Emile Cheysson - Collected Quotes

"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 correspondence or anomalies, to find the causes, to identify the laws." (Émile Cheysson, circa 1877)

"It is this combination of observation at the foundation and geometry at the summit that I wished to express by naming this method Geometric Statistics. It cannot be subject to the usual criticisms directed at the use of pure mathematics in economic matters, which are said to be too complex to be confined within a formula." (Emile Cheysson, "La Statistique géométrique", 1888)

"It then becomes a method of graphical interpolation or extrapolation, which involves hypothetically extending a curve within or beyond the range of known data points, assuming the continuity of its pattern. In this way, one can fill in gaps in past observations and even probe the depths of the future." (Emile Cheysson, "La Statistique géométrique", 1888)

"This method is what I call Geometric Statistics. But despite its somewhat forbidding name-which I’ll explain in a moment - it is not a mathematical abstraction or a mere intellectual curiosity accessible only to a select few. It is intended, if not for all merchants and industrialists, then at least for that elite who lead the masses behind them. Practice is both its starting point and its destination. It was inspired in me more than fifteen years ago by the demands of the profession, and if I’ve decided to present it today, it’s because I’ve since verified its advantages through various applications, both in private industry and in public service." (Emile Cheysson, "La Statistique géométrique", 1888)

"Graphical statistics thus possess a variety of resources that it deploys depending on the case, in order to find the most expressive and visually appealing way to depict the phenomenon. One must especially avoid trying to convey too much at once and becoming obscure by striving for completeness. Its main virtue - or one might say, its true reason for being - is clarity. If a diagram becomes so cluttered that it loses its clarity, then it is better to use the numerical table it was meant to translate." (Emile Cheysson, "Albume de statistique graphique", 1889)

"This method not only has the advantage of appealing to the senses as well as to the intellect, and of illustrating facts and laws to the eye that would be difficult to uncover in long numerical tables. It also has the privilege of escaping the obstacles that hinder the easy dissemination of scientific work - obstacles arising from the diversity of languages and systems of weights and measures among different nations. These obstacles are unknown to drawing. A diagram is not German, English, or Italian; everyone immediately grasps its relationships of scale, area, or color. Graphical statistics are thus a kind of universal language, allowing scholars from all countries to freely exchange their ideas and research, to the great benefit of science itself." (Emile Cheysson, "Albume de statistique graphique", 1889)

"Today, there is hardly any field of human activity that does not make use of graphical statistics. Indeed, it perfectly meets a dual need of our time: the demand for information that is both rapid and precise. Graphical methods fulfill these two conditions wonderfully. They allow us not only to grasp an entire series of phenomena at a glance, but also to highlight relationships or anomalies, identify causes, and extract underlying laws. They advantageously replace long tables of numbers, so that - without compromising the precision of statistics - they broaden and popularize its benefits." (Emile Cheysson, "Albume de statistique graphique", 1889)

"When a law is contained in figures, it is buried like metal in an ore; it is necessary to extract it. This is the work of graphical representation. It points out the coincidences, the relationships between phenomena, their anomalies, and we have seen what a powerful means of control it puts in the hands of the statistician to verify new data, discover and correct errors with which they have been stained." (Emile Cheysson, "Les methods de la statistique", 1890)

Sources: Bibliothéque Nationale de la France [>>] 

A Picture's Worth

"The drawing shows me at a glance what would be spread over ten pages in a book." (Ivan Turgenev, 1862) [2]

"Sometimes, half a dozen figures will reveal, as with a lighting-flash, the importance of a subject which ten thousand labored words with the same purpose in view, had left at last but dim and uncertain." (Mark Twain, "Life on the Mississippi", 1883) 

"One good picture is worth many pages of written description." (William Sproston Caine, 1891) [2]

"One look is worth a thousand words" (Kathleen Caffyn, 1903) 

"Use a picture. It's worth a thousand words." (Arthur Brisbane, The Post-Standard, 1911)

"One Look Is Worth A Thousand Words" ([advertisement] 1913)

"A picture is worth ten thousand words. If you can’t see the truth in these pictures you are among the vast majority that must learn only by experience." (Arthur Brisbane, 1915)

"One picture is worth ten thousand words." (Frederick R Barnard, Printer’s Ink, 1921)

"One Picture Worth Ten Thousand Words" ([Chinese proverb] 1927)

"In many instances, a picture is indeed worth a thousand words. To make this true in more diverse circumstances, much more creative effort is needed to pictorialize the output from data analysis. Naive pictures are often extremely helpful, but more sophisticated pictures can be both simple and even more informative." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"One word is worth a thousand pictures. If it's the right word." (Edward Abbey, "Beyond the Wall: Essays from the Outside", 1984)

"A picture may be worth a thousand words, a formula is worth a thousand pictures." (Edsger Dijkstra, [conference at ETH Zurich] 1994)

"A magnificent picture is never worth a thousand perfect words." (John Dunning, "The Bookman's Wake", 1995)

"A picture tells a thousand words. But you get a thousand pictures from someone's voice." (Paul Fleischman, "Seek", 2001)

"If a picture is worth a thousand words, a metaphor is worth a thousand pictures." (Daniel H Pink, "A Whole New Mind: Why Right-Brainers Will Rule the Future", 2005)

"A picture may be worth a thousand words, but not all pictures are readable, interpretable, meaningful, or relevant." (Kristen Sosulski, "Data Visualization Made Simple: Insights into Becoming Visual", 2018)

"A good metaphor is worth a thousand pictures." (Anon) 

"As the Chinese say, 1001 words is worth more than a picture." (John McCarthy [source]) 

References:
[1] Wikipedia (2024) A picture is worth a thousand words [link]
[2] Quote Investigator (2022) A Picture Is Worth Ten Thousand Words [link] 
[3] SQL-Troubles (2024) Charts vs. Thousand Words [link]

Howard Wainer - Collected Quotes

"Although arguments can be made that high data density does not imply that a graphic will be good, nor one with low density bad, it does reflect on the efficiency of the transmission of information. Obviously, if we hold clarity and accuracy constant, more information is better than less. One of the great assets of graphical techniques is that they can convey large amounts of information in a small space." (Howard Wainer, "How to Display Data Badly", The American Statistician Vol. 38(2), 1984) 

"The essence of a graphic display is that a set of numbers having both magnitudes and an order are represented by an appropriate visual metaphor - the magnitude and order of the metaphorical representation match the numbers. We can display data badly by ignoring or distorting this concept." (Howard Wainer, "How to Display Data Badly", The American Statistician Vol. 38(2), 1984)

"The standard error of most statistics is proportional to 1 over the square root of the sample size. God did this, and there is nothing we can do to change it." (Howard Wainer, "Improving Tabular Displays, With NAEP Tables as Examples and Inspirations", Journal of Educational and Behavioral Statistics Vol 22 (1), 1997)

"[…] a graph is nothing but a visual metaphor. To be truthful, it must correspond closely to the phenomena it depicts: longer bars or bigger pie slices must correspond to more, a rising line must correspond to an increasing amount. If a graphical depiction of data does not faithfully follow this principle, it is almost sure to be misleading. But the metaphoric attachment of a graphic goes farther than this. The character of the depiction ism a necessary and sufficient condition for the character of the data. When the data change, so too must their depiction; but when the depiction changes very little, we assume that the data, likewise, are relatively unchanging. If this convention is not followed, we are usually misled." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"A graphic display has many purposes, but it achieves its highest value when it forces us to see what we were not expecting." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Nothing that had been produced before was even close. Even today, after more than two centuries of graphical experience, Playfair’s graphs remain exemplary standards for clearcommunication of quantitative phenomena. […] Graphical forms were available before Playfair, but they were rarely used to plot empirical information." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Oftentimes a statistical graphic provides the evidence for a plausible story, and the evidence, though perhaps only circumstantial, can be quite convincing. […] But such graphical arguments are not always valid. Knowledge of the underlying phenomena and additional facts may be required." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Placing a fact within a context increases its value greatly. […] . An efficacious way to add context to statistical facts is by embedding them in a graphic. Sometimes the most helpful context is geographical, and shaded maps come to mind as examples. Sometimes the most helpful context is temporal, and time-based line graphs are the obvious choice. But how much time? The ending date (today) is usually clear, but where do you start? The starting point determines the scale. […] The starting point and hence the scale are determined by the questions that we expect the graph to answer." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Simpson’s Paradox can occur whenever data are aggregated. If data are collapsed across a subclassification (such as grades, race, or age), the overall difference observed may not represent what is going on. Standardization can help correct this, but nothing short of random assignment of individuals to groups will prevent the possibility of yet another subclassificatiion, as yet unidentified, changing things around again. But I believe that knowing of the possibility helps us, so that we can contain the enthusiasm of our impulsive first inferences." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The appearance, and hence the perception, of any statistical graphic is massively influenced by the choice of scale. If the scale of the vertical axis is too narrow relative to the scale of the horizontal axis, random meanders look meaningful." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The difficult task of properly setting the scale of a graph remains difficult but not mysterious. There is agreement among experts spanning two hundred years. The default option should be to choose a scale that fills the plot with data. We can deviate from this under circumstances when it is better not to fill the plot with data, but those circumstances are usually clear. It is important to remember that the sin of using too small a scale is venial; the viewer can correct it. The sin of using too large a scale cannot be corrected without access to the original data; it can be mortal." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Usually the effectiveness of a good display increases with the complexity of the data. When there are only a few points, almost anything will do; even a pie chart with only three or four categories is usually comprehensible." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Thus when we look at, or prepare, a time-based statistical graphic, it is important to ask what is the right time scale, the right context, for the questions of greatest interest. The answer to this question is sometimes complex, but the very act of asking it provides us with some protection against surprises." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The only thing we know for sure about a missing data point is that it is not there, and there is nothing that the magic of statistics can do change that. The best that can be managed is to estimate the extent to which missing data have influenced the inferences we wish to draw." (Howard Wainer, "14 Conversations About Three Things", Journal of Educational and Behavioral Statistics Vol. 35(1, 2010)

"For an analyst to willfully avoid learning about the science is akin to malfeasance. Of course, it is likely that a deep understanding both of the science and of data analytic methods does not reside in the same person. When it does not, data analysis should be done jointly. It is my understanding that data mining is not often done as a team. This is unfortunate, for then it is too easy to miss what might have been found." (Howard Wainer, Comment, Journal of Computational and Graphical Statistics Vol. 20(1), 2011)

"Too often there is a disconnect between the people who run a study and those who do the data analysis. This is as predictable as it is unfortunate. If data are gathered with particular hypotheses in mind, too often they (the data) are passed on to someone who is tasked with testing those hypotheses and who has only marginal knowledge of the subject matter. Graphical displays, if prepared at all, are just summaries or tests of the assumptions underlying the tests being done. Broader displays, that have the potential of showing us things that we had not expected, are either not done at all, or their message is not able to be fully appreciated by the data analyst." (Howard Wainer, Comment, Journal of Computational and Graphical Statistics Vol. 20(1), 2011)

William Playfair - Collected Quotes

 "As knowledge increases amongst mankind and transactions multiply, it becomes more and more desirable to abbreviate and facilitate the modes of conveying information from one person to another, and from one individual to the many." (William Playfair, "The Commercial and Political Atlas", 1786)

"As to the propriety and justness of representing sums of money, and time, by parts of space, tho’ very readily agreed to by most men, yet a few seem to apprehend there may possibly be some deception in it, of which they are not aware […]" (William Playfair, "The Commercial and Political Atlas", 1786)

"Figures and letters may express with accuracy, but they can never represent either number or space. A map of the river Thames, or of a large town, expressed in figures, would give but a very imperfect notion of either, though they might be perfectly exact in every dimension." (William Playfair, "The Commercial and Political Atlas", 1786)

"Geography is only a branch of statistics, a knowledge of which is necessary to the well-understanding of the history of nations, as well as their situations relative to each other." (William Playfair, "The Commercial and Political Atlas", 1786)

"Information that is imperfectly acquired, is generally as imperfectly retained; and a man who has carefully investigated a printed table, finds, when done, that he has only a very faint and partial idea of what he has read; and that like a figure imprinted on sand, is soon totally erased and defaced." (William Playfair, "The Commercial and Political Atlas", 1786)

"Men in general are very slow to enter into what is reckoned a new thing; and there seems to be a very universal as well as great reluctance to undergo the drudgery of acquiring information that seems not to be absolutely necessary." (William Playfair, "The Commercial and Political Atlas", 1786) 

"[...] the eye is the best judge of proportion, being able to estimate it with more quickness and accuracy than any other of our organs [...] this mode of representation gives a simple, accurate, and permanent idea, by giving form and shape to a number of separate ideas, which are otherwise abstract and unconnected." (William Playfair, "The Commercial and Political Atlas", 1786)

"To give insight to statistical information it occurred to me, that making an appeal to the eye when proportion and magnitude are concerned, is the best and readiest method of conveying a distinct idea." (William Playfair, "The Commercial and Political Atlas", 1786)

"That I have succeeded in proposing and putting in practice a new and useful mode of stating accounts, has been so generally acknowledged, that it remains only for me to request that those who do not, at the first sight, understand the manner of inspecting the Charts, will read with attention the few lines of directions facing the first Chart, after which they will find all the difficulty entirely vanish, and as much, information may be obtained in five minutes as would require whole days to imprint on the memory, in a lasting manner, by a table of figures." (William Playfair, "The Commercial and Political Atlas", 1786)

"This Chart is different from the others in principle, as it does not comprehend any portion of time, and it is much inferior in utility to those that do; for though it gives the extent of the different branches of trade, it does not compare the same branch of commerce with itself at different periods; nor does it imprint upon the mind that distinct idea, in doing which, the chief advantage of Charts consists: for as it wants the dimension that is formed by duration, there is no shape given to the quantities." (William Playfair, "The Commercial and Political Atlas", 1786)

"In the course of executing that design, it occurred to me that tables are by no means a good form for conveying such information. [...] Making an appeal to the eye when proportion and magnitude are concerned is the best and readiest method of conveying a distinct idea." (William Playfair, "The Statistical Brewery", 1801)

"No study is less alluring or more dry and tedious than statistics, unless the mind and imagination are set to work, or that the person studying is particularly interested in the subject; which last can seldom be the case with young men in any rank of life." (William Playfair, "The Statistical Brewery", 1801)

"Statistical accounts are to be referred to as a dictionary by men of riper years, and by young men as a grammar, to teach them the relations and proportions of different statistical subjects, and to imprint them on the mind at a time when the memory is capable of being impressed in a lasting and durable manner, thereby laying the foundation for accurate and valuable knowledge." (William Playfair, "The Statistical Brewery", 1801)

"Statistical knowledge, though in some degree searched after in the most early ages of the world, has not till within these last 50 years become a regular object of study." (William Playfair, "The Statistical Brewery", 1801)

"The advantages proposed by [the graphical] mode of representation, are to facilitate the attainment of information, and aid the memory in retaining it: which two points form the principal business in what we call learning. Of all the senses, the eye gives the liveliest and most accurate idea of whatever is susceptible of being represented to it; and when proportion between different quantities is the object, then the eye has an incalculable superiority. (William Playfair, "The Statistical Breviary", 1801)

"Regarding numbers and proportions, the best way to catch the imagination is to speak to the eyes." (William Playfair, "Elemens de statistique", 1802)

On Graphics VI: Trivia

"Mathematicians have sought knowledge in figures, Philosophers in systems, Logicians in subtleties, and Metaphysicians in sounds. It is not in any nor in all of these. He that studies only men, will get the body of knowledge without the soul, and he that studies only books, the soul without the body." (Charles C Colton, "Lacon: Many Things in Few Words", 1820)

"[...] we can not readily break up a complicated problem into successive steps which can be taken independently. We have, in fact, to solve the problem first, by determining what are the actual mutual relations of the classes involved, and then to draw the circles to represent this final result; we cannot work step-by-step towards the conclusion by aid of our figures." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"Whereas in meaningful arithmetic equations and inequations are sentences expressing thoughts, in formal arithmetic they are comparable with the positions of chess pieces, transformed in accordance with certain rules without considerations for any sense. For if they were viewed as having sense, the rules could not be arbitrarily stipulated; they would have to be so chosen that from formulas expressing true propositions could be derived only formulas likewise expressing true propositions. Then the standpoint of formal arithmetic would have to be abandoned, which insists that the rules for the manipulation of signs are quite arbitrarily stipulated. Only subsequently may one ask whether the signs can be given a sense compatible with the rules previously laid down. Such matters, however, lie entirely outside formal arithmetic and only arise when applications are to be made. Then, however, they must be considered; for an arithmetic with no thought as its content will also be without possibility of application. Why can no application be made of a configuration of chess pieces? Obviously, because it expresses no thought. If it did so and every chess move conforming to the rules corresponded to a transition from one thought to another, applications of chess would also be conceivable. Why can arithmetical equations be applied? Only because they express thoughts. How could we possibly apply an equation which expressed nothing and was nothing more than a group of figures, to be transformed into another group of figures in accordance with certain rules? Now, it is applicability alone which elevates arithmetic from a game to the rank of a science. So applicability necessarily belongs to it. Is it good, then, to exclude from arithmetic what it needs in order to be a science?"  (Gottlob Frege, "Grundgesetze der Arithmetik" ["Basic Laws of Arithmetic"], 1893)

"[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract. […] I believe that the theory of numbers would be made more accessible, and would awaken more general interest, if it mere presented in connection with graphical elements and appropriate figures.”  (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1908)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"What in fact is the schema of the object? In one essential respect it is a schema belonging to intelligence. To have the concept of an object is to attribute the perceived figure to a substantial basis, so that the figure and the substance that it thus indicates continue to exist outside the perceptual field. The permanence of the object seen from this viewpoint is not only a product of intelligence, but constitutes the very first of those fundamental ideas of conservation which we shall see developing within the thought process." (Jean Piaget, "The Psychology of Intelligence", 1950)

"[Arithmetic] is another of the great master-keys of life. With it the astronomer opens the depths of the heavens; the engineer, the gates of the mountains; the navigator, the pathways of the deep. The skillful arrangement, the rapid handling of figures, is a perfect magician's wand." (Edward Everett)

Graphics IV: Geometry

"Of the figures, the first is especially scientific. The mathematical sciences carry out their demonstrations through it – e.g. arithmetic and geometry and optics – and so do almost all those sciences which inquire into the reason why. For deductions giving the reason why are carried out, either in general or for the most part and in most cases, through this figure. For this reason, then, it is especially scientific; for study of the reason why has most importance for knowledge." (Aristotle, "Posterior Analytics", cca. 350 BC) 

"The method of demonstration is therefore generally feeble and ineffective with regard to facts of nature (I refer to corporeal and changeable things). But it quickly recovers its strength when applied to the field of mathematics. For whatever it concludes in regard to such things as numbers, proportions and figures is indubitably true, and cannot be otherwise. One who wishes to become a master of the science of demonstration should first obtain a good grasp of probabilities. Whereas the principles of demonstrative logic are necessary; those of dialectic are probable." (John of Salisbury, "Metalogicon", 1159)

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." (Galileo Galilei, “The Assayer”, 1623)

"It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth." (George Berkeley, "The Analyst; Or, A Discourse Addressed to an Infidel Mathematician", 1734)

"The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"[…] 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)

"The analytical equations, unknown to the ancients, which Descartes first introduced into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they apply to all phenomena in general. There cannot be a language more universal and more simple, more free from errors and obscurities, that is to say, better adapted to express the invariable relations of nature." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur les Surfaces Élastiques", 1880)

"Certainly it is permitted to anyone to put forward whatever hypotheses he wishes, and to develop the logical consequences contained in those hypotheses. But in order that this work merit the name of Geometry, it is necessary that these hypotheses or postulates express the result of the more simple and elementary observations of physical figures." (Giuseppe Peano, "Sui fondamenti della geometria", 1894)

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"Many theorems are obvious upon looking at a moderately-sized figure; but the reasoning must be such as to convince the mind of their truth when, from excessive increase or diminution of the scale, the figures themselves have past the boundary even of imagination." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"A sign is a thing which is the representative, or deputy, of another thing for the purpose of affecting a mind. […] The utility of icons is evidenced by the diagrams of the mathematician, whether they involve continuity, like geometrical figures, or are arrays of discrete objects like a body of algebraical formulae, all of which are icons. Icons have to be used in all thinking." (Charles S Peirce, [manuscript] 1903)

"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our ideas and prevent our attention from wandering owing to the difficulty of keeping a long chain of syllogisms in our minds, it is excusable if we are apt to forget that we are not in reality reasoning about the objects in the figure, but about objects which ore their idealizations, and of which the objects in the figure are only an imperfect representation. Even if we only visualize, we see the images of more or less gross physical objects, in which various qualities irrelevant for our specific purpose are not entirely absent, and which are at best only approximate images of those objects about which we are reasoning." (Ernest W Hobson, "Squaring the Circle", 1913)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. [...] The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"Exact figures have, in principle, the same role in geometry as exact measurements in physics; but, in practice, exact figures are less important than exact measurements because the theorems of geometry are much more extensively verified than the laws of physics. The beginner, however, should construct many figures as exactly as he can in order to acquire a good experimental basis; and exact figures may suggest geometric theorems also to the more advanced. Yet, for the purpose of reasoning, carefully drawn free-hand figures are usually good enough, and they are much more quickly done." (George Pólya, "How to solve it", 1945)

"Two kinds of sets turn up in geometry. First of all, in geometry we ordinarily talk about the properties of some set of geometric figures. For example, the theorem stating that the diagonals of a parallelogram bisect each other relates to the set of all parallelograms. Secondly, the geometric figures are themselves sets composed of the points occurring within them. We can therefore speak of the set of all points contained within a given circle, of the set of all points within a given cone, etc." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

“We become quite convinced that a theorem is correct if we prove it on the basis of reasonably sound statements about numbers or geometrical figures which are intuitively more acceptable than the one we prove.” (Morris Kline, “Mathematics: The loss of certainty”, 1980)

"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is typical that there is more than one way of looking at a geometrical figure, just as there are many ways of looking at lines of algebra. Perception, 'seeing', is an essential feature of mathematics. This is obvious when we are looking for patterns - how can you possibly 'spot' a pattern if you cannot in some sense 'see' it? But it is just as true when the mathematician is looking for hidden connections, or studying a position in a mathematical game, searching for a tactical sequence, or trying to 'see' the possibilities clearly. Superficially, it might seem that it is only geometry (and related fields of mathematics) that depends on perception, but this is not so. Perception is everywhere in mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Geometry had its origins in the interest of working with lines, figures, and solids that could be imagined in the mind. Algebra had its origins in problems involving number - number hinged by geometric conceptions of iconic figures." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"We feel we are discovering mathematics. The results are there, waiting for us. They have been inherent in the figures all along. We are not inventing them. […] we are discovering facts that already exist, that are inherent in the objects we study. Although we have creative freedom to invent the objects themselves - to create idealizations like perfect spheres and circles and cylinders - once we do, they take on lives of their own." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

Graphics III: Topology

"After all the progress I have made in these matters, I am still not happy with Algebra, because it provides neither the shortest ways nor the most beautiful constructions of Geometry. This is why when it comes to that, I think that we need another analysis which is properly geometric or linear, which expresses to us directly situm, in the same way as algebra expresses magnitudinem. And I think that I have the tools for that, and that we might represent figures and even engines and motion in character, in the same way as algebra represents numbers in magnitude." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"I found the elements of a new characteristic, completely different from Algebra and which will have great advantages for the exact and natural mental representation, although without figures, of everything that depends on the imagination. Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"The use of figures is, above all, then, for the purpose of making known certain relations between the objects that we study, and these relations are those which occupy the branch of geometry that we have called Analysis Situs [that is, topology], and which describes the relative situation of points and lines on surfaces, without consideration of their magnitude." (Henri Poincaré, "Analysis Situs", Journal de l'Ecole Polytechnique 1, 1895)

“In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure.” (David Hilbert, “Geometry and Imagination”, 1952)

"[…] topology, a science that studies the properties of geometric figures that do not change under continuous transformations." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"[…] under plane transformations, like those encountered in the arbitrary stretching of a rubber sheet, certain properties of the figures involved are preserved. The mathematician has a name for them. They are called continuous transformations. This means that very close lying points pass into close lying points and a line is translated into a line under these transformations. Quite obviously, then, two intersecting lines will continue to intersect under a continuous transformation, and nonintersecting lines will not intersect; also, a figure with a hole cannot translate into a figure without a hole or into one with two holes, for that would require some kind of tearing or gluing - a disruption of the continuity." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Roughly speaking, manifolds are geometrical objects obtained by glueing open discs (balls) like a papier-mache is glued of small paper scraps. To this end, one first prepares a clay or plastecine figure which is then covered with several sheets of paper scraps glued onto one another. After the plasticine is removed, there remains a two-dimensional surface." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Two figures which can be transformed into one other by continuous deformations without cutting and pasting are called homeomorphic. […] The definition of a homeomorphism includes two conditions: continuous and one- to-one correspondence between the points of two figures. The relation between the two properties has fundamental significance for defining such a paramount concept as the dimension of space." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"[…] topology is concerned precisely with those properties of geometric figures that do not change when the figures are transformed. Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

On Sampling III

"The fact must be expressed as data, but there is a problem in that the correct data is difficult to catch. So that I always say 'When you see the data, doubt it!' 'When you see the measurement instrument, doubt it!' [...]For example, if the methods such as sampling, measurement, testing and chemical analysis methods were incorrect, data. […] to measure true characteristics and in an unavoidable case, using statistical sensory test and express them as data." (Kaoru Ishikawa, Annual Quality Congress Transactions, 1981)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences", Journal of the American Statistical Association Vol. 84, 1989)

"A little thought reveals a fact widely understood among statisticians: The null hypothesis, taken literally (and that’s the only way you can take it in formal hypothesis testing), is always false in the real world. [...] If it is false, even to a tiny degree, it must be the case that a large enough sample will produce a significant result and lead to its rejection. So if the null hypothesis is always false, what’s the big deal about rejecting it?" (Jacob Cohen,"Things I Have Learned (So Far)", American Psychologist, 1990)

"When looking at the end result of any statistical analysis, one must be very cautious not to over interpret the data. Care must be taken to know the size of the sample, and to be certain the method forg athering information is consistent with other samples gathered. […] No one should ever base conclusions without knowing the size of the sample and how random a sample it was. But all too often such data is not mentioned when the statistics are given - perhaps it is overlooked or even intentionally omitted." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"When the sample size is small or the study is of one organization, descriptive use of the thematic coding is desirable." (Richard Boyatzis, "Transforming qualitative information", 1998)

On Networks (1960-1969)

"Any pattern of activity in a network, regarded as consistent by some observer, is a system, Certain groups of observers, who share a common body of knowledge, and subscribe to a particular discipline, like 'physics' or 'biology' (in terms of which they pose hypotheses about the network), will pick out substantially the same systems. On the other hand, observers belonging to different groups will not agree about the activity which is a system." (Gordon Pask, The Natural History of Networks, 1960)

"Clearly, if the state of the system is coupled to parameters of an environment and the state of the environment is made to modify parameters of the system, a learning process will occur. Such an arrangement will be called a Finite Learning Machine, since it has a definite capacity. It is, of course, an active learning mechanism which trades with its surroundings. Indeed it is the limit case of a self-organizing system which will appear in the network if the currency supply is generalized." (Gordon Pask, "The Natural History of Networks", 1960)

"I am using the term 'network' in a general sense, to imply any set of interconnected and measurably active physical entities. Naturally occurring networks, of interest because they have a, self-organizing character, are, for example, a marsh, a colony of microorganisms, a research team, and a man." (Gordon Pask, "The Natural History of Networks", 1960)

"A NETWORK is a collection of connected lines, each of which indicates the movement of some quantity between two locations. Generally, entrance to a network is via a source (the starting point) and exit from a network is via a sink (the finishing point); the lines which form the network are called links (or arcs), and the points at which two or more links meet are called nodes." (Cecil W Lowe, "Critical Path Analysis by Bar Chart", 1966)

"In a network, one can plot the figures on a plane which has no meaning, and then look for the arrangement which produces the minimum number of intersections, or the simplest figure. After this transformation, the graphic will yield maximum efficiency, based on the discovery of a meaningful order expressed by the plane." (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)

On Diagrams (2000-2019)

"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) 

"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)

"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)

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

"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)

"In mathematics a good diagram is an invaluable aid to clear reasoning, whereas a bad one can seriously mislead. But it is not often that a diagram is good enough to suggest fresh avenues of inquiry altogether." (Anthony W F Edwards, "Cogwheels of the mind: The story of Venn diagrams", 2004)

"Good diagrams clarify. Very good diagrams force the ideas upon the viewer. The best diagrams compellingly embody the ideas themselves." (R W Oldford & W H Cherry, "Picturing Probability: the poverty of Venn diagrams, the richness of Eikosograms", 2006)

"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)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

"Diagrams are information graphics that are made up primarily of geometric shapes, such as rectangles, circles, diamonds, or triangles, that are typically (but not always) interconnected by lines or arrows. One of the major purposes of a diagram is to show how things, people, ideas, activities, etc. interrelate and interconnect. Unlike quantitative charts and graphs, diagrams are used to show interrelationships in a qualitative way." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"What advantages do diagrams have over verbal descriptions in promoting system understanding? First, by providing a diagram, massive amounts of information can be presented more efficiently. A diagram can strip down informational complexity to its core - in this sense, it can result in a parsimonious, minimalist description of a system. Second, a diagram can help us see patterns in information and data that may appear disordered otherwise. For example, a diagram can help us see mechanisms of cause and effect or can illustrate sequence and flow in a complex system. Third, a diagram can result in a less ambiguous description than a verbal description because it forces one to come up with a more structured description." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"[…] 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)

"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) 

On Diagrams (1950-1974)

"The diagrams incorporate a large amount of information. Their use provides extensive savings in space and in mental effort. In the case of many theorems, the setting up of the correct diagram is the major part of the proof. We therefore urge that the reader stop at the end of each theorem and attempt to construct for himself the relevant diagram before examining the one which is given in the text. Once this is done, the subsequent demonstration can be followed more readily; in fact, the reader can usually supply it himself." (Samuel Eilenberg & Norman E. Steenrod, "Foundations of Algebraic Topology", 1952)

"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)

"Diagrams are sometimes used, not merely to convey several pieces of information such as several time series on one chart, but also to provide visual evidence of relationships between the series." (Alfred R Ilersic, "Statistics", 1959)

"It is extremely helpful to the imagination to have a geometric picture available in terms of which we can visualize sets and operations on sets. […] Diagrammatic thought of this kind is admittedly loose and imprecise; nevertheless, the reader will find it invaluable. No mathematics, however abstract it may appear, is ever carried on without the help of mental images of some kind, and these are often nebulous, personal, and difficult to describe." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"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)

"Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows." (Saunders Mac Lane, "Categories for the Working Mathematician", 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)

Jacques Bertin - Collected Quotes

"A graphic should not only show the leaves, it should show the branches as well as the entire tree." (Jacques Bertin, "The Semiology of Graphics", 1967)

"Graphic representation constitutes one of the basic sign-systems conceived by the human mind for the purposes of storing, understanding, and communicating essential information. As a "language" for the eye, graphics benefits from the ubiquitous properties of visual perception. As a monosemic system, it forms the rational part of the world of images. […] Graphics owes its special significance to its double function as a storage mechanism and a research instrument."  (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The aim of the graphic is to make the relationship among previously defined sets appear." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The great difference between the graphic representation of yesterday, which was poorly dissociated from the figurative image, and the graphics of tomorrow, is the disappearance of the congential fixity of the image. […] When one can superimpose, juxtapose, transpose, and permute graphic images in ways that lead to groupings and classings, the graphic image passes from the dead image, the 'illustration,' to the living image, the widely accessible research instrument it is now becoming. The graphic is no longer only the 'representation' of a final simplification, it is a point of departure for the discovery of these simplifications and the means for their justification. The graphic has become, by its manageability, an instrument for information processing." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The plane is the mainstay of all graphic representation. It is so familiar that its properties seem self-evident, but the most familiar things are often the most poorly understood. The plane is homogeneous and has two dimensions. The visual consequences of these properties must be fully explored." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The problem that still remains to be solved is that of the orderable matrix, that needs the use of imagination […] When the two components of a data table are orderable, the normal construction is the orderable matrix. Its permutations show the analogy and the complementary nature that exist between the algorithmic treatments and the graphical treatments." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"There are as many types of questions as components in the information." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 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, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"As with any graphic, networks are used in order to discover pertinent troups of to inform others of the groups and structures dis(Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)overed. It is a good means of displaying structures, However, it ceases to be a means of discovery when the elements are numerous. The figure rapidly becomes complex, illegible and untransformable." (Jacques Bertin, "Graphics and graphic information processing", 1977)

"Computers are able to multiply useless images without taking into account that, by definition, every graphic corresponds to a table. This table allows you to think about three basic questions that go from the particular to the general level. When this last one receives an answer, you have answers for all of them. Understanding means accessing the general level and discovering significant grouping (patterns). Consequently, the function of a graphic is answering the three following questions:

Which are the X,Y, Z components of the data table? (What it’s all about?)

What are the groups in X, in Y that Z builds? (What the information at the general level is?

What are the exceptions?

These questions can be applied to every kind of problem. They measure the usefulness of whatever construction or graphical invention allowing you to avoid useless graphics." (Jacques Bertin, [interview] 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)

"The use of computers shouldn't ignore the objectives of graphics, that are: (1) Treating data to get information. (2) Communicating, when necessary, the information obtained." (Jacques Bertin, [interview] 2003)

"Graphics is the visual means of resolving logical problems." (Jacques Bertin, "Graphics and Graphic Information Processing", 2011)

James H Jeans - Collected Quotes

"The concepts which now prove to be fundamental to our understanding of nature- a space which is finite; a space which is empty, so that one point [of our 'material' world] differs from another solely in the properties of space itself; four-dimensional, seven- and more dimensional spaces; a space which for ever expands; a sequence of events which follows the laws of probability instead of the law of causation - or alternatively, a sequence of events which can only be fully and consistently described by going outside of space and time - all these concepts seem to my mind to be structures of pure thought, incapable of realisation in any sense which would properly be described as material." (James Jeans, "The Mysterious Universe", 1930)

"The final truth about phenomena resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge is complete. We go beyond the mathematical formula at our own risk; we may find a [nonmathematical] model or picture that helps us to understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault." (James Jeans, "The Mysterious Universe", 1930)

"Today there is a wide measure of agreement, which on the physical side of science approaches almost to unanimity, that the stream of knowledge is heading towards a non-mechanical reality; the universe begins to look more like a great thought than like a great machine. Mind no longer appears as an accidental intruder into the realm of matter; we are beginning to suspect that we ought rather to hail it as a creator and governor of the realm of matter [...]" (James Jeans, "The Mysterious Universe", 1930)

"In brief, a mathematical formula can never tell us what a thing is, but only how it behaves; it can only specify an object through its properties. And these are unlikely to coincide in toto with the properties of any single macroscopic object of our everyday life.” (James H Jeans, "The Mysterious Universe", 1930)

"The essential fact is simply that all the pictures which science now draws of nature, and which alone seem capable of according with observational fact, are mathematical pictures." (James H Jeans, "The Mysterious Universe", 1930)

"The making of models or pictures to explain mathematical formulae and the phenomena they describe is not a step towards, but a step away from reality; it is like making graven images of a spirit." (James H Jeans, "The Mysterious Universe", 1930)

"[…] our knowledge of the external world must always consist of numbers, and our picture of the universe - the synthesis of our knowledge - must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols - they do not belong to Nature, but to the parables by which we try to make Nature comprehensible." (James H Jeans, "The New World-Picture of Modern Physics", Supplement to Nature Vol. 134 (3384), 1934)

“A science which confines itself to correlating the phenomena can never learn anything about the reality underlying the phenomena, while a science which goes further than this, and introduces hypotheses about reality can never acquire certain knowledge of a positive kind about reality […]” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“Exhaustive studies by many investigators have shown that the fundamental laws of nature do not control the phenomena directly. We must picture them as operating in a substratum of which we can form no mental picture unless we are willing to introduce a number of irrelevant and therefore unjustifiable suppositions.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“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)

“It was supposed that a model which reproduced all the phenomena of a science, and so made it possible to predict them all, must in some way correspond to the reality underlying the phenomena. But obviously this cannot be so. After one perfect model had been found, a second of equal perfection might appear, and as both models could not correspond to reality, we should have at least one perfect model which did not correspond to reality. Thus we could never be sure that any model corresponded to reality. In brief, we can never have certain knowledge as to the nature of reality.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“[…] many philosophers have found it difficult to accept the hypothesis that an object is just about what it appears to be, and so is like the mental picture it produces in our minds. For an object and a mental picture are of entirely different natures - a brick and the mental picture of a brick can at best no more resemble one another than an orchestra and a symphony. In any case, there is no compelling reason why phenomena - the mental visions that a mind constructs out of electric currents in a brain - should resemble the objects that produced these currents in the first instance.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“Nothing is left in the world but happenings for which no explanation or interpretation is offered or even attempted, and science has now for its single aim the discovery of the laws to which these happenings conform - the pattern of events.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“Physicists who are trying to understand nature may work in many different fields and by many different methods; one may dig, one may sow, one may reap. But the final harvest will always be a sheaf of mathematical formulae. These will never describe nature itself, hut only our observations on nature. Our studies can never put us into contact with reality; we can never penetrate beyond the impressions that reality implants in our minds.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

"[…] physics tries to discover the pattern of events which controls the phenomena we observe. But we can never know what this pattern means or how it originates; and even if some superior intelligence were to tell us, we should find the explanation unintelligible. Our studies can never put us into contact with reality, and its true meaning and nature must be for ever hidden from us." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

“Science usually advances by a succession of small steps, through a fog in which even the most keen-sighted explorer can seldom see more than a few paces ahead. Occasionally the fog lifts, an eminence is gained, and a wider stretch of territory can be surveyed - sometimes with startling results. A whole science may then seem to undergo a kaleidoscopic ‘rearrangement’, fragments of knowledge being found to fit together in a hitherto unsuspected manner. Sometimes the shock of readjustment may spread to other sciences; sometimes it may divert the whole current of human thought.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“The pictures we draw of nature show similar limitations; these are the price we pay for limiting our pictures of nature to the kinds that can be understood by our minds. As we cannot draw one perfect picture, we make two imperfect pictures and turn to one or the other according as we want one property or another to be accurately delineated. Our observations tell us which is the right picture to use for each particular purpose […] . Yet some properties of nature are so far-reaching and general that neither picture can depict them properly of itself. In such cases we must appeal to both pictures, and these sometimes give us different and inconsistent information. Where, then, shall we find the truth?” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

 “Those who are content with a positivist conception of the aims of science will feel that he is in an entirely satisfactory position; he has discovered the pattern of events, and so can predict accurately; what more can he want? A mental picture would be an added luxury, but also a useless luxury. For if the picture did not bear any resemblance at all to the reality it would be valueless, and if it did it would be unintelligible […]” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

“When two hypotheses are possible, we provisionally choose that which our minds adjudge to the simpler on the supposition that this Is the more likely to lead in the direction of the truth.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

 “Whenever a man increases the content of his mind he gains new knowledge, and this occurs each time a new relation is established between the worlds on the two sides of the sense-organs - the world of ideas in an individual mind, and the world of objects existing outside individual minds which is common to us all.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

 “Yet a review of receipt physics has shown that all attempts at mechanical models or pictures have failed and must fail. For a mechanical model or picture must represent things as happening in space and time, while it has recently become clear that the ultimate processes of nature neither occur in, nor admit of representation in, space and time. Thus an understanding of the ultimate processes of nature is for ever beyond our reach: we shall never be able - even in imagination - to open the case of our watch and see how the wheels go round. The true object of scientific study can never be the realities of nature, but only our own observations on nature.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

"All the pictures which science draws of Nature, and which alone seem capable of according with observational facts, are mathematical pictures." (Sir James Jeans)

On Truth (1920-1929)

"The terrible thing about the quest for truth is that you find it." (Rémy de Gourmont, "Philosophic Nights in Paris", 1920)

"It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability." (John M Keynes, "A Treatise on Probability", 1921)

"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)

“We all know that Art is not truth. Art is a lie that makes us realize truth.” (Pablo Picasso, “The Arts”, 1923)

“Science does not aim at establishing immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved.” (Bertrand Russell, “The ABC of Relativity”, 1925)

"Progress in truth - truth of science and truth of religion - is mainly a progress in the framing of concepts, in discarding artificial abstractions or partial metaphors, and in evolving notions which strike more deeply into the root of reality." (Alfred N Whitehead, "Religion in the Making", 1926)

"The scientist is a lover of truth for the very love of truth itself, wherever it may lead." (Luther Burbank, "Why I Am An Infidel", 1926)

“If our so-called facts are changing shadows, they are shadows cast by the light of constant truth.” (Sir Arthur S Eddington, “Science and the Unseen World”, 1929) 

“Try to be conspicuously accurate in everything, pictures as well as text. Truth is not only stranger than fiction, it is more interesting.” (William R Hearst, “Letter of Instruction to Hearst Publishers”, 1929)

On Theorems (1990-1999)

"[…] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation." (Ivars Peterson, "Islands of Truth", 1990)

"A distinctive feature of mathematics, that feature in virtue of which it stands as a paradigmatically rational discipline, is that assertions are not accepted without proof. […] By proof is meant a deductively valid, rationally compelling argument which shows why this must be so, given what it is to be a triangle. But arguments always have premises so that if there are to be any proofs there must also be starting points, premises which are agreed to be necessarily true, self-evident, neither capable of, nor standing in need of, further justification. The conception of mathematics as a discipline in which proofs are required must therefore also be a conception of a discipline in which a systematic and hierarchical order is imposed on its various branches. Some propositions appear as first principles, accepted without proof, and others are ordered on the basis of how directly they can be proved from these first principle. Basic theorems, once proved, are then used to prove further results, and so on. Thus there is a sense in which, so long as mathematicians demand and provide proofs, they must necessarily organize their discipline along lines approximating to the pattern to be found in Euclid's Elements." (Mary Tiles, "Mathematics and the Image of Reason", 1991)

"[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less 'beautiful'. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula." (Jean Dieudonné, "Mathematics – The Music of Reason", 1992)

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics",  1995)

"Mathematics is about theorems: how to find them; how to prove them; how to generalize them; how to use them; how to understand them. […] But great theorems do not stand in isolation; they lead to great theories. […] And great theories in mathematics are like great poems, great paintings, or great literature: it takes time for them to mature and be recognized as being 'great'." (John L Casti, "Five Golden Rules", 1995)

"To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not ‘more’ beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics.” (Calvin C Clawson, “Mathematical Mysteries”, 1996)

"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", 1997)

"Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net." (John Polkinghorne, "Belief in God in an Age of Science", 1998)

"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so – and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist, Sept.–Oct. 1999)

"Let us regard a proof of an assertion as a purely mechanical procedure using precise rules of inference starting with a few unassailable axioms. This means that an algorithm can be devised for testing the validity of an alleged proof simply by checking the successive steps of the argument; the rules of inference constitute an algorithm for generating all the statements that can be deduced in a finite number of steps from the axioms." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"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)

See also:
Theorems I, II, III, IV, V, VI, VII, VIII, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX