How are theorems proved without axioms

axiom

 

A axiom (from Greek ἀξίωμα: "Appreciation, judgment, principle accepted as true") is a principle of a theory, a science or an axiomatic system that is not justified or deductively derived within this system.

Delimitations

Within a theory that can be formalized, a thesis is a proposition that is to be proven. An axiom is a proposition that is not supposed to be proved in theory, but is presupposed without proof. If the chosen axioms of the theory logically independent none of them can be derived from the others. The axioms of this calculus can always be derived within the framework of a formal calculus. In the formal or syntactical sense, this is a proof; From a semantic point of view, it is a circular argument. Otherwise the following applies: "If a derivation is based on the axioms of a calculus or on true statements, then one speaks of a proof."

Axiom is used as the opposite of theorem (in the narrower sense). Theorems and axioms are theorems of a formalized calculus that are connected by derivative relationships. Theorems are theorems that are derived from axioms through formal proofs. Occasionally, however, the terms thesis and theorem are used in a broader sense for all valid propositions of a formal system, i.e. as a generic term that encompasses both axioms and theorems in the original sense.

Axioms can thus be understood as conditions of the complete theory insofar as they can be expressed in a formalized calculus. Within an interpreted formal language, different theories can be distinguished by the choice of axioms. In the case of uninterpreted calculi of formal logic, one speaks of instead of theories logical systems, which are completely determined by axioms and rules of inference. This relativizes the concept of deducibility or provability: it only ever exists in relation to a given system. The axioms and the derived statements belong to the object language, the rules to the metalanguage.

However, a calculus is not necessarily one Axiomatic calculus, which therefore consists of “a set of axioms and the smallest possible set of rules of inference”. There are also proof calculi and tableau calculi.

Immanuel Kant calls axioms “synthetic principles a priori, provided that they are immediately certain” and by this definition excludes them from the realm of philosophy. This is based on concepts which, as abstract images, never have any evidence as an object of immediate intuition. He therefore distinguishes the discursive principles of philosophy from the intuitive principles of mathematics: the former would have to “be comfortable with justifying their authority on account of them by thorough deduction” and therefore do not meet the criteria of an a priori.

Distinctions

The expression axiom is used in three basic meanings. He describes

  1. an immediately obvious principle - the classic (material) axiom,
  2. a law of nature that can be postulated as a principle for empirically well-confirmed rules - the scientific (physical) axiom concept,
  3. a starting sentence that is assumed to be valid in a calculus of a formal language - the modern (formal) concept of axioms.

Classic axiom term

The classic axiom is applied to the elements the geometry of Euclid and the Analytica posteriora of Aristotle. axiom In this view, denotes an immediately illuminating principle or a reference to such a principle. An axiom in this essentialist sense does not need any proof because of its empirical evidence. Axioms were viewed as absolutely true propositions about existing objects that oppose these propositions as objective realities. This importance was prevalent until the 19th century.

At the end of the 19th century there was a “cutting of the cord from reality”. The systematic investigation of different axiom systems for different geometries (Euclidean, hyperbolic, spherical geometry, etc.), which could not possibly all describe the actual world, had to result in the axiom concept being understood more formalistically and axioms as a whole taking on a conventional character in the sense of definitions . David Hilbert's writings on axiomatics proved to be groundbreaking, replacing the postulate of evidence stemming from the empirical sciences with the formal criteria of completeness and consistency. An alternative conception therefore does not simply relate an axiom system to the actual world, but follows the scheme: If some structure fulfills the axioms, then it also fulfills the derivations from the axioms (so-called Theorems). Such views can be located in implicationism, deductivism or eliminative structuralism.

In axiomatized calculi in the sense of modern formal logic, the classic epistemological (evidence, certainty), ontological (reference to ontologically more fundamental things) or conventional (acceptance in a certain context) criteria for marking axioms can be omitted. Axioms differ from theorems only formally in that they are the basis of logical deductions in a given calculus. As a “fundamental” and “independent” principle, they cannot be derived from other source sentences within the axiom system and are therefore not accessible to any proof.

Scientific axiom concept

In the empirical sciences, axioms also refer to fundamental laws that have been empirically confirmed many times. Newton's axioms of mechanics are given as an example.

Scientific theories, especially physics, are also based on axioms. Theories are inferred from these, the theorems and corollaries of which make predictions about the outcome of experiments. If statements of the theory contradict experimental observation, the axioms are adjusted. For example, Newton's axioms only provide good predictions for “slow” and “large” systems and have been replaced or supplemented by the axioms of special relativity and quantum mechanics. Nevertheless, one continues to use Newton's axioms for such systems, since the conclusions are simpler and the results are sufficiently accurate for most applications.

Formal axiom concept

Hilbert (1899) made a formal axiom dominant: An axiom is any inferred proposition. This is a purely formal quality. The evidence or the ontological status of an axiom is irrelevant and is left to a separate interpretation.

A axiom then a basic statement is that

  • Is part of a formalized system of sentences,
  • is accepted without evidence and
  • from which, together with other axioms, all propositions (theorems) of the system are logically derived.

It is sometimes claimed that axioms are completely arbitrary in this understanding: an axiom is "an unproven and therefore not understood sentence", because whether an axiom is based on insight and is therefore "understandable" initially does not matter. It is true that an axiom - related to a theory - is unproven. But that doesn't mean that an axiom has to be unprovable. The quality of being an axiom is relative to a formal system. What is an axiom in one science may be a theorem in another.

One axiom is misunderstood only insofar as its truth is not formally proven, but presupposed. The modern concept of axioms serves to decouple the axiom property from the problem of evidence, but this does not necessarily mean that there is no evidence. However, it is a defining feature of the axiomatic method that in the deduction of the theorems, conclusions are only drawn on the basis of formal rules and no use is made of the interpretation of the axiomatic signs.

Examples of axioms

Traditional logic

  • Theorem of identity
  • Theorem of contradiction
  • Sentence of the excluded third party
  • Theorem of sufficient reason

Classic logic

  • Comprehension axiom: “For every predicate P there is a multitude of all things that fulfill this predicate. "

The original formulation comes from the naive set theory of Georg Cantor and only seemed to clearly express the connection between extension and intention of a term. It was a great shock when it turned out that it was not in the axiomatization by Gottlob Frege free of contradictions could be added to the other axioms, but evoked Russell's antinomy.

mathematics

Axioms are the foundation of mathematics.

In general, terms such as group, ring, body, Hilbert space, topological space etc. are defined in mathematics by a system of axioms. Sometimes there are also axioms Laws called (e.g. the associative law).

  • The field axioms in connection with the arrangement axioms and the completeness axiom define the real numbers.
  • Axiom of parallels: "For every straight line and every point that is not on this straight line, there is exactly one straight line through this point that is parallel to the straight line." This postulate of Euclidean geometry was always considered less plausible than the others. Since its validity has been contested, attempts have been made to derive it from the other definitions and postulates. In the context of the axiomatization of geometry at the turn of the 19th century it turned out that such a derivation is not possible, since it is logical from the axiomatization of the other postulates independently is. This cleared the way for recognition non-Euclidean geometries.
  • “Any natural numbern has exactly one successorn + 1.“Is an axiom of Peano arithmetic, which describes the system of natural numbers with the arithmetic operations addition and multiplication.
  • The term “probability” has been defined exactly implicitly since 1933 by a system of axioms set up by Kolmogorow. This provided all different stochastic schools - French, German, British, frequentists, Bayesians, probabilists and statisticians - with a uniform theory for the first time.

physics

Suggestions for the axiomatization of important sub-areas

Theories of the empirical sciences can also be reconstructed “axiomatically”. In the philosophy of science, however, there are different understandings of what it means to “axiomatize a theory”. Axiomatizations have been proposed for different physical theories. Hans Reichenbach devoted himself, among other things, in three monographs to his proposal for an axiomatic theory of relativity, whereby he was particularly strongly influenced by Hilbert. Alfred Robb and Constantin Carathéodory also submitted axiomatization proposals for the special theory of relativity. For both the special and the general theory of relativity, there are now a large number of attempts at axiomatization discussed in the theory of science and in the philosophy of physics. Patrick Suppes and others have proposed a much discussed axiomatic reconstruction in the modern sense for classical particle mechanics in their Newtonian formulation, and Georg Hamel, a student of Hilbert, and Hans Hermes have already presented axiomatizations of classical mechanics. Günther Ludwig's company is still one of the most popular proposals for an axiomatization of quantum mechanics. The formulation by Arthur Wightman from the 1950s was particularly important for axiomatic quantum field theory. In the field of cosmology, among others, Edward Arthur Milne was particularly influential for approaches to axiomatization. For classical thermodynamics there are axiomatization proposals from Giles, Boyling, Jauch, Lieb and Yngvason, among others. For all physical theories that operate with probabilities, especially statistical mechanics, the axiomatization of probability calculation by Kolmogorow became important.

Relationship between experiment and theory

The axioms of a physical theory can neither be formally proven nor, according to the view that has become commonplace, directly and collectively verified or falsifiable through observation. According to a view of theories and their relationship to experiments and the resulting idiom, which is particularly widespread in epistemological structuralism, tests of a particular theory on reality usually concern statements of the form “this system is a classical particle mechanics”. If a corresponding theory test succeeds, e.g. correct prognoses of measured values ​​were given, this check can possibly be as confirmation it applies that a corresponding system was correctly counted among the intended applications of the corresponding theory, and in the event of repeated failures, the set of intended applications can and should be reduced by corresponding types of systems.

Based on an article in: Wikipedia.de
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Date of the last change: Jena, the: 03.04. 2021