AXIOMATIC METHOD: CHARACTERISTICS, STEPS, EXAMPLES - CULTURE VOCABULARY - 2024

The axiomatic method or also called Axiomatics is a formal procedure used by the sciences by means of which statements or propositions called axioms are formulated, connected to each other by a relation of deductibility and that 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, as early as the 19th century, the appearance of a geometry with axioms different from 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 the 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

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 constructed in an arbitrary way. And others, according to a model in which its guaranteed truth is intuitively considered.

In order to understand specifically what this difference and its consequences consist of, it is necessary to go through the evolution of this method.

Ancient or content axiomatic method

It is the one established in Ancient Greece towards the 5th century BC Its sphere of application is geometry. The fundamental work of this stage is the Elements of Euclid, 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 requiring any logical proof, that is, without the need for proof, since for them they are a self-evident truth.

For his part, Euclid presents five axioms for geometry:

1-Given two points there is a line that contains or joins them.

2-Any segment can be continuously extended in an unlimited line on both sides.

3-You can draw a circle that has a center at any point and any radius.

4-The right angles are all the same.

5-Taking any straight line and any point that is not in it, there is a straight line parallel to it and that contains that point. This axiom is known, later, as the axiom of parallels and has also been enunciated as: a single parallel can be drawn from 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, an attempt is made to deduce the fifth from the other 4, but it is not possible.

This made that already in the XIX century, those who maintained the five were in favor of Euclidean geometry and those who denied the fifth, were those who created the non-Euclidean geometries.

Non-Euclidean axiomatic method

It is precisely Nikolai Ivanovich Lobachevski, János Bolyai and Johann Karl Friedrich Gauss who see the possibility of constructing, without contradiction, a geometry that comes from systems of axioms other than those of Euclid. This destroys the belief in the absolute truth or a priori of the axioms and the theories that derive from them.

Consequently, axioms begin to be conceived as starting points for 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 the axiomatic theory.

In this way, geometric, algebraic and arithmetic theories appear built by means of the axiomatic method.

This stage culminates in the creation of axiomatic systems for arithmetic like Giuseppe Peano's in 1891; David Hubert's geometry in 1899; the statements and predicate calculations of 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 that 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 foundations of geometry" he explains the first example of this methodology. From here on, 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 ancient system the axiomatic theory is based on the evidence of the axioms. While in the foundation of the formal theory it is given by the demonstration of the non-contradiction of its axioms.

Steps

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 needing to be proved.

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 set and allow the introduction of new concepts within 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 theorems: the legs theorem and the height theorem.

Both arise from the observation of this Greek geometer that when the height with respect to the hypotenuse is plotted within a right triangle, two more triangles of 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 homologous sides are proportional.

It can be seen that the congruent angles in the triangles in this way verify the similarity that exists between the three involved triangles according to the AAA similarity criterion. This criterion holds that when two triangles have all the same angles they are similar.

Once it is shown that the triangles are similar, the proportions specified in the first theorem can be established. The same statement that in a right triangle, the measure of each leg is the geometric proportional mean between the hypotenuse and the projection of the leg on it.

The second theorem is that of height. It specifies that any right triangle the height that 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 around the world not only in teaching, but also in engineering, physics, chemistry, and astronomy.

References

1. Giovannini, Eduardo N. (2014) Geometry, formalism and intuition: David Hilbert and the formal axiomatic method (1895-1905). Revista de Filosofía, Vol. 39 No. 2, pp.121-146. Taken from magazines.ucm.es.
2. Hilbert, David. (1918) Axiomatic thought. In W. Ewald, editor, from Kant to Hilbert: a source book in the foundation of mathematics. Volume II, pp 1105-1114. Oxford University Press. 2005 a.
3. Hintikka, Jaako. (2009). What is the axiomatic method? Synthese, November 2011, volume 189, pp.69-85. Taken from link.springer.com.
4. López Hernández, José. (2005). Introduction to contemporary Philosophy of Law. (pp.48-49). Taken from books.google.com.ar.
5. Nirenberg, Ricardo. (1996) The Axiomatic Method, a reading by Ricardo Nirenberg, Fall 1996, the University at Albany, Project Renaissance. Taken from Albany.edu.
6. Venturi, Giorgio. (2015) Hilbert between the formal and the informal side of Mathematics. Manuscript vol. 38 no. 2, Campinas July / Augusto 2015. Taken from scielo.br.