🏛️On Art: On Music & Mathematics (1975-1989)

"The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music." (Scott Buchanan,"Poetry and Mathematics", 1975)

"One of the mysteries is how the human mind can hear a piece of music, a symphony from the beginning to the end, before beginning; or see a sculpture finished all the way round, when it doesn't exist. Now these faculties are the sort faculties which are needed in sciences, math, and medicine and all kind of things. But if one has them, one has to learn to use them." (Barbara Hepworth, "Art Talk, conversations with 15 woman artists", 1975)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"Mathematical physics represents the purest image that the view of nature may generate in the human mind; this image presents all the character of the product of art; it begets some unity, it is true and has the quality of sublimity; this image is to physical nature what music is to the thousand noises of which the air is full […]" (Théophile de Donder, 1977)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. On the aesthetic aspects, as has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive." (Richard E Bellman, 1978)

"Just as one can appreciate the beauty of a Beethoven quartet without being able to read a note of music, it is possible to learn about the scope and power and, yes, beauty of a scientific explanation of nature without solving equations." (Gilbert Shapiro,"Physics Without Math", 1979)

"The physicist […] engages in complex and difficult calculations, involving the manipulating of ideal, mathematical quantities that, at first glance, are wholly lacking in the music of the living world and the beauty of the resplendent cosmos. It would seem as if there exists no relationship between these quantities and reality. Yet these ideal numbers that cannot be grasped by one's senses, these numbers that only are meaningful from within the system itself, only meaningful as part of abstract mathematical functions, symbolize the image of existence. […] As a result of scientific man's creativity there arises an ordered, illumined, determined world, imprinted with the stamp of creative intellect, of pure reason and clear cognition. From the midst of the order and lawfulness we hear a new song, the song of the creature to the Creator, the song of the cosmos to its Maker." (Joseph B Soloveitchik, "Halakhic Man", 1983)

"Music is mathematics, the mathematics of listening, mathematics for the ears." (Karlheinz Stockhausen, "Conversations with Stockhausen", 1987)

🏛️On Art: On Music & Mathematics (1990-1999)

"For a musician, visualization is the process of picturing in our minds eye what we hear in our mind's ear. Visualization is something we all do. In fact, putting a visual form before the mind's eye or forming a mental image is something that precedes most things that we do." (Jerry Bergonzi, "Melodic Structures", 1992)

"To me, that the distribution of prime numbers can be so accurately represented in a harmonic analysis is absolutely amazing and incredibly beautiful. It tells of an arcane music and a secret harmony composed by the prime numbers." (Enrico Bombieri, "PrimeTerritory", The Sciences, 1992)

"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)

"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)

"Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical." (Edward Rothstein,"Emblems of Mind: The Inner Life of Music and Mathematics", 1995)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103" (7), 1996)

"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)

"How deep truths can be defined as invariants – things that do not change no matter what; how invariants are defined by symmetries, which in turn define which properties of nature are conserved, no matter what. These are the selfsame symmetries that appeal to the senses in art and music and natural forms like snowflakes and galaxies. The fundamental truths are based on symmetry, and there’s a deep kind of beauty in that." (K C Cole, The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

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

"Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals." (Györgi Ligeti, [interview] 1999)

"Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it’s not that one approach is right and the other wrong – the methods one chooses to use are largely a matter of taste and training." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"That randomness gives rise to innovation and diversity in nature is echoed by the notion that chance is also the source of invention in the arts and everyday affairs in which naturally occurring processes are balanced between tight organization, where redundancy is paramount, and volatility, in which little order is possible. One can argue that there is a difference in kind between the unconscious, and sometimes conscious, choices made by a writer or artist in creating a string of words or musical notes and the accidental succession of events taking place in the natural world. However, it is the perception of ambiguity in a string that matters, and not the process that generated it, whether it be man-made or from nature at large." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

🥀On Beauty: Beauty & Mathematics (1850-1874)

"The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language. It would defy all the power of Mercury himself to explain to a person ignorant of the science what is meant by the single phrase “functional exponent.” How much more impossible, if we may say so, would it be to explain a whole treatise like Hamilton’s Quaternions, in such a wise as to make it possible to judge of its value! But to one who has learned this language, it is the most precise and clear of all modes of expression. It discloses the thought exactly as conceived by the writer, with more or less beauty of form, but never with obscurity. It may be prolix, as it often is among French writers; may delight in mere verbal metamorphoses, as in the Cambridge University of England; or adopt the briefest and clearest forms, as under the pens of the geometers of our Cambridge; but it always reveals to us precisely the writer’s thought." (Thomas Hill, North American Review 85, 1857)

"The fixed laws of science can supply natural religion with numberless illustrations of the wisdom, the beneficence, the order, the beauty that characterizes the workmanship of God; while they illustrate His infinity by the marvelous complexity of natural combinations, by the variety and order of His creatures, by the exquisite finish alike bestowed on the very greatest and on the very least of His works, as if size were absolutely nothing in His sight." (Frederick Temple, "Present Relations of Science to Religion", 1860)

"In order to depict nature in its exalted sublimity, we must not dwell exclusively on its external manifestations, but we must trace its image, reflected in the mind of man, at one time filling the dreamy land of physical myths with forms of grace and beauty, and at another developing the noble germ of artistic creations." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe" Vol. 2, 1869)

"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance." (James J Sylvester, [Presidential Address to British Association] 1869)

"The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both." (Henry Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

🥀On Beauty: Beauty & Mathematics (1500-1799)

"Geometry existed before the creation. It is co-eternal with the mind of God. It is God himself. Where there is matter there is geometry. [...] It is absolutely necessary that the work of such a creator be of the greatest beauty [...]" (Johannes Kepler, "The Mystery of the Cosmos", 1596)

"Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)

“It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur in producing a given event, or effect, is oftentimes so immensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged, or combined together, it often happens that men, even of the best understandings and the greatest circumspection, ale guilty of that fault in reasoning which the writers on logic call the insufficient, or imperfect enumeration of parts, or cases: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions. It must therefore be acknowledged, that that art which affords a cure to this weakness, or defect, of our understandings, and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted any one arrangement of them that can lead to the object of our inquiry, deserves to be considered as most eminently useful and worthy of our highest esteem and attention. And this is the business of the art, or doctrine of combinations.” (Jakob Bernoulli, “Ars Conjectandi” [“The Art of Conjecturing”], 1713)

"A strong curiosity has prompted men in all times to study nature; every useful art has some connexion with the science; and the unexhausted beauty and variety of things makes it ever agreeable, new and surprising." (Colin Maclaurin, "An Account of Sir Isaac Newton’s Philosophical Discoveries", 1748)

"Beauty has in art the same foundation as does truth in philosophy. What is the truth? The conformity of our judgements with beings. What is the beauty of imitation? The conformity of the image with the thing." (Denis Diderot, "Entretiens sur le fils naturel", 1757)

"The knowledge of mechanics is one of those things that contribute to distinguish civilized nations from barbarians. From it the works of art derive much of their beauty and value; without it we can make very little progress in the knowledge of the works of nature. By this science we are enabled to improve every power and force in nature, and render the motions of the elements water, air, and fi re, subservient to the purposes of life." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 3, 1794)

🥀On Beauty: Beauty & Mathematics (1800-1849)

"Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers" (Carl F Gauss, "Disquisitiones arithmeticae" ["Arithmetical Researches"], 1801)

"At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes" (real values of the argument) and regard the imaginary values as just vestigial - or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications." (Carl F Gauss, [letter to Bessel] 1811)

"It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations." (Carl Friedrich Gauss, 1817)

"The Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things, that has not been enumerated. [...] Besides its uses in mathematical investigations, it not only enables us to form our ideas of the elegant compositions of design, but to contemplate the prodigious variety which constitutes the beauties of nature, and which arises from the combinations of objects, by their number, forms, color, and positions. It has a relation to every species of useful knowledge upon which the mind of man can be employed." (Peter Nicholson, "Essays on the Combinatorial Analysis", 1818)

"By the word symmetry […] one thinks of an external relationship between pleasing parts of a whole; mostly the word is used to refer to parts arranged regularly against one another around a centre. We have […] observed [these parts] one after the other, not always like following like, but rather a raising up from below, a strength out of weakness, a beauty out of ordinariness." (Johann Wolfgang von Goethe, "Zur Naturwissenschaft überhaupt, besonders zur Morphologie", 1820)

"In science, reason is the guide; in poetry, taste. The object of the one is truth, which is uniform and indivisible; the object of the other is beauty, which is multiform and varied." (Charles C Colton,"Lacon", 1820)

"The mathematician is perfect only in so far as he is a perfect man, in so far as he senses in himself the beauty of truth; only then will his work be thorough, transparent, prudent, pure, clear, graceful, indeed elegant." (Johann Wolfgang von Goethe, 1829)

"Beauty cannot be defined by abscissas and ordinates; neither are circles and ellipses created by their geometrical formulas." (Carl von Clausewitz, "On War", 1832)

"Nature builds up by her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (Sir John F W Herschel, "The Cabinet of Natural Philosophy", 1831)

"A mathematician is only perfect insofar as he is a perfect man, sensitive to the beauty of truth." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

🥀On Beauty: Beauty & Mathematics (1875-1899)

"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"But I thoroughly believe myself, and hope to prove to you, that science is full of beautiful pictures, of real poetry, and of wonder-working fairies; and what is more […] though they themselves will always remain invisible, yet you will see their wonderful power at work everywhere around you. […] There is only one gift we must have before we can learn to know them - we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind, of that which is, though it is invisible to us." (Arabella Buckley, Fairyland, 1879)

"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [President’s address] 1883)

"It [astronomy] is, above all others, a science that cultivates the imagination. Yet its theories and distances are based upon rigorous mathematical demonstrations. Thus the study has at once the beauty of poetry and the exactness of Geometry." (Joel D Steele," The Story of the Stars: New Descriptive Astronomy", 1884)

"I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the ‘Law of Frequency of Error’. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along." (Sir Francis Galton, "Natural Inheritance", 1889)

"It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. […] An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence. Some people hate the very name of statistics, but I find them full of beauty and interest. Whenever they are not brutalised, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary. They are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man." (Sir Francis Galton,"Natural Inheritance", 1889)

"I do think [...] that you would find it would lose nothing by omitting the word 'vector' throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, [Letter to Robert B Hayward] 1892) 

"Considering the remarkable elegance, generality, and simplicity of the method [Homer’s Method of finding the numerical values of the roots of an equation], it is not a little surprising that it has not taken a more prominent place in current mathematical textbooks. [...] As a matter of fact, its spirit is purely arithmetical; and its beauty, which can only be appreciated after one has used it in particular cases, is of that indescribably simple kind, which distinguishes the use of position in the decimal notation and the arrangement of the simple rules of arithmetic. It is, in short, one of those things whose invention was the creation of a commonplace." (George Chrystal, "Algebra", 1893)

"We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry. There will come with the greater love of science greater love to one another. Living more nearly to Nature is living farther from the world and from its follies, but nearer to the world's people; it is to be of them, with them, and for them, and especially for their improvement. We cannot see how impartially Nature gives of her riches to all, without loving all, and helping all; and if we cannot learn through Nature's laws the certainty of spiritual truths, we can at least learn to promote spiritual growth while we are together, and live in a trusting hope of a greater growth in the future." (Maria Mitchell, "Maria Mitchell: Life, Letters and Journals", 1896)

🥀On Elegance (1925-1949)

 "Rhythm, symmetry, and a happy combination of elegance and utility - a blend often desired in later days of hope and struggle - these have been fully attained, and with them a delight in quiet communion with Nature, expressing as she does the sense of beauty in orderliness." (Marie-Luise Gothein, "A History of Garden Art", 1928)

"A system of laws may be beautiful, or a mathematical proof may be elegant, although no auditory or visual experience is directly involved in either case. It would seem indeed that all feeling of desirability which is more than mere appetite has some claim to be regarded as aesthetic feeling." (George D Birkhoff, "Aesthetic Measure", 1933)

"It is sometimes said of two expositions of one and the same mathematical proof that the one is simpler or more elegant than the other. This is a distinction which has little interest from the point of view of the theory of knowledge; it does not fall within the province of logic, but merely indicates a preference of an aesthetic or pragmatic character." (Karl R Popper,"The Logic of Scientific Discovery", 1934)

"Mathematicians themselves set up standards of generality and elegance in their exposition which are a bar to understand." (Kenneth E Boulding, "Economic Analysis", 1941)

"One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects ‘elegance’ in its ‘architectural’ structural makeup." (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I (1), 1947])

"As an Art, Mathematics has its own standard of beauty and elegance which can vie with the more decorative arts. In this it is diametrically opposed to a Baroque art which relies on a wealth of ornamental additions. Bereft of superfluous addenda, Mathematics may appear, on first acquaintance, austere and severe. But longer contemplation reveals the classic attributes of simplicity relative to its significance and depth of meaning." (Dudley E Littlewood,"The Skeleton Key of Mathematics", 1949)

On Assumptions V

"Pick the assumptions to pieces till the stuff they are made of is exposed to plain view - this is the cardinal rule for understanding the basis of our beliefs." (Eric T Bell, "The Search for Truth", 1934)

"No step should ever be taken, and no statement should ever be made, without giving a legitimate reason therefor. A 'legitimate' reason means a reference to a definition, to an assumption, or to a proposition previously proved; it may also include reference to a statement in the hypothesis, or to a line arbitrarily drawn at the outset of the demonstration." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"Whenever we discuss a geometric statement in order to prove it, we refer to previously proved statements for 'evidence'. In this process of referring back to earlier statements that have already been established by proof, it is clear that some few of the earliest statements must of necessity remain unproved - that is, they have to be taken for granted, since we must begin our chain of reasoning somewhere. It is not possible to prove every statement. Those statements which are thus taken for granted without proof are called assumptions. Hence in all logical reasoning, two considerations are of prime importance: (1) the meanings of the terms used, and (2) the basic assumptions made." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"A bad model is a combination of assertions, some factual, others conjectural, and others plainly false but convenient. [..]  By definition, a bad model does not give power to see-accurately, deeply, or at all-into the actual situation, but only into the assertions embodied in the model. Thus, if the use of a bad model provides insight, it does so not by revealing truth about the world but by revealing its own assumptions and thereby causing its user to go learn something about the world." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"In everyday language, an axiom is often held to be a 'self-evident truth'. That phrase betrays some awfully sloppy thinking. A truth can be evident, but self-evident? Evidence is what convinces people that something is true. So what on earth is 'self-evident' supposed to mean? A truth that convinces itself? [...] To those who were seeking foundations for mathematics, axioms were much more prosaic. They weren't truths at all, let alone evident ones, and certainly not self-evident ones - assuming that such a slogan means anything at all. Axioms were assumptions. A place to start. A collection of statements that mathematicians agreed to accept. You are free to challenge them if you wish, but even if you do, you won't change mathematics (though you might create some more, branching off in a new direction). From the axiomatic viewpoint, mathematics consists of the deductions that are made, once the axioms are accepted as a starting point. It's like a game. If you want to play football, then you follow the rules of football. Of course you are free to change the rules - but then you're playing a different game." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"When people question assumptions, the map may clarify what they are. When logic is challenged, the map may help. When people want to know how goals and strategies are linked, the map may show how they are. The map does not make the decisions. Rather, it provides a record that preserves complexity, yet organizes and categorizes that complexity in such a way that people can understand and manage it. And if more mapping needs to be done, the map is there as a base on which to build." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

On Definitions VII

"No step should ever be taken, and no statement should ever be made, without giving a legitimate reason therefor. A 'legitimate' reason means a reference to a definition, to an assumption, or to a proposition previously proved; it may also include reference to a statement in the hypothesis, or to a line arbitrarily drawn at the outset of the demonstration." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"To define a term adequately means to describe it in such a way that no doubt or ambiguity arises as to its meaning when that term is used. A good definition should indicate to what larger group of objects the object defined belongs, as well as how that object differs from other objects of that group. The larger group is sometimes called the genus, and the smaller, special group is called the species. Geometric objects, i.e., figures, are for the most part easy to define with precision." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"The definition of a problem and the action taken to solve it largely depend on the view which the individuals or groups that discovered the problem have of the system to which it refers. A problem may thus find itself defined as a badly interpreted output, or as a faulty output of a faulty output device, or as a faulty output due to a malfunction in an otherwise faultless system, or as a correct but undesired output from a faultless and thus undesirable system. All definitions but the last suggest corrective action; only the last definition suggests change, and so presents an unsolvable problem to anyone opposed to change." (Herbert Brün, "Technology and the Composer", 1971)

"A definition in mathematics is an exercise in uncovering the essence of things, one reason that good definitions are so hard to pull off, since a definition brings the essence to light, and the light brings the definition to life." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Definitions pin things down, they limit the prospects for creativity and diversity. A definition, implicitly, attempts to reduce all possible variations of a concept to a single pithy phrase." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Whenever you create or define a mathematical object, it always carries with it the blueprint of its own construction - the defining features that make it what it is and not some other thing." (Paul Lockhart, "Measurement", 2012)

🧊On Geometry: On Curves (2020-2029)

"Skewed data means data that is shifted in one direction or the other. Skewness can cause machine learning models to underperform. Many machine learning models assume normally distributed data or data structures to follow the Gaussian structure. Any deviation from the assumed Gaussian structure, which is the popular bell curve, can affect model performance. A very effective area where we can apply feature engineering is by looking at the skewness of data and then correcting the skewness through normalization of the data." (Anthony So et al, "The Data Science Workshop" 2nd Ed., 2020)

"Correlation does not imply causation: often some other missing third variable is influencing both of the variables you are correlating. […] The need for a scatterplot arose when scientists had to examine bivariate relations between distinct variables directly. As opposed to other graphic forms - pie charts, line graphs, and bar charts - the scatterplot offered a unique advantage: the possibility to discover regularity in empirical data (shown as points) by adding smoothed lines or curves designed to pass 'not through, but among them', so as to pass from raw data to a theory-based description, analysis, and understanding." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

"Standard graphs, like bar and line charts, are so common because they are perceptually more accurate, familiar to people, and easy to create. Nonstandard graphs - those that use circles or curves, for instance - may not allow the reader to most accurately perceive the exact data values. But perceptual accuracy is not always the goal. And sometimes it's not a goal at all. Spurring readers to engage with a graph is sometimes just as important. Sometimes, it's more important. And nonstandard chart types may do just that. In some cases, nonstandard graphs may help show underlying patterns and trends in better ways that standard graphs. In other cases, the fact that these nonstandard graphs are different may make them more engaging, which we may sometimes need to first attract attention to the visualization."  (Jonathan Schwabish, "Better Data Visualizations: A guide for scholars, researchers, and wonks", 2021)

"Ignoring sampling weights can give a misleading presentation of a distribution. Whether for a histogram, bar plot, box plot, two-dimensional contour, or smooth curve, we need to use the weights to get a representative plot." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Side-by-side box plots offer a similar comparison of distributions across groups. The box plot offers a simpler approach that can give a crude understanding of a distribution. Likewise, violin plots sketch density curves along an axis for each group. The curve is flipped to create a symmetric 'violin' shape. The violin plot aims to bridge the gap between the density curve and box plot." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Smoothing and aggregating can help us see important features and relationships, but when we have only a handful of observations, smoothing techniques can be misleading. With just a few observations, we prefer rug plots over histograms, box plots, and density curves, and we use scatterplots rather than smooth curves and density contours. This may seem obvious, but when we have a large amount of data, the amount of data in a subgroup can quickly dwindle. This phenomenon is an example of the curse of dimensionality." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Stacked line plots are even more difficult to read because we have to judge the gap between curves as they jiggle up and down." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)