History of mathematics since the 9th century BC (part 2)

In the seventeenth century, mathematics takes a new form, with the first highlight being René Descartes and Pierre Fermat. René Descartes's great discovery was undoubtedly the "Analytical Geometry" which, in short, consists of the application of algebraic methods to geometry. Pierre Fermat was a leisure lawyer who was busy with math. He developed the theory of prime numbers and solved the important problem of tracing a tangent to any flat curve, thus sowing seeds for what would later be called, in mathematics, the theory of maxima and minima. So we see in the seventeenth century begin to germinate one of the most important branches of mathematics, known as Mathematical Analysis. Physics problems still arise at this time: the study of the movement of a body, previously studied by Galileo Galilei. Such problems give rise to one of the first descendants of Analysis: Differential Calculus.

The Differential Calculus first appears in the hands of Isaac Newton (1643-1727), under the name "calculus of fluxions", and was later rediscovered independently by the German mathematician Gottfried Wihelm Leibniz. Analytical Geometry and Calculus give a great boost to mathematics. Seduced by these new theories, seventeenth- and eighteenth-century mathematicians bravely and carelessly set out to elaborate new analytical theories. But in this momentum they were led more by intuition than by a rational attitude in the development of science. The consequences of such procedures were not delayed, and contradictions began to appear. A classic example of this is the case of infinite sums, such as the sum below:

S = 3 + 3 - 3 + 3…

Assuming you have an infinite number of terms. If we group the neighboring plots we will have:

S = (3 - 3) + (3 - 3) +… = 0 + 0 +… = 0

If we group the neighboring plots, but from the 2nd, not grouping the first:

S = 3 + (- 3 + 3) + (- 3 + 3) +… = 3 + 0 + 0 +… = 3

Which leads to contradictory results. This "carelessness" in working with infinite series was very characteristic of mathematicians of that time, who then found themselves in a "dead end." Such facts led, at the turn of the eighteenth century, to a critical attitude of revising the fundamental facts of mathematics. It can be argued that such a review was the cornerstone of mathematics.This review begins with Analysis, with the French mathematician Louis Cauchy (1789 - 1857), a full professor at the Paris Faculty of Sciences. leaving more than 500 written works, of which we highlight two in the Analysis: “Notes on the Development of Functions in Series” and “Lessons on the Application of Calculus to Geometry.” At the same time, different geometries arise from Euclid's, the so-called non-Euclidean geometries.

Around 1900, the axiomatic method and geometry were influenced by this critical revision attitude, carried out by many mathematicians, among which we highlight D. Hilbert, with his work "Grudlagen der Geometrie". published in 1901. Algebra and Arithmetic take on new impulses. One problem that preoccupied mathematicians was whether or not to solve algebraic equations through formulas that appeared with radicals. It was already known that in 2nd and 3rd degree equations this was possible; Hence the question arose: do the equations of the 4th degree onwards admit solutions by radicals?

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In works published around 1770, Lagrange (1736 - 1813) and Vandermonde (1735-96) began systematic studies of resolution methods. As research developed to find such a resolution, it became clear that this was not possible. In the first third of the nineteenth century, Niels Abel (1802-29) and Evariste de Galois (1811-32) solve the problem by demonstrating that the fourth and fifth degree equations could not be solved by radicals. Galois's work, only published in 1846, gave rise to the so-called "group theory" and the so-called "Modern Algebra," also giving the number theory great impetus.

With respect to number theory we cannot forget the works of R. Dedekind and Gorg Cantor. R. Dedekind defines irrational numbers by the famous notion of "Cut." Georg Cantor begins the so-called Set Theory, and boldly addresses the notion of infinity, revolutionizing it. From the nineteenth century mathematics then begins to branch into several disciplines, which become increasingly abstract.

Currently such abstract theories are developed, which are subdivided into other disciplines. Those who say that we are in the "golden age" of mathematics, and that in the last fifty years so many disciplines, new mathematics, have been created as they had in previous centuries. This rush toward the "Abstract", though not practical at all, is intended to further "Science." History has shown that what seems to us pure abstraction, pure mathematical fantasy, later turns out to be a true granary of practical applications.