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]

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]

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

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]

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]

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

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

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

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]

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

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]

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

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

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

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

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]

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