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 evidence or rebuttal. In fact, this version of knowledge is presented in the form of a deductive structure, which initially includes the logical justification of the content from the foundations - axioms.
This method cannot be a discovery, but is only a classifying concept. It is more suitable for teaching. The basis is the initial position, and the remaining information follows as a logical consequence. Where is the axiomatic method of constructing a theory? It lies in the structure of most modern and established sciences.
The formation and development of the concept of the axiomatic method, the 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 well-established statements, and subsequent theories are derived by logical construction.
This is explained as follows: there are words and concepts that are defined by other concepts. As a result, the researchers came to the conclusion that there are elementary conclusions that are justified and constant - basic, that is, axioms. For example, when proving a theorem, they usually rely on facts that are well-established and do not require refutation.
However, before that they needed to be justified. In the process, it turns out that the unreasonable 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, possess properties that are indicated in constant terms.
Further research on axioms
The method was considered ideal until the nineteenth century. Logical means of searching for basic concepts were not studied back then, but in the Euclidean system one can observe the structure of obtaining meaningful consequences from the axiomatic method. The scientist's studies 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 approved axioms that are true visually.
Merits of ancient Greek minds
Euclid proved many concepts, some of which were justified. However, most attribute this merit to Pythagoras, Democritus and Hippocrates. The latter constituted a complete course in geometry. True, later in Alexandria there was a collection of "Beginning", the author of which was Euclid. Then, it was renamed "Elementary Geometry." After some time, they began to criticize him based on some reasons:
- all values were built only using a ruler and compass;
- geometry and arithmetic were disconnected and proved taking into account reasonable numbers and concepts;
- axioms, some of them, in particular, the fifth postulate, suggested deleting from the general list.
As a result, non-Euclidean geometry arises 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 arrived at deductive construction methods.
The development of mathematical knowledge based on axioms
When the new geometry system began to develop, the axiomatic method also changed. In mathematics, they began to turn more often to a purely deductive construction of a theory. As a result, a whole system of evidence has emerged in modern numerical logic, which is the main section of all science. The mathematical structure began to understand the need for justification.
So, by the end of the century clear tasks and the construction of complex concepts had formed, which from a complex theorem were reduced to a simple 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 of mathematical constructions:
- consistency;
- completeness;
- independence.
In the process, a way of interpretation appeared and successfully developed. This method is described as follows: for each output concept, a mathematical object is set in theory, the totality of which is called a field. Saying about the specified elements can be false or true. As a result, statements get names depending on the conclusions.
Features of the theory of interpretation
As a rule, the field and properties are also considered in a mathematical system, and it, in turn, can become axiomatic. Interpretation proves statements in which there is relative consistency. An additional option is a series of facts in which the theory becomes contradictory.
In fact, the condition is satisfied in some cases. The result is that if the statements of one of the statements contain two false or true concepts, then it is considered negative or positive. Using this method, the consistency of Euclidean geometry was proved. With the interpretation method, the question of the independence of axiom systems can be solved. 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 mistaken.
However, along with successful statements, the method also has weaknesses. Consistency and independence of axiom systems are resolved as issues that get relative results. The only important achievement of the interpretation is the discovery of the role of arithmetic as a structure in which the question of consistency is reduced to a number of other sciences.
The 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 the formal system was clarified. As a result, a common system emerged, and mathematical objects became accurate. In addition, it became possible to solve the substantiation issues. Thus, the formal system is constructed by the exact class in which the subsystems of formulas and theorems are located.
To build this structure, you only need to be guided by technical amenities, because they have no meaning. 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 actual content or deductive inference. The language of numerical science is translated into the formal system, in the process any concrete and meaningful expression is determined by the formula.
Formalization method
In the natural state of things, a similar method will be able to solve such global issues as consistency, and also build the positive essence of mathematical theories according to the derived formulas. And basically all this will be decided by the formal system on the basis of proven statements. Mathematical theories were constantly complicated by justifications, and Gilbert suggested exploring this structure using finite methods. But this program failed. Godel'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;
- insoluble formulas appeared;
- statements are unprovable.
True judgments and reasonable finite support are considered formalized. With this in mind, the axiomatic method has certain and clear boundaries and possibilities within the framework of this theory.
The results of the development of axioms in the works of mathematicians
Despite the fact that some judgments were refuted and did not receive proper development, the way of constant concepts plays a significant role in the formation of 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 solving 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 communication. In this case, we consider various views, methods, theories that should take into account the logical meaning and justification.
The consistency of the formal system indicates a similar refinement of arithmetic, which is based on induction, counting, transfinite number. In the scientific field, axiomatization is the most important tool with irrefutable concepts and assertions taken as a basis.
The essence of the initial statements and their role in theories
Evaluation of the axiomatic method indicates that a certain structure lies in its essence. This system is built with the identification of the fundamental concept and fundamental statements that are undetectable. The same thing happens with theorems that are considered initial and accepted without proof. In the natural sciences, for such statements are the rules, assumptions, laws.
Then there is a process of fixing the established bases for reasoning. As a rule, it is immediately indicated that the other is deduced from one position, and the rest are left in the process, which, in essence, coincide with the deductive method.
Features of the system in modern times
The axiomatic system contains:
- 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 positions. And the theory was interpreted in a unique, natural way. Today, the axiom is a provision that is obvious in itself, and an agreement, and any one, can act as an initial concept that does not require substantiation. As a result, the initial values may be far from obvious. This method requires a creative approach, knowledge of the relationships and the original theory.
Key Findings
The deductively axiomatic method is scientific knowledge, built according to a certain scheme, which is based on correctly conscious hypotheses that derive statements about empirical facts. Such a conclusion is based on logical structures, by hard derivation. Axioms are initially irrefutable statements that do not require evidence.
In deduction, certain requirements apply 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 meaning of truth and falsity, because it will no longer have meaning and value.
If such a 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. The deductively axiomatic method is a way of constructing and substantiating scientific knowledge.
Practical application of the method
The axiomatic method of constructing scientific knowledge has practical application. In fact, this method affects and has global significance in 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 need to formulate the initial value, then you need to know the nature of sets and elements. In essence, the axiomatic method formed the basis of various fields of science.