"But the language of analysis, most perfect of all, being in itself a powerful instrument of discoveries, its notations, especially when they are necessary and happily conceived, are so many germs of new calculi." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1902)
"Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. [...] Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)
"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)
"There can be no doubt that science is in many ways the natural enemy of language. Language, either literary or colloquial, demands a rich store of living and vivid words - words that are 'thought-pictures', and appeal to the senses, and also embody our feelings about the objects they describe. But science cares nothing about emotion or vivid presentation; her ideal is a kind of algebraic notation, to be used simply as an instrument of analysis; and for this she rightly prefers dry and abstract terms, taken from some dead language, and deprived of all life and personality." (Logan P Smith, "The English Language", 1912)
"A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher. [...] a perfect notation would be a substitute for thought." (Bertrand Russell, [introduction at Ludwig Wittgenstein, "Tractatus Logico-Philosophicus"] 1922)
"This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time." (Florian Cajori, "A History of Mathematical Notations", 1928)
"An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation carefully may be repaid by the time we save later by avoiding hesitation and confusion." (George Pólya, "How to Solve It", 1945)
"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."
"The creation of a word or a notation for a class of ideas may be, and often is, a scientific fact of very great importance, because it means connecting these ideas together in our subsequent thought" (Jacques S Hadamard, "Newton and the Infinitesimal Calculus", 1947)
"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […] The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Pólya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])
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