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Table 1 Historical development of mathematical concepts Classic axioms are obvious implications of definitions axioms are conventional theorems are absolute objective truth theorems are implications of the corresponding axioms relationships between points, lines etc.

In ancient Greek mathematics, "space" was a geometric abstraction of the three-dimensional reality observed in everyday life. About BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment.

Translations, rotations and reflections transform a figure into congruent figures; homotheties — into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles.

A third equivalence relation, introduced by Gaspard Monge inoccurs in projective geometry: The relation between the two geometries, Euclidean and projective, [4]: The question "what is the sum of the three angles of a triangle" is meaningful in Euclidean geometry but meaningless in projective geometry.

A different situation appeared in the 19th century: Eugenio Beltrami in and Felix Klein in obtained Euclidean "models" of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory as a logical possibility.

It showed that axioms are not "obvious", nor "implications of definitions". Rather, they are hypotheses.

To what extent do they correspond to an experimental reality? This important physical problem no longer has anything to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths".

These Euclidean objects and relations "play" the non-Euclidean geometry like contemporary actors playing an ancient performance.

Actors can imitate a situation that never occurred in reality. Relations between the actors on the stage imitate relations between the characters in the play.

Likewise, the chosen relations between the chosen objects of the Euclidean model imitate the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not.

The golden age and afterwards[ edit ] The word "geometry" from Ancient Greek: According to Bourbaki, [4]: The original space investigated by Euclid is now called three-dimensional Euclidean space.

These axiom systems describe the space via primitive notions such as "point", "between", "congruent" constrained by a number of axioms.

Analytic geometry made great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. Simultaneously, numbers began to displace geometry as the foundation of mathematics.

Dedekind is careful to note that this is an assumption that is incapable of being proven. Three-dimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product. Also, a three-dimensional projective space is now defined as the space of all one-dimensional subspaces that is, straight lines through the origin of a four-dimensional vector space.Science Books Online lists free science e-books, textbooks, lecture notes, monographs, and other science related documents.

All texts are available for free reading online, or for downloading in various formats. Select your favorite category from the menu on the top left corner of the screen or see all the categories below. Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models, Second Edition (Chapman & Hall/CRC Texts in Statistical Science) 2nd Edition.

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Algebra is a branch of mathematics, as we know maths is queen of science, it plays vital role of developing and flourishing technology, we use all scopes in Published: Mon, 15 May Patterns Within Systems Of Linear Equations.

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Mathematics Itself: Formatics, On the Origin, Nature, Fabrication of Logic and Mathematics