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Topic: abstraction

topics > Group: philosophy



Group:
formalism
Group:
function
Group:
natural language
Group:
philosophy of mathematics
Group:
sets

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

Summary

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)((x.in.x).equiv. ~(x.in.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)


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