Mathematician

“ An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us. ”

“ The concrete mathematical interest of his inquiry sustains him in his labors, and wins for him the sympathy of his fellows. To the questions, "Why consider the ideal structure of just this system of object at all?" "Why study various sorts of numbers, or the properties of functions, or of groups, or the system of points in projective geometry?"—the pure mathematician in general, cares to reply only, that the topic of his special investigation appears to him to possess sufficient mathematical interest. The freedom of his science thus justifies his enterprise. ”

“ “My dear, you are a mathematician, are you not? ”

“ 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 mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed. ”

“ 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. The question, “What is a number?” is the pre-eminent philosophic question in this subject, but it is one which the mathematician as such need not ask, provided he knows enough of the properties of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher’s question. ”

“ As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics. ”

“ It is not necessary that he should be a profound mathematician, or able on every occasion to calculate the value of active and opposing forces. Calculations have been made by much better mathematicians than he is likely to be, and besides, such results are subject to so many exceptions and modifications, arising from the imperfect nature of the materials, that experience at last is found to be the principal guide. ”

“ Through the science of mathematics, the relation between magnitudes only conceivable to the mathematician by his employment of calculative symbols, can be correctly ascertained, and a working hypothesis for practical purposes thereby obtained. ”

“ Thus the mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular. ”

“ As to the method of work used by mathematicians in such cases, where the existence of an object is in question, I again speak quite as a layman in this field; but, so far as I have observed, the mathematicians, in proving the sort of existence of which they speak, proceed very much like students of other types of real Being. To prove the existence of an object whose what is already stated, but whose that is in question, the mathematician may simply produce, as it were, before your eyes, an object of the desired type, and may then let you observe that it meets the requirement. ”

“ When a king asked Euclid, the mathematician, whether he could not explain his art to him in a more compendious manner? he was answered, that there was no royal way to geometry. ”

“ No more than these machines need the mathematician know what he does. ”

“ I maintain that two and two the mathematician would continue to make four, in spite of the whine of the amateur for three, or the cry of the critic for five. ”

“ I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest man thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all incumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run. ”

“ So intimate is the union between mathematics and physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create “for each successive class of phenomena, a new calculus or a new geometry, as the case might be, which 242 might prove not wholly inadequate to the subtlety of nature.” ”

“ The chemist smiles at the childish efforts of alchemists, but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of to-day. ”

“ It should seem that it is easier to square the circle than to get round a mathematician. ”

“ 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. ”

“ Even in those sciences which are not yet under the dominion of mathematics, and perhaps never will be, a working copy of the mathematical process has been made. This is not known to the followers of those sciences who are not themselves mathematicians and who very often exalt their horns against the mathematics in consequence. ”

“ It is not much more awkward, however, than the system pursued by the great mathematicians of antiquity, unacquainted with the brilliant invention which has given a new aspect to mathematical science, of determining the value, in a great measure, by the relative position of the figures. ”

“ It must be remembered, however, that, even among advanced and professed mathematicians, the faculty of solving problems is very unequally distributed—a fact which is openly recognized at the great mathematical University of Cambridge. ”

“ The mathematician has reached the highest rung on the ladder of human thought. ”

“ We all have a natural tendency to exalt our own special branch of study, and this is perhaps desirable. It appears to me that mathematicians are prone to go further than others in this direction; their usual assumption being that whatever has been demonstrated mathematically must be infallibly true. ”

“ Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. ”

“ Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested by form alone. ”

“ 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. ”

“ The abstract part of mathematics is then purely instrumental, and is only an immense and admirable extension of natural logic to a certain class of deductions. On the other hand, geometry and mechanics, which, as we shall see presently, constitute the concrete part, must be viewed as real natural sciences, founded on observation, like all the rest, although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization, which has sometimes caused a misconception of the experimental character of their first principles. ”

“ 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. ”

“ 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. ”

“ Besides its own extreme importance as a means of highly perfecting geometrical science, or, rather, of establishing the whole of it on rational bases, the philosophical study of this admirable conception must have so much the greater interest in our eyes from its characterizing with perfect clearness the general method to be employed in organizing the relations of the abstract to the concrete in mathematics, by the analytical representation of natural phenomena. There is no conception, in the whole philosophy of mathematics which better deserves to fix all our attention. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts. ”

“ MATHEMATICS FOR PRACTICAL MEN: Being a Commonplace Book of Pure and Mixed Mathematics. Designed chiefly for the use of Civil Engineers, Architects and Surveyors. By Olinthus Gregory, L.L.D., F.R.A.S., Enlarged by Henry Law, C.E. 4th Edition, carefully Revised by J. R. Young, formerly Professor of Mathematics, Belfast College. With 13 Plates. 8vo, £1 1s. cloth.
"The engineer or architect will here find ready to his hand rules for solving nearly every mathematical difficulty that may arise in his practice. ”

“ 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. ”

“ 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. ”