William E Story on mathematics at Clark University 1889-1899
The Department of Mathematics
Mathematics occupies a peculiar position relatively to the arts and sciences. It is, par excellence, an art, inasmuch as its chief function is to solve problems, - not such examples as are given in the text-books, and which serve only as exercises in the application of methods, but any problems that may arise in human experience and for whose correct solution sufficient data are at hand. When any line of investigation, to whatever subject it may refer, has been carried so far that exact reasoning may be applied to it, mathematics is the authority to which the results of observation are submitted for the final determination of their consistency and the conclusions that may be drawn from them, and furnishes the means of applying these conclusions to the prediction of phenomena not yet observed. No science and no branch of technology is exact, that is, capable of predicting with certainty what will happen under given conditions, unless it rests upon a mathematical foundation. Astronomy, physics, and applied mechanics already have this foundation to a considerable extent, while the other sciences are still in the inductive stage, in which material is being collected with which, it is to be hoped, such foundation will ultimately be laid.
Mathematics is also a science, inasmuch as it has accumulated a large body of systematic knowledge involving and leading to the methods that it employs in its solutions. These methods are of such a peculiar nature, differing so widely from other methods, that a special course of training is requisite if any one would learn to use them, and their number and variety have become so great that a lifetime would not suffice to acquire familiarity with them all. But new problems are continually arising and demanding new methods, and we need, therefore, a body of men who shall devote themselves especially to the task of supplying this demand. While the colleges are engaged in general liberal education, teaching a variety of subjects that develop the mental faculties (and no subject is more efficient than mathematics for this purpose) and make the student acquainted with his own tastes and powers, thus enabling him to determine the lifework for which he is best fitted, it is the special function of the university to extend the limits of human knowledge, and to train those who have unusual intellectual talents to employ them to the best advantage. We believe this object is best accomplished by an institution devoted solely to it, and whose teachers' energies are not diverted by the lower, though no less important, aims of the college.
When the policy that should characterise this University was under discussion, the first point decided was t hat its work should be strictly post-graduate, and that it should not compete with other institutions in the work that is generally recognised as undergraduate. In accordance with this principle, the mathematical department fixed its standard of admission so as to require such a knowledge of mathematics as can be obtained in the average American college, and laid out upon this foundation a curriculum of its own, as extensive and as thorough as circumstances allowed. In elaborating the details of t his curriculum, we have kept in mind the fact that those who pursue post-graduate studies in pure mathematics almost always look forward to careers as professors in colleges or other higher institutions of learning; and we have taken the view that, other things being equal, the ideal teacher is a master of his subject, not only conversant with the general principles of all its more important branches, the problems that have arisen in each, the methods that have been devised for the solution of these problems, and the results that have been obtained, but also unbiased, ready and sound in judgment, and actively engaged in scientific research. We believe that the training that is best adapted to produce efficient specialists is also the training that is best adapted to produce efficient teachers of specialties.
While desirous of supplying all possible facilities to those who wish to pursue studies in special branches, and to those who, already occupying permanent positions, have but a limited leave of absence, we have made it our chief object to provide a thorough training for those who, having just completed a college course, have not yet entered upon their life-work. This provision consists of such courses of lectures, seminaries, and individual assistance as should enable a faithful student endowed with the proper natural ability to satisfy the requirements for the degree of Doctor of Philosophy at the end of his third year with us. The requirements for this degree have been determined by our conception of the ideal teacher, as already stated. To acquire the necessary breadth of knowledge of mathematics as a whole, the candidate is expected to attend, during his first two years, specified courses of lectures on the general principles, methods, and results of all the more important branches of pure mathematics, to supplement these lectures by private reading, and to take an active part in the seminary.
In the seminary, a special topic, more or less directly connected with the subject of some lecture, is assigned, from time to time, to each student, who is required to read it up and make an oral report upon it before the class. Advanced courses of lectures on special subjects that vary from year to year are also given, and each candidate for the degree is expected to attend a number of such courses. The student spends the greater part of his third year in the original investigation, under the constant personal guidance of one of the instructors, of a topic of his own selection. In preparing for this investigation, he is required to make a practically complete bibliography of the subject, and to read all the more important available articles that have been written on it. The results of the investigation, embodied in a dissertation suitable for printing, must be submitted to the instructor under whose direction the work was done, and must receive his approval before the candidate will be admitted to the final examination for the degree. This approval will not be given unless the dissertation is satisfactory in form and completeness and the results are sufficiently novel and important to constitute a real contribution to science. The dissertation is, in fact, the main criterion by which the candidate is judged, and no amount of other work will compensate for its defects. The ability of our graduates to carry on research and the excellence of the work actually done is assured by the regulation that each dissertation accepted by us as worthy of the degree shall be printed with the explicit approval of a member of our Faculty. It is evident that, whereas anyone that has the necessary preparation and taste for mathematics may profit by the advantages here afforded, only those who have a certain amount of mathematical genius can secure the degree.
In making appointments to fellowships and scholarships we have endeavoured to maintain the same high standard. We are on the lookout for mathematical geniuses; but it is difficult to determine from the evidence of others whether candidates come up to our standard or not; so that we have adopted the general policy of giving the best appointments to those only that have been with us for at least one year, and about whom we are in position to judge for ourselves. Of course, this policy could not be carried out during the earlier years of the University, and its effect is apparent in the fact that, whereas seventy-five per cent of the students that entered the mathematical department during the first three years remained with us but one year, only twenty per cent of those that have been admitted during the last seven years left at the end of their first year. I do not mean to imply that those who left before completing our course were inferior in ability to those who remained three years, but we desire particularly to encourage men who can and will go forward to the degree.
Nearly all of those who have studied mathematics with us have adopted teaching as a profession, two-thirds are now members of college faculties, and one-third are engaged in higher school work. Those who have received the doctor's degree have generally secured at once desirable positions in which to begin their life-work, and most of them have already acquired for themselves, by distinguished ability, very decided influence in the institutions with which they are connected. Of those who have left without the degree fully one-half ought to have continued, and would have done so but for want of pecuniary means; and we have been obliged to turn away many men of very great promise on account of our inability to assist them in providing the means of subsistence during the unproductive period of student life. We could employ for fellowships, with decided advantage, ten times the amount now at our disposal.
Although, as I believe, students will find here a broader post-graduate curriculum in mathematics and greater personal attention from the instructors than at any other university in the country, we need greater facilities to make our course what it ought to be. Four-fifths of the instruction in the department is now given by two men, and we are compelled to give in alternate years lectures on fundamental subjects that ought to be given every year. As I have said, we lay great stress upon the ability of our students to investigate; but this faculty can be fully developed only under the personal guidance of one who is himself in the habit of investigating and who has the facilities and opportunities necessary for such work. A teacher's usefulness is greatly increased by the inspiration that comes from a personal identification with his subject, from the fact that he has ideas of his own about it, and that he has extended it by his individual exertions; and the investigator can have no greater incentive to search for new results than the opportunity to present his thoughts and discoveries to an intelligent and appreciative class in the lecture-room. But the necessity of teaching many subjects simultaneously distracts the mind and is fatal to research.
The ideal conditions for an instructor in an institution like this would be those under which he could teach one subject at a time, and that a subject that he was himself developing, and follow this subject with his class to such no point as to bring into evidence the scope and importance of his own work. To apply this method to the courses that are actually given here would require the services of three additional instructors in mathematics. We are actually labouring under the disadvantage that some of the important branches now taught by us are not of such paramount interest to any one of our instructors as to be the subject of his personal investigation. We are compelled to restrict ourselves to elementary courses in many branches that ought to be carried to a much higher point, and to omit altogether from our consideration applications of mathematics to statistics, to the arts, and to other sciences. Applications to physics receive the attention of the physical department, to be sure, but the mathematical department ought to do much more than it is at present able to do in preparing students for higher work in physics. The number of instructors necessary for such advanced work as we do is not to be determined by the number of our students, but by the number of subjects taught.
Again, every expert investigator finds himself continually obliged to spend much time in details that could just as well be worked out by a younger man, to whom such work would be of immense advantage, not only as an exercise in the practical application of methods, but also as furnishing the opportunity for a prolonged study of the workings of an investigator's mind; and example is worth more than precept in the development of the faculty of investigation. We ought to have the means of retaining our best graduates for a year or two as personal assistants to the instructors, during which period they might also be gaining experience in the class-room by teaching a few hours a week under the supervision of one of the regular instructors. Such work is not drudgery, and would be, I think, sufficiently attractive to an ambitious young man to induce him to remain with us on a moderate stipend while he is waiting for such appointment as may seem to him desirable.
It is almost universally assumed that a mathematician needs no material equipment other than brains, with, possibly, a few books. However true this assumption may have been some decades ago, - and I fancy that its truth then rested solely upon the difficulty of procuring such equipment, - it is not true now, as must be apparent to anyone who studied carefully the German educational exhibit at the World's Fair in Chicago. Ten years ago our department started out with a fair nucleus for a mathematical library and a moderate collection of models, to which we have not been able to make many additions. We have very few of the older mathematical works that illustrate the history of the subject, and we need particularly complete sets of many important mathematical journals and the transactions of learned societies. In these journals and transactions have appeared most of the original investigations to which, as investigators ourselves, we have continual occasion to refer, both for suggestions and to avoid apparent plagiarism and the unnecessary duplication of research. We should also be greatly assisted in our class-work by a more complete collection of models.
In short, what I have in mind as a model mathematical department for post-graduate work would have, say, four professors and assistant professors, each having his personal assistant, and at least two instructors of lower grade for the more elementary work, and would be provided with a complete mathematical library and with all the apparatus that it is now possible to procure, with suitable provision for the purchase of new books and apparatus as they appear in the market.
These schemes are not incapable of realisation, although, perhaps, opposed to the traditions of education in this country. This University has never had any traditions excepting such as were based upon high ideals. Its mathematical department was not modelled after that of any other institution, but was determined by the conception of what would constitute perfection in such a department. We have always lived up to our ideals, in so far as we have done anything, without regard to considerations of material interest. We are not here to do what is done elsewhere, and we do not acknowledge that it would be best for us to do what other institutions, in their experience, have thought wisest. We propose to adopt no temporary policy that we shall sometime want to abandon, confident that the ideal university of the future will be ideal from the very root and not a graft upon inferior stock .
When the doors of the University were first opened to students, in the fall of 1889, the mathematical staff consisted of William E Story, Professor, Oskar Bolza, Associate, and Henry Taber, Docent; a year later it was increased by the appointment of Joseph de Perott, Docent, and Henry S White, Assistant; and in 1892 Drs Bolza and White resigned their positions to accept Associate Professorships in the University of Chicago and Northwestern University, respectively, and Dr Tuber was promoted to an Assistant Professorship, thus leaving the department with practically the same teaching force as it had during the first year.
The instruction has been given by lectures, seminaries, and individual conferences. The number of lectures (of fifty minutes each) was sixteen a week the first year, nineteen and twenty a week in the second and third years, respectively, and about fourteen a week, on the average, each year since. In some years courses of lectures on certain mathematical subjects having important physical applications have been given by Assistant Professor Webster of the Department of Physics.
JOC/EFR May 2019
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