George Abram Miller

“ 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?" ”

“ 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 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”