On Combinatorial Analysis (2010-)

"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)

"Combinatorics with all its various aspects is a broad field of Mathematics which has many applications in areas like Topology, Group Theory and even Analysis. A reason for its wide range of applications might be that Combinatorics is rather a way of thinking than a homogeneous theory, and consequently Combinatorics is quite difficult to define. Nevertheless, let us start with a definition of Combinatorics which will be suitable for our purpose: Combinatorics is the branch of Mathematics which studies collections of objects that satisfy certain criteria, and is in particular concerned with deciding how large or how small such collections might be." (Lorenz J Halbeisen,"Combinatorial Set Theory: With a Gentle Introduction to Forcing", 2011) 

"Galois and Abel independently discovered the basic idea of symmetry. They were both coming at the problem from the algebra of polynomials, but what they each realized was that underlying the solution of polynomials was a fundamental problem of symmetry. The way that they understood symmetry was in terms of permutation groups. A permutation group is the most fundamental structure of symmetry. […] permutation groups are the master groups of symmetry: every kind of symmetry is encoded in the structure of the permutation group." (Mark C Chu-Carroll, "Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation", 2013)

"Combinatorics is the art and science of distilling a complex mathematical structure into simple attributes and developing from this a deeper understanding of the original structure." (Josephine Yu, "Tropical Combinatorics and Applications", 2016)

"Combinatorics is everything. All our worlds, the physical, mathematical, and even spiritual, are inherently finite and discrete, and so-called infinities, be their actual or potential, as well as the ‘continuum’, are ‘optical illusions’." (Doron Zeilberger, [interview with Enumerative Combinatorics and Applications, by Toufik Mansour", 2020)

"If you ask a mathematician what they love most about mathematics, certain answers invariably arise: beauty, abstraction, creativity, logical structure, connection (between disciplines and between people), elegance, applicability, and fun. [..] Combinatorics is the branch of mathematics best situated to embody and illustrate all of these virtues." (Stephen Melczer, "An Invitation to Analytic Combinatorics", 2021) 

On Combinatorial Analysis (1850-1899)

"Partitions constitute the sphere in which analysis lives, moves, and has its being; and no power of language can exaggerate or paint too forcibly the importance of this till recently almost neglected, but vast, subtle, and universally permeating, element of algebraical thought and expression." (James J Sylvester, "On the Partition of Numbers", 1857)

"We have continually to make our choice among different courses of action open to us, and upon the discretion with which we make it, in little matters and in great, depends our prosperity and our happiness. Of this discretion a higher philosophy treats, and it is not to be supposed that Arithmetic has anything to do with it; but it is the province of Arithmetic, under given circumstances, to measure the choice which we have to exercise, or to determine precisely the number of courses open to us." (William A Whitworth, "Choice and Chance", 1870)

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection." (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)

"That immense framework and planking of concepts to which the needy man clings his whole life long in order to preserve himself is nothing but a scaffolding and toy for the most audacious feats of the liberated intellect. And when it smashes this framework to pieces, throws it into confusion, and puts it back together in an ironic fashion, pairing the most alien things and separating the closest, it is demonstrating that it has no need of these makeshifts of indigence and that it will now be guided by intuitions rather than by concepts. There is no regular path which leads from these intuitions into the land of ghostly schemata, the land of abstractions. There exists no word for these intuitions; when man sees them he grows dumb, or else he speaks only in forbidden metaphors and in unheard - of combinations of concepts. He does this so that by shattering and mocking the old conceptual barriers he may at least correspond creatively to the impression of the powerful present intuition." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"Nature is a spectacle continually exhibited to our senses, in which phenomena are mingled in combinations of endless variety and novelty." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The imagination is one of the highest prerogatives of man. By this faculty he unites, independently of the will, former images and ideas, and thus creates brilliant and novel results […] The value of the products of our imagination depends of course on the number, accuracy, and clearness of our impressions; on our judgment and taste in selecting or rejecting the involuntary combinations, and to a certain extent on our power of voluntarily combining them." (Charles Darwin, "The Descent of Man", 1874)

"If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force - and every other representation remains indefinite and therefore useless - it follows that, in the great dice game of existence, it must pass through calculable number of combinations. In infinite time, every possible combination would at some time or another be realized; more: it would be realized an infinite number of times. And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combination conditions of the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demonstrated: the world as a circular movement that has already repeated itself infinitely often and plays its game in infinitum. This conception is not simply a mechanistic conception; for if it were that, it would not condition an infinite recurrence of identical cases, but a final state. Because the world has not reached this, mechanistic theory must be considered an imperfect and merely provisional hypothesis." (Friedrich Nietzsche, "The Will to Power", [notes written 1883-1888] 1901)

"The combinatory analysis in my opinion holds the ground between the theory of numbers and algebra, and is the proper passage between the realms of discontinuous and continuous quantity. It would appear advisable [...] to consider the theory of partitions an important part of combinatory analysis." (Percy A MacMahon, "Combinatory Analysis: A Review of the Present State of Knowledge", Proceedings of the London Mathematical Society Vol. s1-28 No. 1, 1896)

"All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly." (Ludwig Boltzmann, 1899)

On Combinatorial Analysis (1900-1949)

"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)

"To facilitate eyeless observation of his sense-transcending world, the mathematician invokes the aid of physical diagrams and physical symbols in endless variety and combination [...]" (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1907-1908, 1908)

"What, in fact, is mathematical discovery? It does not consist in making new combinations with mathematical entities that are already known. That can be done by anyone, and the combinations that could be so formed would be infinite in number, and the greater part of them would be absolutely devoid of interest. Discovery consists precisely in not constructing useless combinations, but in constructing those that are useful, which are an infinitely small minority. Discovery is discernment, selection." (Henri Poincaré, "Science and Method", 1908)

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system." (Charles P Steinmetz, [New York Times interview] 1911)

"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 game of chess has always fascinated mathematicians, and there is reason to suppose that the possession of great powers of playing that game is in many features very much like the possession of great mathematical ability. There are the different pieces to learn, the pawns, the knights, the bishops, the castles, and the queen and king. The board possesses certain possible combinations of squares, as in rows, diagonals, etc. The pieces are subject to certain rules by which their motions are governed, and there are other rules governing the players. [...] One has only to increase the number of pieces, to enlarge the field of the board, and to produce new rules which are to govern either the pieces or the player, to have a pretty good idea of what mathematics consists." (James B Shaw, "What is Mathematics?", Bulletin American Mathematical Society Vol. 18, 1912)

"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) 

"Combinatory Analysis is well known even to students of elementary algebra, where it figures under the title of Permutations and Combinations; while in theory of equation the subject is intimately connected with symmetric functions. The formal statement of the objects of this branch of mathematics is that it includes the formation, enumeration, and other properties of the different groups of a finite number of elements which are arranged according to prescribed laws. By its subject-matter combinatory analysis is related to some of the most ancient problems which have exercised human ingenuity." (Percy A MacMahon, "Review on 'Combinatory Analysis'", Science Progress in the Twentieth Century Vol. 10 (40), 1916)  

"One of the four grand divisions of what may be called properly static mathematics is the theory of configurations. It includes the construction out of given elements of compound forms under certain given conditions or restrictions [...] These constructions vary from the mere permutation of a linear series of elements up to the complicated trees of chemical combinations studied by Cayley, and in general to all sorts of problems in what has been happily denominated tactics by Cayley, or syntactics by Cournot. [...] The study of configurations usually begins with combinatory analysis. By this is usually meant the study of arrangements along a line of a collection of objects, either as individuals or in groups; arrangements at the nodes of a lattice, combinations of arrangements. Such problems arise not only as matters of tactic, curious problems or puzzles, but in the determination of the number of such arrangements needed in solving problems in the theory of probabilities." (James B Shaw, "Review on 'Combinatory Analysis'", Science Vol. 45 (1165), 1917)

"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)

"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind – as Eddington puts it – the colorful tale of the subjective storyteller mind. The combinatorics of aggregates and complexes deals with some of the simplest such structures imaginable. It is gratifying that combinatorial mathematics is so closely related to the philosophically important problems of individuation and probability, and that it accounts for some of the most fundamental phenomena in inorganic and organic nature. This structural viewpoint occurs in the foundations of quantum mechanics. In a widely different field John von Neumann’s and Oskar Morgenstern’s attempt to found economics on a theory of games is characteristic of the same trend. The network of nerves joining the brain with the sense organs is a subject that by its very nature invites combinatorial investigation. Modern computing machines translate our insight into the combinatorial structure of mathematics into practice by mechanical and electronic devices." (Hermann Weyl, 1947)

"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)

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)

 

"the combinatorial analysis mainly consists in the analysis of complicated developments by means of a priori consideration and collection of the different combinations of terms which can enter the coefficients" (Augustus DeMorgan, "Differential and Integral Calculus", 1842) 

"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)

On Combinatorial Analysis (2000-2009)

"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)

"We don't have enough parallel universes to allow all uses of all junction types - in the absence of quantum computing the combinatorics are not in our favor [...]" (Larry Wall, 2003)

"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." (Timothy Gowers et al, "The Princeton Companion to Mathematics", 2008) 

"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)

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", [Nobel 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)

On Combinatorial Analysis (1950-1974)

“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)

On Combinatorial Analysis (-1799)

“So there arise two kinds of variation: complexion [combinations] and situs [permutations]. And viewed in themselves, both complexion and situs belong to metaphysics, or to the science of whole and parts. If we look at their variability, however, that is, at the quantity of variation, we must turn to numbers and to arithmetic. I am inclined to think that the science of complexions pertains more to pure arithmetic, and that of situs to an arithmetic of figure.” (Gottfried Leibniz, “Dissertatio de arte combinatoria” [“Dissertation on the Art of Combinations”], Leipzig, 1666)

“Even the wisest and most prudent people often suffer from what Logicians call insufficient enumeration of cases.” (Jacob Bernoulli , 1692)

“[This subject] has a relation to almost every species of useful knowledge that the mind of man can be employed upon.” (Jacob Bernoulli, “Ars Conjectandi”[“The Art of Conjecturing”], 1713)

“It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur in producing a given event, or effect, is oftentimes so immensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged, or combined together, it often happens that men, even of the best understandings and the greatest circumspection, ale guilty of that fault in reasoning which the writers on logic call the insufficient, or imperfect enumeration of parts, or cases: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions. It must therefore be acknowledged, that that art which affords a cure to this weakness, or defect, of our understandings, and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted any one arrangement of them that can lead to the object of our inquiry, deserves to be considered as most eminently useful and worthy of our highest esteem and attention. And this is the business of the art, or doctrine of combinations.” (Jakob Bernoulli, “Ars Conjectandi” [“The Art of Conjecturing”], 1713)

"It proceeds indeed upon mathematical principles in calculating the number of the combinations of the things proposed: but by the conclusions that are obtained by it, the sagacity of the natural philosopher, the exactness of the historian, the skill and judgement of the physician, and the prudence and foresight of the politician, may be assisted; because the business of all these important professions is but to form reasonable conjectures concerning the several objects which engage their attention, and all wise conjectures are the results of a just and careful examination of the several different effects that may possibly arise from the causes that are capable of producing them." (Jacob Bernoulli, "Ars conjectandi", 1713)

"We have three principal means: observation of nature, reflection, and experiment. Observation gathers the facts reflection combines them, experiment verifies the result of the combination. It is essential that the observation of nature be assiduous, that reflection be profound, and that experimentation be exact. Rarely does one see these abilities in combination. And so, creative geniuses are not common." (Denis Diderot, "On the Interpretation of Nature", 1753)

"Invention, strictly speaking, is little more than a new combination of those images which have been previously gathered and deposited in the memory: nothing can come of nothing." (Joshua Reynolds, "Discourses on Art", 1769)

"[…] determine the probability of a future or unknown event not on the basis of the number of possible combinations resulting in this event or in its complementary event, but only on the basis of the knowledge of order of familiar previous events of this kind" (Nicolas de Condorcet, "Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix", 1785)

"It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind." (Nicolas de Condorcet, "Outlines Of An Historical View Of The Progress Of The Human Mind", 1795)

🆁Gian-Carlo Rota - Collected Quotes

"We often hear that mathematics consists mainly in ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks." (Gian-Carlo Rota, "Discrete Thoughts", 1981) 

"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 (Ian P Goulden and David M Jackson, "Combinatorial Enumeration", 1983)])

"There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory." (Gian-Carlo Rota, [lecture] 1996)

"Books on paradoxes in statistics are similar to mystery books. They have a faithful readership, and they follow a rigorous sequence in their presentation, like Greek tragedies." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Group theory, like lattice theory, is the whipping boy of mathematicians in need of concealing their feelings of insecurity." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Mathematical logic deals not with the truth but only with the game of truth." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Nature imitates mathematics." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", Synthese, 111(2), 1997)

"[...] the beauty of a piece of mathematics is strongly dependent upon schools and periods of history. A theorem that is in one context thought to be beautiful may in a different context appear trivial. [...] Undoubtedly, many occurrences of mathematical beauty eventually fade or fall into triviality as mathematics progresses." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", Synthese, 111(2), 1997)

On Truth: Geometrical Truth

"The connected course of reasoning by which any Geometrical truth is established is called a demonstration." (Robert Potts, "Euclid's Elements of Geometry", 1845)

"In all cases, however, it must be kept in view that every geometrical truth is deduced by a comparison between two others, which agree, one in one particular part, and the other in another, with the conclusion so deduced." (?,"Miscallanea Mathematica", American Railroad Journal, No. X, 1846)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"Geometrical truth is a product of reason; that makes it superior to empirical truth, which is found through generalization of a great number of instances." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1954)

"Conventionalism as geometrical and mathematical truths are created by our choices, not dictated by or imposed on us by scientific theory. The idea that geometrical truth is truth we create by the understanding of certain conventions in the discovery of non-Euclidean geometries." (Clifford Singer, "Engineering a Visual Field", 1955)

"To enter a temple constructed wholly of invariable geometric proportions is to enter an abode of eternal truth." (Robert Lawlor, "Sacred Geometry", 1982)

"Geometrical truth is (as we now speak) synthetic: it states facts about the world. Such truths are not ordinary truths but essential truths, giving the reality of the empirical world in which they are imperfect embodied." (Fred Wilson, "The External World and Our Knowledge of It", 2008)