# History of mathematics

The history of mathematics goes back to antiquity and the beginnings of counting in the Neolithic Age . Evidence of the first beginnings of counting processes goes back around 50,000 years. The building of the pyramids in ancient Egypt over 4500 years ago with its precisely calculated shapes is a clear indication of the existence of extensive mathematical knowledge. In contrast to the mathematics of the Egyptians, of which only a few sources exist because of the sensitive papyri , there are about 400 clay tablets of Babylonian mathematics in Mesopotamia . The two cultural areas had different number systems , but both knew the four basic arithmetic operations and approximations for the circle number${\ displaystyle \ pi}$ . Mathematical evidence from China is much more recent, as documents were destroyed by fire, and early Indian mathematics is just as difficult to date. In ancient Europe, mathematics was practiced by the Greeks as a science within the framework of philosophy . The orientation towards the task of “purely logical proof” and the first approach to axiomatization , namely Euclidean geometry, date from this time . Persian and Arabic mathematicians took up the Greek, but also Indian insights, which the Romans had rather neglected, and established algebra . From Spain and Italy this knowledge spread to the European monastery schools and universities. The development of modern mathematics (higher algebra, analytical geometry , probability theory , analysis, etc.) took place in Europe from the Renaissance onwards . Europe remained the center of the development of mathematics until the 19th century, the 20th century saw an "explosive" development and internationalization of mathematics with a clear focus on the USA, which, especially after the Second World War, attracted mathematicians from all over the world great demand due to the expansive technological development.

## Mathematics of the ancient Egyptians and Babylonians

### Egypt

The most important of the few surviving sources that give us information about the mathematical abilities of the Egyptians are the Rhind papyrus , the Moscow papyrus and the so-called “leather roll”.

The Egyptians mostly only used mathematics for practical tasks such as calculating wages, calculating the amount of grain for baking bread or calculating area . They knew the four basic arithmetic operations , such as subtraction as the inverse of addition , multiplication was traced back to repeated doubling and division to repeated halving. In order to be able to carry out the division completely, the Egyptians used general fractions of natural numbers, which they represented by the sums of ancestral fractions and the fraction 2/3. You could also solve equations with an abstract unknown . In geometry they were familiar with the calculation of the areas of triangles , rectangles and trapezoids , as an approximation of the number of circles π (pi) and the calculation of the volume of a square truncated pyramid . Archaeological finds of records of mathematical evidence are still missing today. They had their own hieroglyphs for numbers , starting from 1800 BC. They used the hieratic script , which was written with rounded and simplified hieroglyphic characters. ${\ displaystyle \! ^ {{\ left ({\ frac {16} {9}} \ right)} ^ {2}}}$

### Babylon

Babylonian cuneiform tablet YBC 7289 with a sexagesimal approximation for the square root of 2 (on the diagonal)

The Babylonians used a sexagesimal - place value system , albeit with imperfect expression, then the meaning that often only found out of context. The clay tablets obtained are, for example, tables of numbers for multiplication, with reciprocal values ​​(according to their method for division), squares and cubes; Table values ​​that were not available could be determined by linear interpolation and the application of divisibility rules. There are also tables with tasks that correspond, for example, to today's linear systems of equations or compound interest calculations, and explanations of calculation methods. They had an algorithm for calculating square roots (Babylonian root extraction) and could even solve quadratic equations with it. They knew the Pythagorean theorem and used 3 or 3 + 1/8 as an approximation for the circle number π. The Babylonians evidently did not seek a strict argument .

## Mathematics in Greece

The mathematics of ancient Greece is divided into four major periods:

According to a tradition that comes from antiquity but is controversial among science historians, the history of mathematics as a science begins with Pythagoras of Samos . The principle "everything is number" is ascribed to him - albeit wrongly. He founded the school of the Pythagoreans , from which mathematicians such as Hippasus of Metapontus and Archytas of Taranto emerged . In contrast to the Babylonians and Egyptians, the Greeks had a philosophical interest in mathematics. One of the findings of the Pythagoreans is the irrationality of geometric route relationships, which is said to have been discovered by Hippasus. The earlier widespread view that the discovery of irrationality among the Pythagoreans triggered a philosophical "fundamental crisis" because it shook their earlier convictions, is rejected by today's research. The ancient legend that Hippasus betrayed his secrets by making his discovery public is believed to have originated from a misunderstanding.

Mathematics was very popular in the Platonic Academy in Athens. Plato valued it very much because it served to be able to acquire true knowledge. Greek mathematics then developed into a proving science . Aristotle formulated the basics of propositional logic . With the exhaustion method, Eudoxos von Knidos created a rudimentary form of infinitesimal calculus for the first time . However, due to the lack of real numbers and limit values, this method was quite unwieldy. Archimedes expanded this and calculated among other things an approximation for the circle number π.

Euclid summarized a large part of the mathematics (geometry and number theory) known at the time in his textbook elements . Among other things, it proves that there are infinitely many prime numbers. This work is considered a prime example of mathematical proof: from a few specifications, all results are derived with a rigor that should not have existed before. Euclid's “Elements” is still used as a textbook today, after more than 2000 years.

In contrast to the Greeks, the ancient Romans were hardly concerned with higher mathematics, they were more interested in practical applications, for example in surveying and engineering. The Roman surveyors were called Gromatici or Agrimensors ; their writings were summarized in a collective work ( Corpus Agrimensorum ) in the 6th century . Important agrimensors were Sextus Iulius Frontinus , Hyginus Gromaticus and Marcus Iunius Nipsus . Until late antiquity, mathematics remained largely a domain of the Greek-speaking inhabitants of the empire, the focus of mathematical research in Roman times was on Sicily and in North Africa, especially in Alexandria . Pappos made new contributions to geometry (also with first results on projective geometry), Apollonios to conic sections and Diophantus made contributions to a geometrically disguised algebra and number theory (solution of integer equations, later called Diophantine problems after him). The last mathematician in Alexandria known by name was Hypatia , who was killed by a Christian mob in 415.

## Chinese and Indian math

### China

The first extant textbook of Chinese mathematics is the Zhoubi suanjing . It was made during the Han Dynasty , between 206 BC. BC to AD 220, supplemented by Liu Hui , since most of the mathematical records were destroyed and written down from memory as a result of the burning of books and documents during the Qin dynasty . The mathematical knowledge is used until the 18th century BC. Dated. Further additions followed later until 1270 AD. It also contains a dialogue on the calendar between Zhou Gong Dan, Duke of Zhou, and Minister Shang Gao. Almost as old is Jiu Zhang Suanshu ("Nine Chapters on the Art of Mathematics"), which contains 246 problems on various areas; Among other things, the Pythagorean Theorem can be found in it, but without any evidence. The Chinese used a decimal place value system written from horizontal and vertical lines (Suan Zi, "arithmetic with stakes"); around 300 AD, Liu Hui calculated the number 3.14159 as an approximation for π using a 3072 corner .

Chinese mathematics reached its peak in the 13th century. The most important mathematician of this time was Zhu Shijie with his textbook Siyuan Yujian ("Precious Mirror of the Four Elements"), which dealt with algebraic systems of equations and fourteenth-degree algebraic equations and solved them using a kind of Horner method . After this period there was an abrupt break in mathematics in China. Around 1600, the Japanese took up knowledge of Wasan (Japanese mathematics). Your most important mathematician was Seki Takakazu (around 1700). Mathematics was practiced as a secret temple science.

### India

Aryabhata

Dating, according to a bon mot by the Indian scientist W. D. Whitney , is extremely problematic throughout Indian history.

The oldest allusions to the geometric rules for the sacrificial altar can already be found in the Rig Veda . But it was not until several centuries later that the Sulbasutras ("rules of rope", geometric methods for the construction of sacrificial altars) and other doctrinal texts such as the Silpa Sastras (rules for temple building) etc. were created (i.e., canonized) . The Aryabhatiya and various other " Siddhantas " ("systems", mainly astronomical tasks). The Indians developed the familiar decimal position system , that is, the polynomial notation based on base 10 and the associated calculation rules. Written multiplication in Babylonian, Egyptian or Roman number notation was extremely complicated and worked by means of substitution; d. H. with many rules of decomposition and summarization related to the notation, while in Indian texts there are many “elegant” and simple procedures, for example for drawing roots in writing.

Our numerals ( Indian digits ) for the decimal digits are derived directly from the Indian Devanagari . The earliest use of the digit 0 is dated around AD 400; Aryabhata around 500 and Bhaskara around 600 were already using it without hesitation, his contemporary Brahmagupta even calculated with it as a number and knew negative numbers. The naming of the numerals in different cultures is inconsistent: The Arabs call these (adopted Devanagari) digits “Indian numbers”, the Europeans “Arabic numbers” based on medieval reception history and the Japanese for an analogous reason Romaji , that is, Latin or Roman characters (along with the Latin alphabet). Europeans mean “ Roman numerals ” to mean something else.

With the spread of Islam to the east, around 1000 to 1200 at the latest, the Muslim world took over many of the Indian findings; Islamic scholars translated Indian works into Arabic, which also reached Europe via this route. A book by the Persian mathematician Muhammad ibn Musa Chwarizmi was translated into Latin in Spain in the 12th century . The Indian numerals (figurae Indorum) were first used by Italian merchants. Around 1500 they were known in what is now Germany.

Other important mathematicians were Brahmagupta (598–668) and Bhaskara II (1114–1185).

## Mathematics in the heyday of Islam

In the Islamic world, the capital Baghdad was the center of science for mathematics . Muslim mathematicians took over the Indian position arithmetic and the sine and developed the Greek and Indian trigonometry further supplemented the Greek geometry and translated and annotated the mathematical works of the Greeks. The most important mathematical achievement of the Muslims is the establishment of today's algebra. This knowledge came to Europe via Spain, the Crusades and Italian sea trade. In the translation school in Toledo, for example, many of the Arabic scripts were translated into Latin.

The following phases can be distinguished:

## Mayan mathematics

Our knowledge about the mathematics and astronomy (calendar calculation) of the Maya comes mainly from the Dresden Codex . The Mayan numeral is based on base 20. The reason for this is believed to be that the ancestors of the Maya counted with fingers and toes. The Maya knew the number 0 but did not use fractions. To represent numbers, they used dots, lines, and circles, which represented the digits 1, 5, and 0. The mathematics of the Maya was highly developed, comparable to the advanced cultures in the Orient. They used them for calendar calculations and for astronomy. The Mayan calendar was the most accurate of its time.

## Mathematics in Europe

### Mathematics in the Middle Ages

The Middle Ages as an epoch of European history began around the end of the Roman Empire and lasted until the Renaissance . The history of this time was determined by the great migration and the rise of Christianity in Western Europe. The decline of the Roman Empire led to a vacuum that was only compensated in Western Europe by the rise of the Frankish Empire . In the course of the creation of a new political order by the Franks, the so-called Carolingian renaissance came about . Ancient knowledge was first preserved in monasteries. In the later Middle Ages, monastery schools were replaced by universities as centers of learning. An important enrichment of Western European science took place in that the Arabic tradition and further development of Greek mathematics, medicine and philosophy as well as the Arabic adaptation of Indian mathematics and numeric writing became known in the West through translations into Latin. The contacts to Arab scholars and their writings arose on the one hand as a result of the Crusades in the Middle East and on the other hand through contacts with the Arabs in Spain and Sicily, in addition there were trade contacts, especially with the Italians in the Mediterranean area, to whom, for example, Leonardo da Pisa (“ Fibonacci ”) owed some of his mathematical knowledge.

#### Rise of the monastery schools

Boëthius (medieval illustration)

Boëthius (approx. 480-524) stands on the border between the Roman Empire and the beginning of the New . His introduction to arithmetic formed the basis for teaching this subject up to the end of the Middle Ages; Also influential, if to a lesser extent, was his introduction to geometry. In 781 Charlemagne appointed the scholar Alcuin of York (735-804) to head his court school, who was supposed to develop the educational system of the Franconian Empire. He was also called the "Teacher of the West Franconians". A pupil of Alkuin founded the school system in the eastern Franconian Empire, Rabanus Maurus from Mainz . Mathematical teaching content was taught according to the classification of the seven liberal arts in the four subjects of the quadrivium :

• Arithmetic: The properties and types of numbers (e.g. even, odd, prime numbers, area and body numbers) as well as proportions and numerical relationships, in each case according to Boëthius, also basic knowledge of Greek and Latin numerals , basic arithmetic, finger arithmetic and in the 11th-12th . Century abacus , since the 13th century also written arithmetic with Arabic numerals
• Geometry: elements of Euclidean geometry, measurement and surveying, geography and z. T. also history
• Astronomy: Basic knowledge of Ptolemaic astronomy and z. Partly also astrology , use of the astrolab since the 10th century , also computistics to calculate the Easter date and the movable festivals of the church year
• Music: Harmony according to the numerical proportions of the ancient church modes

The following arithmetic books created in monasteries are known: Exercises to sharpen the minds of young people (around 800) (previously attributed to Alcuin von York), the exercises from the Annales Stadenses (Stade Abbey) (around 1180) and the Practica of Algorism Ratisbonensis (Emmeram Abbey, Regensburg ) (around 1450).

#### Calculation of the Easter date

The calculation of the date for Easter , the most important festival in Christianity , played a major role in the development of mathematics in the Middle Ages. Charlemagne decreed that a monk had to deal with computistics in every monastery . This should ensure the knowledge of the calculation of the Easter date . The exact calculation of the date and the development of the modern calendar were further developed by these monks, the basics were taken over by the Middle Ages from Dionysius Exiguus (approx. 470 to approx. 540) and Beda the Venerable (approx. 673-735). In 1171 Reinher von Paderborn published an improved method for calculating the Easter date.

#### Universities

The early medieval monastery schools were only supplemented later in the Middle Ages by the cathedral schools, the schools of the mendicant orders and the universities. They were therefore initially the only bearers of the ancient cultural heritage by ensuring that ancient works were copied and distributed. For a long time, copying, commenting and compiling the teaching material remained the only form of dealing with the topics of mathematics. It was not until the High Middle Ages that the somewhat more critical method of scholasticism developed , with which doctrinal opinions were checked for contradictions in their pro and contra and these were resolved if possible in accordance with the standpoints of the ecclesiastical and ancient authorities, which were regarded as fundamental.

This method was applied to the representations of ancient science, particularly that of Aristotle, from the 12th century. In the 12th century the universities of Paris and Oxford became the European center of scientific activity. Robert Grosseteste (1168–1253) and his student Roger Bacon (1214–1292) drafted a new scientific paradigm . Not the appeal to ecclesiastical or ancient authorities, but the experiment should determine the assessment of correctness significantly. Pope Clement IV asked Roger Bacon in 1266 to convey his views and suggestions for remedying the shortcomings in science. Bacon wrote several books in response, including his Opus Maius . Bacon pointed out the importance of mathematics as the key to science; he dealt in particular with the geometry applied to optics. Unfortunately, the Pope died before the book reached him. Another important contribution by Bacon concerns the calendar reform, which he called for, but which was not implemented until 1582 as the Gregorian calendar reform .

An important methodological development in science was the quantification of qualities as a key for the quantitative description of processes. Nikolaus von Oresme (1323-1382) was one of the first who also dealt further with the change in intensities. Oresme studied various forms of movement. He developed a kind of functional description by plotting speed against time. He classified the different forms of movement and looked for functional connections.

Nikolaus von Kues (Nikolaus Cusanus)

Oresme, but also Thomas Bradwardine (1295–1349), Wilhelm von Ockham (1288–1348), Johannes Buridan (approx. 1300 to approx. 1361) and other scholars from Merton College examined the functional description of the relationships between speed, force and location , in short: they dealt with kinetics . Methodologically important advances have also been made. Grosseteste formulated the principle of the uniformity of nature, according to which bodies of the same nature behave in the same way under the same conditions. Here it becomes clear that even then the scholars were aware that the circumstances under which certain behavior is viewed must be checked if comparisons are to be made. Furthermore, Grosseteste formulated the principle of the economy of description, according to which, under the same circumstances, those arguments are to be preferred that require fewer questions to be answered or fewer assumptions for complete proof. William Ockham was one of the greatest logicians of the time, and Ockham's razor is famous , a principle that says that a theory should always contain as few assumptions and concepts as possible.

The scholars of the time were often theologians too. The preoccupation with spiritual questions such as B. the omnipotence of God led them to questions about the infinite. In this context, Nikolaus von Kues (Nikolaus Cusanus) (1401–1464) should be mentioned, who was one of the first to describe the infinity of the world , even before Galileo or Giordano Bruno . His principle of the coincidentia oppositorum testifies to a profound philosophical preoccupation with the topic of infinity.

#### Practical math

Towards the end of the Middle Ages the cathedrals of Europe emerged, the construction of which made completely new demands on the mastery of statics and challenged technological excellence in this area. In this context, geometric problems were also repeatedly dealt with. An important textbook that deals with the architecture, the site huts book of Villard de Honnecourt .

In the field of surveying geometry, steady progress was made throughout the Middle Ages; the geometry of geodesics in the 11th century , based on a book by Boëthius , and in the 12th century the more conventional Geometria practica by Hugo von St. Victor (1096 -1141). In the 13th century, Levi ben Gershon (1288–1344) described a new measuring device, the so-called Jacob's staff .

#### Start of the money economy

Leonardo da Pisa (Fibonacci), fantasy portrait

With the beginning of an economy based not on exchange of goods but on money, new areas of application of mathematics emerged. This applies in particular to Italy, which at the time was a transshipment point for goods to and from Europe, and whose leading role in finance and banking at the time is still evident today in the use of words such as “account”, “gross” and “net”. affects. In this context, Leonardo da Pisa, called Fibonacci , and his Liber abbaci should be mentioned in particular , which has nothing to do with the abacus as a calculation board, but rather the word abacus or "abbacco" as a synonym for, according to a language that was emerging in Italy at the time Math and arithmetic used. In the mathematics of Fibonacci a synthesis of commercial arithmetic, traditional Greco-Latin mathematics and new methods of Arabic and (Arabic-mediated) Indian mathematics took place, which was unique for the Middle Ages. Mathematically less demanding, but more geared towards the practical requirements of bankers and merchants, were the numerous arithmetic books that had been written in Italian as textbooks on practical and mercantile arithmetic since the 14th century.

### Early Modern Mathematics

Arabic mathematics came via Spain, where the Moors were expelled from Europe in the course of the Reconquista , and through trade relations with Europe and their mathematics subsequently influenced European fundamentally. Terms such as algebra , algorithm and Arabic numerals go back to it. During the Renaissance, the ancient classics and other works became widely available through the printing press. The art of the Renaissance led to the development of perspective (including Albrecht Dürer , Filippo Brunelleschi , Leon Battista Alberti , Piero della Francesca ) and descriptive geometry and the projective geometry associated with it ( Gérard Desargues ) also had its origin in architecture. The voyages of discovery led to developments in cartography and navigation (the long acute longitude problem ) and land surveying ( geodesy ) was important for the development of the territorial states. Practical requirements of engineers (not least of a military nature) such as Simon Stevin (decimal fractions) and astronomers led to improvements in computing technology, especially through the invention of logarithms ( John Napier , Jost Bürgi ).

In Germany, the proverbial Adam Ries (e) explained arithmetic to his compatriots in the local language, and the use of Indian numerals instead of the impractical Roman numerals became popular. In France, René Descartes discovered that geometry, which had been taught by Euclid until then , can also be described algebraically and, conversely, algebraic equations can be interpreted geometrically ( analytical geometry ) after the introduction of a coordinate system . An exchange of letters between Blaise Pascal and Pierre de Fermat in 1654 about problems with games of chance is considered to be the birth of classical probability theory .

Blaise Pascal was also one of the founders of combinatorics ( binomial coefficients , Pascal's triangle ) and built one of the first calculating machines. François Viète systematically used variables (unknowns) in equations. With this the algebra was further formalized. Pierre de Fermat, who was a full-time judge, provided important results for the calculus of variations and in number theory (solving algebraic equations in whole numbers, so-called Diophantine problems ), in particular the “ small Fermat's theorem ” and formulated the “ big Fermat's theorem ”. He claimed that the equation has no positive integer solutions if . In the margin of his edition of the Arithmetica by Diophant of Alexandria he wrote the sentence: "I have found a wonderful proof, but unfortunately the margin is too narrow for it". Mathematicians searched in vain for this alleged proof for 400 years. The proof of the theorem was only possible centuries later (1995) using methods inaccessible to Fermat (see below). In Italy Cardano and Tartaglia found the algebraic formula for the solutions of the cubic equation and the search for further solution formulas for higher equations did not come to an end until the Galois theory in the 19th century. ${\ displaystyle x ^ {n} + y ^ {n} = z ^ {n}}$${\ displaystyle n \ geq 3}$

### Development of the infinitesimal calculus

The problem of determining tangents on curves ( differential calculus ) and areas under curves ( integral calculus ) occupied many mathematicians of the 17th century, with important contributions from, for example, Bonaventura Cavalieri , Johannes Kepler , Gilles de Roberval , Pierre de Fermat , Evangelista Torricelli , René Descartes, Isaac Barrow (who influenced Newton) and Christian Huygens (who especially influenced Leibniz).

Independently of one another, Isaac Newton and Leibniz developed one of the most far-reaching discoveries in mathematics, infinitesimal calculus, and thus the concept of the derivation and connection of differential and integral calculus using the fundamental theorem of analysis . In order to deal with the problem of infinitely small sizes, Newton mainly argued about speeds (fluxions) . Leibniz gave a more elegant formulation of the infinitesimal calculus and justified the name and the integral sign . Between the two mathematicians and their students, there was later a protracted dispute over priority, which also came to a head in a conflict between continental European and English mathematics. The versatile, but rather philosophically interested Leibniz did not come close to Newton, who was personally very difficult and controversial in terms of mathematical abilities (Leibniz had previously been in correspondence with Newton, who saw it in such a way that he was essential to him in this way Submitted results that Newton had not published, but circulated among selected mathematicians), but received support from continental European mathematicians, especially the gifted mathematicians of the Bernoulli family from Switzerland. ${\ displaystyle {\ tfrac {\ mathrm {d}} {\ mathrm {d} x}}}$ ${\ displaystyle \ textstyle \ int}$

At the same time, Isaac Newton laid the foundations of theoretical mechanics and theoretical physics in his famous main work Philosophiae Naturalis Principia Mathematica . He did not use the language of analysis, but formulated his sentences in the classical geometric style, but it was clear to his contemporaries that he had gained them with the help of analysis, and theoretical physics and mechanics were then also used in this language in the 18th century expanded.

Leibniz, in turn, also came up with ideas for a universal algebra, determinants , binary numbers and a calculating machine .

### Mathematics in the 18th century

The methods of calculus were further developed, even if the requirements for mathematical rigor were still very low at that time, which some philosophers such as George Berkeley sharply criticize. One of the most productive mathematicians of that time was the Swiss Leonhard Euler . Much of the “modern” symbolism used today goes back to Euler. In addition to his contributions to analysis, he was the first to introduce the symbol i as a solution to the equation x 2 = −1 , among many other improvements in the notation  . The prehistory of complex numbers went back to Cardano and other Renaissance mathematicians, but this expansion of the number range caused difficulties for the imagination of most mathematicians for a long time and their real breakthrough in mathematics was only achieved in the 19th century, after a geometric interpretation was discovered when two-dimensional vectors ( Caspar Wessel 1799, Jean-Robert Argand , Gauß). Numerous applications of mathematics in physics and mechanics also originate from Euler.

Euler also speculated about what an analysis situs could look like, describing the positional relationships of objects without using a metric (length and angle measurement). This idea was later expanded into the theoretical building of topology . Euler's first contribution to this was the solution of the Königsberg bridge problem and his polyhedron replacement . Another fundamental connection between two distant areas of mathematics, analysis and number theory , also goes back to him. Euler was the first to discover the connection between the zeta function and prime numbers , which Bernhard Riemann made the basis of analytical number theory in the 19th century. Further contributions to the analysis of time and its application came from the Bernoullis (especially Johann I Bernoulli , Daniel Bernoulli ), Lagrange and D'Alembert , in particular the expansion and application of the calculus of variations to the solution of many problems in mechanics. A center of development was France and Paris, where after the French Revolution and under Napoleon, mathematics took off in newly founded engineering schools (especially the Ecole Polytechnique ). Mathematicians like Jakob I Bernoulli at the beginning of the century, Abraham de Moivre , Laplace and Thomas Bayes in England developed the theory of probability .

Lagrange made important contributions to algebra (quadratic forms, equation theory) and number theory, Adrien-Marie Legendre to analysis (elliptical functions, etc.) and number theory and Gaspard Monge to descriptive geometry.

### Mathematics in the 19th century

From the 19th century, the basics of mathematical terms were questioned and substantiated. Augustin-Louis Cauchy established the definition of the limit value . He also laid the foundations of function theory . The close connection between the development of physics and mechanics and analysis from the 18th century remained and many mathematicians were theoretical physicists at the same time, which was not yet separated at that time. An example of the connection is the development of Fourier analysis by Joseph Fourier . One of the central themes of the 19th century was the investigation of special functions, especially elliptical functions and their generalizations ( Niels Henrik Abel and Carl Gustav Jacobi played an important role here ) and algebraic geometry of curves and surfaces with connections to function theory (including Bernhard Riemann with his idea of ​​the Riemann surface , Alfred Clebsch , Felix Klein and the Italian school for algebraic surfaces). An abundance of individual results were discovered in the most varied of areas, but their order and strict justification could often only be established in the 20th century. As in the 18th century, celestial mechanics remained a large field of activity for mathematicians and a source of developments in mathematics . ${\ displaystyle \ epsilon}$

The French Évariste Galois , who was killed young in a duel, used methods of group theory in his Galois theory to investigate the solvability of algebraic equations, which led to the proof of the general insolubility of polynomial equations (degree 5 and higher) by radicals (root operations). This was shown independently by Niels Henrik Abel. With the help of Galois theory, some of the classical problems of antiquity were recognized as unsolvable, namely the trisection of the angle and the doubling of the cube ( Pierre Wantzel succeeded without Galois theory , however ). The squaring of the circle was only done by proving the transcendence of by Ferdinand Lindemann . New geometries emerged, especially projective geometry ( Jean-Victor Poncelet , Jakob Steiner , Karl von Staudt ) was greatly expanded and Felix Klein arranged these and other geometries with the help of the concept of the transformation group ( Erlanger program ). ${\ displaystyle \ pi}$

The algebraics realized that one can not only calculate with numbers; all you need are shortcuts. This idea was formalized in groups (for example Galois, Arthur Cayley , Camille Jordan , Ferdinand Georg Frobenius ), rings , ideals and bodies (including Galois, finite bodies are called Galois bodies after Galois), with algebraists in Germany such as Richard Dedekind , Leopold Kronecker played an important role. The Norwegian Sophus Lie studied the properties of symmetries . Through his theory, algebraic ideas were introduced into analysis and physics. The modern quantum field theories are essentially based on symmetry groups. The vector concept was created (by Hermann Grassmann among others ) and the competing concept of quaternions (by William Rowan Hamilton ), an example of the many newly discovered algebraic structures, as well as the modern theory of matrices (linear algebra).

Two of the most influential mathematicians of the time, Carl Friedrich Gauß and Bernhard Riemann, worked in Göttingen . In addition to fundamental knowledge in analysis, number theory, and function theory, they and others created differential geometry with the concept of curvature and the extensive generalization into higher dimensions by Riemann ( Riemann's geometry ). The Non-Euclidean geometry made the limitations of the centuries learned Euclidean axiom system significantly and was Nikolai Lobachevsky and János Bolyai reasons (their existence was known Gauss, but nothing about it published). With his Disquisitiones Arithmeticae, Gauss laid the foundations of algebraic number theory and proved the fundamental theorem of algebra .

In Berlin, Karl Weierstrass in particular founded a mathematical school based on the strict foundation of analysis and the foundation of function theory on power series, while Riemann founded geometric function theory and emphasized the role of topology. Weierstrasse's student Sofja Wassiljewna Kowalewskaja was one of the first women to play a prominent role in mathematics and the first professor in mathematics.

Georg Cantor surprised with the realization that there can be more than one “infinity”. He defined what is a set for the first time and thus became the founder of set theory . Towards the end of the 19th century, Henri Poincaré took a leading role in mathematics, among other things, he made significant progress in algebraic topology and the qualitative theory of differential equations, which later made him a forerunner of chaos theory .

The newly increased demands on the rigor of proofs and efforts to axiomatize sub-areas of mathematics were represented by Richard Dedekind for real numbers, Giuseppe Peano for natural numbers and David Hilbert for geometry. After thousands of years, the logic has been completely overhauled. Gottlob Frege invented predicate logic , the first innovation in this area since Aristotle . At the same time, his work marked the beginning of the fundamental crisis in mathematics .

France experienced a great upswing in mathematics after the French Revolution, and at the beginning of the century Germany followed suit with the dominant research personality of Gauss, who, however, did not form a school and, like Newton, was in the habit of not publishing even significant new discoveries. The German system of research seminars at universities was first formed in Königsberg and then became a central component of teaching in the mathematical centers in Göttingen and Berlin and then also had an impact, for example, in the USA, for which Germany was formative in mathematics. In Italy, too, mathematics took off after the country's independence, especially in algebraic geometry (Italian school of Francesco Severi , Guido Castelnuovo and Federigo Enriques ) and the foundations of mathematics (Peano). Great Britain had a particular focus on theoretical physics, but its mathematical schools repeatedly tended to take special paths that isolated them from continental Europe, such as the stubborn adherence to Newtonian style of analysis in the 18th century and the emphasis on the role of quaternions at the end of the 19th century Century. Felix Klein, who last worked in Göttingen alongside Hilbert and was well connected, took a leading position in many respects in Germany towards the end of the century and organized an encyclopedia project on mathematics and its applications, which also included French mathematicians. The defeat in the Franco-Prussian War of 1870/71 acted as an incentive for many French mathematicians as in other areas to catch up a supposed deficit to the rising German Empire, which led to a new bloom of French mathematics. The First World War also led to a break in relationships in mathematics.

## Modern math

The 20th century saw an unprecedented expansion of mathematics, both in breadth and depth, which eclipsed the preceding centuries. The number of mathematicians and users of mathematics increased sharply, also with regard to the number of countries of origin and women. America and the Soviet Union took on a leading role in addition to the traditional Central European nations, especially after the Second World War, but also countries like Japan and China after opening to the West. Mathematics became a key discipline due to the great technological advances in the 20th century and especially digitization.

In 1900 Hilbert formulated a number of famous problems ( Hilbert's problems ), which often served as guidelines for further progress and most of which were solved or brought closer to a solution in the course of the 20th century. A concern of modern mathematics was the need to consolidate the foundations of this science once and for all. However, this began with a crisis at the beginning of the 20th century: Bertrand Russell recognized the importance of Frege's work. At the same time, however, he also discovered insoluble contradictions in it, which were connected with paradoxes of the infinite ( Russell's antinomy ). This realization shook all of mathematics. Several attempts to save it were made: Russell and Alfred North Whitehead tried in their several thousand pages work Principia Mathematica to build a foundation with the help of type theory . Alternatively, Ernst Zermelo and Abraham Fraenkel established set theory axiomatically ( Zermelo-Fraenkel set theory ). The latter prevailed because its few axioms are much easier to handle than the difficult presentation of the Principia Mathematica .

David Hilbert , photo from 1886
Kurt Gödel (1925)

The doubts about the basics remained. David Hilbert , who founded a famous school in Göttingen and revolutionized the most varied of mathematical disciplines (from geometry, algebraic number theory, functional analysis with contributions to physics to the fundamentals of mathematics), however, essentially focused on one area in individual creative periods and gave up earlier research areas completely, turned in his last creative phase to the fundamentals of mathematics and the formalization of mathematical proofs. For Hilbert and his formalistic school, evidence was only a series of deductions from axioms, a series of symbols, and according to a famous saying by Hilbert relating to the axiomatization of geometry, points, lines, and planes in formula language should always be through Can replace tables, chairs and beer mugs , only the axioms and rules of derivation were important. Kurt Gödel's Incompleteness Theorem showed, however, that in any formal system that is extensive enough to build up the arithmetic of natural numbers there are theorems that can neither be proven nor disproved. Mathematicians and logicians such as Gerhard Gentzen proved the consistency of sub-areas of mathematics (each with recourse to principles that go beyond these sub-areas). Another direction, which began at the beginning of the century with the intuitionism of Brouwer , who was also one of the founders of set-theoretical topology, tried to build up constructive mathematics based on finite steps , in which, however, one had to forego important mathematical theorems.

In addition to logic, other areas of mathematics were increasingly abstracted and placed on axiomatic foundations, in which David Hilbert and his school played a leading role. French mathematicians such as Henri Lebesgue ( Lebesgue integral ), Jacques Hadamard and Emile Borel (measure theory), the Hilbert School in Göttingen and the Polish school under its leading figure Stefan Banach were centers of the development of functional analysis , i.e. the investigation of infinitely dimensional functional spaces. With the help of the Banach spaces and their dualities , many problems, for example the integral equations , can be solved very elegantly. The Polish school of the interwar period was also a leader in topology and basic mathematical research, and the Russian mathematicians also initially focused on functional analysis ( Lusin School , Andrei Kolmogorow ) and topology (including Pawel Sergejewitsch Alexandrow , Lev Pontryagin ). Mathematics was fertilized by the development of new physical theories, in particular quantum mechanics (with a connection in particular to functional analysis) and the theory of relativity , which promoted the tensor calculus and differential geometry. The distributions ( Laurent Schwartz , Sergei Lwowitsch Sobolew ) of functional analysis were first introduced by Paul Dirac in quantum mechanics. This in turn benefited from the development of the spectral theory of linear operators (linear algebra in an infinite number of dimensions).

Andrei Kolmogorov provided an axiomatic justification for probability . For him, the probability is similar to the area and can be treated with methods of measure theory . This gave this area a secure foundation, even if the disputes over questions of interpretation continued (see also the history of the calculus of probability ). A great source of "useful mathematics" was the development of diverse statistical methods ( Ronald Aylmer Fisher , Karl Pearson , Abraham Wald , Kolmogorow and others) with broad applications in experimentation, medicine, but also in the social sciences and humanities, market research and politics .

The leading role of the Hilbert School ended with National Socialism, which was also reflected in mathematics among the representatives of German mathematics , and the expulsion of the majority of Jewish scientists from their university positions. Many found refuge in the United States and elsewhere and there stimulated the development of mathematics.

During the Second World War, there was a great need to solve specific mathematical problems for military purposes, for example in the development of the atomic bomb, the radar or the decoding of codes. John von Neumann and Alan Turing , who had previously developed the abstract concept of a universal calculating machine in the theory of predictability , worked on specific computer projects. The computer found its way into mathematics. This led to a dramatic advancement in numerical mathematics . With the help of the computer, complex problems that could not be solved by hand can now be calculated relatively quickly, and numerical experimentation made many new phenomena accessible in the first place ( experimental mathematics ).

Abstraction and formalization reached a high point in the work of the author collective Nicolas Bourbaki , which included leading mathematicians in France (and beyond) such as André Weil , Jean-Pierre Serre , Henri Cartan and Claude Chevalley and whose meetings began as early as the late 1930s. After the fall of the Hilbert School and the expulsion of many mathematicians by the National Socialists after the war, from which the USA in particular benefited, they took on a leading role in the structural conception of mathematics, initially consciously following the Göttingen algebraic school wanted to overcome the analysis-oriented curriculum in France, but soon also had an impact far beyond it (with the new mathematics in the school curriculum of the 1960s and 1970s).

Significant in the second half of the 20th century was the fundamental upheaval in algebraic geometry , primarily through the work of Alexander Grothendieck and his school, as well as the broad development of algebraic topology, and - in part with it - the development of category theory . This was a further increase in abstraction after the development of abstract algebra in the first half of the 20th century, especially in the school of Emmy Noether, and provided new approaches and ways of thinking that have become effective in large parts of mathematics. The category theory offered an alternative to set theory as a theory of basic structures.

In addition to the tendency towards abstraction, there has always been a tendency in mathematics to explore concrete objects in detail. These studies were also particularly suitable for bringing the role of mathematics closer to the public (for example, fractals from the 1980s and chaos theory, the catastrophe theory of the 1970s).

Important new developments such as the Atiyah-Singer index rate or the evidence of the Weil conjectures are reflected in the awards of the Fields Medal and the Abel Prize . Many problems, some of them centuries old, were solved in the 20th century, such as the four-color problem , the Kepler conjecture (both with computer assistance), the classification theorem of finite groups , the Mordell conjecture ( Gerd Faltings ), the Poincaré conjecture (by Grigori Perelman 2002) and finally in 1995 of Fermat's theorem by Andrew Wiles . Fermat's statement that the margin of a book page was too narrow for a proof was confirmed: Wiles' proof is over 100 pages long and he needed aids that went far beyond the mathematical knowledge of Fermat's time. Some problems were recognized as principally unsolvable (such as the continuum hypothesis by Paul Cohen ), many new problems were added (such as the abc conjecture ) and the Riemann hypothesis is one of the few problems on the Hilbert list whose proof, despite the great efforts of many mathematicians, continues in seems far away. A list of key unsolved problems in mathematics is the List of Millennium Problems . At the end of the century there was again a strong interaction between mathematics and physics via quantum field theories and string theory with surprising and deep-seated connections in various areas of mathematics (infinite dimensional Lie algebras, supersymmetry, dualities with applications in counting algebraic geometry, knot theory, etc.). Before that, elementary particle physics had benefited from mathematics, in particular through its classification of continuous symmetry groups, the Lie groups, their Lie algebras and their representations ( Elie Cartan , Wilhelm Killing in the 19th century), and Lie groups are also a central, unifying topic of mathematics in the 20th century. Century with a wide variety of applications within mathematics to number theory ( Langlands program ).

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## literature

Biographies of mathematicians can be found in:

Commons : History of Mathematics  - Collection of Images, Videos and Audio Files
Wikisource: Mathematics  - Sources and Full Texts
Wikisource: Arithmetic books  - sources and full texts

## Individual evidence

1. ^ Howard Eves : An Introduction to the History of Mathematics . 6th Edition, 1990 p. 9.
2. Moscow Papyrus
3. ^ Heinz-Wilhelm Alten et al .: 4000 years of algebra . Springer-Verlag, Berlin Heidelberg 2003, ISBN 3-540-43554-9 , p. 49.
4. Ifrah Universal History of Numbers . Two thousand and one chapter 29.
5. "All the data given in Indian literary history are, as it were, knock-down cones" from: Alois Payer: Introduction to the exegesis of Sanskrit texts . Script . Cape. 8: The actual exegesis. Part II: On individual questions of synchronous understanding ( online ).
6. Calculus History, McTutor
7. ^ Moritz Cantor: Lectures on the history of mathematics. Volume 3, 1901, pp. 285–328 ( digital edition Univ. Heidelberg, 2014).
8. Thomas Sonar: The history of the priority dispute between Leibniz and Newton . Springer Verlag, Berlin 2016.
 This version was added to the list of articles worth reading on October 9, 2005 .