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Mathematics in classification systemsby Craig Fraser 1. IntroductionThe classification of mathematical studies is involved in extraordinary difficulties, and so is the classifying of many mathematical books. The relations of the branches are so intricate, so plastic, so recondite, that it is wellnigh impossible to define them or to comprehend them. Insofar as → library science is concerned, classification of mathematical subjects occurred within the larger framework of → library classification, a project which has drawn sustained attention from 1870 to the present. The two American giants in library work in the formative period of classification were Melvil Dewey and Charles Cutter. In 1876 Dewey published the famous Dewey decimal system of classification, while Cutter’s Expansive scheme of 1885 would provide the basis for the Library of Congress system. The latter was established in 1905 by James Hanson and Charles Martel, both European immigrants to the United States. In the early Twentieth century additional classification schemes appeared. Among the more notable of these were the “subject” system formulated by the Englishman James Duff Brown and the “bibliographic” system invented by City College of New York librarian Henry E. Bliss. The practical goal of all classification was information retrieval, allowing for example a user to go to a large library, consult the catalogue, and retrieve a given book of interest. The call number given to a book had to be abstract — it could make no reference to any particular library or to the physical arrangement of the books in any library it happened to belong to [1]. The motivation for classification schemes was the appearance of an increasing number of comprehensive libraries with substantial holdings of books from many subject areas. Some insight into the classification of books may be gleaned by looking briefly at the question of → classification in the physical world. The pioneering English astronomer William Herschel was interested in the → classification of stellar objects. In an article of 1802 on nebulae and star clusters he announced that he had made something of a breakthrough in connecting the classification of these objects to their inherent characteristics. Herschel (1802, 477) wrote: The classification adopted in my [earlier] catalogues is little more than an arrangement of the objects for the convenience of the observer and may be compared to the disposition of books in a library, where the different size of the volumes is often more considered than their contents. But here [in 1802] in dividing the different parts of which the sidereal heavens are composed into proper classes, I shall have to examine the nature of the various celestial objects that have been hitherto discovered, in order to arrange them in a manner most comfortable to their construction. Simon Shaffer (1980, 218) describes this shift in Foucauldian terms as a move from an artificial to a natural system of classification. Of course, the astronomer in his role as natural philosopher is different from the librarian who — as Herschel noted — must take into account physical features or practical necessities related to the handling of books. Furthermore, the classification of books is related to the classification of knowledge, something that is evidently different in character from the classification of objects in the natural world. There is in library classification a tension between epistemological questions related to the organization of knowledge and contingent matters such as collocation, user practice and the storage and retrieval of materials. Library science, as the discipline of book classification and cataloguing had come to be known by the 1920s, was a subject that integrated the theoretical organization of knowledge with the utilitarian function of identifying and retrieving information. Librarians sought philosophical meaning or justification for the schema they employed, and they appealed to principles of the organization of knowledge. Classification pioneers such as Dewey and Cutter were generalists who were not primarily concerned with any particular subject area. To the extent that their interests were focused on particular fields, these were found in humanistic and social subjects, not in the natural sciences. Among all of the major systems of book classification, the Library of Congress (LC) Classification scheme was the one that achieved dominance in North American university and research libraries [2]. In 1870 the US Copyright Office was by legislation placed in the Library of Congress, and the Library received copies of all publications submitted for copyright. The holdings of the Library increased and became more complete than any elsewhere, including the collections of major university libraries and large public libraries. This was particularly true for the smalledition research books and monographs in academic fields. In 1901 the Library established its catalogue card distribution service, which allowed libraries throughout America to receive catalogue cards for all published books. Paul Edlund (1976, 398), an historian of the Library, comments on the significance of this development: By including the card distribution service in its functions, the Library, at that time a reference library to Congress with a small constituency consisting almost exclusively of congressmen and their staff members, was adopting a potentially enormous constituency — that of the total American library community. The importance of the LCC system in the world of classification was apparent in the years following its establishment. While the major university libraries with their specialized collections containing many older and foreignlanguage books continued to maintain a patchwork of local classification systems, the LCC has made steady headway up to the present as the dominant and most widely used system of library classification in North America. The Library of Congress today is also involved in the Dewey decimal system through its Dewey Program. From the website of this Program: In 1930 the Library of Congress began to print DDC numbers on many of its cards, thus making the system immediately available to the nation’s libraries. Today the Dewey Program continues to support the nation’s libraries, especially public and school libraries, as well as many foreign libraries that classify their collections according to DDC. [3] Research libraries such as the University of Illinois at Urbana–Champaign that follow the Dewey system adopt call numbers for new books that are provided by the Decimal Program [4]. Through the international use of the Dewey system the LC plays a central role today in library classification not just in the North America but worldwide. The work of the library cataloguers in the decades around 1900 was carried out against the background of a broad Nineteenthcentury interest in the classification of knowledge. In sections one to four we examine how mathematical subjects were classified, from the most general level down to the specific level of particular subject areas in analysis. Section 5 examines in more detail one of these subject areas, complex analysis, and follows the classification of books in this subject up to the present. The final section is devoted to the Mathematics Subject Classification (MSC), developed in the late 1960s by the American Mathematical Society. The MSC applies to both periodical articles and books. A major revision of the system takes place every ten years. With the arrival of the online reviewing services MathSciNet and zbMath, the MSC has become the worldwide accepted standard for classification of mathematical knowledge as we enter the age of electronic publishing. 2. Place of mathematics in classification schemesPrior to, and concurrent with, the development of library classification systems in the Nineteenth century there was a great deal of interest in the general problem of the classification of knowledge. Although this problem was a venerable one, going back to the Greeks, it was of special concern in the Nineteenth century and was the object of extensive research and discussion of a kind particular to this period. Insofar as the sciences were concerned mathematics occupied a privileged place in this philosophical project. In his book Why Is There Philosophy of Mathematics at All? Ian Hacking explores the prominent place that mathematics has played throughout history in the writings of the great philosophers. In a review of this book Max Siegel (2016, 253) observes that “mathematics has perennially fascinated philosophers, to the point that the philosophy of mathematics is not the philosophy of a special science — like the philosophy of physics or biology — but rather a central field of analytic philosophy.“ In all of the major classification schemes, mathematics is either placed first in a category of general or fundamental science — along with philosophy; or it is put at the beginning of the natural sciences, followed by subjects that follow a presumed reductionist foundational order that exists among them: physics, then chemistry and finally biology. Although the majority of the thinkers at the end of the Nineteenth century interested in the organization of knowledge were neither scientists nor mathematicians, they possessed definite ideas about the classification of scientific subjects. A recurring theme was the coupling of mathematics with philosophy and its separation from subjects in natural science such as physics, astronomy and chemistry. This point of view seems to have reflected the general and fairly widespread influence of philosophical logicism on contemporary scientific thought. The Harvard psychologist Hugo Münsterberg was responsible for the scientific plan of The International Congress of Arts and Science held in 1904 at the St. Louis Exposition. Knowledge was divided into seven divisions, and each division was composed of a collection of departments. The divisions were: ( The philosophical view of the place of mathematics within knowledge never really found favor with librarians who worked on the concrete project of book classification. However, the affinity of mathematics with logic — if not philosophy — and its separation from the natural sciences was a prominent feature of Brown’s 1906 subject classification scheme. Brown held that the classification of books should reflect the classification of knowledge, so that library classification was never a purely contingent project of information retrieval. Brown asserted that logic and mathematics should be grouped together under “Generalia” and should precede all other branches of knowledge, being preliminary to any field of investigation, from physics to economics to philosophy and history, or anything else. Classes of knowledge were given by Brown in the following order: generalia, physical science, biological science, medical science, agriculture and domestic arts, philosophy and religion, social and political science, language and literature, literary forms, and history and geography. Brown's classification was also distinctive in its positioning of practical subjects adjacent to their presumed theoretical counterparts. Following the section on mathematics there would be books on painting and sculpture. There were two reasons for such an arrangement. First, fine and graphic art exemplified the visual point of view of geometry and could be regarded in some sense as the embodiment of geometric ideas. Second, Brown believed that the subject of visual representation was fundamental in character and that familiarity with it was necessary for its use in various applied fields of investigation and work. It may seem odd that the section of the library stacks devoted to mathematics would be followed by books on Flemish art, while books on physics would be located a few aisles over — but such was Brown's rather idiosyncratic notion of book classification! A librarian influenced by Brown was Henry Bliss of the City College of New York. More than Brown, Bliss emphasized the affinity of mathematics with philosophy and logic and its separation from science. He wrote two books on the organization of knowledge: the first (1929) was a general and somewhat philosophical work, while the second (1933) was directed more specifically to the classification of books. Bliss regarded mathematics more as a method than a branch of science, and he believed that a grounding in logic was proper preparation for its study. He observed (1929, 258) that “Logic is usually regarded as a branch of philosophy and the close relation of philosophical thought to mathematical thought is often affirmed”. He had many criticisms of the Library of Congress Classification scheme, including the position of mathematics: “In a broader aspect the separation of Sciences in The conception of Münsterberg and Bliss was implicitly rejected by thinkers concerned only with the organization and classification of subjects in the natural sciences. For them, mathematics clearly had to be included as a subject area, usually at the beginning of the classification, while philosophy did not appear at all. Any viable classification scheme would need to reflect the place of subjects in the real world. Classifying mathematics with philosophy and separating it from physics and engineering might be sensible in the domain of humanistic thought, but it made little sense in actual practice. In the Cutter, Dewey and LC classification schemes mathematics is separated from philosophy and grouped with the natural sciences. In the Dewey system, philosophy is placed near the beginning under Cutter also grouped philosophy near the beginning under the letter The LCC system seems to have been patterned after Cutter and the placement of philosophy with respect to the natural sciences follows this earlier system. Sayers (191516, 135) observes that “The outline of the [LC] classification is almost directly based upon The Expansive system, as a comparative paradigm of the two will demonstrate”. The LCC will be the subject of more detailed study in sections 4 and 5. 3. Scope of mathematics in classification schemesUntil the Nineteenth century mathematics was interpreted broadly to include subjects that today would be regarded as part of astronomy, physics or engineering. But by the second half of the Nineteenth century, when library classification systems were being developed, the scope of mathematics had narrowed substantially. Writers such as Münsterberg and Bliss who viewed logic and mathematics as kindred subjects and grouped mathematics with philosophy adhered to a conception of mathematics that certainly did not include subjects in physics such as mechanics. However, these thinkers did not represent scientists and mathematicians themselves. Among the latter mathematics was a subject that involved traditional logic only very peripherally if at all. Discussions of the scope and relative position of purely scientific subjects in the Nineteenth century focused on what was called the hierarchy of sciences, a notion introduced by Auguste Comte in 1830 in the second lesson of his Cours de philosophie positive. Comte believed that there is a natural progression of scientific subjects, beginning with mathematics, passing through astronomy, physics, chemistry and biology, and ending with sociology. This hierarchy could be justified on methodological or philosophical grounds, and was often taken for granted for practical reasons. The Comtean hierarchy of sciences was accepted by virtually all of the systems of book classification, and survives to the present. Among those writers who were primarily interested in the natural sciences, mathematics was placed within science, at the very beginning of the Comtean hierarchy. An important figure was the French physicist AndréMarie Ampère, who along with Comte and some other French figures of the period was a mathematical empiricist in orientation. These authors separated mathematics completely from philosophy, which tended to occupy a lower position in the overall scheme of knowledge and learning than it had traditionally held. Mechanics was a kind of hybrid subject, part of mathematics and different from subjects in physics, but distinct from arithmetic and geometry in possessing a physical character. Ampère presented a rather detailed and elaborate classification scheme for the sciences in his Essai sur la philosophie des sciences of 1834. The mathematical sciences were made up of arithmetic, geometry, mechanics and astronomy. Arithmetic and geometry were mathematical subjects “proprement dites”, while mechanics and astronomy were physicomathematical in character. The physical sciences included atomic theory and chemistry. Ampère’s point of view was reflected in some later French writers on scientific classification. Thus Charles Renouvier (1859) in his Essais de Critique Générale put rational mechanics and applied mathematics together with mathematical subjects (arithmetic, algebra, mathematical analysis, geometry) in the category of Logical Sciences, which were to be distinguished from Physical Sciences, the latter including astronomy. At the end of the century Edwin Goblot’s Essai sur la classification des sciences (1898) positioned mechanics as part of mathematics and distinguished it from physics. The estimable Scottish authority Robert Flint (1904, 278), in his survey of work on classification, seems to regard mechanics as part of mathematics, writing that mechanics “is as abstract as Geometry, and in its applications is not more concrete”, and “Mechanics is both abstract and concrete, both quantitative and qualitative, and cannot be denied to be on the borderland between mathematical and physical science” [5]. Although Ampère’s understanding of the scope of mathematics was adopted by some later authors, the view that came to be much more common as the century progressed was just the opposite. There was a decided shift away from the view that mechanics was part of mathematics. In the last part of the century both humanistic and scientific thinkers interpreted mathematics as a subject more or less coextensive with what today would be called pure mathematics. This shift is apparent in the writings of the English polymath Herbert Spencer, who published in 1864 The Classification of the Sciences. Spencer opposed Comte's hierarchy, mainly on the grounds of the reductionist ordering of the sciences along a linear sequence that it implied. Spencer was among that group of thinkers who believed that logic and mathematics were closely connected and distinguished by their abstractness from the natural sciences. Mathematics and logic dealt with relations, while the natural sciences dealt with objects. Rather than putting the natural sciences into a sequence he divided them into two distinct groups: the abstractconcrete sciences, consisting of mechanics, physics and chemistry; and the concrete sciences, consisting of astronomy, geology, biology, psychology and sociology. The exclusion of mechanics from mathematics was also advocated by the prominent Austrian physicist Ernst Mach, who published his noted critical and historical account of mechanics in 1883. Although Mach’s philosophy shared similarities with the empiricist outlook of Comte and Ampère, he insisted that mechanics was not part of mathematics. At the beginning of the preface to his book he proclaimed: Mechanics will here be treated, not as a branch of mathematics, but as one of the physical sciences. If the reader's interest is in that side of the subject, if he is curious to know how the principles of mechanics have been ascertained, from what sources they take their origin, and how far they can be regarded as permanent acquisitions, he will find, I hope, in these pages some enlightenment. All this, the positive and physical essence of mechanics, which makes its chief and highest interest for a student of nature, is in existing treatises completely buried and concealed beneath a mass of technical considerations. (Mach 1883, i) [6] Mach's position was influenced by his phenomenological understanding of mechanics and his belief that a priori metaphysical conceptions had no place in physics, a mistake that could arise if mechanics was taken as part of mathematics. There were also important developments in Nineteenthcentury mathematics that influenced scientific thought in the century’s second half. In a footnote toward the end of his book Mach discussed the discovery of nonEuclidean geometry. This discovery showed that geometry was not simply a description of spatial reality, for there were multiple geometries and only one spatial reality. Mathematics including geometry was evidently about intellectual structures, while mechanics was about objects in the external world. NonEuclidean geometries existed, but noninertial physics did not. Mach was opposed to the interpretation of the properties of real space (“die Eigenschaften des gegebenen Raumes”) by what he called “the pseudotheories of geometry that seek to excogitate these properties by metaphysical arguments”. The common view among the classifiers of science in the second half of the century was that mathematics did not include mechanics. This fact is apparent in a broad range of authors discussed by Flint in his 1904 historical survey. William Whewell in 1858 distinguished mathematics (arithmetic, geometry, algebra, differentials) from astronomy and mechanics (Flint 1904, 198). W. D. Wilson in 1856 separated mechanics, which he called a practical science, from mathematical subjects (arithmetic, algebra, geometry, calculus) which made up, with the study of method and ontology, the pure sciences (Flint 1904, 215–216). Eugène de Roberty in an 1881 book on sociology separated mathematics from mechanics, regarding the latter as a descriptive science (Flint 1904, 2634). In his 1887 book Versuch einer concreten Logik, the Prague philosopher Tomáš Masaryk advocated a hierarchal conception of science, placing mathematics first and assigning mechanics to a second group (Flint 1904, 2778) [7]. Masaryk followed Mach in explicitly separating mechanics from mathematics. In 1870 the Scottish philosopher Alexander Bain asserted that mathematics was distinct from mechanics, and placed the latter with physics (Flint 1904, 2412). In Karl Pearson’s Grammar of Science of 1892 logic and mathematics were classified as abstract sciences, while mechanics was one of the concrete sciences. One year later Raoul de La Grasserie followed Herbert Spencer in classifying mathematics as an abstract science and mechanics as abstractconcrete (Flint 1904, 289292). Writing in the early 1930s but expressing longheld views, Henry Bliss (1933, 293) asserted that the possibility of a mathematical treatment of mechanics “should not mislead scientists to admit the claims of some mathematicians that Mechanics is merely a branch of Mathematics. That is not true even or Rational, or Analytic Mechanics, which of course should not be dissevered from the subscience as a whole”. The bookclassification schemes at the end of the century were united in limiting the scope of mathematics, and in either placing mechanics within physics or including it as a subject area in its own right. In Cutter, mechanics was put with physics rather than mathematics, while astronomy was made a distinct subject area, after chemistry. Although Dewey had included some applied subjects in mathematics, mechanics was placed in physics, along with optics, thermodynamics and electromagnetism. In the International Catalogue of Scientific Literature (see Section 4) mechanics received its own subject area, intermediate between mathematics and physics. In the systems of both Brown and Bliss, mechanics is separated from mathematics and classified as a physics subject, along with thermodynamics and electromagnetism. Alone among the major classification systems, the LCC scheme placed mechanics under mathematics, and situated astronomy as a subject field between mathematics and physics. The title of the original LCC volume on mathematics is worded Class QA: Mathematics (Including Analytic Mechanics). It is not entirely clear why the architects of LCC proceeded this way, but the grouping of mechanics within mathematics is a singular feature of the LC classification system that continues to the present. 4. The place of calculus/analysis in classification schemes for mathematicsComte's distinction between abstract mathematics consisting of arithmetic, algebra and calculus, on the one hand, and concrete mathematics, consisting of geometry and mechanics on the other, reflected a classificatory order that placed calculus ahead of geometry. It was also in keeping with the prevailing conception in French mathematics of calculus as a form of “algebraic analysis”, the very title of Augustin Cauchy’s famous textbook of 1821 on the calculus. In his 1834 book Ampère introduced neologisms to designate the various subject areas of mathematics. What he called "arithmologie" was divided into two parts, the first consisting of arithmetic and algebra, and the second consisting of the theory of functions and the theory of probabilities. The theory of functions included calculusrelated parts of mathematics. Geometry was the second major subject area of mathematics, under which Ampère placed synthetic and analytic geometry, as well as the theory of lines and surfaces and something called molecular geometry. The other major subject area of mathematics consisted of the physicomathematical subjects mechanics and astronomy (the latter called “Urinologie” by Ampère). Mechanics in turn was divided into elementary and transcendental mechanics, while astronomy was divided into general astronomy and celestial mechanics. Among the many writers who wrote on classification of science from the 1840s to the end of the century, the predominant tendency was to depart from Comte and Ampère by placing geometry ahead of calculus. Mathematical subjects were placed in the standard order: arithmetic, algebra, geometry and calculus. Whewell (1858) conceived of mathematics as the subjects “Geometry, Arithmetic, Algebra, and Differentials, and based on the ideas of space, time, number, sign, and limit” (Flint 1904, 199). Bain (1870) divided mathematics into arithmetic, algebra, geometry, algebraic geometry and the higher calculus (the latter dealing with incommensurable magnitudes) (Flint 1904, 199). Wilson (1856) gave the order arithmetic, geometry, algebra, calculus, trigonometry and analytic geometry (Flint 1904, 216). Paul Janet (1897) used abstraction as something that distinguished arithmetic, geometry and mechanics from algebra and the differential and integral calculus (Flint 1904, 304). Flint (1904, p. 278) himself wrote that "Arithmetic and Geometry are very different both as to matter and method from Calculus and Kinematics" [8]. With the exception of the Library of Congress, the major library classification schemes around 1900 placed geometry before calculus. Dewey and Cutter both adopted the order arithmetic, algebra, geometry, trigonometry, and calculus, while Brown presented these subjects in the order arithmetic, algebra, geometry, calculus, and trigonometry. The librarians presumably were guided by historical and pedagogical considerations: calculus had originated as a set of methods for the study of curves and surfaces, and calculus was a more advanced teaching subject than elementary geometry and therefore was placed after it. The librarians may also have perceived the natural order to be one of successive abstraction, and calculus and higher analysis were viewed as more abstract than geometry [9]. Although the focus in this section is on the classification of books, it is necessary to look at how periodical mathematical literature was classified by subject in the second half of the Nineteenth century, as this would bear directly on the classification scheme for mathematical books adopted by the LC. Unlike book classification, which was aimed at a broad readership at various levels of engagement with the subject, the practices followed by journals reflected the outlook of active researchers in the field. Insofar as the ordering of subjects is concerned, the point of view was essentially a continuation of the French outlook expressed by Comte and Ampère early in the century. The Zeitschrift für Mathematik und Physik, founded in 1856, was one of the first journals to explicitly divide its contents into subject categories. The latter were presented in the order arithmetic and analysis, geometry, mechanics, optics, electricity and Galvinism, and smaller and miscellaneous subjects. This selection of topics and their ordering was certainly not pedagogical. Even from the viewpoint of its intended audience it was not an altogether natural ordering. The table of contents was based on an implicit understanding of the logical character of mathematics and the relationships that exists among its parts. The grouping of analysis with arithmetic and its placement ahead of geometry reflected the prevailing view of advanced researchers, and indicated more generally the wellknown “arithmetization of analysis” of mathematics in the Nineteenth century. Calculus in its original formulation was known as “fine geometry”, and Eighteenthcentury masters of analysis such as Euler and Lagrange were known as geometers. By the second half of the Nineteenth century the research picture had shifted substantially, and geometry had become something of a subsidiary subject with respect to the primary grounding of mathematics in arithmetic, algebra and analysis. Carl Ohrtmann and Felix Müller were Berlin gymnasium teachers of mathematics who founded in 1871 the reviewing periodical Jahrbuch über die Fortschritte der Mathematik. There was a large increase in the growth of mathematical literature in the Nineteenth century, and a corresponding need to assist researchers in navigating materials published in their fields. Ohrtmann and Muller modelled the Jahrbuch after an abstracting journal for physics that had already been in existence for close to twentyfive years, the Fortschritte der Physik. Although the publications reviewed in the Jahrbuch consisted mainly of periodical literature, books were also included. The Jahrbuch followed Ampère and the Zeitschrift in its presentation of subjects and their ordering: history and philosophy, algebra, number theory, series, differential and integral calculus, function theory (complex functions), pure, elementary and synthetic geometry, analytic geometry, mechanics, mathematical physics (electromagnetism, theory of heat, optics), and geodesy and astronomy [10]. Since there was already a physics reviewing journal, the physics subjects included in the Jahrbuch were ones in which the treatment was highly mathematical. At the end of the century the Royal Society of London established the International Catalogue of Scientific Literature (1902), a major international bibliographic project that was intended to cover both periodical and book literature. In this work mathematics (which was also referred to as “pure mathematics”) was divided into the following subject areas: fundamental concepts, algebra and number theory, analysis, and geometry. This ordering of subjects became canonical in the classification of Twentieth century mathematical literature, at least as this was followed by the LCC and mathematical reviewing services. (It should be noted that the Dewey system continued to place geometry before calculus and analysis up until the late 1960s, at which time its schedules were revised and brought into alignment with the LC.) The classification schedules for mathematical subjects in the original LCC system of 1905 were compiled by J. David Thompson, chief of the science section, under the direction of Martel, head of classification for the whole of LC. Thompson was a native of England who had studied mathematics at the University of Cambridge, graduating 16th Wrangle in 1895. In the preface to the volume on science he (1905, 3) states that he has relied notably on the schedules of the International Catalogue of Scientific Literature. While the overall scheme of the LCC system was patterned on the Cutter system of classification, the organization of scientific subjects followed the ICSL. Insofar as advanced mathematical subject areas were concerned, Thompson followed the ICSL very closely. The 1905 edition of the LC science schedules was republished in multiple later editions, each containing modifications and extensions of the original scheme. In the 1930s there were two new library classification systems, Bliss’s bibliographic classification and S. R. Ranganathan’s (1933) → Colon Classification. Although Bliss presented the three subject areas of mathematics as arithmeticandalgebra, geometry and analysis, he classified these subjects in the order arithmeticandalgebra, analysis and geometry. He made this change for reasons of what he called “collocation”, apparently referring to the usage established by the ICSL and the LCC. Ranganathan also classified mathematics subjects in the order arithmetic, algebra, analysis and geometry, and followed LCC in including mechanics within mathematics. In a departure from all other classification schemes he placed astronomy within mathematics. 5. Analysis in the LCC system for mathematics

MR’s mission statement from 1940 asserts that “MATHEMATICAL REVIEWS will cover not only pure mathematics, but also papers in the borderline fields of applied mathematics”. Applied fields included mechanics, statistics and probability, and some topics in mathematical physics. Nevertheless, the contents of MR in the early years were weighted towards pure mathematics, perhaps more so than was the case for Zentralblatt. The conception of mathematics that was predominate at the middle of the Twentieth century is sometimes characterized as "modern", a term that denotes an emphasis on axiomatic development, a focus on the concept of structure, and a belief in set theory as the appropriate language for expressing mathematics. (For accounts of mathematical modernism see Corry (2003) and Gray (2008).) The modern perspective accentuates the abstract character of mathematics and tends to downplay applications [20]. The viewpoint of modern mathematics is often associated with the French group Bourbaki, but was prevalent among advanced researchers throughout Europe, North America and Asia into the 1960s [21]. Although there was some degree of interest in applications in the 1940 MR, its contents bore the clear imprint of contemporary modernist notions of mathematics.
The subject classification organization employed by MR in 1940 was somewhat more detailed than the schematic overview presented in Table 1. The full table of the contents for the first volume in 1940 of MR is given in Figure 3.
The 1940 table of contents provided the rough template for all classification schemes at MR up to the present. By the late 1950s modifications were made to some of the subject headings. For example, differential equations were split into two separate headings, ordinary differential equations and partial differential equations. Algebraic geometry, which until 1959 was grouped with geometry, was moved to an earlier position in the classification under algebra. Although there were a few other minor adjustments, most of the subject headings were retained.
The twodigit subject codes which later became standard first appeared in early 1960 in the index to volume 20 (1959) of MR. Numbers selected from the range 02
to 98
were attached to the 59 subject headings that had been used in the monthly issues and that had appeared in the table of contents at the beginning of the volume. Up until 1959 the annual alphabetic subject indexes had provided a more detailed subject breakdown than the one given by the subject headings. This more finegrained classification was built into the 1959 coding, with each subject divided into classes, and each class being indicated by a twodigit decimal number. For example, in the period 194058 the category of semigroups was listed under groups in the annual subject indexes. In the 1959 coding, semigroups had the designation 20.80
. Here 20
was the subject number for groups and generalizations, while 80
was for semigroups. The code for ordinary differential equations in Banach spaces was 34.95
. Here 34
was the number for ordinary differential equations, and 95
was for equations in Banach spaces.
The passage from the annual subject index in 1958 to the one in 1959 was not simply a matter of introducing codes. The shift represented the crossing of a divide. Before 1959 all subjects in the index were listed alphabetically, and classes under these subjects were also listed alphabetically. The 1959 index listed subjects by their place in the table of contents, and the classes under each subject were ordered according to some conception of the natural relationships that existed among them.
The final issue of MR for each year from 1960 until the end of the decade consisted of an index that included a table of subject headings with codes, following the template established in volume 20. Although there was a fair degree of stability in the set of the subject headings and the corresponding twodigit codes, the classes assigned under each code were very much in flux. In some cases, codes contained a threedigit decimal suffix, an indication of how detailed the classification scheme was becoming. A certain level of stability was reached by 1968, two years prior to the emergence of MSC1970.
It should be noted that until the late 1970s subject codes did not appear in regular issues of MR and were not a prominent feature of the journal. They arose as a point of interest only in the index volumes at the end of each year and were likely little noticed by an average user of the journal. It is possible that their primary purpose during this period and even later concerned the role they played within the internal operations of MR in classifying the large volumes of literature coming to it for classification.
The 1968 classification system provided the basis for the Mathematics Subject Classification scheme of 1970. MSC1970 was not motivated by any particular interest in classifying the contents of MR. Its origins were more mundane. A reader of an issue of MR could simply consult the table of contents and go to the subject section corresponding to his or her area of mathematical interest. By contrast, in order to efficiently process requests for offprints or titles it was useful to have a formal system of classificatory codes. Hence the motivation to develop the MSC came in the late 1960s from the AMS's Mathematical Offprint Service (MOS) and its successor the Mathematical Title Service (MTS). The AMS stated that the MTS was “a disciplinewide system for selective dissemination of the titles of papers […] The essential factor in the successful operation of MTS is precise and complete classification. To facilitate such classification, the AMS (MOS) Subject Classification Scheme (1970) was developed” (AMS 1972, 73). It was under the auspices of the MOS and MTS that the MSC was created and maintained during its early years. The codes did not accompany the reviews that appeared in the regular issues of MR, but were published by the MTS in the Index of Mathematical Papers published twice annually. According to Pitcher (1988, 145), at some point the functions of the MTS were taken over by MR and the MSC was administered by the editorial staff of MR [22]. The MTS itself was discontinued.
Figure 4 is the table of contents for both the 1968 and 1970 volumes of MR. In the 1968 classification scheme there were 57 subject areas and 900 classes placed under these headings, which evidently required some substantial deliberation about class definition and identity. (For the detailed 1968 Subject Classification see AMS (1968).)
In the 1970 MSC scheme the number of classes and subclasses increased from 900 to 1900, an expansion that must have involved much further classificatory labor. The 1970 scheme maintained the subject whole numbers, but added a letter of the alphabet to indicate a class. Further topic divisions within this class were indicated by a twodigit number [23]. For example, representation theory of symmetric groups now received the code 20C30
. Here 20
as before was the subject area of group theory, C
indicated representation theory of finite groups, and 30
was for representation of symmetric groups. Boundary value problems in nonlinear ordinary differential equations was now designated 34B15
, with 34
for ordinary differential equations, B
for boundary value problems and 15
for nonlinear equations. (The full MSC1970 is given in AMS (1972, 73199) and Fang (1972, 3457).)
A publication under review would have a primary classification code, and possibly also a secondary classification code, and even additional codes if that was appropriate. The contents of each issue of MR were organized under the subject headings in Figure 4. The subject contents were not further divided by classes. Rather the full MSC codes for reviewed publications were bibliographical constructions that were used in the early years by the MTS to connect the research interests of subscribers to the contents of MR.
Twodigit MSC codes appeared in 1977 for the first time in the table of contents of the first volume of MR for that year. They also appeared beside each subject heading and at the top of every page. During this period threedigit codes (twodigit subject number and class letter) were employed in the AMS’s periodical Current Mathematical Publications. In 1980 twodigit subject codes were provided for each review in regular issues of MR and became a more integral part of the journal. By the 1990s every review in an issue of MR was accompanied by its full 5symbol MSC code, according to the latest version of the MSC at the time of publication.
In the first half of the 1980s MR published a series of subject index volumes covering its first 40 years: in 1981, for the period 19731979; in 1983, for 19401959; and in 1985, for 19591972. The organizational principle adopted in these indexes for each period reflected the original subject organization used at the time of publication. The subject index for 19731979 used MSC1970 as the basis for classification and categorized reviews according to their fivesymbol subjectclasssubclass designation. The index for 19401958 employed a very detailed breakdown on an alphabetical basis, presenting the subjects alphabetically with classes under these subjects also listed alphabetically. Presumably this approach was adopted because no system involving subject coding existed between 1940 and 1958. Nevertheless, the classification was remarkably detailed, much more so than the actual subjectheading organization of the original volumes. Finally, for the period 19591972 the 1968 classification was adopted as the standard, apparently because 1968 represented the final product of the evolving annual schemes used from 1959 until then. It should be noted that although the MSC1970 was already in place for the three years 19701972 it was not used as the classification standard in the 1985 index volume for 19591972.
In the 1970s, Zentrablatt also adopted the MSC and the two reviewing agencies have worked together in revising the classification system. Major revisions occurred in 1980, 1991, 2000, 2010 and the latest revision will appear in 2020. The process of revision is a joint project of the editorial staffs of Mathematical Reviews and Zentralblatt in consultation with the mathematical community. The MSC has moved far beyond its initial involvement in titleretrieval to become the dominant system today for classifying mathematics. In contrast to library classification systems, it has no competitors. As more of the book literature appears in electronic form without call numbers, MSC will be the default classifier, and its dominance will continue to grow. Traditional library systems such as LCC and Dewey will increasingly fall by the wayside.
The number of subject headings appearing in MR increased from 37 in 1940 to 60 in 1970, the latter including several subjects that did not exist or were present only in a nascent form in 1940, among them category theory, optimal control, computing machines, operations research and game theory. While the number of subject headings has not increased dramatically in the different revisions of the MSC that have taken place since 1970, there have been large increases in subject classes and divisions within these classes.
A trend that was evident in 1970 and has intensified up to the present is the very wide view taken of the scope of mathematics, far exceeding the range assumed for mathematics by Library of Congress or by what is covered in a university department of mathematics. In addition to the traditional core areas (foundations, algebra and number theory, analysis, and geometry) a considerable part of physics and astronomy is within MR’s domain, and there are also present subjects in chemistry, biology, engineering, medicine, economics and sociology. The science categories in Library of Congress are QA
(mathematics including mechanics), QB
(astronomy), QC
(physics), QD
(chemistry), QE
(geology), QH
(natural history), QK
(botany), QL
(zoology), QM
(human anatomy), QP
(physiology), and QR
(microbiology). Engineering subjects in LCC are classified under T
, medicine is under R
, psychology is under B
, and social sciences (including economics) are under H
. MR reviews publications from all of these subjects, if the work in question makes substantial use of mathematics. The situation has changed from the early days of MR, when virtually all book literature reviewed was in the QA
subject category.
By the mid1960s MR was republishing reviews from journals in ancillary fields, including Applied Mechanics Reviews, Computing Reviews, Electrical and Electronics Abstracts, Physics Abstracts and the Soviet abstracting journal Referativnyi Žurnal Matematika (Mechanika, etc.). In subsequent years Statistical Theory and Method Abstracts was added to this list. Although these reprints were only a very small percentage of the reviews published in MR, they indicated some degree of interaction with sciences allied with mathematics.
The online version of MR, MathSciNet, was established in 1996, and has replaced the printed edition, which was discontinued in 2012 [24]. Similarly, Zentralblatt has been replaced by the electronic reviewing service zbMath. Both MathSciNet and zbMath use the MSC. Entering either the title or author of a publication in MathSciNet, one is taken directly to the review, which also includes the MSC primary and secondary codes for the publication under review. Entering an MSC classification code leads to all reviews for publications with this classification. One can search according to year and publication type, under book, journal, and proceedings, or all three.
The main document on the AMS website giving information concerning the MSC classification system contains statements about literature predating 1968 that must be read carefully [25]. The assertion “The MSC classification has been revised a number of times since 1940” is incorrect because the MSC did not exist until 1970. The classification scheme from 1940 to 1958 represented by subject headings took place without any codes and remained stable during these years. In MR volumes of the mid 1960s individual reviews were not assigned codes, although in the subject classification for the years 19591972 published in 1985 they were retroactively given codes according to the 1968 system. MathSciNet has adopted the 1985 convention in assigning codes for the period 19591972.
The assignment of codes to reviews from 1940 to 1958 apparently posed a challenge for MathSciNet. Here are how subject codes were applied to literature appearing in the first volume in 1940 of MR. If a given subject heading in 1940 was also a subject heading in 1970 then it was assigned the same code as the one in MSC1970. For the majority of subject headings this was true and thus many of the codes are the same as the ones in 1970. In cases where some revision to the subject headings occurred, it was necessary to improvise. As we noted above, until 1955 articles on ordinary differential equations and partial differential equations were grouped under the subject heading “Differential equations”. In 1956 this heading was replaced by the two headings, “Ordinary differential equations” and “Partial differential equations, and in 1959 these two subjects were assigned the codes 34
and 35
respectively. It was apparently not viable to go back to every article pre1956 under the heading “Differential equations” and determine if it was part of ordinary or partial differential equations. The solution was to introduce a new classification number, and thus the 1940 subject heading “Differential equations” was assigned the number 36
, which was not used in MSC1970 and therefore was available.
In most cases differences in the codes assigned to the literature from 1940 to 1958 and MSC1970 are relatively minor, but there are some significant anomalies. For example, algebraic geometry was a subject heading in 1940 and was also one in 1970. The MSC1970 code for algebraic geometry was 14
and so this was the code assigned to this subject in the 1940 volume. However, in 1940 algebraic geometry was placed in geometry, while from 1959 on it was placed in algebra. As one moves through the 1940 table of contents one reaches geometry with numbers 48
and higher, except for the anomalous appearance of algebraic geometry, with the much lower number of 14
. The policy used in assigning codes to pre1959 subjects was not to give a characterization of the older classification on its own terms, but to have codes suitable for retrieving information about this literature in line with the post1970 MSC world of classification.
There are also some curious aspects concerning the coding of older literature. If one enters “09” into the MathSciNet search engine there results almost 4000 reviews from 1940 to 1961 dealing with aspects of algebra, mainly rings and fields. Nevertheless, the code 09
only appeared in the subject index for 1960 and 1961, under the heading universal algebra, and only 10 reviews were ever published with this code. The user today has no way of knowing that 09
exists as a code, much less that it covers literature dealing with algebra. This is also true for the code 36
assigned to differential equations [26]. It seems that there is no information in MathSciNet or MR is about the codes assigned to pre1959 literature. The only way to identify these codes is to take an original review and match the subject heading under which it appeared to the corresponding twodigit code assigned to it by MathSciNet when one performs an author or title search for the review.
MathSciNet also indexes a substantial amount of pre1940 literature, going back to the 1870s. In many cases no classification codes are given, but there is still a fair amount of literature for which they are assigned. Since MR did not exist before 1940 there was no classification scheme to attend to, and so one was free to assign codes at will. MathSciNet took full advantage of this situation. The pre1940 literature is given complete fivesymbol MSC1970 classification codes. For the twodigit part of the code for ordinary differential equations, an article from 1906 is assigned the code 34
, from 19401955 the code 36
and from 1956 on the code 34
[27].
Some further examples from MathSciNet illustrate the shifts in classification that have occurred over the years. We consider the subjects of set theory and celestial mechanics. Set theory went from foundations between 1940 and 1958; then to set theory alone (04
) between 1959 and 1999 [28]. In MSC2000, mathematical logic and foundations were given the code 03
; 02
and 04
were abolished; and set theory was returned to logic and foundations, and given the code 03E
. In MathSciNet, Abraham A. Fraenkel and Yehoshua BarHillel's Foundations of Set Theory (1958) has the code 02
, because it was included under the subject heading of foundations in the issue of MR in which it appeared in 1958 and 02
is the code for foundations in MSC1970. Nicolas Bourbaki's Elements of Mathematics Theory of Sets (1968) has the code 04
because it was included under the subject heading set theory in MR in 1968 and 04
was the code for set theory in 1968.
Celestial mechanics, including mathematical work on the threebody problem, was placed at the founding of MR within astronomy, where it remained until 1970. In that year it was put under dynamics of a system of particles and given the designation 70F15
. Astronomy in MSC1970 has the code 85
. This number is also given to all reviews in MR before 1970 that were included under the subject heading of astronomy at the time of their publication, including celestial mechanics. Theodore E. Sterne's An Introduction to Celestial Mechanics (1960) has the MSC code 85
. The link to 85
in MathSciNet states that this code is for astronomy and astrophysics since 1940; also noted is the fact that the code for celestial mechanics since 1970 is 70F15
.
The call number of a book identifies the part of mathematical knowledge to which the subject of the book belongs. The call number system — whether it be LCC, Dewey, or anything else — provides a representation of the division of mathematical knowledge into subjects. The call number is also an identifier that allows one to retrieve the book from the library's collection. A system of library classification has the dual function of book retrieval and intellectual subject organization. By contrast, the MSC is a bibliographic classification system. The MSC subject classification code is used to assist the reader in identifying the literature corresponding to a given code. It is not the unique identifier of any particular publication. (Of course, each review has an identification code (currently of the form MRxxxxxx), but this code has no inherent classificatory meaning or relation to the item under review.) Instead one uses the author or title and then is led by a search to its review. The MSC classification code is not required to do this. One can be a consistent user of MathSciNet without knowing specific MSC codes, an obvious difference from a library, where call numbers are necessary to find the books.
Beginning in the 1980s and 1990s mathematics books have generally included the MSC codes with the publishing data presented at the beginning of the book. Since the 1950s this data had traditionally included the LC call number and sometimes also the Dewey classification. As we move closer to the present one finds more and more only the MSC codes given. For books that are only available electronically and where there are no library call numbers, the MSC codes are all there is, in terms of situating the book within the framework of mathematical knowledge. In a world in which books and journals are released electronically a traditional library call number is unnecessary, but a classification code remains valuable, even more so given the pace of subject growth and its continual fragmentation into specialties. A system of classification is evidently also required in order for search engines to work effectively. MSC codes have become ubiquitous: they are required on papers submitted for publication, they must be included for a paper presented at a conference, and they are given at the beginning of books. MSC2010 has 63 subject headings and around 2700 subclasses, and that number will likely increase in 2020. While the MSC may have become a ramified system of some complexity, it fulfills a range of functions and will surely loom large in the future.
A key event in the history of the MSC was the transition in 1959 from the traditional subject index that was ordered alphabetically to one that was based on the place of the subject in the organizational framework provided by the table of contents and the associated coding of subjects and classes. There was a shift in emphasis from retrieving knowledge to organizing knowledge, a tendency that has grown more pronounced in the subsequent history of the MSC up to the present.
The prominence of the MSC scheme today is noteworthy in light of the variable role it plays in practice in retrieving literature on any particular subject or by any particular author. On the website announcing the project to revise MSC210 there is the following admission (AMS 2018): "In the decade since the last revision, keyword searching has become increasingly prevalent, with remarkable improvements in searchable databases". The author of the document nonetheless emphasizes the valuable functions that MSC performs. Codes are used by publishers in several different ways, help arXiv in classifying submissions, and are an aid in organizing paper sessions at conferences.
Current interest in the MSC goes beyond its use in information retrieval and is indicative of a deeper disciplinary interest in the question of the classification of mathematical knowledge. Since the broad classificatory framework at the twodigit subject level is fairly established, attention is channeled to the classes under each subject and the further divisions and orderings within these classes. A finegrained sense of the vast field that is presentday mathematics is provided by the landscape of the MSC and the periodic revisions that it undergoes.
Daniel Parrochia (2018, 281282), in a larger study of the place of philosophy in modern mathematics, criticizes the MSC2010 on the ground that is does not reflect underlying connections that exist between different parts of mathematics. Combinatorics has its own subject designation, but combinatorics arises in other subjects, including number theory and theory of Lie groups. Both set theory and category theory are comprehensive theories and for this reason are allied philosophically. Nevertheless, the two theories have separate subject designations, 03
for set theory and 18
for category theory. The Langlands program connects number theory and analysis but this linkage is not apparent in the subject categories of the MSC. Modern thinkers from David Hilbert to Bourbaki have admired the organic unity of mathematics, a unity that is not (in Parrochia’s view) apparent in the MSC.
The MSC was of course not created ex nihilo as an abstract scheme to classify mathematics. Rather it evolved historically and has been modified regularly throughout its history since 1959. Some of the revisions have been substantial, but the core structure of the classification has remained intact. The system of primary, secondary and tertiary classification codes as well as the systematic use of crossreferencing serve to make connections between topics that lie on different subject branches.
Consider the example of set theory and category theory. Subject 18
began as homological algebra in 1959 (homological algebra was not a subject heading until the year before). In 1968, 18
became category theory and homological algebra, where it has remained until today. In MSC2010 there are seven classes in 18
(18A18G
), six of which concern categories and one that concerns homological algebra. Meanwhile, as we saw above, set theory began in foundations in 1940 and migrated to its own subject 04
from 19591999, only to return to foundations as a class within 03
in MSC2000. It would be necessary to restructure the classification fairly radically in order to join a subject affiliated historically and substantively with homological algebra to one whose primary origins and character lie in measure theory and foundations.
It is possible that a reorientation of the nature of mathematics could be achieved along the lines suggested by Parrochia (2018, 282303), making fundamental the theory of categories and sheaves and Alexander Grothendieck’s program. Such a program would be of interest from a foundational and mathematical viewpoint, and is worth pursuing at least on a theoretical level. It might better reflect the organic unity of mathematics but it is not clear that it would provide a basis for the organization of all of mathematical knowledge on the scale of the MSC. As a matter of practical reality, the MSC is so established, extensive and widely used that its central place in the classification of mathematical knowledge is unlikely to be seriously challenged anytime soon.
I am grateful to two referees for their detailed comments on a first draft of the expanded paper.
1. LaMontagne (1952, 208) observes that Charles Cutter “foresaw the continuing growth of the library and knew that each change in the shelving of books entailed the changing of "shelf marks" — a long and expensive process. Cutter therefore decided to abandon fixed location “and to adopt a method which would allow books to be moved without changing the marks on the catalogues”. The part in quotation marks is from (Cutter 1882, 6). Cutter’s “bench marks” are what we refer to today as call numbers.
2. The University System of Georgia (2019) asserts: "Libraries in the United States generally use either the Library of Congress Classification System (LC) or the Dewey Decimal Classification System to organize their books. Most academic libraries use LC, and most public libraries and K12 school libraries use Dewey". This is also true for Canada and some other parts of the British Commonwealth.
3. https://www.loc.gov/aba/dewey/about.html and https://www.loc.gov/aba/dewey/index.html.
4. See endnote 14 below.
5. Flint (1904, 222223 and 308312) gives accounts of the classifications of Renouvier and Goblot.
6. English translation is by Thomas J. McCormack from the 1893 English edition of Mach's book, The Science of Mechanics: A Critical and Historical Account of Its Development (Open Court, Chicago).
7. Flint (1904, 277) mistakenly gives the date of publication of Masaryk’s book as 1866. Masaryk was born in 1850 and entered the University of Vienna in 1872.
8. An exception to the prevailing consensus was Karl Pearson, who in his Grammar of Science (1892) put theory of functions and calculus together with arithmetic and algebra, these subjects dealing with quantity, while geometry was classified as a distinct subject area dealing with space (Flint 1904, 296). Earlier the Paris book seller JacquesCharles Brunet (1814) in his pioneering classification scheme placed mathematical subjects in the order arithmetic, algebra, calculus, and geometry. Brunet was presumably influenced by Comte and Ampère. Brunet's catalogue was exceptional among all classification schemes in placing mathematics at the end of the sciences, following philosophy, physics, chemistry, geology, biology, and medicine.
9. The Universal Decimal Classification (UDC) is a bibliographical and library classification system that was influenced in its origins by Dewey but developed into an important international service of its own that is widely used today. McIlwaine (1997) provides an historical study of the UDC. Its ordering of mathematical subjects follows older Dewey: foundations, number theory, algebra, geometry, analysis. It is also of interest to consider the subject ordering adopted by encyclopedias of mathematics. The influential Japanese Iwanami Sugaku Ziten (Encyclopedic Dictionary of Mathematics) (Japanese Mathematical Society 1954) posits the order arithmetic, algebra, geometry, analysis. Subsequent editions appeared in 1960 and 1968. Fang (1972) devotes chapter 4 of his book to the Iwanami. Fang regards it as a work of “obvious greatness” (17). An English translation was published by MIT Press in 1977.
10. On the formative place of the Jahrbuch in the modern classification of mathematics see Section 6.
11. An interesting graphical illustration of this change in usage is provided by the Google Ngram (Michell et al. 2011) for the terms “theory of functions” and “complex analysis” for the period from 1880 to 2008. See Figure 1.
12. While there is certainly a general conservatism among librarians with respect to classification, in the case of the Dewey there have been revisions of the classification that have been retroactively applied by some libraries to books in their collections. See the discussion below of the University of Illinois at UrbanaChampaign.
13. At least in some quarters today there is still a preference for placing geometry before analysis, in accordance with traditional nineteenth–century notions of mathematical classification. The American Mathematical Society in tracking new PhDs uses the subject order: foundations, algebra/number theory, geometry/topology, analysis. The National Science Foundation in the United States organizes its science programs in the order: algebra/number theory, topology/foundations, geometrical analysis, analysis. The International Mathematics Union divides the lecture sections at its congresses into the categories: foundations, algebra, number theory, algebraic geometry, geometry, topology, Lie theory, analysis. (See Dave Rusin, "The Divisions of Mathematics", http://web.archive.org/web/20150516041021/http://www.math.niu.edu/~rusin/knownmath/index/tour_div.html. See also, http://web.archive.org/web/20150424115620/http://www. mathunion.org/activities/icm/icm2010programstructure/.)
14. I am grateful to Michael Norman, Head of Content Access Information at the UICC, and Tim Cole, Head of the Mathematics Library at UICC, for information about their collection (emails, November 27 and 28, 2016). Tim Cole reports that there were further revisions of the Dewey system with Dewey 22 (2003), but that these revisions were more minor than the ones in schedule 18. The Mathematics Library has not updated books before 2003 to the specifications of schedule 22. There have been further minor revisions to mathematics classification since 2003. Today the UICC relies mainly on the Dewey call numbers for new books provided by the Decimal Classification Division of the Library of Congress. The UICC is also moving in the direction of classifying new books using the LC classification system itself.
15. There was not complete consistency in cataloguing during the transition years between about 1968 and 1972. It should also be noted that books on real analysis were sometimes assigned 517
rather than 517.5
(before 1971) and 515
rather than 515.8
(after 1971).
16. Although there was a great deal of research on dynamical systems at the end of the twentieth century, it did not receive its own twodigit subject heading in AMS’s Mathematics Subject Classification (discussed in the next section) until the year 2000. Literature under this subject heading has expanded greatly since then.
17. An account of reviewing and bibliographical journals for mathematics during this period is given by Pemberton (1969, 5560).
18. The history of the Jahrbuch and Zentrablatt in the 1930s and 1940s is documented by SiegmundSchultze (1993), although the subject of classification is not raised. A short account of Zentralblatt in the first two decades of its existence is given by Ett and Welt (1998).
19. "Die Kapiteleinteilung wurde im Laufe der Jahre geändert und vertieft, um mit aktuellen Entwicklungen auf den verschiedenen Gebieten der Mathematik Schritt zu halten. Später bildeten diese Abschnitte die Grundlage für den Aufbau einer MathematikKlassifikation."
20. SiegmundSchulze (1993, 133136) observes that in the 1930s a more structural approach to mathematics was evident in abstract algebra and functional analysis, a trend that intensified despite the influence of such German mathematicians as Ludwig Bieberbach, who favored a more intuitive and downtoearth "Aryan" approach to mathematics.
21. The American movement in the 1950s and 1960s to overhaul primary and secondary school mathematics education — the socalled new mathematics — was deeply influenced by contemporary modern mathematical perspectives. See Hayden (1981) for a history of the new math movement.
22. Writing apparently on behalf of the AMS, Pitcher (1988, 145) states, "Repeated offers to assist with a revision of the Dewey Decimal Classification of mathematics, universally recognized as ineffective and outdated for research mathematics, have been refused”. At the time the MSC scheme was being developed, Dewey classifiers were already working on schedule 18 which included a reclassification of mathematics books. It is possible that this was the reason the Dewey people were not receptive to the AMS offers. See endnote 10 above.
23. In principle, the 1968 classification allowed for 100x100=10,000 subclasses. By contrast, this number for MSC1970 was 100x26x100=260,000. MCS1970 also had other classificatory advantages connected to the use of letters to designate broad topic categories.
24. We do not in this article consider technical matters related to the creation of MathSciNet or the platform for its application. On this subject, see Gala et al. (2019).
25. This document is at https://mathscinet.ams.org/mathscinet/help/field_help.html#mscp.
26. The user who enters 34
for ordinary differential equations into MathSciNet and discovers the absence of any returns for the fifteenyear period from 1940 to 1955 will view this as a mystery, and will have no way of knowing that entering 36
would produce the missing reviews.
27. There is some literature on ordinary differential equations from the period 19491955 that is given the classification 34
in MathSciNet but the reviews in question appear in volumes of MR for 1956 and 1957. The subject heading "ordinary differential equations" appeared for the first time in volume 17 in 1956. (The MathSciNet search by year goes by the year of publication of the article rather than the year of publication of the review.)
28. In the 1940s MR there is the subject heading “Theory of sets, theory of functions of real variables”. It is a matter here of the use of sets in real analysis and measure theory. Set theory as we understand it appears under the subject heading.
29. The following volumes of MR were consulted for this article: 1, 6, 11, 16(1), 18(2), 19(1), 19(2), 20(1), 20(2), 22(1), 22(2), 28(2), 29(1), 33(1), 34(2), 36(1), 36(2), 37(2), 38(1), 44(1), 45(1), 46(1), 49(1), 50(1), 51(1), 53(1), 80ac, 94km, Index of Math. Papers (1970), Index of Math. Papers (1971), Subject Index 197379 (V. 15), Subject Index 19401958, Subject Index 19591972 (V. 14), Subject Index 198084 (V. 1), Current Math. Publications (V. 9 and 22).
30. Brunet (1814). In an appendix the translator gives a statistical breakdown of the contents of Brunet’s bookshop, using the total number of bench marks or call numbers each subject receives as a percentage of the total number of bench marks overall. Arts and Sciences constituted 22.7% of the contents of the collection, while the mathematical sciences constituted 15.2% of Arts and Sciences. From this one may deduce that the mathematical sciences represented 3.5% of the overall collection.
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Version 1.0, published 20191031, last edited 20191126
Article category: KO in specific domains
The present study is an expanded and revised version of Fraser (2017).
This article (version 1.0) is also published in Knowledge Organization. How to cite it:
Fraser, Craig. 2020. “Mathematics in Library and Review Classification
Systems: An Historical Overview”. Knowledge Organization 47, no. 4: 334356. Also available in ISKO Encyclopedia of Knowledge Organization, eds. Birger Hjørland and Claudio Gnoli, https://www.isko.org/cyclo/mathematics
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