Evidence is not required: an example of an axiom

What is hidden behind the mysterious word "axiom", where did it come from and what does it mean? A 7th grade school student will easily answer this question, since recently, when mastering the basic planimetry course, he already faced the task: "What statements are called axioms, give examples." A similar question for an adult person, most likely, will lead to embarrassment. 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 term

So what statements are called axioms? Examples of axioms are very diverse and are not limited to any one area of science. This term came from the ancient Greek language and in literal translation implies "accepted position".

A strict definition of this term says that the axiom is the basic thesis of any theory that does not need proofs. This concept is widely spread in mathematics (and especially in geometry), logic, philosophy.

Even the ancient Greek Aristotle said that evident evidence is not needed. For example, no one doubts that sunlight is only visible during the day. Another mathematician, Euclid, developed this theory. An example of an axiom about parallel straight lines that never crosses belongs to it.

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 a further theory.

Approvals from the school course

Schoolchildren get acquainted with the postulates that do not require confirmation in the lessons of mathematics. Therefore, when the graduates of the upper grades are given the task: "Give examples of axioms", they often remember the courses of geometry and algebra. Here are examples of frequently encountered answers:

• For a line there are points that refer to it (that is, lie on a line) and do not belong (do not lie on a line);
• A straight line can be drawn through any two points;
• To divide the plane into two half-planes, one must draw a straight line.

Algebra and arithmetic explicitly do not introduce such statements, but an example of an axiom can also be found in these sciences:

• Any number is equal to itself;
• Unit precedes all natural numbers;
• If k = 1, then also l = k.

Thus, more complicated concepts are introduced through simple theses, corollaries are made, and theorems are derived.

The construction of a scientific theory based on axioms

To build a scientific theory (it does not matter which field of research we are talking about), we need a foundation - the bricks from which it will form. The essence of the axiomatic method: a dictionary of terms is created, an example of an axiom is formulated, on the basis of which other postulates are derived.

The scientific glossary should contain elementary concepts, that is, those that can not be determined through others:

• Consistently explaining each term, setting out its meanings, reach the foundations of any science.
• The next step is to identify the basic set of statements, which should be sufficient to prove the remaining statements of the theory. The very basic postulates themselves are accepted without justification.
• The final step is the construction and logical conclusion of the theorems.

Postulates from various sciences

Expressions without evidence are not only in exact sciences, but also in those that are customarily referred to as humanitarian. A vivid example is the philosophy that defines an axiom as an assertion, which one can be known without practical knowledge.

An example of an axiom is also in the legal sciences: "one can not judge one's own deed". Proceeding from this assertion, norms of civil law are deduced - impartiality of legal proceedings, that is, a judge can not consider a case if he is directly or indirectly interested in it.

Not everything is taken for granted

To understand the difference between true axioms and simple expressions that are declared truth, one must analyze the attitude towards them. For example, if we are talking about religion, where everything is taken for granted, there is a principle of complete conviction that something is true, since it is impossible to prove. And in the scientific community they say that it is impossible to check a certain position, accordingly, it will be an axiom. Willingness to doubt, recheck - that's what distinguishes a true scientist.