Business Analysis and Big Data Applications

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."

O brave youth, how good for thee it were couldst thou be made to understand how infinitely precious are thy school years—years when thou hast leisure to grow, when new worlds break in upon thee, and thou fashionest thy being in the light of the ideals of truth and goodness and beauty! If now thou dost not fit thyself to become free and whole, thou shalt, when the doors of this fair mother-house of the mind, close behind thee, be driven into ways that lead to bondage, be compelled to do that which cripples and dwarfs; for the work whereby men gain a livelihood involves mental and moral mutilation, unless it be done in the spirit of religion and culture. Ah! well for thee, canst thou learn while yet there is time that it will profit thee nothing to become the possessor of millions, if the price thou payest is thy manhood.

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".