“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

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

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François Viète 3

French mathematician 1540–1603

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“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)

“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

“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) (1854–1923) scholar

A Short History of Greek Mathematics (1884)

“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.

“But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.”

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

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

“Avarice, the spur of industry, is so obstinate a passion, and works its way through so many real dangers and difficulties, that it is not likely to be scared by an imaginary danger, which is so small, that it scarcely admits of calculation.”

David Hume book Essays, Moral, Political, and Literary

Part I, Essay 12: Of Civil Liberty
Essays, Moral, Political, and Literary (1741-2; 1748)
Context: Avarice, the spur of industry, is so obstinate a passion, and works its way through so many real dangers and difficulties, that it is not likely to be scared by an imaginary danger, which is so small, that it scarcely admits of calculation. Commerce, therefore, in my opinion, is apt to decay in absolute governments, not because it is there less secure, but because it is less honourable.

“That logic, as a science, is susceptible of very wide applications is admitted; but it is equally certain that its ultimate forms and processes are mathematical.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 12; Cited in: William Stanley Jevons (1887) The Principles of Science: : A Treatise on Logic and Scientific Method. p. 155

“In obedience to the feeling of reality, we shall insist that, in the analysis of propositions, nothing "unreal" is to be admitted. But, after all, if there is nothing unreal, how, it may be asked, could we admit anything unreal? The reply is that, in dealing with propositions, we are dealing in the first instance with symbols, and if we attribute significance to groups of symbols which have no significance, we shall fall into the error of admitting unrealities, in the only sense in which this is possible, namely, as objects described.”

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

Source: 1910s, Introduction to Mathematical Philosophy (1919), Ch. 16: Descriptions

“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 (1640–1718) French mathematician

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

“First, you know, a new theory is attacked as absurd; then it is admitted to be true, but obvious and insignificant; finally it is seen to be so important that its adversaries claim that they themselves discovered it.”

William James (1842–1910) American philosopher, psychologist, and pragmatist

Lecture VI, Pragmatism's Conception of Truth
1900s, Pragmatism: A New Name for Some Old Ways of Thinking (1907)

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François Viète 3

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