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