Mathematics

“ The true science of mathematics is the most imaginative and creative of all sciences, but the mere application of mathematics to figures for the construction of engines, ships, or buildings is the dullest sort of drudgery. ”

“ Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. ”

“ How far every boy ought to be made to do mathematics (beyond simple arithmetic) I cannot say. I know that I am extremely grateful for the small amount of mathematics forced into me. I am even thankful for a very mechanical side of the subject, namely, the use of mathematical tables in general, and for being compelled to work out innumerable sums by logarithms, which we had to do in a “neat tabular form” to quote our precise master’s words. ”

“ 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. ”

“ Mathematics is based upon nothing, and, consequently, arises out of nothing.
"Out of nothing, therefore, it is possible for something to arise; for mathematics, consisting of propositions, is something, in relation to 0." ”

“ I dispute the availability, and thus the value, of that reason which is cultivated in any special form other than the abstractly logical. I dispute, in particular, the reason educed by mathematical study. The mathematics are the science of form and quantity; mathematical reasoning is merely logic applied to observation upon form and quantity. The great error lies in supposing that even the truths of what is called pure algebra are abstract or general truths. ”

“ 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. ”

“ Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning. ”

“ The difference between philosophy and mathematics may be illustrated by our present problem, namely, the nature of number. Both start from certain facts about numbers which are evident to inspection. But mathematics uses these facts to deduce more and more complicated theorems, while philosophy seeks, by analysis, to go behind these facts to others, simpler, more fundamental, and inherently more fitted to form the premisses of the science of arithmetic. ”

“ With regard to the peculiar case of mathematics, which is widely applicable because formal and as exact as formal, it seems enough to say that while mathematics has very properly become the ideal of all knowledge, not excluding such sciences as psychology and sociology, the final value, the peculiar applicability of mathematics, lies in its character as a general attitude or method. It is not strictly a science, but the ideal method of science. ”

“ Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect ”

“ The reality of mathematics consists in the universality of its quantities; viz. numbers or figures. Every number, and every thing which belongs to mathematics, can be derived from no other source than zero. ”

“ The definite object of our researches in concrete mathematics being the discovery of the equations which express the mathematical laws of the phenomenon under consideration, and these equations constituting the true starting point of the calculus, which has for its object to obtain from them the determination of certain quantities by means of others, I think it indispensable, before proceeding any farther, to go more deeply than has been customary into that fundamental idea of equation, the continual subject, either as end or as beginning, of all mathematical labours. ”

“ The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples. ”

“ These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate. ”

“ 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 value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines, but of its method. Mathematics will ever remain the most perfect type of the Deductive Method in general; and the applications of mathematics to the deductive branches of physics, furnish the only school in which philosophers can effectually learn the most difficult and important portion of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex. ”

“ And while the mathematical subjects of the quadrivium were taught as science and for their own sakes, the new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write. ”

“ And so the result of this analysis of the bases of the actually developed and significant branches of mathematics, constitutes a sort of empirical revelation of what categories the exact sciences have practically found to be of such significance as to be worthy of exhaustive treatment. Thus the instinctive sense for significant truth, which has all along been guiding the development of mathematics, comes at least to a clear and philosophical consciousness. ”

“ The close study of pure mathematics, by directing the mind to processes of calculation rather than to phenomena, induces that sublime indifference to facts which has characterized the purely mathematical intellect of all ages. ”

“ In mathematics, too, we should be looking in vain, at that stage where the real instruction in mathematics begins, for anything new and more easy, based on the whole previous instruction of the exercises of the second [Pg 272] year up to 20. Where in arithmetic the real difficulties are met with, where it becomes necessary to explain the subject from all its sides to the pupil, as in numeration, in addition, subtraction, division, in the division and multiplication of fractions, you will not find even a shadow of anything easier, any new explanation, but only quotations from old arithmetics. ”

“ The sciences classed as pure mathematics are Arithmetic, Geometry, Algebra, Analytical Geometry and Calculus. Arithmetic is eminently the science of numbers, and treats of, or practically illustrates their nature and uses. ”

“ Mathematics itself were nothing if it had none other than its highest principle zero. In order, therefore, that mathematics may become a real science, it must, in addition to its highest principle, subdivide into a number of details, namely, first of all into numbers, and, finally, into propositions. What is tenable in regard to mathematics must be equally so of all the sciences; they must all resemble mathematics.
34. The first act towards realization or the becoming something, is an origination of Many. All reality can, accordingly, manifest itself only in multiplicity. ”

“ 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. ”

“ 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. ”

“ Mathematics and pure physics eminently represent the larger part of these six elements which I have named. Mathematics demands concentration. Mathematics is, in a sense, the mind giving itself to certain abstract truths. What is X2 but a form of the mind? Mathematics demands clearness of thinking and of statement. Without clearness mathematics is naught. It also represents comprehensiveness. ”

“ The mathematics are either pure or mixed. To the pure mathematics are those sciences belonging which handle quantity determinate, merely severed from any axioms of natural philosophy; and these are two, geometry and arithmetic, the one handling quantity continued, and the other dissevered. Mixed hath for subject some axioms or parts of natural philosophy, and considereth quantity determined, as it is auxiliary and incident unto them. ”

“ According to this definition, then, the science of mathematics—vast and profound as it is with reason reputed to be—instead of being an immense concatenation of prolonged mental labours, which offer inexhaustible occupation to our intellectual activity, would seem to consist of a simple [Pg 19] series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure them, by the aid of operations of similar character to the superposition of lines, as practiced by the carpenter with his rule. ”

“ 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. ”

“ 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. ”

“ So the most fantastic forms of mathematics, which themselves seem to bear no relation to actual phenomena, just because the relations involved in them are the relations that have been derived from dealing with an actual world, may contribute to the solutions of problems in other forms of calculus, or even to the creation of new forms of mathematics. ”