What is hidden behind the mysterious word "axiom", where did it come from and what does it mean? A schoolboy of the 7th-8th grade can easily answer this question, since quite recently, when mastering the basic course of planimetry, he has already faced 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 remember the basics of science. However, the word "axiom" is often used in everyday life.
Term Definition
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 comes from the ancient Greek language and, in literal translation, means “the accepted position.”
The strict definition of this term says that an axiom is the main thesis of any theory that does not need proof. This concept is widespread in mathematics (and especially in geometry), logic, philosophy.
Even the ancient Greek Aristotle said that obvious facts do not need proof. For example, no one doubtsthat sunlight is only visible during the day. This theory was developed by another mathematician - Euclid. 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.
Statements from the school course
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 examples of common responses:
- for a line there are points that belong to it (that is, lie on the line) and do not belong (do not lie on the line);
- a straight line can be drawn through any two points;
- to divide a 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 the axiom can be found in these sciences:
- any number is equal to itself;
- one precedes all natural numbers;
- if k=l, then l=k.
Thus, through simple theses, more complex concepts are introduced, corollaries are made and theorems are derived.
Building a scientific theory based on axioms
To build a scientific theory (no matter what area of research it is), you need a foundation - the bricks from which it iswill add up. The essence of the axiomatic method: a dictionary of terms is created, an example of an axiom is formulated, on the basis of which the remaining postulates are derived.
A scientific glossary should contain elementary concepts, that is, those that cannot be defined through others:
- Sequentially explaining each term, outlining its meaning, 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 basic postulates themselves are accepted without justification.
- The final step is the construction and logical derivation of theorems.
Postulates from various sciences
Expressions without evidence exist not only in the exact sciences, but also in those that are commonly referred to as the humanities. A striking example is philosophy, which defines an 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 derive the norms of civil law - the impartiality of legal proceedings, that is, the judge cannot consider the 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 true, you need to analyze the relationship to them. For example, if speechit is about a religion where everything is taken for granted, there is a widespread principle of complete conviction that something is true, since it cannot be proven. And in the scientific community they talk about the impossibility yet to verify some position, respectively, it will be an axiom. The willingness to doubt, to double-check is what distinguishes a true scientist.
