“There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found.”

— Jacques Ozanam

Source: A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms, 1702, p. 26

Adopted from Wikiquote. Last update June 3, 2021. History

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Jacques Ozanam 10

French mathematician 1640–1718

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“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. Analysis, without a subsequent synthesis, is incomplete; it is a mean cut of from its end. Synthesis, without a previous analysis, is baseless; for synthesis receives from analysis the elements which it recomposes.”

Sir William Hamilton, 9th Baronet (1788–1856) Scottish metaphysician (1788–1856)

Lectures on Metaphysics and Logic: "6th Lecture on Metaphysics", p. 69, ed. 1871, Boston; partly reported in Austin Allibone ed. Prose Quotations from Socrates to Macaulay. (1903), p. 34

“We tend to believe the premises because we can see that their consequences are true, instead of believing the consequences because we know the premises to be true. But the inferring of premises from consequences is the essence of induction; thus the method in investigating the principles of mathematics is really an inductive method, and is substantially the same as the method of discovering general laws in any other science.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

"The Regressive Method of Discovering the Premises of Mathematics" (1907), in Essays in Analysis (1973), pp. 273–274
1900s

“Every other method of resolving this difference of opinion has failed. We are reduced to violence, which is the last resort.”

Jane Rogers book The Testament of Jessie Lamb

Source: The Testament of Jessie Lamb (2011), Chapter 10 (p. 75)

“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 book Opticks, or a Treatise of the Reflections, Refractions, Inflections and Colours of Light

Query 31
Opticks (1704)

“Criticism and self-criticism is a kind of method. It is a method of resolving contradictions among the people and it is the only method.”

Mao Zedong (1893–1976) Chairman of the Central Committee of the Communist Party of China

Talk at an Enlarged Central Work Conference (30 January 1962)
1960s
Original: (zh-CN) 批评和自我批评是一种方法，是解决人民内部矛盾的方法，而且是唯一的方法。

“As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition.”

Isaac Newton book Opticks, or a Treatise of the Reflections, Refractions, Inflections and Colours of Light

Query 31
Opticks (1704)

“I thought fit to… explain in detail in the same book the peculiarity of a certain method, by which it will be possible… to investigate some of the problems in mathematics by means of mechanics. This procedure is… no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards… But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

“If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems.”

David Hilbert Mathematical Problems

Mathematical Problems (1900)

“The method of viewing things which proceeds in accordance with the principle of sufficient reason is the rational method, and it alone is valid and of use in practical life and in science. The method which looks away from the content of this principle is the method of genius, which is only valid and of use in art.”

Arthur Schopenhauer book The World as Will and Representation

Die dem Satz vom Grunde nachgehende ist die vernünftige Betrachtungsart, welche im praktischen Leben, wie in der Wissenschaft, allein gilt und hilft: die vom Inhalt jenes Satzes wegsehende ist die geniale Betrachtungsart, welche in der Kunst allein gilt und hilft.
Die Welt als Wille und Vorstellung, Zweiter Band, Ergänzungen zum dritten Buch, para. 36 (1859)
The World as Will and Representation (1819; 1844; 1859)

“Qualitatively different contradictions can only be resolved by qualitatively different methods.”

Mao Zedong book On Contradiction

On Contradiction (1937)
Original: (zh-CN) 不同质的矛盾，只有用不同质的方法才能解决。

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Jacques Ozanam 10

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