Tietze: Famous Problems of Mathematics
Below we give an extract from the Preface to the volume:
Below we give an extract from the Preface to the volume:
Famous Problems of Mathematics
One sometimes encounters quite curious conceptions as to what studying really means. If only everything is made quite palatable to the student, then, in the opinion of some, it ought to be an easy matter to advance to a mastery of the material, child's play to attain the loftiest heights of knowledge. It is thought to be an error if toil and effort are required, on the way to the heights and that therefore, through the fault of the teacher, the goal is reached by only a few, and only slowly and gradually even by these. Of course everyone realizes that in, say, skiing or horseback riding it is not sufficient to listen to instructions on the motions and carriage of the body, in order to handle oneself properly from the beginning on the skis or in the saddle; rather, many a drop of sweat and an occasional tumble must precede the assurance sought for.
Why should it not be similar in the intellectual field, in the acquisition of a body of knowledge: that genuine success is accorded only to serious, persistent effort along with occasional correction of false conceptions? And that even the best teacher can indeed guide the student and help him to avoid unnecessary obstacles on the way to acquiring skills and knowledge, but neither can nor should spare him persistent work ' ; and that in addition, where talent is lacking, the hardest labour of love must be content with modest results. In this respect the serious study of mathematics is hardly an exception.
When a learned man of antiquity was called before his king to explain mathematics to him, and the king, as soon as things became more difficult, asked whether there was not a simpler way of learning it, the scholar replied, "There is no royal road to mathematics." Whoever wants to climb the heights of this science simply has to work his way to the peak with his own power, gradually exercised and steeled. Not to mention those who are making their first ascent and wish to master a problem which is still unsolved and has resisted all previous efforts. One who has merely looked out from the compartment window of an alpine railway or of a comfortable alpine hotel, will not thereby become a proficient mountaineer. A mathematical work that attempted to glide over all the difficulties of the subject matter would be completely unfit for training a reader in mathematical thinking and giving insight into this special field.
But we would not be willing to dispense with mountain railways either unless there were too many of them. We are not speaking here of those people who use them and take them for granted, scarcely reflecting that the entire business of constructing a railway could be done only by engineers and workmen of high competence in the building of mountain railways; of those who, from the hotel terrace, esteem the alpine scenery as no more than a somewhat altered background for the unaltered horizon of interests they have brought up with them from the lowlands and who are incapable of any inward elevation by the world of the mountains. For many, however, the heart does beat higher, and even though they themselves may not feel the strength to climb mountains, they nevertheless convey to others their love for the mountain world. And some little boy who, has been brought along by his parents, receives an ineradicable impression, and will perhaps some day become an inspired and eminent alpinist.
This book seeks to resemble such a convenient alpine railway up into the world of mathematics. It can give nothing to the man in whom no chord vibrates responsively to the peculiar harmonies of this realm of thought, who accepts with inward indifference the technological and scientific acquisitions which he uses every day of his life, and which were made possible only by the fruitful contributions of mathematics. But we venture to hope that there will also be readers who will not remain entirely untouched by the aura which breathes in this realm that affects most people as being so unapproachable. It is possible, if fate so wills it, that there may even be some young man with unspent energies and with the necessary ability, who will feel the urge to pursue the subject more seriously, spurred by the distant vision offered to him here. For I confess that what is offered here will not be able to satisfy him, nor should it. From the railway route along which we here guide the reader are excluded, because of their difficulty of access, many of the most important mountain heights and lines of advance.
If then our book takes up the task, first of all of giving to outsiders a more accurate and perhaps a more appealing image of the nature of mathematics than one commonly finds; and if the notes offered at the end, part of which are addressed to readers with more advanced knowledge, are intended to supply supplementary material and a bibliography for more detailed study of the subject; at the same time we have had, in publishing this material, an additional goal before us which is of utmost seriousness.
It has to do with the recruitment of successors in our field.
As late as the early 1930's our lecture halls were still overfilled, and even the best students found entry into the profession obstructed for years ahead by the many, all too many graduates of earlier classes. If institutions of higher learning almost took pride in outdoing their own and each other's records on numbers of students enrolled, that was an unhealthy phenomenon (and a difficult one to resist, for reasons which are not under discussion here), in which quality of teaching results gave way to quantity. Nevertheless, in a course with 200 students you might easily count on finding twenty competent students, some of them eminently talented. To be sure, the overflow was afterwards brought under control.
When the same course was offered again later, there were only twenty students instead of 200, and if the consequent selection had brought into the lecture hall the best students of their class-year, one might expect a considerable improvement in the quality of their work. This did not occur. And although the students actually worked harder than earlier classes had done, the average achievement sank instead of rising. First-rate talents which might some day show capacity for original research seemed to be extinct. Instead of being able to cover the material faster because of the elimination of the 180 average and weak students, the teacher had to slow down his tempo from year to year if he wished to maintain contact with his students at all. This could not help but give one pause.
This brings us back to the introductory question: how an extensive, far-reaching field of study is to be mastered. An essential requisite for successful study at the university level is the thorough, firmly fixed propaedeuties acquired in the secondary school-received and inwardly digested with a certain love by those who are talented in the field. Even more important is that the student should have already learned what study at an intellectual level really is. Whoever comes to the university without this must be a failure, like the aspirant Hieronymus Jobs in C A Kortum's satirical epic, which inspired Wilhelm Busch to retell it in illustrated verses.
Jerome is to learn how to study,In a profession like ours the vocation must have been awakened by teachers who have themselves experienced it. Much of the aversion to mathematics comes from instruction given by a teacher who himself was not in complete command of his subject, and was not filled by that warmth for the inner beauty of his specialty which would enable him to communicate it to others. Everyone will concede that a fondness and natural gift for a specific field of knowledge are not always conjoined with pedagogical talent. At the same time, the competence first mentioned must be the driving force for the good teacher of the future in his choice of a profession.
He is to go to the university at Easter.
There are of course cases where it cannot be predicted who will become a competent or pre-eminent research man, even after he has completed the usual university studies, let alone before then. A rapid and thorough advance in the acquisition of professional knowledge is of itself no guarantee of a productive vein. We have seen enough of premature over-estimation on the part of students of their own potential, and enough of falsely optimistic judgments of students by teachers. On the other hand we are also familiar with the tormenting doubts suffered in their student years as to the scope of their own abilities by men who later became distinguished men of research. Only the further development of the individual can be decisive, and it would be quite meaningless to decree in advance who should be trained as a teacher and who should become a research scholar. If we were asked for an opinion we could only recommend that the goal not be set too high, particularly since that permanent economic independence required for the life of the "private scholar" is granted only to the few. It has thus come about that by and large the men who later became our research scholars and our successors in university chairs, have come up mostly from among the men who had prepared themselves first of all to be secondary school teachers.
If one knows that research is necessary, one recognizes the seriousness of the question as to who is to come after us. Certainly the top talents are not rarer than they used to be. It is only necessary to make sure that they find their true assignment. An understanding of the nature of our profession disseminated among wider circles might lay the groundwork for this. This is the aim in publishing this book, which is based on a series of lectures delivered for students of all faculties at the University of Munich.
We are conscious that in doing so we have ventured upon no very easy task. It is like the mutual understanding - or lack of understanding - between age and youth. Many a great young poet or actor succeeds in portraying the feeling or thought of old age, but many older people no longer succeed in transplanting themselves back into that spirit of youth which they themselves have after all once experienced. Thus it happens that a talented disciple of mathematics is able to see beyond the knowledge which his teacher is imparting to him at the moment. But the teacher whose material has become ever more familiar to him in the course of the years - and often, too, the young instructor who has just completed his own advanced professional training - find it difficult to think themselves into the position of the beginner and to realize what could cause him difficulties. At the same time, individual students vary greatly as to what they find difficult, and only after much experience does a teacher become aware of how many misunderstandings can arise. In lectures given before an audience less familiar with the subject, the opportunity for such misunderstandings increases with every word which is current in the specialty but which is used in a different sense from the ordinary one; it increases also with every fact which has become self-apparent to the specialist and which indeed may also be known to the auditor, but which does not occur immediately to the latter.
To counteract this danger I prepared quite a large number of illustrations and tables. In geometric constructions in particular the sketches were designed to appear step by step in their intermediate stages, not merely in their final form, overburdened by all the lines which were added one after the other.
[Many works on mathematical amusements and games have been written for a wider circle of readers, but by their nature these touch only exceptionally upon questions of significance for the main stream of science. As for the older writings on "mathematica delectans," they mostly contain only a collection of dressed-up exercises in equation-solving, on the following general pattern: The boy Henry comes from school and says to his father: "Father, you are now exactly six times as old as I am. When will you be 25 ,times as old?" Father: "That was when you were still small, but in 1 3/4 years I will be only five times as old as you." How old are the two? When was the father 25 times as old as his son?]
The last chapter (Chapter 13) on the curvature of space, lies somewhat outside the framework of the easily understood to which the other questions are confined. It is not so much special preliminary knowledge that is required, as an emancipation from firmly rooted ideas, from ideas, moreover, which an outsider to mathematics usually has not pursued critically enough. If I nevertheless decided to retain this chapter also, it was not from any dread of the number 13, which I have always found ridiculous, but because I simply did not want to evade a subject which was formerly passionately discussed, and about which a great deal has been written, some of it correct, some false, some tenable, some off the track.
Those who have heard me lecture are familiar with my custom of interpolating, on occasion, references to matters which are close to us in time or place, references which serve not only for the enlivenment or illustration of the specialized topics, but are also intended to throw light on general questions of education, the cultivation of science, and culture in general. The elaboration of these lectures in book form, a work which extended over the last war years to about a year after the end of the war, is likewise not free of such interpolations. The proverb, "Out of the abundance of the heart the mouth speaketh," did not of course always enjoy unrestricted validity, and at times intimations had to suffice - and did suffice.
Local colouring and the character of the time have in the meanwhile greatly changed, and from the first outlines for these lectures to the completion of the manuscript and proof-reading, a decade and a half of world-historic events and catastrophes have rolled by, accompanied by many professional and personal experiences and sufferings. Buildings then standing have collapsed or been cleared away. Young students of that time have become mature men - or never returned home. If allusions had been made here and there in the text to conditions which once were the order of the day, much of this now belongs to the past. In the discussion of the disputes over the priority of discovery of the solution of third-degree equations, a reference to the difference between objective and brachial methods of conflict had a certain timeliness in view of the battles-royal that were then frequently "organized" at political meetings. Or when, in composing the section on Gauss, the war-torn era was described in which the first years of Gauss' activity at Göttingen fell, we were again in the midst of an epoch in which the slogan of creating a "new Europe" by military means was being advanced.
If a work by Albert Einstein can now be quoted without further ado, at that time the book's chances of appearing were endangered by mere mention in it of the theory of relativity, which had the effect of a red flag on many in those days when, blinded by the universal propaganda, some scientists really believed that this theory was pure humbug and when, worse still, one of our greatest theoretical physicists was replaced by a man, foisted upon us from on high, who tossed Planck's quantum theory into the same kettle of damnation with the theory of relativity, so that our theoretical physicists had great difficulty in interceding on behalf of continued use of the theory of relativity in research and in teaching.
If, later on, in the summer of 1945 and the winter of 1945-46, work on this book would probably have come to a halt without medical intervention against symptoms of starvation, then a brief remark alludes to my urgent anticipation, now recurring each year, of the harvest. And if (my) faith that through wise institutions the good in man could be led to prevail, was exposed before and afterward to strong doubts, these too found their precipitate in a few scattered remarks on this problem of squaring the human circle.
The conditions reflected in these interpolations were widely separated from each other in time, and may therefore seem to some readers rather dusty, like yellowed old photographs of strangers who no longer interest the living and could be sorted out of the album. The question was whether a revision of the entire work should be undertaken to remove all non-essentials and restrict the content to factual and professional matters. Some who are personally close to the author have recommended retention of the changing snapshots. It will be found understandable that I have followed these recommendations.
The lectures regularly contain a few biographical data on individual mathematicians who have a special relationship with the problem under discussion, or who discovered its solution. I am indebted to colleagues abroad for a quantity of information. More recent historical studies may have made some biographical details out of date. It could not be my task to pursue all that in detail since my project was only the more modest one of giving to the reader an approximate picture of the research scholars and of their times.
Munich, Summer 1959.
JOC/EFR April 2007
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