🦋Science: On Numbers (Quotes)

"Figures are not always facts." (Aesop, "The Widow and the Hen", cca. 6th century BC)

"Things that matter most
Must never be at the mercy of things that matter least.
The first sign we don’t know what we are doing is an obsession with numbers." (Johann Wolfgang von Goethe)

"Round numbers are always false." (Samuel Johnson, [Letter to Thomas Boswell], 1778)

"There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations." (Auguste Comte, “The Positive Philosophy”, 1830)

"There are two aspects of statistics that are continually mixed, the method and the science. Statistics are used as a method, whenever we measure something, for example, the size of a district, the number of inhabitants of a country, the quantity or price of certain commodities, etc. […] There is, moreover, a science of statistics. It consists of knowing how to gather numbers, combine them and calculate them, in the best way to lead to certain results. But this is, strictly speaking, a branch of mathematics." (Alphonse P de Candolle, "Considerations on Crime Statistics", 1833)

"If statistical graphics, although born just yesterday, extends its reach every day, it is because it replaces long tables of numbers and it allows one not only to embrace at glance the series of phenomena, but also to signal the correspondences or anomalies, to find the causes, to identify the laws." (Émile Cheysson, cca. 1877) 

"[…] when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of science." (William T Kelvin, "Electrical Units of Measurement", 1883)

"Statistics may, for instance, be called the science of counting. Counting appears at first sight to be a very simple operation, which any one can perform or which can be done automatically; but, as a matter of fact, when we come to large numbers, e.g., the population of the United Kingdom, counting is by no means easy, or within the power of an individual; limits of time and place alone prevent it being so carried out, and in no way can absolute accuracy be obtained when the numbers surpass certain limits." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may rightly be called the science of averages. […] Great numbers and the averages resulting from them, such as we always obtain in measuring social phenomena, have great inertia. […] It is this constancy of great numbers that makes statistical measurement possible. It is to great numbers that statistical measurement chiefly applies." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics is the name for that science and art which deals with uncertain inferences - which uses numbers to find out something about nature and experience." (Warren Weaver, 1952)

"Extrapolations are useful, particularly in the form of soothsaying called forecasting trends. But in looking at the figures or the charts made from them, it is necessary to remember one thing constantly: The trend to now may be a fact, but the future trend represents no more than an educated guess. Implicit in it is 'everything else being equal' and 'present trends continuing'. And somehow everything else refuses to remain equal." (Darell Huff, "How to Lie with Statistics", 1954)

"Quantitative performance measurements - whether single, multiple, or composite - are seen to have undesirable consequences for over-all organizational performance. The complexity of large organizations requires better knowledge of organizational behavior for managers to make best use of the personnel available to them." (V F Ridgway, "Dysfunctional Consequences of Performance Measurements", Administrative Science Quarterly Vol. 1 (2), 1956)

"The purpose of computing is insight, not numbers […] sometimes […] the purpose of computing numbers is not yet in sight." (Richard Hamming, "Numerical Methods for Scientists and Engineers", 1962)

"A well constructed numerical estimate can be worth a thousand words." (Charles L Schultze, 1967)

"What goes wrong [in long-range planning] is that sensible anticipation gets converted into foolish numbers: and their validity always hinges on large loose assumptions." (Robert Heller, "The Naked Manager: Games Executives Play", 1972)

"[...] be wary of analysts that try to quantify the unquantifiable." (Ralph Keeney & Raiffa Howard, "Decisions with Multiple Objectives: Preferences and Value Trade-offs", 1976)

"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)

"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)

"People often feel inept when faced with numerical data. Many of us think that we lack numeracy, the ability to cope with numbers. […] The fault is not in ourselves, but in our data. Most data are badly presented and so the cure lies with the producers of the data. To draw an analogy with literacy, we do not need to learn to read better, but writers need to be taught to write better." (Andrew Ehrenberg, "The problem of numeracy", American Statistician 35(2), 1981)

“Data in isolation are meaningless, a collection of numbers. Only in context of a theory do they assume significance […]” (George Greenstein, “Frozen Star”, 1983)

"Inept graphics also flourish because many graphic artists believe that statistics are boring and tedious. It then follows that decorated graphics must pep up, animate, and all too often exaggerate what evidence there is in the data. […] If the statistics are boring, then you've got the wrong numbers." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"A final goal of any scientific theory must be the derivation of numbers. Theories stand or fall, ultimately, upon numbers." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"The drudgery of the numbers will make you free." (Harold Geneen, "Managing", 1984)

"The professional's grasp of the numbers is a measure of the control he has over the events that the figures represent." (Harold Geneen, Managing, 1984)

"When you have mastered the numbers, you will in fact no longer be reading numbers, any more than you read words when reading a book. You will be reading meanings." (Harold Geneen & Alvin Moscow, "Managing", 1984)

"Numbers have undoubted powers to beguile and benumb, but critics must probe behind numbers to the character of arguments and the biases that motivate them." (Stephen J Gould, "An Urchin in the Storm: Essays About Books and Ideas", 1987)

"Whenever decisions are made strictly on the basis of bottom-line arithmetic, human beings get crunched along with the numbers." (Thomas R Horton, Management Review, 1987)

"When you are drowning in numbers you need a system to separate the wheat from the chaff." (Anthony Adams, The New York Times, 1988)

"Torture numbers, and they will confess to anything." (Gregg Easterbrook, New Republic, 1989)

"[…] you simply cannot make sense of any number without a contextual basis. Yet the traditional attempts to provide this contextual basis are often flawed in their execution. [...] Data have no meaning apart from their context. Data presented without a context are effectively rendered meaningless." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"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)

"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)

"Not all statistics start out bad, but any statistic can be made worse. Numbers - even good numbers - can be misunderstood or misinterpreted. Their meanings can be stretched, twisted, distorted, or mangled. These alterations create what we can call mutant statistics - distorted versions of the original figures." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"Every number has its limitations; every number is a product of choices that inevitably involve compromise. Statistics are intended to help us summarize, to get an overview of part of the world’s complexity. But some information is always sacrificed in the process of choosing what will be counted and how. Something is, in short, always missing. In evaluating statistics, we should not forget what has been lost, if only because this helps us understand what we still have." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)

"In much the same way, people create statistics: they choose what to count, how to go about counting, which of the resulting numbers they share with others, and which words they use to describe and interpret those figures. Numbers do not exist independent of people; understanding numbers requires knowing who counted what, why they bothered counting, and how they went about it." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)

"Data, reason, and calculation can only produce conclusions; they do not inspire action. Good numbers are not the result of managing numbers." (Ronald J Baker, "Measure what Matters to Customers: Using Key Predictive Indicators", 2006)

"Statistics can certainly pronounce a fact, but they cannot explain it without an underlying context, or theory. Numbers have an unfortunate tendency to supersede other types of knowing. […] Numbers give the illusion of presenting more truth and precision than they are capable of providing." (Ronald J Baker, "Measure what Matters to Customers: Using Key Predictive Indicators", 2006)

"Our culture, obsessed with numbers, has given us the idea that what we can measure is more important than what we can't measure. Think about that for a minute. It means that we make quantity more important than quality." (Donella Meadows, "Thinking in Systems: A Primer", 2008)

"What gets measured gets managed - even when it’s pointless to measure and manage it, and even if it harms the purpose of the organisation to do so." (Simon Caulkin, "The rule is simple: be careful what you measure", 2008) [source]

"What gets measured gets managed - so be sure you have the right measures, because the wrong ones kill." (Simon Caulkin, "The rule is simple: be careful what you measure", 2008) [source]

"By giving numbers a proper shape, by visually encoding them, the graphic has saved you time and energy that you would otherwise waste if you had to use a table that was not designed to aid your mind." (Alberto Cairo, "The Functional Art", 2011) 

"The value of having numbers - data - is that they aren't subject to someone else's interpretation. They are just the numbers. You can decide what they mean for you." (Emily Oster, "Expecting Better", 2013)

"One very common problem in data visualization is that encoding numerical variables to area is incredibly popular, but readers can’t translate it back very well." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)

"As long as measurements are abused as a tool of control, measuring will remain the weakest area in a manager’s performance." (Peter Drucker)

"If the statistics are boring, you've got the wrong numbers." (Edward Tufte)

"Nothing is so fallacious as facts, except figures." (George Canning) [attributed]

"Sometimes the numbers don’t explain everything. The numbers are not the business - they are symbols of the business." (Gerald Deitchle)

"Strategic planning is not strategic thinking. Indeed, strategic planning often spoils strategic thinking, causing managers to confuse real vision with the manipulation of numbers." (Henry Mintzberg)

🦋Science: On Prediction (Quotes)

"The aim of every science is foresight. For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past." (Auguste Compte, "Plan des travaux scientifiques nécessaires pour réorganiser la société", 1822)

"As a science progresses, its power of foresight rapidly increases, until the mathematician in his library acquires the power of anticipating nature, and predicting what will happen in circumstances which the eye of man has never examined." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"No matter how solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not contradict it, if we undertake to verify it . […] It is far better to foresee even without certainty than not to foresee at all." (Henri Poincaré, "The Foundations of Science", 1913)

"In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict facts of the same kind. The  catch in this criterion lies in the world 'simplest'." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"It is characteristic of a good scientific theory that it makes no more assumptions than are needed to explain the facts under consideration and predict a few more." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"[…] the statistical prediction of the future from the past cannot be generally valid, because whatever is future to any given past, is in tum past for some future. That is, whoever continually revises his judgment of the probability of a statistical generalization by its successively observed verifications and failures, cannot fail to make more successful predictions than if he should disregard the past in his anticipation of the future. This might be called the ‘Principle of statistical accumulation’." (Clarence I Lewis, "Mind and the World-Order: Outline of a Theory of Knowledge", 1929)

"Postulate 1. All chance systems of causes are not alike in the sense that they enable us to predict the future in terms of the past. Postulate 2. Constant systems of chance causes do exist in nature. Postulate 3. Assignable causes of variation may be found and eliminated."(Walter A Shewhart, "Economic Control of Quality of Manufactured Product", 1931)

"Rule 1. Original data should be presented in a way that will preserve the evidence in the original data for all the predictions assumed to be useful." (Walter A Shewhart, "Economic Control of Quality of Manufactured Product", 1931)

"Rule 2. Any summary of a distribution of numbers in terms of symmetric functions should not give an objective degree of belief in any one of the inferences or predictions to be made therefrom that would cause human action significantly different from what this action would be if the original distributions had been taken as evidence." (Walter A Shewhart, "Economic Control of Quality of Manufactured Product", 1931)

"Factual science may collect statistics, and make charts. But its predictions are, as has been well said, but past history reversed." (John Dewey, "Art as Experience", 1934)

"It is never possible to predict a physical occurrence with unlimited precision." (Max Planck, "A Scientific Autobiography", 1949)

"To say that observations of the past are certain, whereas predictions are merely probable, is not the ultimate answer to the question of induction; it is only a sort of intermediate answer, which is incomplete unless a theory of probability is developed that explains what we should mean by ‘probable’ and on what ground we can assert probabilities." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The world is not made up of empirical facts with the addition of the laws of nature: what we call the laws of nature are conceptual devices by which we organize our empirical knowledge and predict the future." (Richard B Braithwaite, "Scientific Explanation", 1953)

"The predictions of physical theories for the most part concern situations where initial conditions can be precisely specified. If such initial conditions are not found in nature, they can be arranged." (Anatol Rapoport, "The Search for Simplicity", 1956)

"Predictions, prophecies, and perhaps even guidance – those who suggested this title to me must have hoped for such-even though occasional indulgences in such actions by statisticians has undoubtedly contributed to the characterization of a statistician as a man who draws straight lines from insufficient data to foregone conclusions!" (John W Tukey, "Where do We Go From Here?", Journal of the American Statistical Association, Vol. 55, No. 289, 1960)

"Can there be laws of chance? The answer, it would seem should be negative, since chance is in fact defined as the characteristic of the phenomena which follow no law, phenomena whose causes are too complex to permit prediction." (Félix E Borel, "Probabilities and Life", 1962)

"[…] All predictions are statistical, but some predictions have such a high probability that one tends to regard them as certain." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"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)

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

"It is of course desirable to work with manageable models which maximize generality, realism, and precision toward the overlapping but not identical goals of understanding, predicting, and modifying nature. But this cannot be done."(Richard Levins, "The strategy of model building in population biology", American Scientist Vol. 54 (4), 1966)

"To make a prediction would mean (using the term in the sense we propose) to venture to try to 'guess', among the possible alternatives, the one that will occur. This is an attempt often made, not only by would-be magicians and prophets, but also by experts and such like who are inclined to precast the future in the forge of their fantasies). To make a 'prediction', therefore, would not entail leaving the domain of the logic of certainty, but simply including the statements and data which we assume ourselves capable of guessing, along with the ascertained truths and the collected data." (Bruno de Finetti, "Theory of Probability", 1974)

"The moment you forecast you know you’re going to be wrong, you just don’t know when and in which direction." (Edgar R Fiedler, 1977)

"Prediction of the future is possible only in systems that have stable parameters like celestial mechanics. The only reason why prediction is so successful in celestial mechanics is that the evolution of the solar system has ground to a halt in what is essentially a dynamic equilibrium with stable parameters. Evolutionary systems, however, by their very nature have unstable parameters. They are disequilibrium systems and in such systems our power of prediction, though not zero, is very limited because of the unpredictability of the parameters themselves. If, of course, it were possible to predict the change in the parameters, then there would be other parameters which were unchanged, but the search for ultimately stable parameters in evolutionary systems is futile, for they probably do not exist… Social systems have Heisenberg principles all over the place, for we cannot predict the future without changing it." (Kenneth E Boulding, Evolutionary Economics, 1981)

"Prediction can never be absolutely valid and therefore science can never prove some generalization or even test a single descriptive statement and in that way arrive at final truth." (Gregory Bateson, Mind and Nature: A necessary unity", 1988)

"We can predict only those things we set up to be predictable, not what we encounter in the real world of living and reactive processes." (Bill Mollison, "Permaculture: A Designers' Manual", 1988)

"A model is generally more believable if it can predict what will happen, rather than 'explain' something that has already occurred." (James R Thompson, "Empirical Model Building", 1989)

"The ability of a scientific theory to be refuted is the key criterion that distinguishes science from metaphysics. If a theory cannot be refuted, if there is no observation that will disprove it, then nothing can prove it - it cannot predict anything, it is a worthless myth." (Eric Lerner, "The Big Bang Never Happened", 1991)

"[…] the simplest hypothesis proposed as an explanation of phenomena is more likely to be the true one than is any other available hypothesis, that its predictions are more likely to be true than those of any other available hypothesis, and that it is an ultimate a priori epistemic principle that simplicity is evidence for truth." (Richard Swinburne, "Simplicity as Evidence for Truth", 1997)

"The point is that scientific descriptions of phenomena in all of these cases do not fully capture reality they are models. This is not a shortcoming but a strength of science much of the scientist's art lies in figuring out what to include and what to exclude in a model, and this ability allows science to make useful predictions without getting bogged down by intractable details." (Philip Ball," The Self-Made Tapestry: Pattern Formation in Nature", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"When a system is predictable, it is already performing as consistently as possible. Looking for assignable causes is a waste of time and effort. Instead, you can meaningfully work on making improvements and modifications to the process. When a system is unpredictable, it will be futile to try and improve or modify the process. Instead you must seek to identify the assignable causes which affect the system. The failure to distinguish between these two different courses of action is a major source of confusion and wasted effort in business today." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"When a process displays unpredictable behavior, you can most easily improve the process and process outcomes by identifying the assignable causes of unpredictable variation and removing their effects from your process." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"A smaller model with fewer covariates has two advantages: it might give better predictions than a big model and it is more parsimonious (simpler). Generally, as you add more variables to a regression, the bias of the predictions decreases and the variance increases. Too few covariates yields high bias; this called underfitting. Too many covariates yields high variance; this called overfitting. Good predictions result from achieving a good balance between bias and variance. […] fiding a good model involves trading of fit and complexity." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"The only way to look into the future is use theories since conclusive data is only available about the past." (Clayton Christensen et al, "Seeing What’s Next: Using the Theories of Innovation to Predict Industry Change", 2004)

"Most long-range forecasts of what is technically feasible in future time periods dramatically underestimate the power of future developments because they are based on what I call the 'intuitive linear' view of history rather than the 'historical exponential' view." (Ray Kurzweil, "The Singularity is Near", 2005)

"Complexity carries with it a lack of predictability different to that of chaotic systems, i.e. sensitivity to initial conditions. In the case of complexity, the lack of predictability is due to relevant interactions and novel information created by them." (Carlos Gershenson, "Understanding Complex Systems", 2011)

"The illusion that we understand the past fosters overconfidence in our ability to predict the future." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"[...] things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate." (Steven Strogatz, "The Joy of X: A Guided Tour of Mathematics, from One to Infinity", 2012)

"In common usage, prediction means to forecast a future event. In data science, prediction more generally means to estimate an unknown value. This value could be something in the future (in common usage, true prediction), but it could also be something in the present or in the past. Indeed, since data mining usually deals with historical data, models very often are built and tested using events from the past." (Foster Provost & Tom Fawcett, "Data Science for Business", 2013)

"In data science, a predictive model is a formula for estimating the unknown value of interest: the target. The formula could be mathematical, or it could be a logical statement such as a rule. Often it is a hybrid of the two." (Foster Provost & Tom Fawcett, "Data Science for Business", 2013)

"Under complexity science, the more interacting factors, the more unpredictable and irregular the outcome. To be succinct, the greater the complexity, the greater the unpredictability." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"Without precise predictability, control is impotent and almost meaningless. In other words, the lesser the predictability, the harder the entity or system is to control, and vice versa. If our universe actually operated on linear causality, with no surprises, uncertainty, or abrupt changes, all future events would be absolutely predictable in a sort of waveless orderliness." (Lawrence K Samuels, "Defense of Chaos", 2013)

"The problem of complexity is at the heart of mankind’s inability to predict future events with any accuracy. Complexity science has demonstrated that the more factors found within a complex system, the more chances of unpredictable behavior. And without predictability, any meaningful control is nearly impossible. Obviously, this means that you cannot control what you cannot predict. The ability ever to predict long-term events is a pipedream. Mankind has little to do with changing climate; complexity does." (Lawrence K Samuels, "The Real Science Behind Changing Climate", LewRockwell.com, August 1, 2014) 

"Models are formal structures represented in mathematics and diagrams that help us to understand the world. Mastery of models improves your ability to reason, explain, design, communicate, act, predict, and explore." (Scott E Page, "The Model Thinker", 2018)

🏷️Knowledge Representation: On Diagrams (Quotes)

"[…] the speculative propositions of mathematics do not relate to facts; […] all that we are convinced of by any demonstration in the science, is of a necessary connection subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical lines, I might affirm of this individual figure, that its three angles are equal to two right angles; but as the imperfection of my senses puts it out of my power to be, in any case, certain of the exact correspondence of the diagram which I delineate, with the definitions given in the elements of geometry, I never can apply with confidence to a particular figure, a mathematical theorem. On the other hand, it appears from the daily testimony of our senses that the speculative truths of geometry may be applied to material objects with a degree of accuracy sufficient for the purposes of life; and from such applications of them, advantages of the most important kind have been gained to society." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"They [diagrams] are designed not so much to allow of reference to particular numbers, which can be better had from printed tables of figures, as to exhibit to the eye the general results of large masses of figures which it is hopeless to attack in any other way than by graphical representation." (William S Jevons, [letter to Richard Hutton] 1863)

"[…] it must be noticed that these diagrams do not naturally harmonize with the propositions of ordinary life or ordinary logic. […] The great bulk of the propositions which we commonly meet with are founded, and rightly founded, on an imperfect knowledge of the actual mutual relations of the implied classes to one another. […] one very marked characteristic about these circular diagrams is that they forbid the natural expression of such uncertainty, and are therefore only directly applicable to a very small number of such propositions as we commonly meet with." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream, - not any particular existence, and yet not general. At that moment we are contemplating an icon." (Charles S Peirce, "On The Algebra of Logic : A Contribution to the Philosophy of Notation" in The American Journal of Mathematics 7, 1885)

“Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth.” (Charles S Peirce, “Kinds of Reasoning”, cca. 1896)

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"We imagine cases, place mental diagrams before our mind's eye, and multiply these cases, until a habit is formed of expecting that always to turn out the case, which has been seen to be the result in all the diagrams. To appeal to such a habit is a very different thing from appealing to any immediate instinct of rationality. That the process of forming a habit of reasoning by the use of diagrams is often performed there is no room for doubt. It is perfectly open to consciousness." (Charles S Peirce,"Fallibility of Reasoning and the Feeling of Rationality", cca. 1902)

"Arithmetical symbols are written diagrams and geometrical figures are graphic formulas." (David Hilbert, Bulletin of the American Mathematical Society, Mathematical Problems Vol. 8, 1902)

"A diagram is a representamen [representation] which is predominantly an icon of relations and is aided to be so by conventions. Indices are also more or less used. It should be carried out upon a perfectly consistent system of representation, founded upon a simple and easily intelligible basic idea." (Charles S Peirce, 1903)

"A diagram is an icon or schematic image embodying the meaning of a general predicate; and from the observation of this icon we are supposed to construct a new general predicate." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904) 

"A theorem […] is an inference obtained by constructing a diagram according to a general precept, and after modifying it as ingenuity may dictate, observing in it certain relations, and showing that they must subsist in every case, retranslating the proposition into general terms." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory; and there would soon be such an advance in logic that every science would feel the benefit of it." (Charles S Peirce, "Prolegomena to an Apology for Pragmaticism", Monist 16(4), 1906)

"To facilitate eyeless observation of his sense-transcending world, the mathematician invokes the aid of physical diagrams and physical symbols in endless variety and combination [...]" (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1907-1908, 1908)

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)

"We rise from the conception of form to an understanding of the forces which gave rise to it. [...] in the representation of form we see a diagram of forces in equilibrium, and in the comparison of kindred forms we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other." (D'Arcy Wentworth Thompson, "On Growth and Form" Vol. II, 1917)

"The preliminary examination of most data is facilitated by the use of diagrams. Diagrams prove nothing, but bring outstanding features readily to the eye; they are therefore no substitutes for such critical tests as may be applied to the data, but are valuable in suggesting such tests, and in explaining the conclusions founded upon them." (Sir Ronald A Fisher, "Statistical Methods for Research Workers", 1925)

"What the diagram has in common with the symbolic schema is the fact that the diagram spatially represents an abstract and unextended object. But there is here nothing other than a determinate location in space. This location serves as a mooring, an attachment, an orientation for our memory, but does not play any role in our thought." (Jean-Paul Sartre, "The Imaginary: A phenomenological psychology of the imagination", 1940)

"[…] statistical literacy. That is, the ability to read diagrams and maps; a 'consumer' understanding of common statistical terms, as average, per cent, dispersion, correlation, and index number." (Douglas Scates, “Statistics: The Mathematics for Social Problems”, 1943)

"When I undertake some geometrical research, I have generally a mental view of the diagram itself, though generally an inadequate or incomplete one, in spite of which it affords the necessary synthesis - a tendency which, it would appear, results from a training which goes back to my very earliest childhood." (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field”, 1949)

"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […] The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Polya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"The diagrams and circles aid the understanding by making it easy to visualize the elements of a given argument. They have considerable mnemonic value […] They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power." (Martin Gardner, "Logic Machines and Diagrams", 1958) 

"An information retrieval system is therefore defined here as any device which aids access to documents specified by subject, and the operations associated with it. The documents can be books, journals, reports, atlases, or other records of thought, or any parts of such records - articles, chapters, sections, tables, diagrams, or even particular words. The retrieval devices can range from a bare list of contents to a large digital computer and its accessories. The operations can range from simple visual scanning to the most detailed programming." (Brian C Vickery, "The Structure of Information Retrieval Systems", 1959)

"A model is a qualitative or quantitative representation of a process or endeavor that shows the effects of those factors which are significant for the purposes being considered. A model may be pictorial, descriptive, qualitative, or generally approximate in nature; or it may be mathematical and quantitative in nature and reasonably precise. It is important that effective means for modeling be understood such as analog, stochastic, procedural, scheduling, flow chart, schematic, and block diagrams." (Harold Chestnut, "Systems Engineering Tools", 1965) 

"A diagram thus enables us to discover the internal categorization which characterizes the information being processed in a much shorter time than does a map. […] A diagram permits the rapid and precise internal processing of information having three components, but it does not permit introducing the information into a universal system of visual memorization and geographic comparison. It is a closed graphic system, limited solely to the information being processed. […] In a diagram, one begins by attributing a meaning to the planar dimensions, then one plots the correspondences." (Jacques Bertin, "Semiology of graphics", 1967) 

"A graphic is a diagram when correspondences on the plane can be established among all elements of another component." (Jacques Bertin, "Semiology of graphics", 1967) 

"To analyse graphic representation precisely, it is helpful to distinguish it from musical, verbal and mathematical notations, all of which are perceived in a linear or temporal sequence. The graphic image also differs from figurative representation essentially polysemic, and from the animated image, governed by the laws of cinematographic time. Within the boundaries of graphics fall the fields of networks, diagrams and maps. The domain of graphic imagery ranges from the depiction of atomic structures to the representation of galaxies and extends into the spheres of topography and cartography."  (Jacques Bertin, "Semiology of graphics", 1967) 

"When the correspondences on the plane can be established between: - all the divisions of one component - and all the divisions of another component, the construction is a DIAGRAM." (Jacques Bertin, "Semiology of graphics", 1967) 

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"A diagram is worth a thousand proofs." (Carl E Linderholm, "Mathematics Made Difficult", 1971)

"The result of the implementation, the logical design, is traditionally shown as a series of block diagrams. These blocks represent in effect a series of statements, Actually, a direct presentation of these statements is more suitable and, although less familiar, more easily understood. The Harvard Mark IV was to large degree designed and described by such statements, as has been the case with several subsequent developments." (Gerrit Blaauw, "Computer Architecture", 1972) 

"Whether or not a given conceptual model or representation of a physical system happens to be picturable, is irrelevant to the semantics of the theory to which it eventually becomes attached. Picturability is a fortunate psychological occurrence, not a scientific necessity. Few of the models that pass for visual representations are picturable anyhow. For one thing, the model may be and usually is constituted by imperceptible items such as unextended particles and invisible fields. True, a model can be given a graphic representation - but so can any idea as long as symbolic or conventional diagrams are allowed. Diagrams, whether representational or symbolic, are meaningless unless attached to some body of theory. On the other hand theories are in no need of diagrams save for psychological purposes. Let us then keep theoretical models apart from visual analogues."  (Mario Bunge, "Philosophy of Physics", 1973)

"Pencil and paper for construction of distributions, scatter diagrams, and run-charts to compare small groups and to detect trends are more efficient methods of estimation than statistical inference that depends on variances and standard errors, as the simple techniques preserve the information in the original data." (W Edwards Deming, "On Probability as Basis for Action", American Statistician, Volume 29, Number 4, November 1975)

"The formalist makes a distinction between geometry as a deductive structure and geometry as a descriptive science. Only the first is regarded as mathematical. The use of pictures or diagrams, or even mental imagery, all are non- mathematical. In principle, they should be unnecessary. Consequently. he regards them as inappropriate in a mathematics text, perhaps even in a mathematics class." (Philip J Davis & Reuben Hersh, "The Mathematical Experience", 1981)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"[The diagram] is only an heuristic to prompt certain trains of inference; [...] it is dispensable as a proof-theoretic device; indeed, [...] it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array." (Neil Tennant, "The withering away of formal semantics", Mind and Language Vol. 1 (4), 1986)

"We distinguish diagrammatic from sentential paper-and-pencil representations of information by developing alternative models of information-processing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, like the propositions in a text. Diagrammatic representations are indexed by location in a plane. Diagrammatic representations also typically display information that is only implicit in sentential representations and that therefore has to be computed, sometimes at great cost, to make it explicit for use. We then contrast the computational efficiency of these representations for solving several. illustrative problems in mathematics and physics." (Herbert A. Simon, "Why a diagram is (sometimes) worth ten thousand words", 1987) 

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

"The value of diagram techniques even at this rudimentary level should be clear by now: it is easier to visualize where simplifications may be found in a complicated network by searching for a reducible linkage than by examining a complicated algebraic expression."(Geoffrey E Stedman, "Diagram Techniques in Group Theory", 1990)

"Diagrams are physical situations. They must be, since we can see them. As such, they obey their own set of constraints. […] By choosing a representational scheme appropriately, so that the constraints on the diagrams have a good match with the constraints on the described situation, the diagram can generate a lot of information that the user never need infer. Rather, the user can simply read off facts from the diagram as needed." (Jon Barwise & John Etchemendy, "Visual information and valid reasoning", [in "Visualization in Teaching and Learning Mathematics"], 1991)

"It has been said that the art of geometry is to reason well from false diagrams." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. We will use diagrams to depict mental models for a variety of concepts, and it is important to keep in mind that any diagram, or even a non-diagrammatic representation that represents the same essential relations would be equally effective." (Lyn D English & Graeme S Halford," "Mathematics Education: Models and Processes", 1995) 

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 1995)

"Given particular tasks of reasoning, different types of diagrams show different degrees of suitedness. For example, Euler diagrams are superior in handling certain problems concerning inclusion and membership among classes and individuals, but they cannot be generally applied to such problems without special provisos. Diagrams make many proofs in geometry shorter and more intuitive, while they take certain precautions of the reasoner's to be used validly. […] Mathematicians experience that coming up with the 'right' sorts of diagrams is more than half-way to the solution of most complicated problems." (Atsushi Shimojima, "Operational Constraints in Diagrammatic Reasoning" , [in "Logical Reasoning with Diagrams"], 1996)

"Making a good choice of representational conventions is always important in solving a problem, but especially true of charts. This sensitivity of type of chart to the particularities of the task at hand makes a very general logic of charts useless." (Jon Barwise & Eric Hammer, "Diagrams and the Concept of Logical System", [in "Logical Reasoning with Diagrams"], 1996)

"Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic.  In short, pictures can prove theorems." (James R Brown, "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures", 1999)

"Concept maps have long provided visual languages widely used in many different disciplines and application domains. Abstractly, they are sorted graphs visually represented as nodes having a type, name and content, some of which are linked by arcs. Concretely, they are structured diagrams having discipline- and domain-specific interpretations for their user communities, and, sometimes, formally defining computer data structures. Concept maps have been used for a wide range of purposes and it would be useful to make such usage available over the World Wide Web." (Brian R. Gaines, "WebMap: Concept Mapping on the Web", 2001) 

"Always remember that the model is not the diagram. The diagram’s purpose is to help communicate and explain the model. The code can serve as a repository of the details of the design." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)

"Data is transformed into graphics to understand. A map, a diagram are documents to be interrogated. But understanding means integrating all of the data. In order to do this it’s necessary to reduce it to a small number of elementary data. This is the objective of the 'data treatment' be it graphic or mathematic." (Jacques Bertin [interview], 2003)

"Diagrams are a means of communication and explanation, and they facilitate brainstorming. They serve these ends best if they are minimal. Comprehensive diagrams of the entire object model fail to communicate or explain; they overwhelm the reader with detail and they lack meaning." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. [… Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"Graphical design notations have been with us for a while [...] their primary value is in communication and understanding. A good diagram can often help communicate ideas about a design, particularly when you want to avoid a lot of details. Diagrams can also help you understand either a software system or a business process. As part of a team trying to figure out something, diagrams both help understanding and communicate that understanding throughout a team. Although they aren't, at least yet, a replacement for textual programming languages, they are a helpful assistant." (Martin Fowler," UML Distilled: A Brief Guide to the Standard Object Modeling", 2004)

"A diagram is a graphic shorthand. Though it is an ideogram, it is not necessarily an abstraction. It is a representation of something in that it is not the thing itself. In this sense, it cannot help but be embodied. It can never be free of value or meaning, even when it attempts to express relationships of formation and their processes. At the same time, a diagram is neither a structure nor an abstraction of structure." (Peter Eisenman, "Written Into the Void: Selected Writings", 1990-2004, 2007)

"A modeling language is usually based on some kind of computational model, such as a state machine, data flow, or data structure. The choice of this model, or a combination of many, depends on the modeling target. Most of us make this choice implicitly without further thinking: some systems call for capturing dynamics and thus we apply for example state machines, whereas other systems may be better specified by focusing on their static structures using feature diagrams or component diagrams. For these reasons a variety of modeling languages are available." (Steven Kelly & Juha-Pekka Tolvanen, "Domain-specific Modeling", 2008)

"[…] a conceptual model is a diagram connecting variables and constructs based on theory and logic that displays the hypotheses to be tested." (Mary Wolfinbarger Celsi et al, "Essentials of Business Research Methods", 2011)

"Diagrams furnish only approximate information. They do not add anything to the meaning of the data and, therefore, are not of much use to a statistician or research worker for further mathematical treatment or statistical analysis. On the other hand, graphs are more obvious, precise and accurate than the diagrams and are quite helpful to the statistician for the study of slopes, rates of change and estimation, (interpolation and extrapolation), wherever possible." (S C Gupta & Indra Gupta, "Business Statistics", 2013)

"Systems archetypes thus provide a good starting theory from which we can develop further insights into the nature of a particular system. The diagram that results from working with an archetype should not be viewed as the 'truth', however, but rather a good working model of what we know at any point in time." (Daniel H Kim, "Systems Archetypes as Dynamic Theories", The Systems Thinker Vol. 24 (1), 2013)

"As mathematics gets more abstract, diagrams become more and more prominent as the ways that things fit together abstractly become both more subtle and more important. Moreover, the diagram often sums up the situation more succinctly than the explanation in words, [..]" (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Sometimes mathematical advances happen by just looking at something in a slightly different way, which doesn’t mean building something new or going somewhere different, it just means changing your perspective and opening up huge new possibilities as a result. This particular insight leads to calculus and hence the understanding of anything curved, anything in motion, anything fluid or continuously changing." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Models are formal structures represented in mathematics and diagrams that help us to understand the world. Mastery of models improves your ability to reason, explain, design, communicate, act, predict, and explore." (Scott E Page, "The Model Thinker", 2018)