Abstract
This chapter is an attempt to show how mathematical thought has changed in the last two centuries. In fact, with the discovery of the so-called non-Euclidean Geometries, mathematical thinking changed profoundly. With the negation of the postulate for “antonomasia,” that is the uniqueness of the parallel for Euclid, and the construction of a geometric theory equally valid on the logical and coherence plane, called non-Euclidean geometry, the meaning of the word “postulate” or “axiom” changes radically. The axioms of a theory do not necessarily have to be dictated by real evidence. On this basis the constructions of arithmetic and geometry are built. The axiomatic-deductive method becomes the mathematical method. It will also highlight the constant link between mathematics and the reality that surrounds us, which tends to make itself explicit through an artificial, abstract language and with clear and certain grammatical rules. Finally, you will notice the connection with the existing technology, that is the new electronic and digital technology.
“Mathematics is the main language of science and technology, as such, it is the key to understanding and shaping the world around us.” ~C. F. Gauss
Top2. The Crisis Of Geometry And The Birth Of Geometries
During the nineteenth century one of the main moments of the evolution of Mathematics was certainly the invention of the non-Euclidean Geometries by Gauss (1777-1855), Schwelkart (1778-1859), Lobachevski (1792-1856), Bolyal (1802-1860) and Riemann (1826-1866), together with the demonstration of their intrinsic coherence (Boyer, 1968). The first logical consequence was the radical change in the way in which we conceive of Geometry. In fact, if we suppose that there exists an entity external to us, which is the object of this science, it could not be described or studied with contradictory doctrines among them, such as Euclidean Geometry and any of the non-Euclidean ones. This is of course rests on the principle that every scientific theory is based on the internal coherence of the reality being studied. To better understand what we want to affirm, it is opportune to reflect on the “Elements” of Euclid (Fitzpatrick, 2007), the first scientific treatise in the history of humanity. It is a sequence of propositions; some of which describe the entities that will be discussed later. Other propositions express some truths that are considered incontrovertible and for this reason Euclid calls them “common notions”. Finally, other propositions are presented as requests for consent, which is why they are called “postulates”. The postulates concern properties of geometrical entities, or of the operations that we can do on them. These include the so-called postulate “par excellence” which basically affirms the uniqueness of a parallel to a given straight line for a point outside of it. Statements, common notions and postulates are given without proof while the subsequent propositions are rigorously demonstrated with procedures and reasoning based on the previously expressed statements.
Key Terms in this Chapter
Axiomatic Method: A procedure by which an entire system (e.g., a science) is generated in accordance with specified rules by logical deduction from certain basic propositions (axioms or postulates), which in turn are constructed from a few terms taken as primitive.
Technology: Is the sum of techniques, skills, methods, and processes used in the production of goods or services or in the accomplishment of objectives, such as scientific investigation.
Symbolic Language: The symbolic language of mathematics is a special-purpose language. It has its own symbols and rules of grammar that are quite different from those of natural languages.
Non-Euclidean Geometry: Non-Euclidean geometry consists of geometries based on axioms closely related to those specifying Euclidean geometry. The traditional non-Euclidean geometries are: hyperbolic geometry and elliptic geometry, where the parallel postulate is replaced with an alternative one.
Mathematization of Reality: Use of any mathematical object to formalize some relationships between the variables that describe a real-world phenomenon. We see, also, that we are using a mathematical model of this phenomenon.