Topic: abstraction

topics > Group: philosophy

natural language
philosophy of mathematics

abstraction as part of language
abstraction by common attributes
abstraction by name
abstraction by resemblance
abstract data type
abstraction in programming
abstraction in programming language
database as a model of reality
decomposition of a system into levels
hierarchical structures
localized understanding
machine independent programming
metaphysics and epistemology
mathematical proof
models of reality
object-oriented design
science as measurement
sensitivity of software to change
symbolic representation
virtual machine
what is a number


Abstraction is the process of forming concepts from particulars. The Platonic ideal is that these concepts already exist; that particulars are only shadows of what really is. This view is common among mathematicians.

In formal settings, a concept must have sharply defined limits. This is possible if the concept is about other concepts, for example numbers, taxonomies, and invented structures.

In less formal settings, abstraction is a way to base language and thought in other worlds than the sensible one of day to day experience. (cbb 4/94)

Subtopic: abstract forms up

Quote: Plato's theory of forms postulates the existence of abstract forms [»shocJF9_1979]
Quote: measurement belongs to science while eternal truth belongs to faith; e.g., Ptolemy's astronomy vs. Plato's philosophy
Quote: natural phenomena are approximations to Plato's ideal forms; ideal forms exist in artificial worlds such as math and computer science [»wegnP10_1986]
Quote: mathematics is an objective, ideal reality that is neither subjective nor physical [»daviPJ_1981]
Quote: an individual is an instance of something general; pure particulars do not exist [»stawPF_1954]
Quote: something is one if it has a form; e.g., the parts of a shoe is not a shoe if scrambled together [»aris_322a]
Quote: definition concerns forms and their parts; must distinguish parts that belong to concrete objects but not their form [»aris_322a]
Quote: for Aristotle, Form provides the capacity to function without actually doing so
Quote: there is no science but the science of the general
Quote: the true, profound analogies arise from the mathematical spirit that disdains matter and embraces form
Quote: Panini wanted a self-contained science that perfected grammatical description, without recourse to extraneous definition [»misrVN_1966]
Quote: Panini provides a full description of organized data from a living speech; algebraic without justification or extralinguistic definition; e.g., definition by ommission [»misrVN_1966]

Subtopic: eternal vs. tangible existence up

Quote: abstractions either have an "eternal" existence (values, concepts) or a temporary, tangible existence (objects) [»taivA4_1993]
Quote: classes are abstract entities, universals, but not aggregates or collections; e.g., a heap of stones is not a class, What is its size? [»quinWV2_1947]
Quote: for Plato, beings are ideas, sensible objects are defined via ideas, and sensible objects participate in the ideas that designate them [»aris_322a]
Quote: Plato thought it impossible to find a common definition for sensible things; they change too much
Quote: a program is a concrete representation of the ideas in a problem solution. Its structure should match the ideas [»stroB_1991]
Quote: consider universal concepts instead of instances of concepts; e.g., 'gold' as a concept is richer than 'metal', while metal covers more instances [»leibGW4_1679]
Quote: formal reasoning concerns statements that do not mention actual things; e.g., if all alphas are betas and x is an alpha than x is a beta [»russB_1919, OK]
Quote: a universal property need not be indicated [»browGS_1972]

Subtopic: abstraction as innate up

Quote: all intelligence must have a subcognitive substrate to compose categories; otherwise need to define all possible attributes of each category in every context [»frenRM1_1990]
Quote: the ideas of movements and figures are innate in us [»chomN_1965]
Quote: humans manipulate symbol sequences to alter their meaning; a combinatoric system not available to animals

Subtopic: abstraction as concepts, sharp limits up

Quote: an abstraction is a concept with more than one possible realization; solving a problem as an abstraction, solves many problems at once [»parnDL3_1976]
Quote: things are primarily one when their primary being is one
Quote: for logic and rigorous proof, a concept must have sharp limits, i.e., clear membership determination [»fregG_1884]
Note: 'Men are mortal' subordinates the concept 'men' to the concept 'mortal'; not an attribute
Quote: concepts can accurately represent man-made structures that originated as concepts; e.g., chess [»sowaJF_1984]
Quote: principle of abstraction--given any condition, there is a class of those members satisfying the condition [»quinWV2_1937]
Quote: a category is a number of objects which are considered equivalent; e.g., dog, animal [»roscE7_1976]
Quote: the greater the INTERESTS of an ORGANIZATION the sharper the boundaries [»holtAW_1997]

Subtopic: language as abstraction up

Quote: we are led by our thought beyond the scope of our imagination
Quote: we cannot abandon magic; our language assumes the existence of abstract entities; so with mankind and gravity comes Lady Luck and fate [»mitcR_1979]
Quote: all languages are anchored in other worlds rather than in sensible experience [»mitcR_1979]
Quote: using an abstraction is same as using a basic element; but within abstraction have full power of the language [»hehnEC_1977a, OK]

Subtopic: neutral terms up

Quote: use neutral names for new concepts in new fields; avoid words such as 'name,', 'address', 'value', or 'set' [»straC8_1967]

Subtopic: abstraction as understanding complex structures up

Quote: principles of structuring -- transparency of meaning and purpose, recursive application, narrow interface, manifestness of structure [»hoarCA_1974]
Quote: understand X+Y by recursively understanding each part independently; manifest structure, clear meaning and purpose
Quote: the interface between the parts of X+Y is a single number

Subtopic: abstraction as use pattern up

Quote: use patterns, like C++ expressions, are more abstract and readable than an equivalent set of operations and auxiliary types [»dosrG1_2006]

Subtopic: mathematical abstraction up

Quote: mathematical definitions are names or abbreviations of speech to remove tedious drudgery [»galiG_1638]
Quote: Viete introduced "analytics" and the use of letters in the theory of equations; e.g., x^2+ax+b=0 [»vietF_1591]
Quote: mathematical analysis uses the art and rules of logic to obtain equations of species (e.g., 'x') instead of numbers [»vietF_1591]
Quote: rules of arithmetic for calculation by species or forms of things, as, for instance, the letters of the alphabet [»vietF_1591]

Subtopic: compositionality up

Quote: operational semantics requires execution sequence; must have compositionality or definitions become ad hoc [»degaP6_2001]

Subtopic: abstraction as number up

Quote: number applies to concepts that are abstracted from things; this explains number's wide range of applicability [»fregG_1884]
Quote: statements such as "All whales are mammals" and of number concern concepts instead of objects; an indefinite object is really a concept [»fregG_1884]
Quote: classical mathematics requires and assumes universals as values of its bound variables; e.g., numbers [»quinWV2_1947]
Quote: arithmetic expressions are highly efficient because of the extreme narrowness of the interface; a single number stored in a high-speed register [»hoarCA_1974]
Quote: the sole natural object of mathematical thought is the whole number [»poinH_1905, OK]

Subtopic: abstraction as pointing up

Note: pointing is the abstraction of recognition; concrete yet elusive; what are you pointing at? [»cbb_2000, OK]

Subtopic: logical abstraction up

Quote: all uses of the variable 'x' can be transformed into qualification, classes, functions, or combinators; e.g., singular description [»quinWV2_1947]

Subtopic: abstraction as relevance up

Quote: abstraction is ignoring the irrelevant aspects of a subject in order to concentrate solely on the relevant aspects [»taivA4_1993]

Subtopic: abstraction by taxonomy, hierarchy, category up

Quote: a taxonomy is a hierarchy of categories; most abstract is the root [»roscE7_1976]

Subtopic: abstraction by relationships up

Quote: a concept is not isolated; it derives its power from its relationships to related concepts [»stroB_1991]
Quote: things are one when they do, undergo, have, or relate to something in common [»aris_322a]
Quote: an individual is the direct source of another individual; there are no general explanatory factors; may share a common formula [»aris_322a]

Subtopic: Russell's paradox up

Quote: abstraction leads to Russell's paradox, (.thereExists.x)(( ~(
Quote: avoid Russell's paradox by restricting abstraction to stratified conditions that do not reference the abstract class [»quinWV2_1937]
Quote: avoid Russell's paradox by separate rules for class existence and elementhood
Quote: Scott's type theory defines types on a universal set while avoiding Russell's paradox
Quote: Russell's theory of types avoids Russell's paradox by stratified formulas; e.g., .in. only occurs in contexts of the form n .in. n+1

Related Topics up

Group: formalism   (9 topics, 478 quotes)
Group: function   (12 topics, 232 quotes)
Group: natural language   (16 topics, 539 quotes)
Group: philosophy of mathematics   (11 topics, 330 quotes)
Group: sets   (7 topics, 148 quotes)

Topic: abstraction as part of language (18 items)
Topic: abstraction by common attributes (19 items)
Topic: abstraction by name (29 items)
Topic: abstraction by resemblance (13 items)
Topic: abstract data type (64 items)
Topic: abstraction in programming (67 items)
Topic: abstraction in programming language (47 items)
Topic: classification (65 items)
Topic: database as a model of reality (25 items)
Topic: decomposition of a system into levels (49 items)
Topic: entities (20 items)
Topic: hierarchical structures (46 items)
Topic: localized understanding (43 items)
Topic: logic (84 items)
Topic: machine independent programming (13 items)
Topic: metaphysics and epistemology (99 items)
Topic: mathematical proof (23 items)
Topic: models of reality (33 items)
Topic: object-oriented design (30 items)
Topic: reductionism (51 items)
Topic: science as measurement (36 items)
Topic: sensitivity of software to change (44 items)
Topic: symbolic representation (26 items)
Topic: virtual machine (13 items)
Topic: what is a number
(55 items)

Updated barberCB 5/05
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