Garry Kasparov - Collected Quotes

"A tactician feels at home reacting to threats and seizing opportunities on the battlefield. When your opponent has blundered, a winning tactic can suddenly appear and serve as both means and end. […] Every time you make a move, you must consider your opponent’s response, your answer to that response, and so on. A tactic ignites an explosive chain reaction, a forceful sequence of moves that carries the players along on a wild ride. You analyze the position as deeply as you can, compute the dozens of variations, the hundreds of positions. If you don’t immediately exploit a tactical opportunity, the game will almost certainly turn against you; one slip and you are wiped out. But if you seize the opportunities that your strategy creates, you’ll play your game like a Grandmaster." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Acquired patterns and the logic to employ them combine with our inherent qualities to create a unique decision-maker. As time goes by, experience and knowledge are focused through the prism of talent, which can itself be sharpened, focused, and polished. This mix is the source of intuition, an absolutely unique tool that each of us possesses and that we can continuously hone into an ever-finer instrument." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Against solid strategy, diversionary tactics will either be insufficient, or flawed. If they are insufficient, you can and should ignore them, continuing along your path. If they are radical enough to force you from your path, they are likely flawed in some way - unless you have blundered. Often an opponent is so eager to get you to change your course that he fatally weakens his own position in the attempt." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Any discipline in which access to information is nearly unlimited but time is a major factor has a strong intuitive element. Stock analysts search for visual patterns in stock charts, shapes such as 'teacups' and 'rising wedges, the way chess players look for checkmating patterns. Intuition tells us not just what and how, but also when." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Effective tactics result from alertness and speed, this is obvious, but they also require an understanding of all the possibilities at hand. Experience allows us to instantly apply the patterns we have successfully used in the past." (Garry Kasparov, "How Life Imitates Chess", 2007)

"In chess we see many cases of good strategy failing due to bad tactics and vice versa. A single oversight can undo the most brilliant concepts. Even more dangerous in the long run are cases of bad strategy succeeding due to good tactics, or due to sheer good fortune. This may work once, but rarely twice. This is why it is so important to question success as vigorously as you question failure." (Garry Kasparov, "How Life Imitates Chess", 2007)

"It’s true that to be a great chess player you must have a good memory, but it is much harder to explain what, exactly, we are remembering. Patterns? Numbers? Mental pictures of the board and pieces? The answer seems to be 'all of the above'." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Knowing a solution is at hand is a huge advantage; it’s like not having a 'none of the above' option. Anyone with reasonable competence and adequate resources can solve a puzzle when it is presented as something to be solved. We can skip the subtle evaluations and move directly to plugging in possible solutions until we hit upon a promising one. Uncertainty is far more challenging." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Opposite pairs working in harmony: this has become a theme of our quest to perfect decision-making. Calculation and evaluation. Patience and opportunism, intuition and analysis, style and objectivity. At the performance level these elements come together in management and vision, strategy and tactics, planning and reaction. Success comes from balancing these forces and harnessing their inherent power." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Setbacks and losses are both inevitable and essential if you're going to improve and become a good, even great, competitor. The art is in avoiding catastrophic losses in the key battles." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Solving new problems is what keeps us moving forward as individuals and as a society, so don't back down." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Sometimes the hardest thing to do in a pressure situation is to allow the tension to persist. The temptation is to make a decision, any decision, even if it is an inferior choice." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Tactics involve calculations that can tax the human brain, but when you boil them down, they are actually the simplest part of chess and are almost trivial compared to strategy. Think of tactics as forced, planned responses, basically a series of  'if-then' statements that would make a computer programmer feel right at home." (Garry Kasparov, "How Life Imitates Chess", 2007)

"The middle game requires alertness in general and alertness to patterns in particular. These are general ideas that anyone can learn with practice; the more you play, the better you become at recognizing the patterns and applying the solutions. That is, to find similarities to positions you have seen before and then to recall what worked (or what didn’t work) in that situation. There is still potential for great creativity, if you are able to relate known patterns to new positions to find the unique solution: the best move." (Garry Kasparov, "How Life Imitates Chess", 2007)

[tactics:] "The means of effecting a strategic plan. Every move in a chess game has some tactical components. Tactics require calculation and are the foundation of combinations." (Garry Kasparov, "How Life Imitates Chess", 2007)

"The worst enemy of the strategist is the clock. Time trouble, as we call it in chess, reduces us all to pure reflex and reaction, tactical play. Emotion and instinct cloud our strategic vision when there is no time for proper evaluation. A game of chess can suddenly seem a lot like a game of chance. Even the finest sense of intuition can’t flourish in the long term without accurate calculations." (Garry Kasparov, "How Life Imitates Chess", 2007)

"There is still a great deal of uncharted territory in the opening phase of the game. New ideas, new concepts, new plans in old and forgotten variations, there is still much to discover in the opening. The tactical patterns and strategic concepts of the middle game have been well mapped out by generations of Grandmasters, although there are occasional fresh twists. In the endgame, however, the plans and possibilities are open and known to all, an almost mathematical exercise. This isn’t to say that everything is predetermined. With flawless play from both sides, the endgame will advance toward a predictable conclusion. But since humans are flawed, damage can be inflicted or repaired. Even if one player is at a clear disadvantage, he may simply outplay his opponent." (Garry Kasparov, "How Life Imitates Chess", 2007)

"This obligation to move can be a burden to a player without strategic vision." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Whereas strategy is abstract and based on long-term goals, tactics are concrete and based on finding the best move right now. Tactics are conditional and opportunistic, all about threat and defense. No matter what pursuit you’re engaged in - chess, business, the military, managing a sports team - it takes both good tactics and wise strategy to be successful." (Garry Kasparov, "How Life Imitates Chess", 2007)

"But chess is a limited game and every position will have patterns and markers our intuition can interpret. Each of the estimated tens of thousands of positions a strong master has imprinted in memory can also be broken down into component parts, rotated, twisted, and still be useful. Outside of the opening sequences that are indeed memorized, strong human players don’t rely on recall as much as on a super-fast analogy engine." (Garry Kasparov, "Deep Thinking", 2017)

"Machines that replace physical labor have allowed us to focus more on what makes us human: our minds. Intelligent machines will continue that process, taking over the more menial aspects of cognition and elevating our mental lives toward creativity, curiosity, beauty, and joy. These are what truly make us human, not any particular activity or skill like swinging a hammer - or even playing chess." (Garry Kasparov, "Deep Thinking", 2017)

"The human mind isn’t a computer; it cannot progress in an orderly fashion down a list of candidate moves and rank them by a score down to the hundredth of a pawn the way a chess machine does. Even the most disciplined human mind wanders in the heat of competition. This is both a weakness and a strength of human cognition. Sometimes these undisciplined wanderings only weaken your analysis. Other times they lead to inspiration, to beautiful or paradoxical moves that were not on your initial list of candidates." (Garry Kasparov, "Deep Thinking", 2017)

"The main problem of chess programming is the very large number of possible continuations involved, what is called the 'branching factor'. Right from the start, the sheer number of possibilities was enough to stress the resources of the fastest computers then conceivable." (Garry Kasparov, "Deep Thinking", 2017)

On Patterns (2000-2009)

"In a linear world of equilibrium and predictability, the sparse research into an evidence base for management prescriptions and the confused findings it produces would be a sign of incompetence; it would not make much sense. Nevertheless, if organizations are actually patterns of nonlinear interaction between people; if small changes could produce widespread major consequences; if local interaction produces emergent global pattern; then it will not be possible to provide a reliable evidence base. In such a world, it makes no sense to conduct studies looking for simple causal relationships between an action and an outcome. I suggest that the story of the last few years strongly indicates that human action is nonlinear, that time and place matter a great deal, and that since this precludes simple evidence bases we do need to rethink the nature of organizations and the roles of managers and leaders in them." (Ralph D Stacey, "Complexity and Organizational Reality", 2000)

"The central proposition in [realistic thinking] is that human actions and interactions are processes, not systems, and the coherent patterning of those processes becomes what it becomes because of their intrinsic capacity, the intrinsic capacity of interaction and relationship, to form coherence. That emergent form is radically unpredictable, but it emerges in a controlled or patterned way because of the characteristic of relationship itself, creation and destruction in conditions at the edge of chaos." (Ralph D Stacey et al, "Complexity and Management: Fad or Radical Challenge to Systems Thinking?", 2000)

"Although the detailed moment-to-moment behavior of a chaotic system cannot be predicted, the overall pattern of its 'random' fluctuations may be similar from scale to scale. Likewise, while the fine details of a chaotic system cannot be predicted one can know a little bit about the range of its 'random' fluctuation." (F David Peat, "From Certainty to Uncertainty", 2002)

"There are endless examples of elaborate structures and apparently complex processes being generated through simple repetitive rules, all of which can be easily simulated on a computer. It is therefore tempting to believe that, because many complex patterns can be generated out of a simple algorithmic rule, all complexity is created in this way." (F David Peat, "From Certainty to Uncertainty", 2002)

"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Didier Sornette, "Why Stock Markets Crash: Critical events in complex financial systems", 2003)

"Learning is the process of creating networks. Nodes are external entities which we can use to form a network. Or nodes may be people, organizations, libraries, web sites, books, journals, database, or any other source of information. The act of learning (things become a bit tricky here) is one of creating an external network of nodes - where we connect and form information and knowledge sources. The learning that happens in our heads is an internal network (neural). Learning networks can then be perceived as structures that we create in order to stay current and continually acquire, experience, create, and connect new knowledge (external). And learning networks can be perceived as structures that exist within our minds (internal) in connecting and creating patterns of understanding." (George Siemens, "Knowing Knowledge", 2006)

"Some number patterns, like even and odd numbers, lie on the surface. But the more you learn about numbers, both experimentally and theoretically, the more you discover patterns that are not so obvious. […] After a hidden pattern is exposed, it can be used to find more hidden patterns. At the end of a long chain of patterned reasoning, you can get to very difficult theorems, exploring facts about numbers that you otherwise would not know were true." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Still, in the end, we find ourselves drawn to the beauty of the patterns themselves, and the amazing fact that we humans are smart enough to prove even a feeble fraction of all possible theorems about them. Often, greater than the contemplation of this beauty for the active mathematician is the excitement of the chase. Trying to discover first what patterns actually do or do not occur, then finding the correct statement of a conjecture, and finally proving it - these things are exhilarating when accomplished successfully. Like all risk-takers, mathematicians labor months or years for these moments of success." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"There is a big debate as to whether logic is part of mathematics or mathematics is part of logic. We use logic to think. We notice that our thinking, when it is valid, goes in certain patterns. These patterns can be studied mathematically. Thus, logic is a part of mathematics, called 'mathematical logic'." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006) 

"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate."  (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity" 5th Ed. , 2007)

"Perception requires imagination because the data people encounter in their lives are never complete and always equivocal. [...] We also use our imagination and take shortcuts to fill gaps in patterns of nonvisual data. As with visual input, we draw conclusions and make judgments based on uncertain and incomplete information, and we conclude, when we are done analyzing the patterns, that out picture is clear and accurate. But is it?" (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008)

"Why is the human need to be in control relevant to a discussion of random patterns? Because if events are random, we are not in control, and if we are in control of events, they are not random. There is therefore a fundamental clash between our need to feel we are in control and our ability to recognize randomness. That clash is one of the principal reasons we misinterpret random events."  (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008)

"In emergent processes, the whole is greater than the sum of the parts. A mathematical phenomenon that appears in certain dynamic systems also occurs within biological systems, from molecular interactions within the cells to the cognitive processes that we use to move within society. [...] Emergent patterns of ideas, beauty, desires, or tragicomedy wait, ready to trap the next traveler in their complex domain of neatly patterned squares - the never-ending world of chess metaphors." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought." (Clifford A Pickover, "The Math Book", 2009)

"The master of chess is deeply familiar with these patterns and knows very well the position that would be beneficial to reach. The rest is thinking in a logical way (calculating) about how each piece should be moved to reach the new pattern that has already taken shape in the chess player’s mind. This way of facing chess is closely related to the solving of theorems in mathematics. For example, a mathematician who wishes to prove an equation needs to imagine how the terms on each side of the equal sign can be manipulated so that one is reduced to the other. The enterprise is far from easy, to judge by the more than two hundred years that have been needed to solve theorems such as that of Fermat (z^n = x^n + y^n), using diverse tricks to prove the equation." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

Diego Rasskin-Gutman - Collected Quotes

"A chess hypothesis is basically the equivalent to drawing up a strategic plan. Experimentation in chess is equivalent to the moves that are found to carry out each plan. Throughout the history of chess, both the plans (the hypotheses) as well as the moves (the experiments) have been evolving (thanks to results from the practice of the game and from analyses), and this knowledge is the patrimony of professional players." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"A second class of metaphors - mathematical algorithms, heuristics, and models - brings us closer to the world of computer science programs, simulations, and approximations of the brain and its cognitive processes." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"An algorithm refers to a successive and finite procedure by which it is possible to solve a certain problem. Algorithms are the operational base for most computer programs. They consist of a series of instructions that, thanks to programmers’ prior knowledge about the essential characteristics of a problem that must be solved, allow a step-by-step path to the solution." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Any scientific hypothesis springs from knowledge that was previously generated by observations of facts in the real world. In addition, hypotheses produce predictions that need to be tested. For some, scientific definitions are limited to natural phenomena (although this definition would require mathematics to stop being a science since it deals with ideal objects)." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"As art, chess speaks to us of the personal decisions that are made in the course of a game. Looking at this facet of the game, the essential protagonist is the aesthetic sense rather than the capacity for calculation, which thus moves us closer to the human dimension and farther from mathematical algorithms." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"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)

"Chess also offers a modality that includes an exercise of totally free creation - compositions. These artificial positions are created for didactic reasons to illustrate a certain subject or to propose a problem that has to be solved following a series of indications" (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Chess is human communication. Each player, in each move, must understand the opponent’s message or soon fall into difficulties. In this way, the creative act is united with the capacity to understand the opponent’s intentions, resulting in a fight of ideas, wills, and creative imagination." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Cognitive psychology has followed a different direction, considering intelligence as a set of mental representations and a series of processes that operate on these representations that allows the individual to adapt to the changing conditions of the environment. This type of approach is connected with information theory. The intelligent mind operates by processing information that it collects from the environment, and the better and faster this information is processed, the more intelligence is demonstrated." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Finally, chess has a science - like special attraction since it lets the player first propose hypotheses of different strategic plans that are based on the game rules and possible moves of the pieces and then refute those hypotheses after careful investigation of the different lines of play. This process is analogous to the everyday work of a scientist." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"From its mystical origins as a dialogue with the supernatural powers to a metaphor for war, chess passes through a period as a representation of order in the universe until it becomes the game-art-science that millions of people all over the world are passionate about and that has developed into a testing ground for the sciences of artificial intelligence and cognitive psychology." (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)

"Game theory proposes a method called minimization-maximization (minimax) that determines the best possibility that is available to a player by following a decision tree that minimizes the opponent’s gain and maximizes the player’s own. This important algorithm is the basis for generating algorithms for chess programs." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"If intelligence is a capacity that is gradually acquired as a result of development and learning, then a machine that can learn from experience would have, at least in theory, the capacity to carry out intelligent behavior. [...] Humans have created machines that imitate us - that provide mirrors to see ourselves and measure our strength, our intellect, and even our creativity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"In emergent processes, the whole is greater than the sum of the parts. A mathematical phenomenon that appears in certain dynamic systems also occurs within biological systems, from molecular interactions within the cells to the cognitive processes that we use to move within society. [...] Emergent patterns of ideas, beauty, desires, or tragicomedy wait, ready to trap the next traveler in their complex domain of neatly patterned squares - the never-ending world of chess metaphors." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"In fact, H [entropy] measures the amount of uncertainty that exists in the phenomenon. If there were only one event, its probability would be equal to 1, and H would be equal to 0 - that is, there is no uncertainty about what will happen in a phenomenon with a single event because we always know what is going to occur. The more events that a phenomenon possesses, the more uncertainty there is about the state of the phenomenon. In other words, the more entropy, the more information." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Many terms that are used to comment on games are aesthetic allusions, indicating that among chess players it is hard to separate out the game’s creative and analytic aspects. Terms that are frequently used include subtlety, depth, beauty, surprise, vision, brilliance, elegance, harmony, and symmetry." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"On the surface, chess is a game that has a winner and a loser. However, a deeper look reveals that perhaps chess is not just a game but a line of communication between two brains. [...] chess is a communication device. As with any other act of communication, it is necessary to have someone who sends the message, a transmission medium, and someone who receives the message. Players are both the communicators and receivers; the board and the chess pieces are the transmission medium. In an exchange of messages, ideas, attitudes, and personal positions about the uncertainty of our world, however, where is the win, and where is the loss?" (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Pattern perception (that is, the perception of similarities in spatial or temporal configurations) has a fundamental role in playing chess [...] The two essential components in decision making in chess are recognizing patterns stored in long-term memory (which requires an exhaustive knowledge database) and searching for a solution within the problem space. The first component uses perception and long-term memory, and the second leans mainly on the calculation of variations, which in turn has its foundations in logical reasoning." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The brain and its cognitive mental processes are the biological foundation for creating metaphors about the world and oneself. Artificial intelligence, human beings’ attempt to transcend their biology, tries to enter into these scenarios to learn how they function. But there is another metaphor of the world that has its own particular landscapes, inhabitants, and laws. The brain provides the organic structure that is necessary for generating the mind, which in turn is considered a process that results from brain activity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The master of chess is deeply familiar with these patterns and knows very well the position that would be beneficial to reach. The rest is thinking in a logical way (calculating) about how each piece should be moved to reach the new pattern that has already taken shape in the chess player’s mind. This way of facing chess is closely related to the solving of theorems in mathematics. For example, a mathematician who wishes to prove an equation needs to imagine how the terms on each side of the equal sign can be manipulated so that one is reduced to the other. The enterprise is far from easy, to judge by the more than two hundred years that have been needed to solve theorems such as that of Fermat (z^n = x^n + y^n), using diverse tricks to prove the equation." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The mind creates a metaphor of ourselves and of the world that surrounds us. And it is so skillful that it has created machines that are capable of simulating human beings’ own creativity in a series of 1s and 0s [...]" (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The complexities of the universe are reflected in the complexities of our brains and in that natural, intimate and solitary activity that we call mind. In this process of matching up and representing, the inexhaustible human curiosity accepts the ancestral challenge of exploring the enormity of what we have yet to know. Chess, a world of fixed rules but with almost infinite borders, is an approachable model of that profound and endless human search." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The problem of identifying the subset of good moves is much more complicated than simply counting the total number of possibilities and falls completely into the domain of strategy and tactics of chess as a game." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The simplest basic architecture of an artificial neural network is composed of three layers of neurons - input, output, and intermediary (historically called perceptron). When the input layer is stimulated, each node responds in a particular way by sending information to the intermediary level nodes, which in turn distribute it to the output layer nodes and thereby generate a response. The key to artificial neural networks is in the ways that the nodes are connected and how each node reacts to the stimuli coming from the nodes it is connected to. Just as with the architecture of the brain, the nodes allow information to pass only if a specific stimulus threshold is passed. This threshold is governed by a mathematical equation that can take different forms. The response depends on the sum of the stimuli coming from the input node connections and is 'all or nothing'." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Vision is a capacity to understand a position and to generate solid strategic plans. And a good base of chess knowledge is needed to understand what it means to play with brilliance or elegance." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"With Kurt Gödel, we fi nd in the twentieth century the idea that formal systems are incomplete, a concept that is perhaps important to chess theory. If undecidable statements exist in chess, then it is impossible to solve them completely with a computer chess program." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

On Chess VI: Trivia III

"The chemists who uphold dualism are far from being agreed among themselves; nevertheless, all of them in maintaining their opinion, rely upon the phenomena of chemical reactions. For a long time the uncertainty of this method has been pointed out: it has been shown repeatedly, that the atoms put into movement during a reaction take at that time a new arrangement, and that it is impossible to deduce the old arrangement from the new one. It is as if, in the middle of a game of chess, after the disarrangement of all the pieces, one of the players should wish, from the inspection of the new place occupied by each piece, to determine that which it originally occupied." (Auguste Laurent, "Chemical Method", 1855)

"And as the number of combinations that can be made on the chess-board, is so great that probably no two games exactly alike were ever played; so no two games which the student plays with nature to wrest from her hidden truths, which were worth playing at all, ever made use of quite the same methods in quite the same way." (Alfred Marshall, "Principles of Economics", 1890)

"Imagine that [...] the world is something like a great chess game being played by the gods, and we are observers of the game. [...] If we watch long enough, we may eventually catch on to a few of the rules [...]. However, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited [...]. We must limit ourselves to the more basic question of the rules of the game. If we know the rules, we consider that we 'understand' the world." (Richard P. Feynman, "The Feynman Lectures on Physics", 1964)

"If arithmetical skill is the measure of intelligence, then computers have been more intelligent than all human beings all along. If the ability to play chess is the measure, then there are computers now in existence that are more intelligent than any but a very few human beings. However, if insight, intuition, creativity, the ability to view a problem as a whole and guess the answer by the “feel” of the situation, is a measure of intelligence, computers are very unintelligent indeed. Nor can we see right now how this deficiency in computers can be easily remedied, since human beings cannot program a computer to be intuitive or creative for the very good reason that we do not know what we ourselves do when we exercise these qualities." (Isaac Asimov, "Machines That Think", 1983)

"Chess is a unique cognitive nexus, a place where art and science come together in the human mind and are then refined and improved by experience." (Garry Kasparov, "How Life Imitates Chess: Making the Right Moves, from the Board to the Boardroom", 2007)

"Finally, chess has a science - like special attraction since it lets the player first propose hypotheses of different strategic plans that are based on the game rules and possible moves of the pieces and then refute those hypotheses after careful investigation of the different lines of play. This process is analogous to the everyday work of a scientist." (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)

"The complexities of the universe are reflected in the complexities of our brains and in that natural, intimate and solitary activity that we call mind. In this process of matching up and representing, the inexhaustible human curiosity accepts the ancestral challenge of exploring the enormity of what we have yet to know. Chess, a world of fixed rules but with almost infinite borders, is an approachable model of that profound and endless human search." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Playing chess involves following the rules of the game (ingenuity), and it also seems to require insight (intuition) into which rules to choose given different positions on the game board. To win at chess, it is not enough to apply the rules; you have to know which rules to select in the first place." (Erik J Larson, "The Myth of Artficial Intelligence: Why computers can't think the way we do", 2021)

"If chess permits a virtually infinite variety of games, the rules of nature  surely do. Science may be immortal after all." (John Horgan)

On Chess V: Chess and Mathematics III

"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)

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub." (Richard J Trudeau, "Dots and Lines", 1976)

"There is still a great deal of uncharted territory in the opening phase of the game. New ideas, new concepts, new plans in old and forgotten variations, there is still much to discover in the opening. The tactical patterns and strategic concepts of the middle game have been well mapped out by generations of Grandmasters, although there are occasional fresh twists. In the endgame, however, the plans and possibilities are open and known to all, an almost mathematical exercise. This isn’t to say that everything is predetermined. With flawless play from both sides, the endgame will advance toward a predictable conclusion. But since humans are flawed, damage can be inflicted or repaired. Even if one player is at a clear disadvantage, he may simply outplay his opponent." (Garry Kasparov, "How Life Imitates Chess", 2007)

"As art, chess speaks to us of the personal decisions that are made in the course of a game. Looking at this facet of the game, the essential protagonist is the aesthetic sense rather than the capacity for calculation, which thus moves us closer to the human dimension and farther from mathematical algorithms." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"For all their wealth of content, for all the sum of history and social institution invested in them, music, mathematics, and chess are resplendently useless (applied mathematics is a higher plumbing, a kind of music for the police band). They are metaphysically trivial, irresponsible. They refuse to relate outward, to take reality for arbiter. This is the source of their witchery." (George Steiner, George Steiner at The New Yorker, 2009)

"In emergent processes, the whole is greater than the sum of the parts. A mathematical phenomenon that appears in certain dynamic systems also occurs within biological systems, from molecular interactions within the cells to the cognitive processes that we use to move within society. [...] Emergent patterns of ideas, beauty, desires, or tragicomedy wait, ready to trap the next traveler in their complex domain of neatly patterned squares - the never-ending world of chess metaphors." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The master of chess is deeply familiar with these patterns and knows very well the position that would be beneficial to reach. The rest is thinking in a logical way (calculating) about how each piece should be moved to reach the new pattern that has already taken shape in the chess player’s mind. This way of facing chess is closely related to the solving of theorems in mathematics. For example, a mathematician who wishes to prove an equation needs to imagine how the terms on each side of the equal sign can be manipulated so that one is reduced to the other. The enterprise is far from easy, to judge by the more than two hundred years that have been needed to solve theorems such as that of Fermat (z^n = x^n + y^n), using diverse tricks to prove the equation." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Chess is a perfect arena for just such an exerted exploration of the possible. Its chequered sea is very deep indeed. The mathematics behind the game’s complexity are staggering. […] For all its immensity, chess is a finite game. It is therefore at least conceivable that a machine might one day be programmed with the knowledge, deep down in its nodes, of every possible sequence of moves for every possible game. No combination, however ingenious, would ever surprise it; every board position would be as familiar as a face." (Daniel Tammet, "Thinking in Numbers" , 2012) 

"Chess, with its straightforward rules and tiny Cartesian playing field, is a game tailor-made for computers." (John Horgan, "The End of Science", 2015)

On Chess IV: Trivia II

"Education in Chess has to be an education in independent thinking and judging. Chess must not be memorized […]" (Emanuel Lasker, "Lasker's Manual of Chess", 1925)

"Human affairs are like a chess game. Only those who do not take it seriously can be called good players." (Hong Zicheng, "A Chinese Garden of Serenity: Epigrams from the Ming Dynasty", 1959)

"Truth derives its strength not so much from itself as from the brilliant contrast it makes with what is only apparently true. This applies especially to Chess, where it is often found that the profoundest moves do not much startle the imagination." (Emanuel Lasker, "Common Sense in Chess", 1965)

"Chess problems demand from the composer the same virtues that characterize all worthwhile art: originality, invention, conciseness, harmony, complexity, and splendid insincerity." (Vladimir Nabokov, "Poems and problems", 1971)

"Clinging to any form of conservatism can be dangerous. Become too conservative and you are unprepared for surprises. You cannot depend on luck. Logic is blind and often knows only its own past. Logic is good for playing chess but is often too slow for the needs of survival." (Frank Herbert," Chapterhouse: Dune", 1986)

"Chess is infinite, and one has to make only one ill-considered move, and one's opponent's wildest dreams will become reality." (David Bronstein, "200 Open Games", 1991)

"Independence of thought is a most valuable quality in a chess-player, both at the board and when preparing for a game." (David Bronstein, "200 Open Games", 1991) 

"The laws of chess are as beautiful as those governing the universe - and as deadly." (Katherine NevilleA Calculated Risk, 1992)

"Chess reflects the real world in miniature. Endeavor, struggle, success, and defeat - they are part of each game ever played." (Bruce Pandolfini, "Pandolfini's Ultimate Guide to Chess", 2008)

"Chess is a game by its form, an art by its content and a science by the difficulty of gaining mastery in it. Chess can convey as much happiness as a good book or work of music can. However, it is necessary to learn to play well and only afterwards will one experience real delight." (Tigran Petrosian)

On Chess III: Chess and Mathematics II

"Observe, finally, that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science: the different moves of the game do not resemble one another." (Henri Poincaré, "On the Nature of Mathematical Reasoning", 1894)

"Chess problems are the hymn-tunes of mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"I will say only that if a chess problem is, in the crude sense, 'useless', then that is equally true of most of the best mathematics; that very little of mathematics is useful practically, and that the little [that is] is comparatively dull." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"There are three intellectual pursuits, and, so far as I am aware, only three, in which human beings have performed major feats before the age of puberty. They are music, mathematics, and chess." (George Steiner, "Extraterritorial", 1971)

"Geniuses of certain kinds - mathematicians, chess players, computer programmers - seem, if not mad, at least lacking in the social skills most easily identified with sanity." (James Gleick, "Genius: the life and science of Richard Feynman", 1992)

"Chess is not Mathematics, where ten is always more than one; in chess the King with a pawn can beat opponent's King with all pieces if they are placed badly." (Ashot Nadanian, [Interview at S'pore Chess News], 2010)

"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)

"Chess is the art that expresses the science of logic." (Mikhail Botvinnik)

"Every good mathematician should also be a good chess player and vice versa." (Henri Poincaré)

"Mathematics, like chess, requires too direct and personal a confrontation to allow graceful defeat." (Alfred Adler)

On Chess II: Trivia I

"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)

"The chess board is the world, the pieces are the phenomena of the universe, the rules of the game are what we call the laws of Nature. The player on the other side is hidden from us. We know that his play is always fair, just, and patient. But also we know, to our cost, that he never overlooks a mistake, or makes the smallest allowance for ignorance." (Thomas H Huxley, "A Liberal Education", 1868)

"Chess represents a contest between two forces, and consequently the general principles which govern a battle in its widest significance, are also applicable to the game of Chess." (Dr. Max Euwe, "Strategy & Tactices in chess", 1937)

"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)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system (the mind) can be mapped onto the other (the computer)." (George Johnson, Machinery of the Mind: Inside the New Science of Artificial Intelligence, 1986)

"An internal model allows a system to look ahead to the future consequences of current actions, without actually committing itself to those actions. In particular, the system can avoid acts that would set it irretrievably down some road to future disaster ('stepping off a cliff'). Less dramatically, but equally important, the model enables the agent to make current 'stage-setting' moves that set up later moves that are obviously advantageous. The very essence of a competitive advantage, whether it be in chess or economics, is the discovery and execution of stage-setting moves." (John H Holland, 1992)

"Visual thinking is often driven more strongly by the conceptual knowledge we use to organize our images than by the contents of the images themselves. Chess masters are known for their remarkable memory for the pieces on a chessboard. But it's not because people with photographic memories become chess masters. The masters are no better than beginners when remembering a board of randomly arranged pieces. Their memory captures meaningful relations among the pieces, such as threats and defenses, not just their distribution in space."(Steven Pinker, "How the Mind Works", 1997)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"So everyone has and uses mental representations. What sets expert performers apart from everyone else is the quality and quantity of their mental representations. Through years of practice, they develop highly complex and sophisticated representations of the various situations they are likely to encounter in their fields - such as the vast number of arrangements of chess pieces that can appear during games. These representations allow them to make faster, more accurate decisions and respond more quickly and effectively in a given situation. This, more than anything else, explains the difference in performance between novices and experts." (Anders Ericsson & Robert Pool," "Peak: Secrets from  the  New  Science  of  Expertise" , 2016)

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)

5 Books 10 Quotes III: Beauty and Symmetry III

James R Newman, "The World of Mathematics Vol. I", 1956
	"In the everyday sense symmetry carries the meaning of balance, proportion, harmony, regularity of form. Beauty is sometimes linked with symmetry, but the relationship is not very illuminating since beauty is an even vaguer quality than symmetry."
"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection." (Herman Weyl, "Symmetry")
James R Newman, "The World of Mathematics Vol. II", 1956
	"Mathematicians study their problems on account of their intrinsic interest, and develop their theories on account of their beauty." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?")
"If we seek a cause wherever we perceive symmetry, it is not that we regard a symmetrical event as less possible than the others, but, since this event ought to be the effect of a regular cause or that of chance, the first of these suppositions is more probable than the second." (Pierre-Simon de Laplace, "Concerning Probability")
James R Newman, "The World of Mathematics Vol III", 1956
	"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")
"The world of ideas which it discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance." (James J Sylvester, "The Study That Knows Nothing of Observation")
James R Newman, "The World of Mathematics Vol IV", 1956
	"[...] what are the mathematic entities to which we attribute this character of beauty and elegance, and which are capable of developing in us a sort of esthetic emotion? They are those whose elements are harmoniously disposed so that the mind without effort can embrace their totality while realizing the details. This harmony 'is at once a satisfaction of our esthetic needs and an aid to the mind, sustaining and guiding." (Henri Poincare, "Mathematical Creation")
"When, for instance, I see a symmetrical object, I feel its pleasurable quality, but do not need to assert explicitly to myself, ‘How symmetrical!’. This characteristic feature may be explained as follows. In the course of individual experience it is found generally that symmetrical objects possess exceptional and desirable qualities. Thus our own bodies are not regarded as perfectly formed unless they are symmetrical. Furthermore, the visual and tactual technique by which we perceive the symmetry of various objects is uniform, highly developed, and almost instantaneously applied. It is this technique which forms the associative 'pointer.' In consequence of it, the perception of any symmetrical object is accompanied by an intuitive aesthetic feeling of positive tone." (George D Birkhoff, "Mathematics of Aesthetics")
K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997
	"Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith."
"How deep truths can be defined as invariants – things that do not change no matter what; how invariants are defined by symmetries, which in turn define which properties of nature are conserved, no matter what. These are the selfsame symmetries that appeal to the senses in art and music and natural forms like snowflakes and galaxies. The fundamental truths are based on symmetry, and there’s a deep kind of beauty in that." Previous Post <<||>> Next Post

On Chess I: Chess and Mathematics I

"A chess problem is genuine mathematics, but it is in some way ‘trivial’ mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful –‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men." (Friedrich Waismann & Karl Menger, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1951)

"It [mathematics] is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst." (Norbert Wiener, "Ex-prodigy: My Childhood and Youth", 1953)

"The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation." (Norbert Wiener, "Ex-Prodigy: My Childhood and Youth", 1953)

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

"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)

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"[…] mathematics can never prove anything. No mathematics has any content. All any mathematics can do is – sometimes – turn out to be useful in describing some aspects of our so-called ‘physical universe’. That is a bonus; most forms of mathematics are as meaning-free as chess." (Robert A Heinlein, "The Number of the Beast", 1980)

"[…] mathematics is not best learned passively; you don’t sop it up like a romance novel. You’ve got to go out to it, aggressive, and alert, like a chess master pursuing checkmate." (Robert Kanigel, "The Man Who Knew Infinity: A Life of the Genius Ramanujan", 1991)

"Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, "The Mathematical Experience", 1995)