“It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments.” (Martin Gardner, “Logic Machines and Diagrams”, 1958)
"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one’s craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful." (Jerome S Bruner,"On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)
"Combinatorial mathematics cuts across the many subdivisions of mathematics, and this makes a formal definition difficult. But by and large it is concerned with the study of the arrangement of elements into sets. The elements are usually finite in number, and the arrangement is restricted by certain boundary conditions imposed by the particular problem under investigation. Two general types of problems appear throughout the literature. In the first the existence of the prescribed configuration is in doubt, and the study attempts to settle this issue. These we call existence problems. In the second the existence of the configuration is known, and the study attempts to determine the number of configurations or the classification of these configurations according to types. These we call enumeration problems. This monograph stresses existence problems, but many enumeration problems appear from time to time." (Herbert J Ryser, "Combinatorial Mathematics", 1963)
"The modern era has uncovered for combinatorics a wide range of fascinating new problems. These have arisen in abstract algebra, topology, the foundations of mathematics, graph theory, game theory, linear programming, and in many other areas. Combinatorics has always been diversified. During our day this diversification has increased manifold. Nor are its many and varied problems successfully attacked in terms of a unified theory. Much of what we have said up to now applies with equal force to the theory of numbers. In fact, combinatorics and number theory are sister disciplines. They share a certain intersection of common knowledge, and each genuinely enriches the other." (Herbert J Ryser, "Combinatorial Mathematics", 1963)"The field of combinatorial analysis has been shockingly and incredibly neglected of late; too many mathematicians, blinded by their own overspecialized work, regard it with an air of snobbish condescension which betrays a lurking fear of anything that may "rock the boat." Yet, the need for more systematic understanding of combinatorial phenomena is now making itself clearly felt in all branches of mathematics, and even more so in the natural sciences. From the physics of elementary particles to genetics, from communication theory to computing, the need is increasing for the study of discrete structures. In view of this situation, one may predict with fair probability that the next decades will witness an explosion of combinatorial activity not unlike the development of topology since the beginning of this century. In the words of the author, "we believe that the greatest truths of combinatorial analysis are yet to be revealed." (Gian-Carlo Rota, "Review on Combinatorial mathematics, by H.J. Ryser", MAA Monthly, 1965)
"Combinatorial theory is the name now given to a subject formerly called ‘combinatorial analysis’ or ‘combinatorics’, though these terms are still used by many people. Like many branches of Mathematics, its boundaries are not clearly defined, but the central problem may be considered that of arranging objects according to specified rules and finding out in how many ways this may be done. If the specified rules are very simple, then the chief emphasis is on the enumeration of the number of ways in which the arrangement may be made. If the rules are subtle or complicated, the chief problem is whether or not such arrangements exist, and to find methods for constructing the arrangements. An intermediate area is the relationship between related choices, and a typical theorem will assert that the maximum for one kind of choice is equal to the minimum for another kind." (Marshall Hall, "Combinatorial Theory", 1969)
"Combinatorial analysis, or – as it coming to be called, combinatorial theory – is both the oldest and one of the least developed branches of mathematics. [...] Combinatorial problems are found nowadays in increasing numbers in every branch of science, even in those where mathematics is rarely used. [...] Combinatorial theory has been slowed in its theoretical development by the very success of the few men who have solved some of the outstanding combinatorial problems of their day, for, just as the man of action feel little need to philosophize, so the successful problem-solver in mathematics feels little need for designing theories, that would unify, ant therefore enable the less-talented worker to solve, problems of comparable and similar difficulty. But the sheer number and the rapidly increasing complexity of combinatorial problems has made the situation no longer tolerable. It is doubtful that one man alone can solve any of the major combinatorial problems of our day." (Gian-Carlo Rota, "Discrete Thoughts", 1969)
"Combinatorial analysis, or combinatorial theory, as it has come to be called, is currently enjoying an outburst of activity. This can be partly attributed to the abundance of new and highly relevant problems brought to the fore by advances in discrete applied mathematics, and partly to the fact that only lately has the field ceased to be the private preserve of mathematical acrobats, and attempts have been made to develop coherent theories, thereby bringing it closer to the mainstream of mathematics." (Gian-Carlo Rota, "Combinatorial theory, old and new", Proceedings of the International Congress of Mathematicians Nice, 1970)
"Though combinatorics has been successfully applied to many branches of mathematics these can not be compared neither in importance nor in depth to the applications of analysis in number theory or algebra to topology, but I hope that time and the ingenuity of the younger generation will change this." (Paul Erdős, "On the application of combinatorial analysis", Proceedings of the International Congress of Mathematicians Nice, 1970)
"Combinatorial analysis, or combinatorial theory, as it has come to be called, is currently enjoying an outburst of activity. This can be partly attributed to the abundance of new and highly relevant problems brought to the fore by advances in discrete applied mathematics, and partly to the fact that only lately has the field ceased to be the private preserve of mathematical acrobats, and attempts have been made to develop coherent theories, thereby bringing it closer to the mainstream of mathematics." (Gian-Carlo Rota, "Combinatorial theory, old and new", Proceedings of the International Congress of Mathematicians Nice, 1970)
"Broadly speaking combinatorial analysis is now taught in two parts which I will label: The first classical, the second important. Classical combinatorics is concerned with counting problems. [...] As a mathematician, I like classical combinatorics. It is full of interesting devices: permutations, combinations, generating functions, amusing identities, etc. Relevant, it is not, except as a possible supplement to a basic course in probability. [...] Classical combinatorics is sometimes useful in preventing people from using an exhaustive procedure on the computer such as listing all combinations or examining all the cases. [...] The part of combinatorial analysis which I have labeled ‘important’ is concerned with selecting the best combination out of all the combinations. This is what linear programming is all about." (George B Dantzig, "On the relation of operations research to mathematics", [panel talk before AMS], 1971)
"Another criticism of combinatorics is that it "lacks abstraction." The implication is that combinatorics is lacking in depth and all its results follow from trivial, though possible elaborate, manipulations. This argument is extremely misleading and unfair. It is precisely the "lack of abstraction," i.e., the concrete visualization of the concepts involved, which helps to make combinatorics so appealing to its adherents. On the other hand, the "depth" of the subject is rapidly increasing as it increasingly draws upon more and more techniques and concepts from other branches of mathematics, such as group representation theory, statistical mechanics, harmonic analysis, homological algebra, and algebraic topology, to say nothing of the increasing sophistication of various new purely combinatorial techniques." (Richard P Stanley, "Book Review of Principles of Combinatorics, by Claude Berge", Bull. AMS, 1971)
"The current resurgence of combinatorics (also known as combinatorial analysis and combinatorial theory) is by now recognized by all mathematicians. Scoffers regard combinatorics as a chaotic realm of binomial coefficients, graphs, and lattices, with a mixed bag of ad hoc tricks and techniques for investigating them. In reality, there has been a tremendous unifying drive to combinatorics in recent years. We now have a broad and sophisticated understanding of such standard combinatorial concepts as inversion, composition, generating functions, finite differences, and incidence relations." (Richard P Stanley, "Book Review of Principles of Combinatorics, by Claude Berge", Bull. AMS, 1971)
"Every hard problem in mathematics has something to do with combinatorics." (Lennart Carleson, cca. 1974)
"For a long time the aim of combinatorial analysis was to count the different ways of arranging objects under given circumstances. Hence, many of the traditional problems of analysis or geometry which are concerned at a certain moment with finite structures, have a combinatorial character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathematics, but exclusively for finite sets." (Louis Comtet, "Advanced Combinatorics", 1974)
"Since mechanically obtained randomness contains all kinds of possible permutations, including the most regular ones, it cannot be relied upon always to exhibit a pervasive irregularity." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)
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