Mathematical analysis

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."

"In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered."

"I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in certain order, or that that order, has been to great extent necessary – being determined by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for the evolution had arrived."

"THEY who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance ; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation."

"The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it.""

"The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions."

"Mathematical Analysis is... the true rational basis of the whole system of our positive knowledge."

"Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena. . . . but if the employment of mathematical analysis should ever become so preponderant in chemistry (an aberration which is happily almost impossible) it would occasion vast and rapid retrogradation...."

"Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression."

"So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically."

"Machine-held strings of binary digits can simulate a great many kinds of things, of which numbers are just one kind. For example, they can simulate automobiles on a freeway, chess pieces, electrons in a box, musical notes, Russian words, patterns on a paper, human cells, colors, electrical circuits, and so on. To think of a computer as made up essentially of numbers is simply a carryover from the successful use of mathematical analysis in studying models. Most of this series of lectures has been devoted to applications of computers, and this is not the time to give details about their usefulness. I merely wish to point out certain types of things being done with computers today that could not have been done in 1945. Some of these are technological, some are intellectual."

"Mathematical analysis is co-extensive with nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures ; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. Its chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them. If matter escapes us, as that of air and light because of its extreme tenuity, if bodies are placed far from us in the immensity of space, if man wishes to know the aspect of the heavens at successive periods separated by many centuries, if gravity and heat act in the interior of the solid earth at depths which will forever be inaccessible, mathematical analysis is still able to trace the laws of these phenomena. It renders them present and measurable, and appears to be the faculty of the human mind destined to supplement the brevity of life and the imperfection of the senses, and what is even more remarkable, it follows the same course in the study of all phenomena; it explains them in the same language, as if in witness to the unity and simplicity of the plan of the universe, and to make more manifest the unchangeable order which presides over all natural causes."

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

"Mathematical Analysis is as extensive as nature herself."

"Perhaps the least inadequate description of the general scope of modern Pure Mathematics I will not call it a definition would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations."

"[D]uring the last half-century, number and measurable quantity have been separated... the idea of number alone has been recognized as the foundation upon which Mathematical Analysis rests, and the theory of extensive magnitude is now regarded as a separate department in which the methods of Analysis are applicable, but as no longer forming part of the foundation upon which Analysis itself rests."

"I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i.e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion. I have succeeded in finding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufficiently simple and effective for their solution to be made completely achievable under practical conditions."

"Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity."

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"The analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist."

"L'analyse mathématique, n'est elle donc qu'un vain jeu d'esprit? Elle ne peut pas donner au physicien qu'un langage commode; n'est-ce pa là un médiocre service, dont on aurait pu se passer à la rigueur; et même n'est il pas à craindre que ce langage artificiel ne soit pas un voile interposé entre la réalité at l'oeil du physicien? Loin de là, sans ce langage, la pluspart des anaologies intimes des choses nous seraient demeurées à jamais inconnues; et nous aurions toujours ignoré l'harmonie interne du monde, qui est, nous le verrons, la seule véritable réalité objective."

"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself."

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and to write."

"As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler."

"Le développement naturel de cette étude conduisit bientôt les géomètres à embrasser dans leurs recherches les valeurs imaginaires de la variable aussi bien que les valeurs réelles. La théorie de la série de Taylor, celle des fonctions elliptiques, la vaste doctrine de Cauchy firent éclater la fécondité de cette généralisation. Il apparut que, entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe."

"This is often quoted in English in the following form:"

"At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the Riemann hypothesis holds."

"The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes."

"Mathematical techniques to achieve numerical solutions for partial differential equations began to appear about the turn of the century. The first definitive work was carried out by Richardson, who in a paper delivered to the in London in 1910 introduced a finite-difference technique for numerical solution of . Called a "relaxation technique," that approach is still used today to obtain numerical solutions for so-called s (the equations that govern inviscid subsonic flows are such equations). However, modern numerical analysis is usually considered to have begun in 1928, when Courant, Friedrichs, and Lewy published a definitive paper on the numerical solution of so-called s (the equations that govern inviscid compressible flow are such equations)."

"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."

"Numerical analysis is often considered neither beautiful nor, indeed, profound. Pure mathematics is beautiful if your heart goes after the joy of abstraction, applied mathematics is beautiful if you are excited by mathematics as a means to explain the mystery of the world around us. But numerical analysis? Surely, we compute only when everything else fails, when mathematical theory cannot deliver an answer in a comprehensive, pristine form and thus we are compelled to throw a problem onto a number-crunching computer and produce boring numbers by boring calculations. This, I believe, is nonsense."

"One of the most important theorems in calculus is the Mean Value Theorem (MVT), which is used to prove many theorems of both differential and integral calculus, as well as other subjects such as numerical analysis. MVT is said to be the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance. The proof of the Mean-Value Theorem is based on a special case of it known as Rolle’s Theorem."

"In the 1950s and 1960s, the founding fathers of the field discovered that inexact arithmetic can be a source of danger, causing errors in results that "ought" to be right. The source of such problems is numerical instability: that is, the amplification of rounding errors from microscopic to macroscopic scale by certain modes of computation. These men, including von Neumann, Wilkinson, Forsythe, and Henrici, took great pains to publicize the risks of careless reliance on machine arithmetic. These risks are very real, but the message was communicated all too successfully, leading to the current widespread impression that the main business of numerical analysis is coping with rounding errors. In fact, the main business of numerical analysis is designing algorithms that converge quickly; rounding-error analysis, while often a part of the discussion, is rarely the central issue. If rounding errors vanished, 90% of numerical analysis would remain."

"The subject of numerical analysis has ancient roots, and it has had periods of intense development followed by long periods of consolidation. In many cases, the new developments have coincided with the introduction of new forms of computing machines. For example, many of the basic theorems about computing solutions of ordinary s were proved soon after desktop adding machines became common at the turn of the 20th century. The emergence of the digital computer in the mid-20th century spurred interest in solving partial differential equations and large systems of linear equations, as well as many other topics. The advent of parallel computers similarly stimulated research on new classes of algorithms. However, many fundamental questions remain open, and the subject is an active area of research today."

"A numerical equation is said to be analysed as soon as we discover the several limits, or pairs of numbers, within which all its unequal real roots lie individually, and its equal roots in distinct groups; that is, as soon as these unequal roots, and groups of equal roots, are all separated and severally enclosed, each between two assignable numbers."

"There are three main tools in nonstandard analysis. One is the transference principle, which roughly states that the same assertions of the formal language are true in the standard universe as in the nonstandard universe. It is typically used by proving a desired result in the nonstandard universe, and then, noting that the result is expressible in the language, concluding that it holds in the standard universe as well. Another technique is concurrence. This is a logical technique that guarantees that the extended structure contains all possible completions, compactifications, and so forth. The third technique is internality. A set s of elements of the nonstandard universe is internal if s itself is an element of the nonstandard universe; otherwiise, s is external. A surprislingly useful method of proof is one by reductio ad absurdum in which the contradiction is that some set one knows to be external would in fact be internal under the assumption being refuted."

"Nonstandard analysis has proved to be a natural framework for studying the local properties of Banach spaces. The central construction in this approach is the nonstandard hull, introduced by Luxemburg ... . Not only is this a useful tool in studying the local theory of Banach space geometry, but also nonstandard hulls arise naturally in many other places within nonstandard analysis."

"We describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondifferentiability of fractals and their infinite length forbid a complete description based on usual real numbers. We show that using nonstandard analysis it is possible to solve this problem: A class of nonstandard curves (whose standard part is the usual fractal) is defined so that a curvilinear coordinate along the fractal can be built, this being the first step towards the possible definition and study of a fractal space. We mention fields of physics to which such a formalism could be applied in the future."

"It was in 1966 that A. Robinson's book ... on nonstandard analysis ... appeared. In it, a first rigorous foundation of the theory of infinitesmals was developed. In fact, A. Robinson had been using theorems in mathematical logic in the fifties to derive known mathematical results in a neoclassical way. His methods were based on the theory of models and in particular on the Löwenheim-Skolem theorem."

"... in the author's opinion this is the most important advantage of nonstandard analysis over standard analysis: To have convenient (almost "explicit") representations of certain objects like Hahn-Banach limits for which by standard methods more or less only their mere existence can be proved with the axiom of choice."