10 November 2018

On Graphics I: Go, Figures!

"If one, with the scientist, studies the works of nature, which are made up of elements or matter and form, his reasoning is dependent on the data provided by sense-experience. And if one, with the mathematician, abstracts figures or calculates numerically, he must, in order to gain assent, accurately adduce many examples of both differentiated plurality and quantitative extension. The like holds true of the philosopher, whose domain is [abstract] reasoning, and who is the client of both the scientist and the mathematician. For the philosopher, too, begins with those things which are based on the evidence of the senses and contribute to the knowledge of immaterial intelligibles." (John of Salisbury, "Metalogicon", 1159)

"A mathematician, as good as he may be, without the support of a good drawing, is nothing but a half-mathematician, but also a man without eyes." (Lodovico Cardi, [letter to Galileo Galilei] 1611)

"The process by which I propose to accomplish this is one essentially graphical; by which term I understand not a mere substitution of geometrical construction and measurement for numerical calculation, but one which has for its object to perform that which no system of calculation can possibly do, by bringing in the aid of the eye and hand to guide the judgment, in a case where judgment only, and not calculation, can be of any avail. (John F W Herschel, "On the investigation of the orbits of revolving double stars: Being a supplement to a paper entitled 'micrometrical measures of double stars'", Memoirs of the Royal Astronomical Society  1833)

"Diagrams are of great utility for illustrating certain questions of vital statistics by conveying ideas on the subject through the eye, which cannot be so readily grasped when contained in figures." (Florence Nightingale, "Mortality of the British Army", 1857)

"The dominant principle which characterizes my graphic tables and my figurative maps is to make immediately appreciable to the eye, as much as possible, the proportions of numeric results. […] Not only do my maps speak, but even more, they count, they calculate by the eye." (Chatles D Minard, "Des tableaux graphiques et des cartes figuratives", 1862) 

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

"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)

"The graphical method has considerable superiority for the exposition of statistical facts over the tabular. A heavy bank of figures is grievously wearisome to the eye, and the popular mind is as incapable of drawing any useful lessons from it as of extracting sunbeams from cucumbers." (Arthur B Farquhar & Henry Farquhar, "Economic and Industrial Delusions", 1891)

"The visible figures by which principles are illustrated should, so far as possible, have no accessories. They should be magnitudes pure and simple, so that the thought of the pupil may not be distracted, and that he may know what features of the thing represented he is to pay attention to." (National Education Association, 1894)

"[…] geometry is the art of reasoning well from badly drawn figures; [...]" (Henri Poincaré, 1895)

"Nothing is so illuminating as a set of properly proportioned diagrams." (Allan C Haskell, "How to Make and Use Graphic Charts", 1919)

"Mathematics is not a compendium or memorizable formula and magically manipulated figures." (Scott Buchanan, "Poetry and Mathematics", 1929)

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

"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)

"Most people tend to think of values and quantities expressed in numerical terms as being exact figures; much the same as the figures which appear in the trading account of a company. It therefore comes as a considerable surprise to many to learn that few published statistics, particularly economic and sociological data, are exact. Many published figures are only approximations to the real value, while others are estimates of aggregates which are far too large to be measured with precision." (Alfred R Ilersic, "Statistics", 1959)

"When using estimated figures, i.e. figures subject to error, for further calculation make allowance for the absolute and relative errors. Above all, avoid what is known to statisticians as 'spurious' accuracy. For example, if the arithmetic Mean has to be derived from a distribution of ages given to the nearest year, do not give the answer to several places of decimals. Such an answer would imply a degree of accuracy in the results of your calculations which are quite un- justified by the data. The same holds true when calculating percentages." (Alfred R Ilersic, "Statistics", 1959)

"Figures alone prove or disprove nothing. Only the conclusions drawn from the collected material can do this. And these are theoretical." (Ludwig von Mises) 

"Figures and symbols are closely connected with mathematical thinking, their use assists the mind. […] At any rate, the use of mathematical symbols is similar to the use of words. Mathematical notation appears as a sort of language, une langue bien faite, a language well adapted to its purpose, concise and precise, with rules which, unlike the rules of ordinary grammar, suffer no exception." (George Pólya, "How to solve it", 1945)

"Geometry is the science of correct reasoning on incorrect figures." (George Pólya, "How to Solve It", 1945)

"Many people think of mathematics itself as a static art - a body of eternal truth that was discovered by a few ancient, shadowy figures, and upon which engineers and scientists can draw as needed." (Paul R Halmos, "Innovation in Mathematics", Scientific American Vol. 199 (3) , 1958)

"The art of using the language of figures correctly is not to be over-impressed by the apparent air of accuracy, and yet to be able to take account of error and inaccuracy in such a way as to know when, and when not, to use the figures. This is a matter of skill, judgment, and experience, and there are no rules and short cuts in acquiring this expertness." (Ely Devons, "Essays in Economics", 1961)

"If it can’t be expressed in figures, it is not science; it is opinion." (Robert Heinlein, "Time Enough For Love: the Lives of Lazarus Long", 1973)

"To remember simplified pictures is better than to forget accurate figures." (Otto Neurath, "Empiricism and Sociology", 1973)

"The greatest value of a picture is when it forces us to notice what we never expected to see." (John W Tukey, "Exploratory Data Analysis", 1977) 

"In information graphics, what you show can be as important as what you hide." (Alberto Cairo, "The Functional Art", 2011)

"[geometry is] the art of reasoning well from ill-drawn figures"  (Henri Poincaré)

"It is very helpful to represent these things in this fashion since nothing enters the mind more readily than geometric figures." (René Descartes)

"Statistics are figures used as arguments." (Leonard L Levison)

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