Mathematics

"When the body degenerates into a material straight line the number of degrees of freedom is reduced to five; and when this straight line is constrained to move parallel to some fixed plane the number of degrees of freedom is still further reduced to four."

"[F]or the bare hyperboloid, both real and Lambertian sources are in the same plane. This is a limitation in certain applications; however, we can manipulate these positions with lenses to place the sources at more convenient locations. ...this is an ideal concentrator. ...we have postulated an ideal lens while the hyperboloid is an ideal concentrator, albeit operating on a virtual source. The design considerations readily follow from the geometry of the hyperbola."

"A surface which may be generated by a moving straight line is called a '. The plane, the cone, and the cylinder are simple examples... the hyperbolic paraboloid is a . ...[T]he unparted hyperboloid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{ c^2} = 1 is a ruled surface having two sets of rectilinear generators, i.e., ...through every point of it two straight lines may be drawn, each of which shall lie entirely on the surface."

"If we write the equation of the hyperboloid in the form \frac{x^2}{ a^2} - \frac{z^2}{c^2} = 1 - \frac{y^2}{b^2}, \qquad (1)It is evident that (1) is the product of two equations\begin{array}{lcl} \frac{x}{a} - \frac{z}{c} = k_1(1 - \frac{y}{b}), \\ \frac{x}{a} + \frac{z}{c} = \frac{1}{k_1}(1 + \frac{y}{b}), \qquad (2) \end{array}for any value of k_1. But (2) are the equations of a straight line... Moreover this straight line lies entirely on the surface, since the coordinates of every point of it satisfy (2) and hence (1). As different values are assigned to k_1, we obtain a series of straight lines lying entirely on the surface. Conversely if P_1( x_1, y_1, z_1) is any point of (1), \frac{\frac{x_1}{a} - \frac{z_1}{c}}{1 - \frac{y_1}{b}} = \frac{1 + \frac{y_1}{b}}{\frac{x_1}{a} + \frac{z_1}{c}}Therefore P_1 determines the same value of k_1 from both equations (2). Hence every point of (1) lies in one and only one line (2). We may also regard (1) as the product of the two equations\begin{array}{lcl} \frac{x}{a} - \frac{z}{c} = k_2(1 + \frac{y}{b}), \\ \frac{x}{a} + \frac{z}{c} = \frac{1}{k_2}(1 - \frac{y}{b}), \qquad (3) \end{array}whence it is evident that there is a second set of straight lines lying entirely on the surface, one and only one of which may be drawn through any point of the surface. Equations (2) and (3) are the equations of the rectilinear generators, and every point of the surface may be regarded as the point of intersection of one line from each set."

"The equation of the tangent plane at the point (a, 0, 0) of the conicoid \frac{x^2}{a^2} \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 1 is x = a; this meets the surface in straight lines whose projection on the plane x = 0 are given by the equation \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 0. These lines are clearly real when the surface is an hyperboloid of one sheet, and imaginary when the surface is an , or an hyperboloid of two sheets. Hence the hyperboloid of one sheet is a . The hyperbolic paraboloid is a particular case of the hyperboloid of one sheet; hence the hyperbolic paraboloid is also a ruled surface. This can be proved at once from the equation of the paraboloid. For, the tangent plane at the origin is z = 0, and this meets the paraboloid ax^2 + by^2 + 2z = 0 in the straight lines given by the equations ax^2 = by^2 = 0, z = 0; the lines are clearly real when a and b have different signs, and are imaginary when a and b have the same sign. Hence an hyperboloid of one sheet (including an hyperbolic paraboloid as a particular case) is the only ruled conicoid in addition to a cone, a cylinder, and a pair of planes."

"Suppose we start with a disk radiator and we place a mirror in the form of a hyperboloid of revolution coincident with a set of flow lines... We truncate the mirror at some distance so the open end is a circle... Considering the inside as a mirror, this forms a nonimaging concentrator with unusual properties. The foci of the hyperbolas in this section are at... the ends of the diameter of the original disk. Then all rays entering the aperture... and pointing somewhere inside the disk will be reflected by the mirror so as to strike, eventually, the inner disk... Thus, the concentrator takes all rays from the virtual source... which can pass the entry aperture... and concentrate them into an exit aperture. This result is easily proved for rays in the meridional section... the extreme angle rays emerge from the exit aperture but only after an infinite number of reflections. ...rays at angles inside the extreme angles all emerge. Thus, in the meridional plane this is a concentrator of maximal theoretical concentration. This property holds for skew rays, although this is not quite so obvious. ...When used in reverse, the same design produces a virtual ring that fills the space between a Lambertian source and the larger diameter... [disk]. The visual effect produced is striking."

"The only general of revolution which can degenerate into a cylinder, a cone, or a plane is the hyperboloid."

"Rotation of a ... forms a hyperboloid in which the hyperbola becomes a meridian of this surface..."

"All screws of a given pitch belonging to a system of the third order are the generators of a certain hyperboloid. There is... a different hyperboloid for each pitch. ...all these hyperboloids are concentric."

"[F]or the hyperboloid of one sheet and the hyperbolic paraboloid the curvature is negative at every point."

"Upon the hyperboloid of one sheet, and likewise upon the hyperbolic paraboloid, the two lines of striction coincide."

"When an hyperboloid of revolution of one sheet is deformed into another ruled surface, the circle of gorge becomes a Bertrand curve and the generators are parallel to the corresponding binormals of the conjugate Bertrand curve."

"The hyperboloid of revolution (generated as a surface of revolution) is a special case of the elliptical hyperboloid... The shape of a small portion of the hyperboloid of revolution around the equator is similar to that of the , but larger portions are quite different, as are the s. The image of the Gauss map of the whole hyperboloid of revolution omits disks around the north and south poles, whereas the image of the Gauss map of the whole catenoid omits only the north and south poles."

"One way to deal with the matters... is to work with a set of numerical values. I can do this with the help of Ball's Cartesian equation for the hyperboloids of reguli of the 3-system (1900). This equation is quoted and proved by Hunt (1978)... I shall choose the three principle screws of a 3-system of motion... I have, (a) accorded with convention, (b) ensured that the pitch quadric will be real... Ball's equation... clearly represents a series of concentric quadric surfaces... This circumstance of there being none of the concentric hyperboloids coaxial with one of the principal axes is a characteristic of the 3-system. ...within a certain, central zone of the system, the intersections among the hyperboloids are complicated and not easy to understand. Outside that zone, however... the hyperboloids appear in relation to one another... Each successive hyperboloid is wholly 'outside' its predecessor (or wholly 'inside' as the case may be), and no intersections are apparent. ...outside a certain, central zone, only one real screw can be found to pass through a generall chosen point..."

"Dynamical systems theory began in the late nineteenth century with the work of Henri Poincaré (1854–1912). He solved an important problem in celestial mechanics by using techniques of dynamical systems. In modern times, Stephen Smale, Adrien Douady, John Milnor, Lennart Carleson, and many other notable mathematicians have helped to develop this theory into a powerful mathematical tool. The subject is exciting in that it integrates analysis, geometry, and computer graphics into a whole that is greater than the sum of its parts."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect."

"Bell's theorem is the most profound discovery of science."

"Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics."

"It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel."

"There is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"The theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head."

"Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet."

"Theta functions pervade all of mathematics ranging from the theory of partial differential equations, mathematical physics, to algebraic geometry, number theory and more recently to representation theory."

"In the winter of 2008, Jenifer and I visited Chennai Mathematical Institute. This remarkable Institute is the creation of Seshadri. It is a unique blend of an American style liberal arts college with traditional Indian guru one-on-one teaching, adding physics, computer science, history and music to its maths curriculum. Only in India could an intellectual with no business or management experience, who spends all his spare time singing classical south Indian music, have been the catalyst for such a unique educational experiment."

"I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for... science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. ...the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics."

"My decision to leave applied mathematics for economics was in part tied to the widely-held popular belief in the 1960s that macroeconomics had made fundamental inroads into controlling business cycles and stopping dysfunctional unemployment and inflation."

"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. 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."

"From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics."

"Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations."

"Today, ring theory is a fertile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential operators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups)."

"Category theory plays somewhat the same role in algebra and topology that set theory plays in analysis."

"The cornerstone of Category Theory is the Yoneda lemma. It asserts that a category C may be embedded in the category C^\wedge of all contravariant functors from this category to the category Set of sets, the morphisms in Set being the usual maps. This allows us, in some sense, to reduce Category Theory to Set Theory. The Yoneda lemma naturally leads to the notion of representable functor, and in particular to that of adjoint functor."

"The purpose of sheaf theory is quite general: it is to obtain global information from local information, or else to define “obstructions” which characterize the fact that a local property does not hold globally any more: for example a manifold is not always orientable, or a differential equation can be locally solvable, but not globally."

"If geometry is dressed in a suit coat, topology dons jeans and a T-shirt."

"Topologists are interested not only in finite-dimensional spaces (for example, subspaces of Rn), but also in infinite-dimensional ones, such as the spaces occurring in quantum field theory."

"In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."

"Presentday topology consists of two distinct parts: point set topology and algebraic topology. The first has mainly been the prerogative of Poland plus a strong American component: the school of R. L. Moore (of Austin, Texas)."

"In Euler’s investigation of the bridges of Königsberg, he discovered that it was the general arrangement of features that was important, not their exact locations. This observation led to the creation of graph theory, one of the earliest incarnations of topology."

"In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the s: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the ... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the ... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology."

"Num. xxxiii. 1. 'Build me here seven alters and prepare me here seven oxen and seven rams.' The ancients were very superstitious about certain numbers, supposing that God delighted in odd numbers.Around his waxen image first I wind Three woollen fillets, of three colours joined; Thrice bind about his thrice devoted head, Which round the sacred alter thrice is led. Unequal numbers please the gods."

"Leibniz discovered his future with the help of a Nuremberg society of alchemists. ...In private ...he exhibited an avid interest in the subject throughout his life—as did many of his contemporaries, such as Isaac Newton. Indeed, he was so certain that he would one day soon discover the means to turn lead into gold that at one point he fretted that the resulting oversupply of the yellow metal might drive down the price and thus deprive him of hard-earned profits."

": Prithee, no more prattling; go; I'll hold. This is the third time; I hope good luck lies in odd numbers. Away, go! They say there is divinity in odd numbers, either in nativity, chance, or death. Away!"

"One thing that particularly interested Plato was the mysticism of numbers. In his Republic (Book VIII) he speaks in an obscure fashion of a certain mystic number, but he does not make clear what this number is. He calls it "the lord of better and worse births"... One theory is that 60, or 12,960,000 is the Platonic number. This number played an important part in the mysticism of the Hindus and the Babylonians, and it is possible that Pythagoras found it on the banks of the Euphrates, if he really studied there, and that he took it with him to Crotona, passing it on to his disciples, who, in turn, told it to Plato and his followers. ...More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch..."

"Among the first of those who bade adieu to the Scholastic creed was the Cardinal Nicolas Cusanus, a man of rare sagacity and an able mathematician; who arranged and republished the Pythagorean Ideas, to which he was much inclined, in a very original manner, by the aid of his Mathematical knowledge. He considered God as the unconditional Maximum, which at the same time, as Absolute Unity, is also the unconditional Minimum, and begets of Himself and out of Himself, Equality and the combination of Equality with Unity (Son and Holy Ghost). According to him, it is impossible to know directly and immediately this Absolute Unity (the Divinity); because we can make approaches to the knowledge of Him only by the means of Number or Plurality. Consequently he allows us only the possession of very imperfect notions of God, and those by mathematical symbols. It must be admitted that the Cardinal did not pursue this thought very consequently, and that his view of the universe which he connected with it, and which represented it as the Maximum condensed, and thus become finite, was very obscure. Nor was he more successful in his view of the one-ness of the Creator and of Creation, or in his attempt to explain the mysteries of the Trinity and Incarnation, by means of this Pantheistic Theism. Nevertheless, numerous profound though undeveloped observations on the faculty of cognition, are found in his writings, interspersed with his prevailing Mysticism. For instance, he observes, that the principles of knowledge possible to us are contained in our ideas of Number (ratio explicata) and their several relations; that absolute knowledge is unattainable to us (precisio veritatis inattingibilis, which he styled docta ignorantia), and that all which is attainable to us is a probable knowledge (conjectura). With such opinions he expressed a sovereign contempt for the Dogmatism of the Schools."

"It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosophers, except in France. Philosophical works on the subject, such as Cohen's Princip der Infinitesimalmethode und seine Geschichte, are vitiated, as regards the constructive theory, by an undue mysticism inherited from Kant, and leading to such results as the identification of intensive magnitude with the extensive infinitesimal."

"An equation for me has no meaning unless it expresses a thought of God."

"It seems necessary to say something at the outset in justification of the scientific as against the mystical attitude. Metaphysics, from the first, has been developed by the union or the conflict of these two attitudes. Among the earliest Greek philosophers, the Ionians were more scientific and the Sicilians more mystical. But among the latter, Pythagoras, for example, was in himself a curious mixture of the two tendencies: the scientific attitude led him to his proposition on right-angled triangles, while his mystic insight showed him that it is wicked to eat beans. Naturally enough, his followers divided into two sects, the lovers of right-angled triangles and the abhorrers of beans; but the former sect died out, leaving, however, a haunting flavour of mysticism over much Greek mathematical speculation, and in particular over Plato's views on mathematics. Plato, of course, embodies both the scientific and the mystical attitudes in a higher form than his predecessors, but the mystical attitude is distinctly the stronger of the two, and secures ultimate victory whenever the conflict is sharp. Plato, moreover, adopted from the the device of using logic to defeat common sense, and thus to leave the field clear for mysticism—a device still employed in our own day by the adherents of the classical tradition. The logic used in defence of mysticism seems to me faulty as logic..."

"Mathematics has ceased to seem to me non-human in its subject matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal. I think that the timelessness of mathematics has none of the sublimity that it once seemed to me to have, but consists merely in the fact that the pure mathematician is not talking about time. I can no longer find any mystical satisfaction in the contemplation of mathematical truth. ... One effect of the War was to make it impossible to go on living in a world of abstraction. ... I have no longer the feeling that intellect is superior to sense, and that only Plato's world of ideas gives access to the 'real' world. I used to think of sense, and of thought which is built on sense, as a prison... I now think of sense, and of thoughts built on sense, as windows... I think that we can, however imperfectly, mirror the world, like Leibniz's monads; and I think it is the duty of the philosopher to make himself as undistorting a mirror as he can. ...to recognize such distortions ...Of these, the most fundamental is that we view the world from the here and now, not with that large impersonality which theists attribute to the Deity. To achieve such impartiality is impossible for us, but we can travel a certain distance towards it. To show the road to this end is the supreme duty of the philosopher."

"[T]he soul has its essence in mathematical ideas, and it has a prior knowledge of them. .. and brings of them to light when it is set free of the hindrances that arise from sensation. For our sense-perceptions engage the mind with divisible things... and...every divisible thing is an obstacle to our returning upon ourselves. ... Consequently when we remove these hindrances. . we become knowers in actuality..."