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Axiomatic system AI simulator
(@Axiomatic system_simulator)
Hub AI
Axiomatic system AI simulator
(@Axiomatic system_simulator)
Axiomatic system
In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.
A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed in natural language, which is normal in books and technical papers, the nouns are intended as placeholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof. In a fully formal setting, a logical system such as predicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.
The reduction of a body of propositions to a particular collection of axioms underlies mathematical research. This dependence was very prominent, and contentious, in the mathematics of the first half of the twentieth century, a period to which some major landmarks of the axiomatic method belong. The probability axioms of Andrey Kolmogorov, from 1933, are a salient example. The approach was sometimes attacked at "formalism", because it cut away parts of the working intuitions of mathematicians, and those applying mathematics. In historical context, this alleged formalism is now discussed as deductivism, still a widespread philosophical approach to mathematics.
Major axiomatic systems were developed in the nineteenth century. They included non-Euclidean geometry, Georg Cantor's abstract set theory, and Hilbert's revisionist axioms for Euclidean geometry.
David Hilbert "was the first who explicitly adopted the axiomatic method as an investigative framework for the study of the foundations of mathematics". For Hilbert, a major foundational issue was the logical status of Cantor's set theory. In his list of 23 unsolved problems in mathematics from 1900, Hilbert made the continuum hypothesis the first problem on the list.
Hilbert's sixth problem asked for "axiomatization of all branches of science, in which mathematics plays an important part". He had in mind at least major areas in mathematical physics and probability. Of the effect on science, Giorgio Israel has written:
Founded by mathematician Felix Klein ... the Göttingen School, under the influence of David Hilbert, turned its efforts towards ... set theory, functional analysis, quantum mechanics and mathematical logic. It did so by taking on as its methodical principle the axiomatic method that was to revolutionise the science of [the twentieth century], from the theory of probabilities to theoretical physics.
Israel comments also on cultural resistance, at least in France and Italy, to this "German model" and its international scope. The initial International Congress of Mathematicians had heard the views of Henri Poincaré from France on mathematical physics; Hilbert's list was a submission to the second Congress. The Italian school of algebraic geometry took a different attitude to axiomatic work in theory building and pedagogy.
Axiomatic system
In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.
A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed in natural language, which is normal in books and technical papers, the nouns are intended as placeholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof. In a fully formal setting, a logical system such as predicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.
The reduction of a body of propositions to a particular collection of axioms underlies mathematical research. This dependence was very prominent, and contentious, in the mathematics of the first half of the twentieth century, a period to which some major landmarks of the axiomatic method belong. The probability axioms of Andrey Kolmogorov, from 1933, are a salient example. The approach was sometimes attacked at "formalism", because it cut away parts of the working intuitions of mathematicians, and those applying mathematics. In historical context, this alleged formalism is now discussed as deductivism, still a widespread philosophical approach to mathematics.
Major axiomatic systems were developed in the nineteenth century. They included non-Euclidean geometry, Georg Cantor's abstract set theory, and Hilbert's revisionist axioms for Euclidean geometry.
David Hilbert "was the first who explicitly adopted the axiomatic method as an investigative framework for the study of the foundations of mathematics". For Hilbert, a major foundational issue was the logical status of Cantor's set theory. In his list of 23 unsolved problems in mathematics from 1900, Hilbert made the continuum hypothesis the first problem on the list.
Hilbert's sixth problem asked for "axiomatization of all branches of science, in which mathematics plays an important part". He had in mind at least major areas in mathematical physics and probability. Of the effect on science, Giorgio Israel has written:
Founded by mathematician Felix Klein ... the Göttingen School, under the influence of David Hilbert, turned its efforts towards ... set theory, functional analysis, quantum mechanics and mathematical logic. It did so by taking on as its methodical principle the axiomatic method that was to revolutionise the science of [the twentieth century], from the theory of probabilities to theoretical physics.
Israel comments also on cultural resistance, at least in France and Italy, to this "German model" and its international scope. The initial International Congress of Mathematicians had heard the views of Henri Poincaré from France on mathematical physics; Hilbert's list was a submission to the second Congress. The Italian school of algebraic geometry took a different attitude to axiomatic work in theory building and pedagogy.