abstraction-in-math
Abstraction in mathematics
Abstraction
The acts of the mind, wherein it exerts its power over simple ideas, are chiefly these three:
Combining several simple ideas into one compound one, and thus all complex ideas are made.
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.
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.
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 calledconcreta(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
existence inside or outside space-time
whether they have causes and effects or not
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
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