Axiomatization
"The axiomatization of a theory means its presentation in such a way that certain sentences of this theory, the axioms , are placed at the beginning and further sentences are derived from them through logical deduction ."
Axiomatization in general
The concept of axiomatization goes back to Euclid (approx. 350-300 BC), who validly applied this procedure for the first time to a mathematical discipline, geometry. Axiomatization has therefore been understood since Euclid to mean the generation and construction of true propositions (derivatives, theorems) from general principles (definitions, postulates, axioms) which as a whole result in a complete and consistent system. It was not until the end of the 19th century that mathematicians succeeded in axiomatising further areas of their subject area (e.g. the arithmetic of natural numbers by Giuseppe Peano in 1899 or set theory by Ernst Zermelo in 1908).
“As a first approximation, axiomatization consists in bringing order into the set of statements that apply in a certain area. This order consists in trying to isolate as small a subset of all these true statements as possible, which has the property that the other statements follow logically from them; in this (small) subset of statements, the axioms, the entire knowledge about the area is represented. "
“The axiomatic representation of an area provides three things: on the one hand, an economic representation of the knowledge about the area, then a clarification of the interdependencies between the statements of the area, and finally a clear definition of the justification obligations that are assumed with the assertion of statements in the area. "The axiomatization of a theory promotes its" clarity and verifiability. "
“One of the most important consequences of axiomatization is definition”. To build a deductive science one has to start from undefined basic concepts, the meaning of which cannot be explained. At the same time, a new term must not be introduced without being traced back to the basic terms or otherwise explicated.
Axiomatization is also seen as a method which - in order to avoid infinite regress when determining basic concepts for a research area - selects a small group of expressions and uses them in an undefined manner, but uses all other expressions only when they are based on them selected basic concepts or axioms are determined.
Axiomatization in Mathematics
Axiomatization refers to the attempt to reduce mathematical facts to axioms. Historically, this process has been accompanied by increasing formalization . In the wake it comes to the modern universal dispute .
While axioms have been used since ancient times, it wasn't until the late 19th century that serious efforts were made to put all mathematics on an axiomatic basis. David Hilbert added this goal to his list of 23 unsolved problems and initiated the Hilbert program in 1920 to create such a system of axioms.
From 1934 onwards, the French mathematicians group Bourbaki attempted a systematic axiomatization of all mathematics.
In 1931 the Austrian mathematician Kurt Gödel showed that all mathematics cannot be axiomatized ( incompleteness theorem ).
See also
swell
- ^ Carnap: Introduction to Symbolic Logic. 3rd ed., 1968, p. 172.
- ↑ Hoyningen-Huene: Logic. 1998, p. 240 f.
- ↑ Hoyningen-Huene: Logic. 1998, p. 241
- ^ Bußmann: Lexicon of Linguistics. 3rd ed., 2002 / Axiom.
- ↑ Bochenski: The contemporary methods of thinking. 10th ed., 1993, p. 78.
- ↑ Tarski: Introduction to Mathematical Logic. 5th ed., 1977, p. 127.