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57-640: ZbMATH Open , formerly Zentralblatt MATH , is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics , produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH. Editors are the European Mathematical Society , FIZ Karlsruhe, and the Heidelberg Academy of Sciences . zbMATH is distributed by Springer Science+Business Media . It uses

114-468: A quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof . Pure mathematics, according to a view that can be ascribed to the Bourbaki group , is what is proved. "Pure mathematician" became a recognized vocation, achievable through training. The case was made that pure mathematics

171-524: A cylinder looped back through itself to join with its other end from the "inside". It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional Möbius strip , with one method of construction being the attachment of the edges of two Möbius strips . During the 1890s, Klein began studying mathematical physics more intensively, writing on

228-523: A few exceptions and initially contained 880 references per year (1868) and up to 7000 references in the later phase (around 1930). Some of the mathematical abstracts were written by famous mathematicians such as Felix Klein , Sophus Lie , Richard Courant , or Emmy Noether . During WW II publication of the Jahrbuch was stopped. The Jahrbuch' s founding concept was characterized by its documentary completeness. The Jahrbuch only appeared when all papers in

285-517: A foreign member of the Royal Netherlands Academy of Arts and Sciences . Around 1900, Klein began to become interested in mathematical instruction in schools. In 1905, he was instrumental in formulating a plan recommending that analytic geometry , the rudiments of differential and integral calculus , and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around

342-424: A geometric way, connecting potential theory and conformal mappings . This work drew on notions from fluid dynamics . Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on methods of Charles Hermite and Leopold Kronecker , he produced similar results to those of Brioschi and later completely solved

399-399: A grand uniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. He came up with his proof during an asthma attack at 2:30 A.M. on 23 March 1882. Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over

456-400: A mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use. Hardy considered some physicists, such as Einstein and Dirac , to be among

513-479: A prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology , and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers , called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and

570-595: A professorship at the University of Göttingen in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's prime center for mathematics research. However, he never managed to transfer from Leipzig to Göttingen his own leading role as developer of geometry . He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular, mechanics and potential theory . The research facility Klein established at Göttingen served as model for

627-444: A sharp divergence from physics , particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold , as too much Hilbert , not enough Poincaré . The point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central. Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of

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684-476: A systematic use of axiomatic methods . This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics. Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science . A famous early example

741-730: A year had been completely processed. This was later paid for with a great loss of relevance. In addition, there was since 1931 the Zentralblatt MATH , which surpassed the Jahrbuch in terms of speed of publication. The Zentralblatt MATH abstracting service provides reviews (brief accounts of contents) of current articles, conference papers, books and other publications in mathematics, its applications, and related areas. The reviews are predominantly in English, with occasional entries in German and French. Reviewers are volunteers invited by

798-447: Is Isaac Newton 's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections , geometrical curves that had been studied in antiquity by Apollonius . Another example is the problem of factoring large integers , which is the basis of the RSA cryptosystem , widely used to secure internet communications. It follows that, presently,

855-410: Is difficult to appreciate their novelty when first presented, and understand the fact that they were not immediately accepted by all his contemporaries. Klein saw his work on complex analysis as his major contribution to mathematics, specifically his work on: Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate the plane. In 1879, he examined

912-516: Is offered by American mathematician Andy Magid : I've always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of commutative ring theory and non-commutative ring theory . An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a non-commutative ring is a not-necessarily-commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by

969-563: Is useful in engineering education : One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following: Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition. As

1026-647: The Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn , 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868. Plücker died in 1868, leaving his book concerning

1083-567: The Mathematics Subject Classification codes for organising reviews by topic. Mathematicians Richard Courant , Otto Neugebauer , and Harald Bohr , together with the publisher Ferdinand Springer, took the initiative for a new mathematical reviewing journal. Harald Bohr worked in Copenhagen. Courant and Neugebauer were professors at the University of Göttingen . At that time, Göttingen was considered one of

1140-645: The Rockefeller Foundation to donate a large amount of money for the construction. The service was founded in 1931, by Otto Neugebauer as Zentralblatt für Mathematik und ihre Grenzgebiete . It contained the bibliographical data of all recently published mathematical articles and book, together with peer reviews done by mathematicians over the world. In the preface to the first volume, the intentions of Zentralblatt are formulated as follows: Zentralblatt für Mathematik und ihre Grenzgebiete aims to publish—in an efficient and reliable manner—reviews of

1197-941: The Royal Society in 1885, and was awarded its Copley Medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for several further years. Klein was one of ninety-three signatories of the Manifesto of the Ninety-Three , a document penned in support of the German invasion of Belgium in the early stages of World War I . He died in Göttingen in 1925. Klein's dissertation, on line geometry and its applications to mechanics , classified second degree line complexes using Weierstrass 's theory of elementary divisors. Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie , he discovered

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1254-472: The gyroscope with Arnold Sommerfeld . During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften . This enterprise, which endured until 1935, provided an important standard reference of enduring value. In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On

1311-482: The "real" mathematicians, but at the time that he was writing his Apology , he considered general relativity and quantum mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well. Another insightful view

1368-408: The 200,000 entries of the earlier similar publication Jahrbuch über die Fortschritte der Mathematik from 1868 to 1942, added in 2003. As of January 2021, the complete database is accessible for free. Previously, only the first three records in a search were available without a subscription. Pure mathematics While pure mathematics has existed as an activity since at least ancient Greece ,

1425-473: The Progress of Mathematics) was internationally the first comprehensive journal of abstracts in the history of mathematics . It contains information about almost all of the most important publications in mathematics and their areas of application from the period 1868 to 1942. The Jahrbuch was written in 1868 by the mathematicians Carl Ohrtmann (1839–1885) and Felix Müller (1843–1928). It appeared annually with

1482-607: The So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric . This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, believing it to be circular. Klein's synthesis of geometry as

1539-585: The Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig University . His colleagues included Walther von Dyck , Rohn, Eduard Study and Friedrich Engel . Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed and he battled with depression for the next two years. Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period. Klein accepted

1596-415: The accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry. Currently, the significance of Klein's contributions to geometry is evident. They have become so much part of mathematical thinking that it

1653-543: The action of PSL(2,7) , considered as an image of the modular group , and obtained an explicit representation of a Riemann surface now termed the Klein quartic . He showed that it was a complex curve in projective space , that its equation was x y  +  y z  +  z x  = 0, and that its group of symmetries was PSL(2,7) of order 168. His Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale (1882) treats complex analysis in

1710-455: The art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being." Euclid of Alexandria , when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns." The Greek mathematician Apollonius of Perga

1767-523: The basis of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes , and thus became acquainted with Alfred Clebsch , who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits to Berlin and Paris. In July 1870, at the beginning of the Franco-Prussian War , he was in Paris and had to leave

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1824-454: The best mathematical journals in the world. Founded by Clebsch, it grew under Klein's management, to rival, and eventually surpass Crelle's Journal , based at the University of Berlin . Klein established a small team of editors who met regularly, making decisions in a democratic spirit. The journal first specialized in complex analysis , algebraic geometry , and invariant theory . It also provided an important outlet for real analysis and

1881-430: The best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein recruited David Hilbert from the University of Königsberg . This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932. Under Klein's editorship, Mathematische Annalen became one of

1938-452: The central places for mathematical research, having appointed mathematicians like David Hilbert , Hermann Minkowski , Carl Runge , and Felix Klein , the great organiser of mathematics and physics in Göttingen. His dream of a building for an independent mathematical institute with a spacious and rich reference library was realised four years after his death. The credit for this achievement is particularly due to Richard Courant , who convinced

1995-449: The concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable , and Russell's paradox ). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with

2052-562: The country. For a brief time he served as a medical orderly in the Prussian army before being appointed Privatdozent (lecturer) at Göttingen in early 1871. The University of Erlangen appointed Klein professor in 1872, when he was only 23 years old. For this, he was endorsed by Clebsch, who regarded him as likely to become the best mathematician of his time. Klein did not wish to remain in Erlangen, where there were very few students, and

2109-511: The distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics. Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory , and "logistic", now called arithmetic . Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn

2166-488: The editors based on their published work or a recommendation by an existing reviewer. Zentralblatt MATH is provided both over the Web and in printed form. The service reviews more than 2,300 journals and serials worldwide, as well as books and conference proceedings. Zentralblatt MATH is now edited by the European Mathematical Society , FIZ Karlsruhe , and the Heidelberg Academy of Sciences . The database also incorporates

2223-516: The entire world literature in mathematics and related areas in issues initially appearing monthly. As the name suggests, the main focus of the journal is mathematics. However, those areas that are closely related to mathematics will be treated as seriously as the so-called pure mathematics. Zentralblatt and the Jahrbuch über die Fortschritte der Mathematik had in essence the same agenda, but Zentralblatt published several issues per year. An issue

2280-760: The first president of the International Commission on Mathematical Instruction in 1908 at the Fourth International Congress of Mathematicians in Rome. Felix Klein was born on 25 April 1849 in Düsseldorf , to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province . His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended

2337-562: The fundamental properties of the asymptotic lines on the Kummer surface . They later investigated W-curves , curves invariant under a group of projective transformations . It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan . Klein devised the " Klein bottle " named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space , but it may be immersed as

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2394-412: The idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders, however imperfect in form, must have been examined. Like all other sciences, mathematics arose out of the needs of men...But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from

2451-495: The kind between pure and applied . In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis ) started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method , strongly influenced by David Hilbert 's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of

2508-406: The latter we mean not-necessarily-applied mathematics ... [emphasis added] Friedrich Engels argued in his 1878 book Anti-Dühring that "it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been invented from any source other than the world of reality". He further argued that "Before one came upon

2565-484: The most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy 's 1940 essay A Mathematician's Apology . It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry , Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in

2622-633: The new group theory . In 1893, Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the World's Columbian Exposition . Due partly to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman, by Grace Chisholm Young , an English student of Arthur Cayley 's, whom Klein admired. In 1897, Klein became

2679-517: The preface of the fifth book of Conics that the subject is one of those that "...seem worthy of study for their own sake." The term itself is enshrined in the full title of the Sadleirian Chair , "Sadleirian Professor of Pure Mathematics", founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of

2736-434: The problem by means of the icosahedral group . This work enabled him to write a series of papers on elliptic modular functions . In his 1884 book on the icosahedron , Klein established a theory of automorphic functions , associating algebra and geometry. Poincaré had published an outline of his theory of automorphic functions in 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove

2793-607: The real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform." Felix Klein Felix Christian Klein ( German: [klaɪn] ; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator , known for his work in group theory , complex analysis , non-Euclidean geometry , and the associations between geometry and group theory . His 1872 Erlangen program classified geometries by their basic symmetry groups and

2850-447: The study of functions , called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more abstract mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in abstraction was seen mid 20th century. In practice, however, these developments led to

2907-457: The study of the properties of a space that is invariant under a given group of transformations , known as the Erlangen program (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become

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2964-603: The world. In 1908, Klein was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians . Under his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany. The London Mathematical Society awarded Klein its De Morgan Medal in 1893. He was elected a member of

3021-530: Was an influential synthesis of much of the mathematics of the time. During his tenure at the University of Göttingen , Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. His seminars covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction, and promoted mathematics education reform at all grade levels in Germany and abroad. He became

3078-460: Was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted, They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason. And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in

3135-404: Was later shortened to Zentralblatt MATH. In addition to the print issue, the services were offered online under the name zbMATH since 1996. Since 2004 older issues were incorporated back to 1826. The printed issue was discontinued in 2013. Since January 2021, the access to the database is now open under the name zbMATH Open . The Jahrbuch über die Fortschritte der Mathematik (Yearbook on

3192-531: Was pleased to be offered a professorship at the Technische Hochschule München in 1875. There he and Alexander von Brill taught advanced courses to many excellent students, including Adolf Hurwitz , Walther von Dyck , Karl Rohn , Carl Runge , Max Planck , Luigi Bianchi , and Gregorio Ricci-Curbastro . In 1875, Klein married Anne Hegel, granddaughter of the philosopher Georg Wilhelm Friedrich Hegel . After spending five years at

3249-547: Was published as soon as sufficiently many reviews were available, in a frequency of three or four weeks. In the late 1930s, it began rejecting some Jewish reviewers and a number of reviewers in England and United States resigned in protest. Some of them helped start Mathematical Reviews , a competing publication. The electronic form was provided under the name INKA-MATH ( acronym for In formation System Ka rlsruhe-Database on Math ematics) since at least 1980. The name

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