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Popular Science Monthly/Volume 73/October 1908/The Classification of Mathematics

By Professor G. A. MILLER


HERR VALENTIN, of Berlin, who has been working on a general mathematical bibliography for more than twenty years, estimates that the total number of different mathematical works is about 35,000 and that about 95,000 mathematical articles have appeared in the various periodicals.[1] The present rate of growth of this literature is so rapid that, without increasing the amount per year, the next fifty years would produce more than the total produced from the earliest records to the present time. Without some means of classification this vast store of knowledge would have little value from the difficulty of finding what is wanted. Before entering upon a description of any details of classification I shall make a few remarks on some of the terms of classification which are familiar to all; viz., arithmetic, algebra and geometry.

About three years ago Sir Oliver Lodge published an unusual work under the unusual title "Easy mathematics, chiefly arithmetic, being a collection of hints to teachers, parents, self-taught students, and adults; and containing a summary or indication of most things in elementary mathematics useful to be known." This title is followed by a no less unusual preface, whose tenor may be inferred from the following quotation: "The mathematical ignorance of the average educated person has always been complete and shameless, and recently I have become so impressed with the unedifying character of much of the arithmetical teaching to which ordinary children are liable to be exposed that I have ceased to wonder at the widespread ignorance, and have felt impelled to try and take some step towards supplying a remedy." The main reason for referring to this work in this connection is to call attention to what appears to be a very common use of the word arithmetic, as including most but not all of the mathematics which the average educated man should know.

Efforts to arrive at a much more accurate definition of the term arithmetic are apt to meet with disappointment. On the one hand, we meet with contradictory classifications among works of the highest authority. The great mathematical encyclopedia which is being published almost simultaneously in German and French includes determinants under arithmetic, while the International catalogue of scientific literature places determinants under algebra. If one were inclined to adopt the common definition that arithmetic is the science of the relations existing between numbers, one would be perplexed by the fact that the theory of groups of finite order is classed with arithmetic in the encyclopedia mentioned above, while it might be difficult to name any other mathematical subject which makes less direct use of numbers than this theory does.

Although these conflicting uses of the term arithmetic preclude the possibility of formulating a definition which is in accord with the usage of all of the prominent mathematicians, yet this term presents very much less serious difficulties than that of algebra from the standpoint of giving an acceptable definition. All are agreed that the four fundamental operations with natural numbers constitute a part of arithmetic. In fact, all that is generally studied in the elementary schools under the title of arithmetic is now universally regarded as a part of this subject, even if the Greeks called it logistica and dignified what is now generally known as higher arithmetic, or number theory, by the term arithmetic. While it might be difficult to find anything which was included under the term arithmetic during the entire historic period of mathematics, it is not difficult to find things which are now universally accepted as parts of this subject.

When we come to the term algebra, on the contrary, it seems impossible to find any common ground. If we think of algebra as a generalized arithmetic in which numbers are replaced by symbols which may have any numerical value, we are perplexed by such statements as "In arithmetic it is customary to represent any number whatever by a letter, it being understood that this letter represents the same number as long as the same subject is under consideration."[2] On the other hand, if one were inclined to consider the elements of the theory of equations as the peculiar sphere of algebra, the recent standard encyclopedia of elementary mathematics by Weber and Wellstein,[3] in which simple and quadratic equations are classed under arithmetic, would imply that such usage was not universal among eminent authorities.

Coming to the term geometry, we encounter scarcely less trouble. On the one hand, we find it advocated that geometry should be recognized as a science independent of mathematics, just as psychology is gradually being recognized as an independent science and not as a branch of philosophy,[4] while, on the other, we find that the Paris Academy of Sciences uses the term geometry as a synonym for pure mathematics. In the one case, the term geometry is used for what is not regarded as mathematics, and, in the other, it is supposed to comprise all that is generally included under the term mathematics. With such a wide range of usage among eminent authorities it is evident that an acceptable definition is hopeless.

These instances appear sufficient to emphasize the fact that the terms arithmetic, algebra and geometry have no definite meanings in mathematical literature. They may be compared with the names of the constellations, which attract the attention of the amateur but are not generally taken very seriously by the professional astronomer since their boundaries are not defined with clearness. Just as it may be difficult to establish a connection between the figures represented by the names of some of the constellations and the arrangement of the brighter stars in them, so it is difficult to see much connection between the meaning of the terms arithmetic, algebra and geometry, and some of the subjects classed under these heads. In a growing science it is very desirable to have some elastic terms—terms to which we assign broader and perhaps even different meanings as our knowledge advances. In fact, the term mathematics is itself preeminently one whose meaning is a matter of slow development, even if we accept such brief definitions as mathematics is the science of saving thought, or "mathematics is the science of drawing necessary conclusions."

The fact that many things which appear unrelated when studied superficially exhibit the most intimate connections when viewed from a higher standpoint has doubtless been a potent cause of the variety of usage as regards general terms of classification. There are no natural lines of division in mathematics. In fact, one of the most attractive phases in the development of mathematics is the discovery of the relations existing between what was supposed to be unrelated. In other words, the unifying of mathematical truths is one of the chief concerns of many of the workers in this domain. Although the elements of arithmetic, algebra and geometry appear sufficiently distinct to the beginner, the marks of distinction vanish one by one as one proceeds in following up the ideas starting from these centers, as is evidenced by the term analytic geometry, since analysis and algebra were synonyms for Newton, Euler and Lagrange.

Notwithstanding the fact that there are no natural lines of division in mathematics, classification is essential and need not be entirely artificial; for, marks of differences which are only superficial are, nevertheless, worthy of note and frequently furnish convenient centers for groups of very closely related ideas. Both subject-matter and method offer many such superficial marks of difference which are utilized for the sake of classification. As we go away from these centers we naturally reach facts which seem equally closely related to more than one center, and in such cases it is necessary to have either a duplicate or an artificial classification. For some purposes, such as arranging books on shelves in a library, the former is not feasible, and hence arises the constant opportunity of complaint on the part of those who use libraries with a view to obtain all available facts along certain lines. This opportunity is inherent in the subject and hence must exist under the most ideal conditions, but it is sometimes made more apparent by the fact that books are not always classified by those who are as familiar with the subject-matter as their authors were at the time of writing.

Although the present active period in mathematical development has exhibited many relations between subjects which appeared to be unrelated, yet it has been still richer in exhibiting new centers of developments which promise to be useful, and hence it has called for a great extension of classification headings, as may be seen from the 1908 edition of l'Index du répertoire bibliographique des sciences mathématiques. In order that a method of classification should give promise of usefulness for a long period of time, it must therefore be so constructed as to admit readily of indefinite extensions. This is a characteristic property of the two important methods of classification which have been adopted after international conferences, viz., l'Index du répertoire just mentioned and the International Catalogue mentioned above. The former of these provides for an indefinite extension of its fundamental headings by using the capital letters of the Roman alphabet with various exponents to represent these headings. On the other hand, the International Catalogue divides all mathematics into four parts, in addition to a general heading for history, periodical, general treatises, etc.

In the first five annual issues of the International Catalogue these four parts into which all mathematics is divided bore the following headings: Fundamental notions, algebra and theory of numbers, analysis, and geometry. In the last issue of this catalogue the first of these headings is replaced by arithmetic and algebra, in accordance with the decision of the international convention of 1905. The term algebra now appears in two of the four headings, and, if it is remembered that the theory of numbers is higher arithmetic, this term is implicitly in two of these four headings. This is another evidence of the vagueness of the terms arithmetic and algebra as used in some of the best mathematical literature of the present day, and seems to imply that these terms, especially the former, are more and more devoted to those fundamental notions which are most prominent in the later developments, or have the most frequent uses in related sciences. As the science grows some things which are now classed with analysis or geometry will naturally be put under other headings.

While it might be impossible to advance good reasons for dividing mathematics into exactly four grand divisions, yet a small number of divisions offers advantages by furnishing names which will generally be remembered and by emphasizing the connection between extensive developments. It is true that the names of these grand divisions do not have a very definite meaning, but they have some meaning, and they exhibit something of the tenor of the various branches whose names are too numerous and appear too erudite to the average educated man. Instead of simply saying that one is working on Ausdehnungslehre it may be some satisfaction to add for the benefit of the uninitiated that this is a kind of algebra, and thus established language contact even if thought contact is out of question.

A question of more general interest is the number of parts into which mathematics is divided in the final classifications. The answer to this question gives some idea of the fractional part of the entire literature which must be examined by one who is seeking all that is known along a particular line. The last issue of the International Catalogue contains only about two hundred headings, so that one would have to look over one two-hundredth of the total publications of the year in order to find all that had been written during the year on a subject comprised under a single heading. In this respect l'Index du répertoire is much superior. In fact, the last number of the Revue semestrielle des publications mathématiques, which follows this index, classifies the publications under about seven hundred headings, and, as a large number of headings have no entries during one of the periods of six months, it would frequently be possible to get at all the literature which appeared on a particular subject during a period of years by examining less than a thousandth part of the total mathematical literature of the period.

The preceding paragraph relates to the classification of current literature. The classification of the total literature is in a much less satisfactory condition. The magnitude of this work may be inferred from the facts that Müller's Mathematisches Vocabularium contains more than ten thousand technical terms used in pure and applied mathematics and that it is not exhaustive. As most of these terms relate to concepts which either are or may become the centers of a series of closely related developments, we can predict no limit to the number of headings under which the mathematics of the future will be treated. In fact, if we adopt the view that mathematics consists of creations as well as of discoveries, considerations as to limits become very vague even if they do not lack interest.

Professor Sylvester once called himself the mathematical Adam in the proud consciousness of having named a large number of algebraic concepts and that these names had become more or less current among his colleagues. While technical terms are useful for the sake of classification yet the number of such terms may become so large as to justify the criticism of Hankel that the stately mathematical structure resembled the tower of Babel. The proper time for the introduction of new technical terms and new heads of classification must depend upon the good judgment of the workers in this field, it being remembered that terms and classifications are secondary matters, although by no means useless, and that the main thing is to extend the domain of knowledge, especially where the beauty or usefulness of the results assure them a permanent place in the intellectual wealth of the world.

A large variety of classifications may prove serviceable to the investigator. Sometimes an author's catalogue may render the best service, while at another the grouping together of a large number of related things as is done in the Jahrbuch über die Fortschritte der Mathematik, where pure mathematics is divided into only fifty parts, renders the most valuable service. At still another time, the dictionary arrangement under thousands of terms, as it appears in the indexes of large works, especially in the incomplete encyclopedia to which we referred above, offers the most convenient method of arriving at the desired information. Fortunately the current mathematical literature is now being classified according to several different methods, each possessing peculiar advantages for the different needs of the scholar.

The history of classification in mathematics is very old as may be seen from the fact that the mathematical handbook of Ahmes, which was written about 1700 B. C., already contains the divisions into arithmetic, and plane and solid geometry. As this work bears the title "Directions for obtaining a knowledge of all dark things," it would appear that the observance of the distinction between arithmetic and geometry may have been older than of that which separates mathematics from the other sciences. In fact, even at the time of Plato, the term mathematics included all scientific instruction and its more restrictive meaning seems to have had its origin in the Peripatetic School. The main divisions of the mathematical sciences during the Greek period were: logistica, arithmetic, plane and solid geometry, music and astronomy. One of the best known classifications of mathematics is the Quadrivium of Boethins, viz., arithmetic, music, geometry and astronomy; and it is an interesting coincidence that the International Catalogue should have established a quadrivium of pure mathematics, as was observed above.

Among the general terms of classification, that of analysis is probably the least familiar to the non-mathematician. All have some idea of arithmetic, algebra and geometry from the context of elementary text-books bearing these names, but it is not customary to place the term analysis on the cover of an elementary text-book in the English language. Perhaps this is due, in part, to the fact that our secondary mathematics does not generally involve the concept of derivative, which was introduced into French secondary instruction in 1902, and has proved successful in that country as well as in some others where it has been tried. From a historical point of view analysis is merely an extension of algebra. It has already been observed that for Newton, Euler and Lagrange analysis and algebra were synonyms, and the present significance of the term analysis may be inferred from the fact that the most familiar subjects that are now generally comprised under it are differential and integral calculus and the theory of functions.

It may be desirable to add that the aim of the present article was to convey some idea of the extent and nature of the modern mathematical developments, from the standpoint of classification. While this standpoint reveals only the outside yet this side is worth knowing and is apt to awaken some thoughts in regard to what may be within. As the use of machines has revolutionized the material world, so the use of mathematics has revolutionized the efficiency of thought in regard to problems whose solution is sufficiently elementary to come under known developments. As these developments proceed they provide for greater efficiency and wider applications. The analogy between mathematics as a machine of thought and an ordinary labor saving device known as a machine can be traced through many points of interest. It should, however, be emphasized that the real mathematician is the inventor of such machines, but not the machine itself.

  1. Felix Mueller, Bibliotheca Mathematica, Vol. 7 (1907), p. 416.
  2. "Encyclopédie des sciences mathématiques" (1904), p. 22.
  3. Published by B. G. Teubner, Leipzig, Germany.
  4. Bôcher, Bulletin of the American Mathematical Society, Vol. 11 (1904), p. 124.