“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 (scholar)

A Short History of Greek Mathematics (1884)

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scholar 1854–1923

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“There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered; Theon named it analysis, and defined it as the assumption of that which is sought as if it were admitted and working through its consequences to what is admitted to be true. This is opposed to synthesis, which is the assuming what is admitted and working through its consequences to arrive at and to understand that which is sought.”

François Viète (1540–1603) French mathematician

Source: In artem analyticem Isagoge (1591), Ch. 1 as quoted by Douglas M. Jesseph, Squaring the Circle: The War Between Hobbes and Wallis (1999) p. 225

“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 of Alexandria (290–350) Greek mathematician of Antiquity

Source: The Thirteen Books of Euclid's Elements (1908), Ch. IX. §6

“Analysis… takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (ɣϵɣονός) and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards”

Pappus of Alexandria (290–350) Greek mathematician of Antiquity

άνάπαλɩν λὐσɩν
The Thirteen Books of Euclid's Elements (1908)

“Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.”

Lucio Russo (1944) Italian historian and scientist

2.4, "Discrete Mathematics and the Notion of Infinity", p. 45
The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (2004)

“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 (1540–1603) French mathematician

Source: In artem analyticem Isagoge (1591), Ch. 1 as quoted by Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1934-1936) Appendix.

“Knowledge is also inferred from what is accepted as established knowledge, with new knowledge being based on the best explanation. This includes possible truths subject to proof.”

Nayef Al-Rodhan (1959) philosopher, neuroscientist, geostrategist, and author

Source: Sustainable History and the Dignity of Man (2009), p.108

“The so called άναλυόμϵνος ('Treasury of Analysis') is… a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.”

Pappus of Alexandria (290–350) Greek mathematician of Antiquity

Source: The Thirteen Books of Euclid's Elements (1908), Ch. IX. §6

“Christianity claims that the supernatural is as reasonable as the natural, that man himself is supernatural as truly as he is natural, and that the Bible is so clearly the word of God by proofs that are unanswerable, that it is unreasonable to disbelieve its divine truths.”

Abbott Eliot Kittredge (1834–1912) American minister

Source: Dictionary of Burning Words of Brilliant Writers (1895), P. 35.

“That is why a criminal trial is not a search for truth. Scientists search for truth. Philosophers search for morality. A criminal trial searches for only one result: proof beyond a reasonable doubt.”

Alan M. Dershowitz (1938) American lawyer, author

[Alan, Dershowitz, http://online.wsj.com/article/SB10001424052702303544604576429783247016492.html?mod=WSJ_Opinion_LEADTop, Casey Anthony: The System Worked, The Wall Street Journal, July 7, 2011, July 7, 2011] published 2011-07-07

“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

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