Higher mathematics

“ For instance, let us look at a course of modern mathematics, as represented in the books taken in for university honours. A student living in Queen Elizabeth’s time would have had no infinitesimal calculus to study, hardly even algebraic geometry, for what is now called the higher mathematics was invented since then. Going back into the Middle Ages, we come to the time when algebra had been just brought in, a novelty due to the Hindu mathematicians and their scholars, the Arabs ”

“ The extreme imperfection of algebra, with respect to the resolution of equations, has led analysts to occupy themselves with a new class of questions, whose true character should be here noted. They have busied themselves in filling up the immense gap in the resolution of algebraic equations of the higher degrees, by what they have named the numerical resolution of equations. ”

“ And, not to pass by unanswered an objection which might be raised, I would remark that, owing to the flourishing state of mathematical instruction in our universities for a long time past, the present masters of mathematics in the higher classes of our gymnasia are, almost without exception, qualified to teach analytical geometry, and would even be glad were they authorized to teach that branch. ”

“ It is admitted that a higher branch of mathematics often throws much light on the lower branches, but these side-lights should be employed for the purpose of leading the learner onward to broader generalizations. Unless one sees the lower clearly, the higher is obscure. Build solidly the foundation on arithmetic—written and mental—and the higher branches will be more easily mastered and time saved. ”

“ Everywhere it is "capacity of adaptation to facts" that is the criterion of a branch of mathematics, except, of course, that in mathematics the facts are not always physical facts. Mathematics has successfully accomplished a generalization whereby its own methods furnish the material for higher generalizations. The imaginary number and the hyper-dimensional or non-Euclidian geometries may be absurd if measured by the standard of physical reality, but they nevertheless have something real about them in relation to certain mathematical processes on a lower level. ”

“ And they do not stop at Arithmetic; they include Algebra, Geometry, and Calculus, even; and they confidently maintain that these higher imaginative portions of mathematics are parts of a practical education. Now, leaving out some of the technical professions like engineering, how many professions can you name that use any mathematics above arithmetic? ”

“ Thirdly, the tremendous future effect of mathematical knowledge on the lives of men, on their daily avocations, on their habitual thoughts, on the organization of society, must have been even more completely shrouded from the foresight of those early thinkers. Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. ”

“ We have seen that, before we can attain to 'applied mathematics,' which constitute the basis of those sciences by which we dominate the material world, we have to begin with 'pure mathematics.' In that region of study we have to idealize and speak of things, not as they are in our experience, but as they might be if certain tendencies that we see around us could be infinitely—yes, and we must add, impossibly—extended. Yet in the end, if we go patiently onward, we find that our 'pure mathematics' lead us to conclusions of immense practical importance. ”

“ The great inventions of the seventeenth century—Analytical Geometry and the Infinitesimal Calculus—were so fruitful in new results that mathematicians had neither time nor inclination to examine their foundations. Philosophers, who should have taken up the task, had too little mathematical ability to invent the new branches of mathematics which have now been found necessary for any adequate discussion. ”

“ Surely the claim of mathematics to take a place among the liberal arts must now be admitted as fully made good. Whether we look at the advances made in modern geometry, in modern integral calculus, or in modern algebra, in each of these three a free handling of the material employed is now possible, and an almost unlimited scope is left to the regulated play of fancy. ”

“ In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science. ”

“ As one of the chief triumphs of modern mathematics consists in having discovered what mathematics really is, a few more words on this subject may not be amiss. It is common to start any branch of mathematics—for instance, Geometry—with a certain number of primitive ideas, supposed incapable of definition, and a certain number of primitive propositions or axioms, supposed incapable of proof. Now the fact is that, though there are indefinables and indemonstrables in every branch of applied mathematics, there are none in pure mathematics except such as belong to general logic. ”

“ And as there is no loss whatever in rigidity of proof, the natural consequence is surely that of two pupils who have devoted an equal period of time to mathematics during their school training, the one employing in his geometry the modern methods and the other the Euclidean, the former must be more prepared for the intelligent use of mathematical formulæ and reasoning in the university or the technical school than the latter. ”

“ But to make any progress in the sciences the study of particular problems is, of course, indispensable." In fact, a thorough understanding of its special theories is necessary to the successful treatment of the foundations of the science. Only that architect is in the position to lay a sure foundation for a structure who knows its purpose thoroughly and in detail. So we turn now to the special problems of the separate branches of mathematics and consider first arithmetic and algebra. ”

“ The prejudice lasted long, and was upheld by the misunderstood authority of Lord Bacon; until the prodigious applications of mathematics to physical science—to the discovery of the laws of external nature—slowly and tardily restored the reasoning process to the place which belongs to it as a source of real knowledge. Mathematics, pure and applied, are still the great conclusive example of what can be done by reasoning. Mathematics also habituates us to several of the principal precautions for the safety of the process. Our first studies in geometry teach us two invaluable lessons. ”

“ In the course of a comprehensive historical survey the lecturer characterized, in a way I found utterly convincing, the present mathematical interpretation of nature as a transitional stage of human consciousness - a kind of knowing which is on the way from a past pre-mathematical to a future post-mathematical form of cognition. The importance of mathematics, whether as a discipline of the human spirit or as an instrument of natural science, was not for a moment undervalued. ”

“ I shall show you that it is not learning, but a little learning which is a dangerous thing; and that if our work is thorough, the broader the culture, the profounder the piety.
For our purpose this morning, learning may be divided into four provinces: literature, history, science and philosophy, to which must be added in any complete topography of the realm, pure mathematics. By pure mathematics I mean arithmetic, algebra, geometry, logarithms, the calculus and the like. But pure mathematics is simply an instrument by which the scientific mind reaches certain results. ”

“ One of the main contentions of the agitators is that our mathematical instruction should be more concrete and inductive. This is frequently expressed by the term 'the laboratory method of teaching mathematics' and several of our leading universities have announced courses to be taught by this method. This gives evidence of a profound movement in methods of mathematical instruction, which will doubtless effect many reforms even if it can not be expected that the extreme views will find general adoption.
The teaching of elementary geometry has perhaps been most severely attacked. ”

“ But when at last they have been found, they are seen to be few in number, and to be the sole source of everything in pure mathematics. The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole ”

“ ALL objects in the world which can be made the subjects of our contemplation are subordinate to the conditions of Space, Time, and Number; and on this account, the doctrines of pure mathematics have most numerous and extensive applications in every department of our investigations of nature. And there is a peculiarity in these Ideas, which has caused the mathematical sciences to be, in all cases, the first successful efforts of the awakening speculative powers of nations at the commencement of their intellectual progress. ”

“ The aims of college mathematics can perhaps be most clearly understood by recalling the fact that mathematics constitutes a kind of intellectual shorthand and that many of the newer developments in a large number of the sciences tend toward pure mathematics. In particular, "there is a constant tendency for mathematical physics to be absorbed in pure mathematics." [8] As sciences grow, they tend to require more and more the strong methods of intellectual penetration provided by pure mathematics. ”

“ Side by side with language study is the study of mathematics in the schools, claiming the second place in importance of all studies. It has been pointed out that mathematics concerns the laws of time and space—their structural form, so to speak—and hence that it formulates the logical conditions of all matter both in rest and in motion. Be this as it may, the high position of mathematics as the science of all quantity is universally acknowledged. The elementary branch of mathematics is arithmetic, and this is studied in the primary and grammar schools from six to eight years, or even longer. ”

“ Pure mathematics first gave us this conception; applied mathematics extends it to the realm of physical nature. Applied mathematics shews us that not only the truths of abstract number and extension, but the external facts of the universe, which we apprehend by our senses, form, at least in a large part of all nature, a web similarly held together. We are able, by reasoning from a few fundamental truths, to explain and predict the phenomena of material objects: and what is still more remarkable, the fundamental truths were themselves found out by reasoning ”

“ The study of the pure mathematics will of course not be extensively pursued in an institution, which, like this, has a direct practical tendency and aim. But it is still to be remembered, that pure mathematics lie at the foundation of mechanical philosophy, and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation. ”

“ Much progress has been made toward the saner view of letting secondary mathematics build its little structure into the air with some view to harmony and proportion, and of requiring college mathematics to build on as well as upon the work done by the secondary schools. The fruitful and vivifying notions of function, derivative, and group are slowly making their way into secondary mathematics, and the graphic methods have introduced some of the charms of analytic geometry into the same field. ”

“ Behind and before all these sciences, however, lies the great science of mathematics—the most powerful instrument the mind can employ in the investigation of natural law—and the science of mathematics must be divided into abstract mathematics or the calculus, and concrete mathematics embracing general geometry and rational mechanics. We have thus really six great sciences.
Mathematics. Mathematics may be defined briefly as the indirect measurement of magnitudes and the determination of magnitudes by each other. It is the business of concrete mathematics to discover the equations of phenomena ”

“ Not only is mathematics independent of us and our thoughts, but in another sense we and the whole universe of existing things are independent of mathematics. The apprehension of this purely ideal character is indispensable, if we are to understand rightly the place of mathematics as one among the arts. It was formerly supposed that pure reason could decide, in some respects, as to the nature of the actual world: geometry, at least, was thought to deal with the space in which we live. ”

“ I have known men, who when they were at school, never could see the good of mathematics, but who, when in after life they made this discovery, not only became eminent as scientific engineers, but made considerable progress in the study of abstract mathematics. ”

“ However, granting that mathematics practically constitute an absolutely certain and objective science, how is it with the rest of the sciences? Undoubtedly the most certain among them are those "exact sciences" whose principles are to be directly proved by mathematics; thus, in the first place, a great part of physics. ”