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Mathematical analysis

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Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

Mathematical analysis or just analysis is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions.

CONTENT : A - F , G - L , M - R , S - Z , See also , External links

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Quotes are arranged alphabetically by author

A - F

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  • My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many can have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes. It is really interesting to seek the cause.
  • 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.
  • 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.
    • Joseph Fourier, Theory of Heat, [Freeman], (Cambridge, 1878), Chap. 1, p. 12.

G - L

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  • The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "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." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths.
  • Analysis and synthesis, though commonly treated as two different methods, are, if properly understood, only the two necessary parts of the same method. Each is the relative and correlative of the other.
    • Sir W. Hamilton, reported in Austin Allibone ed. Prose Quotations from Socrates to Macaulay. (1903), p. 34
  • The Mss. [of Euclid’s Elements] contain a curious addition to XIII. I-5 in the shape of analyses and syntheses for each proposition prefaced by the heading:
    "What is analysis and what is synthesis.
    "Analysis is the assumption of that which is sought as if it were admitted <and the arrival> by means of its consequences at something admitted to be true.
    "Synthesis is an assumption of that which is admitted <and the arrival> by means of its consequences at something admitted to be true."
    There must apparently be some corruption in the text; it does not, in the case of synthesis, give what is wanted. B and V have, instead of "something admitted to be true," the words "the end or attainment of what is sought." ...
    the addition is altogether alien from the plan and manner of the Elements. The interpolation took place before Theon's time, and the probability is that it was originally in the margin, whence it crept into the text of P after XIII. 5. Heiberg... cited the remark of Pappus at the beginning of his "comparisons of the five [regular solid] figures which have an equal surface," to the effect that he will not use "the so-called analytical investigation by means of which some of the ancients effected their demonstrations." More recently Heiberg conjectures that the author is Heron, on the ground that the sort of analysis and synthesis recalls Heron's remarks on analysis and synthesis in his commentary on the beginning of Book II. and his quasi-algebraical alternative proofs of propositions in that Book.
  • 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.
    • E. W. Hobson "Presidential Address British Association for the Advancement of Science, Section A, (1910}" in: Nature, Vol. 84, p. 287
  • Vieta presented his analytic art as "the new algebra" and took its name from the ancient mathematical method of "analysis", which he understood to have been first discovered by Plato and so named by Theon of Smyrna. Ancient analysis is the 'general' half of a method of discovering the unknown in geometry; the other half, "synthesis", being particular in character. The method was defined by Theon like this: analysis is the "taking of the thing sought as granted and proceeding by means of what follows to a truth that is uncontested"'. Synthesis, in turn, is "taking the thing that is granted and proceeding by means of what follows to the conculsion and comprehension of the thing sought" (Vietae 1992: 320). The transition from analysis to synthesis was called "conversion", depending on whether the discovery of the truth of a geometrical theorem or the solution ("construction") to a geometrical problem was being demonstrated, the analysis was called respectively "theoretical" or "problematical".
    • Burt C. Hopkins, "Nastalgia and Phenomenon: Hussel and Patočka on the End of the Ancient Cosmos," The Phenomenological Critique of Mathematisation and the Question of Responsibility: Formalisation and the Life-World (2015) ed., Ľubica Učník, Ivan Chvatík, Anita Williams, p. 71, Contributions to Phenomenology 76
  • 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.
    • Laplace, A Philosophical Essay on Probabilities, [Truscott and Emory] (New York 1902), p. 176.
  • Functions are the bread and butter of modern scientists, statisticians, and economists. Once many repeated... experiments and observations produce the same functional interrelationships, those may acquire the... status of laws of nature—mathematical descriptions... Descartes' ideas... opened the door for a systematic mathematization of everything—the very essence of the notion that God is a mathematician. ...[B]y establishing the equivalence of two perspectives of mathematics (algebraic and geometric) previously considered disjoint, Descartes expanded the horizons of mathematics and paved the way to the modern era of analysis, which allows [us] to comfortably cross from one mathematical discipline to another.

M - R

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  • A great part of the progress of formal thought... has been due to the invention of what we may call stenophrenic, or short-mind, symbols. These... disengage the mind from the consideration of ponderous and circuitous mechanical operations and economise its energies for the performance of new and unaccomplished tasks of thought. And the advancement of those sciences has been most notable which have made the most extensive use of these... Here mathematics and chemistry stand pre-eminent. The ancient Greeks... even admitting that their powers were more visualistic than analytic, were yet so impeded by their lack of short-mind symbols as to have made scarcely any progress whatever in analysis. Their arithmetic was a species of geometry. They did not possess the sign for zero, and also did not make use of position as an indicator of value. ...The historical calculations of Archimedes, his approximation to the value of π, etc., owing to this lack of appropriate... symbols, entailed enormous and incredible labors, which, if they had been avoided, would... have led to [even] great[er] discoveries.
    • Thomas J. McCormack, "Joseph Louis Lagrange. Biographical Sketch" (1898) in his translation of Joseph Louis Lagrange, Lectures on Elementary Mathematics (1898); 2nd edition (1901) p. vii.
  • [A]t the close of the Middle Ages, when the so-called Arabic figures became established throughout Europe with the symbol 0 and the principle of local value, immediate progress was made in the art of reckoning. The problems... led up to the general solutions of equations of the third and fourth degree by the Italian mathematicians of the sixteenth century. Yet even these discoveries were made in somewhat the same manner as problems in mental arithmetic are now solved in common schools; for the present signs of plus, minus, and equality, the radical and exponential signs, and especially the systematic use of letters for denoting general quantities in algebra, had not yet become universal. The last step was definitively due to... Vieta... and the mighty advancement of analysis resulting therefrom can hardly be measured or imagined.
    • Thomas J. McCormack, "Joseph Louis Lagrange. Biographical Sketch" (1898) in his translation of Joseph Louis Lagrange, Lectures on Elementary Mathematics (1898); 2nd edition (1901) p. viii.
  • The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
    • John von Neumann, originally published in Works of the Mind (edited by Robert B. Heywood, 1947) and reproduced in von Neumann, Collected Works (edited by A. H. Taub, 1961), Vol. I, "The Mathematician", pp. 1–10.
  • By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis: and the Synthesis consists in assuming the Causes discover'd, and establish'd as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations.
  • The investigation of difficult things by the method of analysis ought ever to precede the method of composition.
    • Isaac Newton, reported in Austin Allibone ed. Prose Quotations from Socrates to Macaulay. (1903), p. 34.
  • Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.
  • 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.
    • So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the intimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.
  • 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.
    • Bertrand Russell, Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 5

S - Z

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  • In mathematics there is a certain way of seeking the truth, a way which Plato is said first to have discovered and which was called "analysis" by Theon and was defined by him as "taking the thing sought as granted and proceeding by means of what follows to a truth which is uncontested"; so, on the other hand, "synthesis" is "taking the thing that is granted and proceeding by means of what follows to the conclusion and comprehension of the thing sought." And although the ancients set forth a twofold analysis, the zetetic and the poristic, to which Theon's definition particularly refers, it is nevertheless fitting that there be established also a third kind, which may be called rhetic or exegetic, so that there is a zetetic art by which is found the equation or proportion between the magnitude that is being sought and those that are given, a poristic art by which from the equation or proportion the truth of the theorem set up is investigated, and an exegetic art by which from the equation set up or the proportion, there is produced the magnitude itself which is being sought. And thus, the whole threefold analytic art, claiming for itself this office, may be defined as the science of right finding in mathematics. ...the zetetic art does not employ its logic on numbers—which was the tediousness of the ancient analysts—but uses its logic through a logistic which in a new way has to do with species [of number]...
  • François Viète, In Artem Aanalyticem Isagoge (1591) Ch. 1, as quoted by Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1934-1936) Appendix
  • 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.
    • H. G. Wells. Mankind in the Making, (London, 1904), pp. 191-192.
  • 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.

See also

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Wikipedia
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Wikipedia has an article about:



Mathematics
Mathematicians
(by country)

Abel • Anaxagoras • Archimedes • Aristarchus of Samos • Aryabhata • Averroes • Arnold • Banach • Bhāskara I • Bhāskara II • Brahmagupta • Cantor • Cartan • Chern • Cohen • Descartes • Diophantus • Erdős • Euclid • Euler • Fourier • Gauss • Gödel • Grassmann • Grothendieck • Hamilton • Hilbert • Hypatia • Lagrange • Laplace • Leibniz • Milnor • Newton • von Neumann • Noether • Penrose • Perelman • Poincaré • Pólya • Pythagoras • Ramanujan • Riemann • Russell • Schwartz • Serre • Tao • Tarski • Thales • Turing • Weil • Weyl • Wiles • Witten

Numbers

1 • 23 • 360 • e • π • Fibonacci numbers • Fraction • Irrational number • Negative number • Number • Prime number • Quaternion • Octonion • Sedenion

Concepts

Abstraction • Algorithms • Axiomatic system • Completeness • Deductive reasoning • Differential equation • Dimension • Ellipse • Elliptic curve • Exponential growth • Infinity • Integration • Geodesic • Induction • Proof • Partial differential equation • Principle of least action • Prisoner's dilemma • Probability • Randomness • Theorem • Topological space • Wave equation

Results

Euler's identity • Fermat's Last Theorem

Pure math

Abstract algebra • Algebra • Analysis • Algebraic geometry (Sheaf theory) • Algebraic topology • Arithmetic • Calculus • Category theory • Combinatorics • Commutative algebra • Complex analysis • Differential calculus • Differential geometry • Differential topology • Ergodic theory • Foundations of mathematics • Functional analysis • Game theory • Geometry • Global analysis • Graph theory • Group theory • Harmonic analysis • Homological algebra • Invariant theory • Logic • Non-Euclidean geometry • Nonstandard analysis • Number theory • Numerical analysis • Operations research • Representation theory • Ring theory • Set theory • Sheaf theory • Statistics • Symplectic geometry • Topology

Applied math

Computational fluid dynamics • Econometrics • Fluid mechanics • Mathematical physics • Science

History of math

Algorism • Ancient Greek mathematics • Euclid's Elements • Hindu–Arabic numeral system • History of algebra • History of calculus • History of logarithms • Indian mathematics • Principia Mathematica

Other

Mathematics and mysticism • Mathematical Platonism • Mathematics education • Mathematics, from the points of view of the Mathematician and of the Physicist • Philosophy of mathematics • Unification in science and mathematics


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