Axiom

An axiom is a principle of a theory of a science or an axiomatic system that is not justified within this system or derived deductively.

• 3.1 Traditional logic
• 3.2 Classical Logic
• 3.3 Mathematics
• 3.4 Physics 3.4.1 Ratio of experiment and theory

• 4.1 Article in subject-specific encyclopedias and dictionaries
• 4.2 monographs

Accruals

Within a formalized theory, a theory is a set which is to be proved. An axiom is a proposition which is not to be proved in theory but is beweislos provided. If the selected axioms of the theory are logically independent, so none of them can be derived from the other. As part of a formal calculus, the axioms of this calculus are always derived. It is in the formal or syntactic meaning to a proof; considered semantically it is a circular argument. Otherwise: " If a derivation from the axioms of a calculus or of true statements, so it is called a proof. "

Axiom is used as an antonym of Theorem (in the narrow sense). Theorems as axioms are sentences of a formal calculus, which are connected by derivation relations. Theorems are a few sentences that are derived by formal proof courses of axioms. Sometimes, the terms thesis and Theorem but are used in a broad sense for all valid sentences of a formal system, ie as the generic term that includes both axioms and theorems in the original sense.

Axioms can thus be understood as conditions of the complete theory, in that they are expressible in a formal calculus. Within an interpreted formal language different theories can be distinguished by the choice of axioms. For non - interpreted calculi of formal logic, however, we speak instead of theories of logical systems which are completely determined by the axioms and rules of inference. This puts the notion of derivability or provability: it consists only in relation to a given system. The axioms and the derived statements belong to the object language, the rules in the metalanguage.

However, calculus is not necessarily an axiomatic calculus, ie consisting " of a set of axioms and the smallest possible amount of final rules." There is also proof calculi and tableau calculi.

Distinctions

The term axiom is used in three basic meanings. He referred

Classic Axiom term

The classic axiom term is attributed to the elements of the geometry of Euclid and the Analytica posteriora of Aristotle. Axiom called a classic immediately plausible or conventionally accepted principle or a reference to such a. This meaning was dominant until the 19th century. An axiom is understood to mean not a proof, nor is it a proof accessible. Axioms were thereby considered true statements about existing objects that are outside these sentences. In the 19th century, however, began a study of different axiom systems for different geometries ( Euclidean, hyperbolic, spherical geometry, etc.), all of which could not possibly describe the actual world. This is considered as a motive that the axiom concept was understood formalistic. Therefore, an alternative mode of conception relates an axiom system not simply to the actual world, but follows the scheme: If any structure satisfies the axioms, then it also satisfies the derivations from the axioms (called theorems ). This view is often referred to as Implikationismus, deductivism or eliminative structuralism.

In axiomatized calculus in the sense of modern formal logic, the classical epistemological (evidence, certainty ), ontological (reference to ontologically more fundamental ) or conventional can be omitted criteria for the award of axioms ( acceptance in a particular context ). Axioms differ from theorems then only formally by the fact that they are the basis of logical derivations in a given calculus.

Substantive axiom term

In the empirical sciences are called axioms, fundamental laws, which have often been confirmed empirically. As an example, Newton 's laws of mechanics are called.

Even scientific theories, especially physics, based on axioms. From these theories be concluded meet their theorems and corollaries predictions about the outcome of experiments. Can claims of the theory in contradiction to the experimental observation will adapt axioms. For example, provide the Newtonian axioms only for " slow" and "large " systems good predictions and have been replaced or supplemented by the axioms of special relativity and quantum mechanics. Nevertheless, one uses the Newton 's axioms more of such systems, because the consequences are easy and, for most applications, the results are sufficiently accurate.

Formal Axiom term

Through Hilbert ( 1899), a formal axiom term was dominant: An axiom is any unabgeleitete statement. This is a purely formal property. The evidence or the ontological status of an axiom does not matter and remains one left to separately to be considered interpretation.

An axiom is then a fundamental statement that

• Is part of a formalized system of sets,
• Is assumed without proof and
• Be derived from the other axioms together with all sentences ( theorems ) of the system logical.

Partly it is claimed, in this understanding are completely arbitrary axioms: Axiom A is "an unproven and therefore misunderstood sentence", because whether an axiom is based on insight and therefore is " understandable ", first does not matter. True it is that an axiom - based on a theory - is unproven. But that does not mean that an axiom must be unprovable. The property of being an axiom is relative to a formal system. What is an axiom in science, a theorem may be in another.

An axiom is not understood only insofar as its truth is not formally proven, but is provided. The modern term axiom is used to decouple the Axiom property of the evidence issue, but that does not necessarily mean that there is no evidence. However, it is a defining characteristic of the axiomatic method, that is closed in the deduction of the theorems only on the basis of formal rules and is not made by the interpretation of the axiomatic character use.

Axioms are based only in a pointed formulation to arbitrariness. In a formal system, the demands for consistency, independence and completeness of the axiomatic system are usually placed on them.

Examples of axioms

Traditional logic

• Principle of identity
• Law of contradiction
• Law of excluded middle
• Principle of sufficient reason

Classical logic

• Comprehension axiom: " There is for every predicate P the set of all things that satisfy this predicate. " The original wording was derived from naive set theory Georg Cantor and seemed only the relationship between extension and intension of a concept clearly pronounce. It was a great shock when it turned out that it could not be added consistent with the other axioms in the axiomatization by Gottlob Frege, but Russell's antinomy caused.

Mathematics

Generally in mathematics concepts such as group, ring, body Hilbert space, topological space, etc. by a system of axioms can be defined.

• The field axioms in connection with the arrangement of axioms and the axiom of completeness define the real numbers.
• Parallel axiom: "To every line and every point which does not lie on this line, there is exactly one line parallel to the line through this point. " This postulate of Euclidean geometry was always considered less plausible than the others. Since its validity was contested, an attempt was made to derive it from the other definitions and postulates. As part of the axiomatization of geometry around the turn of the 19th century, it became clear that such a deduction is not possible, since it is logically independent of the axiomatization of the other postulates. Thus the way was free for the recognition of non-Euclidean geometries.
• "Every natural number n has exactly one successor n 1. " Is an axiom of Peano arithmetic, which describes the system of natural numbers with the arithmetic operations addition and multiplication.
• The term " probability " is precisely defined implicitly since 1933 by a situated Kolmogorov axioms. So were all the different stochastic schools - French, German, British, frequentists, Bayesian, probabilists and statisticians - the first time supplied with a unified theory.

Physics

Even theories of empirical science can be " axioms " reconstruct. In the philosophy of science, however, there are different views on what it even means to make a " axiomatization of a theory ." For different physical theories axiomatizations have been proposed. Hans Reichenbach dedicated to, inter alia, in three monographs his proposal a axiomatic theory of relativity, in which he was particularly influenced by Hilbert. Even Alfred Robb and Constantin Carathéodory Axiomatisierungsvorschläge laid before the special theory of relativity. Exists both for the special as for the general theory of relativity in the meantime a lot of discussion in the philosophy of science and in the philosophy of physics Axiomatisierungsversuchen. Patrick Suppes and others have suggested a much debated axiomatic reconstruction in the modern sense as for the classical particle mechanics in its Newtonian formulation, as already submitted Georg Hamel, a student of Hilbert, and Hans Hermes axiomatization of classical mechanics. One of the most talked about proposals a axiomatization of quantum mechanics still the company is one of Günther Ludwig. For the Axiomatic quantum field theory esp. the formulation of Arthur Wightman from the 1950s was important. In the field of cosmology for an axiomatization approaches, inter alia, Edward Arthur Milne was particularly influential. For classical thermodynamics Axiomatisierungsvorschläge exist, inter alia, by Giles, Boyling, Jauch, Lieb and Yngvason. For all physical theories that operate with probabilities, especially statistical mechanics, the axiomatization of probability theory by Kolmogorov was important.

Ratio of experiment and theory

The axioms of a physical theory are neither provable nor formal, so now the usual point of view, direct and total by observations verifiable or falsifiable. One especially popular in the epistemological structuralism perspective of theories and their relation to experiments and the resulting speech According to relate testing a particular theory to reality rather usually statements of the form " this system is a classical particle mechanics ". Succeed, a corresponding theory test were given, for example, correct predictions of measured values, this check can possibly be considered as confirmation that a similar system was counted properly be among the intended applications of the corresponding theory can at repeated failures and should the amount of intended applications can be reduced by appropriate types of systems.