Quotes about notation

A collection of quotes on the topic of notation, use, language, mathematics.

Quotes about notation

“The development of Indian trigonometry, based on sine as against chord of the Greeks, a necessity for astronomical calculations with his own concise notation which expresses the full sine table in just one couplet for easy remembrance. One of the two methods suggested by him for the sine table is based on the property that the second order sine differences were proportional to sines themselves.”

Aryabhata (476–550) Indian mathematician-astronomer

In, P.245.
Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures

“If the task of philosophy is to break the domination of words over the human mind […], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers […] I believe the cause of logic has been advanced already by the invention of this concept notation.”

Gottlob Frege book Begriffsschrift

Begriffsschrift (1879) Preface to the Begriffsschrift

“His work, called Aryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is an alphabetical system in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, …., 100 etc. The other mathematical parts of Aryabhatiya consists of rules without examples. Another alphabetic system prevailed in Southern India, the numbers 1-19 being designated by consonants, etc.”

Aryabhata (476–550) Indian mathematician-astronomer

Florian Cajori in: A History of Mathematical Notations http://books.google.co.in/books?id=_byqAAAAQBAJ&pg=PT961&dq=Notations&hl=en&sa=X&ei=Wz65U5WYDIKulAW1qIGYDA&ved=0CBwQ6AEwAA#v=onepage&q=Notation&f=false, Courier Dover Publications, 26 September 2013, p. 47.

Mathematics, Parting

“Freiherr Hugo Reiss made a notation on his pad. Broach subject with SS General Otto Skorzeny, or better yet Otto Ohlendorf at Amt III of the Reichssicherheitshauptamt. Didn’t Ohlendorf head Einsatzgruppe D?”

Philip K. Dick book The Man in the High Castle

The Man in the High Castle

“The difficulty, as in all this work, is to find a notation which is both concise and intelligible to at least two people of whom one may be the author.”

Paul Taunton Matthews (1919–1987) British scientist

with Abdus Salam. [1951, October, The Renormalization of Meson Theories, Reviews of Modern Physics, 23, 4, 311-314] About the difficulty to express renormalization in quantum field theories. Also known as the Salam criterion.

People, Intelligence

“The Unified Modeling Language (UML) is a general-purpose visual modeling language that is used to specify, visualize, construct, and document the artifacts of a software system. It captures decisions and understanding about systems that must be constructed. It is used to understand, design, browse, conﬁgure, maintain, and control information about such systems. It is intended for use with all development methods, lifecycle stages, application domains, and media. The modeling language is intended to unify past experience about modeling techniques and to incorporate current software best practices into a standard approach. UML includes semantic concepts, notation, and guidelines. It has static, dynamic, environmental, and organizational parts. It is intended to be supported by interactive visual modeling tools that have code generators and report writers. The UML speciﬁcation does not deﬁne a standard process but is intended to be useful with an iterative development process. It is intended to support most existing object-oriented development processes.”

James Rumbaugh (1947) Computer scientist, software engineer

James Rumbaugh, Ivar Jacobson & Grady Booch (1998) The Unified Modeling Language Reference Manual. p. 1

Understanding, Support, Decision, Past

“The first period embraces the time between the first records of empirical determinations of the ratio of the circumference to the diameter of a circle until the invention of the Differential and Integral Calculus, in the middle of the seventeenth century. This period, in which the ideal of an exact construction was never entirely lost sight of, and was occasionally supposed to have been attained, was the geometrical period, in which the main activity consisted in the approximate determination of π by the calculation of the sides or the areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork of the method was the Greek method of Exhaustions. In the earlier part of the period the work of approximation was much hampered by the backward condition of arithmetic due to the fact that our present system of numerical notation had not yet been invented; but the closeness of the approximations obtained in spite of this great obstacle are truly surprising. In the later part of this first period methods were devised by which the approximations to the value of π were obtained which required only a fraction of the labour involved in the earlier calculations. At the end of the period the method was developed to so high a degree of perfection that no further advance could be hoped for on the lines laid down by the Greek Mathematicians; for further progress more powerful methods were required.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), pp. 10-11

Hope, Time, Parting, Perfection

“Object-oriented design is a method of design encompassing the process of object-oriented decomposition and a notation for depicting both logical and physical as well as static and dynamic models of the system under design.”

Grady Booch (1955) American software engineer

Source: Object-oriented design: With Applications, (1991), p. 37

“The utility of a language as a tool of thought increases with the range of topics it can treat, but decreases with the amount of vocabulary and the complexity of grammatical rules which the user must keep in mind. Economy of notation is therefore important.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§1.4
Notation as a Tool of Thought (1979)

“These computing machines had already been designed, and some built, by {[w|Vannevar Bush}}, Norbert Wiener, and others, and were almost ready-made for the job. These scientists, as well as others such as von Neumann, Shannon and Bigelow, were in a position to see that machines of an electronic kind were ideally suited to carry out the whole of the operations of range-finding and location without any human intervention whatever.
These electronic computing machines were already developed to a very high degree of efficiency for the solution of mathematical equations, and some technical difficulties had led to the suggestion that a process of scanning, similar to that used in television, might be incorporated into the computer. Another innovation was the use of binary notation rather than decimal notation as in the early Harvard Mark I computer.”

Frank Honywill George (1921–1997) British psychologist

Source: The Brain As A Computer (1962), p.18

Mathematics

“All form is a process of notation.”

Dick Higgins (1938–1998) English composer and poet

An Exemplativist Manifesto (1976)

“With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable "native speaker" of mathematical notation.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

Source: Math for the Layman (1999), Ch. 10, §D

Mathematics, Knowledge, Light

“But in our opinion truths of this kind should be drawn from notions rather than from notations.”

Carl Friedrich Gauss book Disquisitiones Arithmeticae

About the proof of Wilson's theorem. Disquisitiones Arithmeticae (1801) Article 76

Truth

“…it was Darwin’s chief contribution, not only to Biology but to the whole of natural science, to have brought to light a process by which contingencies a priori improbable, are given, in the process of time, an increasing probability, until it is their non-occurrence rather than their occurrence which becomes highly improbable. … Let the reader … attempt to calculate the prior probability that a hundred generations of his ancestry in the direct male line should each have left at least one son. The odds against such a contingency as it would have appeared to his hundredth ancestor (about the time of King Solomon) would require for their expression forty- four figures of the decimal notation; yet this improbable event has certainly happened.”

Ronald Fisher (1890–1962) English statistician, evolutionary biologist, geneticist, and eugenicist

Retrospect of criticisms of the theory of natural selection. In Evolution as a Process, eds. J.S.Huxley, A.C.Hardy and E.B.Ford, London: Allen and Unwin, 1954.
1950s

Nature, Light, Appearance, Science

“In the course of the fifteenth century, the sexagesimal division of the radius, in terms of which cords and goniometrical line-segments were expressed, was generally superseded, though not immediately replaced, by a decimal system of positional notation. Instead, mathematicians sought to avoid fractions by taking the Radius equal to a number of units of length of the form {\displaystyle 10^{n}} {\displaystyle 10^{n}}…The first to apply this method was the German astronomer Regiomontanus… the second half of the sixteenth and the first decades of the seventeenth century… observed of a gradual development of this method of Regiomontanus into a complete system of decimal positional fractions. Yet none of the steps taken by… writers is comparable in importance and scope with the progress achieved by Stevin in his De Thiende.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: Simon Stevin: Science in the Netherlands around 1600, 1970, p. 17-18

“Oh, I have a special way of writing the lyrics when it is dictated to me. No matter what the language of the song, even if it is Bengali, I write it in Hindi. I have certain notations and markings to indicate the way it should be pronounced. I feel the Devanagari script is the closest to the phonetics of the language. English alphabets are not very good for that purpose. Moreover, I listen carefully and try to grasp as much as possible when the lyrics are read to me.”

Shreya Ghoshal (1984) Indian playback singer

Getting it right (Singing) http://www.thehindu.com/features/metroplus/queen-of-the-charts/article390455.ece

Way, Feeling, Reading

“The eulogy of the non-lover in the speech of Lysias … stresses the fact that the non-lover … does not commit extreme acts under the influence of passions. Since he acts from calculation, he never has occasion for remorse. … The non-lover demonstrates his superiority through prudence and objectivity. … We must now observe how these points of superiority correspond to those of “semantically purified” speech, … the kind of speech approaching pure notation in the respect that it communicates abstract intelligence without impulsion. It is a simple instrumentality, showing no affection for the object of its symbolizing and incapable of inducing bias in the hearer. In its ideal conception, it would have less power to move than 2 + 2 = 4, since it is generally admitted that mathematical equations may have the beauty of elegance, and hence are not above suspicion where beauty is suspect. But this neuter language will be an unqualified medium of transmission of meanings from mind to mind, and by virtue of it minds can remain in an unprejudiced relationship to the world and also to other minds. … Instead of passion, it offers the serviceability of objectivity … It distrusts any departure from the literal and prosaic.”

Richard M. Weaver (1910–1963) American scholar

“The Phaedrus and the Nature of Rhetoric,” pp. 6-7.
The Ethics of Rhetoric (1953)

World, Beauty, Intelligence, Communication

“I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen’s, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell’s equations.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

"Kenneth E. Iverson" http://keiapl.info/rhui/autobio.htm, autobiographical sketch from an unfinished work (ca. 2004), on his experience at Harvard with "a Masters program in Automatic Data Processing in 1955; in effect, the first computer science program."

Mathematics, Sense, Learning

“The properties of executability and universality associated with programming languages can be combined, in a single language, with the well-known properties of mathematical notation which make it such an effective tool of thought.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§5
Notation as a Tool of Thought (1979)

Mathematics, Universe

“It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B. C. In Liu Hui's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position… and an arrangement of terms in a kind of determinant notation. The rules given by Liu Hui form a kind of rhetorical algebra.
The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving indeterminate equations. …Sun-tzï solved such problems by analysis and was content with a single result…
The Chinese certainly knew how to solve quadratics as early as the 1st century B. C., and rules given even as early as the K'iu-ch'ang Suan-shu… involve the solution of such equations.
Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should…
By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).
The culmination of Chinese is found in the 13th century. …numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner”

David Eugene Smith (1860–1944) American mathematician

1819
Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Way, Question, Problem, Appearance

“The Uniﬁed Modeling Language (UML) is a general-purpose visual modeling language that is used to specify, visualize, construct, and document the artifacts of a software system. It captures decisions and understanding about systems that must be constructed. It is used to understand, design, browse, conﬁgure, maintain, and control information about such systems. It is intended for use with all development methods, lifecycle stages, application domains, and media. The modeling language is intended to unify past experience about modeling techniques and to incorporate current software best practices into a standard approach. UML includes semantic concepts, notation, and guidelines. It has static, dynamic, environmental, and organizational parts. It is intended to be supported by interactive visual modeling tools that have code generators and report writers. The UML speciﬁcation does not deﬁne a standard process but is intended to be useful with an iterative development process. It is intended to support most existing object-oriented development processes.”

Ivar Jacobson (1939) Swedish computer scientist

James Rumbaugh, Ivar Jacobson, & Grady Booch (1999) The Unified Modeling Language Reference Manual. p. 1.

Understanding, Support, Decision, Past

“It has happened that I have begun a canvas in a brilliant tonality, going on to finish it in a grey notation. (1898)”

Albert Marquet (1875–1947) French artist

As quoted by J. E. Müller, Le Fauvisme, Paris, Hazan, 1956, p. 92

Grady Booch (1955) American software engineer

James Rumbaugh, Ivar Jacobson, & Booch (1999) The Unified Modeling Language Reference Manual. p. 1

Understanding, Support, Decision, Past

“Mechanical drawings and blueprints are not mere pictures, but a complete and rich language. In blueprint language, scientific, mathematical, and geometric formulations, notations, mensurations, and naming do not merely describe an object or process, they actually model it. Because of broad differences in subject, purpose, roles, and the needs of the people who use them, many forms of blueprint have evolved, but all rigorously present well structured information in understandable form.”

Douglas T. Ross (1929–2007) American computer scientist

Source: Structured analysis (SA): A language for communicating ideas (1977), p. 16.

People, Mathematics, Understanding, Drawing

“The Dirac notation, though originally applied to the propagation of single particles, also applies to describing the propagation of ensembles of coherent, or indistinguishable, photons.”

F. J. Duarte (1954) Chilean-American physicist

in Dirac Optics, [F. J. Duarte, Tunable Laser Optics, Elsevier Academic, 2003, 0-12-222696-8, 25]

“Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.”

Antoine Augustin Cournot Researches into the Mathematical Principles of the Theory of Wealth

Source: Researches into the Mathematical Principles of the Theory of Wealth, 1897, p. 4; Cited in: Moritz (1914, 197): About mathematics as language

Pain, Understanding, Reading

“One of the distinct features of XP is the lack of any documentation whatsoever, except for the code itself. This is a contraposition to the modeling techniques like the Unified Modeling Language (UML), which strongly focus on documentation. XP takes an extreme position there, not even documenting the architecture of the system. Often, it is very difficult to extract the overall structure, behavior or interactions with the environment from the code. The code is a rather detailed and fragile representation of the system’s tasks. Even though the code contains all necessary information about the system, this information is often burdened with details and it is tedious to extract the aspects one is interested in. Therefore, it would be useful to have a more compact system representation. The UML does provide a number of notations that are suited for this purpose. However, the tools so far are not capable of supporting UML in such a manner that it can be well-integrated with the approach of Extreme Programming.”

Bernhard Rumpe (1967) German computer scientist

Source: Executable Modeling with UML. A vision or a Nightmare (2002), p. 698

Behavior, Support

“What matters is not what you paint but what your hand dances. The painting is only a notation of the dance of the hand, of the gesture of love.”

John Diamond (doctor) (1934) Australian doctor

Source: Stillpoints: An Introductory Guide to Haiku Painting (2008), p. 20

Love, Dance, Painting

“Designed as a companion volume to the acclaimed Object-Oriented Analysis, this book focuses on the middle part of the software lifecycle: the activity of design. It shows readers how to apply object-oriented design, and how to tailor and expand the method to suit specific organization and project needs. Readers will explore the major issues in OOD; the role of OOD in the systems lifecycle; how to use graphical notation; strategies for creating design; and hints for evaluating the efficiency of a design created with OOD. For software engineers and other users undertaking real-world systems development projects and designing overall software architecture for systems will find this reference approach to improving systems design indispensable.”

Ed Yourdon (1944–2016) American software engineer and pioneer in the software engineering methodology

Abstract.
Object-oriented design (1991)

Books, World, Parting

“The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 161; Cited in: Moritz (1914, 263); Arithmetics

“The over-publicized bit about expression (or non-expression) was simply a way of saying that music is supra-personal and super-real and as such beyond verbal meanings and verbal descriptions. It was aimed against the notion that a piece of music is in reality a transcendental idea "expressed in terms of" music, with the reductio ad absurdum implication that exact sets of correlatives must exist between a composer's feelings and his notation. It was offhand and annoyingly incomplete, but even the stupider critics could have seen that it did not deny musical expressivity, but only the validity of a type of verbal statement about musical expressivity. I stand by the remark, incidentally, though today I would put it the other way around: music expresses itself.”

Igor Stravinsky (1882–1971) Russian composer, pianist and conductor

Igor Stravinsky and Robert Craft (1962). Expositions and Developments.
1960s

Stupidity, Music, Way, Feeling

“The precision provided (or enforced) by programming languages and their execution can identify lacunas, ambiguities, and other areas of potential confusion in conventional [mathematical] notation.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

Source: Math for the Layman (1999), Ch. 10, §D

Mathematics

“I asked him what, if anything, got him down about teaching. He said he didn't think that anything about it got him exactly down, but there was one thing, he thought, that frightened him: reading the pencilled notations in the margins of books in the college library.”

Jerome David Salinger book Raise High the Roof Beam, Carpenters and Seymour: An Introduction

Raise High the Roof Beam, Carpenters and Seymour: An Introduction (1963), Seymour: An Introduction (1959)

Books, Thinking, Reading

“Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. […] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. […] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

Source: Math for the Layman (1999), Ch. 10, §D

Mathematics

“When I came to the United States [in 1898], I filled my sketchbook with drawings, very much as any educated girl of my generation might have kept a diary... My American sketches were private notations of visual experiences which I wanted to fix on paper as a personal 'memento.”

Gabriele Münter (1877–1962) German painter

Source: Interview by Edouard Roditi (1958), p. 114

Education, Girl, Drawing, Personality

“Most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§0
Notation as a Tool of Thought (1979)

Way, Mathematics

“Here, in the last pages of the "Abegg" Variations, Schumann plays the motto theme A-B-E-G-G (B in German notation is the English Bb) not by sounding the last four notes but by taking them away, one by one, from, the chord of Bb-E-G. This is the first time in history that a melody is signified not by the attack but by the release of a series of notes. The motto, however, ends with a repeated final G. If the motto is played by releasing each successive note, we are faced with a paradox: when the G is released once on the piano, it is no longer there to be released again-the motto is not only unplayable as conceived but unimaginable. Schumann signifies as much by another paradox: he adds accents to the sustained notes.”

Charles Rosen (1927–2012) American pianist and writer on music

Source: The Romantic Generation (1995), Ch. 1 : Music and Sound

History, Time, Success

“The initial motive for developing APL was to provide a tool for writing and teaching. Although APL has been exploited mostly in commercial programming, I continue to believe that its most important use remains to be exploited: as a simple, precise, executable notation for the teaching of a wide range of subjects.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

"A Personal View of APL", IBM Systems Journal, 30 (4), 1991

Motivation

“An edition is… still wanted which shall, while in some places adhering… to the original text, at the same time be so entirely remodelled by the aid of accepted modern notation as to be thoroughly readable by any competent mathematician, and this want it is the object of the present work to supply.”

Thomas Little Heath (1861–1940) British civil servant and academic

Preface, p. ix
Apollonius of Perga (1896)

Time

“LISP is now the second oldest programming language in present widespread use (after FORTRAN)… Its core occupies some kind of local optimum in the space of programming languages given that static friction discourages purely notational changes. Recursive use of conditional expressions, representation of symbolic information externally by lists and internally by list structure, and representation of program in the same way will probably have a very long life.”

John McCarthy (1927–2011) American computer scientist and cognitive scientist

John McCarthy (1979) " History of Lisp http://www-formal.stanford.edu/jmc/lisp/lisp.html," as quoted in: Avron Barr, Edward Feigenbaum. The Handbook of artificial intelligence, Volume 2. Addison-Wesley, 1986. p. 5 <br class="br">1970s

Life, Way, Change

“He loved hitherto-unthought-of, thereafter-unthinkable combinations of instruments. When some extraordinary array of players filed half-proudly, half-sheepishly on to the stage, looking like the Bremen Town Musicians—if those were, as I think they were, a rooster, a cat, a dog, and a donkey—you could guess beforehand that it was to be one of Gottfried’s compositions. His Joyous Celebration of the Memory of the Master Johann Sebastian Bach had a tone-row composed of the notes B, A, C, and H (in the German notation), of these inverted, and of these transposed; and there were four movements, the first played on instruments beginning with the letter b, the second on instruments beginning with the letter a, and so on. After the magnificent group that ushered in the piece (bugle, bass-viol, bassoon, basset-horn, bombardon, bass-drum, baritone, and a violinist with only his bow) it was sad to see an Alp horn and an accordion come in to play the second movement. Gottfriend himself said about the first group: “Vot a bunch!””

Randall Jarrell book Pictures from an Institution

When I asked him how he had thought of it he said placidly: “De devil soldt me his soul.”
Source: Pictures from an Institution (1954) [novel], Chapter 4: “Constance and the Rosenbaums”, p. 136

Love, Sadness, Dogs, Soul

“That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, √-1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out of question.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

In Theoria residiorum biquadraticorum, Commentatio secunda; Werke, Bd. 2 (Goettingen, 1863), p.177. As quoted by Robert Edouard Moritz in Memorabilia mathematica: the philomath's quotation book (1914) p. 282.

Question

“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 (1918–2010) French geographer and cartographer

Source: Semiology of graphics (1967/83), p. 2

Law, Music, Mathematics, Time

“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 North Whitehead (1861–1947) English mathematician and philosopher

Source: 1910s, An Introduction to Mathematics (1911), ch. 5.

Problem

“In all engineering disciplines nowadays, software engineering excluded, there exists an established engineering process to develop a system, which is accompanied by a number of suited modeling description techniques. Software engineering, being a rather new field, has not as yet established any clear methodical guidance or a fully standardized modeling notation.”

Bernhard Rumpe (1967) German computer scientist

Source: Executable Modeling with UML. A vision or a Nightmare (2002), p. 697

“The Rule of Three, or Golden Rule of Arithmeticall whole Numbers. Be the three termes given 2 3 4. …To finde their fourth proporcionall Terme: that is to say, in such Reason to the third terme 4, as the second terme 3, is to the first terme 2 [Modern notation: \frac{x}{4} = \frac{3}{2}]. …Multiply the second terme 3, by the third terme 4, & giveth the product 12: which dividing by the first terme 2, giveth the Quotient 6: I say that 6 is the fourth proportional terme required.”

Simon Stevin (1548–1620) Flemish scientist, mathematician and military engineer

Disme: the Art of Tenths, Or, Decimall Arithmetike (1608)

“I've rediscovered the part of my brain that can't decode anything, that can't add, that can't work from a verbalized concept, that doesn't care about stylish notation, that makes melodies that have pitch and rhythm, that doesn't know anything about zen eternity and gets bored and changes, that isn't worried about being commercial or avant-garde or serial or any other little category. Beauty is enough.”

Beth Anderson (1950) American neo-romantic composer

Variant quotes:
I've rediscovered the part of my brain that can't decode anything, that can't add, that can't work from a verbalized concept, that doesn't know anything about Zen eternity and gets bored and changes, that isn't worried about being commercial or avant-garde or serial or any other little category. Beauty is enough.
Beauty is Revolution (1980)
Source: Jane Weiner LePage (1983) Women composers, conductors, and musicians of the twentieth century: selected biographies. p. 14

Beauty, Change, Parting

“The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 29.
Theory of Numbers, 1892

Science

“It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. …Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.”

Thomas Little Heath (1861–1940) British civil servant and academic

Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)

Problem, Sense, Time

“I counted to ten slowly, using binary notation.”

Robert A. Heinlein (1907–1988) American science fiction author

Source: The Door Into Summer (1957), Chapter 8

“It is important to distinguish the difficulty of describing and learning a piece of notation from the difficulty of mastering its implications. […] Indeed, the very suggestiveness of a notation may make it seem harder to learn because of the many properties it suggests for exploration.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§5.2
Notation as a Tool of Thought (1979)

Learning

“The idea that beauty is revolution is a revelation to me. I once believed that the concept of the music was more important than the sound, that the politics of the notation was more important than the time limits of the rehersals and therefore, more important than the sound of the performance”

Beth Anderson (1950) American neo-romantic composer

Cited (earlier) in: American Women Composers (1979) AWC news. Volumes 2-3. p. 41
Beauty is Revolution (1980)

Beauty, Music, Idea, Time

“The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, "The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), pp. 1-2; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/198/mode/2up, (1914) p. 90; Study and research in mathematics

Men, Secret, Way, Error

“During the process of stepwise refinement, a notation which is natural to the problem in hand should be used as long as possible.”

Niklaus Wirth (1934) Swiss computer scientist

Program Development by Stepwise Refinement (1971)

Nature, Problem

“Feynman uses Dirac's notation to describe the quantum mechanics of stimulated emission… he applies that physics to… dye molecules… In this regard, Feynman could have predicted the existence of the tunable laser.”

F. J. Duarte (1954) Chilean-American physicist

in Introduction to Lasers, [F. J. Duarte, Tunable Laser Optics, Elsevier Academic, 2003, 0-12-222696-8, 3] (while discussing The Feynman Lectures on Physics).

“The folksong collectors Bartok (Hungary) and Plicka (Czechoslovakia) notated the perceived interval deviations with a plus or minus sign (+,-). Older folksong collectors had notated all the intervals in the system of semitones, not observing the finer modifications of intervals. This first gained currency from the phongraph and tape recordings of folk music from various lands…. From folksong I learned to perceive melodic intervals a little smaller or greater than those of the semitone-system…. In my youth it often happened that the folk singers, who during the intermissions of dance festivals sang songs of "their kind," deviating from the semitone system, demanded that the first violinist of the Wisowitzer Kapelle play "their" melody just as they sang it. Once a tempermental singer threatened to strike the double-bass player with a beer mug if he didn't "play along" with the song, exactly as he sang it.”

Alois Hába (1893–1973) Czech music educator and composer

Mein Weg zur Viertel- und Sechsteltonmusik (1971) Düsseldorf: Verlag der Gesellschaft zur Förderung der systematische Musikwissenschaft, 12, 14; translated by and printed in Modernism and Music: An Anthology of Sources(2004) by Daniel Albright ISBN 0226012670 .

Youth, Music, Beer, Learning

“By the way, the = notation was invented by Robert Recorde (1510-1558). He choose two parallel lines as a symbol of equality " because noe 2 thynges can be moare equalle."”

Brian Hayes (scientist) (1900) American scientist, columnist and author

Source: Group Theory in the Bedroom (2008), Chapter 11, Identity Crisis, p. 203

Way

“If it is to be effective as a tool of thought, a notation must allow convenient expression not only of notions arising directly from a problem, but also of those arising in subsequent analysis, generalization, and specialization.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§1.1
Notation as a Tool of Thought (1979)

Problem

“\frac {dy}{dx} = \frac {\omega^2x}{g}…The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form
\frac {dy}{dx}…Leibniz's notation …is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation… need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. …it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure… it brought mathematics into disrepute… some of the best minds… such as… Berkeley, complained that calculus is incomprehensible. …\frac {dy}{dx} is the limit of a ratio of dy to dx… Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio… and multiply by dx to achieve the separation of variables. We get
{dy} = \frac {\omega^2x}{g}xdx”

George Pólya (1887–1985) Hungarian mathematician

Mathematical Methods in Science (1977)

Mathematics, Time

“My monograph on visible spectroscopy using molecular orbital notations eventually resulted into a book in 1960.”

C. N. R. Rao (1934) Indian chemist

How I made it: CNR Rao, Scientist (2010)

Books

“It would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress. Not only does the area widen but the subjects of study increase rapidly in number, and the work of the mathematician tends to become more and more specialized. It is, of course, merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician, but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted, even in a general way, with the progress of any of the branches of mathematics except those which form the field of his own labours. I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained…. I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject”

James Whitbread Lee Glaisher (1848–1928) English mathematician and astronomer

Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 466 : On the expansion of the field of mathematics, and on the importance of a well-chosen notation

Truth, Faith, Way, Exercises

“[The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine…Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.”

Ada Lovelace (1815–1852) English mathematician, considered the first computer programmer

As quoted by Menabrea, Luigi (1842). Sketch of the Analytical Engine invented by Charles Babbage Esq.. Scientific Memoirs (Richard Taylor): 694.

Music, Science

“We each have to say it, to say it in our own way. Not of our own will, but as it comes through us. Good or bad, great or little: that isn't what human creation is about. It is that we have to try; to put it down in pigment, or words, or musical notations, or we die.”

Madeleine L'Engle (1918–2007) American writer

Section 1.9 
The Crosswicks Journal, A Circle of Quiet (1972)
Context: My husband is my most ruthless critic. … Sometimes he will say, "It's been said better before." Of course. It's all been said better before. If I thought I had to say it better than anyone else, I'd never start. Better or worse is immaterial. The thing is that it has to be said; by me; ontologically. We each have to say it, to say it in our own way. Not of our own will, but as it comes through us. Good or bad, great or little: that isn't what human creation is about. It is that we have to try; to put it down in pigment, or words, or musical notations, or we die.

Music, Way

“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… Of these graphical notations, the UML's importance comes from its wide use and standardization within the OO development community. The UML has become not only the dominant graphical notation within the OO world but also a popular technique in non-OO circles.”

Martin Fowler (1963) British programmer

Source: UML Distilled: A Brief Guide to the Standard Object Modeling, 2004, p. xxvi

World, Communication, Idea, Understanding