On Combinatorial Analysis (1975-1999)

"Combinatorial analysis, or combinatorics, is the study of how things can be arranged. In slightly less general terms, combinatorial analysis embodies the study of the ways in which elements can be grouped into sets subject to various specified rules, and the properties of those groupings. […] Combinatorial analysis often asks for the total number of different ways that certain things can be combined according to certain rules." (Martin Gardner, "Aha! Insight", 1978) 

"Every branch of mathematics has its combinatorial aspects […] There is combinatorial arithmetic, combinatorial topology, combinatorial logic, combinatorial set theory-even combinatorial linguistics, as we shall see in the section on word play. Combinatorics is particularly important in probability theory where it is essential to enumerate all possible combinations of things before a probability formula can be found." (Martin Gardner, "Aha! Insight", 1978)

"It is now generally recognized that the field of combinatorics has, over the past years, evolved into a fully-fledged branch of discrete mathematics whose potential with respect to computers and the natural sciences is only beginning to be realized. Still, two points seem to bother most authors: The apparent difficulty in defining the scope of combinatorics and the fact that combinatorics seems to consist of a vast variety of more or less unrelated methods and results. As to the scope of the field, there appears to be a growing consensus that combinatorics should be divided into three large parts: (i) Enumeration, including generating functions, inversion, and calculus of finite differences; (ii) Order Theory, including finite sets and lattices, matroids, and existence results such as Hall's and Ramsey's; (iii) Configurations, including designs, permutation groups, and coding theory." (Martin Aigner, "Combinatorial Theory", 1979)

"Having vegetated on the fringes of mathematical science for centuries, combinatorics has now burgeoned into one of the fastest growing branches of mathematics – undoubtedly so if we consider the number of publications in this field, its applications in other branches of mathematics and in other sciences, and also the interest of scientists, economists and engineers in combinatorial structures. The mathematical world was attracted by the successes of algebra and analysis and only in recent years has it become clear, due largely to problems arising from economics, statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty." (László Lovász, "Combinatorial Problems and Exercises", 1979)

"Perhaps combinatorics is no longer deemed to be the slum of topology but it still has a remarkable polarising effect on mathematicians. The practitioners of combinatorics tend to idolise it as the only truly interesting branch of mathematics, while people not active in combinatorics are likely to have no respect for it and dismiss it as a collection of scattered results and trivial artificial problems. This highly unsatisfactory situation cannot be blamed entirely on the youth of the subject, though it is certainly one of the reasons. Those of us who work in combinatorics are also at e efault, for most of our journals do publish more than their fair share of below par papers. Furthermore, as combinatorics fails to command the respect of the majority of the mathematical community, some combinatorialists feel entitled to disregard the huge developments in the main branches of mathematics." (Béla Bollobás, "Book Review of Combinatorial Problems and Exercises by László Lovász", Bull. AMS, 1981)

"We have not begun to understand the relationship between combinatorics and conceptual mathematics." (Jean Dieudonné, "A Panorama of Pure Mathematics: As Seen by N. Bourbaki", 1982)

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"Combinatorics can be classified into three types: enumerative, existential, and constructive. Enumerative combinatorics deals with the counting of combinatorial objects. Existential combinatorics studies the existence or nonexistence of combinatorial configurations. Constructive combinatorics deals with methods for actually finding specific configurations (as opposed to merely demonstrating their existence theoretically). [...] In constructive combinatorics, the problem is usually one of finding a solution efficiently, [..] using a reasonable length of time." (George Pólya et al, "Notes on Introductory Combinatorics", 1983) 

"The progress of mathematics can be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example, the theory of enumeration appear to be boundless. Rules for the enumeration of sets subject to various conditions, or combinatorial objects as they are often called, appear to obey an indefinite variety of and seem to lead to a welter of generating functions. We are at first led to suspect that the class of objects with a common property that may be enumerated is indeed infinite and unclassifiable." (Gian-Carlo Rota, [Preface to Combinatorial Enumeration by I.P. Goulden and D.M. Jackson], 1983)

"Combinatorics can be classified into three types: enumerative, existential, and constructive. Enumerative combinatorics deals with the counting of combinatorial objects. Existential combinatorics studies the existence or nonexistence of combinatorial configurations. Constructive combinatorics deals with methods for actually finding specific configurations (as opposed to merely demonstrating their existence theoretically). [...] In constructive combinatorics, the problem is usually one of finding a solution efficiently, [...] using a reasonable length of time." (George Pólya, Robert E Tarjan & Donald R Woods, "Notes on Introductory Combinatorics", 1983)

"Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing: you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea - combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques." (Gian-Carlo Rota, "Mathematics, Philosophy and Artificial Intelligence", Los Alamos Science No. 12, 1985)

"Much combinatorics of our day came out of an extraordinary coincidence. Disparate problems in combinatorics. ranging from problems in statistical mechanics to the problem of coloring a map, seem to bear no common features. However, they do have at least one common feature: their solution can be reduced to the problem of finding the roots of some polynomial or analytic function. The minimum number of colors required to properly color a map is given by the roots of a polynomial, called the chromatic polynomial; its value at N tells you in how many ways you can color the map with N colors. Similarly, the singularities of some complicated analytic function tell you the temperature at which a phase transition occurs in matter. The great insight, which is a long way from being understood, was to realize that the roots of the polynomials and analytic functions arising in a lot of combinatorial problems are the Betti numbers of certain surfaces related to the problem, Roughly speaking, the Betti numbers of a surface describe the number of different ways you can go around it. We are now trying to understand how this extraordinary coincidence comes about. If we do, we will have found a notable unification in mathematics." (Gian-Carlo Rota, "Mathematics, Philosophy and Artificial Intelligence", Los Alamos Science No. 12, 1985)

"The mathematical theories generally called 'mathematical theories of chance' actually ignore chance, uncertainty and probability. The models they consider are purely deterministic, and the quantities they study are, in the final analysis, no more than the mathematical frequencies of particular configurations, among all equally possible configurations, the calculation of which is based on combinatorial analysis. In reality, no axiomatic definition of chance is conceivable." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)

"One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the modern combinatorial techniques, including Topological methods, Algebraic methods, and Probabilistic methods, merely supply existence proofs and do not yield efficient (deterministic or randomized) algorithms for the corresponding problems." (Noga Alon, "Non-Constructive Proofs in Combinatorics", Proceedings of the International Congress of Mathematicians Kyoto, 1990)

"Does the heart of mathematics lie in the building of structures or in the solving of individual problems? Not an either – or question, to be sure, but one that is particularly effective in splitting the ranks of combinatorialists. Use of algebraic structure to explain discrete phenomena will be central to some, to others grotesque. A clever argument is beautiful to the problem-solver, a curiosity to a structuralist. The very term "combinatorial methods", has to this author, an oxymoronic character. It is the brilliant proofs, those that expand and/or transcend known technologies, which express the soul of the subject." (Joel Spencer, "Probabilistic methods", Handbook of Combinatorics Vol. 2, 1995)

"Combinatorics belongs to those areas of mathematics having experienced a most impressive growth in recent years. This growth has been fueled in large part by the increasing importance of computers, the needs of computer science and demands from applications where discrete models play more and more important roles. But also more classical branches of mathematics have come to recognize that combinatorial structures are essential components of many mathematical theories." (Ronald Graham, Martin Grötschel & László Lovász, "Handbook of Combinatorics" Vol. 1, 1995) 

"Combinatorics is special. Most mathematical topics which can be covered in a lecture course build towards a single, well-defined goal, such as the Prime Number Theorem. Even if such a clear goal doesn’t exist, there is a sharp focus (e.g. finite groups). By contrast, combinatorics appears to be a collection of unrelated puzzles chosen at random. Two factors contribute to this. First, combinatorics is broad rather than deep. Second, it is about techniques rather than results. [...] Combinatorics could be described as the art of arranging objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and if so, in how many different ways it can be done. If the rules are simple, the existence of an arrangement is clear, and we concentrate on the counting problem. But for more involved rules, it may not be clear whether the arrangement is possible at all." (Peter J Cameron, "Combinatorics: topics, techniques, algorithms", 1995)

"It is not surprising that computer science is perhaps the most important field of applications of combinatorial ideas. Modern computers operate in a discrete fashion both in time and space, and much of classical mathematics must be "discretized" before it can be implemented on computers as, for example, in the case of numerical analysis. The connection between combinatorics and computer science might be even stronger than suggested by this observation; each field has profited from the other. Combinatorics was the first field of mathematics where the ideas and concepts of computer science, in particular complexity theory, had a profound impact." (L. Lovász et al, "Combinatorics in computer science", [in "Handbook of Combinatorics" Vol. 2)] 1995) 

"The field normally classified as algebra really consists of two quite separate fields. Let us call them Algebra One and Algebra Two for want of a better language. Algebra One is the algebra whose bottom lines are algebraic geometry or algebraic number theory. Algebra One has by far a better pedigree than Algebra Two, and has reached a high degree of sophistication and breadth. Commutative algebra, homological algebra, and the more recent speculations with categories and topoi are exquisite products of Algebra One.
Algebra Two has had a more accidented history. [...] In the beginning Algebra Two was largely cultivated by invariant theorists. Their objective was to develop a notation suitable to describe geometric phenomena which is independent of the choice of a coordinate system. In pursuing this objective, the invariant theorists of the nineteenth century were led to develop explicit algorithms and combinatorial methods. [...] Algebra Two has recently come of age. In the last twenty years or so, it has blossomed and acquired a name of its own: algebraic combinatorics. Algebraic combinatorics, after a tortuous history, has at last found its own bottom line, together with a firm place in the mathematics of our time." (Gian-Carlo Rota, "Combinatorics, Representation Theory and Invariant Theory, in Indiscrete Thoughts", 1997)