Zero-Sum Games

"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)

"Zero is where it all begins, the clean slate. We speak of zero-sum games (in which anyone who wins anything does so only at the equal expense of the losers), zero hour (the time at which a military operation begins), ground zero (the impact point of a bomb, particularly a nuclear one), to zero in on something (getting it precisely in the cross hairs), zero degrees of temperature-which, depending on the scale you use, can be the freezing point of water (Centigrade), fortified wine (Fahrenheit), or the universe (Kelvin); the last, a bit chillier than - 2730 C or - 459' F, is aptly called absolute zero." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"In a mathematical sense a zero-sum game is a loser's game when it is valued in terms of utility. The best decision for both is to refuse to play this game." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"A game is a situation of strategic interdependence: the outcome of your choices (strategies) depends upon the choices of one or more other persons acting purposely. The decision makers involved in a game are called players, and their choices are called moves. The interests of the players in a game may be in strict conflict; one person’s gain is always another’s loss. Such games are called zero-sum. More typically, there are zones of commonality of interests as well as of conflict and so, there can be combinations of mutually gainful or mutually harmful strategies. Nevertheless, we usually refer to the other players in a game as one’s rivals." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Chess, as a game of zero sum and total information is, theoretically, a game that can be solved. The problem is the immensity of the search tree: the total number of positions surpasses the number of atoms in our galaxy. When there are few pieces on the board, the search space is greatly reduced, and the problem becomes trivial for computers’ calculation capacity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Game theory postulates rational behavior for each participant. Each player is conscious of the rules and behaves in accordance with them, each player has sufficient knowledge of the situation in which he or she is involved to be able to evaluate what the best option is when it comes to taking action (a move), and each player takes into account the decisions that might be made by other participants and their repercussions with respect to his or her own decision. Game theory about zero-sum games with two participants is relevant for chess. In this type of situation, each action that is favorable to one participant (player) is proportionally unfavorable for the opponent. Thus, the gain of one represents the loss of the other." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"In the context of a zero-sum game, opposing tendencies are formulated in two distinct ways. First, conflicting tendencies are conceptualized as two mutually exclusive, discrete entities. The conflicts are treated as dichotomies that are usually expressed as X or NX. If X is right then NX has to be wrong. This represents an or relationship, a win/lose struggle with a moral obligation to win. The loser, usually declared wrong, is eliminated. Second, opposing tendencies are formulated in such a way that they can be represented by a continuum. Between black and white are a thousand shades of gray." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"The real challenge dto building a viable social system is the ability to create unity in diversity, meeting the varying interests of independent members operating in an interdependent whole. […] In the long run, the society and the individual either stand together or fall separately. A win/win relationship is achieved not through zero-sum or even compromise. For both of them to win requires reconceptualization of the nature and the relationship of the whole and the parts." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

John F Nash - Collected Quotes

"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)

"One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other." (John F Nash, "Non-cooperative Games", Annals of Mathematics Vol. 54 (2), 1951)

"Rather than solve the two-person cooperative game by analyzing the bargaining process, one can attack the problem axiomatically by stating general properties that 'any reasonable solution' should possess. By specifying enough such properties one excludes all but one solution. " (John F Nash, "Two-Person Cooperative Games", 1953)

"The information obtained by discovering dominances for one player may be of relevance to the others, insofar as the elimination of classes of mixed strategies as possible components of an equilibrium point is concerned. For the t's whose components are all undominated are all that need be considered and thus eliminating some of the strategies of one player may make possible the elimination of a new class of strategies for another player." (John F Nash, "Two-Person Cooperative Games", 1953)

"Instead of having a single control unit sequencing the operations of the machine in series (except for certain subsidiary operations as certain input and output functions) as is now done, the idea is to decentralize control with several different control units capable of directing various simultaneous operations and interrelating them when appropriate." (John F Nash, "Parallel Control", 1954)

"It is interesting to consider what a thinking machine will be like. It seems clear that as soon as the machines become able to solve intellectual problems of the highest difficulty which can be solved by humans they will be able to solve most of the problems enormously faster than a human." (John F Nash, "Parallel Control", 1954)

"We could define the intelligence of a machine in terms of the time needed to do a typical problem and the time needed for the programmer to instruct the machine to do it." (John F Nash, "Parallel Control", 1954)

"Successful treatment of non-linear partial differential equations generally depends on 'a priori' estimates controlling the behavior of solutions. These estimates are themselves theorems about linear equations with variable coefficients, and they can give a certain compactness to the class of possible solutions." (John F Nash, "Continuity of Solutions of Parabolic and Elliptic Equations", 1958)

"If you're going to develop exceptional ideas, it requires a type of thinking that is not simply practical thinking [...]"  (John F Nash)

"What truly is logic? Who decides reason? […] It's only in the mysterious equations of love that any logical reasons can be found." (John F Nash)

On Puzzles (Unsourced)

"It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle." (Albert Einstein)

"Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man." (Maria Goeppert-Mayer)

"Science is a game - but a game with reality, a game with sharpened knives [..] If a man cuts a picture carefully into 1000 pieces, you solve the puzzle when you reassemble the pieces into a picture; in the success or failure, both your intelligences compete. In the presentation of a scientific problem, the other player is the good Lord. He has not only set the problem but also has devised the rules of the game - but they are not completely known, half of them are left for you to discover or to deduce. The experiment is the tempered blade which you wield with success against the spirits of darkness - or which defeats you shamefully. The uncertainty is how many of the rules God himself has permanently ordained, and how many apparently are caused by your own mental inertia, while the solution generally becomes possible only through freedom from its limitations." (Erwin Schrödinger)

"The art of simplicity is a puzzle of complexity." (Douglas Horton)

"Throughout science there is a constant alternation between periods when a particular subject is in a state of order, with all known data falling neatly into their places, and a state of puzzlement and confusion, when new observations throw all neatly arranged ideas into disarray." (Sir Hermann Bondi)

"While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what anyone man will be up to, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician." (Sir Arthur C Doyle)

On Strategy I

"It is possible to learn strategic flexibility [...] however, that it is difficult to teach it. It is not a matter of learning a few readily grasped general principles, but of learning a lot of small, 'local' rules, each of which is applicable in a limited area. The point is not to learn how to drive a steamroller with which one can flatten all problems in the same way, but to learn the adroitness of a puppeteer, who at one time holds many strings in his hands and who is able to adapt his movements to the given circumstances in the most sophisticated ways." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London (B), 1990)

"[…] a rule for choosing an action is termed a strategy. If the rule says to always take the same action, it's called a pure strategy; otherwise, the strategy is called mixed. A solution to a game is simply a strategy for each player that gives each of them the best possible payoff, in the sense of being a regret-free choice." (John L Casti, "Five Golden Rules", 1995)

"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)

"A strategy is usually expressed by a set of heuristic rules. The heuristic rules ease the process of searching for an optimal solution. The process is usually iterative and at one step either the global optimum for the whole problem (state) space is found and the process stops, or a local optimum for a subspace of the state space of the problem is found and the problem continues, if it is possible to improve." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"Strategy in complex systems must resemble strategy in board games. You develop a small and useful tree of options that is continuously revised based on the arrangement of pieces and the actions of your opponent. It is critical to keep the number of options open. It is important to develop a theory of what kinds of options you want to have open." (John H Holland, [presentation] 2000)

"[...] a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration." Yu-Chi Ho & David L Pepyne, "Simple explanation of the no-free-lunch theorem and its implications", Journal of Optimization Theory and Applications 115, 2002)

"[...] the System may be so thoroughly organized around the familiar response strategy that a new response would require extensive restructuring - something that Systems do with the greatest reluctance and difficulty." (John Gall, "Systemantics: The Systems Bible", 2002)

"We can find the minimax strategy by exploiting the game’s symmetry. Roughly speaking, the minimax strategy must have the same kind of symmetry." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"A heuristic is a strategy we derive from previous experience with a similar problem." (Darius Foroux, "Think Straight", 2017)

On Nonlinearity VI

"Up until now most economists have concerned themselves with linear systems, not because of any belief that the facts were so simple, but rather because of the mathematical difficulties involved in nonlinear systems [... Linear systems are] mathematically simple, and exact solutions are known. But a high price is paid for this simplicity in terms of special assumptions which must be made." (Paul A Samuelson, "Foundations of Economic Analysis", 1966)

"Linear relationships are easy to think about: the more the merrier. Linear equations are solvable, which makes them suitable for textbooks. Linear systems have an important modular virtue: you can take them apart and put them together again - the pieces add up. Nonlinear systems generally cannot be solved and cannot be added together. [...] Nonlinearity means that the act of playing the game has a way of changing the rules. [...] That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems." (James Gleick, "Chaos: Making a New Science", 1987)

"Never in the annals of science and engineering has there been a phenomenon so ubiquitous‚ a paradigm so universal‚ or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg‚ for beneath it lies a much finer structure of immense complexity‚ a geometric labyrinth of endless convolutions‚ and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore‚ thereby forfeiting their only chance to grapple with reality." (Leon O Chua, "Editorial", International Journal of Bifurcation and Chaos, Vol. l (1), 1991) 

"It remains an unhappy fact that there is no best method for finding the solution to general nonlinear optimization problems. About the best general procedure yet devised is one that relies upon imbedding the original problem within a family of problems, and then developing relations linking one member of the family to another. If this can be done adroitly so that one family member is easily solvable, then these relations can be used to step forward from the solution of the easy problem to that of the original problem. This is the key idea underlying dynamic programming, the most flexible and powerful of all optimization methods." (John L Casti, "Five Golden Rules", 1995)

"When it comes to modeling processes that are manifestly governed by nonlinear relationships among the system components, we can appeal to the same general idea. Calculus tells us that we should expect most systems to be 'locally' flat; that is, locally linear. So a conservative modeler would try to extend the word 'local' to hold for the region of interest and would take this extension seriously until it was shown to be no longer valid." (John L Casti, "Five Golden Rules", 1995)

"A system at a bifurcation point, when pushed slightly, may begin to oscillate. Or the system may flutter around for a time and then revert to its normal, stable behavior. Or, alternatively it may move into chaos. Knowing a system within one range of circumstances may offer no clue as to how it will react in others. Nonlinear systems always hold surprises." (F David Peat, "From Certainty to Uncertainty", 2002)

"In a linear system a tiny push produces a small effect, so that cause and effect are always proportional to each other. If one plotted on a graph the cause against the effect, the result would be a straight line. In nonlinear systems, however, a small push may produce a small effect, a slightly larger push produces a proportionately larger effect, but increase that push by a hair’s breadth and suddenly the system does something radically different." (F David Peat, "From Certainty to Uncertainty", 2002)

"Complex systems are full of interdependencies - hard to detect - and nonlinear responses." (Nassim N Taleb, "Antifragile: Things That Gain from Disorder", 2012)

Statistical Tools III: Cards

"In short, absolute, so-called mathematical factors never find a firm basis in military calculations. From the very start there is an interplay of possibilities, probabilities, good luck and bad that weaves its way throughout the length and breadth of the tapestry. In the whole range the human activities war most closely resembles a game of cards." (Carl von Clausewitz, "On War", 1832)

"The law of large numbers is noted in events which are attributed to pure chance since we do not know their causes or because they are too complicated. Thus, games, in which the circumstances determining the occurrence of a certain card or certain number of points on a die infinitely vary, can not be subjected to any calculus. If the series of trials is continued for a long time, the different outcomes nevertheless appear in constant ratios. Then, if calculations according to the rules of a game are possible, the respective probabilities of eventual outcomes conform to the known Jakob Bernoulli theorem. However, in most problems of contingency a prior determination of chances of the various events is impossible and, on the contrary, they are calculated from the observed result." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"As an instrument for selecting at random, I have found nothing superior to dice. It is most tedious to shuffle cards thoroughly be- tween each successive draw, and the method of mixing and stirring up marked balls in a bag is more tedious still. A teetotum or some form of roulette is preferable to these, but dice are better than all. When they are shaken and tossed in a basket, they hurtle so variously against one another and against the ribs of the basket-work that they tumble wildly about, and their positions at the outset afford no perceptible clue to what they will be after even a single good shake and toss." (Francis Galton, Nature vol. 42, 1890) 

"Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent." (Arthur M Marshall, 1894)

"If you take a pack of cards as it comes from the maker and shuffle it for a few minutes, all trace of the original systematic order disappears. The order will never come back however long you shuffle. Something has been done which cannot be undone, namely, the introduction of a random element in place of the arrangement." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"It seems hard to sneak a look at God's cards. But that He plays dice and uses 'telepathic' methods [...] is something that I cannot believe for a single moment." (Albert Einstein, [Letter to Cornel Lanczos] 1942)

"We must emphasize that such terms as 'select at random', 'choose at random', and the like, always mean that some mechanical device, such as coins, cards, dice, or tables of random numbers, is used." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"A thorough understanding of game theory, should dim these greedy hopes. Knowledge of game theory does not make one a better card player, businessman or military strategist." (Anatol Rapoport, "The Use and Misuse of Game Theory," 1962)

"Life is like a game of cards. The hand that is dealt you represents determinism. The way you play it is free will." (Jawaharlal Nehru, Saturday Review, 1967)

"There may be such a thing as habitual luck. People who are said to be lucky at cards probably have certain hidden talents for those games in which skill plays a role. It is like hidden parameters in physics, this ability that does not surface and that I like to call 'habitual luck'." (Stanislaw Ulam, "Adventures of a Mathematician", 1976)

"Gambling was the place where statistics and profound human consequences met most nakedly, after all, and cards, even more than dice or the numbers on a roulette wheel, seemed able to define and perhaps even dictate a player's... luck." (Tim Powers, "Last Call", 1992)

"An example, which, like tossing a coin, is intimately associated with games of chance, is the shuffling of a deck of cards. […] the process is not completely random, if by what happens next we mean the outcome of the next single riffle, since one riffle cannot change any given order of the cards in the deck to any other given order. In particular, a single riffle cannot completely reverse the order of the cards, although a sufficient number of successive riffles, of course, can produce any order." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Whether we shuffle cards or roll dice, chance is only a result of our human lack of deftness: we don't have enough control to immobilize a die at will or to individually direct the cards in a deck. The comparison is an important one nonetheless, and highlights the limits of this method of creating chance - it doesn't matter who rolls the dice, but we wouldn't let just anyone shuffle the cards." (Ivar Ekeland, "The Broken Dice, and Other Mathematical Tales of Chance", 1993)

"For several centuries that we know of, and probably for many centuries before that, flipping a coin (or rolling a die) has been the epitome of probability, the paradigm of randomness. You flip the coin (or roll the die), and nobody can accurately predict how it will fall. Nor can the most powerful computer predict correctly how it will fall, if it is flipped energetically enough. This is why cards, dice, and other gambling aids crop up so often in literature both directly and as metaphors. No doubt it is also the reason for the (perhaps excessive) popularity of gambling as entertainment. If anyone had any idea what numbers the lottery would show, or where the roulette ball will land, the whole industry would be a dead duck." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"We cannot really have a perfectly shuffled pack of perfect cards; this ‘collection of equally likely hands’ is actually a fiction. We create the idea, and then use the rules of arithmetic to calculate the required chances. This is characteristic of all mathematics, which concerns itself only with rules defining the behaviour of entities which are themselves undefined (such as ‘numbers’ or ‘points’)." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"To look at the development of physics since Newton is to observe a struggle to define the limits of science. Part of this process has been the intrusion of scientific methods and ideas into domains that have traditionally been the province of metaphysics or religion. In this conflict, Hawking’s phrase ‘to know the Mind of God’ is just one example of a border infringement. But by playing the God card, Hawking has cleverly fanned the flames of his own publicity, appealing directly to the popular allure of the scientist-as-priest." (Peter Coles, "Hawking and the Mind of God", 2000)

"In contrast, the system may be a pack of cards, and the dynamic may be to shuffle the pack and then take the top card. Imagine that the current top card is the ace of spades, and that after shuffling the pack the top card becomes the seven of diamonds. Does that imply that whenever the top card is the ace of spades then the next top card will always be the seven of diamonds? Of course not. So this system is random."(Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"In modelling terms, the difference between randomness and determinacy is clear enough. The randomness in the pack of cards arises from our failure to prescribe unique rules for getting from the current state to the next one. There are lots of different ways to shuffle a pack. The determinism of the cannonball is a combination of two things: fully prescribed rules of behaviour, and fully defined initial conditions. Notice that in both systems we are thinking on a very short timescale: it is the next state that matters - or, if time is flowing continuously, it is the state a tiny instant into the future. We don't need to consider long-term behaviour to distinguish randomness from determinacy."(Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"The randomness of the card-shuffle is of course caused by our lack of knowledge of the precise procedure used to shuffle the cards. But that is outside the chosen system, so in our practical sense it is not admissible. If we were to change the system to include information about the shuffling rule – for example, that it is given by some particular computer code for pseudo-random numbers, starting with a given ‘seed value’ – then the system would look deterministic. Two computers of the same make running the same ‘random shuffle’ program would actually produce the identical sequence of top cards."(Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Players must accept the cards dealt to them. However, once they have those cards in hand, they alone choose how they will play them. They decide what risks and actions to take." (John C Maxwell, "The Difference Maker: Making Your Attitude Your Greatest Asset", 2006)

"It's a game of a million inferences. There are a lot of things to draw inferences from - cards played and not played. These inferences tell you something about the probabilities. It's got to be the best intellectual exercise out there. You're seeing through new situations every ten minutes. Bridge is about weighing gain/loss ratios. You're doing calculations all the time." (Warren Buffett)

"The card-player begins by arranging his hand for maximum sense. Scientists do the same with the facts they gather." (Isaac Asimov)

Adam Smith - Collected Quotes

"Every man, as the Stoics used to say, is first and principally recommended to his own care; and every man is certainly, in every respect, fitter and abler to take care of himself than of any other person. Every man feels his own pleasures and his own pains more sensibly than those of other people. The former are the original sensations; the latter the reflected or sympathetic images of those sensations. The former may be said to be the substance; the latter the shadow. (Adam Smith, "The Theory of Moral Sentiments", 1759)

"The man of system, on the contrary, is apt to be very wise in his own conceit; and is often so enamoured with the supposed beauty of his own ideal plan of government, that he cannot suffer the smallest deviation from any part of it. He goes on to establish it completely and in all its parts, without any regard either to the great interests, or to the strong prejudices which may oppose it. He seems to imagine that he can arrange the different members of a great society with as much ease as the hand arranges the different pieces upon a chess-board. He does not consider that the pieces upon the chess-board have no other principle of motion besides that which the hand impresses upon them; but that, in the great chess-board of human society, every single piece has a principle of motion of its own, altogether different from that which the legislature might choose to impress upon it. If those two principles coincide and act in the same direction, the game of human society will go on easily and harmoniously, and is very likely to be happy and successful. If they are opposite or different, the game will go on miserably, and the society must be at all times in the highest degree of disorder." (Adam Smith, "The Theory of Moral Sentiments", 1759)

"Every system which endeavours, either, by extraordinary encouragements, to draw towards a particular species of industry a greater share of the capital of the society than what would naturally go to it; or, by extraordinary restraints, to force from a particular species of industry some share of the capital which would otherwise be employed in it; is in reality subversive of the great purpose which it means to promote. It retards, instead of accelerating, the progress of the society towards real wealth and greatness; and diminishes, instead of increasing, the real value of the annual produce of its land and labour. All systems either of preference or of restraint, therefore, being thus completely taken away, the obvious and simple system of natural liberty establishes itself of its own accord. Every man, as long as he does not violate the laws of justice, is left perfectly free to pursue his own interest his own way, and to bring both his industry and capital into competition with those of any other man, or order of men." (Adam Smith, "The Wealth of Nations", 1776)

"It is unjust that the whole of society should contribute towards an expence of which the benefit is confined to a part of the society." (Adam Smith, 1776)

"Systems in many respects resemble machines. A machine is a little system, created to perform, as well as to connect together, in reality, those different movements and effects which the artist has occasion for.  A system is an imaginary machine invented to connect together in the fancy those different movements and effects which are already in reality performed. […] The machines that are first invented to perform any particular movement are always the most complex, and succeeding artists generally discover that, with fewer wheels, with fewer principles of motion, than had originally been employed, the fame effects may be more easily produced. The first systems, in the fame manner, are always the most complex, and a particular connecting chain, or principle, is generally thought necessary to unite every two seemingly disjointed appearances: but it often happens, that one great connecting principle is afterwards found to be sufficient to bind together all the discordant phænomena that occur in a whole species of things." (Adam Smith, "The Wealth of Nations", 1776)

"That the chance of gain is naturally over-valued, we may learn from the universal success of lotteries." (Adam Smith, "An Inquiry Into the Nature and Causes of the Wealth of Nations", 1776)

"I suspect that even if the random walkers announced a perfect mathematic proof of randomness I would go on believing that in the long run future earnings influence present value, and that in the short run the dominant factor is the elusive Australopithecus, the temper of the crowd." (Adam Smith, "The Money Game", 1968)

On Randomness IX (Probabilities)

"The most important application of the theory of probability is to what we may call 'chance-like' or 'random' events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe - after many unsuccessful attempts - that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"The classical theory of probability was devoted mainly to a study of the gamble's gain, which is again a random variable; in fact, every random variable can be interpreted as the gain of a real or imaginary gambler in a suitable game." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

"To every event defined for the original random walk there corresponds an event of equal probability in the dual random walk, and in this way almost every probability relation has its dual." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

“The epistemological value of probability theory is based on the fact that chance phenomena, considered collectively and on a grand scale, create non-random regularity.” (Andrey Kolmogorov, “Limit Distributions for Sums of Independent Random Variables”, 1954)

"The urn model is to be the expression of three postulates: (1) the constancy of a probability distribution, ensured by the solidity of the vessel, (2) the random-character of the choice, ensured by the narrowness of the mouth, which is to prevent visibility of the contents and any consciously selective choice, (3) the independence of successive choices, whenever the drawn balls are put back into the urn. Of course in abstract probability and statistics the word 'choice' can be avoided and all can be done without any reference to such a model. But as soon as the abstract theory is to be applied, random choice plays an essential role."(Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)

"Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments. Possibility theory, on the other hand, is ideal for formalizing incomplete information expressed in terms of fuzzy propositions." (George Klir, "Fuzzy sets and fuzzy logic", 1995)

"Often, we use the word random loosely to describe something that is disordered, irregular, patternless, or unpredictable. We link it with chance, probability, luck, and coincidence. However, when we examine what we mean by random in various contexts, ambiguities and uncertainties inevitably arise. Tackling the subtleties of randomness allows us to go to the root of what we can understand of the universe we inhabit and helps us to define the limits of what we can know with certainty." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"The subject of probability begins by assuming that some mechanism of uncertainty is at work giving rise to what is called randomness, but it is not necessary to distinguish between chance that occurs because of some hidden order that may exist and chance that is the result of blind lawlessness. This mechanism, figuratively speaking, churns out a succession of events, each individually unpredictable, or it conspires to produce an unforeseeable outcome each time a large ensemble of possibilities is sampled."  (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"Chance is just as real as causation; both are modes of becoming.  The way to model a random process is to enrich the mathematical theory of probability with a model of a random mechanism. In the sciences, probabilities are never made up or 'elicited' by observing the choices people make, or the bets they are willing to place.  The reason is that, in science and technology, interpreted probability exactifies objective chance, not gut feeling or intuition. No randomness, no probability." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006) 

"[...] according to the quantum theory, randomness is a basic trait of reality, whereas in classical physics it is a derivative property, though an equally objective one. Note, however, that this conclusion follows only under the realist interpretation of probability as the measure of possibility. If, by contrast, one adopts the subjectivist or Bayesian conception of probability as the measure of subjective uncertainty, then randomness is only in the eye of the beholder." (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)

Mathematics as Game I

"So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality." (Henri Poincaré, "The Value of Science", 1905)

"Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." (David Hilbert, "Natur und Mathematisches Erkennen", 1919–20) 

"A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules and syllogisms, without motivation or goal." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1956) 

"To the average mathematician who merely wants to know that his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency." (Paul Cohen, "Axiomatic set theory, American Mathematical Society", 1971)

"There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it." (Imre Lakatos, "Proofs and Refutations", 1976)

"The way the mathematics game is played, most variations lie outside the rules, while music can insist on perfect canon or tolerate a casual accompaniment." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"If doing mathematics or science is looked upon as a game, then one might say that in mathematics you compete against yourself or other mathematicians; in physics your adversary is nature and the stakes are higher." (Mark Kac, "Enigmas Of Chance", 1985)

"Mathematicians are used to game-playing according to a set of rules they lay down in advance, despite the fact that nature always writes her own. One acquires a great deal of humility by experiencing the real wiliness of nature." (Philip W Anderson, "More and Different: Notes from a Thoughtful Curmudgeon", 2011)

"Often the key contribution of intuition is to make us aware of weak points in a problem, places where it may be vulnerable to attack. A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)

Game Theory II

"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution." (Richard A Epstein, "The Theory of Gambling and Statistical Logic" [Revised Edition], 1977)

"Game theory is a collection of mathematical models designed to study situations involving conflict and/or cooperation. It allows for a multiplicity of decision makers who may have different preferences and objectives. Such models involve a variety of different solution concepts concerned with strategic optimization, stability, bargaining, compromise, equity and coalition formation." (Notices of the American Mathematical Society Vol. 26 (1), 1979) 

"A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play." (James P Cars, "Finite and Infinite Games: A Vision of Life as Play and Possibility", 1986)

"Game theory is a theory of strategic interaction. That is to say, it is a theory of rational behavior in social situations in which each player has to choose his moves on the basis of what he thinks the other players' countermoves are likely to be." (John Harsanyi, "Games with Incomplete Information", The American Economic Review Vol. 85 (3), 1997)

"In principle, every social situation involves strategic interaction among the participants. Thus, one might argue that proper understanding of any social situation would require game-theoretic analysis. But in actual fact, classical economic theory did manage to sidestep the game-theoretic aspects of economic behavior by postulating perfect competition, i. e., by assuming that every buyer and every seller is very small as compared with the size of the relevant markets, so that nobody can significantly affect the existing market prices by his actions." (John Harsanyi, "Games with Incomplete Information" 1997)

"Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises." (Thomas Flanagan, "Game Theory and Canadian Politics", 1998)

"I think game theory creates ideas that are important in solving and approaching conflict in general." Robert Aumann, 2005)

"The players in a game are said to be in strategic equilibrium (or simply equilibrium) when their play is mutually optimal: when the actions and plans of each player are rational in the given strategic environment - i. e., when each knows the actions and plans of the others." (Robert Aumann, "War and Peace", 2005)

"An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

"Game theory brings to the chaos–theory table the idea that generally, societies are not designed, and that most situations don't come with a rulebook. Instead, people have their own plans and designs on how things should fit together. They want to determine how the game is played, and they see societal designers as myopic busybodies who would imprison them with their theories." (Lawrence K Samuels, "In Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

Game Theory I

"The implication of game theory, which is also the implication of the third image, is, however, that the freedom of choice of any one state is limited by the actions of the others." (Kenneth Waltz, "Man, the State, and War", 1959)

"At present game theory has, in my opinion, two important uses, neither of them related to games nor to conflict directly. First, game theory stimulates us to think about conflict in a novel way. Second, game theory leads to some genuine impasses, that is, to situations where its axiomatic base is shown to be insufficient for dealing even theoretically with certain types of conflict situations... Thus, the impact is made on our thinking process themselves, rather than on the actual content of our knowledge. (Anatol Rapoport, "Fights, games, and debates", 1960)

"It is the shortcomings of game theory (as originally formulated) which force the consideration of the role of ethics, of the dynamics of social structure, and of social structure and of individual psychology in situations of conflict." (Anatol Rapoport, "Fights, games, and debates", 1960)

"Although the drama of games of strategy is strongly linked with the psychological aspects of the conflict, game theory is not concerned with these aspects. Game theory, so to speak, plays the board. It is concerned only with the logical aspects of strategy." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"Game theory applies to a very different type of conflict, now technically called a game. The well-known games such as poker, chess, ticktacktoe and so forth are games in the strict technical Bark and counterbark sense. But what makes parlor games is not their entertainment value or detachment from real life." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"Whether game theory leads to clear-cut solutions, to vague solutions, or to impasses, it does achieve 

one thing. In bringing techniques of logical and mathematical analysis gives men an opportunity to bring conflicts up from the level of fights, where the intellect is beclouded by passions, to the level of games, where the intellect has a chance to operate." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not." (Anatol Rapoport, "Prisoner's dilemma: A study in conflict and cooperation", 1965)

"Evolutionary game theory is a way of thinking about evolution at the phenotypic level when the fitnesses of particular phenotypes depend on their frequencies in the population." (John M Smith, "Evolution and the Theory of Games", 1973)

"Strategy in complex systems must resemble strategy in board games. You develop a small and useful tree of options that is continuously revised based on the arrangement of pieces and the actions of your opponent. It is critical to keep the number of options open. It is important to develop a theory of what kinds of options you want to have open." (John H Holland, [presentation] 2000)

"Game theory postulates rational behavior for each participant. Each player is conscious of the rules and behaves in accordance with them, each player has sufficient knowledge of the situation in which he or she is involved to be able to evaluate what the best option is when it comes to taking action (a move), and each player takes into account the decisions that might be made by other participants and their repercussions with respect to his or her own decision. Game theory about zero-sum games with two participants is relevant for chess. In this type of situation, each action that is favorable to one participant (player) is proportionally unfavorable for the opponent. Thus, the gain of one represents the loss of the other." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

On Equilibrium (2000-2009)

"An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory." (Ivar Ekeland, "Le meilleur des mondes possibles" ["The Best of All Possible Worlds"], 2000)

"Economic theory in general ignores processes which take time to occur, and instead assumes that everything occurs in equilibrium." (Steve Keen, "Naked Emperor Of The Social Sciences", 2001)

"The obsession with equilibrium has imposed enormous costs on economics." (Steve Keen, "Debunking Economics - The Naked Emperor Of The Social Sciences", 2001)

"In this crucial sense, the theory of punctuated equilibrium adopts a very conservative position. The theory asserts no novel claim about modes or mechanisms of speciation; punctuated equilibrium merely takes a standard microevolutionary model and elucidates its expected expression when properly scaled into geological time." (Stephen J Gould, "The Structure of Evolutionary Theory", 2002)

"Positive feedbacks, when unchecked, can produce runaways until the deviation from equilibrium is so large that other effects can be abruptly triggered and lead to ruptures and crashes." (Didier Sornette, "Why Stock Markets Crash - Critical Events in Complex Systems", 2003)

"Stock market bubbles don't grow out of thin air. They have a solid basis in reality - but reality as distorted by a misconception. Under normal conditions misconceptions are self-correcting, and the markets tend toward some kind of equilibrium. Occasionally, a misconception is reinforced by a trend prevailing in reality, and that is when a boom-bust process gets under way. Eventually the gap between reality and its false interpretation becomes unsustainable, and the bubble bursts." (George Soros, [interview] 2004)

"The players in a game are said to be in strategic equilibrium (or simply equilibrium) when their play is mutually optimal: when the actions and plans of each player are rational in the given strategic environment - i. e., when each knows the actions and plans of the others." (Robert Aumann, "War and Peace", 2005)

"When defining living systems, the term dynamic equilibrium is essential. It does not imply something which is steady or stable. On the contrary, it is a floating state characterized by invisible movements and preparedness for change. To be in dynamic equilibrium is adapting adjustment to balance. Homeostasis stands for the sum of all control functions creating the state of dynamic equilibrium in a healthy organism. It is the ability of the body to maintain a narrow range of internal conditions in spite of environmental changes." (Lars Skyttner, "General Systems Theory: Problems, Perspective, Practice", 2005)

"The second law of thermodynamics states that in an isolated system, entropy can only increase, not decrease. Such systems evolve to their state of maximum entropy, or thermodynamic equilibrium. Therefore, physical self-organizing systems cannot be isolated: they require a constant input of matter or energy with low entropy, getting rid of the internally generated entropy through the output of heat ('dissipation'). This allows them to produce ‘dissipative structures’ which maintain far from thermodynamic equilibrium. Life is a clear example of order far from thermodynamic equilibrium." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"The invisible hand is an emergent property of this system, which never reaches an optimal equilibrium, but instead is fundamentally dynamic and unstable, with complex effects on society."(David Orrell, "The Other Side of the Coin", 2008)

"Perhaps the most significant deviations from the concept of linear change are the concepts of equilibrium and randomness. For most intents and purposes, the concept of equilibrium has been used to describe places or times when change stops occurring. Randomness suggests that the changes are unpredictable and not explicable by any known concepts or predictors. " (Stephen J Guastello & Larry S Liebovitch, "Introduction to Nonlinear Dynamics and Complexity" [in "Chaos and Complexity in Psychology"], 2009)

Anatol Rapoport - Collected Quotes

"The first attempts to consider the behavior of so-called ‘random neural nets’ in a systematic way have led to a series of problems concerned with relations between the 'structure' and the ‘function’ of such nets. The ‘structure’ of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of "this" neuron synapsing on ‘that’ one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport. "Cycle distributions in random nets." The Bulletin of Mathematical Biophysics 10 (3), 1948)

"A fundamental value in the scientific outlook is concern with the best available map of reality. The scientist will always seek a description of events which enables him to predict most by assuming least. He thus already prefers a particular form of behavior. If moralities are systems of preferences, here is at least one point at which science cannot be said to be completely without preferences. Science prefers good maps." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)

"No map contains all the information about the territory it represents. The road map we get at the gasoline station may show all the roads in the state, but it will not as a rule show latitude and longitude. A physical map goes into details about the topography of a country but is indifferent to political boundaries. Furthermore, the scale of the map makes a big difference. The smaller the scale the less features will be shown." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)

"[…] theoretical science is essentially disciplined exploitation of metaphor." (Anatol Rapoport, "Operational Philosophy", 1953)

"Scientific metaphors are called models. They are made with the full knowledge that the connection between the metaphor and the real thing is primarily in the mind of the scientist. And they are made with a clearly definable purpose - as starting points of a deductive process. […] Like every other aspect of scientific procedure, the scientific metaphor is a pragmatic device, to be used freely as long as it serves its purpose, to be discarded without regrets when it fails to do so." (Anatol Rapoport, "Operational Philosophy", 1954) 

"The predictions of physical theories for the most part concern situations where initial conditions can be precisely specified. If such initial conditions are not found in nature, they can be arranged. Such arrangements are considerably easier to realize with inanimate than with animate matter, because the properties of animate matter are much more sensitive to being tampered with than inanimate matter. In particular, living tissue in vitro may behave quite differently than in situ. Controlled biological experiments are, of course, possible, but they are more difficult and their scope is more limited than that of physical experiments. For this reason, biology has had to depend to a greater extent than physics on theories of larger speculative scope, in which reasoning by imaginative analogy plays a more important role." (Anatol Rapoport, "The Search for Simplicity", 1956)

"A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions" (Anatol Rapoport, "Various meanings of theory", American Political Science Review 52, 1958)

"A thorough understanding of game theory, should dim these greedy hopes. Knowledge of game theory does not make one a better card player, businessman or military strategist." (Anatol Rapoport, "The Use and Misuse of Game Theory", 1962)

"Although the drama of games of strategy is strongly linked with the psychological aspects of the conflict, game theory is not concerned with these aspects. Game theory, so to speak, plays the board. It is concerned only with the logical aspects of strategy." (Anatol Rapoport, "The Use and Misuse of Game Theory", 1962)

"The outstanding feature of behavior is that it is often quite easy to recognize but extremely difficult or impossible to describe with precision." (Anatol Rapoport. "An Essay on Mind", General Systems, 1962)

"[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not." (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)

"The usefulness of the models in constructing a testable theory of the process is severely limited by the quickly increasing number of parameters which must be estimated in order to compare the predictions of the models with empirical results" (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)

Martin Gardner - Collected Quotes

"Chess combines the beauty of mathematical structure with the recreational delights of a competitive game." (Martin Gardner, "Mathematics, Magic, and Mystery", 1956)

"No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are 'undecideable'. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)

"In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." (Martin Gardner, "Mathematical Games", Scientific American, 1973)

"Surreal numbers are an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other 'real' value does. The system is truly 'surreal.'" (Martin Gardner, "Mathematical Magic Show", 1977)

“All mathematical problems are solved by reasoning within a deductive system in which basic laws of logic are embedded.” (Martin Gardner, “Aha! Insight”, 1978)

"At the heart of mathematics is a constant search for simpler and simpler ways to prove theorems  and solve problems. [...] The sudden hunch, the creative leap of the mind that ‘sees’ in a flash how to solve a problem in a simple way, is something quite different from general intelligence." (Martin Gardner, "Aha! Insight", 1978)

“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)

"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

“Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics.” (Martin Gardner, "Aha! Insight", 1978)

“Graph theory is the study of sets of points that are joined by lines.” (Martin Gardner, “Aha! Insight”, 1978)

“The great revolutions in science are almost always the result of unexpected intuitive leaps. After all, what is science if not the posing of difficult puzzles by the universe? Mother Nature does something interesting, and challenges the scientist to figure out how she does it. In many cases the solution is not found by exhaustive trial and error […] or even by a deduction based on the relevant knowledge.”  (Martin Gardner, "Aha! Insight", 1978)

“The word ‘induction’ has two essentially different meanings. Scientific induction is a process by which scientists make observations of particular cases, such as noticing that some crows are black, then leap to the universal conclusion that all crows are black. The conclusion is never certain. There is always the possibility that at least one unobserved crow is not black." (Martin Gardner, “Aha! Insight”, 1978)

"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics.” (Martin Gardner, “Aha! Insight”, 1978)

"The external world exists; the structure of the world is ordered; we know little about the nature of the order, nothing at all about why it should exist." (Martin Gardner, "Order and Surprise", 1983)

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

"Besides being essential in modern physics, the complex-number field provides pure mathematics with a multitude of brain-boggling theorems. It is worth keeping in mind that complex numbers, although they include the reals.as a subset, differ from real numbers in startling ways. One cannot, for example, speak of a complex number as being either positive or negative: those properties apply only to the reals and the pure imaginaries. It is equally meaningless to say that one complex number is larger or smaller than another." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)

"The seemingly preposterous assumption that there is a square root of -1 was justified on pragmatic grounds: it simplified certain calculations and so could be used as long as 'real' values were obtained at the end. The parallel with the rules for using negative numbers is striking. If you are trying to determine how many cows there are in a field (that is, if you are working in the domain of positive integers), you may find negative numbers useful in the calculation, but of course the final answer must be in terms of positive numbers because there is no such thing as a negative cow." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)

"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)

"[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime." (Martin Gardner, "When You Were a Tadpole and I Was a Fish", 2009)

"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" (Martin Gardner, The Dover Math and Science Newsletter, 2011)

"All mathematicians share […] a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)

"Mathematics is not only real, but it is the only reality. [The] entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." (Martin Gardner)

"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)

"There are some traits all mathematicians share. An obvious one is a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)