por What is the axiomatic method?
He axiomatic method It is a formal procedure used by the sciences through which statements or propositions called axioms are formulated, connected to each other by a deductibility relationship and which are the basis of the hypotheses or conditions of a certain system.
This general definition must be framed within the evolution that this methodology has had throughout history. In the first place, there is an ancient or content method, born in Ancient Greece from Euclid and later developed by Aristotle.
Second, already in the 19th century, the appearance of a geometry with axioms other than those of Euclid. And finally, the formal or modern axiomatic method, whose greatest exponent was David Hilbert.
Beyond its development over time, this procedure has been the basis of the deductive method being used in geometry and logic where it originated. It has also been used in physics, chemistry, and biology.
And it has even been applied within legal science, sociology and political economy. However, currently its most important sphere of application is mathematics and symbolic logic and some branches of physics such as thermodynamics, mechanics, among other disciplines.
Characteristics of the axiomatic method
Although the fundamental characteristic of this method is the formulation of axioms, these have not always been considered in the same way.
There are some that can be defined and built in an arbitrary way. And others, according to a model in which their intuitively guaranteed truth is considered.
In order to understand specifically what this difference consists of and its consequences, it is necessary to go through the evolution of this method.
Old or content axiomatic method
It is established in Ancient Greece around the 5th century BC. Its sphere of application is geometry. The fundamental work of this stage is Euclid’s Elements, although it is considered that before him, Pythagoras, had already given birth to the axiomatic method.
Thus the Greeks take certain facts as axioms, without the need for any logical proof, that is, without the need for demonstration, since for them they are a self-evident truth.
For his part, Euclid presents five axioms for geometry:
Given two points there is a line that contains or joins them.
Any segment can be extended continuously in a straight line unlimited on both sides.
You can draw a circle that has a center at any point and any radius.
Right angles are all equal.
Taking any straight line and any point not on it, there exists a straight line parallel to it and containing that point. This axiom is later known as the axiom of parallels and has also been stated as: a single parallel can be drawn through a point outside a line.
However, both Euclid and later mathematicians agree that the fifth axiom is not as intuitively clear as the other 4. Even during the Renaissance, attempts were made to deduce the fifth from the other 4, but it was not possible.
This meant that already in the 19th century, those who maintained the five were supporters of Euclidean geometry and those who denied the fifth were the ones who created non-Euclidean geometries.
Non-Euclidean axiomatic method
It is precisely Nikolai Ivánovich Lobachevski, János Bolyai and Johann Karl Friedrich Gauss who see the possibility of building, without contradiction, a geometry that comes from systems of axioms other than those of Euclid. This destroys the belief in the absolute or a priori truth of the axioms and the theories that derive from them.
Consequently, the axioms begin to be conceived as starting points of a given theory. Also, both his choice and the problem of its validity in one sense or another begin to be related to facts outside of axiomatic theory.
Thus appear geometric, algebraic and arithmetic theories built by means of the axiomatic method.
This stage culminates with the creation of axiomatic systems for arithmetic such as that of Giuseppe Peano in 1891; David Hubert’s geometry in 1899; the statements and calculations of predicates by Alfred North Whitehead and Bertrand Russell, in England in 1910; Ernst Friedrich Ferdinand Zermelo’s axiomatic theory of sets in 1908.
Modern or formal axiomatic method
It is David Hubert who initiates the conception of a formal axiomatic method and who leads to its culmination, David Hilbert.
It is precisely Hilbert who formalizes scientific language, considering its statements as formulas or sequences of signs that have no meaning in themselves. They only acquire meaning in a certain interpretation.
In «The basics of geometry” explains the first example of this methodology. From here geometry becomes a science of pure logical consequences, which are extracted from a system of hypotheses or axioms, better articulated than the Euclidean system.
This is because in the old system the axiomatic theory is founded on the evidence of the axioms. While the foundation of the formal theory is given by the demonstration of the non-contradiction of its axioms.
Steps of the axiomatic method
The procedure that carries out an axiomatic structuring within scientific theories recognizes:
A-The choice of a certain number of axioms, that is, a number of propositions of a certain theory that are accepted without the need to be demonstrated.
B-The concepts that are part of these propositions are not determined within the framework of the given theory.
C-The rules of definition and deduction of the given theory are fixed and allow new concepts to be introduced into the theory and logically deduce some propositions from others.
D-The other propositions of the theory, that is, the theorem, are deduced from A on the basis of C.
examples
This method can be verified through the proof of the two best known Euclid’s theorems: the theorem of the legs and the one of the height.
Both arise from the observation of this Greek geometer that when the height is plotted with respect to the hypotenuse within a right triangle, two more triangles than the original appear. These triangles are similar to each other and at the same time similar to the triangle of origin. This assumes that their respective counterpart sides are proportional.
It can be seen that congruent angles in triangles of this shape verify the similarity that exists between the three triangles involved according to the AAA similarity criterion. This criterion holds that when two triangles have all their angles equal they are similar.
Once the triangles are shown to be similar, the proportions specified in the first theorem can be established. The same states that in a right triangle, the measure of each leg is the geometric mean proportional between the hypotenuse and the projection of the leg on it.
The second theorem is that of height. It specifies that any right triangle whose height is drawn according to the hypotenuse is the geometric proportional mean between the segments that are determined by said geometric mean on the hypotenuse.
Of course, both theorems have numerous applications throughout the world, not only in the field of education, but also in engineering, physics, chemistry, and astronomy.