On Functions IV

"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)

"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)

"In mathematics, as in the world about us, when one quantity depends on a second quantity, or when the value of one symbol depends on the value of another symbol, the first is said to be a function of the second. If the second quantity, or the second symbol, is thought of as taking on a number of arbitrary values (e.g., the angle A when it increases or decreases), it is called an independent variable and the function which depends on it is called a dependent variable. It may happen that a function depends on more than one independent variable." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"It is sometimes said that mathematics is the study of sets and functions. Naturally, this oversimplifies matters; but it does come as close to the truth as an aphorism can." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up, onto the highlands of mathematics itself, where these concepts are indispensable in almost all of pure mathematics as it is today." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Continuity is the rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable - say x - is associated with a value of a dependent variable - say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question 'How close?' is asked, difficulties arise." (Erik Gregersen [Ed.], "Math Eplained: The Britannica Guide to Analysis and Calculus", 2011)

"Functions are the most basic way of mathematically representing a relationship. [...] In mathematics, it is useful to think of a function as an action; a function takes a number as input, does something to it, and outputs a new number. [...] We are now in a position to refine our definition of a function. A function is a rule that assigns an output value f(x) to every input x. This is consistent with the everyday use of the word function: the output f(x) is a function of the input x. The output depends on the input [...]" (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"Thus, much of physics can be seen as a iterative process: an object or a bunch of objects have some initial condition or seed. [...] Iterated functions are an example of what mathematicians call dynamical systems. A dynamical system is just a generic name for some variable or set of variables that change over time. There are many different types of dynamical systems - the iterated functions introduced above are just one type among many. Dynamical systems is now generally recognized as a branch of applied mathematics that studies properties of how systems change over time." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012

Functions: Zeta

"No one has ever been able to prove, for example, that every even number greater than two can be expressed as the sum of two primes. Yet this is as well established by observation as any of the laws of physics. It is known that this and various other theorems are true if a certain hypothesis about the Zeta function, enunciated by Riemann nearly a century ago, is correct. No one has been able to prove this hypothesis. It has only been shown that all the consequences deducible if it is true are so far verified by experience. But any day a computer with little knowledge of pure mathematics may disprove it. Here then is a possible triumph for the mathematical amateur." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"At this point, it is not possible to remain silent on what is probably the most intriguing unsolved problem in the theory of the zeta function and actually in all of number theory - and most likely even one of the most important unsolved problems in contemporary mathematics, namely the famous Riemann hypothesis. [...] Still, the problem is open and fascinates and teases the best contemporary minds." (Emil Grosswald, "Topics in the Theory of Numbers", 1966)

"The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. [...] The main interest comes from trying to improve the Prime Number Theorem, i.e., getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fare, and whose proof has since then become the single most desirable achievement for a mathematician." (Martin C Gutzwiller, "Chaos in Classical and Quantum Mechanics", 1990)

"I am firmly convinced that the most important unsolved problem in mathematics today is the truth or falsity of a conjecture about the zeros of the zeta function, which was fi rst made by Riemann himself. [...] Even a single exception to Riemann’s conjecture would have enormous ly strange consequences for the distribution of prime numbers [...] If the Riemann hypothesis turns out to be false, there will be huge oscillations in the distribution of primes. In an orchestra, that would be like one loud instrument that drowns out the others - an aesthetically distasteful situation." (Enrico Bombieri, "Prime Territory: Exploring the Infinite Landscape at the Base of the Number System", The Sciences, 1992)

"It is intriguing that any of the various new expansions and associated observations relevant to the critical zeros arise from the fi eld of quantum theory, feeding back, as it were, into the study of the Riemann zeta function. But the feedback of which we speak can move in the other direction, as techniques attendant on the Riemann zeta function apply to quantum studies." (Jonathan M Borwein et al, "Computational Strategies for the Riemann Zeta Function", Journal of Computational and Applied Mathematics Vol. 121, 2000

"Riemann had found a passageway from the familiar world of numbers into a mathematics which would have seemed utterly alien to the Greeks who had studied prime numbers two thousand years before. He had innocently mixed imaginary numbers with his zeta function and discovered, like some mathematical alchemist, the mathematical treasure emerging from this admixture of elements that generations had been searching for. He had crammed his ideas into a ten-page paper, but was fully aware that his ideas would open up radically new vistas on the primes. (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"[...] in one of those unexpected connections that make theoretical physics so delightful, the quantum chorology of spectra turns out to be deeply connected to the arithmetic of prime numbers, through the celebrated zeros of the  Riemann zeta function: the zeros mimic quantum energy levels of a classically chaotic system. The connection is not only deep but also tantalizing, since its basis is still obscure - though it has been fruitful for both mathematics and physics." (Michael V. Berry)

"How could it be that the Riemann zeta function so convincingly mimics a quantum system without being one?" (Michael V Berry) [33] 

"It’s a whole beautiful subject and the Riemann zeta function is just the fi rst one of these, but it’s just the tip of the iceberg. They are just the most amazing objects, these L-functions - the fact that they exist, and have these incredible properties are tied up with all these arithmetical things - and it’s just a beautiful subject. Discovering these things is like discovering a gemstone or something. You’re amazed that this thing exists, has these properties and can do this." (J Brian Conrey) 

On Functions III

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The course of the values of a continuous function is determined at all points of an interval, if only it is determined for all rational points of this interval." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The most general definition of a function that we have reached in modern mathematics starts by fixing the values that the independent variable x can take on. We define that x must successively pass through the points of a certain 'point set'. The language used is therefore geometric […]." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The modern definition of a function in the context of real numbers is that it is a relationship, usually a formula, by which a correspondence is established between two sets A (the domain) and B (the range) of real numbers in such a manner that to every number in set A there corresponds only one number in set B." (Alan Jeffrey, "Mathematics for Engineers and Scientists", 1969)

"If you start with a number and form its square root, you get another number. The term for such an 'object' is function. You can think of a function as a mathematical rule that starts with a mathematical object-usually a number-and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as 'function' is transformation: the rule transforms the first object into the second. […] Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"If each change of a certain quantity results in a corresponding change of another quantity, we can say that there exists a functional relationship between those two quantities. Viewed in this manner, the idea of functions expands endlessly. The concept of functions is truly comprehensive, but while it is all encompassing, it is not fathomless; at least, not with respect to our current subject of manifolds. You might feel that linear functions or quadratic functions are far too specific and that you are sinking into the depths of the ocean called functions. However, you will be rescued from the ocean depths by understanding of the functions that are needed to describe manifolds. These functions are continuous, analytic, and differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])

"When graphing a function, the width of the line should be inversely proportional to the precision of the data." (Marvin J Albinak)

On Functions II: Homeomorphism

"From the point of view of general topology, homeomorphic spaces are the same. That is to say, the properties that interest us are those that, when true for one space, are true for all spaces homeomorphic to it." (Andrew H Wallace, "Differential Topology: First Steps", 1968)

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)

"Showing that two spaces are homeomorphic is a geometrical problem, involving the construction of a specific homeomorphism between them. The techniques used vary with the problem. […] Attempting to prove that two spaces are not homeomorphic to one another is a problem of an entirely different nature. We cannot possibly examine each function between the two spaces individually and check that it is not a homeomorphism. Instead we look for 'topological invariants' of spaces: an invariant may be a geometrical property of the space, a number like the Euler number defined for the space, or an algebraic system such as a group or a ring constructed from the space. The important thing is that the invariant be preserved by a homeomorphism- hence its name. If we suspect that two spaces are not homeomorphic, we may be able to confirm our suspicion by computing some suitable invariant and showing that we obtain different answers." (Mark A Armstrong, "Basic Topology", 1979)

"Topology has to do with those properties of a space which are left unchanged by the kind of transformation that we have called a topological equivalence or homeomorphism. But what sort of spaces interest us and what exactly do we mean by a 'space? The idea of a homeomorphism involves very strongly the notion of continuity [...]"  (Mark A Armstrong, "Basic Topology", 1979)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Topology studies the properties of geometrical objects that remain unchanged under transformations called homeomorphisms and deformations." (Victor V Prasolov, "Intuitive Topology", 1995)

"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)

"Two figures which can be transformed into one other by continuous deformations without cutting and pasting are called homeomorphic. […] The definition of a homeomorphism includes two conditions: continuous and one- to-one correspondence between the points of two figures. The relation between the two properties has fundamental significance for defining such a paramount concept as the dimension of space." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Intuitively, two spaces that are homeomorphic have the same general shape in spite of possible deformations of distance and angle. Thus, if two spaces are not homeomorphic, they will tend to look distinctly different. Our job is to specify the difference. To do this rigorously, we need to define some property of topological spaces and show that the property is preserved under transformations by any homeomorphism. Then if one space has the property and the other one does not have the property, there is no way they can be homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"The definition of homeomorphism was motivated by the idea of preserving the general shape or configuration of a geometric figure. Since path components are significant characteristics of a space, it is certainly reasonable that a homeomorphism will preserve the decomposition of a space into path components. […] Suppose we are given two geometric figures that we suspect are not topologically equivalent. If both of the figures are path-connected, counting components will not distinguish the spaces. However, we might be able to remove a special subset of one of the figures and count the number of components of the remainder. If no comparable set can be removed from the other space to leave the same number of components, we will then know that the two spaces are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"The easiest way to show two figures are homeomorphic is often to construct an explicit homeomorphism between them. But what if two figures are not homeomorphic? Surely we cannot be expected to check every function between the sets and show that it is not a homeomorphism. One of the goals of the field of topology is to discover easier ways of detecting the differences between spaces that are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"A topological property is, therefore, any property that is preserved under the set of all homeomorphisms. […] Homeomorphisms generally fail to preserve distances between points, and they may even fail to preserve shapes." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)

On Functions I

"Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e.g., a quality. For whatever ratio is found to exist between intensity and intensity, in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa." (Nicole Oresme, "Tractatus de configurationibus qualitatum et motuum" ["A treatise on the uniformity and difformity of intensities"], 1352) [definition of a functional relationship between two variables]

"Here, we call function of a variable magnitude, a quantity formed in whatever manner with that variable magnitude and constants." (Johann I Bernoulli, 1718)

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. […] Functions are divided into algebraic and transcendental. The former are those made up from only algebraic operations, the latter are those which involve transcendental operations."(Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

"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 take the word 'mapping' in the widest possible sense; any point of the spherical surface is represented on the plane by any desired rule, so that every point of the sphere corresponds to a specified point in the plane, and inversely." (Leonhard Euler, "On the representation of Spherical Surfaces onto the Plane", 1777)

"When variable quantities are so tied to each other that, given the values of some of them, we can deduce the values of all the others, we usually conceive these various quantities expressed in terms of several of them, which then bear the name independent variables; and the remaining quantities expressed in terms of the independent variables, are what we call functions of these same variables." (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"[…] a function of the variable x will be continuous between two limits a and b of this variable if between two limits the function has always a value which is unique and finite, in such a way that an infinitely small increment of this variable always produces an infinnitely small increment of the function itself." (Augustin-Louis Cauchy, "Mémoire sur les fonctions continues" ["Memoir on continuous functions"], 1844)

"If we designate by z a variable magnitude, which may take successively all possible real values, then, if to each of these values corresponds a unique value of the indeterminate magnitude w, we say that w is a function of z. […] This definition does not stipulate any law between the isolated values of the function, this is evident, because after this function has been dealt with for a given interval, the way it is extended outside this interval remains quite arbitrary." (Bernhard Riemann, 1851)

Calculus I: Differential Calculus I

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

"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)

"Everyone 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 analytical geometry." (Felix Klein, Jahresbericht der Deutsche Mathematiker Vereinigung Vol. 1, 1902)

"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)

"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)

"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) 

"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)

"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)

"Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity." (Mary Somerville)

On Differential Equations IV

"Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduced […] to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations […] contain the chief results of the theory; they express, in the most general and concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect forever with mathematical science one of the most important branches of natural  philosophy." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)

"Men have fallen in love with statues and pictures. I find it easier to imagine a man falling in love with a differential equation, and I am inclined to think that some mathematicians have done so. Even in a nonmathematician like myself, some differential equations evoke fairly violent physical sensations to those described by Sappho and Catallus when viewing their mistresses. Personally, I obtain an even greater 'kick' from finite difference equations, which are perhaps more like those which an up-to-date materialist would use to describe human behavior." (John B S Haldane, "The Inequality of Man and Other Essays", 1932)

"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 semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid V Kantorovich, "On one class of functional equations", 1936)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"The study of changes in the qualitative structure of the flow of a differential equation as parameters are varied is called bifurcation theory. At a given parameter value, a differential equation is said to have stable orbit structure if the qualitative structure of the flow does not change for sufficiently small variations of the parameter. A parameter value for which the flow does not have stable orbit structure is called a bifurcation value, and the equation is said to be at a bifurcation point." (Jack K Hale & Hüseyin Kocak, "Dynamics and Bifurcations", 1991)

"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Following the traditional classification in the field of control systems, a system that describes the input-output behavior in a way similar to a mathematical mapping without involving a differential operator or equation is called a static system. In contrast, a system described by a differential operator or equation is called a dynamic system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

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On Tensors I

"The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem 'linear' by reverting to infinitely small quantities." (Hermann Weyl, "Space - Time - Matter", 1922)

"The field equation may [...] be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium - in complete analogy with […] the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish." (Howard P Robertson, "Geometry as a Branch of Physics", 1949)

"The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction." (Hilary Putnam, "Mathematics, matter, and method", 1975)

"Maxwell's equations […] originally consisted of eight equations. These equations are not 'beautiful'. They do not possess much symmetry. In their original form, they are ugly. […] However, when rewritten using time as the fourth dimension, this rather awkward set of eight equations collapses into a single tensor equation. This is what a physicist calls 'beauty', because both criteria are now satisfied.  (Michio Kaku, "Hyperspace", 1995)

 "(…) the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object - one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical - including you." (Max Tegmark, "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality", 2014)

"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two- dimensional surface embedded in three- dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015) 

"In geometric and physical applications, it always turns out that a quantity is characterized not only by its tensor order, but also by symmetry." (Hermann Weyl, 1925)

"Ultra-modern physicists [are tempted to believe] that Nature in all her infinite variety needs nothing but mathematical clothing [and are] strangely reluctant to contemplate Nature unclad. Clothing she must have. At the least she must wear a matrix, with here and there a tensor to hold the queer garment together." (Sydney Evershed)

On Differentiability I

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"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)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] differentiation is performed by focusing on the behavior of a function near one point. A quantity obtained in this manner is essentially a local quantity. Is it possible that such local quantities can show us something very basic about global properties such as smoothness? Does there exist a place in mathematics which would enable us to study the relationship between local and global quantities?" (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To consider differentiable functions, we must introduce a coordinate system on the plane and thereby to concentrate on the world of numbers.[...] a continuous function defined on a plane can be differentiable or nondifferentiable depending on the choice of coordinates. [...] the choice of coordinates on the plane determines which functions among the continuous functions should be selected as differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future)." (Benoît Mandelbrot, "The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward", 2004)?

"Roughly speaking, a function defined on an open set of Euclidean space is differentiable at a point if we can approximate it in a neighborhood of this point by a linear map, which is called its differential (or total derivative). This differential can be of course expressed by partial derivatives, but it is the differential and not the partial derivatives that plays the central role." (Jacques Lafontaine, "An Introduction to Differential Manifolds", 2010)

"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])

Catastrophe Theory III

"On the plane of philosophy properly speaking, of metaphysics, catastrophe theory cannot, to be sure, supply any answer to the great problems which torment mankind. But it favors a dialectical, Heraclitean view of the universe, of a world which is the continual theatre of the battle between 'logoi', between archetypes." (René F Thom, "Catastrophe Theory: Its Present State and Future Perspectives", 1975)

"At the large scale where many processes and structures appear continuous and stable much of the time, important changes may occur discontinuously. When only a few variables are involved, as well as an optimizing process, the event may be analyzed using catastrophe theory. As the number of variables in- creases the bifurcations can become more complex to the point where chaos theory becomes the relevant approach. That chaos theory as well as the fundamentally discontinuous quantum processes may be viewed through fractal eyeglasses can also be admitted. We can even argue that a cascade of bifurcations to chaos contains two essentially structural catastrophe points, namely the initial bifurcation point at which the cascade commences and the accumulation point at which the transition to chaos is finally achieved." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"Catastrophe theory is a local theory, telling us what a function looks like  in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...]  involve extrapolating these rock-solid, local results to regions that may  well be distant in time and space from the singularity." (John L Casti, "Five Golden Rules", 1995)

"Chaos and catastrophe theories are among the most interesting recent developments in nonlinear modeling, and both have captured the interests of scientists in many disciplines. It is only natural that social scientists should be concerned with these theories. Linear statistical models have proven very useful in a great deal of social scientific empirical analyses, as is evidenced by how widely these models have been used for a number of decades. However, there is no apparent reason, intuitive or otherwise, as to why human behavior should be more linear than the behavior of other things, living and nonliving. Thus an intellectual movement toward nonlinear models is an appropriate evolutionary movement in social scientific thinking, if for no other reason than to expand our paradigmatic boundaries by encouraging greater flexibility in our algebraic specifications of all aspects of human life." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"[...] chaos and catastrophe theories per se address behavioral phenomena that are consequences of two general types of nonlinear dynamic behavior. In the most elementary of behavioral terms, chaotic phenomena are a class of deterministic processes that seem to mimic random or stochastic dynamics. Catastrophe phenomena, on the other hand, are a class of dynamic processes that exhibit a sudden and large scale change in at least one variable in correspondence with relatively small changes in other variables or, in some cases, parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Chaos and catastrophe theories directly address the social scientists' need to understand classes of nonlinear complexities that are certain to appear in social phenomena. The probabilistic properties of many chaos and catastrophe models are simply not known, and there is little likelihood that general procedures will be developed soon to alleviate the difficulties inherent with probabilistic approaches in such complicated settings." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Fundamental to catastrophe theory is the idea of a bifurcation. A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed. This occurs when an attractor in the system changes in response to change in the value of a parameter. A catastrophe is one type of bifurcation. The broader framework within which catastrophes are located is called dynamical bifurcation theory." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Probably the most important reason that catastrophe theory received as much popular press as it did in the mid-1970s is not because of its unchallenged mathematical elegance, but because it appears to offer a coherent mathematical framework within which to talk about how discontinuous behaviors - stock market booms and busts or cellular differentiation, for instance - might emerge as the result of smooth changes in the inputs to a system, things like interest rates in a speculative market or the diffusion rate of chemicals in a developing embryo. These kinds of changes are often termed bifurcations, and playa central role in applied mathematical modeling. Catastrophe theory enables us to understand more clearly how - and why - they occur." (John L Casti, "Five Golden Rules", 1995)

"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)

"The reason catastrophe theory can tell us about such abrupt changes in a system's behavior is that we usually observe a dynamical system when it's at or near its steady-state, or equilibrium, position. And under various assumptions about the nature of the system's dynamical law of motion, the set of all possible equilibrium states is simply the set of critical points of a smooth function closely related to the system dynamics. When these critical points are nondegenerate, Morse's Theorem applies. But it is exactly when they become degenerate that the system can move sharply from one equilibrium position to another. The Thorn Classification Theorem tells when such shifts will occur and what direction they will take." (John L Casti, "Five Golden Rules", 1995)

On Manifolds II (Geometry II)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system." (Charles P Steinmetz, [New York Times interview] 1911)

"That branch of mathematics which deals with the continuity properties of two- (and more) dimensional manifolds is called analysis situs or topology. […] Two manifolds must be regarded as equivalent in the topological sense if they can be mapped point for point in a reversibly neighborhood-true (topological) fashion on each other." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"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)

"In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?" (Sir John A Thomson, "The System of Animate Nature", 1920)

"An 'empty world', i. e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy." (Howard P Robertson, "On Relativistic Cosmology", 1928)

"Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but 'logical relations' or 'artificial manifolds'. They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"We must [...] maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"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)

Kenji Ueno - Collected Quotes

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Analyticity is the property of a differentiable function y = f(x) that can be represented by the infinite series for all x near each point x0." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"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)

"[…] continuity appears when we try to mathematically express continuously changing phenomena, and differentiability is the result of expressing smoothly changing phenomena."  (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Continuous functions can change freely, while analytic functions are rigid. In this sense, we can say that continuous and analytic functions are antipodal." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] differentiation is performed by focusing on the behavior of a function near one point. A quantity obtained in this manner is essentially a local quantity. Is it possible that such local quantities can show us something very basic about global properties such as smoothness? Does there exist a place in mathematics which would enable us to study the relationship between local and global quantities?" (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Even though there are no methods to represent each differentiable function by an 'equation', we can still investigate differentiable functions by various analytic methods. Because of this, we can say that differentiable functions have more mathematical reality than continuous functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Here we have the following question: 'Which concept is closer to the concept of differentiability -continuity or analyticity?' The answer depends upon the point of view. Our point of view is that continuity appears when we try to mathematically express continuously changing phenomena, and differentiability is the result of expressing smoothly changing phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If each change of a certain quantity results in a corresponding change of another quantity, we can say that there exists a functional relationship between those two quantities. Viewed in this manner, the idea of functions expands endlessly. The concept of functions is truly comprehensive, but while it is all encompassing, it is not fathomless; at least, not with respect to our current subject of manifolds. You might feel that linear functions or quadratic functions are far too specific and that you are sinking into the depths of the ocean called functions. However, you will be rescued from the ocean depths by understanding of the functions that are needed to describe manifolds. These functions are continuous, analytic, and differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"In abstract mathematics, special attention is given to particular properties of numbers. Then those properties are taken in a very pure (and primitive) form. Those properties in pure form are then assigned to a given set. Therefore, by studying in details the internal mathematical structure of a set, we should be able to clarify the meaning of original properties of the objects. Likewise, in set theory, numbers disappear and only the concept of sets and characteristic properties of sets remain." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"In view of the developments of abstract mathematics, the first thing mathematicians studied was how to extract the property of 'nearness' from the set of numbers. If the property of nearness could be extracted using a few axioms, and if it was possible to associate the extracted property with a set, then the resulting set would provide an abstract scene to study 'nearness'." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] moving from the concept of continuity to differentiability, and then to analyticity, we are moving from weaker properties to stronger ones. Therefore, the relations between the corresponding properties of functions can be expressed as follows: {continuous functions} > {differentiate functions} > {analytic functions}." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Similarly to the graphs of continuous functions, graphs of differentiable (smooth) functions which coincide in a neighborhood of a point P can branch off outside of the neighborhood. Because of this property, differentiable functions can represent smoothly changing natural phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"The best example to indicate the rigidity of analytic functions is a soap film (with little viscosity) on a wire frame (think of a bubble blower). The soap film, which is created by the surface tension, stretches across the wire frame and is known to have analyticity. Therefore, if we try to change a certain region of the film by tapping it with a stick, then the film loses analyticity and will immediately brake." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] the only characteristic property that continuous functions have is that near objects are sent to corresponding near objects, that is, a convergent sequence is mapped to the corresponding convergent sequence. It is reasonable to say that we cannot expect to extract from that property neither numerical consequences, nor a method to extensively study continuity. On the contrary, analytic functions can be represented by equations (precisely speaking, by infinite series). Compared to analytic functions, continuous functions, in general, are difficult to represent explicitly, although they exist as a concept." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"The property of smoothness includes the property of continuity. The notion of a topological space was born from the development of abstract algebra as a universal notion for the property of continuity." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To consider differentiable functions, we must introduce a coordinate system on the plane and thereby to concentrate on the world of numbers.[...] a continuous function defined on a plane can be differentiable or nondifferentiable depending on the choice of coordinates. [...] the choice of coordinates on the plane determines which functions among the continuous functions should be selected as differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To describe the property of smoothness, differentiable functions should be specified first. To do so, coordinates need to be introduced on the topological space. Those coordinates can be local coordinates such as the ones used by Gauss. Once coordinates are introduced around a point a in a topological space, differentiable functions near the point a are distinguished from the continuous functions in the region near a. If different coordinates are chosen, then a different set of differentiable functions is distinguished. In other words, the choice of local coordinates determines the notion of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"When the absolute concept of coordinate systems introduced by Descartes shifted to the relative concept of coordinate systems introduced by Gauss, a clear differences between continuity and differentiability emerged."  (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"When we study the concept of continuity by itself, numbers are not necessary as long as you are dealing with objects which have the property of 'nearness' . Therefore, if we can introduce the notion of 'nearness' detached from numbers from a purely abstract point of view, then we can discuss topics related to continuity based upon this notion. This approach enables us to become familiar with the science we call mathematics." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

On Continuity I (Calculus)

"Since [...] nature is a principle of motion and mutation [...] it is necessary that we should not be ignorant of what motion is [...] But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. [..] frequently [...] those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous." (Aristotle, "Physics", cca. 350 BC)

"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD) 

"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)

"There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)

"Every quantity which, keeping the same expression, increases or diminishes continually (non per saltum), is called a variable, and that which, with the same expression, keeps the same value, is called fixed or constant." (Pierre Varignon, "Eclaircissemens sur l'Analyse des Infinimens Petits", 1725)

"In fact, a similar principle of hardness cannot exist; it is a chimera which offends that general law which nature constantly observes in all its operations; I speak of that immutable and perpetual order, established since the creation of the Universe, that can be called the LAW OF CONTINUITY, by virtue of which everything that takes place, takes place by infinitely small degrees. It seems that common sense dictates that no change can take place at a jump; natura non operatur per saltion; nothing can pass from one extreme to the other without passing through all the degrees in between." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"The Law of Continuity, as we here deal with it, consists in the idea that [...] any quantity, in passing from one magnitude to another, must pass through all intermediate magnitudes of the same class. The same notion is also commonly expressed by saying that the passage is made by intermediate stages or steps; [...] the idea should be interpreted as follows: single states correspond to single instants of time, but increments or decrements only to small areas of continuous time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium", 1758)

"Any change involves at least two conditions, one preceding and one following, which are distinct from one another in such a way that the difference between the former and the latter can be established. Now the law of continuity prohibits the thing which is being changed to transcend abruptly from the former to the latter. It must pass through an intermediate condition which is as little distinct from the previous as from the subsequent one. And because the difference between this intermediate condition and the previous condition can be established still, there must be an intermediate condition between these two as well, and this must continue in the same way, until the difference between the previous condition and the one immediately succeeding it vanishes. As long as the set of these intermediate conditions can be established, every difference between one and the next can be established as well: hence their set must become larger than any given set if these differences shall vanish, and thus we imagine infinitely many conditions where one differs from the next to an infinitely small degree." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" [Beginnings of the Analysis of the Infinite"], 1766)

"It is held because of this law in particular, that no change may occur suddenly, but rather that every change always passes by infinitely small stages, of which the trajectory of a point in a curved line provides a first example." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"Whoever wishes to extend this law [of continuity] to the real must justify his inferences by a law other than that, the suspicion remaining that he took images for things." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"[continuity] could be only appearance, and in this case Euler’s entire argument against the atoms would disappear; for one would be justified to apply the law of continuity only where experience shows that it agrees with the phenomena. [...]. The law of continuity thus belongs to the clothes of things which we must need rely on wherever reality seems impenetrably cloaked with it, but which we do not consider to be reality itself, and which we may still less cloak with things which do not serve us to see them." (Johann S T Gehler, "Physikalisches Wörterbuch" Bd. 4, 1798)

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be." (Bertrand Russell, "Mysticism and Logic and Other Essays", cca. 1910)

"The course of the values of a continuous function is determined at all points of an interval, if only it is determined for all rational points of this interval." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"A common and very powerful constraint is that of continuity. It is a constraint because whereas the function that changes arbitrarily can undergo any change, the continuous function can change, at each step, only to a neighbouring value." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"As a simple trick, the discrete can often be carried over into the continuous, in a way suitable for practical purposes, by making a graph of the discrete, with the values shown as separate points. It is then easy to see the form that the changes will take if the points were to become infinitely numerous and close together." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Analysis is primarily concerned with limit processes and continuity, so it is not surprising that mathematicians thinking along these lines soon found themselves studying (and generalizing) two elementary concepts: that of a convergent sequence of real or complex numbers, and that of a continuous function of a real or complex variable." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future)." (Benoît Mandelbrot, "The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward", 2004)

"Continuity is the rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable - say x - is associated with a value of a dependent variable - say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question 'How close?' is asked, difficulties arise." (Erik Gregersen [Ed.], "Math Eplained: The Britannica Guide to Analysis and Calculus", 2011)

Keith J Devlin - Collected Quotes

"In fact, the answer to the question 'What is mathematics?' has changed several times during the course of history [...] It was only in the last twenty years or so that a definition of mathematics emerged on which most mathematicians agree: mathematics is the science of patterns." (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"Mathematics is a product - a discovery - of the human mind. It enables us to see the incredible, simple, elegant, beautiful, ordered structure that lies beneath the universe we live in. It is one of the greatest creations of mankind - if it is not indeed the greatest." (Keith Devlin, "Life By the Numbers", 1998)

"At this stage you might be thinking that there is no justification for calling something of the form a+bi a number, even if you are prepared to countenance i = √-1 in the first place. But remember, it is not what numbers are that matters, but how they behave. Provided the complex numbers have a workable and useful (either in mathematics itself or possibly in a wider context) arithmetic, possibly forming a field, then they have as much right to be called 'numbers' as do any others." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"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)

"Sometimes the solution of a long-standing problem marks the end (or the beginning of the end) of a mathematical era or field, the culmination of years of effort. On other occasions it may open up an entire new area of research, possibly previously undreamt of." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"The reason why a 'crude', experimental approach is not adequate for determining mathematical truth lies in the nature of what mathematics is and is intended to be. Though its roots lie in the physical world, mathematics is a precise and idealized discipline. The 'points', 'lines', 'planes', and other ideal constructs of mathematics have no exact counterpart in reality. What the mathematician does is to take a totally abstract, idealized view of the world, and reason with his abstractions in an entirely precise and rigorous fashion." (Keith Devlin, "Mathematics: The New Golden Age", 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)

"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)

"A neural network is a particular kind of computer program, originally developed to try to mimic the way the human brain works. It is essentially a computer simulation of a complex circuit through which electric current flows." (Keith J Devlin & Gary Lorden, "The Numbers behind NUMB3RS: Solving crime with mathematics", 2007)

"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence." (Keith J Devlin)

Joseph-Louis de Lagrange - Collected Quotes

"I consider it as one of the most important steps made by Analysis in the last period, that of not being bothered any more by imaginary quantities, and to be able to submit them to calculus, in the same way as the real ones." (Joseph-Louis de Lagrange, [letter to Antonio Lorgna] 1777)

"It seems to me that the mine [of mathematics] has already gone too deep, and that unless someone discovers new veins it will be necessary sooner or later to abandon it. Physics and chemistry now offer more brilliant riches and are more easily exploited; even the taste of the century seems to have entirely turned in this direction, and it is not impossible that the place of geometry in the academies will one day become like that which is currently occupied by the chairs of Arabic in the universities." (Joseph-Louis de Lagrange, [letter to D'Alembert] 1781)

"Statics is the science of the equilibrium of forces. In general, force or power is the cause, whatever it may be, which induces or tends to impart motion to the body to which it is applied. The force or power must be measured by the quantity of motion produced or to be produced. In the state of equilibrium, the force has no apparent action. It produces only a tendency for motion in the body it is applied to. But it must be measured by the effect it would produce if it were not impeded. By taking any force or its effect as unity, the relation of every other force is only a ratio, a mathematical quantity, which can be represented by some numbers or lines. It is in this fashion that forces must be treated in mechanics." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of  shortening and simplifying our proofs." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of  arithmetic, to translate it into lines necessitates the use of geometry." (Joseph-Louis de Lagrange, "Leçons Élémentaires de Mathématiques", 1795)

"As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection." (Joseph-Louis de Lagrange, "Leçons Élémentaires de Mathématiques", 1795)

"I regarded as quite useless the reading of large treatises of pure analysis: too large a number of methods pass at once before the eyes. It is in the works of application that one must study them; one judges their utility there and appraises the manner of making use of them." (Joseph-Louis de Lagrange)

"In general, nothing measurable can be measured except by fractions expressing the result of the measurement, unless the measure be contained an exact number of times in the thing to be measured." (Joseph-Louis de Lagrange, "Leçons Élémentaires de Mathématiques", 1795)

"The ordinary operations of algebra suffice to resolve problems in the theory of curves." (Joseph-Louis de Lagrange, "Théorie des fonctions analytiques", 1797)