The axiomatic method is a way of constructing scientific theories that are already established. It is based on arguments, facts, statements that do not require proof or refutation. In fact, this version of knowledge is presented in the form of a deductive structure, which initially includes a logical substantiation of the content from the fundamentals - axioms.
This method cannot be a discovery, but is only a classifying concept. It is more suitable for teaching. The basis contains the initial provisions, and the rest of the information follows as a logical consequence. Where is the axiomatic method of constructing a theory? It lies at the core of most modern and established sciences.
Formation and development of the concept of axiomatic method, definition of the word
First of all, this concept arose in Ancient Greece thanks to Euclid. He became the founder of the axiomatic method in geometry. Today it is common in all sciences, but most of all in mathematics. This method is formed on the basis of established statements, and subsequent theories are derived by logical construction.
This is explained as follows: there are words and concepts thatdefined by other terms. As a result, the researchers came to the conclusion that there are elementary conclusions that are justified and are constant - basic, that is, axioms. For example, when proving a theorem, they usually rely on facts that are already well-established and do not require refutation.
However, before that, they needed to be substantiated. In the process, it turns out that an unreasoned statement is taken as an axiom. Based on a set of constant concepts, other theorems are proved. They form the basis of planimetry and are the logical structure of geometry. The established axioms in this science are defined as objects of any nature. They, in turn, have properties that are specified in constant concepts.
Further exploration of the axioms
The method was regarded as ideal until the nineteenth century. The logical means of searching for basic concepts were not studied back in those days, but in the Euclid system one can observe the structure of obtaining meaningful consequences from the axiomatic method. The scientist's research showed the idea of how to get a complete system of geometric knowledge based on a purely deductive path. They were offered a relatively small number of stated axioms that are demonstrably true.
Merit of ancient Greek minds
Euclid proved many concepts, and some of them were justified. However, the majority ascribes these merits to Pythagoras, Democritus and Hippocrates. The latter compiled a complete course of geometry. True, later in Alexandria came outcollection "Beginning", the author of which was Euclid. Then, it was renamed to "Elementary Geometry". After a while, they began to criticize him based on some reasons:
- all values were built only with a ruler and a compass;
- geometry and arithmetic were separated and proved with valid numbers and concepts;
- axioms, some of them, in particular, the fifth postulate, were proposed to be deleted from the general list.
As a result, non-Euclidean geometry appears in the 19th century, in which there is no objectively true postulate. This action gave impetus to the further development of the geometric system. Thus, mathematical researchers came to deductive construction methods.
Development of mathematical knowledge based on axioms
When a new system of geometry began to develop, the axiomatic method also changed. In mathematics, they began to turn more often to a purely deductive theory construction. As a result, a whole system of proofs arose in modern numerical logic, which is the main section of all science. In the mathematical structure began to understand the need for justification.
Thus, by the end of the century, clear tasks and the construction of complex concepts were formed, which from a complex theorem were reduced to the simplest logical statement. Thus, non-Euclidean geometry stimulated a solid foundation for the further existence of the axiomatic method, as well as for solving problems of a general nature.mathematical constructions:
- consistency;
- fullness;
- independence.
In the process, a method of interpretation emerged and was successfully developed. This method is described as follows: for each output concept in the theory, a mathematical object is set, the totality of which is called a field. The statement about the specified elements can be false or true. As a result, statements are named depending on the conclusions.
Features of the theory of interpretation
As a rule, the field and properties are also considered in the mathematical system, and it, in turn, can become axiomatic. The interpretation proves statements in which there is relative consistency. An additional option is a number of facts in which the theory becomes contradictory.
In fact, the condition is fulfilled in some cases. As a result, it turns out that if there are two false or true concepts in the statements of one of the statements, then it is considered negative or positive. This method was used to prove the consistency of Euclid's geometry. Using the interpretative method, one can solve the question of the independence of systems of axioms. If you need to refute any theory, then it is enough to prove that one of the concepts is not derived from the other and is erroneous.
However, along with successful statements, the method also has weaknesses. Consistency and independence of systems of axioms are solved as questions that get results that are relative. The only important achievement of interpretation isdiscovery of the role of arithmetic as a structure in which the question of consistency is reduced to a number of other sciences.
Modern development of axiomatic mathematics
The axiomatic method began to develop in the work of Gilbert. In his school, the very concept of theory and formal system was clarified. As a result, a general system arose, and mathematical objects became precise. In addition, it became possible to resolve the issues of justification. Thus, a formal system is constructed by an exact class, which contains subsystems of formulas and theorems.
To build this structure, you only need to be guided by technical convenience, because they have no semantic load. They can be inscribed with signs, symbols. That is, in fact, the system itself is built in such a way that the formal theory can be applied adequately and fully.
As a result, a specific mathematical goal or task is poured into a theory based on factual content or deductive reasoning. The language of numerical science is transferred to a formal system, in the process any concrete and meaningful expression is determined by the formula.
Formalization method
In the natural state of things, such a method will be able to solve such global issues as consistency, as well as build a positive essence of mathematical theories according to the derived formulas. And basically all this will be solved by a formal system based on proven statements. Mathematical theories were constantly complicated by justifications, andGilbert proposed to investigate this structure using finite methods. But this program failed. Gödel's results already in the twentieth century led to the following conclusions:
- natural consistency is impossible due to the fact that formalized arithmetic or other similar science from this system will be incomplete;
- unsolvable formulas appeared;
- claims are unprovable.
True judgments and reasonable finite finishing are considered formalizable. With this in mind, the axiomatic method has certain and clear boundaries and possibilities within this theory.
Results of the development of axioms in the works of mathematicians
Despite the fact that some judgments have been refuted and not developed properly, the method of constant concepts plays a significant role in shaping the foundations of mathematics. In addition, interpretation and the axiomatic method in science have revealed the fundamental results of consistency, independence of choice statements and hypotheses in multiple theory.
In addressing the issue of consistency, the main thing is to apply not only established concepts. They also need to be supplemented with ideas, concepts and means of finite finishing. In this case, various views, methods, theories are considered, which should take into account the logical meaning and justification.
The consistency of the formal system indicates a similar finishing of arithmetic, which is based on induction, counting, transfinite number. In the scientific field, axiomatization is the most importanta tool that has irrefutable concepts and statements that are taken as a basis.
The essence of initial statements and their role in theories
Evaluation of an axiomatic method indicates that some structure lies in its essence. This system is built from the identification of the underlying concept and fundamental statements that are undefined. The same thing happens with theorems that are considered original and are accepted without proof. In the natural sciences, such statements are supported by rules, assumptions, laws.
Then the process of fixing the established reasoning bases takes place. As a rule, it is immediately indicated that another is deduced from one position, and in the process the rest come out, which, in essence, coincide with the deductive method.
Features of the system in modern times
The axiomatic system includes:
- logical conclusions;
- terms and definitions;
- partially incorrect statements and concepts.
In modern science, this method has lost its abstractness. Euclidean geometric axiomatization was based on intuitive and true propositions. And the theory was interpreted in a unique, natural way. Today, an axiom is a provision that is obvious in itself, and an agreement, and any agreement, can act as an initial concept that does not require justification. As a result, the original values may be far from descriptive. This method requires creativity, knowledge of relationships and underlying theory.
Basic principles of deriving conclusions
Deductively axiomatic method is scientific knowledge, built according to a certain scheme, which is based on correctly realized hypotheses, deriving statements about empirical facts. Such a conclusion is built on the basis of logical structures, by hard derivation. Axioms are initially irrefutable statements that do not require proof.
During deduction, certain requirements are applied to the initial concepts: consistency, completeness, independence. As practice shows, the first condition is based on formal logical knowledge. That is, the theory should not have the meanings of truth and falsity, because it will no longer have meaning and value.
If this condition is not met, then it is considered incompatible and any meaning is lost in it, because the semantic load between truth and falsehood is lost. Deductively, the axiomatic method is a way of constructing and substantiating scientific knowledge.
Practical application of the method
The axiomatic method of constructing scientific knowledge has a practical application. In fact, this way influences and has a global significance for mathematics, although this knowledge has already reached its peak. Examples of the axiomatic method are as follows:
- affine planes have three statements and a definition;
- equivalence theory has three proofs;
- binary relations are divided into a system of definitions, concepts and additional exercises.
If you want to formulate the original meaning, you need to know the nature of sets and elements. In essence, the axiomatic method formed the basis of various fields of science.
