What is hidden behind the mysterious word "axiom", where did it come from and what does it mean? A 7–8th grade student will easily answer this question, because more recently, when mastering the basic planimetric course, he was already faced with the task: “What statements are called axioms, give examples.” A similar question from an adult is likely to lead to difficulty. The more time passes from the moment of study, the more difficult it is to recall the basics of science. However, the word "axiom" is often used in everyday life.
Definition of the term
So what statements are called axioms? Examples of axioms are very diverse and are not limited to any one area of science. The mentioned term came from the ancient Greek language and in literal translation means "accepted position".
A strict definition of this term states that the axiom is the main thesis of a theory that does not need proof. This concept is widespread in mathematics (and especially in geometry), logic, and philosophy.
Even the ancient Greek Aristotle stated that evidence was not needed for obvious facts. For example, no one doubts that sunlight is visible only during the day. Another mathematician, Euclid, developed this theory. An example of the axiom about parallel lines that never intersect belongs to him.
Over time, the definition of the term has changed. Now the axiom is perceived not only as the beginning of science, but also as some intermediate result obtained , which serves as a starting point for further theory.
School course statements
Schoolchildren get acquainted with postulates that do not require confirmation in mathematics lessons. Therefore, when high school graduates are given the task: “Give examples of axioms,” they most often recall courses in geometry and algebra. Here are some common answers:
- for a straight line there are points that relate to it (that is, lie on a straight line) and do not relate (do not lie on a straight line);
- a line can be drawn through any two points;
- to split the plane into two half-planes, you need to draw a straight line.
Algebra and arithmetic do not explicitly introduce such statements, but an example of an axiom can be found in these sciences:
- any number is equal to itself;
- the unit precedes all natural numbers;
- if k = l, then l = k.
So, through simple theses more complex concepts are introduced, corollaries are made and theorems are deduced.
The construction of a scientific theory based on axioms
To build a scientific theory (no matter what area of research is being discussed), you need a foundation - the bricks from which it will be composed. The essence of the axiomatic method: a glossary of terms is created, an example of an axiom is formulated, on the basis of which the remaining postulates are derived.
The scientific glossary should contain elementary concepts, that is, those that cannot be defined through others:
- Successively explaining each term, setting out its meanings, they reach the foundations of any science.
- The next step is to identify a basic set of statements, which should be sufficient to prove the remaining statements of the theory. The basic tenets themselves are accepted without justification.
- The final step is the construction and inference of theorems.
Postulates from various sciences
Expressions without evidence are not only in the exact sciences, but also in those that are commonly referred to as humanitarian. A vivid example is the philosophy that defines the axiom as a statement that can be known without practical knowledge.
There is an example of an axiom in the legal sciences: “one cannot judge one’s own deed”. Based on this statement, they deduce the norms of civil law - impartiality of legal proceedings, that is, a judge cannot consider a case if he is directly or indirectly interested in it.
Not everything is taken on faith
To understand the difference between true axioms and simple expressions that are declared truth, you need to analyze the attitude towards them. For example, if we are talking about religion, where everything is taken on faith, the principle of the full conviction that something is true is widespread, since it is impossible to prove. But in the scientific community they talk about the impossibility of verifying a situation for now, respectively, it will be an axiom. Willingness to doubt, double-check - this is what distinguishes a true scientist.