On Phase Spaces

 "Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"[…] physicists have come to appreciate a fourth kind of temporal behavior: deterministic chaos, which is aperiodic, just like random noise, but distinct from the latter because it is the result of deterministic equations. In dynamic systems such chaos is often characterized by small fractal dimensions because a chaotic process in phase space typically fills only a small part of the entire, energetically available space." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"When a system has more than one attractor, the points in phase space that are attracted to a particular attractor form the basin of attraction for that attractor. Each basin contains its attractor, but consists mostly of points that represent transient states. Two contiguous basins of attraction will be separated by a basin boundary." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"One of the reasons we deal with the pendulum is that it is easy to plot its motion in phase space. If the amplitude is small, it's a two-dimensional problem, so all we need to specify it completely is its position and its velocity. We can make a two-dimensional plot with one axis (the horizontal), position, and the other (the vertical), velocity." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing - seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Roughly spoken, bifurcation theory describes the way in which dynamical system changes due to a small perturbation of the system-parameters. A qualitative change in the phase space of the dynamical system occurs at a bifurcation point, that means that the system is structural unstable against a small perturbation in the parameter space and the dynamic structure of the system has changed due to this slight variation in the parameter space." (Holger I Meinhardt, "Cooperative Decision Making in Common Pool Situations", 2012)

"The impossibility of predicting which point in phase space the trajectory of the Lorenz attractor will pass through at a certain time, even though the system is governed by deterministic equations, is a common feature of all chaotic systems. However, this does not mean that chaos theory is not capable of any predictions. We can still make very accurate predictions, but they concern the qualitative features of the system’s behavior rather than the precise values of its variables at a particular time. The new mathematics thus represents the shift from quantity to quality that is characteristic of systems thinking in general. Whereas conventional mathematics deals with quantities and formulas, nonlinear dynamics deals with qualities and patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point."  (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

On Bifurcations III

"At the large scale where many processes and structures appear continuous and stable much of the time, important changes may occur discontinuously. When only a few variables are involved, as well as an optimizing process, the event may be analyzed using catastrophe theory. As the number of variables in- creases the bifurcations can become more complex to the point where chaos theory becomes the relevant approach. That chaos theory as well as the fundamentally discontinuous quantum processes may be viewed through fractal eyeglasses can also be admitted. We can even argue that a cascade of bifurcations to chaos contains two essentially structural catastrophe points, namely the initial bifurcation point at which the cascade commences and the accumulation point at which the transition to chaos is finally achieved." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"The idea of one description of a system bifurcating from another also provides the key to begin unlocking one of the most important, and at the same time perplexing, problems of system theory: characterization of the complexity of a system." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Whenever patterns are perceived in a process, there is the possibility of extrapolation. Whatever the nature of the pattern, it provides a handle for grasping something about the way it will unfold in the future." (Ervin László, "Vision 2020: Reordering Chaos for Global Survival", 1994)

"In many nonlinear systems, however, small changes of certain parameters may produce Dramatic changes in the basic characteristics of the phase portrait. Attractors may disappear, or change into one another, or new attractors may suddenly appear. Such systems are said to be structurally unstable, and the critical points of instability are called 'bifurcation points', because they are points in the system’s evolution where a fork suddenly appears and the system branches off in a new direction. Mathematically, bifurcation points mark sudden changes in the system’s phase portrait. Physically, they correspond to points of instability at which the system changes abruptly and new forms of order suddenly appear." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point."  (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

Hiroki Sayama - Collected Quotes

"A giant component is a connected component whose size is on the same order of magnitude as the size of the whole network. Network percolation is the appearance of such a giant component in a random graph, which occurs when the average node degree is above 1." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A good model is simple, valid, and robust. Simplicity of a model is really the key essence of what modeling is all about. The main reason why we want to build a model is that we want to have a shorter, simpler description of reality. [...] Validity of a model is how closely the model’s prediction agrees with the observed reality. This is of utmost importance from a practical viewpoint. If your model’s prediction doesn’t reasonably match the observation, the model is not representing reality and is probably useless. [...] finally, robustness of a model is how insensitive the model’s prediction is to minor variations of model assumptions and/or parameter settings. This is important because there are always errors when we create assumptions about, or measure parameter values from, the real world. If the prediction made by your model is sensitive to their minor variations, then the conclusion derived from it is probably not reliable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A model is a simplified representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point."  (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Chaos can be understood as a dynamical process in which microscopic information hidden in the details of a system’s state is dug out and expanded to a macroscopically visible scale (stretching), while the macroscopic information visible in the current system’s state is continuously discarded (folding)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Chaos is a long-term behavior of a nonlinear dynamical system that never falls in any static or periodic trajectories. [It] looks like a random fluctuation, but still occurs in completely deterministic, simple dynamical systems. [It] exhibits sensitivity to initial conditions. [It] occurs when the period of the trajectory of the system’s state diverges to infinity. [It] occurs when no periodic trajectories are stable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Complex systems are networks made of a number of components that interact with each other, typically in a nonlinear fashion. Complex systems may arise and evolve through self-organization, such that they are neither completely regular nor completely random, permitting the development of emergent behavior at macroscopic scales." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Emergence is a nontrivial relationship between the properties of a system at microscopic and macroscopic scales. Macroscopic properties are called emergent when it is hard to explain them simply from microscopic properties." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Equilibrium points are important for both theoretical and practical reasons. Theoretically, they are key points in the system’s phase space, which serve as meaningful references when we understand the structure of the phase space. And practically, there are many situations where we want to sustain the system at a certain state that is desirable for us. In such cases, it is quite important to know whether the desired state is an equilibrium point, and if it is, whether it is stable or unstable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Mean-field approximation is a technique that ignores spatial relationships among components. It works quite well for systems whose parts are fully connected or randomly interacting with each other. It doesn’t work if the interactions are local or non-homogeneous, and/or if the system has a non-uniform pattern of states. In such cases, you could still use mean-field approximation as a preliminary, “zeroth-order” approximation, but you should not derive a final conclusion from it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"[...] nonlinearity means that the outputs of a system are not given by a linear combination of the inputs. In the context of system behavior, the inputs and outputs can be the current and next states of the system, and if their relationship is not linear, the system is called a nonlinear system." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"One of the unique features of typical CA [ cellular automata] models is that time, space, and states of cells are all discrete. Because of such discreteness, the number of all possible state-transition functions is finite, i.e., there are only a finite number of “universes” possible in a given CA setting. Moreover, if the space is finite, all possible configurations of the entire system are also enumerable. This means that, for reasonably small CA settings, one can conduct an exhaustive search of the entire rule space or phase space to study the properties of all the 'parallel universes'." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Self-organization is a dynamical process by which a system spontaneously forms nontrivial macroscopic structures and/or behaviors over time." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"The challenge in developing a model becomes particularly tough when it comes to the modeling of complex systems, because their unique properties (networks, nonlinearity, emergence, self-organization, etc.) are not what we are familiar with. We usually think about things on a single scale in a step-by-step, linear chain of reasoning, in which causes and effects are clearly distinguished and discussed sequentially. But this approach is not suitable for understanding complex systems where a massive amount of components are interacting with each other interdependently to generate patterns over a broad range of scales. Therefore, the behavior of complex systems often appears to contradict our everyday experiences." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"The sensitivity of chaotic systems to initial conditions is particularly well known under the moniker of the 'butterfly effect', which is a metaphorical illustration of the chaotic nature of the weather system in which 'a flap of a butterfly’s wings in Brazil could set off a tornado in Texas'. The meaning of this expression is that, in a chaotic system, a small perturbation could eventually cause very large-scale difference in the long run." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"There are several reasons why reaction-diffusion systems have been a popular choice among mathematical modelers of spatio-temporal phenomena. First, their clear separation between non-spatial and spatial dynamics makes the modeling and simulation tasks really easy. Second, limiting the spatial movement to only diffusion makes it quite straightforward to expand any existing non-spatial dynamical models into spatially distributed ones. Third, the particular structure of reaction-diffusion equations provides aneasy shortcut in the stability analysis (to be discussed in the next chapter). And finally, despite the simplicity of their mathematical form, reaction-diffusion systems can show strikingly rich, complex spatio-temporal dynamics. Because of these properties, reaction-diffusion systems have been used extensively for modeling self-organization of spatial patterns." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Trajectories of a deterministic dynamical system will never branch off in its phase space (though they could merge), because if they did, that would mean that multiple future states were possible, which would violate the deterministic nature of the system. No branching means that, once you specify an initial state of the system, the trajectory that follows is uniquely determined too. You can visually inspect where the trajectories are going in the phase space visualization. They may diverge to infinity, converge to a certain point, or remain dynamically changing yet stay in a confined region in the phase space from which no outgoing trajectories are running out. Such a converging point or a region is called an attractor." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Variable rescaling is a technique to eliminate parameters from your model without losing generality. The basic idea is this: Variables that appear in your model represent quantities that are measured in some kind of units, but those units can be arbitrarily chosen without changing the dynamics of the system being modeled. This must be true for all scientific quantities that have physical dimensions - switching from inches to centimeters shouldn’t cause any change in how physics works! This means that you have the freedom to choose any convenient unit for each variable, some of which may simplify your model equations." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

Statistical Tools IV: Urns

"The early experts in probability theory were forever talking about drawing colored  balls out of 'urns' . This was not because people are really interested in jars or boxes full of a mixed-up lot of colored balls, but because those urns full of balls could often be designed so that they served as useful and illuminating models of important real situations. In fact, the urns and balls are not themselves supposed real. They are fictitious and idealized urns and balls, so that the probability of drawing out any one ball is just the same as for any other." (Warren Weaver, "Lady Luck: The Theory of Probability". 1963) 

"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)

"Specifically, it seems to me preferable to use, systematically: 'random' for that which is the object of the theory of probability […]; I will therefore say random process, not stochastic process. 'stochastic' for that which is valid 'in the sense of the calculus of probability': for instance; stochastic independence, stochastic convergence, stochastic integral; more generally, stochastic property, stochastic models, stochastic interpretation, stochastic laws; or also, stochastic matrix, stochastic distribution, etc. As for 'chance', it is perhaps better to reserve it for less technical use: in the familiar sense of'by chance', 'not for a known or imaginable reason', or (but in this case we should give notice of the fact) in the sense of, 'with equal probability' as in 'chance drawings from an urn', 'chance subdivision', and similar examples." (Bruno de Finetti, "Theory of Probability", 1974)

"Statisticians talk about populations. In probability books, the equivalent concept is an urn with numbered balls as a prototype for a population. In fact, when sampling from populations, it is customary to number the population and pretend the population is an urn from which we are drawing the sample." (Juana Sánchez, "Probability for Data Scientists", 2020)

"Many people mistakenly think that the defining property of a simple random sample is that every unit has an equal chance of being in the sample. However, this is not the case. A simple random sample of n units from a population of N means that every possible col‐lection of n of the N units has the same chance of being selected. A slight variant of this is the simple random sample with replacement, where the units/marbles are returned to the urn after each draw. This method also has the property that every sample of n units from a population of N is equally likely to be selected. The difference, though, is that there are more possible sets of n units because the same marble can appear more than once in the sample." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Several key assumptions enter into this urn model, such as the assumption that the vaccine is ineffective. It’s important to keep track of the reliance on these assumptions because our simulation study gives us an approximation of the rarity of an outcome like the one observed only under these key assumptions." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"The urn model is a simple abstraction that can be helpful for understanding variation.This model sets up a container (an urn, which is like a vase or a bucket) full of identical marbles that have been labeled, and we use the simple action of drawing marbles from the urn to reason about sampling schemes, randomized controlled experiments, and measurement error. For each of these types of variation, the urn model helps us estimate the size of the variation using either probability or simulation." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

On Stories (From Fiction to Science-Fiction)

"One mark of a second-rate mind is to be always telling stories." (Jean de La Bruyère, "Les Caractères" Aphorism 52, 1688)

"If the story-tellers could ha' got decency and good morals from true stories, who'd have troubled to invent parables?" (Thomas Hardy, "Under the Greenwood Tree", 1872)

"All stories, if continued far enough, end in death, and he is no true-story teller who would keep that from you. Especially do all stories of monogamy end in death, and your man who is monogamous while he often lives most happily, dies in the most lonely fashion." (Ernest Hemingway, "Death in the Afternoon", 1932)

"As if there could be true stories: things happen in one way, and we retell them in the opposite way." (Jean-Paul Sartre, "Nausea", 1938)

"A story must be told or there'll be no story, yet it is the untold stories that are most moving." (John R R Tolkien, [Letter to his son Christopher] 1945)

"Science fiction is no more written for scientists than ghost stories are written for ghosts." Brian Aldiss, Penguin Science Fiction, 1961)

"Almost all serious stories in the world are stories of failure with a death in it. But there is more lost paradise in them than defeat." (Orson Welles, "Chimes at Midnight", 1965)

"Unless physical action reflects psychic action, unless the deeds express the person, I get very bored with adventure stories; often it seems that the more action there is, the less happens." (Ursula K Le Guin, "Vaster Than Empires and More Slow", 1971)

"Only library books speak with such wordless eloquence of the power good stories hold over us." (Stephen King, "Salem's Lot", 1975)

"Individual science fiction stories may seem as trivial as ever to the blinder critics and philosophers of today - but the core of science fiction, its essence, the concept around which it revolves, has become crucial to our salvation if we are to be saved at all." (Isaac Asimov, "My Own View" [in "The Encyclopedia of Science Fiction"], 1978)

"No story comes from nowhere; new stories are born from old." (Salman Rushdie, "Haroun and the Sea of Stories", 1990)

"Nothing comes from nothing, [...] no story comes from nowhere; new stories are born from old—it is the new combinations that make them new."  (Salman Rushdie, "Haroun and the Sea of Stories", 1990)

"People think that stories are shaped by people. In fact, it's the other way around." (Terry Pratchett, "Witches Abroad", 1991)

"To read fiction means to play a game by which we give sense to the immensity of things that happened, are happening, or will happen in the actual world. By reading narrative, we escape the anxiety that attacks us when we try to say something true about the world. This is the consoling function of narrative - the reason people tell stories, and have told stories from the beginning of time." (Umberto Eco, "Six Walks in the Fictional Woods", 1994)

"Don't worry about trying to develop a style. Style is what you can't help doing. If you write enough, you draw enough, you'll have a style, whether you want it or not. Don't worry about whether you're "commercial". Tell your own stories, draw your own pictures. Let other people follow you." (Neil Gaiman, "Gods & Tulips", 1999)

"There are no stories without meaning. And I am one of those men who can find it even when others fail to see it. Afterwards the story becomes the book of the living, like a blaring trumpet that raises from the tomb those who have been dust for centuries." (Umberto Eco, "Baudolino", 2000)

"Stories are artifacts, not really made things which we create and can take credit for, but pre-existing objects which we dig up." (Stephen King, "Everything's Eventual: 14 Dark Tales", 2002)

"No story sits by itself. Sometimes stories meet at corners and sometimes they cover one another completely, like stones beneath a river." (Mitch Albom,"The Five People You Meet in Heaven", 2003)

"Stories come to us as wraiths requiring precise embodiments." (Joyce Carol Oates, "The Faith of a Writer", 2003)

"I think telling stories is like pushing something. Pushing against uncreation itself, maybe." (Stephen King, "The Dark Tower VI: Song of Susannah", 2004)

"Other people's stories may become part of your own, the foundation of it, the ground it goes on." (Ursula K Le Guin, "Gifts", 2004)

"Stories are webs, interconnected strand to strand, and you follow each story to the center, because the center is the end. Each person is a strand of the story." (Neil Gaiman, "Anansi Boys", 2005)

"People wanted the world to be a story, because stories had to sound right and they had to make sense. People wanted the world to make sense." (Terry Pratchett, "Wintersmith", 2006)

"True stories are the ones that lie open at the border, allowing a crossing, a further frontier. The final frontier is just science fiction - don't believe it. Like the universe, there is no end." (Jeanette Winterson, "The Stone Gods", 2007)

"Arithmetic is the death of story." (Jincy Willett, "The Writing Class", 2008)

"Do you know why teachers use me? Because I speak in tongues. I write metaphors. Every one of my stories is a metaphor you can remember. The great religions are all metaphor. We appreciate things like Daniel and the lion's den, and the Tower of Babel. People remember these metaphors because they are so vivid you can't get free of them and that's what kids like in school." (Ray Bradbury, The Paris Review, [interview] 2010)

"Old stories have a habit of being told and retold and changed. Each subsequent storyteller puts his or her mark upon it. Whatever truth the story once had is buried in bias and embellishment. The reasons do not matter as much as the story itself." (Erin Morgenstern, "The Night Circus", 2011)

"Don't be satisfied with stories, how things have gone with others. Unfold your own myth." (Rumi)

"[...] out of monuments, names, words, proverbs, traditions, private records and evidences, fragments of stories, passages of books, and the like, we do save and recover somewhat from the deluge of time." (Francis Bacon) 

"Of course all children's literature is not fantastic, so all fantastic books need not be children's books. It is still possible, even in an age so ferociously anti-romantic as our own, to write fantastic stories for adults: though you will usually need to have made a name in some more fashionable kind of literature before anyone will publish them." (Clive S Lewis)

On Stories (-1949)

 "A likely impossibility is always preferable to an unconvincing possibility. The story should never be made up of improbable incidents; there should be nothing of the sort in it." (Aristotle, "Poetics", cca. 335 BC)

"In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure" (Hermann Hankel, "Die Entwicklung der Mathematik in den letzten Jahrhunderten, 1884)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature 62 develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor, "A Theory of Motives, Ideals and Values in Education", 1907)

"The story of scientific discovery has its own epic unity - a unity of purpose and endeavour - the single torch passing from hand to hand through the centuries; and the great moments of science when, after long labour, the pioneers saw their accumulated facts falling into a significant order - sometimes in the form of a law that revolutionised the whole world of thought - have an intense human interest, and belong essentially to the creative imagination of poetry." (Alfred Noyes, "Watchers of the Sky", 1922)

"The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole; to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction this discovery comes with all the overwhelming force of a revelation; like a palace emerging from the autumn mist as the traveler ascends an Italian hill-side, the stately stories of the mathematical edifice appear in their due order and proportion, with a new perfection in every part." (Bertrand A W Russell, "Mysticism and Logic and Other Essays", 1925)

"This is the reason why mechanical explanations are better understood than stories, even though they are more difficult to reproduce. The exposition, even if it is faulty, excites analogous schemas already existing in the listener’s mind; so that what takes place is not genuine understanding, but a convergence of acquired schemas of thought. In the case of stories, this convergence is not possible, and the schemas brought into play are usually divergent." (Jean Piaget, "The Language and Thought of the Child", 1926)

On Stories (2020-2029)

"Numbers are ideal vehicles for promulgating bullshit. They feel objective, but are easily manipulated to tell whatever story one desires. Words are clearly constructs of human minds, but numbers? Numbers seem to come directly from Nature herself. We know words are subjective. We know they are used to bend and blur the truth. Words suggest intuition, feeling, and expressivity. But not numbers. Numbers suggest precision and imply a scientific approach. Numbers appear to have an existence separate from the humans reporting them." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

"So what does it mean to tell an honest story? Numbers should be presented in ways that allow meaningful comparisons." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

"To tell an honest story, it is not enough for numbers to be correct. They need to be placed in an appropriate context so that a reader or listener can properly interpret them." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

"[...] scatterplots had advantages over earlier graphic forms: the ability to see clusters, patterns, trends, and relations in a cloud of points. Perhaps most importantly, it allowed the addition of visual annotations (point symbols, lines, curves, enclosing contours, etc.) to make those relationships more coherent and tell more nuanced stories." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

"Scatterplots are valuable because, without having to inspect each individual point, we can see overall aggregate patterns in potentially thousands of data points. But does this density of information come at a price - just how easy are they to read? [...] The truth is such charts can shed light on complex stories in a way words alone - or simpler charts you might be more familiar with - cannot." (Alan Smith, "How Charts Work: Understand and explain data with confidence", 2022)

"Data becomes more useful once it’s transformed into a data visualization or used in a data story. Data storytelling is the ability to effectively communicate insights from a dataset using narratives and visualizations. It can be used to put data insights into context and inspire action from your audience. Color can be very helpful when you are trying to make information stand out within your data visualizations." (Kate Strachnyi, "ColorWise: A Data Storyteller’s Guide to the Intentional Use of Color", 2023)

"Data storytelling is a method of communicating information that is custom-fit for a specific audience and offers a compelling narrative to prove a point, highlight a trend, make a sale, or all of the above. [...] Data storytelling combines three critical components, storytelling, data science, and visualizations, to create not just a colorful chart or graph, but a work of art that carries forth a narrative complete with a beginning, middle, and end." (Kate Strachnyi, "ColorWise: A Data Storyteller’s Guide to the Intentional Use of Color", 2023)

"When the colors are dull and neutral, they can communicate a sense of uniformity and an aura of calmness. Grays do a great job of mapping out the context of your story so that the more sharp colors highlight what you’re trying to explain. The power of gray comes in handy for all of our supporting details such as the axis, gridlines, and nonessential data that is included for comparative purposes. By using gray as the primary color in a visualization, we automatically draw our viewers’ eyes to whatever isn’t gray. That way, if we are interested in telling a story about one data point, we can do so quite easily." (Kate Strachnyi, "ColorWise: A Data Storyteller’s Guide to the Intentional Use of Color", 2023)

On Storytelling

"Storytelling reveals meaning without committing the error of defining it." (Hannah Arendt, "Men in Dark Times", 1968)

"Scientific practice may be considered a kind of storytelling practice [...]" (Donna Haraway, "Primate Visions", 1989)

"Storytelling is the art of unfolding knowledge in a way that makes each piece contribute to a larger truth." (Philip Gerard, "Writing a Book That Makes a Difference", 2000)

"The human mind is a wanton storyteller and even more, a profligate seeker after pattern. We see faces in clouds and tortillas, fortunes in tea leaves and planetary movements. It is quite difficult to prove a real pattern as distinct from a superficial illusion." (Richard Dawkins, "A Devil's Chaplain", 2003))

"We have, as human beings, a storytelling problem. We're a bit too quick to come up with explanations for things we don't really have an explanation for." (Malcolm Gladwell, "Blink: The Power of Thinking Without Thinking", 2005)

"There is an extraordinary power in storytelling that stirs the imagination and makes an indelible impression on the mind." (Brennan Manning, "The Ragamuffin Gospel: Good News for the Bedraggled, Beat-Up, and Burnt Out", 2008)

"Mostly we rely on stories to put our ideas into context and give them meaning. It should be no surprise, then, that the human capacity for storytelling plays an important role in the intrinsically human-centered approach to problem solving, design thinking." (Tim Brown, "Change by Design: How Design Thinking Transforms Organizations and Inspires Innovation", 2009)

"The purpose of a storyteller is not to tell you how to think, but to give you questions to think upon." (Brandon Sanderson, "The Way of Kings", 2010)

"All good design is storytelling. All good storytelling is design." (Steven Heller, "Writing and Research for Graphic Designers: A Designer's Manual to Strategic Communication and Presentation", 2012)

"Nonetheless, storytelling and narrative are essential to the design writing process. Without story - or plot, if you will - what have you got? Even a factual business report can tell a tale, albeit often in a neutral manner. Not all stories have to be dramatic or melodramatic. Storytelling is simply the expres sion of something you, as the writer, believe is of interest to you, as the reader. Indeed, you may well be representative of your average reader." (Steven Heller, "Writing and Research for Graphic Designers: A Designer's Manual to Strategic Communication and Presentation", 2012)

"The storytelling mind is allergic to uncertainty, randomness, and coincidence. It is addicted to meaning. If the storytelling mind cannot find meaningful patterns in the world, it will try to impose them. In short, the storytelling mind is a factory that churns out true stories when it can, but will manufacture lies when it can't." (Jonathan Gottschall, "The Storytelling Animal: How Stories Make Us Human", 2012)

"We are, as a species, addicted to story. Even when the body goes to sleep, the mind stays up all night, telling itself stories." (Jonathan Gottschall, "The Storytelling Animal", 2012)

"Good visualization is a winding process that requires statistics and design knowledge. Without the former, the visualization becomes an exercise only in illustration and aesthetics, and without the latter, one of only analyses. On their own, these are fine skills, but they make for incomplete data graphics. Having skills in both provides you with the luxury - which is growing into a necessity - to jump back and forth between data exploration and storytelling." (Nathan Yau, "Data Points: Visualization That Means Something", 2013)

"The fact of storytelling hints at a fundamental human unease, hints at human imperfection. Where there is perfection there is no story to tell." (Ben Okri, "A Way of Being Free", 2014)

"There is no such thing as a fact. There is only how you saw the fact, in a given moment. How you reported the fact. How your brain processed that fact. There is no extrication of the storyteller from the story." (Jodi Picoult, "Small Great Things", 2016)

"Data storytelling provides a bridge between the worlds of logic and emotion. A data story offers a safe passage for your insights to travel around emotional pitfalls and through analytical resistance that typically impede facts." (Brent Dykes, "Effective Data Storytelling: How to Drive Change with Data, Narrative and Visuals", 2019)

"Data becomes more useful once it’s transformed into a data visualization or used in a data story. Data storytelling is the ability to effectively communicate insights from a dataset using narratives and visualizations. It can be used to put data insights into context and inspire action from your audience. Color can be very helpful when you are trying to make information stand out within your data visualizations." (Kate Strachnyi, "ColorWise: A Data Storyteller’s Guide to the Intentional Use of Color", 2023)

"Data storytelling is a method of communicating information that is custom-fit for a specific audience and offers a compelling narrative to prove a point, highlight a trend, make a sale, or all of the above. [...] Data storytelling combines three critical components, storytelling, data science, and visualizations, to create not just a colorful chart or graph, but a work of art that carries forth a narrative complete with a beginning, middle, and end." (Kate Strachnyi, "ColorWise: A Data Storyteller’s Guide to the Intentional Use of Color", 2023

On Stories (-1974)

"An average does not tell the full story. It is hardly fully representative of a mass unless we know the manner in which the individual items scatter around it. A further description of the series is necessary if we are to gauge how representative the average is." (George Simpson & Fritz Kafka, "Basic Statistics", 1952)

"The construction of an economic model, or of any model or theory for that matter (or the writing of a novel, a short story, or a play) consists of snatching from the enormous and complex mass of facts called reality, a few simple, easily-managed key points which, when put together in some cunning way, become for certain purposes a substitute for reality itself." (Evsey Domar, "Essays in the Theory of Economic Growth", 1957)

"In imagination there exists the perfect mystery story. Such a story presents all the essential clews, and compels us to form our own theory of the case. If we follow the plot carefully we arrive at the complete solution for ourselves just before the author’s disclosure at the end of the book. The solution itself, contrary to those of inferior mysteries, does not disappoint us; moreover, it appears at the very moment we expect it." (Leopold Infeld, "The Evolution of Physics", 1961)

"Of course, you all know the old story that some people use statistics the way an inebriate uses a lamppost - for support rather than for illumination. It is not really that bad at all times. Statistics are indeed used for illumination, the difficulty is that everybody is trying to illuminate a different point." (Hyman L Lewis, [in Gerhard Bry's "Business Cycle Indicators for States and Regions"] 1961)

"The two most important characteristics of the language of statistics are first, that it describes things in quantitative terms, and second, that it gives this description an air of accuracy and precision. The limitations, as well as the advantages, of the statistical approach arise from these two characteristics. For a description of the quantitative aspect of events never gives us the whole story; and even the best statistics are never, and never can be, completely accurate and precise. To avoid misuse of the language we must, therefore, guard against exaggerating the importance of the elements in any situation that can be described quantitatively, and we must know sufficient about the error and inaccuracy of the figures to be able to use them with discretion." (Ely Devons, "Essays in Economics", 1961)

"Taken as a story of human achievement, and human blindness, the discoveries in the sciences are among the great epics." (J Robert Oppenheimer, "Reflections on the resonances of physics history" , 1972)

"Our inability to measure important factors does not mean either that we should sweep those factors under the rug or that we should give them all the weight in a decision. Some important factors in some problems can be assessed quantitatively. And even though thoughtful and imaginative efforts have sometimes turned the 'unmeasurable' into a useful number, some important factors are simply not measurable. As always, every bit of the investigator's ingenuity and good judgment must be brought into play. And, whatever un- knowns may remain, the analysis of quantitative data nonetheless can help us learn something about the world - even if it is not the whole story." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

On Stories (1975-1999)

"Exploratory data analysis can never be the whole story, but nothing else can serve as the foundation stone - as the first step." (John W. Tukey, "Exploratory Data Analysis", 1977)

"There are certain basic, known principles about how people's minds go about the business of understanding, and communicating understanding by means of language, which have been known and used for many centuries. No matter how these principles are addressed, they always end up with hierarchic decomposition as being the heart of good storytelling." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

"A model […] is a story with a specified structure: to explain this catch phrase is to explain what a model is. The structure is given by the logical and mathematical form of a set of postulates, the assumptions of the model. The structure forms an uninterpreted system, in much the way the postulates of a pure geometry are now commonly regarded as doing. The theorems that follow from the postulates tell us things about the structure that may not be apparent from an examination of the postulates alone." (Allan Gibbard & Hal R. Varian, "Economic Models", The Journal of Philosophy, Vol. 75, No. 11, 1978)

"However, for most of us, science functions like myth in that we have no personal experience in the matter. We put our trust in the scientific view given us by our culture and enshrined in its myths. If asked why leaves are green, most of us would probably mutter something about “chlorophyll.” But unless we were specialists, we would simply be repeating the story of someone else’s experience." (Wallace B Clift, "Jung and Christianity", 1982)

"Scientific theories (I have said) begin as imaginative constructions. The begin, if you like, as stories, and the purpose of the critical or rectifying episode in scientific reasoning is precisely to find out whether or not these stories are stories about real life. Literal or empiric truthfulness is not therefore the starting-point of scientific enquiry, but rather the direction in which scientific reasoning moves. If this is a fair statement, it follows that scientific and poetic or imaginative accounts of the world are not distinguishable in their origins. They start in parallel, but diverge from one another at some later stge. We all tell stories, but the stories differ in the purposes we expect them to fulfil and in the kinds of evaluations to which they are exposed." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Myth is the system of basic metaphors, images, and stories that in-forms the perceptions, memories, and aspirations of a people; provides the rationale for its institutions, rituals and power structure; and gives a map of the purpose and stages of life." (Sam Keen, "The Passionate Life", 1983)

"Mental models are the images, assumptions, and stories which we carry in our minds of ourselves, other people, institutions, and every aspect of the world. Like a pane of glass framing and subtly distorting our vision, mental models determine what we see. Human beings cannot navigate through the complex environments of our world without cognitive ‘mental maps’; and all of these mental maps, by definition, are flawed in some way." (Peter M Senge, "The Fifth Discipline Fieldbook: Strategies and Tools for Building a Learning Organization", 1994)

"The story of π has been extensively told, no doubt because its history goes back to ancient times, but also because much of it can be grasped without a knowledge of advanced mathematics." (Eli Maor, "e: The Story of a Number", 1994)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"The story of calculus brings out two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction. These are the external and internal aspects of mathematics, often referred to as applied and pure mathematics." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The most erroneous stories are those we think we know best - and therefore never scrutinize or question." (Stephen Jay Gould, "Life's Grandeur: The Spread of Excellence From Plato to Darwin", 1996)

"When scientists need to explain difficult points of theory, illustration by hypothetical example - rather than by total abstraction - works well (perhaps indispensably) as a rhetorical device. Such cases do not function as speculations in the pejorative sense - as silly stories that provide insight into complex mechanisms - but rather as idealized illustrations to exemplify a difficult point of theory." (Stephen Jay Gould, "Leonardo's Mountain of Clams and the Diet of Worms", 1998)