Joseph Mazur - Collected Quotes

"All of this could have been said using notation that kept √-1 instead of the new representative i, which has the same virtual meaning. But i isolates the concept of rotation from the perception of root extraction, offering the mind a distinction between an algebraic result and an extension of the idea of number." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"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)

"In mathematics, the symbolic form of a rhetorical statement is more than just convenient shorthand. First, it is not specific to any particular language; almost all languages of the world use the same notation, though possibly in different scriptory forms. Second, and perhaps most importantly, it helps the mind to transcend the ambiguities and misinterpretations dragged along by written words in natural language. It permits the mind to lift particular statements to their general form." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"In natural language, even the most carefully chosen words drag along concealed meanings that have the power to manipulate reasoning. [...] Symbols of mathematics too sometimes have concealed meanings, but their purpose is to bring along pure thought. It is possible to learn what a mathematical symbol stands for by context. We learn the meanings of mathematical symbols mostly from their definitions: Mostly, because in formal mathematics not everyone easily grasps definitions that are not linked to the familiar properties of experience." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"It may come as a surprise that the symbol i (even though it is just an abbreviation of the word 'imaginary') has a marked advantage over √-1. In reading mathematics, the difference between a + b√-1 and a + bi is the difference between eating a strawberry while holding your nose, missing the luscious taste, and eating a strawberry while breathing normally." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"On a deeper level, the word 'symbol' suggests that, when the familiar is thrown together with the unfamiliar, something new is created. Or, to put it another way, when an unconscious idea fits a conscious one, a new meaning emerges. The symbol is exactly that: meaning derived from connections of conscious and unconscious thoughts." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"One of the wonderful things about mathematics is that - by its best symbols - its progression expands its vision." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Symbolic language surely promotes its own concealed meanings that come from imaginative glimpses into the subconscious, but the best symbols are those that pinpoint meaning and yet permit the mind to quickly roam its databank of similar contextual patterns to compare, to transmit, and to creatively link what is unknown with what is known." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Symbols transcend the medium of communication. They are ubiquitous in our language, and play a sizable role (though perhaps not a central one) in mathematical imagery linking the conscious and subconscious, the familiar and unknown, to give us cultural/emotional predispositions to meaning, all to enhance the creative process."(Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Unlike symbols in poetry, mathematical symbols begin as deliberate designs created by mathematicians. That does not stop symbols from performing the same function that a poem would: to make connections between experience and the unknown and to transfer metaphorical thoughts capable of conveying meaning. As in poetry, there are archetypes in mathematics. If there are such things as self-evident truths, then there probably are things we know about the world that come with the human package at birth." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"When it comes to algebra, visual conception is beyond any similarities in the physical world. That’s okay; as we’ve noted, it’s not the job of mathematics to be concerned with the physical world, nor with what we call 'reality'. Symbolic consistency and meaning are essentials of mathematics. So is certainty. So is imagination. So is the creative process. So is hypothesis. So is belief beyond experience. So is adventure of knowledge. And, in today’s complexity, there is no better way to do the job of mathematics than by symbolic envisagement." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Yet there is a distinct difference between the writer’s art and the mathematician’s. Whereas the writer is at liberty to use symbols in ways that contradict experience in order to jolt emotions or to create states of mind with deep-rooted meanings from a personal life’s journey, the mathematician cannot compose contradictions, aside from the standard argument that establishes a proof by contradiction. Mathematical symbols have a definite initial purpose: to tidily package complex information in order to facilitate understanding." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

On Notation (2010-2019)

 "Notational complexity almost always results in informational inefficiency." (Joel Katz, "Designing Information: Human factors and common sense in information design", 2012) 

"In mathematics, the symbolic form of a rhetorical statement is more than just convenient shorthand. First, it is not specific to any particular language; almost all languages of the world use the same notation, though possibly in different scriptory forms. Second, and perhaps most importantly, it helps the mind to transcend the ambiguities and misinterpretations dragged along by written words in natural language. It permits the mind to lift particular statements to their general form." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"When we use algebraic notation in statistical models, the problem becomes more complicated because we cannot 'observe' a probability and know its exact number. We can only estimate probabilities on the basis of observations." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"Again, classical statistics only summarizes data, so it does not provide even a language for asking [a counterfactual] question. Causal inference provides a notation and, more importantly, offers a solution. As with predicting the effect of interventions [...], in many cases we can emulate human retrospective thinking with an algorithm that takes what we know about the observed world and produces an answer about the counterfactual world." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

"[...] the relations between the two [mathematics and music] disciplines were never truly symmetric. Yes, there are many similarities between the two. For example, mathematics and music both depend on an efficient system of notation - a set of written symbols that convey a precise, unambiguous meaning to its practitioners (although in music this is augmented by a large assortment of verbal terms to indicate the more emotional aspects of playing)." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)

"Programming is the process of taking an algorithm and encoding it into a notation that the computer can execute." (Bradley N Miller et al, "Python Programming in Context", 2019)

On Notation (2000-2009)

"A seemingly modest change of notation may suggest a radical shift in viewpoint. Any new notation may ask new questions." (Barry Mazur, "Imagining Numbers", 2003)

"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)

"An essential feature of mathematics and statistics, particularly at a higher level, is the use of shorthand notation for a variety of concepts and measures. While this can be a strength in terms of providing conciseness and precision, statistical notation often proves to be an obstacle for learners in the early stages of learning." (Alan Graham, "Developing Thinking in Statistics", 2006)

"The notation is more important than the sound. Not the exactitude and success with which a notation notates a sound; but the musicalness of the notation in its notating." (Cornelius Cardew, 2006)

"Understand the data. What is given in the problem? Usually, a question talks about a number of objects which satisfy some special requirements. To understand the data, one needs to see how the objects and requirements react to each other. This is important in focusing attention on the proper techniques and notation to handle the problem." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006) 

"Write down what you know in the notation selected; draw a diagram. Putting everything down on paper helps in three ways: (a) you have an easy reference later on; (b) the paper is a good thing to stare at when you are stuck; (c) the physical act of writing down of what you know can trigger new inspirations and connections." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"But in mathematics there is a kind of threshold effect, an intellectual tipping point. If a student can just get over the first few humps, negotiate the notational peculiarities of the subject, and grasp that the best way to make progress is to understand the ideas, not just learn them by rote, he or she can sail off merrily down the highway, heading for ever more abstruse and challenging ideas, while an only slightly duller student gets stuck at the geometry of isosceles triangles." (Ian Stewart, "Why Beauty is Truth: A history of symmetry", 2007)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre 0 Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

On Notation (1975-1999)

"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, "Structured analysis (SA): A language for communicating ideas", 1977)

"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, "Notation as a Tool of Thought", 1979)

"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, "Notation as a Tool of Thought", 1979)

"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, "Notation as a Tool of Thought", 1979)

"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, "Notation as a Tool of Thought", 1979)

"This difficulty lead very gradually to the recognition of the need for a shorthand to make the sequence of operations easily comprehensible: here we have the problem of notation, which crops up again after every introduction of new objects, and which will probably never cease to torment mathematicians." (Jean Dieudonné, "Mathematics: The Music of Reason", 1992)

"This ambiguity is another example of a growing problem with mathematical notation: There aren't enough squiggles to go around." (Jim Blinn, Jim Blinn's Corner: Dirty Pixels", 1996)

"Mathematical notation is for the scientist what musical notation is for the composer." (John Holland, "Emergence: From Chaos to Order", 1998)

"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, "Math for the Layman", 1999)

"Mathematicians have always appreciated clever notations; but symbolism is usually seen as a tool - it's what the tool does that we really care about. Fair enough. But if we want a richer appreciation of mathematics, we should focus some of our energy on this remarkable tool - notation. Besides mathematics, poetry alone works wonders with it." (James R Brown, "Philosophy of Mathematics", 1999)

"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, "Math for the Layman", 1999)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, "Philosophy of Mathematics", 1999)

On Notation (1900-1949)

"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." (George Pólya, "How to solve it", 1945)

"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])

On Notation (1950-1974)

"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 T Matthews & Abdus Salam, "The Renormalization of Meson Theories", Reviews of Modern Physics 23 (4), 1951)

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. [...] The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it [...] the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study." (Paul R Halmos, "Naive Set Theory", 1960)

"We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact,mathematics is, to a large extent, invention of better notations." (Richard Feynman, "The Feynman Lectures on Physics" Vol 1, 1963)

"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)

"How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature? Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really reflects the interconnectedness of Nature’s laws at different levels." (Murray Gell-Mann, 1969)

"Many cumbersome developments in the standard treatments of mechanics can be simplified and better understood when formulated with modern conceptual tools, as in the well-known case of the use of the 'universal' definition of tensor products of vector spaces to simplify some of the notational excesses of tensor analysis as traditionally used in relativity theory." (Saunders Mac Lane, "Hamiltonian Mechanics and Geometry", The American Mathematical Monthly Vol. 77 (6), 1970)

"The natural world seems a marvel of complexity, requiring a vastly intricate intelligence to create and govern it, just because we have represented it to ourselves in the clumsy 'notation' of thought." (Alan Watts, "Nature, Man, and Woman", 1970)

"The complexity of the universe is beyond expression in any possible notation." (Michael Frayn, "Constructions", 1974)

On Notation (1850-1899)

"I look upon it [mechanical notation] as one of the most important additions I have made to human knowledge. It has placed the construction of machinery in the rank of a demonstrative science. The day will arrive when no school of mechanical drawing will be thought complete without teaching it." Charles Babbage, "Passages From the Life of a Philosopher", 1864)

"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, "Begriffsschrift", 1879)

"The origin of a science is usually to be sought for not in any systematic treatise, but in the investigation and solution of some particular problem. This is especially the case in the ordinary history of the great improvements in any department of mathematical science. Some problem, mathematical or physical, is proposed, which is found to be insoluble by known methods. This condition of insolubility may arise from one of two causes: Either there exists no machinery powerful enough to effect the required reduction, or the workmen are not sufficiently expert to employ their tools in the performance of an entirely new piece of work. The problem proposed is, however, finally solved, and in its solution some new principle, or new application of old principles, is necessarily introduced. If a principle is brought to light it is soon found that in its application it is not necessarily limited to the particular question which occasioned its discovery, and it is then stated in an abstract form and applied to problems of gradually increasing generality. [/] Other principles, similar in their nature, are added, and the original principle itself receives such modifications and extensions as are from time to time deemed necessary. The same is true of new applications of old principles; the application is first thought to be merely confined to a particular problem, but it is soon recognized that this problem is but one, and generally a very simple one, out of a large class, to which the same process of investigation and solution are applicable. The result in both of these cases is the same. A time comes when these several problems, solutions, and principles are grouped together and found to produce an entirely new and consistent method; a nomenclature and uniform system of notation is adopted, and the principles of the new method become entitled to rank as a distinct science." (Thomas Craig, "A Treatise on Projections", 1880)

"Boole's work is not so much an attempt (as used to be commonly said) to 'reduce logic to mathematics', as the employment of symbolic language and notation in a wide generalisation of purely logical processes. His fundamental process is really that of continued dichotomy, or subdivision, in respect of all the class terms which enter into the system of propositions in question. [...] This process in its priori form furnishes us with a complete set of possibilities, which, however, the conditions involved in the statement of the assigned propositions necessary necessarily?) reduce to a limited number of actualities: Boole's system being essentially one for displaying the solution of the problem in the form of a complete enumeration of these actualities." (John Venn, [in "Dictionary of National Biography"], 1886)

"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." (James W L Glaisher, British Association for the Advancement of Science Nature, Section A Vol. 42, [presidential address], 1089)

"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 W L Glaisher, British Association for the Advancement of Science Nature, Section A Vol. 42, [presidential address] 1089)

"The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms." (Florian Cajori, "A History of Mathematics", 1894)

"A mathematical argument is, after all, only organized common sense, and it is well that men of science should not always expound their work to the few behind a veil of technical language, but should from time to time explain to a larger public the reasoning which lies behind their mathematical notation." (George Darwin, "The Tides and Kindred Phenomena in the Solar System", 1898)

"There are now two systems of notations, giving the same formal results, one of which gives them with self-evident force and meaning, the other by dark and symbolic processes. The burden of proof is shifted, and it must be for the author or the supporter of the dark system to show that it is in some way superior to the evident system." (William S Jevons)

On Notation (-1849)

"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)

"But in our opinion truths of this kind should be drawn from notions rather than from notations." (Carl F Gauss, "Disquisitiones Arithmeticae" ["Arithmetical Investigations", 1801)

"Pure mathematics is not concerned with magnitude. It is merely the doctrine of notation of relatively ordered thought operations which have become mechanical." (Friederich von Hardenberg [Novalis], "Philosophical Writings", 1802)

"By use of the symbol v-1 and of the forms proved to obtain in the combination of real quantities, a mode of notation is obtained, by which we may express sines and cosines, relatively to their arc." (Robert Woodhouse,"On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)

"Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them." (J B Joseph Fourier, "The Analytical Theory of Heat", 1822)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If we call +1, -1, and v-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." (Carl F Gauss, "Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)

"Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains." (Antoine-Augustin Cournot, "Recherches sur les Principes Mathématiques de la Théorie des Richesses", 1838)

"The question undoubtedly is, or soon will be, not whether or no we shall employ notation in chemistry, but whether we shall use a bad and incongruous, or a consistent and regular notation." (William Whewell, 1838)

"Again, it [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." (Augusta Lovelace, 1842)

"The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter." (Augustus De Morgan, "Calculus of Functions" Encyclopaedia of Pure Mathematics, 1847)

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)

On Analogy (1825-1849)

"It is true that of far the greater part of things, we must content ourselves with such knowledge as description may exhibit, or analogy supply; but it is true likewise, that these ideas are always incomplete, and that at least, till we have compared them with realities, we do not know them to be just. As we see more, we become possessed of more certainties, and consequently gain more principles of reasoning, and found a wider base of analogy." (Samuel Johnson, 1825)

"Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many." (Sir John Herschel, "A Preliminary Discourse on the Study of Natural Philosophy" , 1830) 

"Whilst chemical pursuits exalt the understanding, they do not depress the imagination or weaken genuine feeling; whilst they give the mind habits of accuracy, by obliging it to attend to facts, they likewise extend its analogies; and, though conversant with the minute forms of things, they have for their ultimate end the great and magnificent objects of nature." (Sir Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher", 1830)

"Science is nothing but the finding of analogy, identity, in the most remote parts." (Ralph W Emerson, 1837)

"It is frequently analogy which guides the experienced to what are called good guesses." (Francis W Newman, "Lectures on Logic", 1838)

"On the whole, Analogy is to be regarded as a step towards satisfactory proof, much in advance of first presumptions, if skillfully applied; though if the excessive vagueness of the word like be not checked, arguments from analogy may be of the wildest and silliest kind." (Francis W Newman, "Lectures on Logic", 1838)

"To reason from analogy is often dangerous, but to illustrate by a fanciful analogy is sometimes a means by which we light an idea, as it were, into the understanding of another." (Anna B Jameson, "Studies, Stories, and Memoirs", 1838)

“The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter.” (Augustus De Morgan, “Calculus of Functions” Encyclopaedia of Pure Mathematics, 1847) 

On Art: Poetry and Mathematics II

"The true spirit of delight, the exaltation...which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry."  (Bertrand Russell, “Mysticism and Logic”, 1917)

"Mathematics is one component of any plan for liberal education. Mother of all the sciences, it is a builder of the imagination, a weaver of patterns of sheer thought, an intuitive dreamer, a poet. The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)

"Mathematics in this sense is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process." (Jacob Bronowski, "Science and Human Values", 1956)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, “Philosophy of Mathematics”, 1999)

"Mathematicians have always appreciated clever notations; but symbolism is usually seen as a tool - it's what the tool does that we really care about. Fair enough. But if we want a richer appreciation of mathematics, we should focus some of our energy on this remarkable tool - notation. Besides mathematics, poetry alone works wonders with it." (James R Brown, “Philosophy of Mathematics”, 1999)

"[...] mathematics bears on poetry not only by analogy, but directly through metrics. Metrics is the science of poetry, and it would be healthy for poetry if that science were more widely and astutely studied." (Kurt Brown, “The Measured Word: On Poetry and Science”, 2001)

"What could mathematics and poetry share, except that the mention of either one is sometimes enough to bring an uneasy chill into a conversation? [...] Both fields use analogies - comparisons of all sorts - to explain things, to express unknown or unknowable concepts, and to teach." (Marcia Birken & Anne C Coon, “Discovering Patterns in Mathematics and Poetry”, 2008)

"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)

"You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things." (Jonathan D Farley)

"Proofs are to mathematics what spelling is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters." (Vladimir Arnold)

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