# abstraction-in-math

### Abstraction in mathematics

### Abstraction

The acts of the mind, wherein it exerts its power over simple ideas, are chiefly these three:

1. Combining several simple ideas into one compound one, and thus all complex ideas are made.
2. The second is bringing two ideas, whether simple or complex, together, and setting them by one another so as to take a view of them at once, without uniting them into one, by which it gets all its ideas of relations.
3. The third is separating them from all other ideas that accompany them in their real existence: this is called abstraction, and thus all its general ideas are made. -- John Locke, _An Essay Concerning Human Understanding_ (1690)

### Abstraction in general

https://en.wikipedia.org/wiki/Abstraction

The concept of abstraction in general.

### Abstraction

More generally, **abstraction** is the process in which, the previously studied and well-understood details of the studied subject, are undergoing the deliberate simplification as the process of understanding moves from the lower to the higher details, i.e. from particularity toward totality.

### Abstract and concrete

https://en.wikipedia.org/wiki/Abstract\_and\_concrete

* In metaphysics, _abstract_ and _concrete_ are classifications that denote whether the object that a term (reference) describes has physical _referents_.

```
denote, denotation, denotational
term, terminology
reference, referee, referent, referral, referential
```

* _Abstract objects_ have no physical existence, whereas _concrete objects_ do.
* Abstract objects are most commonly used in philosophy and semantics.
* Abstract objects are sometimes called `abstracta` (sing. abstractum) and concrete objects are sometimes called `concreta` (sing. concretum).
* There's no consensus on the characteristics of concreteness and abstractness; one possible approach, is to define the distinction, in terms of the difference, between

1. existence inside or outside space-time
2. whether they have causes and effects or not
3. having contingent or necessary existence

* It is often held that an _abstract object_ is an object that does not exist at any particular time or place, but rather exists as a type of thing, as an (abstract, generalization) idea.
* The study of abstract objects is called _abstract object theory (AOT)_. According to AOT, some objects (the ordinary concrete ones around us, like tables and chairs) exemplify properties, while others (abstract objects like numbers, and what others would call "non-existent objects", like the round square, and the mountain made entirely of gold) merely encode them (this is also known as the dual copula strategy).

### Abstraction in mathematics

https://en.wikipedia.org/wiki/Abstraction\_(mathematics) https://mathvault.ca/math-glossary/

**Abstraction in mathematics** is the process of extracting the underlying structure and patterns from a _mathematical object (object or concept)_. By extracting its essential properties, we are also removing any dependencies it might had with other objects, thereby obtaining a more general version of that object; a version that will, presumably, have a wider range of application, and allow us to make comparisions between many abstractions.

By generalizing the mathematical objects, we can apply the

The process of extracting the underlying structures, patterns, or properties of some mathematical objects, with the intention of generalizing these findings to a broader class of objects.

These include, for example:

* Finding the shared properties of similar polynomials
* Formalizing the general pattern of a sequence
* Establishing a one-to-one correspondence between two sets
* Constructing an alternate axiomatic system for Euclidean geometry

At first, abstraction can seem a bit unappealing due to its higher cognitive demand and lack of connection to real-world phenomena. However, it can often turn out to be an integral force in bringing about a broaden applicability to a field - along with some unification in between the fields.

In math, **abstraction** is the process of extracting the underlying _essence_ of a _mathematical concept_ (by removing any dependence on real world objects with which it might originally have been connected) and _generalizing_ it so that it has wider applications among other abstract descriptions and phenomena.

> In philosophy, **essence** is the property that makes an entity what it fundamentally is, which it has by necessity, without which it loses its identity. Essence is contrasted with **accident** - a property that the entity has contingently, without which it still retains its identity.

Major abstractions:

* Zero
* Hindu-Arabic numerals
* Positional numeral system
* Arithmetic operations: 1 + 3 - 2 ⨯ 4 ÷ 2
* Basic algebra has abstracted numbers: a + b - c ⨯ d ÷ c
* Abstract algebra has abstracted operations: a 🚌 b 🚏 c 🐀 d 👜 c
* Functions

sets -> structures -> groups, filed, ring -> categories

### References

https://en.wikipedia.org/wiki/Abstraction\_(mathematics) https://en.wikipedia.org/wiki/Abstraction\_(computer\_science)

"I've sure loved playing with them toy trucks, although I've strongly preferred analyzing them into pieces, in search of their essence. The accompanying phase, the synthesis of the pieces, was merely an afterthought, if that". -- Nesaac Iwton
