"Arithmetical symbols are written diagrams and geometrical figures are graphic formulas." (David Hilbert, Bulletin of the American Mathematical Society Mathematical Problems Vol. 8, 1902)
"The object of geometry is the study of a particular 'group'; but the general concept of group preexists in our minds, at least potentially. It is imposed on us not as a form of our sensitiveness, but as a form of our understanding; only, from among all possible groups, we must choose one that will be the standard, so to speak, to which we shall refer natural phenomena." (Henri Poincaré, "Science and Hypothesis", 1902)
"The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space." (Paul Carus, Science Vol. 18, 1903)
"Surely, among the most important goals of every geometrical instruction is the strengthening of the faculty for spatial imaging and the power for spatial modelling." (Arturo M Schoenflies, 1908)
"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word?" (Henri Poincaré , "Intuition and Logic in Mathematics", 1905)
"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word?" (Henri Poincaré , "Intuition and Logic in Mathematics", 1905)
"But in the mathematical or pure sciences, - geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves, - we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline." (Joshua Fitch,"Lectures on Teaching", 1906)
"Geometry, accordingly, consists of the application of mathematics to experiences concerning space. Like mathematical physics, it can become an exact deductive science only on the condition of its representing the objects of experience by means of schematizing and idealizing concepts." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)
"Architecture is geometry made visible in the same sense that music is number made audible." (Claude F Bragdon, "The Beautiful Necessity: Seven Essays on Theosophy and Architecture", 1910)
"Architecture is geometry made visible in the same sense that music is number made audible." (Claude F Bragdon, "The Beautiful Necessity: Seven Essays on Theosophy and Architecture", 1910)
"Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligation with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!" (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])
"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)
"Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul." (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])
"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our ideas and prevent our attention from wandering owing to the difficulty of keeping a long chain of syllogisms in our minds, it is excusable if we are apt to forget that we are not in reality reasoning about the objects in the figure, but about objects which ore their idealizations, and of which the objects in the figure are only an imperfect representation. Even if we only visualize, we see the images of more or less gross physical objects, in which various qualities irrelevant for our specific purpose are not entirely absent, and which are at best only approximate images of those objects about which we are reasoning." (Ernest W Hobson, "Squaring the Circle", 1913)
"Geometers usually distinguish two kinds of geometry, the first of which they qualify as metric and the second as projective. Metric geometry is based on the notion of distance; two figures are there regarded as equivalent when they are 'congruent' in the sense that mathematicians give to this word. Projective geometry is based on the notion of straight line; in order for two figures considered there to be equivalent, it is not necessary that they be congruent; it suffices that one can pass from one to the other by a projective transformation, that is, that one be the perspective of the other. This second body of study has often been called qualitative geometry, and in fact it is if one opposes it to the first; it is clear that measure and quantity play a less important role. This is not entirely so, however. The fact that a line is straight is not purely qualitative; one cannot assure himself that a line is straight without making measurements, or without sliding on this line an instrument called a straightedge, which is a kind of instrument of measure.” (Henri Poincaré, “Dernières pensées”, 1913)
"Geometers usually distinguish two kinds of geometry, the first of which they qualify as metric and the second as projective. Metric geometry is based on the notion of distance; two figures are there regarded as equivalent when they are 'congruent' in the sense that mathematicians give to this word. Projective geometry is based on the notion of straight line; in order for two figures considered there to be equivalent, it is not necessary that they be congruent; it suffices that one can pass from one to the other by a projective transformation, that is, that one be the perspective of the other. This second body of study has often been called qualitative geometry, and in fact it is if one opposes it to the first; it is clear that measure and quantity play a less important role. This is not entirely so, however. The fact that a line is straight is not purely qualitative; one cannot assure himself that a line is straight without making measurements, or without sliding on this line an instrument called a straightedge, which is a kind of instrument of measure.” (Henri Poincaré, “Dernières pensées”, 1913)
"On the other side of the subject, Geometry is an abstract rational Science which deals with the relations of objects that are no longer physical objects, although these ideal objects, points, straight lines, circles, &c., are called by the same names by which we denote their physical counterparts. At the base of this rational Science there lies a set of definitions and postulations which specify the nature of the relations between the ideal objects with which the Science deals. These postulations and definitions were suggested by our actual spatial perceptions, but they contain an element of absolute exactness which is wanting in the rough data provided by our senses. The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry." (Ernest W Hobson, "Squaring the circle", 1913)
"A much more natural and adequate comparison would, it seems to me, liken Coordinate Geometry to a steam-hammer, which an expert may employ on any destructive or constructive work of one general kind, say the cracking of an eggshell, or the welding of an anchor. But you must have your expert to manage it, for without him it is useless. He has to toil amid the heat, smoke, grime, grease, and perpetual din of the suffocating engine-room. The work has to be brought to the hammer, for it cannot usually be taken to its work. And it is not in general, transferable; for each expert, as a rule, knows, fully and confidently, the working details of his own weapon only. Quaternions, on the other hand, are like the elephant’s trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere - alike in the trackless jungle and in the barrack square - directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one." (Peter G Tait [in Alexander MacFarlane's "Lectures on Ten British Mathematicians", 1916])
"Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint - an enchanted realm where thought is double and flows throughout in parallel streams." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint - an enchanted realm where thought is double and flows throughout in parallel streams." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s ‘blank form of a universe’ without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition." (Lawrence J Henderson, "The Order of Nature: An Essay", 1917)
"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)
"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)
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