Informal axiomatic method

An axiomatic method that does not fix rigidly the applicable language and so does not fix the limits of a meaningful understanding of an object, but requires an axiomatic definition of all special concepts for the given object of study. This term does not have a single universally accepted interpretation.

The historical development of the axiomatic method is characterized by an ever increasing degree of formalization. The informal axiomatic method is a stage in this process.

Euclid's original axiomatic construction of geometry was distinguished by the deductive nature of the presentation in which at the bases were definitions (explanations) and axioms (evident assertions). From them consequences were deduced, relying on common sense and evidence. In the process of deduction, sometimes propositions of a geometric nature were used implicitly that were not laid down in the axioms, especially concerning motion in space and the mutual disposition of lines and points. As a consequence geometric concepts appeared, as well as axioms regulating their application, which were used implicitly by Euclid and his followers. Here arises the question: Have all the axioms, in fact, been discovered? A guiding principle for the answer to this question was formulated by D. Hilbert: "One must be able to say at all times: instead of points, straight lines and planes — tables, chairs and beer mugs" . If a proof does not cease to be convincing after such a replacement, then in fact all special propositions used in this proof are fixed in the axioms. The degree of formalization that can be achieved in this approach is a level of formalization characteristic for the informal axiomatic method. Here Hilbert's classic work [1] can serve as a standard.

The informal axiomatic method can be applied not only to lend a certain completeness to an axiomatic exposition of a specific theory. It is a genuine tool of mathematical research. When in the study of a system of objects their specific character or "nature" is not used, the propositions proved carry over to any system of objects satisfying the axioms in question. According to the informal axiomatic method, the axioms are implicit definitions of the initial concepts (rather than "evident truths" ). What the objects to be studied are, is unimportant. Everything one has to know about them is laid down in the axioms. The object of study of an axiomatic theory is any interpretation of it.

The informal axiomatic method, apart from the indispensable axiomatic definition of all special concepts, also has another characteristic feature. This is the free use, uncontrolled by the axioms and based on a meaningful understanding, of ideas and concepts that can be applied to any sensible interpretation, irrespective of its content. In particular, set-theoretical and logical concepts and principles are widely used, as well as concepts connected with the idea of counting, and others. The penetration of reasoning based on a meaningful understanding and common sense, and not on the axioms, into the axiomatic method stems from the non-fixed language in which the properties of axiomatically given systems of objects are stated and proved. Fixing the language leads to the notion of a formal axiomatic system (see Axiomatic method) and creates a material basis for the clarification and precise description of the admissible logical principles, and for the controlled usage of set-theoretical and other general concepts or of such concepts that are not special for the relevant domain. If in the language there are no means (words) for the transmission of set-theoretical concepts, then this eliminates all proofs based on the use of such means. If the language has the means for expressing certain set-theoretical concepts, then their use in proofs can be confined to definite rules or axioms.

Fixing the language in different ways results in different theories of the basic object of study. For example, by considering the language of narrow predicate calculus for the theory of groups one obtains an elementary theory of groups in which one cannot state any assertion about the totality of all subgroups. If one goes over to the language of second-order predicate calculus, then it becomes possible to consider properties in which quantification over the concept of a subgroup occurs. The transition to the language of the Zermelo–Fraenkel system with its axiomatics serves as a formalization of the informal axiomatic method in the theory of the groups.

References

[1]	D. Hilbert, "Grundlagen der Geometrie" , Springer (1913)
[2]	D. Hilbert, P. Bernays, "Grundlagen der Mathematik" , 1–2 , Springer (1968–1970)