On Combinatorial Analysis (1800-1849)

"Algebra is a species of short-hand writing; a language, or system of characters or signs, invented for the purpose of facilitating the comparison and combination of ideas." (Robert Woodhouse, "On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations." (Carl Friedrich Gauss, 1817)

"The Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things, that has not been enumerated. [...] Besides its uses in mathematical investigations, it not only enables us to form our ideas of the elegant compositions of design, but to contemplate the prodigious variety which constitutes the beauties of nature, and which arises from the combinations of objects, by their number, forms, color, and positions. It has a relation to every species of useful knowledge upon which the mind of man can be employed." (Peter Nicholson, "Essays on the Combinatorial Analysis", 1818)

"It can happen to but few philosophers, and but at distant intervals, to snatch a science, like Dalton, from the chaos of indefinite combination, and binding it in the chains of number, to exalt it to rank amongst the exact. Triumphs like these are necessarily 'few and far between’." (Charles Babbage, "Reflections on the Decline of Science in England, and on Some of Its Causes", 1830)

"Every mathematical method has its inverse, as truly, and for the same reason, as it is impossible to make a road from one town to another, without at the same time making one from the second to the first. The combinatorial analysis is analysis by means of combinations; the calculus of generating functions is combination by means of analysis." (Augustus de Morgan, "The Differential and Integral Calculus", 1836)

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection" (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering" (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex" (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"A successful attempt to express logical propositions by symbols, the laws of whose combinations should be founded upon the laws of the mental processes which they represent, would, so far, be a step towards a philosophical language." (George Boole, "The Mathematical Analysis of Logic", 1847)