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glossary..A

abstract
not tangible or concrete
abstract algebra
The generalisation of algebraic methods originally concerned with number systems to deal with arbitrary algebraic structures over arbitrary domains.
abstraction
the process or result of forming some abstract idea from a number of more particular or concrete examples
(in set theory)
The process of forming a set, typically by binding a free variable in a formula which expresses the truth condition, for membership in the set, of the value denoted by the variable. Comprehension and separation are particular kinds of set abstraction. Abstraction to properties (predicates or propositional functions) is analogous.
(functional)
Forming a function, typically by binding a free variable in an expression which denotes the value of the function for the argument whose value is denoted by the variable.
ad
the Latin preposition meaning "to" or "towards"
ad hoc
for a particular purpose. An inelegant feature in an otherwise well structured system to fix a particular problem. A hack.
ad hoc polymorphism
(after [CardelliTDP]) a kind of polymorphism in programming languages in which a function taking a polymorphic parameter will execute different code according to the type of the parameter supplied (by contrast with universal polymorphism in which the same code is used). Special cases include overloading and inclusion polymorphism.
algebra
the systematic study of number systems using symbolic formulae involving variables
see also:
abstract algebra
computer algebra

altruism
unselfishness, concern for others
analytic
(logic)
a proof which proceeds by analysis of the desired conclusion showing that it is derivable from accepted premises (from classical Greece). Such proofs are now sometimes known as a backward or goal oriented proofs.
(philosophy)
expressing a relationship between concepts. A statement or proposition which lacks empirical content and is true in virtue of its meaning.
(philosophy)
a kind of philosophy particularly concerned with logical or linguistic analysis (see: Varieties of Philosophical Analysis).
(recursion theory)
definable in second order arithmetic
anarchism
  1. the doctrinal abhorrence of coercion, usually including advocacy of the abolition of the state
  2. violent opposition to established authorities, especially those considered oppressive
antinomy
a brittle silvery white metalic element
(philosophy)
a paradox or contradiction
antinomian
relating to the belief that christians are absolved from the obligation to observe the moral law.
antinomianism
A religious sect in sixteenth century Germany which held to the antinomian belief.
a posteriori
knowable or justifiable only on the basis of experience
a priori
knowable or justifiable prior to experience based on purely rational considerations
arithmetization
the process of translating statements or problems from their usual domain into the language of arithmetic, usually so that the methods of arithmetic or logic can be brought to bear on the problem.
arithmetization of analysis
the reduction of the theory of real numbers to that of arithmetic, accomplished by defining a real number as some aggregate of rational numbers, e.g. a Dedkind cut or a Cauchy sequence.
see also:
gödelization

axiom
in a logic or an axiomatic theory an axiom is a sentence which is accepted as true without demonstration. The axioms are the starting points for the derivations of all other theorems.
axiomatic method
a method of doing mathematics in which subject areas are presented and studied as axiomatic theories
axiomatic semantics (computing)
a semantics for a programming language given by defining axioms which permit reasoning about the effects of execution of the various parts of a program.
axiomatic theory
a mathematical (or other) theory presented as a system of axioms.
axiomatisation
The process of formalising some subject as an axiomatic theory.


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