On Symmetry: On the Symmetry of Time

"Every equation and every explanation used in physics must be compatible with the symmetry of time. Thus we can no longer regard effect as subsequent to cause. If we think of the present as pushed into existence by the past, we must in precisely the same sense think of it pulled into existence by the future." (Gilbert N Lewis, "The Symmetry of Time in Physics", Science, 1930)

"Everywhere […] in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since […] these laws are time symmetric- they permit […] time-reverse. […] The initial conditions play a decisive role in endowing the world with its sense of temporal direction. […] some prescription for initial conditions is crucial if we are to understand […]"  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"More generally, thermodynamics shows that there is an irreversible flow of time. Rather than there being time symmetry and indeed a reversibility of time as postulated in classical physics, a clear distinction is drawn between the past and future. An arrow of time results within open systems in the loss of organization and an increase in randomness or disorder over time. This accumulation of disorder or positive entropy results from the Second Law of Thermodynamics." (John Urry, "Global Complexity", 2003)

"Eternal constancy of the laws of physics is a symmetry. What we see as we look back in time, or we peer through telescopes out into space, or we look through our powerful microscopes (particle accelerators), is the same system of laws of physics governing the whole universe at all times and at all places. These are the basic symmetries of the structure of our universe and its contents and, at a deeper level, the symmetries of the laws that govern the universe themselves. Indeed, the symmetries we uncover are the basic principles that define our laws of nature and the laws of physics, hence those that control our universe." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The space and time of the universe that we humans inhabit contain symmetries. These are almost obvious yet subtle, even mysterious. Space and time form the stage upon which the dynamics - that is, the motion and interactions of the physical systems, atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of space and time control the dynamics of the physical interactions of matter." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The concept of symmetry is used widely in physics. If the laws that determine relations between physical magnitudes and a change of these magnitudes in the course of time do not vary at the definite operations (transformations), they say, that these laws have symmetry (or they are invariant) with respect to the given transformations. For example, the law of gravitation is valid for any points of space, that is, this law is invariant with respect to the system of coordinates." (Alexey Stakhov et al, "The Mathematics of Harmony", 2009)

"Many statistical procedures perform more effectively on data that are normally distributed, or at least are symmetric and not excessively kurtotic (fat-tailed), and where the mean and variance are approximately constant. Observed time series frequently require some form of transformation before they exhibit these distributional properties, for in their 'raw' form they are often asymmetric." (Terence C Mills, "Applied Time Series Analysis: A practical guide to modeling and forecasting", 2019)

On Patterns (From Fiction to Science-Fiction)

"Throughout recorded time, and probably since the end of the Neolithic Age, there have been three kinds of people in the world, the High, the Middle, and the Low. They have been subdivided in many ways, they have borne countless different names, and their relative numbers, as well as their attitude toward one another, have varied from age to age; but the essential structure of society has never altered. Even after enormous upheavals and seemingly irrevocable changes, the same pattern has always reasserted itself, just as a gyroscope will always return to equilibrium, however far it is pushed one way or the other." (George Orwell, "1984", 1949)

"There is no reason to assume that the universe has the slightest interest in intelligence -  or even in life. Both may be random accidental by-products of its operations like the beautiful patterns on a butterfly's wings. The insect would fly just as well without them […]" (Arthur C Clarke, "The Lost Worlds of 2001", 1972)

"If a book were written all in numbers, it would be true. It would be just. Nothing said in words ever came out quite even. Things in words got twisted and ran together, instead of staying straight and fitting together. But underneath the words, at the center, like the center of the Square, it all came out even. Everything could change, yet nothing would be lost. If you saw the numbers you could see that, the balance, the pattern. You saw the foundations of the world. And they were solid." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife", 1979)

"There are only patterns, patterns on top of patterns, patterns that affect other patterns. Patterns hidden by patterns. Patterns within patterns. If you watch close, history does nothing but repeat itself. What we call chaos is just patterns we haven't recognized. What we call random is just patterns we can't decipher. what we can't understand we call nonsense. What we can't read we call gibberish. There is no free will. There are no variables." (Chuck Palahniuk, "Survivor", 1999)

"Because the question for me was always whether that shape we see in our lives was there from the beginning or whether these random events are only called a pattern after the fact. Because otherwise we are nothing." (Cormac McCarthy, "All the Pretty Horses", 2010)

On Patterns (-1899)

"Engage people with what they expect; it is what they are able to discern and confirms their projections. It settles them into predictable patterns of response, occupying their minds while you wait for the extraordinary moment - that which they cannot anticipate." (Sun Tzu, "The Art of War", cca. 5th century BC)

"For, among the world's incertitudes, this thing called arithmetic is established by a sure reasoning that we comprehend as we do the heavenly bodies. It is an intelligible pattern, a beautiful system, that both binds the heavens and preserves the earth. For is there anything that lacks measure, or transcends weight? It includes all, it rules all, and all things have their beauty because they are perceived under its standard." (Cassiodorus,"Variae epistolae", cca. 538–540s)

"God forbid that we should give out a dream of our own imagination for a pattern of the world." (Francis Bacon, "The Great Instauration", 1620)

"The nature, mother of the eternal diversities, or the divine spirit, are zaelous of her variety by accepting one and only one pattern for all things, By these reasons she has invented this elegant and admirable proceeding. This wonder of Analysis, prodigy of the universe of ideas, a kind of hermaphrodite between existence and non-existence, which we have named imaginary root?" (Gottfried W Leibniz, "De Bisectione Latereum", 1675)

"Some there are who make a distinction betwixt primary and secondary qualities: by the former, they mean extension, figure motion, rest, solidity or impenetrability and number: by the latter they denote all other sensible qualities, as colours, sounds, tastes, and so forth. The ideas we have of these they acknowledge not to be the resemblances of anything existing without the mind or unperceived; but they will have our ideas of the primary qualities to be patterns or images of things which exist without the mind, in an unthinking substance which they call matter. By matter therefore we are to under-stand an inert, senseless substance, in which extension, figure, and motion, do actually subsist." (George Berkeley, "Principles of Human Knowledge", 1710)

"The function of theory is to put all this in systematic order, clearly and comprehensively, and to trace each action to an adequate, compelling cause. […] Theory should cast a steady light on all phenomena so that we can more easily recognize and eliminate the weeds that always spring from ignorance; it should show how one thing is related to another, and keep the important and the unimportant separate. If concepts combine of their own accord to form that nucleus of truth we call a principle, if they spontaneously compose a pattern that becomes a rule, it is the task of the theorist to make this clear." (Carl von Clausewitz, "On War", 1832)

"Most surprising and far-reaching analogies revealed themselves between apparently quite disparate natural processes. It seemed that nature had built the most various things on exactly the same pattern; or, in the dry words of the analyst, the same differential equations hold for the most various phenomena." (Ludwig Boltzmann, "On the methods of theoretical physics", 1892)

On Patterns (2020-)

"Each decision about what data to gather and how to analyze them is akin to standing on a pathway as it forks left and right and deciding which way to go. What seems like a few simple choices can quickly multiply into a labyrinth of different possibilities. Make one combination of choices and you’ll reach one conclusion; make another, equally reasonable, and you might find a very different pattern in the data." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"The world is full of patterns that are too subtle or too rare to detect by eyeballing them, and a pattern doesn’t need to be very subtle or rare to be hard to spot without a statistical lens." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers [...]. The two merge when mathematicians look for number-theoretic patterns hidden in those sequences." (Kelsey Houston-Edwards, "Mathematicians Set Numbers in Motion to Unlock Their Secrets", Quanta Magazine, 2021)

"While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers [.] The two merge when mathematicians look for number-theoretic patterns hidden in those sequences." (Kelsey Houston-Edwards, "Mathematicians Set Numbers in Motion to Unlock Their Secrets", Quanta Magazine, 2021).

"Chaos theory is a branch of mathematics focusing on the study of chaos - dynamical systems whose random states of disorder and irregularities are governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of complex, chaotic systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state" (meaning that there is a sensitive dependence on initial conditions)." (Nima Norouzi, "Criminal Policy, Security, and Justice in the Time of COVID-19", 2022)

On Patterns (Unsourced)

"Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable." (Alfred N Whitehead)

"Discovery in mathematics is not a matter of logic. It is rather the result of mysterious powers which no one understands, and in which unconscious recognition of beauty must play an important part. Out of an infinity of designs, a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse)

"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff)

"Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse)

"[...] mathematics is the science of patterns." (Lynn A Steen)

"Mathematics is often defined as the science of space and number [...] it was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns." (Lynn A Steen)

"Organizations are not systems but the ongoing patterning of interactions between people. Patterns of human interaction produce further patterns of interaction, not some thing outside of the interaction. We call this perspective complex responsive processes of relating." (Ralph Stacey)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland)

"The reason why we do maths is because it's like poetry. It's about patterns, and that really turned me on. It made me feel that maths was in tune with the other things I liked doing." (Marcus du Sautoy)

"Topology is the science of fundamental pattern and structural relationships of event constellations." (R Buckminster Fuller)

On Patterns (1930-1939)

"The stream of human knowledge is heading towards a non-mechanical reality. The universe begins to look more like a great thought than a great machine. Mind no longer appears to be an accidental intruder into the realm of matter. We are beginning to suspect that we ought rather to hail it as the creator and governor of this realm." (Sir James Jeans, "The Mysterious Universe", 1930)

"In pure mathematics the maximum of detachment appears to be reached: the mind moves in an infinitely complicated pattern, which is absolutely free from temporal considerations. Yet this very freedom - the essential condition of the mathematician’s activity - perhaps gives him an unfair advantage. He can only be wrong – he cannot cheat." (Kytton Strachey, "Portraits in Miniature", 1931)

"[while] the traditional way is to regard the facts of science as something like the parts of a jig-saw puzzle, which can be fitted together in one and only one way, I regard them rather as the tiny pieces of a mosaic, which can be fitted together in many ways. A new theory in an old subject is, for me, a new mosaic pattern made with the pieces taken from an older pattern. [...] Theories come into fashion and theories go out of fashion, but the facts connected with them stay." (William H George, "The Scientist in Action", 1936)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T. Sanders, "Mathematics", National Mathematics Magazine, 1937)

"The laws of science are the permanent contributions to knowledge - the individual pieces that are fitted together in an attempt to form a picture of the physical universe in action. As the pieces fall into place, we often catch glimpses of emerging patterns, called theories; they set us searching for the missing pieces that will fill in the gaps and complete the patterns. These theories, these provisional interpretations of the data in hand, are mere working hypotheses, and they are treated with scant respect until they can be tested by new pieces of the puzzle." (Edwin P Whipple, "Experiment and Experience", [Commencement Address, California Institute of Technology] 1938)

Cassiodorus - Collected Quotes

"Few men are controlled by reason, and few are pleased by a right purpose. The mob, rather, is led to what was plainly invented for oblivion of its cares. For it supposes that whatever serves its pleasure must also be linked to the happiness of the age. Therefore, let us grant the expenses, and not be forever giving from rational considerations. Sometimes it is useful to play the fool, and so control the joys the people long for." (Cassiodorus,"Variae epistolae", cca. 538–540s

"For, among the world's incertitudes, this thing called arithmetic is established by a sure reasoning that we comprehend as we do the heavenly bodies. It is an intelligible pattern, a beautiful system, that both binds the heavens and preserves the earth. For is there anything that lacks measure, or transcends weight? It includes all, it rules all, and all things have their beauty because they are perceived under its standard." (Cassiodorus,"Variae epistolae", cca. 538–540s)

"For the school of grammar has primacy: it is the fairest foundation of learning, the glorious mother of eloquence." (Cassiodorus,"Variae epistolae", cca. 538–540s

"For what is more glorious than music, which modulates the heavenly system with its sonorous sweetness, and binds together with its virtue the concord of nature which is scattered everywhere? For any variation there may be in the whole does not depart from the pattern of harmony. Through this we think with efficiency, we speak with elegance, we move with grace. Whenever, by the natural law of its discipline, it reaches our ears, it commands song. The artist changes men's hearts as they listen; and, when this artful pleasure issues from the secret place of nature as the queen of the senses, in all the glory of its tones, our remaining thoughts take to flight, and it expels all else, that it may delight itself simply in being heard." (Cassiodorus,"Variae epistolae", cca. 538–540s

"Grammar is the mistress of words, the embellisher of the human race; through the practice of the noble reading of ancient authors, she helps us, we know, by her counsels. The barbarian kings do not use her; as is well known, she remains unique to lawful rulers. For the tribes possess arms and the rest; rhetoric is found in sole obedience to the lords of the Romans." (Cassiodorus,"Variae epistolae", cca. 538–540s

“Mathematical science […] has these divisions: arithmetic, music, geometry, astronomy. Arithmetic is the discipline of absolute numerable quantity. Music is the discipline which treats of numbers in their relation to those things which are found in sound.” (Cassiodorus, cca. 6th century)

"It is given to us to live for the most part under the guidance of mathematics.... It is impossible to distinguish from other living creatures anyone who does not understand how to quantify." (Cassiodorus) 

On Time: The Arrow of Time

"Let us draw an arrow arbitrarily. If as we follow the arrow we find more and more of the random element in the world, then the arrow is pointing towards the future; if the random element decreases the arrow points towards the past. [...] I shall use the phrase 'time's arrow' to express this one-way property of time which has no analogue in space. (Arthur Eddington, "The Nature of the Physical World", 1928)

"So far as physics is concerned, time's arrow is a property of entropy alone." (Arthur S Eddington, "The Nature of the Physical World", 1928)

"On a microscopic level there is no preferred direction for time. The equations of motion don’t give a damn whether time moves forward or backward." (L Eisenberg, “The Time of His Life”, The Magazine of Fantasy & Science Fiction, 1968) 

"Time goes forward because energy itself is always moving from an available to an unavailable state. Our consciousness is continually recording the entropy change in the world around us. [...] we experience the passage of time by the succession of one event after another. And every time an event occurs anywhere in this world energy is expended and the overall entropy is increased. To say the world is running out of time then, to say the world is running out of usable energy. In the words of Sir Arthur Eddington, 'Entropy is time's arrow'." (Jeremy Rifkin & Ted Howard, "Entropy: A New World View", 1980)

 "The increase of disorder or entropy with time is one example of what is called an arrow of time something that gives a direction to time and distinguishes the past from the future. There are at least three different directions of time. First, there is the thermodynamic arrow of time - the direction of time in which disorder or entropy increases. Second, there is the psychological arrow of time. This is the direction in which we feel time passes - the direction of time in which we remember the past, but not the future. Third, there is the cosmological arrow of time. This is the direction of time in which the universe is expanding rather than contracting." (Stephen W. Hawking, "The Direction of Time", New Scientist 46, 1987)

"At another level, market crashes constitute beautiful examples of events that we would all like to forecast. The arrow of time is inexorably projecting us toward the undetermined future. Predicting the future captures the imagination of all and is perhaps the greatest challenge." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)

"More generally, thermodynamics shows that there is an irreversible flow of time. Rather than there being time symmetry and indeed a reversibility of time as postulated in classical physics, a clear distinction is drawn between the past and future. An arrow of time results within open systems in the loss of organization and an increase in randomness or disorder over time. This accumulation of disorder or positive entropy results from the Second Law of Thermodynamics." (John Urry, "Global Complexity", 2003)

"The arrow of time, through the defi ning role it plays in everyday life and its intimate link with the origin of the universe, lies at a singular threshold between the reality we experience and the more refi ned reality cutting-edge science seeks to uncover." (Brian Greene, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", 2004)

"We often wonder why the more complex systems seem to indicate a preferred direction of time, or an arrow of time, whereas their elementary counterparts do not. […] This has to do with the if-then nature of physics questions. Anything we observe involves laws of motion but also particular initial conditions. […] The initial conditions are what make a situation look peculiar when we time reverse it." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"In the elementary equations of the world, the arrow of time appears only where there is heat. The link between time and heat is therefore fundamental: every time a difference is manifested between the past and the future, heat is involved. In every sequence of events that becomes absurd if projected backward, there is something that is heating up." (Carlo Rovelli, "The Order of Time", 2018)

On Numbers (From Fiction to Science-Fiction)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"One microscopic glittering point; then another; and another, and still another; they are scarcely perceptible, yet they are enormous. This light is a focus; this focus, a star; this star, a sun; this sun, a universe; this universe, nothing. Every number is zero in the presence of the infinite." (Victor Hugo,"The Toilers of the Sea", 1874)

"Revolution is everywhere, in everything. It is infinite. There is no final revolution, no final number. The social revolution is only one of an infinite number of numbers; the law of revolution is not a social law, but an immeasurably  greater one. It is a cosmic, universal law - like the laws of the conservation of energy and of the dissipation of energy (entropy)." (Yevgeny Zamiatin, ‘‘On Literature, Revolution, Entropy, and Other Matters’’, 1923)

"Words and numbers are of equal value, for, in the cloak of knowledge, one is warp and the other woof. It is no more important to count the sands than it is to name the stars." (Norton Juster, "The Phantom Tollbooth", 1961)

"Why, numbers are the most beautiful and valuable things in the world.’" (Norton Juster, "The Phantom Tollbooth", 1961)

"Science is not a sacred cow - but there are a large number of would-be sacred cowherds busily devoting quantities of time, energy and effort to the task of making it one, so they can be sacred cowherds." (John W. Campbell Jr., "Prologue to Analog", [introduction] 1962)

"Beyond a critical point within a finite space, freedom diminishes as numbers increase. This is as true of humans as it is of gas molecules in a sealed flask. The human question is not how many can possibly survive within the system, but what kind of existence is possible for those who do survive." (Frank Herbert, "Dune", 1965)

"If a book were written all in numbers, it would be true. It would be just. Nothing said in words ever came out quite even. Things in words got twisted and ran together, instead of staying straight and fitting together. But underneath the words, at the center, like the center of the Square, it all came out even. Everything could change, yet nothing would be lost. If you saw the numbers you could see that, the balance, the pattern. You saw the foundations of the world. And they were solid." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"What do such machines really do? They increase the number of things we can do without thinking. Things we do without thinking - there's the real danger." (Frank Herbert, "God Emperor of Dune", 1984)

John M Bryson - Collected Quotes

"A causal map is a word-and-arrow diagram in which ideas and actions are causally linked with one another through the use of arrows. The arrows indicate how one idea or action leads to another. Causal mapping makesit possible to articulate a large number of ideas and their interconnections in such a way that people can know what to do in an area of concern, how to do it and why, because the arrows indicate the causes and consequences of an idea or action. Causal mapping is therefore a technique for linking strategic thinking and acting, helping make sense of complex problems, and communicating to oneself and others what might be done about them." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"A statement of strategic aims is powerful when it represents the interaction between all of the goals [Strategy]: the goals are seen as a system where each goal helps deliver other high-level goals and may in turn be helped by the delivery of subordinate goals." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Causal mapping is a simple and useful technique for addressing situations where thinking– as an individual or as a group– matters. A causal map is a word-and-arrow diagram in which ideas and actions are causally linked with one another through the use of arrows. The arrows indicate how one idea or action leads to another. Causal mapping makes it possible to articulate a large number of ideas and their interconnections in such a way that people can know what to do in an area of concern, how to do it and why, because the arrows indicate the causes and consequences of an idea or action." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Causal mapping is [...] a technique for linking strategic thinking and acting, helping make sense of complex problems, and communicating to oneself and others what might be done about them. With practice, the use of causal mapping can assist you in moving from 'winging it' when thinking matters to a more concrete and rigorous approach that helps you and others achieve success in an easy and far more reliable way" (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Causal mapping makes it possible to articulate a large number of ideas and their interconnections in such a way that we can better understand an area of concern. Causal mapping also helps us know what to do about the issue, what it would take to do those things, and what we would like to get out of having done so. Causal mapping is therefore a particularly powerful technique for making sense of complex problems, linking strategic thinking and acting, and helping to communicate to others what might or should be done. " (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Linking focuses attention not on topics but on action: what results might be achieved by doing something, and what things needed to be done to make other things happen." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Positive feedback loops are very important. Even when afeedback loop is made up of competencies, because they are self-sustaining they may be very important resources for the future of the business [Strategy]. A positive loop means that each of the competencies in the loop feeds all of the others. When the feedback loop is distinctive, this will be even more important because the distinctiveness is self-sustaining." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"The process of constructing a map thus allows you to reflect on your own thinking, or that of someone else. Mapping helps you become conscious of your reasoning and to understand clearly what another person is thinking. In turn, that consciousness and understanding can help you reaffirm what you think or else change your mind. Shared understanding and mutual agreement also become possible." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"When an individual uses causal mapping to help clarify their own thinking, we call this technique cognitive mapping, because it is related to personal thinking or cognition. When a group maps their own ideas, we call it oval mapping, because we often use oval-shaped cards to record individuals’ ideas so that they can be arranged into a group’s map. Cognitive maps and oval maps can be used to create a strategic plan, because the maps include goals, strategies and actions, just like strategic plans." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"When people question assumptions, the map may clarify what they are. When logic is challenged, the map may help. When people want to know how goals and strategies are linked, the map may show how they are. The map does not make the decisions. Rather, it provides a record that preserves complexity, yet organizes and categorizes that complexity in such a way that people can understand and manage it. And if more mapping needs to be done, the map is there as a base on which to build." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)