Book Files: Leibniz on the Art of Discovery in a distinction between synthesis and analysis
Synthesis is achieved when we begin from principles and run through truths in good order, thus discovering certain progressions and setting up tables, or sometimes general formulas, in which the answers to emerging questions can later be discovered. Analysis goes back to the principles in order to solve the given problems only, just as if neither we nor others had discovered anything before. It is more important to establish syntheses, because this work is of permanent value, while we often do work that has already been done in beginning the analysis of a particular problem.
But it is a lesser art to use syntheses already set up by others and theorems already discovered, than to achieve everything through one's own work, by carrying out analyses, especially since we do not always remember or have at hand the truths which we ourselves or others have already discovered. Analysis is of two kinds. The common type advances by leaps and is used in algebra. The other is special and far more elegant but less well known; I call it 'reductive' analysis. Analysis is more necessary in practice, in order to solve problems that are given to us.
Whoever is capable of more theoretical pursuits will be content to practice analysis only far enough to master the art but will then prefer to synthesize and will willingly tackle only such questions to which he is led by the order of research itself.
In this way he will always progress pleasantly and easily and will not feel any difficulties or be disappointed in the outcome, for in a short time he will achieve much more than he could ever have hoped at the start.
But ordinarily people destroy the fruits of their thinking through undue haste and attack too difficult problems at a leap, thus achieving nothing despite great effort.
I have often observed that of the great geniuses of discovery, some are more inclined to analysis, others to the art of combinations. Combination or synthesis is the better means for discovering the use or application of something, as for example, given the magnetic needle, to think of its application in the compass. Analysis, on the contrary, is best suited for discovering the means when the thing to be discovered or the proposed end is given.
The art of combinations in particular, as I take it … is that science in which are treated the forms or formulas of things in general, that is, quality in general or similarity and dissimilarity; in the same way that ever new formulas arise from the elements a, b, c themselves when combined with each other, whether these elements represent quantities or something else.
Gottfried Wilhelm Leibniz, ‘On Universal Synthesis and Analysis, or the Art of Discovery and Judgment’,1679, in Gottfried Wilhelm Leibniz Philosophical Papers and Letters, A Selection translated and edited by Leroy E. Loemker, 2nd ed. Klueger Academic 1989 [pp. 232-233]