31 March 2025

On Mistakes, Blunders and Errors X: Data Science

"Measurement, we have seen, always has an element of error in it. The most exact description or prediction that a scientist can make is still only approximate." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"[…] it is not enough to say: 'There's error in the data and therefore the study must be terribly dubious'. A good critic and data analyst must do more: he or she must also show how the error in the measurement or the analysis affects the inferences made on the basis of that data and analysis." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Many scientists who work not just with noise but with probability make a common mistake: They assume that a bell curve is automatically Gauss's bell curve. Empirical tests with real data can often show that such an assumption is false. The result can be a noise model that grossly misrepresents the real noise pattern. It also favors a limited view of what counts as normal versus non-normal or abnormal behavior. This assumption is especially troubling when applied to human behavior. It can also lead one to dismiss extreme data as error when in fact the data is part of a pattern." (Bart Kosko, "Noise", 2006

"In bagging, generating complementary base-learners is left to chance and to the unstability of the learning method. In boosting, we actively try to generate complementary base-learners by training the next learner boosting on the mistakes of the previous learners." (Ethem Alpaydin, "Introduction to Machine Learning" 2nd Ed, 2010)

"When data is not normal, the reason the formulas are working is usually the central limit theorem. For large sample sizes, the formulas are producing parameter estimates that are approximately normal even when the data is not itself normal. The central limit theorem does make some assumptions and one is that the mean and variance of the population exist. Outliers in the data are evidence that these assumptions may not be true. Persistent outliers in the data, ones that are not errors and cannot be otherwise explained, suggest that the usual procedures based on the central limit theorem are not applicable." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"There are two kinds of mistakes that an inappropriate inductive bias can lead to: underfitting and overfitting. Underfitting occurs when the prediction model selected by the algorithm is too simplistic to represent the underlying relationship in the dataset between the descriptive features and the target feature. Overfitting, by contrast, occurs when the prediction model selected by the algorithm is so complex that the model fits to the dataset too closely and becomes sensitive to noise in the data." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"[...] data often has some errors, outliers and other strange values, but these do not necessarily need to be individually identified and excluded. It also points to the benefits of using summary measures that are not unduly affected by odd observations [...] are known as robust measures, and include the median and the inter-quartile range." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"Statistical models have two main components. First, a mathematical formula that expresses a deterministic, predictable component, for example the fitted straight line that enables us to make a prediction [...]. But the deterministic part of a model is not going to be a perfect representation of the observed world [...] and the difference between what the model predicts, and what actually happens, is the second component of a model and is known as the residual error - although it is important to remember that in statistical modelling, ‘error’ does not refer to a mistake, but the inevitable inability of a model to exactly represent what we observe." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"There are many ways for error to creep into facts and figures that seem entirely straightforward. Quantities can be miscounted. Small samples can fail to accurately reflect the properties of the whole population. Procedures used to infer quantities from other information can be faulty. And then, of course, numbers can be total bullshit, fabricated out of whole cloth in an effort to confer credibility on an otherwise flimsy argument. We need to keep all of these things in mind when we look at quantitative claims. They say the data never lie - but we need to remember that the data often mislead." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

On Mistakes, Blunders and Errors IX: Artificial Intelligence

"Because the subject matter of cybernetics is the propositional or informational aspect of the events and objects in the natural world, this science is forced to procedures rather different from those of the other sciences. The differentiation, for example, between map and territory, which the semanticists insist that scientists shall respect in their writings must, in cybernetics, be watched for in the very phenomena about which the scientist writes. Expectably, communicating organisms and badly programmed computers will mistake map for territory; and the language of the scientist must be able to cope with such anomalies." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"To expect truth to come from thinking signifies that we mistake the need to think with the urge to know." (Hannah Arendt, "The Life of the Mind", 1977)

"There is a tendency to mistake data for wisdom, just as there has always been a tendency to confuse logic with values, intelligence with insight. Unobstructed access to facts can produce unlimited good only if it is matched by the desire and ability to find out what they mean and where they lead." (Norman Cousins, "Human Options : An Autobiographical Notebook", 1981)

"The other buzzword that epitomizes a bias toward substitution is 'big data'. Today’s companies have an insatiable appetite for data, mistakenly believing that more data always creates more value. But big data is usually dumb data. Computers can find patterns that elude humans, but they don’t know how to compare patterns from different sources or how to interpret complex behaviors. Actionable insights can only come from a human analyst (or the kind of generalized artificial intelligence that exists only in science fiction)." (Peter Thiel & Blake Masters, "Zero to One: Notes on Startups, or How to Build the Future", 2014)

"Artificial Intelligence is not just learning patterns from data, but understanding human emotions and its evolution from its depth and not just fulfilling the surface level human requirements, but sensitivity towards human pain, happiness, mistakes, sufferings and well-being of the society are the parts of the evolving new AI systems." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"There are two kinds of mistakes that an inappropriate inductive bias can lead to: underfitting and overfitting. Underfitting occurs when the prediction model selected by the algorithm is too simplistic to represent the underlying relationship in the dataset between the descriptive features and the target feature. Overfitting, by contrast, occurs when the prediction model selected by the algorithm is so complex that the model fits to the dataset too closely and becomes sensitive to noise in the data."(John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"The chief weakness of the machine is that it will not learn by its mistakes. The only way to improve its play is by improving the program. Some thought has been given to designing a program that would develop its own improvements in strategy with increasing experience in play. Although it appears to be theoretically possible, the methods thought of so far do not seem to be very practical. One possibility is to devise a program that would change the terms and coefficients involved in the evaluation function on the basis of the results of games the machine had already played. Small variations might be introduced in these terms, and the values would be selected to give the greatest percentage of wins." (Claude E Shannon)

On Mistakes, Blunders and Errors VIII: Physics

"In time they [physicists] hoped to devise a model which would reproduce all the phenomena of physics, and so make it possible to predict them all. […] To-day we not only have no perfect model, but we know that it is of no use to search for one - it could have no intelligible meaning for us. For we have found out that nature does not function in a way that can be made comprehensible to the human mind through models or pictures. […] Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"In physics it is usual to give alternative theoretical treatments of the same phenomenon. We construct different models for different purposes, with different equations to describe them. Which is the right model, which the 'true' set of equations? The question is a mistake. One model brings out some aspects of the phenomenon; a different model brings out others. Some equations give a rougher estimate for a quantity of interest, but are easier to solve. No single model serves all purposes best." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Steven Weinberg, "The First Three Minutes", 1977)

"In physics it is usual to give alternative theoretical treatments of the same phenomenon. We construct different models for different purposes, with different equations to describe them. Which is the right model, which the 'true' set of equations? The question is a mistake. One model brings out some aspects of the phenomenon; a different model brings out others. Some equations give a rougher estimate for a quantity of interest, but are easier to solve. No single model serves all purposes best." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"It is a testimony to the power of education that classical mechanics could operate for so long under a mistaken conception. Teaching and research concentrated on integrable systems, each feeding the other, until in the end we had no longer the tools nor the interest for studying nonintegrable systems." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

“This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Heinrich Hertz)

"There is a fundamental error in separating the parts from the whole, the mistake of atomizing what should not be atomized. Unity and complementarity constitute reality." (Werner K Heisenberg)

30 March 2025

On Mistakes, Blunders and Errors VII: NLP

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931)

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Since we have no systematic way to avoid all the inconsistencies of commonsense logic, each person must find his own way by building a private collection of 'cognitive censors' to suppress the kinds of mistakes he has discovered in the past." (Marvin Minsky, "Jokes and their Relation to the Cognitive Unconscious", 1980)

"All our language is composed of brief little dreams; and the wonderful thing is that we sometimes make of them strangely accurate and marvelously reasonable thoughts. […] What should we be without the help of that which does not exist? Very little. And our unoccupied minds would languish if fables, mistaken notions, abstractions, beliefs, and monsters, hypotheses, and the so-called problems of metaphysics did not people with beings and objectless images our natural depths and darkness. Myths are the souls of our actions and our loves. We cannot act without moving towards a phantom. We can love only what we create." (Paul Valéry, "The Outlook for Intelligence", 1962)

"To expect truth to come from thinking signifies that we mistake the need to think with the urge to know." (Hannah Arendt, "The Life of the Mind", 1977)

"To assume you know someone well enough that you can and do predict their behavior and mental perspective is a gross and often tragic mistake, for it eliminates that person's freedom to create his or her own opinion and drastically affects the emerging picture of the relationship." (Meredith L Young-Sowers, "Agartha: a journey to the stars", 1984) 

"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"Trust is fundamental to leading others into the dark, since trust enables fear to be 'actionable' as courage rather than actionable as anger. Since the bedrock of trust is faith that all will be OK within uncertainty, leaders’ fundamental role is to ultimately lead themselves. Research has found that successful leaders share three behavioral traits: they lead by example, admit their mistakes, and see positive qualities in others. All three are linked to spaces of play. Leading by example creates a space that is trusted - and without trust, there is no play. Admitting mistakes is to celebrate uncertainty. Seeing qualities in others is to encourage diversity." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

On Mistakes, Blunders and Errors VI: Mind

"Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them." (J B Joseph Fourier, "The Analytical Theory of Heat", 1822)

"It has often been said that, to make discoveries, one must be ignorant. This opinion, mistaken in itself, nevertheless conceals a truth. It means that it is better to know nothing than to keep in mind fixed ideas based on theories whose confirmation we constantly seek, neglecting meanwhile everything that fails to agree with them." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"All experience attests the strength of the tendency to mistake mental abstractions, even negative ones, for substantive realities; and the Permanent Possibilities of sensation which experience guarantees arc so extremely unlike in many of their properties to actual sensations, that since we are capable of imagining something which transcends sensations, there is a great natural probability that we should suppose these to be it." (Hippolyte Taine, "On intelligence", 1871)

"Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent. (William S Jevons, "The Principles of Science", 1874)

"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931)

"Since we have no systematic way to avoid all the inconsistencies of commonsense logic, each person must find his own way by building a private collection of 'cognitive censors' to suppress the kinds of mistakes he has discovered in the past." (Marvin Minsky, "Jokes and their Relation to the Cognitive Unconscious", 1980)

"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite, The Mathematical Intelligencer, Vol. 5, No. 4, 1983)

"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)

"A border is a completely imaginary line on a paper or cybernetic map that has no genuine counterpart in the real world. Do not mistake it for a property line. It is possible, in some instances, for a border to be congruent with a property line, but they are not the same thing at all. One represents the geographical limit of a military and political claim to authority over a given territory. The other is part of the description of something - in this case, land - lawfully owned by an individual or a voluntary and contractual association of individuals." (L Neil Smith, "Only Nixon", 2010)

"The human mind delights in finding pattern - so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 2010)

"Mental models represent possibilities, and the theory of mental models postulates three systems of mental processes underlying inference: (0) the construction of an intensional representation of a premise’s meaning – a process guided by a parser; (1) the building of an initial mental model from the intension, and the drawing of a conclusion based on heuristics and the model; and (2) on some occasions, the search for alternative models, such as a counterexample in which the conclusion is false. System 0 is linguistic, and it may be autonomous. System 1 is rapid and prone to systematic errors, because it makes no use of a working memory for intermediate results. System 2 has access to working memory, and so it can carry out recursive processes, such as the construction of alternative models." (Sangeet Khemlania & P.N. Johnson-Laird, "The processes of inference", Argument and Computation, 2012)

On Mistakes, Blunders and Errors V: From Fiction to Science-Fiction

"As a net is made up of a series of ties, so everything in this world is connected by a series of ties. If anyone thinks that the mesh of a net is an independent, isolated thing, he is mistaken. It is called a net because it is made up of a series of a interconnected meshes, and each mesh has its place and responsibility in relation to other meshes." (Gautama Buddha)

"The truth of voice perishes with the sound; truth latent in the mind is hidden wisdom and invisible treasure; but the truth which illuminates books desires to manifest itself to every disciplinable sense. Let us consider how great a commodity of doctrine exists in books, - how easily, how secretly, how safely, they expose the nakedness of human ignorance without putting it to shame. These are the masters that instruct us without rods and ferules, without hard words and anger, without clothes or money. If you approach them, they are not asleep; if, investigating, you interrogate them, they conceal nothing; if you mistake them, they never grumble; if you are ignorant, they cannot laugh at you." (Richard de Burry, "Philobiblon", 1344)

"From a caprice of nature, not from the ignorance of man. Not a mistake has been made in the working. But we cannot prevent equilibrium from producing its effects. We may brave human laws, but we cannot resist natural ones." (Jules Verne, "Twenty Thousand Leagues Under the Sea", 1870)

"Experience was of no ethical value. It was merely the name men gave to their mistakes." (Oscar Wilde, "The Picture of Dorian Gray", 1891)

 "Mistakes live in the neighborhood of truth and therefore delude us." (Rabindranath Tagore, "Fireflies", 1928)

"The specialist is one who never makes small mistakes while moving towards the grand fallacy." (Marshall McLuhan, "Understanding Media", 1964)

"These machines had become old and worn-out, had begun making mistakes; therefore they began to seem almost human." (Philip K Dick & Ray Nelson, "The Ganymede Takeover", 1967)

"Intelligence takes chances with limited data in an arena where mistakes are not only possible but also necessary." (Frank Herbert, "Chapterhouse: Dune", 1985)

"Human beings are very conservative in some ways and virtually never change numerical conventions once they grow used to them. They even come to mistake them for laws of nature." (Isaac Asimov, "Foundation and Earth", 1986)

"An ordinary mistake is one that leads to a dead end, while a profound mistake is one that leads to progress. Anyone can make an ordinary mistake, but it takes a genius to make a profound mistake." (Frank Wilczek,"The Lightness of Being – Mass, Ether and the Unification of Forces", 2008) 

"Men are more apt to be mistaken in their generalizations than in their particular observations." (Niccolo Machiavelli)

On Mistakes, Blunders and Errors IV: Systems Thinking

"Because the subject matter of cybernetics is the propositional or informational aspect of the events and objects in the natural world, this science is forced to procedures rather different from those of the other sciences. The differentiation, for example, between map and territory, which the semanticists insist that scientists shall respect in their writings must, in cybernetics, be watched for in the very phenomena about which the scientist writes. Expectably, communicating organisms and badly programmed computers will mistake map for territory; and the language of the scientist must be able to cope with such anomalies." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"However, and conversely, our models fall far short of representing the world fully. That is why we make mistakes and why we are regularly surprised. In our heads, we can keep track of only a few variables at one time. We often draw illogical conclusions from accurate assumptions, or logical conclusions from inaccurate assumptions. Most of us, for instance, are surprised by the amount of growth an exponential process can generate. Few of us can intuit how to damp oscillations in a complex system." (Donella H Meadows, "Limits to Growth", 1972)

"[...] the influence of a single butterfly is not only a fine detail-it is confined to a small volume. Some of the numerical methods which seem to be well adapted for examining the intensification of errors are not suitable for studying the dispersion of errors from restricted to unrestricted regions. One hypothesis, unconfirmed, is that the influence of a butterfly's wings will spread in turbulent air, but not in calm air." (Edward N Lorenz, [talk] 1972)

"A diverse community is a resilient community, capable of adapting to changing situations. However, diversity is a strategic advantage only if there is a truly vibrant community, sustained by a web of relationships. If the community is fragmented into isolated groups and individuals, diversity can easily become a source of prejudice and friction. But if the community is aware of the interdependence of all its members, diversity will enrich all the relationships and thus enrich the community as a whole, as well as each individual member. In such a community information and ideas flow freely through the entire network, and the diversity of interpretations and learning styles-even the diversity of mistakes-will enrich the entire community." (Humberto Maturana & Francisco J Varela, "The Tree of Knowledge", 1987)

"It is important to emphasize the value of simplicity and elegance, for complexity has a way of compounding difficulties and as we have seen, creating mistakes. My definition of elegance is the achievement of a given functionality with a minimum of mechanism and a maximum of clarity." (Fernando J Corbató, "On Building Systems That Will Fail", 1991)

"A model for simulating dynamic system behavior requires formal policy descriptions to specify how individual decisions are to be made. Flows of information are continuously converted into decisions and actions. No plea about the inadequacy of our understanding of the decision-making processes can excuse us from estimating decision-making criteria. To omit a decision point is to deny its presence - a mistake of far greater magnitude than any errors in our best estimate of the process." (Jay W Forrester, "Policies, decisions and information sources for modeling", 1994)

"This distinction is familiar in natural science, where one is not expected to mistake, say, the cardiovascular system for the circulation of the blood or the brain with mental processes. But it is unusual in social studies. [...] Mechanism is to system as motion is to body, combination (or dissociation) to chemical compound, and thinking to brain. [In the systemic view], agency is both constrained and motivated by structure, and in turn the latter is maintained or altered by individual action. In other words, social mechanisms reside neither in persons nor in their environment – they are part of the processes that unfold in or among social systems. […] All mechanisms are system-specific: there is no such thing as a universal or substrate-neutral mechanism." (Mario Bunge, "The Sociology-philosophy Connection", 1999)

"A depressing corollary of the butterfly effect (or so it was widely believed) was that two chaotic systems could never synchronize with each other. Even if you took great pains to start them the same way, there would always be some infinitesimal difference in their initial states. Normally that small discrepancy would remain small for a long time, but in a chaotic system, the error cascades and feeds on itself so swiftly that the systems diverge almost immediately, destroying the synchronization. Unfortunately, it seemed, two of the most vibrant branches of nonlinear science - chaos and sync - could never be married. They were fundamentally incompatible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Adaptive systems learn by enlightened trial and error. The system can take a long time to learn well just as it can take a human a long time to learn to properly swing a golf club even with the help of the best golf instructor. But this iterative learning can also produce solutions that we could not find or at least could not find easily by pure mathematical analysis."  (Bart Kosko, "Noise", 2006)

"We forget - or we willfully ignore - that our models are simplifications of the world. We figure that if we make a mistake, it will be at the margin. In complex systems, however, mistakes are not measured in degrees but in whole orders of magnitude." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)

"[…] humans make mistakes when they try to count large numbers in complicated systems. They make even greater errors when they attempt - as they always do - to reduce complicated systems to simple numbers." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"With a linear growth of errors, improving the measurements could always keep pace with the desire for longer prediction. But when errors grow exponentially fast, a system is said to have sensitive dependence on its initial conditions. Then long-term prediction becomes impossible. This is the philosophically disturbing message of chaos." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

On Mistakes, Blunders and Errors III: Mathematics

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations." (Niels H Abel, "Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree", 1824)

"The man of science, who, forgetting the limits of philosophical inquiry, slides from these formulæ and symbols into what is commonly understood by materialism, seems to me to place himself on a level with the mathematician, who should mistake the x's and y's with which he works his problems for real entities - and with this further disadvantage, as compared with the mathematician, that the blunders of the latter are of no practical consequence, while the errors of systematic materialism may paralyse the energies and destroy the beauty of a life." (Thomas H Huxley, "Method and Results", 1893)

"The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect, - the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae." (John T Merz, "History of European Thought in the 19th Century" Vol. 1, 1904)

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality." (Cassius J Keyser,"The Humanization of the Teaching of Mathematics", 1912)

"The mistake from which todays’ science suffers is that the theoreticians are concerned too unilaterally with precision mathematics, while the practitioners use a sort of approximate mathematics, without being in touch with precision mathematics through which they could reach a real approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death." (Eric T Bell, "The Development of Mathematics", 1940)

"But, really, mathematics is not religion; it cannot be founded on faith. And what was most important, the methods yielding such remarkable results in the hands of the great masters began to lead to errors and paradoxes when employed by their less talented students. The masters were kept from error by their perfect mathematical intuition, that subconscious feeling that often leads to the right answer more quickly than lengthy logical reasoning. But the students did not possess this intuition […]" (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Now a mathematician has a matchless advantage over general scientists, historians, politicians, and exponents of other professions: He can be wrong. A fortiori, he can also be right. [...] A mistake made by a mathematician, even a great one, is not a 'difference of a point of view' or 'another interpretation of the data' or a 'dictate of a conflicting ideology', it is a mistake. The greatest of all mathematicians, those who have discovered the greatest quantities of mathematical truths, are also those who have published the greatest numbers of lacunary proofs, insufficiently qualified assertions, and flat mistakes." (Clifford Truesdell, "Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton's Principia" [in "Essays in the History of Mechanics"], 1968)

"The mistakes made by a great mathematician are of two kinds: first, trivial slips that anyone can correct, and, second, titanic failures reflecting the scale of the struggle which the great mathematician waged. Failures of this latter kind are often as important as successes, for they give rise to major discoveries by other mathematicians. One error of a great mathematician has often done more for science than a hundred impeccable little theorems proved by lesser men." (Clifford Truesdell, "Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton's Principia" [in "Essays in the History of Mechanics"], 1968)

"Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Steven Weinberg, "The First Three Minutes", 1977)

"Is catastrophe theory correct? In its mathematics, yes; in the natural philosophy that inspired it and the scientific applications that flow from it, the only possible answer is that it's too soon to say. There is always a chance of error whenever we try to capture any aspect of reality in mathematical symbols, and another chance of error when (after working with the symbols) we use them to generate descriptions or predictions of reality." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Mathematics is not a branch of aesthetics. The mistake, which is common enough, probably stems from the requirement of aesthetic unity but is not identical with that unity." (J K Feibleman,"Assumptions of Grand Logics", 1979)

"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)

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

"So, when trying to solve a problem in mathematics we have to watch out for subtle mistakes, otherwise, we can easily get the wrong solution." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)

"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)

"One mistake that students commonly make early on is that they assume that, in a topological space, any set is either open or closed. This is like meeting a blonde person and a brunette and assuming therefore that all people are either blonde or brunette. [...] It is in fact possible for a set to be both open and closed." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"In mathematics, two angles that are said to coincide fit together perfectly. The word 'coincidence' does not describe luck or mistakes. It describes that which fits together perfectly." (Wayne Dyer, "The Essential Wayne Dyer Collection", 2013)

"The problem of teaching is the problem of introducing concepts and conceptual systems. In this crucial task the procedures of formal mathematical argument are of little value. The way we reason in formal mathematics is itself a conceptual system - deductive logic - but it is a huge mistake to identify this with mathematics. [...] Mathematics lives in its concepts and conceptual systems, which need to be explicitly addressed in the teaching of mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

On Mistakes, Blunders and Errors XIII: Science

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Assert nothing till after repeated experiments and examinations, in all lights, and in all positions. Truth alone is the matter that you are in search after; and if you have been mistaken, let not vanity seduce you to persist in your mistake." (Henry Baker, "The Microscope Made Easy", 1742)

"The only method of preventing such errors from taking place, and of correcting them when formed, is to restrain and simplify our reasoning as much as possible. This depends entirely upon ourselves, and the neglect of it is the only source of our mistakes." (Antoine-Laurent de Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

"Hypotheses are scaffoldings erected in front of a building and then dismantled when the building is finished. They are indispensable for the workman; but you mustn't mistake the scaffolding for the building." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

"Experimental science hardly ever affords us more than approximations to truth; and whenever many agents are concerned we are in great danger of being mistaken." (Sir Humphry Davy, cca. 1836)

"We learn wisdom from failure much more than from success. We often discover what will do, by finding out what will not do; and probably he who never made a mistake never made a discovery." (Samuel Smiles, "Facilities and Difficulties", 1859)

"It has often been said that, to make discoveries, one must be ignorant. This opinion, mistaken in itself, nevertheless conceals a truth. It means that it is better to know nothing than to keep in mind fixed ideas based on theories whose confirmation we constantly seek, neglecting meanwhile everything that fails to agree with them." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"It is surprising to learn the number of causes of error which enter into the simplest experiment, when we strive to attain rigid accuracy." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent. (William S Jevons, "The Principles of Science", 1874)

"There cannot be a greater mistake than that of looking superciliously upon the practical applications of science. The life and soul of science is its practical application; and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind." (William T Kelvin, "Electrical Units of Measurement", 1883)

"It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors." (Thomas H Huxley, "The Progress of Science", 1887)

"The man of science, who, forgetting the limits of philosophical inquiry, slides from these formulæ and symbols into what is commonly understood by materialism, seems to me to place himself on a level with the mathematician, who should mistake the x's and y's with which he works his problems for real entities - and with this further disadvantage, as compared with the mathematician, that the blunders of the latter are of no practical consequence, while the errors of systematic materialism may paralyse the energies and destroy the beauty of a life." (Thomas H Huxley, "Method and Results", 1893)

"The scientific spirit is of more value than its products, and irrationally held truths may be more harmful than reasoned errors." (Thomas H Huxley, "Darwiniana", 1893–94)

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)

"It is a great mistake to suppose that the mind of the active scientist is filled with pro-positions which, if not proved beyond all reasonable cavil, are at least extremely probable. On the contrary, he entertains hypotheses which are almost wildly incredible, and treats them with respect for the time being. Why does he do this? Simply because any scientific proposition whatever is always liable to be refuted and dropped at short notice. A hypothesis is something which looks as if it might be true and were true, and which is capable of verification or refutation by comparison with facts. The best hypothesis, in the sense of the one most recommending itself to the inquirer, is the one which can be the most readily refuted if it is false." (Charles S Peirce, 1896)

"The scientific value of truth is not, however, ultimate or absolute. It rests partly on practical, partly on aesthetic interests. As our ideas are gradually brought into conformity with the facts by the painful process of selection, - for intuition runs equally into truth and into error, and can settle nothing if not controlled by experience, - we gain vastly in our command over our environment. This is the fundamental value of natural science" (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)

"It would be a mistake to suppose that a science consists entirely of strictly proved theses, and it would be unjust to require this. […] Science has only a few apodeictic propositions in its catechism: the rest are assertions promoted by it to some particular degree of probability. It is actually a sign of a scientific mode of thought to find satisfaction in these approximations to certainty and to be able to pursue constructive work further in spite of the absence of final confirmation." (Sigmund Freud, "Introductory Lectures on Psycho-Analysis", 1916)

"The mistake from which todays’ science suffers is that the theoreticians are concerned too unilaterally with precision mathematics, while the practitioners use a sort of approximate mathematics, without being in touch with precision mathematics through which they could reach a real approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of problems and the entities considered. Carried to an extreme, this procedure justifies the well-known joke that a mathematician is a scientist who knows neither what he is talking about nor whether whatever he is talking about exists or not." (Élie Cartan, 1940) 

"Men of science have made abundant mistakes of every kind; their knowledge has improved only because of their gradual abandonment of ancient errors, poor approximations, and premature conclusions." (George Sarton, "A History of Science" Vol 2, 1952)

"Nature is more subtle, more deeply intertwined and more strangely integrated than any of our pictures of her - than any of our errors. It is not merely that our pictures are not full enough; each of our pictures in the end turns out to be so basically mistaken that the marvel is that it worked at all." (Jacob Bronowski, "The Common Sense of Science", 1953) 

"Men of science have made abundant mistakes of every kind; their knowledge has improved only because of their gradual abandonment of ancient errors, poor approximations, and premature conclusions." (George Sarton, "A History of Science" Vol. 2, 1959)

"[…] the progress of science is a little like making a jig-saw puzzle. One makes collections of pieces which certainly fit together, though at first it is not clear where each group should come in the picture as a whole, and if at first one makes a mistake in placing it, this can be corrected later without dismantling the whole group." (Sir George Thomson, "The Inspiration of Science", 1961)

"All discoveries in art and science result from an accumulation of errors." (Marshall McLuhan, "Culture Is Our Business", 1970)

"Early scientific thinking was holistic, but speculative - the modern scientific temper reacted by being empirical, but atomistic. Neither is free from error, the former because it replaces factual inquiry with faith and insight, and the latter because it sacrifices coherence at the altar of facticity. We witness today another shift in ways of thinking: the shift toward rigorous but holistic theories. This means thinking in terms of facts and events in the context of wholes, forming integrated sets with their own properties and relationships." (Ervin László, "Introduction to Systems Philosophy", 1972)

"The point is that every experiment involves an error, the magnitude of which is not known beforehand and it varies from one experiment to another. For this reason, no matter what finite number of experiments have been carried out, the arithmetic mean of the values obtained will contain an error. Of course, if the experiments are conducted under identical conditions and the errors are random errors, then the error of the mean will diminish as the number of experiments is increased, but it cannot be reduced to zero for a finite number of experiments. […] The choice of entities for an experiment must be perfectly random, so that even an apparently inessential cause could not lead to erroneous conclusions." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"[…] in most sciences the question Why? is forbidden and the answer is actually to the question, How? Science is much better in explaining than in understanding, but it likes to mistake one for the other." (Erwin Chargaff, "Voices in the Labyrinth", Perspectives in Biology and Medicine Vol. 18, 1975)

"Early scientific thinking was holistic, but speculative - the modern scientific temper reacted by being empirical, but atomistic. Neither is free from error, the former because it replaces factual inquiry with faith and insight, and the latter because it sacrifices coherence at the altar of facticity. We witness today another shift in ways of thinking: the shift toward rigorous but holistic theories. This means thinking in terms of facts and events in the context of wholes, forming integrated sets with their own properties and relationships." (Ervin László, "Introduction to Systems Philosophy", 1972)

"Whatever humans have learned had to be learned as a consequence only of trial and error experience. Humans have learned only through mistakes." (Buckminster Fuller, "R Buckminster Fuller on education", 1979)

"What shows a theory to be inadequate or mistaken is not, as a rule, the discovery of a mistake in the information that led us to propound it; more often it is the contradictory evidence of a new observation which we were led to make because we held that theory." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Science usually amounts to a lot more than blind trial and error. Good statistics consists of much more than just significance tests; there are more sophisticated tools available for the analysis of results, such as confidence statements, multiple comparisons, and Bayesian analysis, to drop a few names. However, not all scientists are good statisticians, or want to be, and not all people who are called scientists by the media deserve to be so described." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Humans may crave absolute certainty; they may aspire to it; they may pretend, as partisans of certain religions do, to have attained it. But the history of science - by far the most successful claim to knowledge accessible to humans - teaches that the most we can hope for is successive improvement in our understanding, learning from our mistakes, an asymptotic approach to the Universe, but with the proviso that absolute certainty will always elude us. We will always be mired in error. The most each generation can hope for is to reduce the error bars a little, and to add to the body of data to which error bars apply." (Carl Sagan, "The Demon-Haunted World: Science as a Candle in the Dark", 1995)

"It is important to distinguish between the scientific concept of law as a generalization, and the social concept of law which is prescriptive and normative. A desire for tolerance in respecting the laws of different social systems must not lead us into the mistake of attributing volition to the entities of science or relativism to scientific laws." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"The human mind delights in finding pattern - so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 2010)

"No definition of a social problem is perfect, but there are two principal ways such definitions can be flawed. On the one hand, we may worry that a definition is too broad, that it encompasses more than it ought to include. That is, broad definitions identify some cases as part of the problem that we might think ought not to be included; statisticians call such cases false positives (that is, they mistakenly identify cases as part of the problem). On the other hand, a definition that is too narrow excludes cases that we might think ought to be included; these are false negatives (incorrectly identified as not being part of the problem)." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"One should never mistake pattern for meaning." (Iain Banks, "The Hydrogen Sonata",  2012)

"A theory is a supposition which we hope to be true, a hypothesis is a supposition which we expect to be useful; fictions belong to the realm of art; if made to intrude elsewhere, they become either make-believes or mistakes." (G Johnstone Stoney)

"Science with its strict analysis of the facts, its persevering search for new, more consummate truths, and its relentless struggle against discovered mistakes and prejudices - science must saturate all or technics, our culture, and everyday life." (Abram F Joffe)

On Mistakes, Blunders and Errors II: Statistics and Probabilities

“It is a capital mistake to theorize before you have all the evidence. It biases the judgment.” (Sir Arthur C Doyle, “A Study in Scarlet”, 1887)

“It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.” (Sir Arthur C Doyle, “The Adventures of Sherlock Holmes”, 1892)

"What real and permanent tendencies there are lie hid beneath the shifting superfices of chance, as it were a desert in which the inexperienced traveller mistakes the temporary agglomerations of drifting sand for the real configuration of the ground" (Francis Y Edgeworth, 1898)

"Some of the common ways of producing a false statistical argument are to quote figures without their context, omitting the cautions as to their incompleteness, or to apply them to a group of phenomena quite different to that to which they in reality relate; to take these estimates referring to only part of a group as complete; to enumerate the events favorable to an argument, omitting the other side; and to argue hastily from effect to cause, this last error being the one most often fathered on to statistics. For all these elementary mistakes in logic, statistics is held responsible." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"If the chance of error alone were the sole basis for evaluating methods of inference, we would never reach a decision, but would merely keep increasing the sample size indefinitely." (C West Churchman, "Theory of Experimental Inference", 1948)

"There are instances of research results presented in terms of probability values of ‘statistical significance’ alone, without noting the magnitude and importance of the relationships found. These attempts to use the probability levels of significance tests as measures of the strengths of relationships are very common and very mistaken." (Leslie Kish, "Some statistical problems in research design", American Sociological Review 24, 1959)

"Poor statistics may be attributed to a number of causes. There are the mistakes which arise in the course of collecting the data, and there are those which occur when those data are being converted into manageable form for publication. Still later, mistakes arise because the conclusions drawn from the published data are wrong. The real trouble with errors which arise during the course of collecting the data is that they are the hardest to detect." (Alfred R Ilersic, "Statistics", 1959)

"The rounding of individual values comprising an aggregate can give rise to what are known as unbiased or biased errors. [...]The biased error arises because all the individual figures are reduced to the lower 1,000 [...] The unbiased error is so described since by rounding each item to the nearest 1,000 some of the approximations are greater and some smaller than the original figures. Given a large number of such approximations, the final total may therefore correspond very closely to the true or original total, since the approximations tend to offset each other. [...] With biased approximations, however, the errors are cumulative and their aggregate increases with the number of items in the series." (Alfred R Ilersic, "Statistics", 1959)

"While it is true to assert that much statistical work involves arithmetic and mathematics, it would be quite untrue to suggest that the main source of errors in statistics and their use is due to inaccurate calculations." (Alfred R Ilersic, "Statistics", 1959)

"No observations are absolutely trustworthy. In no field of observation can we entirely rule out the possibility that an observation is vitiated by a large measurement or execution error. If a reading is found to lie a very long way from its fellows in a series of replicate observations, there must be a suspicion that the deviation is caused by a blunder or gross error of some kind. [...] One sufficiently erroneous reading can wreck the whole of a statistical analysis, however many observations there are." (Francis J Anscombe, "Rejection of Outliers", Technometrics Vol. 2 (2), 1960)

"The most important and frequently stressed prescription for avoiding pitfalls in the use of economic statistics, is that one should find out before using any set of published statistics, how they have been collected, analysed and tabulated. This is especially important, as you know, when the statistics arise not from a special statistical enquiry, but are a by-product of law or administration. Only in this way can one be sure of discovering what exactly it is that the figures measure, avoid comparing the non-comparable, take account of changes in definition and coverage, and as a consequence not be misled into mistaken interpretations and analysis of the events which the statistics portray." (Ely Devons, "Essays in Economics", 1961)

"The problem of error has preoccupied philosophers since the earliest antiquity. According to the subtle remark made by a famous Greek philosopher, the man who makes a mistake is twice ignorant, for he does not know the correct answer, and he does not know that he does not know it." (Félix Borel, "Probability and Certainty", 1963)

"He who accepts statistics indiscriminately will often be duped unnecessarily. But he who distrusts statistics indiscriminately will often be ignorant unnecessarily. There is an accessible alternative between blind gullibility and blind distrust. It is possible to interpret statistics skillfully. The art of interpretation need not be monopolized by statisticians, though, of course, technical statistical knowledge helps. Many important ideas of technical statistics can be conveyed to the non-statistician without distortion or dilution. Statistical interpretation depends not only on statistical ideas but also on ordinary clear thinking. Clear thinking is not only indispensable in interpreting statistics but is often sufficient even in the absence of specific statistical knowledge. For the statistician not only death and taxes but also statistical fallacies are unavoidable. With skill, common sense, patience and above all objectivity, their frequency can be reduced and their effects minimised. But eternal vigilance is the price of freedom from serious statistical blunders." (W Allen Wallis & Harry V Roberts, "The Nature of Statistics", 1965)

"The calculus of probability can say absolutely nothing about reality [...] We have to stress this point because these attempts assume many forms and are always dangerous. In one sentence: to make a mistake of this kind leaves one inevitably faced with all sorts of fallacious arguments and contradictions whenever an attempt is made to state, on the basis of probabilistic considerations, that something must occur, or that its occurrence confirms or disproves some probabilistic assumptions." (Bruno de Finetti, "Theory of Probability", 1974)

"Mistakes arising from retrospective data analysis led to the idea of experimentation, and experience with experimentation led to the idea of controlled experiments and then to the proper design of experiments for efficiency and credibility. When someone is pushing a conclusion at you, it's a good idea to ask where it came from - was there an experiment, and if so, was it controlled and was it relevant?" (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"There are no mistakes. The events we bring upon ourselves, no matter how unpleasant, are necessary in order to learn what we need to learn; whatever steps we take, they’re necessary to reach the places we’ve chosen to go." (Richard Bach, "The Bridge across Forever", 1984)

"Correlation and causation are two quite different words, and the innumerate are more prone to mistake them than most." (John A Paulos, "Innumeracy: Mathematical Illiteracy and its Consequences", 1988)

"When you want to use some data to give the answer to a question, the first step is to formulate the question precisely by expressing it as a hypothesis. Then you consider the consequences of that hypothesis, and choose a suitable test to apply to the data. From the result of the test you accept or reject the hypothesis according to prearranged criteria. This cannot be infallible, and there is always a chance of getting the wrong answer, so you try and reduce the chance of such a mistake to a level which you consider reasonable." (Roger J Barlow, "Statistics: A guide to the use of statistical methods in the physical sciences", 1989)

"Exploratory regression methods attempt to reveal unexpected patterns, so they are ideal for a first look at the data. Unlike other regression techniques, they do not require that we specify a particular model beforehand. Thus exploratory techniques warn against mistakenly fitting a linear model when the relation is curved, a waxing curve when the relation is S-shaped, and so forth." (Lawrence C Hamilton, "Regression with Graphics: A second course in applied statistics", 1991)

"Most statistical models assume error free measurement, at least of independent (predictor) variables. However, as we all know, measurements are seldom if ever perfect. Particularly when dealing with noisy data such as questionnaire responses or processes which are difficult to measure precisely, we need to pay close attention to the effects of measurement errors. Two characteristics of measurement which are particularly important in psychological measurement are reliability and validity." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"We can consider three broad classes of statistical pitfalls. The first involves sources of bias. These are conditions or circumstances which affect the external validity of statistical results. The second category is errors in methodology, which can lead to inaccurate or invalid results. The third class of problems concerns interpretation of results, or how statistical results are applied (or misapplied) to real world issues." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"This notion of 'being due' - what is sometimes called the gambler’s fallacy - is a mistake we make because we cannot help it. The problem with life is that we have to live it from the beginning, but it makes sense only when seen from the end. As a result, our whole experience is one of coming to provisional conclusions based on insufficient evidence: read ing the signs, gauging the odds." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Big numbers warn us that the problem is a common one, compelling our attention, concern, and action. The media like to report statistics because numbers seem to be 'hard facts' - little nuggets of indisputable truth. [...] One common innumerate error involves not distinguishing among large numbers. [...] Because many people have trouble appreciating the differences among big numbers, they tend to uncritically accept social statistics (which often, of course, feature big numbers)." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"Compound errors can begin with any of the standard sorts of bad statistics - a guess, a poor sample, an inadvertent transformation, perhaps confusion over the meaning of a complex statistic. People inevitably want to put statistics to use, to explore a number's implications. [...] The strengths and weaknesses of those original numbers should affect our confidence in the second-generation statistics." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

 "A major problem with many studies is that the population of interest is not adequately defined before the sample is drawn. Don’t make this mistake. A second major source of error is that the sample proves to have been drawn from a different population than was originally envisioned." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"The difference between 'statistically significant' and 'not statistically significant' is not in itself necessarily statistically significant. By this, I mean more than the obvious point about arbitrary divisions, that there is essentially no difference between something significant at the 0.049 level or the 0.051 level. I have a bigger point to make. It is common in applied research–in the last couple of weeks, I have seen this mistake made in a talk by a leading political scientist and a paper by a psychologist–to compare two effects, from two different analyses, one of which is statistically significant and one which is not, and then to try to interpret/explain the difference. Without any recognition that the difference itself was not statistically significant." (Andrew Gelman, "The difference between ‘statistically significant’ and ‘not statistically significant’ is not in itself necessarily statistically significant", 2005)

"[…] an outlier is an observation that lies an 'abnormal' distance from other values in a batch of data. There are two possible explanations for the occurrence of an outlier. One is that this happens to be a rare but valid data item that is either extremely large or extremely small. The other is that it isa mistake – maybe due to a measuring or recording error." (Alan Graham, "Developing Thinking in Statistics", 2006)

"Many scientists who work not just with noise but with probability make a common mistake: They assume that a bell curve is automatically Gauss's bell curve. Empirical tests with real data can often show that such an assumption is false. The result can be a noise model that grossly misrepresents the real noise pattern. It also favors a limited view of what counts as normal versus non-normal or abnormal behavior. This assumption is especially troubling when applied to human behavior. It can also lead one to dismiss extreme data as error when in fact the data is part of a pattern." (Bart Kosko, "Noise", 2006) 

"A naive interpretation of regression to the mean is that heights, or baseball records, or other variable phenomena necessarily become more and more 'average' over time. This view is mistaken because it ignores the error in the regression predicting y from x. For any data point xi, the point prediction for its yi will be regressed toward the mean, but the actual yi that is observed will not be exactly where it is predicted. Some points end up falling closer to the mean and some fall further." (Andrew Gelman & Jennifer Hill, "Data Analysis Using Regression and Multilevel/Hierarchical Models", 2007)

"If there is an outlier there are two possibilities: The model is wrong – after all, a theory is the basis on which we decide whether a data point is an outlier (an unexpected value) or not. The value of the data point is wrong because of a failure of the apparatus or a human mistake. There is a third possibility, though: The data point might not be an actual  outlier, but part of a (legitimate) statistical fluctuation." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"In error analysis the so-called 'chi-squared' is a measure of the agreement between the uncorrelated internal and the external uncertainties of a measured functional relation. The simplest such relation would be time independence. Theory of the chi-squared requires that the uncertainties be normally distributed. Nevertheless, it was found that the test can be applied to most probability distributions encountered in practice." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"Another kind of error possibly related to the use of the representativeness heuristic is the gambler’s fallacy, otherwise known as the law of averages. If you are playing roulette and the last four spins of the wheel have led to the ball’s landing on black, you may think that the next ball is more likely than otherwise to land on red. This cannot be. The roulette wheel has no memory. The chance of black is just what it always is. The reason people tend to think otherwise may be that they expect the sequence of events to be representative of random sequences, and the typical random sequence at roulette does not have five blacks in a row." (Jonathan Baron, "Thinking and Deciding" 4th Ed, 2008)

"[…] humans make mistakes when they try to count large numbers in complicated systems. They make even greater errors when they attempt - as they always do - to reduce complicated systems to simple numbers." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"There is a growing realization that reported 'statistically significant' claims in statistical publications  are routinely mistaken. Researchers typically express the confidence in their data in terms of p-value: the probability that a perceived result is actually the result of random variation. The value of p (for 'probability') is a way of measuring the extent to which a data set provides evidence against a so-called null hypothesis. By convention, a p- value below 0.05 is considered a meaningful refutation of the null hypothesis; however, such conclusions are less solid than they appear." (Andrew Gelman & Eric Loken, "The Statistical Crisis in Science", American Scientist Vol. 102(6), 2014)

"Using a sample to estimate results in the full population is common in data analysis. But you have to be careful, because even small mistakes can quickly become big ones, given that each observation represents many others. There are also many factors you need to consider if you want to make sure your inferences are accurate." (John H Johnson & Mike Gluck, "Everydata: The misinformation hidden in the little data you consume every day", 2016)

"The central limit conjecture states that most errors are the result of many small errors and, as such, have a normal distribution. The assumption of a normal distribution for error has many advantages and has often been made in applications of statistical models." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"Variance is error from sensitivity to fluctuations in the training set. If our training set contains sampling or measurement error, this noise introduces variance into the resulting model. [...] Errors of variance result in overfit models: their quest for accuracy causes them to mistake noise for signal, and they adjust so well to the training data that noise leads them astray. Models that do much better on testing data than training data are overfit." (Steven S Skiena, "The Data Science Design Manual", 2017)

"Statistical models have two main components. First, a mathematical formula that expresses a deterministic, predictable component, for example the fitted straight line that enables us to make a prediction [...]. But the deterministic part of a model is not going to be a perfect representation of the observed world [...] and the difference between what the model predicts, and what actually happens, is the second component of a model and is known as the residual error - although it is important to remember that in statistical modelling, ‘error’ does not refer to a mistake, but the inevitable inability of a model to exactly represent what we observe." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"If we don’t understand the statistics, we’re likely to be badly mistaken about the way the world is. It is all too easy to convince ourselves that whatever we’ve seen with our own eyes is the whole truth; it isn’t. Understanding causation is tough even with good statistics, but hopeless without them. [...] And yet, if we understand only the statistics, we understand little. We need to be curious about the world that we see, hear, touch, and smell, as well as the world we can examine through a spreadsheet." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"Premature enumeration is an equal-opportunity blunder: the most numerate among us may be just as much at risk as those who find their heads spinning at the first mention of a fraction. Indeed, if you’re confident with numbers you may be more prone than most to slicing and dicing, correlating and regressing, normalizing and rebasing, effortlessly manipulating the numbers on the spreadsheet or in the statistical package - without ever realizing that you don’t fully understand what these abstract quantities refer to. Arguably this temptation lay at the root of the last financial crisis: the sophistication of mathematical risk models obscured the question of how, exactly, risks were being measured, and whether those measurements were something you’d really want to bet your global banking system on." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)


"Always expect to find at least one error when you proofread your own statistics. If you don’t, you are probably making the same mistake twice." (Cheryl Russell)

29 March 2025

On Assertions

"Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"It would be a mistake to suppose that a science consists entirely of strictly proved theses, and it would be unjust to require this. […] Science has only a few apodeictic propositions in its catechism: the rest are assertions promoted by it to some particular degree of probability. It is actually a sign of a scientific mode of thought to find satisfaction in these approximations to certainty and to be able to pursue constructive work further in spite of the absence of final confirmation." (Sigmund Freud, "Introductory Lectures on Psycho-Analysis", 1916)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, “Mathematics in Western Culture”, 1953)

"For foundations it is important to know what we are talking about; we make the subject as specific as possible. In this way we have a chance to make strong assertions. For practice, to make a proof intelligible, we want to eliminate all properties which are not relevant to the result proved, in other words, we make the subject matter less specific." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"Most people imagine mathematics to be a deductive science in which all theorems, results and facts are obtained via logical reasoning by proceeding from certain starting axioms, primal assertions, assumed to be self-evident or not requiring any proof." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"The concepts a person uses are represented as points, and the causal links between these concepts are represented as arrows between these points. This gives a pictorial representation of the causal assertions of a person as a graph of points and arrows. This kind of representation of assertions as a graph will be called a cognitive map. The policy alternatives, all of the various causes and effects, the goals, and the ultimate utility of the decision maker can all be thought of as concept variables, and represented as points in the cognitive map. The real power of this approach ap pears when a cognitive map is pictured in graph form; it is then relatively easy to see how each of the concepts and causal relation ships relate to each other, and to see the overall structure of the whole set of portrayed assertions." (Robert Axelrod, "The Cognitive Mapping Approach to Decision Making" [in "Structure of Decision: The Cognitive Maps of Political Elites"], 1976)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)

"Virtually all mathematical theorems are assertions about the existence or nonexistence of certain entities. For example, theorems assert the existence of a solution to a differential equation, a root of a polynomial, or the nonexistence of an algorithm for the Halting Problem. A platonist is one who believes that these objects enjoy a real existence in some mystical realm beyond space and time. To such a person, a mathematician is like an explorer who discovers already existing things. On the other hand, a formalist is one who feels we construct these objects by our rules of logical inference, and that until we actually produce a chain of reasoning leading to one of these objects they have no meaningful existence, at all." (John L Casti, "Reality Rules: Picturing the world in mathematics" Vol. II, 1992)

"Sometimes in math the best way to make progress is to introduce simplifications [...]. The simplifications are not assertions about the outside world: they are ways to restrict the domain of discourse, to keep it manageable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"A theory is an organizational form of scientific knowledge about a certain set of objects, representing a system of interconnected assertions and proofs and containing methods of explanation and prediction of phenomena and processes in a given problem domain, i.e., of all phenomena and processes described by this theory." (Dmitry A Novikov, "Cybernetics: From Past to Future", 2016)

28 March 2025

Mental Models LXVII: On Causal Maps

"Causal maps are representations of individuals (or groups) beliefs about causal relations. They include elements, with only two kinds of properties. The first property is 'relevance'. The second  is the possibility of being in one (of two) 'influence relationships' (positive or negative) with one (of three) strengths (weak. moderate, or strong)." (Kivia Markoczy & Jeff Goldberg, "A method for eliciting and comparing causal maps", 1995)

"Short-term memory can hold 7 ± 2 chunks of information at once. This puts a rather sharp limit on the effective size and complexity of a causal map. Presenting a complex causal map all at once makes it hard to see the loops, understand which are important, or understand how they generate the dynamics. Resist the temptation to put all the loops you and your clients have identified into a single comprehensive diagram." (John D Sterman, "Business Dynamics Systems Thinking and Modeling for a Complex World", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"A causal map is an abstract representation of the causal relationships among kinds of objects and events in the world. Such relationships are not, for the most part, directly observable, but they can often be accurately inferred from observations. This includes both observations of patterns of contingency and correlation among events as well as observations of the effects of experimental interventions. We can think of everyday theories and theory-formation processes as cognitive systems that allow us to recover an accurate causal map of the world." (Alison Gopnik & Clark Glymour, "Causal maps and Bayes nets: a cognitive and computational account of theory-formation" [in "The cognitive basis of science"], 2002)

"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)

"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)

"Causal maps include elements called nodes, which are allowed to have causal relationships of different strengths of positive or negative loading depicted with a number, usually in the range of from 1 (weak) to 3 (strong). The relationships of the nodes are depicted with arcs or links labeled with the assumed polarity and loading factor or strength of causality, Links with positive polarity refer to dependency (when A increases B increases proportionally to the loading factor) and negative to inverse dependency (when A increases, B decreases)." (Hannu Kivijärvi et al, "A Support System for the Strategic Scenario Process", Encyclopedia of Decision Making and Decision Support Technologies, 2008)

"Fifth principle: (a) in finding solutions for systemic problems do not be content with symptomatic solutions but look for systemic-structural levers that can produce the more incisive effect; (b) if there are several systemic levers, choose the most efficient, that which produces the maximum effects with the minimum effort; (c) to activate the chosen structural lever identify the most effective decisional lever (action variable) taking into account the time necessary to produce the desired effect; (d) the choice of structural and decisional levers, as well as the intensity of the actions to modify their values, must follow from a careful construction, interpretation and assessment of the system’s causal map." (Piero Mella, "Systems Thinking: Intelligence in Action", 2012)

"(1) The causal maps are only models of a world of variables and processes; (2) They are models suitable for depicting that world only if they represent a logical image; (3) A logical image is made up of a network of arrows that depict the cause and effect connections among the variables and processes in the world; this network cannot be in contradiction to the world; (4) This depiction of the world relates to the boundaries between the represented and the external systems; the causal maps always depict a portion of a vaster world;" (Piero Mella, "Systems Thinking: Intelligence in Action", 2012)

"In constructing causal maps, whatever technique is adopted, there is always the problem of identifying or defining the system’s boundaries, either if we zoom in or broaden our perspective by zooming out." (Piero Mella, "Systems Thinking: Intelligence in Action", 2012)

"A Causal Map is hierarchical in structure (linking means to ends) and built with a focus on achieving goals. The process of creating the maps is ideally a group process and this in itself will add lots of value to a collective understanding of goals around EDI, what is required to achieve these and some of the potential challenges around this." (Nicola Morrill, "Supporting Your Efforts on Diversity", 2021)

Mental Models LXVI: On Cognitive Maps

"[…] learning consists not in stimulus-response connections but in the building up in the nervous system of sets which function like cognitive maps […] such cognitive maps may be usefully characterized as varying from a narrow strip variety to a broader comprehensive variety." (Edward C Tolman, "Cognitive maps in rats and men", 1948)

"A person is changed by the contingencies of reinforcement under which he behaves; he does not store the contingencies. In particular, he does not store copies of the stimuli which have played a part in the contingencies. There are no 'iconic representations' in his mind; there are no 'data structures stored in his memory'; he has no 'cognitive map' of the world in which he has lived. He has simply been changed in such a way that stimuli now control particular kinds of perceptual behavior." (Burrhus F Skinner, "About behaviorism", 1974)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"The concepts a person uses are represented as points, and the causal links between these concepts are represented as arrows between these points. This gives a pictorial representation of the causal assertions of a person as a graph of points and arrows. This kind of representation of assertions as a graph will be called a cognitive map. The policy alternatives, all of the various causes and effects, the goals, and the ultimate utility of the decision maker can all be thought of as concept variables, and represented as points in the cognitive map. The real power of this approach ap pears when a cognitive map is pictured in graph form; it is then relatively easy to see how each of the concepts and causal relation ships relate to each other, and to see the overall structure of the whole set of portrayed assertions." (Robert Axelrod, "The Cognitive Mapping Approach to Decision Making" [in "Structure of Decision: The Cognitive Maps of Political Elites"], 1976)

"The cognitive map is not a picture or image which 'looks like' what it represents; rather, it is an information structure from which map-like images can be reconstructed and from which behaviour dependent upon place information can be generated." (John O'Keefe & Lynn Nadel, "The Hippocampus as a Cognitive Map", 1978)

"A fuzzy cognitive map or FCM draws a causal picture. It ties facts and things and processes to values and policies and objectives. And it lets you predict how complex events interact and play out. [...] Neural nets give a shortcut to tuning an FCM. The trick is to let the fuzzy causal edges change as if they were synapses in a neural net. They cannot change with the same math laws because FCM edges stand for causal effect not signal flow. We bombard the FCM nodes with real data. The data state which nodes are on or off and to which degree at each moment in time. Then the edges grow among the nodes."  (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"Under the label 'cognitive maps', mental models have been conceived of as the mental representation of spatial aspects of the environment. A mental model, in this sense, comprises the topology of an area, including relevant districts, landmarks, and paths." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"Bounded rationality simultaneously constrains the complexity of our cognitive maps and our ability to use them to anticipate the system dynamics. Mental models in which the world is seen as a sequence of events and in which feedback, nonlinearity, time delays, and multiple consequences are lacking lead to poor performance when these elements of dynamic complexity are present. Dysfunction in complex systems can arise from the misperception of the feedback structure of the environment. But rich mental models that capture these sources of complexity cannot be used reliably to understand the dynamics. Dysfunction in complex systems can arise from faulty mental simulation-the misperception of feedback dynamics. These two different bounds on rationality must both be overcome for effective learning to occur. Perfect mental models without a simulation capability yield little insight; a calculus for reliable inferences about dynamics yields systematically erroneous results when applied to simplistic models." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Even if our cognitive maps of causal structure were perfect, learning, especially double-loop learning, would still be difficult. To use a mental model to design a new strategy or organization we must make inferences about the consequences of decision rules that have never been tried and for which we have no data. To do so requires intuitive solution of high-order nonlinear differential equations, a task far exceeding human cognitive capabilities in all but the simplest systems."  (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Eliciting and mapping the participant's mental models, while necessary, is far from sufficient [...] the result of the elicitation and mapping process is never more than a set of causal attributions, initial hypotheses about the structure of a system, which must then be tested. Simulation is the only practical way to test these models. The complexity of the cognitive maps produced in an elicitation workshop vastly exceeds our capacity to understand their implications. Qualitative maps are simply too ambiguous and too difficult to simulate mentally to provide much useful information on the adequacy of the model structure or guidance about the future development of the system or the effects of policies." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"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)

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