"Combinatorics appears to many to consist of a large number of isolated problems and results, and therefore to be at a disadvantage in this respect. Each result individually may well require enormous ingenuity, but ingenious people exist, especially in Hungary, and future generations of combinatorialists will not have the time or inclination to read and admire more than a tiny fraction of their output." (Timothy Gowers, "The two cultures of mathematics", 2000)
"Let me attempt to answer this criticism. It is certainly rare in combinatorics for somebody to and a very general statement which suddenly places a large number of existing results in their proper context. It is also true that many of the results proved by combinatorialists are somewhat isolated and will be completely forgotten (but this does not distinguish combinatorics from any other branch of mathematics). However, it is not true that there is no structure at all to the subject. The reason it appears to many mathematicians as though combinatorics is just a miscellaneous collection of individual problems and results is that the organizing principles are less explicit." (Timothy Gowers, "The two cultures of mathematics", 2000)
"Combinatorics could be described as the study of arrangements of 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. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)
"There are various ways in which one can try to define combinatorics. None is satisfactory on its own, but together they give some idea of what the subject is like. A first definition is that combinatorics is about counting things. [...] Combinatorics is sometimes called ‘discrete mathematics’ because it is concerned with ‘discrete’ as opposed to ‘continuous’ structures. Roughly speaking, an object is discrete if it consists of points that are isolated from each other and continuous if you can move from one point to another without making sudden jumps. [...] There is a close affinity between combinatorics and theoretical computer science (which deals with the quintessentially discrete structure of sequences of 0s and 1s), and combinatorics is sometimes contrasted with analysis, though in fact there are several connections between the two. A third definition is that combinatorics is concerned with mathematical structures that have ‘few constraints’. This idea helps to explain why number theory, despite the fact that it studies (among other things) the distinctly discrete set of all positive integers, is not considered a branch of combinatorics." (Timothy Gowers, June Barrow-Green & Imre Leader, "The Princeton Companion to Mathematics", 2008)
"Combinatorics could be described as the study of arrangements of 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. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)
"There are various ways in which one can try to define combinatorics. None is satisfactory on its own, but together they give some idea of what the subject is like. A first definition is that combinatorics is about counting things. [...] Combinatorics is sometimes called ‘discrete mathematics’ because it is concerned with ‘discrete’ as opposed to ‘continuous’ structures. Roughly speaking, an object is discrete if it consists of points that are isolated from each other and continuous if you can move from one point to another without making sudden jumps. [...] There is a close affinity between combinatorics and theoretical computer science (which deals with the quintessentially discrete structure of sequences of 0s and 1s), and combinatorics is sometimes contrasted with analysis, though in fact there are several connections between the two. A third definition is that combinatorics is concerned with mathematical structures that have ‘few constraints’. This idea helps to explain why number theory, despite the fact that it studies (among other things) the distinctly discrete set of all positive integers, is not considered a branch of combinatorics." (Timothy Gowers, June Barrow-Green & Imre Leader, "The Princeton Companion to Mathematics", 2008)
"Combinatorics seems to be the most fruitful source of easy-to-understand but hard-to-prove theorems, and it is also seems to be the place where insights from the infinite world most clearly illuminate the finite world." (John Stillwell, "Mathematics and Its History", 2010)
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