History of mathematics - SpeedyLook encyclopedia

The history of mathematics extends over several millenia and in many areas from the sphere going from China in the Central America. Insofar as historically research in mathematics concentrated in various places of the sphere, in this article we propose an approach by geographical small islands until the 17th century. For 19th and 20th centuries, knowledge and work, are much more numerous and much geographically less localized, cutting in geographical small islands is not justified any more, so much one witnesses a universalization of knowledge.

Prehistory

See also: mathematical Knowledge during the protohistoire

Before the appearance of the writing, drawings reflect the first mathematical knowledge. In a cave in South Africa, paleontologists found paintings ochers decorated with geometrical figures, going back to 70.000 before our era. Prehistoric artefacts were found, dating among 37.000 and of 20.000 before our era attest first attempts to measure time. The concepts of a , two and much were known.

The Os of Ishango going back to 20.000 years before our era is generally quoted to be the first proof of the knowledge of the first prime numbers and the multiplication, but this interpretation remains prone to discussions. It is known as that the megaliths in Egypt in thousand-year-old Ve before our era or in England in thousand-year-old IIIe would incorporate geometrical ideas like the Cercle S, the ellipse S and the triplets pythagoricians. In 2 600 before our era, Egyptian constructions attest of a precise and considered knowledge of the geometry.

The Ethnomathématiques is a field of research at the border of anthropology, ethnology and mathematics which aims inter alia including/understanding the rise of mathematics in the first civilizations starting from the objects, instruments, paintings, and other found documents.

De Sumer in Babylon

One in general locates the beginnings of the writing at Sumer, in the basin of the Tigre and the Euphrate or Mésopotamie. This writing, known as Wedge-shaped, is born from the need to organize the irrigation and the trade. Jointly with the birth of the writing are born the first utility mathematics (economy, calculations of surface). The first positional numerical system appears: the sexagesimal System. During nearly two thousand years, mathematics will develop in the area of Sumer, Akkad then Babylon. The shelves going back to this period consist of numerical tables and directions for use. Thus with Nippur (to a hundred km of Baghdad), were discovered at the 19th century of the school shelves dating from the time paléo-Babylonian (2000 av. J. - C.). One thus knows that they knew the four operations but launched out in more complex calculations with a very high degree of accuracy, like algorithms of extraction of square roots, cubic roots, the resolution of quadratic equations. As they made divisions by multiplication by the reverse, the tables of reverse played a great part. One found some with opposite for numbers with six sexagesimal digits, which indicates a very high degree of accuracy. One also found shelves on which are reproduced the list of the squares of entirety, the list of the cubes and the list of the triplets pythagoricians showing that they knew the property of the right-angled triangles more than 1.000 years before Pythagore. Shelves were also found describing algorithms to solve complex problems.

They were able to use linear interpolations for calculations of the intermediate values not appearing in their tables. The richest period concerning this mathematics is the period of Hammurabi (18th front century J. - C.). Towards 1000 av. J. - C., one observes a development of calculation towards the mathematical Astronomie.

Egypt

China

The oldest primary source of our knowledge on Chinese mathematics comes from the manuscript of Zhoubi Suanjing or the nine chapters on mathematical art , which gathers knowledge dating from the previous time (). It is discovered there that the Chinese had developed methods of calculating and of demonstration which were clean for them: Arithmetic, Fraction S, extraction of the square roots and cubic, way of calculating of the surface of the Circle, volume of the pyramid and Method of the pivot of Gauss. Their development of the algorithms of calculation is remarkably modern. But one also finds, on bones of sheep and oxen, engravings proving that as of 1300 before J. - C., they used a positional Decimal system (Chinese Numération). They are also at the origin of abacuses helping them to calculate. Chinese mathematics before our era is mainly turned towards utility calculations. They develop then in a clean way between the 1st and the 7th century after J. - C. then between 10th and the 13th century.

Civilizations précolombiennes

The Maya Civilization extends from 2600 before J. - C. until 1500 years after J. - C. with an apogee at the time traditional of the 3rd century at the 9th century. Mathematics is mainly numerical and rounds towards the calendar comput and astronomy. The Mayas use a positional Numbering system basic twenty (Maya Numération). The Maya sources result mainly from the Codex (written around the 13th century). But those in great majority were destroyed by the Inquisition and there remain only four codices nowadays (that of Dresden, of Paris, Madrid and Grolier) whose last is perhaps a forgery.

Civilization INCA (1400-1530) developed a positional numbering system bases 10 of them (thus similar to that used today). Not knowing the writing, they used Quipus “to write” the statistics of the State. A quipu is a encordage whose cords present three types of nodes respectively symbolizing the unit, ten and the hundred. A fitting of the nodes on a cord gives a number between 1 and 999; additions of cords allowing to pass to the thousand, the million, etc

India

Ancient Greece

See also: Mathematical of ancient Greece

Concerning Greek mathematics, no original work reached us. There remain only copies, translations and comments via mathematics of Arab language. One can thus think that only major works of this time reached us.

The great innovation of Greek mathematics it is that they leave the field of the utility to return in that of the thought. Mathematics becomes a branch of the Philosophie. philosophical argumentation rises the mathematical argumentation. It is not enough any more to apply, it is necessary to prove and convince: it is the birth of the Démonstration. The other aspect of this new mathematics relates to their object of study. Instead of working on methods, mathematics studies objects, imperfect representations of perfect objects, one does not work on a Cercle but on the idea of a circle.

The great figures of this new mathematics are Thalès (-625 - -547), Pythagore (-580 - -490) and the school pythagorician, Hippocrates (-470 - -410) and the school of Chios, Eudoxe de Cnide (-408 - -355) and the school of Cnide, Théétète of Athens (-415 - -369) then Euclide.

Of its voyages in Egypt, Thalès brings back in Greece knowledge in geometry, works on the isosceles triangles and the triangles registered in a circle.

school pythagorician, we can retain that all is number. The two privileged branches of study are the Arithmétique and the Géométrie. The search for perfect objects leads the Greeks to initially accept like numbers only the rational numbers materialized by the concept commensurable lengths : two lengths are commensurable if there exists a unit in which these two lengths are whole. The failure of this selection materialized by the irrationality of the square Root of two the conduit to accept only the constructible numbers with the rule and the compass. They run up then against the three problems which will cross the history: the Quadrature of the circle, the Trisection of the angle and the Duplication of the cube. Into arithmetic, they set up the concept of even Nombre, odd, perfect and illustrated. Their calculations are all the more remarkable as the Greek Numération is certainly decimal, but additive. In geometry, they study the regular polygons with a preference for the regular Pentagone.

Hippocrates de Chios seeking to solve the problem set up by Pythagore discovers the squaring of the lunules and improves the principle of the demonstration by introducing the concept of equivalent problems.

Eudoxe de Cnide works on the theory of the proportions thus agreeing to handle reports/ratios of irrational numbers. It is probably at the origin of the formalization of the Méthode of exhaustion for calculation by successive approximations of surfaces and volumes.

Théétète works on the regular polyhedral .

But the most important revolution comes from the Éléments of Euclide. The geometrical objects must be defined: they are not any more imperfect objects but of the perfect idea the objects. In its Elements , Euclide launches out in the first formalization of the mathematical thought. It defines the geometrical objects (right-hand sides, circles, angles), it defines space by a series of axioms, it shows by implication the properties which result from this and establishes the formal link between number and length.

After Euclide, two great names clarify Greek mathematics: Archimedes which improves the method of Eudoxe, and Apollonius whose treaty on the Conique S is regarded as the top of the Greek geometry.

Mathematics migrates then to Alexandria then do not finish being melted in mathematics of Arab language.

Civilizations of Arab language

Occident

During the Middle Ages

The role of the Middle Ages was essential for the extension of the field of the numbers. It is during the Middle Ages that the application of the algebra to the trade brought in the East the everyday usage of the irrational numbers, a use which will be transmitted then to Europe. It is as during the Middle Ages, but in Europe, as for the first time of the negative solutions were accepted in problems. It is finally shortly after the end of the Middle Ages which one considered the complex quantities, which made it possible to highlight real solutions of certain cubic equations.

During the European rebirth

As of the 12th century is undertaken in Italy a translation of the Arab texts and, consequently, the redécouverte of the Greek texts. Tolède, old arts center of the Moslem Spain, becomes, following the Reconquista, one of the principal centers of translation, thanks to the work of intellectuals like Gerard de Crémone or Adélard de Bath.

The economic advancement and commercial that Europe knows then, with the opening of new trade route in particular towards the Moslem East, also makes it possible the commercial mediums to be familiarized with the techniques inherited Arabic. Thus, Léonard of Pisa, with its Liber abaci in 1202, largely contributes to make redécouvrir mathematics in Europe. Parallel to the development of sciences, concentrates a mathematical activity in Germany, Italy and Poland at the 14th century and 15th century. One attends a significant development of the Italian school with Cardan, Ferrari, Tartaglia, Scipione del Ferro, Bombelli, school mainly turned towards the resolution of the equations. This tendency is strongly related to the development in the Italian cities of the teaching of mathematics either with a purely theoretical aim such as it could the being in the Quadrivium but with fine practices, in particular intended for the merchants. This teaching is diffused in botteghe of abbaco or “schools the abbaques ones” or maestri teach the arithmetic one, the calculative geometry and methods with future merchants through entertaining, known problems thanks to several “treaties of abbaque” that these Masters left us and studied that recently by the historians of sciences.

It is following work of Cardan joint and Bombelli that the complex numbers were introduced. It is the work undertaken by Jerome Cardan, Viète, Descartes which strongly develops the algebra in Europe.

Until the 16th century, the solution to problem was mainly rhetoric. François Viète introduces the Calcul symbolic system with specific notations for the constants and the variables.

At the XVIIe century

See also: Mathematical in Europe at the XVIIe century

Mathematics carries their glance on physical aspects and techniques. Wire of two fathers, Isaac Newton and Gottfried Leibniz, the Infinitesimal calculus inserts mathematics in the era of the Analyze (Dérivée, Intégrale, differential equation).

The 18th century

Japan

19th century

August 1st

The mathematical history of the 19th century is rich. Too much rich so that under a reasonable test of size one can cover the totality of work of this century. Also should one await this part only the projecting points of work of this century. The XIXe century saw appearing several new theories and the achievement of the work undertaken at the previous century. The tendency to rigor starts at the beginning of the 19th century but will not see its achievement that at the beginning of the 20th century by the questioning of many a priori.

Reviews of mathematics

There existed since the end of the 17th century some academies which published their work and of the annual summaries. Moreover some newspapers had flowered, such as the Acta Eruditorum published by Otto Mencke in Leipzig or the comments of Petersbourg made famous for Euler. But these newspapers or reviews were not specialized in mathematics and accommodated memories of philosophy, history, of botany, as well as of mathematics. The beginning of 19th will see appearing reviews which will specialize in the publication of mathematics. The editors of these reviews are Ferussac (for the general and universal Bulletin of the advertisements and new the scientists ), Gergonne (for the Annales of mathematics pure and applied ), Crelle (for the Journal für die queen und angewandte Mathematik ), Liouville (for the Journal of mathematics pure and applied ) to give only four of them before 1840. They will be followed soon by a crowd of other reviews which each a little famous university enjoys to finance, the such Acta Mathematica of Mittag-Leffler in 1882.

Mechanics

the mechanics of Newton makes its revolution. Using the principle (variational) of less action of Maupertuis, Lagrange states the conditions of first order optimality that Euler had found in any general information and thus finds the equations of mechanics which bear its name. Thereafter, Hamilton, on the steps of Lagrange, expresses these same equations in a form equivalent. They bear also its name. The theory incipient from spaces of Riemann will make it possible to generalize them conveniently.
Delaunay, in an extraordinary calculation, makes a theory of the insurpassée Moon. Faye is expressed thus with its funeral (1872): “Enormous Work, that most qualified judged impossible before him, and where we admire at the same time simplicity in the method and the power in the application”. It solved to make calculation with the 7th order where its precursors (Clairaut, Poisson, Lubbock,…) had stopped with 5th.
It is the Glassmaker who applies the Newtonian theory to the irregularities of Uranus that had just discovered Herschel. It determines by calculation the position and the mass of an unknown planet, which bears today the name of Neptune.
the movement of a solid around a fixed point admits three integrals first algebraic and a last multiplier equalizes to 1. The problem of formal integration by squaring of the movement requires a fourth integral first. This one had been discovered in a particular case by Euler. The question is taken again by Lagrange, Poisson and Poinsot. Lagrange and Poisson discover a new case where this fourth integral is algebraic.
the two traditional cases from now on of the movement of Euler-Poinsot and the movement of Lagrange-Poisson are supplemented, in 1888, by a new case discovered by Sophie Kovalevskaïa. Poincaré had shown that it could not exist case again if the ellipsoid of inertia relating to the point of suspension is not revolution.
Mach states a principle which will be central in the motivations of the relativity of Einstein.
In spite of its successes, mechanics will have evil to find, in teaching, a place which mathematics does not want to yield to him and Flaubert will be able to present as an generally accepted idea that it is a '' “lower part of mathematics” ''.

Mathematical physics

Theory of the numbers

Three major problems will clarify the century: the quadratic law of reciprocity, the distribution of the prime numbers and the great theorem of Fermat. The XIXe century offers considerable progresses on its three questions thanks to the developments of a true theory taking the name of arithmetic or theory of the numbers and resting about abstract and sophisticated tools.

By completely ignoring the work of Euler published in 1784 on the quadratic law of reciprocities, Legendre (1785) and Gauss (1796) finds it by induction. This last ends up giving of it a long complete demonstration in its arithmetic research. The demonstration is simplified in the current of the XIXe century, for example by Zeller in 1852 when it makes only two pages! The quadratic law of reciprocity is promised with a bright future by various generalizations.
Eisenstein shows the cubic law of reciprocity.
Since 1798 Legendre works with its theory of the numbers. It (in 1808) has shown the theorem of rarefaction of the prime numbers and has just proposed a formula approached for $\ pi (X)$ , the number of prime numbers smaller than X. Its research led it to reconsider the screen of Eratosthène. The formula which it obtains is the first element of a method which will take all its direction at the century according to, the method of the screen. Thereafter, in 1830, little before its death, it states a conjecture according to which N ² enters and (n+1) ² exists at least a prime number. This conjecture remains not shown.
the demonstration of Euler of the infinitude of the prime numbers inspires Lejeune-Dirichlet which shows a conjecture of Legendre: there exists an infinity of prime numbers in any arithmetic continuation of the form an+b if has and B are first between them. For that he invents the concept of character arithmetic and the series " of Dirichlet".
the conjecture of Legendre about the distribution of the prime numbers is supported per Gauss and is the subject of work of Tchebyschev in 1850. It shows a framing of $\ pi (X)$ in conformity with the conjecture and it shows the postulate of Bertrand according to which there exists a prime number between N and 2n. But the conjecture of Legendre will be shown only in 1896, by Hadamard and De la Vallee poussin independently.
the most important result is the report of Riemann of 1859 which remains still today the report of the 19th generally quoted century. Riemann studies in this memory the function $\ zeta (S)$ " of Riemann". This function introduced by Euler into its study of the problem of Mengoli is extended to the values complex of S except for 1 which is a pole of residue 1 (theorem of Dirichlet). Riemann states the conjecture, called Hypothèse of Riemann, according to which all zero not realities are of real part equal to 1/2. The demonstrations of Riemann for the majority are only outlined. They are completely shown, except the conjecture of Riemann, by Hadamard and Von Mangold, after 1892.
the great theorem of Fermat, which had already occupied Euler at the previous century is the object of new research by Dirichlet and Legendre (n=5), Dirichlet (n=14), Lamé (n=7), demonstration simplified by Lebesgue. Kummer shows that the great theorem of Fermat is true for the prime numbers regular in 1849. Unfortunately there exist prime numbers irregular and they are even of infinite number.

Logic

Boole launches out in work which will lead to the Boolean algebra, to the symbolic logic and the set theory while wanting to show the existence of God. The calculation of the proposals was born. Augustus De Morgan states the laws which bear its name. Logic leaves philosophy definitively.
Frege poses the bases of the formal logic and Cantor that of the set theory. Neither one nor the other are included/understood per many mathematicians and they cause many concerns. The question of the bases is put. It will be partially solved only tardily at the 20th century. Already point the paradoxes, such that of Burali-Forti, that of Russell or that of Richard in the attempt at set theory of Frege.

Geometry

the century begin with the invention from the descriptive geometry by Gaspard Monge.
Delaunay classified surfaces of revolution of constant curve average, which today bear its name: surface of Delaunay.
Heir to the previous centuries, the century will see to achieve the resolution of the major Greek problems by the negative one. The Trisection of the angle to the rule and the compass is impossible in general. It is the same of the Quadrature of the circle and the Duplication of the cube. Concerning the quadrature of the circle, the 18th century had shown that $\ pi$ was irrational. Liouville, defining the transcendent numbers in 1844, opens the way being studied of the transcendence whose two monuments of the 19th century remain the theorems of Hermite (1872) on the transcendence of E and Lindemann (1881) on that of $\ pi$ , making impossible the quadrature of the circle by the rule and the compass. It is at the end of the century that day the conjecture is made, that will show the century according to in the Théorème of Gelfond-Schneider, that has and exp (a) cannot be simultaneously algebraic.
the other heritage relates to the postulate of Euclide. The problem had in fact quasi solved by Saccheri but this one had not seen that it was close to the goal. Work of Gauss on surfaces brings János Bolyai and Nicolaï Lobatchevsky to call into question the postulate of the parallels. They thus invent a new geometry where the postulate is not true any more, a nonEuclidean geometry. Riemann, after them, will offer a new nonEuclidean solution, before the unit does not form the theory of spaces of Riemann, if important in theory of relativity.
By generalizing the concept of space and distance, Oscar Xavier Schlömilch manages to determine the exact number of regular polyhedrons according to the dimension of space.

Algebra

the results of Welshman and Kummer show that an major advance in algebraic theory of the numbers supposes the comprehension of subtle structures: rings of algebraic entireties subjacent with algebraic extensions. The least complex case is that of the finished and abelian algebraic extensions. It seems simple, the result corresponds to the structures which Gauss at the beginning of the century had studied to solve the problems of the antiquity of construction to the rule and the compass: cyclotomic extensions associated with the polynomials of the same name. One needs nevertheless 50 years and three great names of the algebra to come to end at the end of the century: Kronecker, Weber and Hilbert. It opens the door being studied of the general abelian algebraic extensions, i.e. not finished. Hilbert opens the voice of this chapter of mathematics which represents one of most beautiful the challenge of the future century, the theory of the bodies of class.

Probability and statistics

Legendre in 1805 1811 then Gauss in 1809 introduces, on problems of astronomy, the method of the Least squares, together of methods which will become fundamental in Statistiques.

Pierre-Simon Laplace inserts the analysis in the theory of probability in its analytical theory of the probabilities of 1812 which will remain a long time a monument. Its book gives a first version of the central limit theorem which does not apply whereas for a variable in two states, for example pile or face but not a die has 6 faces. 1901 will have to be waited until to see some appearing the first general version by Liapounov. It is as in this treaty as the method of Laplace for the asymptotic evaluation of certain integrals appears.

Under the impulse of Quételet, which opens in 1841 the first statistical office the Superior council of Statistics, the Statistiques develop and become a field with whole share of mathematics which is based on the probabilities but do not form of it any more part.

the modern theory of the probabilities takes really its essort only with the concept of Mesure and measurable units that Emile Borel introduces in 1897.

Real analysis

At the end it 18th century, to make mathematics consists in writing equalities, sometimes a little doubtful, but without that shocking the reader. Lacroix for example does not hesitate to write

1-1+1-1+… = \ frac12

under the only justification of the development in Taylor series of 1 (1+x). The mathematicians still believe, for little time, that the infinite sum of continuous functions is continuous, and (for longer) than any continuous function a derivative admits…

It is Cauchy which puts a little order in all that while showing that to guarantee that the infinite sum of continuous functions is continuous it is necessary that the series is uniformly convergent.

It is still Cauchy which refuses to consider the sum of divergent series, contrary to the mathematicians of the 18th century whose Lacroix is one of the heirs.

Bolzano shows the first this principle, implicit in the authors of the 18th century, that a function continues which takes values of different signs in an interval is cancelled there, opening the way with topology by the theorem of the intermediate values.

It is necessary almost to await the mileu century so that finally one is interested in the inequalities. Tchebyschev, in its elementary demonstration of the postulate of Bertrand, is one of the first to use them.
, Bessel and Parseval, while dealing with the trigonometrical series what is called show today the inequalities of Bessel-Parseval.
the great application of the trigonometrical series remains the theory of the heat of Fourier, even if this last does not show the convergence of the series that it uses. It will be necessary to await the end of the century so that the question is really clarified by Fejér,…
Poincaré takes part in the contest of king de Suède concerning the solutions of the system of the three bodies. In the report of Stockholm (1889), it gives the first example of chaotic situation. It is expressed as follows:

A very small cause, which escapes to us, determines a considerable effect which we cannot not see, and then we say that this effect is due randomly. If we know exactly the natural laws and the situation of the universe at the initial moment, we could predict exactly the situation of this same universe at one later moment…

Ca was only with regret that one had given up the divergent series at the beginning of the century under the impulse of Cauchy and with an aim primarily of rigor. The divergent series remake, at the end of the century, their appearance. it is a question, in certain case, of giving a sum to such series. The process of summation of Césaro is one of the first. Borel provides his, more sophisticated it. That quickly will become an important subject of study that the 20th century will prolong.

Complex analysis

the theory of the functions of the complex variable, the great subject of all the 19th century, takes its source in work of Cauchy, although interview by Poisson. Definite Cauchy concept of integral of way. It thus manages to state the remainder theorem and the principal properties of the integral " of Cauchy". and in particular the integral Formula of Cauchy.

It thus justifies the development in Taylor series and finds the formula integral of the coefficients while deriving under the sign $\ int$ .Il shows the inequalities " of Cauchy" who will intensely be used, in the theory of the differential equations in particular.

Cauchy published thereafter many applications of its theory in collections of exercises, in particular with the evaluation of real integrals, which he does not hesitate to what is called generalize in today the principal part of Cauchy, a little less than one century before Hadamard for of requires in its resolution of the partial derivative equations by the finished parts of Hadamard and that Schwartz does not come to the distributions.

the theory of the analytical functions develops quickly. Cauchy defines the ray of convergence of a whole series starting from the formula which Hadamard in its thesis will explain perfectly, following work of Wood-Reymond which gave a clear definition of the higher limit.
This makes it possible Liouville to show its theorem and to deduce a news and elementary demonstration from it from the theorem of Alembert-Gauss which one had had so much difficulty to show at the front century.

With died of Cauchy, the torch already passed to Riemann (Theorem of the application in conformity, integral of Riemann replacing the design of Cauchy,…) and Weierstrass which will clear up the concept of essential singular point and of analytical prolongation (although Emile Borel showed thereafter that some of the designs of the " maître" were erroneous).

the theory of Cauchy comes just at the right moment to solve the question of the elliptic integrals finally, theory started with Legendre at the previous century. It is Abel which with the idea of the inversion of the elliptic integrals and discovered the elliptic functions thus that one hastened to study. The very beautiful theory of the elliptic functions is finally completed when appear the treaty of Briot and Bouquet, theory of the elliptic functions, 2nd edition, 1875 and the treaty of Alphen in four volumes (unfortunately stopped by the death of the author).
the most difficult result of the theory remains the theorem of Emile Picard which specifies the theorem of Weierstrass. The first demonstration, with the modular function, is well quickly simplified by Emile Borel at the end of the century.
the century was concerned also much with the theory of the differential equations and in particular of the theory of the potential, of the harmonic functions. Fuchs studies the singularities of the solutions of the linear ordinary differential equations. Emile Picard discovers the process of integration of the differential equations by recurrence, which makes it possible to prove the existence and the unicity of the solutions. That will lead to the study of the integral equations (Fredholm, Vito Volterra…).
Although initiated by Laplace and used sporadically by others during the century, an English electrician, Oliver Heaviside, is put, without another justification, to solve the differential equations by considering the operator of derivation as a noted algebraic quantity p. the theory of the Transformation of Laplace was born. But it will be fully justified only by work of Lerch, Carson, Bromwich, Wagner, Mellin and well of others, at the next century. Oltramare will give also a " calculation of généralisation" based on a close idea.
Emile Borel begins the study of the functions whole and definite the concept of an exponential nature for a whole function. Its goal is to elucidate the behavior of the module of a whole function and in particular to show the bond between the maximum of the module of F on the circle of radius R and the coefficients of the Taylor series of F. Darboux shows that the coefficients of Taylor are written according to the singularities. Others, like Meray, Leau, Fabry, Lindelöf, study the position of the singular points on the circle of convergence or the analytical prolongation of the Taylor series.
Poincaré defines and studies the functions automorphes starting from the geometry hyperbolic.
Schwarz and Christoffel discovers the transformation in conformity which bears their names. It will be intensively used the century according to by average data processing (Driscoll for example).
the apotheosis is reached by the demonstration of the theorem of the prime numbers, in 1896, by Hadamard and De la Vallee poussin independently one of the other.

20th century

The 20th century will have been one century extraodinairement fertile from the mathematical point of view. Three great theorems dominate all the others. on the one hand the Theorem of Gödel, on the other hand the demonstration of the conjecture of Tanyama-Shimura which involved the demonstration of the great theorem of Fermat, finally the demonstration of the conjectures of Weil by Pierre Deligne. New fields of research were born or developed: the dynamic Systems following work of Poincaré, the Probabilities, the Topology, the differential Geometry, the logical , the algebraic Geometry following work of Grothendieck,…

The mathematical community explodes

the trade of Mathématicien really started with professionnaliser at the end of the 19th century. Thanks to universalization, with progress of transport, and average the electronics of communication, mathematical research is not localized on a country or a continent. Since the end of the E century of many conferences, congresses, seminars are held at an intensive pace even annually.
Except two congresses which was held at the 19th century, twenty and one international congress of mathematics was held at the 20th century, almost every four years in spite of the interruptions due to the world wars.
the appearance of the Ordinateur appreciably modified the work conditions of the mathematicians as from the years 1980.

the mathematical development exploded since 1900. At the 19th century, one estimates that one published approximately 900 memories per annum. Currently more: 15000. The number of the mathematicians thus passed from a few hundreds or thousands to more than one million and half in less than one century.
One supported 292 theses of state of mathematics between 1810 and 1901 in France. At the end of the 20th century, it is the number of annually supported theses.

Mechanics

Edouard Husson, in his thesis supported in 1906, solves the problem of the integrals definitively first traditional mechanics for the movement of a solid around a fixed point. There are only four integrals first possible, the fourth appearing only in three particular cases, the movement of Euler-Poinsot, that of Lagrange-Poisson and finally that of Sophie Kowaleski. Complete integration by squaring is thus possible in these three cases. However Goriatchoff shows that integration is also possible in the case of particular initial conditions, and a second case is indicated by Nicolaus Kowalevski in 1908.
the mechanics, which had only little changed since Newton, becomes the object of thorough studies. Einstein publishes its mechanics which contains Newtonian mechanics only while making there tighten celerity C of the light towards the infinite one. The transformation of Galileo leaves his place to the transformation of Lorentz. And a new generalization takes the name of general theory of relativity, between 1909 and 1916.
the cosmological speculation now takes a completely unexpected turning by a sophisticated mathematisation. The static universe of Einstein and that of Sitter are accompanied soon by universes in evolution governed by the equations by Friedman, helped by the searchs for Hubble and Humason which have just discovered the shift towards the red. This spectacular progress is however moderated by the discovery of quantum mechanics. If all goes well on this side until the years 1924, the thesis of Broglie calls all into question. The school of Copenhagen interprets the relations of uncertainties of the Heisenberg like an invitation to regard the wave $\ Psi$ only as one probability of presence, breaking with a total determinism who were the prerogative of the mechanics of Newton and whose Einstein will be the defender baited in the paradox Einstein-Podolski-Rosen. The mechanics of Einstein, which one checked the agreement with the observations, agrees very well to the experimental facts with large scales. Quantum mechanics, of its east coast the queen on an atomic and molecular scale. And two mechanics does not agree. The various attempts at unification are as many failures so much so that one despairs to find this theory unit which would reconcile the two worlds. The pentadimensionnelle theory of Kaluza-Klein, the theory of Einstein of 1931, the theory of the double solution of Broglie, the kinematic theory of Milne, the speculations of Eddington on number 137, the theory of Leaped and Gold,… bring each one an novel idea but which do not solve the problem of the incompatibility of two mechanics. The authors, especially of the physicists, launch out to body lost in a algebrisation of their theories which lead to the theory of the cords, the theory M,… which is still far from solving all the put questions. The unit theory, the great unification is not for this century.
Whereas Einstein had done one of its motivations of them to propose relativity, Goedel shows that the principle of Mach is not registered in the equations of general relativity.

Analyzes

the century starts with the thesis of Lebesgue " integral, length, aire" who constitutes really the beginning of the theory of measurement. Thereafter, of new integrals are create on the traces of Lebesgue (integral of Denjoy, of Perron and Henstock),…). The theory of measurement ends up joining the theory of probability which is axiomatized in 1933 by Kolmogorov.
the theory of Lebesgue leads to the study of spaces $L^p$ . And on traces of Hilbert, Riesz (author of the famous theorem of representation which bears its name), Banach, the differential operators are studied. It is the occasion to create the theory of the distributions, whose premises had been given by Hadamard which had introduced the parts finished into a problem of hydrodynamics. Guelfand, Chilov, Schwartz, Vekua are illustrated thus. The study of the conditions of regularity of the solutions of the partial derivative equations allows Sobolev and its continuators to define its spaces of functions and the theorems of trace according to the geometrical properties of the field.

Theory of the groups

the theory of the groups occupies many people. In particular the sporadic finished groups whose study will be completed only in the years 1980. The study of the groups of Dregs continues and the algebrisation of physics becomes a major stake.

Topology

Poincaré states in 1904 the conjecture which bears its name: “Let us consider a variety compacts V simply related, with 3 dimensions, without edge. Then V is homeomorphic with a hypersphère of dimension 3”. It will be shown in 2003 by Grigori Perelman.

Differential equations

In the study of the differential equations, Painlevé discovers transcendent news. Its study is continued by Gambier.
the thesis of Dulac, constant in 1903, contains a statement which will succitera many complementary work before becoming the theorem of Dulac. Following the example number of theorems " démontrés" , the demonstration was disputed. That of Dulac comprised " trous" highlighted by counterexamples. The theorem of Dulac became the conjecture of Dulac. Then the proof was supplemented and conjectures it of Dulac found its statute of theorem.

Theory of the numbers

the thesis of Cahen (1894) had been the subject of many criticisms. It was the occasion of new studies in the series of Dirichlet and the theory of the functions L, particularly by Mandelbrojt.
One attempted to simplify the evidence of the theorem of the prime numbers (Pram, Erdös and Selberg,…) and those of the theorem of Picardy (Borel). The function zeta of Riemann, with an aim of showing the assumption of Riemann, is the object of very many searchs for Hardy and Littlewood, Speiser, Bohr, Hadamard,… without for all this the mystery is not solved.
the great theorem of Fermat, which narguait the mathematical community since 1637 had taken only modest steps towards the resolution. In a quasi total secrecy, Andrew Wiles attacks the conjecture of Tanyama-Shimura after the demonstration of the conjecture epsilon by Ken Ribet. The latter expressed a bond between the modular functions, the elliptic curves and the great theorem of Fermat so that with the demonstration of the conjecture epsilon it was enough (!) to show the conjecture of Tanyama-Shimura to show the great theorem of Fermat. At the end of seven years, Wiles announces in 1993, conference series on the elliptic curves and their representations during a seminar. It gives to it the essential points of its demonstration of the conjecture of Tanyama-Shimura for the semi-stable elliptic curves without seeing the problem which will delay it more than one year before being solved: the great theorem of Fermat is shown. A little later the conjecture of Tanyama-Shimura is completely shown.
the Problème of Waring is partially solved by Hilbert into 1909 which shows the existence of G (K) while Wieferich attacks the determination of smallest G (K) for an entirety K given. The problem of the determination of G (K) is started with Hardy and Littlewood which state even a not yet shown conjecture. Increases of G (K) given by Vinogradov were improved by Heilbronn (1936), Karatusba (1985), Wooley (1991). One knows the values of G (K) for K ranging between 2 and 20 by work of Pram, Dickson, Wieferich, Hardy and Littlewood,…
the Conjecture of Goldbach is almost solved by Vinogradov in 1936 by showing that almost all the odd integers are written like summons of three prime numbers (weak conjecture of Goldbach).
Pierre Deligne shows, against any waiting, the conjecture of Weil about the eigenvalues of the endomorphisms of Frobenius in algebraic geometry

Complex analysis

On another side, Bieberbach, at the beginning of the century, will emit a conjecture which will be definitively solved only by Louis de Brange de Bourcia, after 70 years of research.
After the First World War, the French mathematical community, which had lost many of its members, was folded up on its favorite subject: complex analysis and the theory of the analytical functions of which it was the principal instigator.
the theory of the whole functions of an infinite nature is the work of Otto Blumenthal about 1913.
the importance of the formula of Jensen is affirmed in the theory of growth initiated by Emile Borel.

Logic and set theory

On the question of the bases, the mathematicians dispute briskly, and of the branches appear under the impulse of Brouwer, of Henri Poincaré,… Cependant the majority of the mathematical community adheres to the Axiome of the choice whose Kurt Gödel will show, in 1930, qu ' it is independent of the other axioms: it is a indécidable proposal. It is the same of the Hypothèse of continuous the (Paul Cohen, 1963). The demonstrations of noncontradictions flower (subject to nonthe contradiction of the set theory and axiomatic of Zermelo-Fraenkel).
In the theory of the demonstration, one will note work of Herbrand (1930) and of Gentzen, too quickly died, the first in 1931, the second in 1945.
One had wondered whether any true proposal, in an axiomatic data, could be shown. The answer is not. The Théorème of incomplétude of Gödel (1931) states that any theory not contradiction able to formalize the arithmetic one does not make it possible to show all the true proposals. In other words, there exist tautologies indémontrables in any theory able to formalize arithmetic… the
Church invents the Lambda calculation and states its thesis, Turing invents the machine which bears its name and Kleene specifies the definition of the recursive functions. The calculable concept of function is invented. Matiyasevich shows that there does not exist Algorithme which makes it possible to say if an equation diophantienne is resolvable, thus giving a negative answer to the Tenth problem of Hilbert. The automata theory and the Théorie of the languages appear.
Donald Knuth publishes his encyclopedia on the art of the programming and creates a new discipline the Analyze of algorithms.

Probabilities

the concept of measurement developed by Emile Borel in 1897 is supplemented by Henri Leon Lebesgue and its theory of the Intégration. This notion of analysis is used by the probabilists for a more rigorous definition of the Probabilité and amongst other things of the Densité of probability

the first modern version of the central limit theorem is given by Alexandre Liapounov in 1901 and the first proof of the modern theorem given by Paul Levy in 1910.

In 1902 Andrei Markov introduces the Chaînes of Markov to undertake a generalization of the law of the great numbers for a continuation of experiments depending the ones on the others. These chains of Markov will know many applications, amongst other things to model the diffusion or for the indexing of Web sites on google.

In 1933 the theory of probability leaves a whole of methods and examples various and becomes a true theory, axiomatized by Kolmogorov.

Kiyoshi Itô sets up a theory and a lemma which bears its name in the years 1940. Those make it possible to connect the stochastic Calcul and the partial derivative equations thus establishing the link between analyzes and probabilities. The mathematician Wolfgang Doeblin had on his side outlined a similar theory before committing suicide with the defeat of his battalion in June 1940. Its work was sent in a fold sealed to the Academy of Science which was open only in 2000.

Numerical analysis

Richard Running introduces the finite elements into 1940 which are used for the numerical solution of partial derivative equations.
the Méthode of Monte Carlo develops, under the impulse of John von Neumann and Stanislas Ulam in particular, at the time of the Second world war and research on the manufacture of the atomic bomb. They are thus called by allusion to the partiqués games of chance with Monte Carlo. These probabilistic methods are used for the numerical resolution of partial derivative equations and stochastic differential equations.

Apparent paradoxes and curiosities

If the acceptance of the axiom of the choice makes it possible to show the existence of bases in the vector spaces of infinite size, in particular spaces of Hilbert, that has also stranger consequences, like the paradox of Banach and Tarski: there exists a cutting of a perfect sphere in five pieces such as with the pieces one can reconstitute two perfect spheres of the same diamêtre as the first.
Of other curiosities, as the theorem of the reversal of the sphere of Smale (which uses the axiom of the choice) are shown.

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