Popular Science Monthly/Volume 75/August 1909/The Future of Mathematics

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By Professor G. A. MILLER


PROFESSOR A. VOSS, of the University of Munich, recently made the following statement: "Our entire present civilization, as far as it depends upon the intellectual penetration and utilization of nature, has its real foundation in the mathematical sciences."[1] He adds that this truth finds expression in the ever-increasing appreciation of the educational value of mathematics, notwithstanding the fact that it is the most unpopular of all the sciences. This unpopularity is natural since "unpopularity is an essential feature of a real science," because such a science can be comprehended only through tireless and continued efforts.

An intelligent expression as regards the future of mathematics must be based not only upon the past and present state of this science, but also upon its real essence. One of those elements which mathematics has in common with some of the other sciences, but which are more prominent in mathematics than in any of the others, is the tendency to use thought in the most economical manner. When one considers the extent to which efforts to simplify methods, theorems and formulas direct mathematical endeavor, one must admit that the statement "Mathematics is the science of saving thought" expresses a great truth, even if it is too sweeping to serve as a definition.

That mathematics is the science which is preeminently devoted to the discovery and mapping of routes along which thought may ascend securely and with the greatest ease, is supported by the fact that it has the oldest and the most extensive symbolical language. In the introduction to his classic history of mathematics, Moritz Cantor asks, "Why has mathematics, since the remotest times, found support, simplification and advancement by means of word symbols, whether these are number symbols or other mathematical symbols?" Although the oldest of these word symbols are probably relics of a very ancient picture language, yet it is of great interest that in mathematics the picture language was retained and used side by side with an alphabetic and syllabic language, while the latter displaced the former elsewhere. Even those who have mastered only the elements of algebra and the differential calculus are in position to appreciate the value of mathematical symbols for the purpose of centralizing and intensifying thought.

It is true that some of the roads which mathematical thought has made through great difficulties have been practically abandoned and that the popularity of many of the others has changed from time to time. Among the former we may class the results of investigations recorded at the beginning of the oldest extensive mathematical work that has been deciphered, viz., the formulas relating to unit fractions which are found in the nearly four thousand-years-old work of Ahmes. A subject which appears to have been placed at the very beginning of advanced mathematical instruction four thousand years ago is now entirely abandoned in our courses, except when the history of the development of the science is under consideration. While mathematics presents a number of other roads which are now of interest only to the historian, yet there are also many which have been known for centuries and which have been pursued with profit and pleasure by great minds in all the civilized nations. The latter class includes all the longer ones leading gradually to points of view from which the connection between many natural phenomena may be clearly discerned.

The intellectual heights reached by means of a long series of connected mathematical theorems do not always reveal their greatest lesson to the first explorers. For instance, the large body of facts relating to conic sections, developed by Apollonius and other Greek geometers, became a much greater glory to the human mind through the discovery, nearly two thousand years later, that the bodies of the solar system describe conic sections. Such experiences in the past tend to justify the fact that a large number of men are devoting their lives to the discovery of abstract results irrespective of applications, and they tend to explain why the largest prize (about twenty-five thousand dollars) ever offered for a mathematical theorem is being offered for a theorem in number theory, which is not expected to have any application to subjects outside of pure mathematics.

There seems to be a general impression abroad to the effect that mathematics and the ancient languages constituted the main parts of the curriculum of our colleges and universities a century or two ago. As regards mathematics this is quite contrary to fact, as may be seen from a few historical data. Less than two centuries ago the students in Harvard College began the study of arithmetic in their senior year. In fact, no knowledge of any mathematics was required to enter Harvard before 1803, and it was not until 1816 that the whole of arithmetic was required for entrance. In other American institutions the mathematical situation was generally worse, and in Europe the improvements were not very much earlier. It is during comparatively recent years that mathematics has made most of its gains towards being recognized as a fundamental science, and the study of advanced mathematics in our universities had a still later origin.

The rapid recent advances in various fields of mathematics have given rise to a very optimistic spirit as to the future. Although we still hold in high esteem the brilliant discoveries of the Greeks, we are inclined to give much more thought and attention to recent work, as may be seen from the references in the extensive German and French mathematical encyclopedias which are in the process of being published. The history of mathematics furnishes many instances of the vanishing of apparently insurmountable barriers. We need only recall the barrier created by the Greek custom of confining oneself to the rule and circle in the most acceptable geometric constructions, and the very formidable barrier furnished by the imaginary, and even by the negative and the irrational roots of a quadratic equation.

Those who fixed their attention upon these barriers in the past have naturally been led to think that the days of important advances in mathematics were about ended and that it only remained to fill in details. Such predictions had few supporters when new methods led over these barrier and turned them into steps to richer mathematical domains. As this process has been repeated so often it has gradually reduced the number of those to whom the future of mathematics looked dark. In fact, Poincaré, in his address[2] before the Fourth International Congress of Mathematicians, which was held at Rome, in April, 1908, said that all those who held these views are dead.

These facts seem to justify a very hopeful spirit as regards future progress, but it is necessary to examine them with great care in order to deduce from them any helpful suggestions as to the probable nature of this progress. Such prognostications clearly demand a mind that can deal with big problems as well as a thorough acquaintance with the past and the present developments in mathematics, to insure that the results obtained by a kind of extrapolation may be worthy of confidence. It is doubtful whether any living mathematician would be more generally regarded as qualified to make reliable predictions along this line than Poincaré, of Paris. The address to which we referred in the preceding paragraph was devoted to this subject and we proceed to give some of the main results.

The objects of mathematical thought are so numerous that we cannot expect to exhaust them. This appears the more evident since the mathematician creates new concepts from the elements which are presented to him by nature. Hence there must be a choice of subject matter, but who is to do the choosing? Some are inclined to think that the mathematician should confine himself to those problems which may be set for him by the physicist or the engineer. If he had done this in the past he would not have created the instruments necessary to solve such problems, and hence it is unreasonable to make such restrictions as to the future.

If the physicists of the eighteenth century had abandoned the study of electricity because it seemed to serve no useful end, we should not have had the many useful applications of electricity during the nineteenth century. Similarly, if the mathematician had abandoned the study of negative and imaginary numbers because they seemed to point only to impossibilities, we should not have had the many powerful instruments of thought which enable us to cope more successfully with many problems of nature. Just as the physicist is largely guided in his work by those facts which seem to point to general laws, so the mathematician is guided in his work by the desire to discover extensive relations and laws having a wide range of application. Millions of isolated facts present themselves to the investigator, some of which are of striking interest to the initiated, but they are of practically no value in the development of mathematics except that they may sometimes serve as an exercise in secondary instruction.

At a first thought the statement that "Mathematics is the art of giving the same name to different things" may appear to be entirely contrary to fact, but from a certain standpoint this statement conveys a very fundamental truth. It should be borne in mind that these different things must have in common the property to which this common name refers, and that it is the duty of the mathematician to discover and exhibit this common property. By way of illustration we may recall the use of x for various unknowns in algebra and the (1,1) correspondence between the two series of operators. When the language has been properly chosen it is often surprising to find that the demonstrations, as regards a known object, apply immediately to a large number of new objects without even a change of name.

Just as the boundaries between the elementary subjects of mathematics—arithmetic, algebra and geometry—vanish when the knowledge of these subjects is sufficiently extended, so the boundaries between subjects in pure and applied mathematics are disappearing, and it is exactly in these bordered lands, or in this common territory of two or more subjects, where the greatest recent progress has been made and where the greatest future activity may be expected. The work in this common territory is made possible by observing similarity of form where there is dissimilarity of matter, or by observing some other common properties which admit mathematical treatment.

In Poincaré's address some of these general observations were illustrated by numerous examples chosen from various fields of higher mathematics. On the contrary, we shall confine our illustrative examples to elementary subjects. Our first effort will be directed towards exhibiting some territory which is common to each of the four subjects—arithemtic, geometry, algebra and trigonometry. By observing common properties we shall not only see a bond connecting these fund a* mental subjects, but we shall also be led to general methods which make it unnecessary to study the same properties in different forms. The thing to be emphasized is that these four elementary subjects have in common fundamental notions which not only connect them, but also establish contact between them and many other subjects. Such a fundamental notion is a group of order 8, known as the octic group. Some of the properties of this group may be easily seen by considering the possible movements of space which transform a square into itself.

The period or order of a movement represents the number of times the movement must be made in order to arrive at the identity, or at the original position. It is clear that the eight movements of the square include two of period four, five of period two, and the identity A profound study of these eight movements would disclose many interesting facts. For instance, it would be seen that only two of them (the square of these of period four and the identity) are commutative with each one of others, while each one of the remaining six is commutative with only four of the possible eight movements. Although a profound study of this group of eight movements would be necessary to exhibit the fundamental role which it plays in the various subjects, it is not necessary to enter deeply into its properties in order to see that it is common to the four subjects mentioned above.

At a first thought it might appear as if these eight movements had nothing in common with trigonometry, but a very fundamental connection may be seen as follows: If the vertex of the angle A is the center of a square and the initial line of A coincides with a line of symmetry of the square, the operations of taking the complement and the supplement of A correspond to movements transforming the square into itself. Hence the eight angles which may be obtained from a given angle by a repetition of finding supplement and complement may be placed in a (1, 1) correspondence with the eight movements of the square. As these eight angles play such a fundamental role in elementary trigonometry, it has been suggested that our ordinary school trigonometry might appropriately be called the trigonometry of the octic group, or the trigonometry of the group of movements of the square.

Although the eight operations of the octic group do not occupy such an important place in elementary arithmetic as in geometry and trigonometry, yet these operations serve to explain some facts which present themselves in the most elementary arithmetic processes. For instance, the operations of subtracting from 2 and dividing 2 lead, in general, to eight distinct numbers. Starting with 5, these eight numbers are

5,—3, 25,—23, 85, 83, 54, 34.

No new number is obtained by dividing 2 by any of these numbers or by subtracting any of them from 2. The proof of the fact that the eight operations by means of which each one of these eight numbers may be derived from any one of them have the same properties in relation to each other as the eight movements of the square is not difficult, but it involves details which may be omitted in a popular exposition.

An instance where the octic group plays an important role in algebra is furnished by the three-valued function xy + zw, which is fundamental in the theory of the general equation of the fourth degree. On account of the existence of this function the solution of the general equation of the fourth degree may be made to depend upon the solution of the general equation of the third degree. This function is transformed into itself by eight substitutions, and we may arrange its letters separately on the vertices of a square in such a way that the eight substitutions transforming the function into itself correspond to the eight movements which transform the square into itself. Such an arrangement exhibits the intimate relations between this function and the movements of a square, and the preceding examples illustrate the fact that the octic group finds application in each of the elementary subjects—arithmetic, algebra, geometry and trigonometry, and that it forms a part of the domain common to all of these disciplines.

In a similar manner other groups could be traced through these elementary subjects of mathematics and it could be shown that the theory of these groups may be used to clarify many fundamental points and to exhibit deep-seated contact. If the common domains will furnish the most active fields of future investigations in accord with the predictions of Poincaré, and if we may expect the greatest future progress to be based upon the modeling of the less advanced science upon the one which has made the more progress, it is reasonable to expect that a subject like group theory will grow in favor, and that some of the elements of this subject will become a part of the ordinary courses in secondary mathematics. In support of this view we may quote a recent statement by Professor Bryan, President of the Mathematical Association, which is as follows: "I believe Professor Perry will get some very good material for applications out of the theory of groups, when explorers have first made their discoveries, and when the colonists have been over it and surveyed it, and discovered means for cultivating it. We do not know anything about its practical applications now."[3]

The future of mathematics appears bright, both for the investigator and for the teacher. When a country which has such an enlightened educational system as France increased the amount of time devoted to secondary mathematics so recently as 1902 and again in 1906, it furnishes one of the strongest possible encouragements to the teacher who may have been troubled by the thought that the educational value of mathematics was not being as fully appreciated as in earlier years. Naturally we may expect that there will be local changes of view as regards the value of mathematics as an educational subject, and these changes will not always be for the better, but the civilized world, as a whole, is learning to appreciate more and more the fundamental importance of early mathematical training, so that we should not be too much perturbed by local steps backwards, but we should move ahead with the assurance that we are engaged in a work of the highest pedagogical importance.

The boundless confidence in the importance of early and extensive mathematical training should, however, not blind us to the need of changes and new adaptations. As an important function of mathematical training is the furnishing of the most useful and the most powerful tools of thought, it is evident that the choice of these tools will vary with the advancement of general knowledge. All admit that the concept of a derivative is one of the most useful elementary tools of thought, and in a number of countries this concept has been introduced into secondary mathematics and used with success. At the last International Mathematical Congress, held at Rome, M. Borel, of Paris, reported that the notion of derivative had been introduced into French secondary education in 1902 and that it had led to satisfactory results. At the same meeting M. Beke, of Budapest, stated that this notion, together with the notion of function and graph, had been introduced into the courses of secondary education in Hungary.

At the recent joint conference of the Mathematical Association and the Federated Association of London Non-Primary Teachers, the chairman remarked: "I have always thought that a mathematician was a man who when he wants to find anything out, uses his brains for that purpose, whereas a physicist, when he wants to find out anything, resorts to experiment." Although this statement is not to be construed literally, yet it does involve a great partial truth and it calls attention to elements which insure mathematical appreciation as long as there is scientific thought. "It is the mind that sees as well as the eye," and the mind sees some of the greatest truths most clearly by means of mathematical symbolism. In fact, mathematical symbols serve both as a telescope and also as a microscope for mental vision, and as long as such vision is demanded the teacher of mathematics will be appreciated.

  1. Voss, "Ueber das Wesen der Mathematik." Rede gehalten om 11. Marz, 1008, in der oeffentlicben Sitzung der k. bayerischen Akademie der Wissenscnaften; Teubner, 1908, p. 4.
  2. Bulletin des Sciences Mathématiques, Vol. 32 (1908), p. 168.
  3. The Mathematical Gazette, January, 1909, p. 17.