below is an excerpt from my paper https://arxiv.org/abs/cs/9906010.

  The main concepts of OOP are the concepts of object and
class. An object is a k-tuple of values (k ³ 1). A value can be, for
example, an integer or
real number, a character or a string of characters or another object. Any
object is an
instance of a class. A class is a description (in an object-oriented
programming language)
of the structure of its objects, some conditions (called class invariant in
the Eifell
language) that all objects of the class must satisfy and some operations on
its objects. A
class is to an object (of its class) as a blueprint of a real object (a
bicycle, for example) is
to the real object.

Note the similarity between the concept of object in OOP and the concept of
mathematical object. A mathematical object, such as a group, a graph and so
on, is also
usually defined as a k-tuple of sets, functions or other mathematical
objects. For example,
a group is a pair( or a 2-tuple) <G, f> where G is the carrier set and f is
a function from
G´G to G, satisfying certain conditions – group axioms. Note that a
concrete group is a
model of the group theory and, more generally, a concrete mathematical
object is a model
of the corresponding theory (excluding primitive mathematical objects such
as, for
example, integer numbers). So we may say that a class is to an object (of
the class) as a
mathematical theory is to a model of the theory (and the class invariant is
the analogue of
the axioms of a theory).

As mathematical theories are the main way of structuring of the mathematical
knowledge, so classes are the main way of structuring of the “programming
knowledge”.
A very important feature of OOP is the possibility to build class
hierarchies by defining a
class as a heir of other class (simple inheritance) or classes (multiple
inheritance). In
mathematics, a similar method is used when we define, for example, the
theory of linear
ordered groups referring to the theory of linear order and the group theory.

Victor Makarov
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