# Analysis

**Analysis** is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development.

## Quotes[edit]

- 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.
- Niels Henrik Abel, Letter to Professor Christoffer Hansteen (1826)
*Oeuvres Complètes de N. H. Abel, mathematician, Nouvelle edition*(1881) ed., Peter Ludwig Mejdell Sylow & Sophus Lie, Vol. 2, pp. 263-265, as quoted by Øystein Ore,*Niels Henrik Abel: Mathematician Extraordinary*(1957) p. 113.

- Niels Henrik Abel, Letter to Professor Christoffer Hansteen (1826)

- Philosophers hasten too much from the analytic to the synthetic method ; that is, they draw general conclusions from too small a number of particular observations and experiments.
- Lord Bolingbroke, reported in Austin Allibone ed.
*Prose Quotations from Socrates to Macaulay.*(1903), p. 34

- Lord Bolingbroke, reported in Austin Allibone ed.

- 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."- Florian Cajori,
*A History of Mathematics*(1893). p. 30

- Florian Cajori,

- The idea that God may be approached and understood through intellectual analysis is uniquely Christian. ...It is probably not an accident that modern science grew explosively in Christian Europe and left the rest of the world behind.
- Freeman Dyson,
*The Scientist As Rebel*(2006)

- Freeman Dyson,

- 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.- James Gow,
*A Short History of Greek Mathematics*(1884) p. 177

- James Gow,

- 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

- Sir W. Hamilton, reported in Austin Allibone ed.

- 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.- Thomas Little Heath, The Thirteen Books of Euclid's Elements, Vol. 3 Book XIII, Prop I, footnote, p. 442.

- 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

- Burt C. Hopkins, "Nastalgia and Phenomenon: Hussel and Patočka on the End of the Ancient Cosmos,"

- Fortunately analysis is not the only way to resolve inner conflicts. Life itself still remains a very effective therapist... The therapy effected by life itself is not, however, within one's control. Neither hardships nor friendships nor religious experience can be arranged to meet the needs of the particular individual. Life as a therapist is ruthless; circumstances that are helpful to one neurotic may entirely crush another.
- Karen Horney
*Our Inner Conflicts*(1945)

- Karen Horney

- Government, in the last analysis, is organized opinion. Where there is little or no public opinion, there is likely to be bad government, which sooner or later becomes autocratic government.
- William Lyon Mackenzie King,
*Message of the Carillon*(1927)

- William Lyon Mackenzie King,

- 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.

- Laplace,

- Functions are the bread and butter of 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.- Mario Livio,
*Is God a Mathematician?*(2009)

- Mario Livio,

- 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.

- Thomas J. McCormack, "Joseph Louis Lagrange. Biographical Sketch" (1898) in his translation of Joseph Louis Lagrange,

- [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.

- Thomas J. McCormack, "Joseph Louis Lagrange. Biographical Sketch" (1898) in his translation of Joseph Louis Lagrange,

- 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.
- Isaac Newton,
*Opticks*(1704) {pp. 380-381 in 4th edition (1730)}

- Isaac Newton,

- 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

- Isaac Newton, reported in Austin Allibone ed.

- 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.- Pappus, (c. 330 AD) as quoted by Thomas Little Heath,
*The Thirteen Books of Euclid's Elements*(1908) Vol. 1, Ch. IX. §6.

- Pappus, (c. 330 AD) as quoted by Thomas Little Heath,

- 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

- Bertrand Russell,

- 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 word Analysis signifies the general and particular heads of a discourse, with their mutual connections, both co-ordinate and subordinate, drawn out into one or more tables.
- Isaac Watts, reported in Austin Allibone ed.
*Prose Quotations from Socrates to Macaulay.*(1903), p. 34

- Isaac Watts, reported in Austin Allibone ed.