On Calculus: On Fluxions

"And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?" (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"And yet in the calculus differentialis, which method serves to all the same intents and ends with that of fluxions, our modern analysts are not content to consider only the differences of finite quantities: they also consider the differences of those differences, and the differences of the differences of the first differences. And so on ad infinitum. That is, they consider quantities infinitely less than the least discernible quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth, &c. fluxions, so there are differences, first, second, third, fourth, &c., in an infinite  progression towards nothing, which you still approach and never arrive at. And (which is most strange) although you should take a million of millions of these infinitesimals, each whereof is supposed infinitely greater than some other real magnitude, and add them to the least given quantity, it shall never be the bigger. For this is one of the modest postulata of our modern mathematicians, and is a cornerstone or ground-work of their speculations."  (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"He who can digest a second or third fluxion, a second or third difference, need not, we think, be squeamish about any point of divinity." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"The foreign mathematicians are supposed by some, even of our own, to proceed in a manner less accurate, perhaps, and geometrical, yet more intelligible. Instead of flowing quantities and their fluxons, they consider the variable finite quantities as increasing or diminishing by the continual addition or subduction of infinitely small quantities. Instead of the velocities wherewith increments are generated, they consider the increments or decrements themselves, which they call differences, and which are supposed to be infinitely small. The difference of a line is an infinitely little line; of a plane an infinitely little plane. 

"The method of Fluxions is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"Newton started out from another principle; and one can say that the metaphysics of this great mathematician on the calculus of fluxions is very exact and illuminating, even though he allowed us only an imperfect glimpse of his thoughts. He never considered the differential calculus as the study of infinitely small quantities, but as the method of first and ultimate ratios, that is to say, the method of finding the limits of ratios. Thus this famous author has never differentiated quantities but only equations; in fact, every equation involves a relation between two variables and the differentiation of equations consists merely in finding the limit of the ratio of the finite differences of the two quantitiescontained in the equation." (Jean LeRond D'Alembert, "Differentiel" ["Differentials", 1754)

"It appears that Fermat, the true inventor of the differential calculus, considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order [...] Newton, through his method of fluxions, has since rendered the calculus more analytical, he also simplified and generalized the method by the invention of his binomial theorem. Leibnitz has enriched the differential calculus by a very happy notation." (Pierre-Simon Laplace, "Exposition du système du monde" ["Exposition of the System of the World"], 1796)

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials." (Pierre-Simon Laplace, "Essai philosophique sur le calcul des probabilities", 1812)

"Every man is ready to join in the approval or condemnation of a philosopher or a statesman, a poet or an orator, an artist or an architect. But who can judge of a mathematician? Who will write a review of Hamilton’s Quaternions, and show us wherein it is superior to Newton’s Fluxions?" (Thomas Hill, 'Imagination in Mathematics', North American Review 85, 1857)

"In reality the origin of the notion of derivatives is in the vague feeling of the mobility of things, and of the greater or less speed with which phenomena take place; this is well expressed by the terms fluent and fluxion, which were used by Newton and which we may believe were borrowed from the ancient mathematician Heraclitus." (Émile Picard, [address to the section of Algebra and Analysis] 1904)

On Calculus (-1799)

"[…] it is not necessary that these hypotheses should be true, or even probably; but it is enough if they provide a calculus which fits the observations […]" (Andrew Osiander, "On the Revolutions of the Heavenly Spheres", 1543)

"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)

"The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment. The sciences must proceed to first causes, above all mathematics where one cannot assume, as in physics, principles that are unknown to us. For there is in mathematics, so to speak, only what we have placed there […] If, however, mathematics always has some essential obscurity that one cannot dissipate, it will lie, uniquely, I think, in the direction of the infinite; it is in that direction that mathematics touches on physics, on the innermost nature of bodies about which we know little." (Bernard Le Bovier de Fontenelle, "Elements de la géométrie de l'infini", 1727)

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire, "Letters on the English", 1733)

"And yet in the calculus differentialis, which method serves to all the same intents and ends with that of fluxions, our modern analysts are not content to consider only the differences of finite quantities: they also consider the differences of those differences, and the differences of the differences of the first differences. And so on ad infinitum. That is, they consider quantities infinitely less than the least discernible quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth, &c. fluxions, so there are differences, first, second, third, fourth, &c., in an infinite  progression towards nothing, which you still approach and never arrive at. And (which is most strange) although you should take a million of millions of these infinitesimals, each whereof is supposed infinitely greater than some other real magnitude, and add them to the least given quantity, it shall never be the bigger. For this is one of the modest postulata of our modern mathematicians, and is a cornerstone or ground-work of their speculations."  (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"What concerns us most here is the metaphysics of the differential calculus. This metaphysics, of which so much has been written, is even more important and perhaps more difficult to explain than the rules of this calculus themselves: various mathematicians, among them Rolle, who were unable to accept the assumption concerning infinitely small quantities, have rejected it entirely, and have held that the principle was false and capable of leading to error. Yet in view of the fact that all results obtained by means of ordinary Geometry can be established similarly and much more easily by means of the differential calculus, one cannot help concluding that, since this calculus yields reliable, simple, and exact methods, the principles on which it depends must also be simple and certain." (Jean LeRond D'Alembert, "Differentiel" ["Differentials", 1754)

"Those quantities that depend on others in this way, namely, those that undergo a change when others change, are called functions of these quantities. This definition applies rather widely and includes all ways in which one quantity could be determined by another." (Leonhard Euler, "Foundations of differential calculus, with applications to finite analysis and series", 1755)

"I see with much pleasure that you are working on a large work on the integral Calculus [...] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics." (Pierre-Simon Laplace [letter to Sylvestre F Lacroix] 1792)

"It appears that Fermat, the true inventor of the differential calculus, considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order [...] Newton, through his method of fluxions, has since rendered the calculus more analytical, he also simplified and generalized the method by the invention of his binomial theorem. Leibnitz has enriched the differential calculus by a very happy notation." (Pierre-Simon Laplace, "Exposition du système du monde" ["Exposition of the System of the World"], 1796)

On Calculus: On Integrals (1800-1849)

"The integral ∫ϕx.dx will always have the same value along two different paths if it is never the case that ϕx = ∞ in the space between the curves representing the paths. This is abeautiful theorem whose not-too-difficult proof I will give at a suitable opportunity [. . . ]. In any case this makes it immediately clear why a function arising from an integral ϕx.dx can have many values for a single value of x, for one can go round a point where ϕx = ∞ either not at all, or once, or several times. For example, if one defines logx by  dx x , starting from x = 1, one comes to logx either without enclosing the point x = 0 or by going around it once or several times; each time the constant +2πi or −2πi enters; so the multiple of logarithms of any number are quite clear." (Carl FriedrichGauss, [letter to Bessel] 1821)

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon [...] when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms." (Jean-Baptiste-Joseph Fourier, "Théorie Analytique de la Chaleur", 1822)

"I have throughout introduced the Integral Calculus in connexion with the Differential Calculus [...] Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold [...]" (Augustus De Morgan, "The Differential and Integral Calculus", 1836)

"A definite Integral always presupposes numeric values; consequently equations in which definite integrals occur are seldom or never correct as general (formal) equations, but can only be admitted as numeric equations; consequently the convergence of any infinite series which may occur in them is an indispensable condition, whereas the condition of convergence with respect to a general series in general investigations, such as must be necessarily first established as the foundation of the possibility of any calculation, is quite as absurd [...]" (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

On Calculus: On Integrals (1850-1899)

"Therefore, one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginery quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann, "Nachlass", cca. 1852-1853)

"Since one could directly derive the expansion in series of algebraic functions according to the powers of an increment, the derivatives, and the integral, one not only held that it was possible to assume the existence of such a series, derivative, and integral for all functions in general, but one never even had the idea that herein lay an assertion, whether it now be an axiom or a theorem - so self-evident did the transfer of the properties of algebraic functions to transcendental ones seem in the light of the geometrical view of curves representing functions. And examples in which purely analytic functions displayed singularities that were clearly different from those of algebraic functions remained entirely unnoticed." (Hermann Hankel, 1870)

"Vectors which are referred to unit of length I shall call Forces, using the word in a somewhat generalized sense, as we shall see. The operation of taking the integral of the resolved part of a force in the direction of a line for every element of that line, has always a physical meaning. In certain cases the result of the integration is independent  of the path of the line between its extremities. The result is then called a Potential." (James Clerk-Maxwell, "Remarks on the Mathematical Classification of Physical Quantities", 1871)

"The splendid creations of this theory have excited the admiration of mathematicians mainly because they have enriched our science in an almost unparalleled way with an abundance of new ideas and opened up heretofore wholly unknown fields to research. The Cauchy integral formula, the Riemann mapping theorem and the Weierstrass power series calculus not only laid the groundwork for a new branch of mathematics but at the same time they furnished the first and till now the most fruitful example of the intimate connections between analysis and algebra. But it isn't just the wealth of novel ideas and discoveries which the new theory furnishes; of equal importance on the other hand are the boldness and profundity of the methods by which the greatest of difficulties are overcome and the most recondite of truths, the mysteria functiorum, are exposed tothe brightest." (Richard Dedekind, "Stetigkeit und Irrationale Zahlen" ["Continuity and Irrational Numbers, 1872)

"The principal advantage arising from the use of hyperbolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions." (William E Byerly, "Elements of the Integral Calculus", 1881)

"The foregoing account of my researches in the theory of manifolds has reached a point where further progress depends on extending the concept of true integral number beyond the previous boundaries; this extension lies in a direction which, to my knowledge, no one has yet attempted to explore. [...] My dependence on this extension of number concept is so great, that without it I should be unable to take freely the smallest step further in the theory of sets." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"There is perhaps the same relation between the action of natural selection during one generation and the accumulated result of a hundred thousand generations, that there exists between differential and integral. How seldom are we able to follow completely this latter relation although we subject it to calculation. Do we on that account doubt the correctness of our integrations?" (Emil du Bois-Reymond, "Reden" Bd. 1, 1885) 

"If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, "Sur les Equations aux Dérivées Partielles de la Physique Mathématique", American Journal of Mathematics Vol. 12, 1890)

"Strange as it may sound, the power of mathematics rests upon its evasion of all unnecessary thought and on its wonderful saving of mental operation. Even those arrangement-signs which we call numbers are a system of marvelous simplicity and economy. When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations; when we resolve new integral expressions into familiar old integrals; we see in this simply a feeble reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with the keen discernment of a great military commander, substituted new operations for whole hosts of old ones. No one will dispute me when I say that the most elementary as well as the highest mathematics are economically-ordered experiences of counting, put in forms ready for use." (Ernst Mach, "Popular Scientific Lectures", 1895)

On Calculus: On Integrals (1900-1974)

"Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests, is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as of analytical geometry." (Felix Klein, Jahresbericht der Deutschen Mathematiker Vereinigung Vol. 11, 1902)

"Energy really is only an integral; now, what we want to have is a substantial definition, like that of Leibniz, and this demand is justifiable to a certain degree, since our very conviction of the conservation of energy rests in great part on this foundation. [..] And so the manuals of physics contain really two discordant definitions of energy, the first which is verbal, intelligible, capable of establishing our conviction, and false; and the second which is mathematical, exact, but lacking verbal expression." (Emile Meyerson, "Identity & Reality", 1908)

"It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock." (James J Sylvester, "Collected Mathematical Papers" Vol. 2, 1908)

"Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar, - we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one." (Ernst Mach,"Populär-wissenschafliche Vorlesungen", 1908)

"The underlying notion of the integral calculus is also that of finding a limiting value, but this time it is the limiting value of a sum of terms when the number of terms increases without bound at the same time that the numerical value of each term approaches Zero. The area bounded by one or more curves is found as the limiting value of a sum of small rectangles; the length of an arc of a curve is found as the limiting value of a sum of lengths of straight lines (chords of the arc); the volume of a solid bounded by one or more curved surfaces is found as the limiting value of a sum of volumes of small solids bounded by planes; etc." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of the semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid Kantorovich, "On one class of functional equations", 1936)

"Despite two centuries of study, the integrals of general dynamical systems remain covered with darkness. To save the classical thermostatics, the practical success of which is shown by the wide use to which it has been put, we must find a way out. That is, we must find some mathematical connection between time averages of the functions of physical interest and the corresponding simple canonical averages." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"Specifically, it seems to me preferable to use, systematically: 'random' for that which is the object of the theory of probability […]; I will therefore say random process, not stochastic process. 'stochastic' for that which is valid 'in the sense of the calculus of probability': for instance; stochastic independence, stochastic convergence, stochastic integral; more generally, stochastic property, stochastic models, stochastic interpretation, stochastic laws; or also, stochastic matrix, stochastic distribution, etc. As for 'chance', it is perhaps better to reserve it for less technical use: in the familiar sense of'by chance', 'not for a known or imaginable reason', or" (but in this case we should give notice of the fact) in the sense of, 'with equal probability' as in 'chance drawings from an urn', 'chance subdivision', and similar examples." (Bruno de Finetti, "Theory of Probability", 1974)

On Calculus: On Integrals (1975-1999)

"Apart from its own intrinsic interest, a knowledge of differentiable manifolds has become useful-ven mandatory-in an ever-increasing number of areas of mathematics and of its applications. This is not too surprising, since differentiable manifolds are the underlying, if unacknowledged, objects of study in much of advanced calculus and analysis. Indeed, such topics as line and surface integrals, divergence and curl of vector fields, and Stokes's and Green's theorems find their most natural setting in manifold theory. But however natural the leap from calculus on domains of Euclidean space to calculus on manifolds may be to those who have made it, it is not at all easy for most students." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"In the path-integral formulation, the essence of quantum physics may be summarized with two fundamental rules: (1). The classical action determines the probability amplitude for a specific chain of events to occur, and (2) the probability that either one or the other chain of events occurs is determined by the probability amplitudes corresponding to the two chains of events. Finding these rules represents a stunning achievement by the founders of quantum physics." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)

"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations" (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

On Calculus: On Integrals (2000-)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Although one speaks nowadays of the determinant of a matrix, the two concepts had different origins. In particular, determinants appeared before matrices, and the early stages in their history were closely tied to linear equations. Subsequent problems that gave rise to new uses of determinants included elimination theory (finding conditions under which two polynomials have a common root), transformation of coordinates to simplify algebraic expressions (e.g., quadratic forms), change of variables in multiple integrals, solution of systems of differential equations, and celestial mechanics." (Israel Kleiner, "A History of Abstract Algebra", 2007)

"Linear algebra is a very useful subject, and its basic concepts arose and were used in different areas of mathematics and its applications. It is therefore not surprising that the subject had its roots in such diverse fields as number theory (both elementary and algebraic), geometry, abstract algebra (groups, rings, fields, Galois theory), anal ysis (differential equations, integral equations, and functional analysis), and physics. Among the elementary concepts of linear algebra are linear equations, matrices, determinants, linear transformations, linear independence, dimension, bilinear forms, quadratic forms, and vector spaces. Since these concepts are closely interconnected, several usually appear in a given context (e.g., linear equations and matrices) and it is often impossible to disengage them." (Israel Kleiner, "A History of Abstract Algebra", 2007)

"Physics deals with both the large and the small. Its domain of study includes the interior of the nucleus of an atom as well as the exterior of a galaxy. It is, therefore, natural for the scope of physical theories to switch between global, or large-scale, and local, or small-scale regimes. Such an interplay between the local and the global has existed ever since Newton and others discovered the mathematical translation of this interplay: Derivatives are defined as local objects while integrals encompass global properties." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden"qualitative" or topological change in its behavior. Bifurcations can occur in both continuous systems" (described by ODEs, DDEs, or PDEs) and discrete systems" (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)

"The integrands of the integrals on a differential manifold (on its curves, surfaces, etc.; all of them oriented) constitute a structure called an algebra. To be more specific, it is a graded algebra. [...] The subspaces constituted by all the elements of a graded algebra thathave the same grade constitute modules (generalization of the concept of vector space)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity. Whether we are working with derivatives, integrals, differential equations, or any of the other tools of calculus, exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

On Calculus (2010-)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"As geometers study shape, the student of calculus examines change: the mathematics of how an object transforms from one state into another, as when describing the motion of a ball or bullet through space, is rendered pictorial in its graphs’ curves." (Daniel Tammet, "Thinking in Numbers" , 2012)

"It has been said that the three most effective problem-solving devices in mathematics are calculus, complex variables, and the Fourier transform." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)

"Mathematics is a technique, a tool, albeit a sophisticated one. Theory is something different. Theory lies in the discovery, understanding, and explaining of phenomena present in the world. Mathematics facilitates this - enormously - but then so does computation. Naturally, there is a difference. Working with equations allows us to follow an argument step by step and reveals conditions a solution must adhere to, whereas computation does not. But computation - and this more than compensates - allows us to see phenomena that equilibrium mathematics does not. It allows us to rerun results under different conditions, exploring when structures appear and don’t appear, isolating underlying mechanisms, and simplifying again and again to extract the bones of a phenomenon. Computation in other words is an aid to thought, and it joins earlier aids in economics - algebra, calculus, statistics, topology, stochastic processes - each of which was resisted in its time. The computer is an exploratory lab for economics, and used skillfully, a powerful generator for theory." (W Brian Arthur, "Complexity and the Economy", 2015)

"[…] the usefulness of mathematics is by no means limited to finite objects or to those that can be represented with a computer. Mathematical concepts depending on the idea of infinity, like real numbers and differential calculus, are useful models for certain aspects of physical reality." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015) 

"Calculus is the study of things that are changing. It is difficult to make theories about things that are always changing, and calculus accomplishes it by looking at infinitely small portions, and sticking together infinitely many of these infinitely small portions." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Sometimes mathematical advances happen by just looking at something in a slightly different way, which doesn’t mean building something new or going somewhere different, it just means changing your perspective and opening up huge new possibilities as a result. This particular insight leads to calculus and hence the understanding of anything curved, anything in motion, anything fluid or continuously changing." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme - all the way out to infinity." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"If real numbers are not real, why do mathematicians love them so much? And why are schoolchildren forced to learn about them? Because calculus needs them. From the beginning, calculus has stubbornly insisted that everything - space and time, matter and energy, all objects that ever have been or will be - should be regarded as continuous. Accordingly, everything can and should be quantified by real numbers. In this idealized, imaginary world, we pretend that everything can be split finer and finer without end. The whole theory of calculus is built on that assumption. Without it, we couldn’t compute limits, and without limits, calculus would come to a clanking halt." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"In mathematics, pendulums stimulated the development of calculus through the riddles they posed. In physics and engineering, pendulums became paradigms of oscillation. […] In some cases, the connections between pendulums and other phenomena are so exact that the same equations can be recycled without change. Only the symbols need to be reinterpreted; the syntax stays the same. It’s as if nature keeps returning to the same motif again and again, a pendular repetition of a pendular theme. For example, the equations for the swinging of a pendulum carry over without change to those for the spinning of generators that produce alternating current and send it to our homes and offices. In honor of that pedigree, electrical engineers refer to their generator equations as swing equations." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity. Whether we are working with derivatives, integrals, differential equations, or any of the other tools of calculus, exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"We are accustomed to intellectual diffusion taking place from the natural and physical sciences into the social sciences; certainly that is the direction taken for both calculus and the scientific method. But statistical graphics in particular, and statistics in general, took the reverse route." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

On Calculus (2000-20009)

"Arithmetic and number theory study patterns of number and counting. Geometry studies patterns of shape. Calculus allows us to handle patterns of motion. Logic studies patterns of reasoning. Probability theory deals with patterns of chance. Topology studies patterns of closeness and position." (Keith Devlin, "The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip", 2000)

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"At present, high school education in some countries covers only little, if any, statistical thinking. Algebra, geometry, and calculus teach thinking in a world of certainty - not in the real world, which is uncertain. [...] Furthermore, in the medical and social sciences, data analysis is typically taught as a set of statistical rituals rather than a set of methods for statistical thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"[…] the branch of mathematics which is most concerned with change is calculus. The key idea of calculus is in fact not so much change itself, but rather the rate at which change occurs." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"The calculus is a theory of continuous change - processes that move smoothly and that do not stop, jerk, interrupt themselves, or hurtle over gaps in space and time. The supreme example of a continuous process in nature is represented by the motion of the planets in the night sky as without pause they sweep around the sun in elliptical orbits; but human consciousness is also continuous, the division of experience into separate aspects always coordinated by some underlying form of unity, one that we can barely identify and that we can describe only by calling it continuous." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

On Calculus (1975-1999)

"Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"I would therefore urge that people be introduced to [the logistic equation] early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems. Not only in research but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties." (Robert M May, "Simple Mathematical Models with Very Complicated Dynamics", Nature Vol. 261 (5560), 1976)

"In comparison with Predicate Calculus encoding s of factual knowledge, semantic nets seem more natural and understandable. This is due to the one-to-one correspondence between nodes and the concepts they denote, to the clustering about a particular node of propositions about a particular thing, and to the visual immediacy of 'interrelationships' between concepts, i.e., their connections via sequences of propositional links." (Lenhart K Schubert, "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Every discovery, every enlargement of the understanding, begins as an imaginative preconception of what the truth might be. The imaginative preconception - a ‘hypothesis’ - arises by a process as easy or as difficult to understand as any other creative act of mind; it is a brainwave, an inspired guess, a product of a blaze of insight. It comes anyway from within and cannot be achieved by the exercise of any known calculus of discovery. " (Sir Peter B Medawar, "Advice to a Young Scientist", 1979)

"The invention of the differential calculus was based on the recognition that an instantaneous rate is the asymptotic limit of averages in which the time interval involved is systematically shrunk. This is a concept that mathematicians recognized long before they had the skill to actually compute such an asymptotic limit." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"Calculus is the mathematics of change. The mathematics you have learned up to this point has served mainly to describe static (unchanging) situations; the calculus handles dynamic (changing) situations. Change is characteristic of the world." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Calculus systematically evades a great deal of numerical calculation." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Continuous distributions are basic to the theory of probability and statistics, and the calculus is necessary to handle them with any ease." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Increasingly [...] the application of mathematics to the real world involves discrete mathematics... the nature of the discrete is often most clearly revealed through the continuous models of both calculus and probability. Without continuous mathematics, the study of discrete mathematics soon becomes trivial and very limited. [...] The two topics, discrete and continuous mathematics, are both ill served by being rigidly separated." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring Mathematics All Around You", 1986)

"Central to the development of the calculus were the concepts of convergence and limit, and with these concepts at hand it became at last possible to resolve the ancient paradoxes of infinity which had so much intrigued Zeno." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)

"The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right." (David Berlinski, "A Tour of the Calculus", 1995)

"The story of calculus brings out two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction. These are the external and internal aspects of mathematics, often referred to as applied and pure mathematics." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"When it comes to modeling processes that are manifestly governed by nonlinear relationships among the system components, we can appeal to the same general idea. Calculus tells us that we should expect most systems to be 'locally' flat; that is, locally linear. So a conservative modeler would try to extend the word 'local' to hold for the region of interest and would take this extension seriously until it was shown to be no longer valid." (John L Casti, "Five Golden Rules", 1995)

"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996

"While calculus is the mathematical key to an understanding of Nature, its roots lie really in problems of geometry." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"Mathematics, in the common lay view, is a static discipline based on formulas taught in the school subjects of arithmetic, geometry, algebra, and calculus. But outside public view, mathematics continues to grow at a rapid rate, spreading into new fields and spawning new applications. The guide to this growth is not calculation and formulas but an open-ended search for pattern." (Lynn A Steen, "The Future of Mathematics Education", 1998)

"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

On Calculus (1950-1974)

"In the mathematical theory of the maximum and minimum problems in calculus of variations, different methods are employed. The old classical method consists in finding criteria -as to whether or not for a given curve the corresponding number assumes a maximum or minimum. In order to find such criteria a considered curve is a little varied, and it is from this method that the name 'calculus of variations' for the whole branch of mathematics is derived." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"While the minimum and maximum problems of calculus of variations correspond to the problem in the ordinary calculus of finding peaks and pits of a surface, the minimax problems correspond to the problem of finding the saddle points of the surface (the passes of a mountain)."(Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"As the sensations of motion and discreteness led to the abstract notions of the calculus, so may sensory experience continue thus to suggest problem for the mathematician, and so may she in turn be free to reduce these to the basic formal logical relationships involved. Thus only may be fully appreciated the twofold aspect of mathematics: as the language of a descriptive interpretation of the relationships discovered in natural phenomena, and as a syllogistic elaboration of arbitrary premise." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Just as no thing or organism exists on its own, it does not act on its own. Furthermore, every organism is a process: thus the organism is not other than its actions. To put it clumsily: it is what it does. More precisely, the organism, including its behavior, is a process which is to be understood only in relation to the larger and longer process of its environment. For what we mean by 'understanding' or 'comprehension' is seeing how parts fit into a whole, and then realizing that they don't compose the whole, as one assembles a jigsaw puzzle, but that the whole is a pattern, a complex wiggliness, which has no separate parts. Parts are fictions of language, of the calculus of looking at the world through a net which seems to chop it up into bits. Parts exist only for purposes of figuring and describing, and as we figure the world out we become confused if we do not remember this all the time." (Alan Watts, "The Book on the Taboo Against Knowing Who You Are", 1966)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The two problems of tangent construction and area evaluation, which previously bore a relation to each other no closer than that of a similarity of type, were now twins, linked by an 'inversion principle'; the powerful algebraic calculus allowed the mathematician to move easily along a whole chain of integrations and differentiations of a function according to his needs. But with power there is always responsibility; and in this case the limitation was that every operation must take place on a function which obeyed a 'law of continuity' (that is, of differentiability). Thus the calculus was understood to operate validly only on those functions which fulfilled these conditions, and they were the differentiable functions: polynomials, trigonometric and exponential functions, and all such algebraic expressions which yielded a definite result from each operation of the calculus." (Ivor Grattan-Guinness, "The Development of the Foundations of Mathematical Analysis from Euler to Riemann", 1970)

"Specifically, it seems to me preferable to use, systematically: 'random' for that which is the object of the theory of probability […]; I will therefore say random process, not stochastic process. 'stochastic' for that which is valid 'in the sense of the calculus of probability': for instance; stochastic independence, stochastic convergence, stochastic integral; more generally, stochastic property, stochastic models, stochastic interpretation, stochastic laws; or also, stochastic matrix, stochastic distribution, etc. As for 'chance', it is perhaps better to reserve it for less technical use: in the familiar sense of'by chance', 'not for a known or imaginable reason', or (but in this case we should give notice of the fact) in the sense of, 'with equal probability' as in 'chance drawings from an urn', 'chance subdivision', and similar examples." (Bruno de Finetti, "Theory of Probability", 1974)

"The calculus of probability can say absolutely nothing about reality [...] We have to stress this point because these attempts assume many forms and are always dangerous. In one sentence: to make a mistake of this kind leaves one inevitably faced with all sorts of fallacious arguments and contradictions whenever an attempt is made to state, on the basis of probabilistic considerations, that something must occur, or that its occurrence confirms or disproves some probabilistic assumptions." (Bruno de Finetti, "Theory of Probability", 1974)

On Calculus (1925-1949)

"The rational concept of probability, which is the only basis of probability calculus, applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time. Using the language of physics, we may say that in order to apply the theory of probability we must have a practically unlimited sequence of uniform observations." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"The result of each calculation appertaining to the field of probability is always, as far as our theory goes, nothing else but a probability, or, using our general definition, the relative frequency of a certain event in a sufficiently long (theoretically, infinitely long) sequence of observations. The theory of probability can never lead to a definite statement concerning a single event. The only question that it can answer is: what is to be expected in the course of a very long sequence of observations? It is important to note that this statement remains valid also if the calculated probability has one of the two extreme values 1 or 0." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"The most important application of the theory of probability is to what we may call 'chance-like' or 'random' events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe - after many unsuccessful attempts - that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"An incontestable claim of mathematics to importance in our civilization is that it is indispensable in a scientific explanation of what we observe in nature, i.e., the phenomena of nature. Of the several fields of elementary mathematics, the calculus may be called the motion-picture machine of mathematics which catches natural phenomena in the act of changing, or, as Newton called it, in a state of flux. Other fields of mathematics are to be likened to the camera which shows a still picture (of nature) as it appears at a given instant without regard to the possible appearance the following instant." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The underlying notion of the integral calculus is also that of finding a limiting value, but this time it is the limiting value of a sum of terms when the number of terms increases without bound at the same time that the numerical value of each term approaches Zero. The area bounded by one or more curves is found as the limiting value of a sum of small rectangles; the length of an arc of a curve is found as the limiting value of a sum of lengths of straight lines (chords of the arc); the volume of a solid bounded by one or more curved surfaces is found as the limiting value of a sum of volumes of small solids bounded by planes; etc." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the […] development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes." (Julian L Coolidge, "A History of Geometrical Methods", 1940)

"The curves treated by the calculus are normal and healthy; they possess no idiosyncrasies. But mathematicians would not be happy merely with simple, lusty configurations. Beyond these their curiosity extends to psychopathic patients, each of whom has an individual case history resembling no other; these are the pathological curves in mathematics." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I, 1947])

"Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution." (Richard Curand [forward to Carl B Boyer’s "The History of the Calculus and Its Conceptual Development", 1949])

ℵ₀On Calculus (1900-1924)

"Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests, is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as of analytical geometry." (Felix Klein, Jahresbericht der Deutschen Mathematiker Vereinigung Vol. 11, 1902)

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions." (Mary E Boole, "Lectures on the Logic of Arithmetic", 1903)

"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)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"The very name calculus of probabilities is a paradox. Probability opposed to certainty is what we do not know, and how can we calculate what we do not know?" (Henri Poincaré, "The Foundations of Science", 1913)

"The question whether any branch of science can ever become purely deductive is easily answered. It cannot. If science deals with the external world, as we believe it does, and not merely with the relations of propositions then no branch of science can ever be purely deductive. Deductive reasoning by itself can never tell us about facts. The use of deduction in science is to serve as a calculus to make our observations go further, not to take the place of observation." (Arthur D Ritchie, "Scientific Method: An Inquiry into the Character and Validity of Natural Laws", 1923)

On Calculus (1850-1899)

"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore, the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind." (James C Maxwell, 1850)

"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)

"The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments." (Thomas Hill, "The Imagination in Mathematics", North American Review Vol. 85, 1857)

"There are no limits in mathematics, and those that assert there are, are infinite ruffians, ignorant, lying blackguards. There is no differential calculus, no Taylor's theorem, no calculus of variations, [...] in mathematics. There is no quackery whatever in mathematics." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"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 phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma." (Henry J S Smith, Nature, Volume 8, 1873)

"The invention of a new symbol is a step in the advancement of civilisation. Why were the Greeks, in spite of their penetrating intelligence and their passionate pursuit of Science, unable to carry Mathematics farther than they did? and why, having formed the conception of the Method of Exhaustions, did they stop short of that of the Differential Calculus? It was because they had not the requisite symbols as means of expression. They had no Algebra. Nor was the place of this supplied by any other symbolical language sufficiently general and flexible; so that they were without the logical instruments necessary to construct the great instrument of the Calculus." (George H Lewes "Problems of Life and Mind", 1873)

"The rules of Arithmetic operate in Algebra; the logical operations supposed to be peculiar to Ideation operate in Sensation, There is but one Calculus, but one Logic; though for convenience we divide the one into Arithmetic the calculus of values, and Algebra the calculus of relations; the other into the Logic of Feeling and the Logic of Signs." (George H Lewes "Problems of Life and Mind", 1873)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, "Sur les Equations aux Dérivées Partielles de la Physique Mathématique", American Journal of Mathematics Vol. 12, 1890)

"Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define." (Giuseppe Peano, "Geometric Calculus", 1895)

On Calculus (1800-1849)

"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae” [“Arithmetical Researches”], 1801)

"The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which of times they are unable to account." (Pierre-Simon Laplace, "Analytical Theory of Probability, 1812)

"Thus, differential calculus has all the exactitude of other algebraic operations. Pierre-Simon Laplace, A Philosophical Essay on Probabilities, 1814)

"If a hypothesis contains all that is part of the problem, if it can be regarded as a true definition, it suffices to introduce this hypothesis into the calculus, in order to obtain all the analytical consequences that belong to the solution of the same problem."(Sophie Germain, 1821) 

"Thus, let us be persuaded that there are other truths than those of algebra, realities other than sensible objects. Let us cultivate ardently the mathematical sciences, without trying to extend them beyond their domain; and let us not imagine that one can tackle history with formulae, or use theorems of algebra or of differential calculus as an assent to morals.16 (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others." (Auguste Comte, "Course of Positive Philosophy", 1830)

"The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others." (Auguste Comte, "Course of Positive Philosophy", 1830)

"Every mathematical method has its inverse, as truly, and for the same reason, as it is impossible to make a road from one town to another, without at the same time making one from the second to the first. The combinatorial analysis is analysis by means of combinations; the calculus of generating functions is combination by means of analysis." (Augustus de Morgan, "The Differential and Integral Calculus", 1836)

"Algebra, as an art, can be of no use to any one in the business of life; certainly not as taught in the schools. I appeal to every man who has been through the school routine whether this be not the case. Taught as an art it is of little use in the higher mathematics, as those are made to feel who attempt to study the differential calculus without knowing more of the principles than is contained in books of rules." (Augustus de Morgan, "Elements of Algebra", 1837)

"The calculus of probability is equally applicable to things of all kinds, moral and physical and, if only in each case observations provide the necessary numerical data, it does not at all depend on their nature." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The law of large numbers is noted in events which are attributed to pure chance since we do not know their causes or because they are too complicated. Thus, games, in which the circumstances determining the occurrence of a certain card or certain number of points on a die infinitely vary, can not be subjected to any calculus. If the series of trials is continued for a long time, the different outcomes nevertheless appear in constant ratios. Then, if calculations according to the rules of a game are possible, the respective probabilities of eventual outcomes conform to the known Jakob Bernoulli theorem. However, in most problems of contingency a prior determination of chances of the various events is impossible and, on the contrary, they are calculated from the observed result." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experimentalist, and the statesman." (François Arago, "Biographies of Distinguished Scientific Men", [Eulogy on Laplace] 1859)

"There seems to me to be something analogous to polarized intensity in the pure imaginary part; and to unpolarized energy (indifferent to direction) in the real part of a quaternion: and thus we have some slight glimpse of a future Calculus of Polarities. This is certainly very vague […]" (Sir William R Hamilton, "On Quaternions; or on a new System of Imaginaries in Algebra", 1844)

Sylvestre-François Lacroix - Collected Quotes

"Every quantity whose value depends on one or more other quantities is called a function of these latter, whether one knows or is ignorant of what operations it is necessary to use to arrive from the latter to the first." (Sylvestre-François Lacroix, "Traité de calcul differéntiel et du calcul intégral", 1797-1798)

"There exists a manner of viewing geometry that could be called géométrie analytique, and which would consist in deducing the properties of extension from the least possible number of principles, and by truly analytic methods." (Sylvestre-François Lacroix, "Traité de calcul differéntiel et du calcul intégral", 1797-1798) [first use of the "analyic geometry" expression)

"In order to indicate that a quantity depends on one or several others, either by operations of any kind, or by other relations, which it is impossible to assign algebraically, but whose existence is determined by certain conditions, we call the first quantity a function of the others." (Sylvestre-François Lacroix, "An elementary treatise on the differential and integral calculus", 1816)

"The subject of this branch of Analysis is the passage of one or more quantities through different states of magnitude, and the changes which consequently take place in other quantities, whose value depends on that of these first." (Sylvestre-François Lacroix, "An elementary treatise on the differential and integral calculus", 1816)

"The words positive and negative are general terms, that indicate the different states a quantity can be in, and that in special cases will have interpretations such as capital and debt, east and west, right and left, up and down, ascending and descending, winning and losing, etc. In each particular case it is up to us to choose which of the two states we wish to call positive, and thereby denote with the + sign, but once this is determined, we must consistently call the other state negative, and indicate it by the sign −." (Sylvestre-François Lacroix, "Beginselen der Stelkunst", 1821)

"[algebra and geometry] should be treated separately, as far apart as they can be, and that the results in each should serve for mutual clarification, corresponding, so to speak, to the text of a book and its translation." (Sylvestre-François Lacroix) 

David Acheson - Collected Quotes

"It is sometimes possible to infer a great deal about a dynamical system simply by finding its equilibrium states and determining which of these are stable to small disturbances and which are unstable. This can also help explain sudden or 'catastrophic' jumps from one state to another as some parameter is gradually varied." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"Real dynamical problems typically involve nonlinear differential equations of second order, but these often simplify greatly if we investigate small oscillations about a position of equilibrium. Coupled oscillators are particularly interesting, an early example being the double pendulum, first studied by Euler and Daniel Bernoulli in the 1730s." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"Systems which exhibit chaotic oscillations typically do so for some ranges of the relevant parameters but not for others, so one matter of obvious interest is how the chaos appears (or disappears) as one of the parameters is gradually varied." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"While calculus is the mathematical key to an understanding of Nature, its roots lie really in problems of geometry." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)

"Differential equations provide, then, some of the deepest links between mathematics and the physical world." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"[…] it is all too easy in mathematics to jump to the wrong conclusion. And it is particularly dangerous to jump to some general conclusion on the basis of a few special cases." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002) 

"So, when trying to solve a problem in mathematics we have to watch out for subtle mistakes, otherwise, we can easily get the wrong solution." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"[…] the branch of mathematics which is most concerned with change is calculus. The key idea of calculus is in fact not so much change itself, but rather the rate at which change occurs." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"This, then, is the essence of chaos: irregular, erratic motion which is extremely sensitive to the initial conditions. […] A hallmark of chaos: two almost imperceptibly different starting conditions lead to two completely different outcomes, within a relatively short space of time." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"This general kind of behaviour, where a gradual change in some parameter can lead to a sudden and unexpected large change in the system as a whole, is known as a catastrophe." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

 "This strange number e pops up – like π – in all sorts of different places in mathematics. And it arises, in particular, in connection with a fundamental question involving the rate at which things change." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"Today, the whole subject of geometry extends way beyond the world of right-angled triangles, circles and so on. There are even branches of the subject in which the ideas of length, angle and area don’t really feature at all. One of these is topology – a sort of rubber-sheet geometry – where a recurring question is whether some geometric object can be deformed ‘smoothly’ into another one." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"While any one of us is fully entitled – of course – to a quite different opinion, this amazing connection between e, i and π is viewed by many mathematicians as, quite simply, the most stunning result in the whole subject … so far." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

David Perkins - Collected Quotes

"A function acts like a set of rules for turning some numbers into others, a machine with parts that we can manipulate to accomplish anything we can imagine." (David Perkins, "Calculus and Its Origins", 2012)

"Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built." (David Perkins, "Calculus and Its Origins", 2012)

"If we wish the word ‘continuous’ to prohibit jumps in a function, its definition must somehow control the vertical change of the function at a sort of microscopic level. That is, at any point on a ‘continuous’ function, the nearby points ought to be as ‘close’ as possible." (David Perkins, "Calculus and Its Origins", 2012)

"Mathematicians approach problems the way rock climbers do cliffs: the more difficult the pitch, the more exhilarating the ascent. After a climb has been solved, others look for new routes, or try equipment that no one else has used, simply for the joy of pioneering." (David Perkins, "Calculus and Its Origins", 2012)

"One trick to seeing beauty in mathematics is to nurture this sense of 'odd as it may seem' while at the same time understanding the subject well enough to know that oddities arise despite our attempts to set the subject on a simple, straightforward footing." (David Perkins, "Calculus and Its Origins", 2012)

"Ever since the discovery of irrational numbers fractured the Greek belief that all numbers were proportions, mathematicians have sorted numbers into categories and hunted for numbers that defied existing categories." (David Perkins, "φ, π, e & i", 2017)

"Imagine that each proof in this book is like a painting that one sees upon entering a gallery full of artwork, in which each work presents an artist’s unique vision of the same theme." (David Perkins, "φ, π, e & i", 2017) 

"Mathematicians linger on cherished topics, illuminating them from a variety of viewpoints, much like artists and poets try over and over to capture truths about the human condition. Re-proving something important in a new way brings joy to both the discoverer and the audience." (David Perkins, "φ, π, e & i", 2017)

"Much of the final resistance to complex numbers faded as it became clear that their behavior posed no threat to the rules and operations of algebra. On the contrary, quite often the complex realm opened paths that made already existing results easier to prove." (David Perkins, "φ, π, e & i", 2017)