"Let us subjugate a collection of objects taking into account their qualities and differences each from another, then we are lead, in our mathematical perspective, to the study of integers and their connecting operations, that is, we are lead to Number Theory. [...] If we, however, disregard the qualities of each individual object and only account for the difference between two objects insofar that they are different, then we are lead to investigations which are concerned with the position, the order, the choosing of these objects. This branch of mathematics is called Combinatorics." (Eugen Netto, "Lehrbuch der Combinatorik", 1901)
"The Combinatorial Analysis, as it was understood up to the end of the 18th century, was of limited scope and restricted application. [...] Writers on the subject seemed to recognize fully that it was in need of cultivation, that it was of much service in facilitating algebraical operations of all kinds, and that it was the fundamental method of investigation in the theory of Probabilities." (Percy A MacMahon, "Combinatorial Analysis", Encyclopædia Britannica 11th Ed., 1911)
"The combinatory analysis as considered in this work occupies the ground between algebra, properly so called, and the higher arithmetic. The methods employed are distinctly algebraical and not arithmetical. The essential connecting link between algebra and arithmetic is found in the circumstance that a particular case of algebraic multiplication involves arithmetical addition. [...] This link was forged by Euler for use in the theory of partitions of numbers. It is used here for the most general theory of combinations of which the partition of numbers is a particular case." (Percy A MacMahon, "Combinatory Analysis" Vol. 1, 1915)
"The theory of the partition of numbers belongs partly to algebra and partly to the higher arithmetic. The former aspect is treated here. It is remarkable that in the international organization of the subject-matter of mathematics ‘Partitions’ is considered to be a part of the Theory of Numbers which is an alternative name for the Higher Arithmetic, whereas it is essentially a subdivision of Combinatory Analysis which is not considered to be within the purview of the Theory of Numbers. The fact is that up to the point of determining the real and enumerating Generating Functions the theory is essentially algebraical [..]" (Percy A MacMahon, "Combinatory Analysis" Vol. 1, 1915)
"The term 'combinatorial analysis' hardly admits of exact definition, and is not used in the International Schedule of pure mathematics. Broadly speaking, it has come to mean the discussion of problems which involve selections from, or arrangements of, a finite number of objects; or combinations of these two operations. For the purpose of this article it will be convenient to use Sylvester's term ‘tactic’ as a synonym for 'combinatorial analysis’." (George B Mathews, "Tactic", Science Progress in the Twentieth Century Vol. 16 No. 61, 1921)
"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1947)
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