Popular Science Monthly/Volume 79/November 1911/American Mathematics

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By Professor G. A. MILLER


ABOUT a dozen years ago a well-known French mathematician wrote as follows in reference to our mathematical situation:[1] "Mathematics in all its forms and in all its parts is taught in numerous [American] universities, treated in a multitude of publications, and cultivated by scholars who are in no respect inferior to their fellow mathematicians of Europe. It is no longer an object of import from the old world but it has become an essential article of national production, and this production increases each day both in importance and in quantity."

Taken by itself this assertion looks good and it is doubtless more nearly true to-day than it was at the time of publication. If we turn our eyes away from this statement and rest them upon the mathematical book shelves of a good library, we can not fail to notice that our accomplishments do not seem to be in accord with the complimentary statement noted above. This disaccord will become still more evident if we look through the pages of some of the standard works of reference, such as the great mathematical encyclopedias which are now in the course of publication.

If a student of the history of mathematics would make a list of the leading mathematicians of the world during the last two or three centuries, arranging the names in order of eminence, he would have a fairly long list before reaching the name of an American. Such names as those of Euler, Cauchy, Gauss, Lagrange, Galois, Abel and Cayley have no equals in the history of American mathematics; and, among living mathematicians, probably all students would agree that there are no American names which should be placed on a mathematical equality with those of Poincaré, Klein, Hilbert, Frobenius, Jordan, Picard and Darboux. Both of these lists of names could be considerably extended without any danger of being unfair to the mathematicians of this country, but they suffice to establish the fact that our mathematical situation is not yet satisfactory, notwithstanding our remarkable progress during recent decades.

This unsatisfactory situation is reflected in many of our standard books of reference. For instance, under such an important word as "matrix" one finds in Webster's New International Dictionary (1910) the following statement: "The matrix has also significance apart from its development into a determinant." In view of the fact that a matrix, as commonly used by mathematicians, has no possible development, it is clear that the given sentence does not convey any information. In fact, the other remarks under this term are almost equally objectionable, and they raise the question whether a philosopher should be selected to define the mathematical terms of a standard work.

It is not implied that such an extensive work as a large dictionary could be expected to be free from defects, but there is always a limit to the number and the type of those which appear excusable. When one reads in such a dictionary that "algebra is that branch of mathematics which treats the relations and properties of quantity by means of letters and symbols," and then turns to page 22 of volume 1 of the large French mathematical encyclopedia and reads that "it is convenient, in arithmetic, to represent any number by a letter, it being understood that this letter denotes a single and the same number whenever one remains in the same subject," it becomes evident that the given definition of algebra is not supported by some of the highest authorities.

In fact, such terms as arithmetic, algebra and geometry are used with such a wide range of meanings by eminent authorities that it seems impossible to give satisfactory definitions of them, and our dictionaries would convey more reliable information about mathematical terms by stating this fact, together with some indication of what broad subjects are generally classed under these terms, than by giving categorical definitions which can be accepted only by those who have a meager knowledge of mathematics.

Without implying that Webster's New International Dictionary is any less reliable with respect to mathematical matters than most others, we shall refer to one more instance of misleading statements in this work. On page 2547 we read as follows: "The cipher was originally a dot, used as a mere arbitrary sign to mark place or local value." Such a definite statement seems strange in view of the fact that the origin of zero is one of the unsettled questions of the history of mathematics. It is of interest to note in this connection that Cantor changed his view with respect to the origin of this concept and this symbol, in the third edition of Volume I. of his classic "Vorlesungen ueber Geschichte der Mathematik," where he states that the symbol for zero and the positional arithmetic are probably due to the Babylonians instead of to the Hindus, as he had stated in the earlier editions of this work, and as is stated in a large number of other works.

Our encyclopedias also frequently exhibit careless editing along the line of mathematical terms, and the choice of editors for such work often seems to indicate that the general editor regarded the choice of the mathematical editors as a matter of little consequence. Possibly those who would be best qualified to render excellent service along this line are unwilling to undertake it in view of the large amount of labor which it involves.

As an instance of a decided misstatement in one of the best of these encyclopedias we may cite the following: "Sylow (1872) was the first to treat the subject [substitutions] apart from its applications to equations."[2] Very little reading along the line of the development of the theory of substitutions would reveal the absurdity of this statement. Nearly all of Cauchy's fundamental work along the line of substitutions was no more intimately connected with the theory of equations than the articles by Sylow. Similar remarks apply to most of that part of Jordan's work which antedates Sylow's fundamental article, and also to the work of a number of other authors.

Judging from the following words of Sir Oliver Lodge; "the mathematical ignorance of the average educated person has always been complete and shameless,"[3] one could not expect to find very much better conditions in England. In fact, in consulting the large Murray English Dictionary, published at Oxford, England, the writer found under the first mathematical term which he consulted, viz., the word "group," not only an incomplete definition, but also the following incorrect statement: "The idea of group as applied to permutations or substitutions is due to Galois." As a matter of fact, the idea of permutation groups was clearly developed by Kufrini about thirty years before Galois, not to mention the still earlier work by Lagrange and the early publications of Cauchy and Abel.

One of the most direct inferences from what precedes is the fact that there is too much mathematical indifference. If more vigorous protests against the inaccuracies in our standard books of reference would be made, publishers and general editors would doubtless exercise greater caution in the selection of their mathematical editors. This mathematical indifference is perhaps still more disastrous when it exists among university administrators. Judging from several of the recent appointments in leading universities, it would appear that we are not moving as rapidly towards high mathematical ideals as one might wish.

The English-speaking pure mathematicians constitute more nearly a terra incognita than the workers in any other large field of knowledge. This is partly due to the nature of the subject and partly to the fact that there are so few mathematical works of reference in the English language. There never has been a good mathematical encyclopedia or other work of general reference in this language, while the French and Germans have had several such works in addition to the great encyclopedias which are now in the process of publication. All large mathematical histories have appeared in foreign languages.

As a result of this lack of intermediate mathematical literature comparatively few of our people know what constitutes a mathematician of high order. The time has been when even the educated public seemed to believe that the author of a successful series of elementary text-books had necessarily gained a place among the great mathematicians of the world. As several of our most popular recent series of text-books were edited by men of remarkably low mathematical attainments, this view is no longer so generally held, but it is questionable whether it has been replaced by a more correct one on the part of the majority of those who feel entitled to express an opinion on the work of mathematicians.

In looking over the work of the fourteen great mathematicians mentioned above one finds that all of them published mathematical articles and that a majority of them also published treatises. Two of them, Abel and Galois, died at an early age, before they had time to develop sufficiently the fields in which they were interested to write extensive treatises. This is especially true of Galois, who published only five papers during his short lifetime of only twenty years, but several of his other papers appeared later.

The extent of the publications of the mathematicians mentioned above varies from the comparatively few brief articles by Galois to the voluminous publications by Euler, which are just now appearing in a collected form and are expected to fill forty large volumes. Judging from the great mathematicians of the recent past, it would appear that publication of original articles is one essential of greatness, but greatness is not measured so much by the number and the extent of such publications as by their merits. It should, however, be observed that nearly all of the great mathematicians of the recent past have published a large number of research papers. In the case of Cayley, who is the only Englishman in the given lists, the number of these papers is about one thousand.

America has never had a mathematician who published as extensively as some of the European mathematicians, and the average extent of our publications is much below the average of the leading mathematical countries of Europe, if we exclude the elementary textbooks. It is doubtless true that the most important consideration at present is the improvement of the quality of our publications, but we are also in need of more mathematical journals to insure more rapid publication of good research material. If the crowded condition of our research journals would induce a larger number to assist in bringing out more good intermediate mathematical literature, it would doubtless be of great importance for the future advancement of the science.

One of the leading agencies in bringing about rapid mathematical advances during the last few decades is the American Mathematical Society, which is now one of the strongest mathematical societies of the world and has probably a larger income than any other similar organization. It publishes two journals and its frequent meetings furnish favorable opportunities to renew zeal and to cooperate in the more important advances. These meetings serve also as a good medium to spread reliable information in reference to young men of promise and to secure for them more prompt recognition than would otherwise be possible.

One of the most hopeful signs as regards American mathematics is the fact that our students are in close contact with several of the mathematical centers of Europe. It is no longer true that nearly all Americans who go abroad for the purpose of studying mathematics locate in the same institution or in the same country. In recent years, Italy has grown rapidly in favor, while the leading universities of Germany and France continue to attract a considerable number of our best students. The rapid interchange of ideas resulting from the scattering of our mathematical students in foreign countries is doing much to dispel prejudices, to make American mathematics cosmopolitan, and to awaken a keener appreciation of the advantages and the disadvantages of our own institutions.

If one bears in mind the facts that our library facilities were very poor until recent years and that no locality offers in itself any special inducements for mathematical study, one should perhaps be surprised by the rapid mathematical advances during the last few decades rather than by the fact that we have not yet attained to greater national eminence. It remains to be seen whether we shall ever be on an equality with the leading mathematical nations of the world. The rapidity with which we have obtained respectful recognition and the American eminence in some of the other sciences might reasonably awaken the hope that we may be not far from the time when we shall deserve, in the strictest sense, the position pictured in the first paragraph in such a friendly spirit.

  1. Laisant, "La Mathematique, Philosophie-enseignement," 1898, p. 143.
  2. New International Encyclopedia (1904), under "Substitutions."
  3. "Easy Mathematics," 1906, preface.