On Equations II

"I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all incumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run. " (Henry David Thoreau)

"Equations are the mathematician's way of working out the value of some unknown quantity from circumstantial evidence. ‘Here are some known facts about an unknown number: deduce the number.’ An equation, then, is a kind of puzzle, centered upon a number. We are not told what this number is, but we are told something useful about it. Our task is to solve the puzzle by finding the unknown number." (Ian Stewart, “Why Beauty Is Truth”, 2007)

“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding" (William Paul Thurston)

"It often happens that understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solution." (Freeman J Dyson)

”I consider that I understand an equation when I can  predict the properties of its solutions, without actually solving it.” (Paul A M Dirac)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain ‘recipes’ for calculating things." (Ian Stewart, “Why Beauty Is Truth”, 2007)

“No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect.” (Arthur C Clarke, “Profiles of the Future”, 1973)

“[…] equations are like poetry: They speak truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend.” (Michael Guillen, “Five Equations That Changed the World”, 1995)

"To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas." (Ivars Peterson, “The Mathematical Tourist”, 1988)

"In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." (Michael Atiyah, “The Art of Mathematics” [in “Art in the Life of Mathematicians”])

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." (Nikola Tesla)

On Equations I: Nature I (Unsourced)

”An equation for me has no meaning unless it expresses a thought of God.” (Srinivasa Ramanujan)

“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." (William P Thurston)

"Our sunsets have been reduced to wavelengths and frequencies. The complexities of the universe have been shredded into mathematical equations. Even our self-worth as human beings has been destroyed." (Dan Brown)

“The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.” (Auguste Comte)

"The equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle." (Edward Witten)

“The way physics explains Nature is to speak in terms of the consequences of a few very basic equations.” (Ekkehard Peik)

“This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Heinrich Hertz)

"To most outsiders, modern mathematics is unknown territory.  Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts.  Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.” (Ivars Peterson)

On Chaos I

“Every deep thinker and observer of the Natural Laws is convinced that Nature is an orderly arrangement of matter and forces; that, in a word, Nature is not chaos, but cosmos.” (Frederick Hovenden, “What is Life?”, 1899) 

"[…] there is a God precisely because Nature itself, even in chaos, cannot proceed except in an orderly and regular manner." (Immanuel Kant) “There is no such thing as chaos, it tacitly asserts, in the sidereal world or outside of it. For chaos is the negation of law, and law is the expression of the will of God.” (Agnes M Clerke, “Problems in Astrophysics”,  1903)

“Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals - such were its abstract characteristics.” (Herbert E Read, “Icon and Idea”, 1955)

“Chaos is but unperceived order; it is a word indicating the limitations of the human mind and the paucity of observational facts. The words ‘chaos’, ‘accidental’, ‘chance’, ‘unpredictable’ are conveniences behind which we hide our ignorance.” (Harlow Shapley, “Of Stars and Men”, 1958) 

“One of mankind’s earliest intellectual endeavors was the attempt to gather together the seemingly overwhelming variety presented by nature into an orderly pattern. The desire to classify - to impose order on chaos and then to form patterns out of this order on which to base ideas and conclusions - remains one of our strongest urges.” (Roger L Batten, 1959) 

“The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos.” (Herbert A Simon, “The Sciences of the Artificial”, 1969)

”Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the fluctuations of the wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities.” (James Gleick, “Chaos”, 1987)

“The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done.” (Ian Stewart, “Does God Play Dice?”, 1989)

“We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty.” (Ivars Peterson, “Newton’s Clock”, 1993)

“The voyage of discovery into our own solar system has taken us from clockwork precision into chaos and complexity. This still unfinished journey has not been easy, characterized as it is by twists, turns, and surprises that mirror the intricacies of the human mind at work on a profound puzzle. Much remains a mystery. We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty.” (Ivars Peterson, “Newton’s Clock”, 1993)

Nature and Mathematics V (Mathematics as the Language of Nature)

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." (Galileo Galilei, “The Assayer”, 1623)

”The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.” (Johannes Kepler)

“Whether man is disposed to yield to nature or to oppose her, he cannot do without a correct understanding of her language.” (Jean Rostand)

"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff)

"The laws of nature are drawn from experience, but to express them one needs a special language: for, ordinary language is too poor and too vague to express relations so subtle, so rich, so precise. Here then is the first reason why a physicist cannot dispense with mathematics: it provides him with the one language he can speak […]. Who has taught us the true analogies, the profound analogies which the eyes do not see, but which reason can divine? It is the mathematical mind, which scorns content and clings to pure form.” (Henri Poincare, Analysis and Physics)

“The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” (Eugene P Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” 1960)

“Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music.” (Northrop Frye, “The Educated Imagination”, 2002)

"Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework." (Melvin Schwartz, "Principles of Electrodynamics", 1972)

"Mathematics is a way of expressing natural laws, it is the easiest and best way to describe a general law or the flow of a phenomenon, it is the most perfect language in which one can narrate a natural phenomenon." (Gheorghe Ţiţeica)

“To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. […] If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” (Richard P Feynman, “The Character of Physical Law”, 1967)

”Nature responds only to questions posed in mathematical language, because nature is the domain of measure and order.” (Alexandre Koyré)

Mathematics as Language I

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." (Eugene Wigner)

"I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language." (Werner Heisenberg)

"Such is the advantage of a well-constructed language that its simplified notation often becomes the source of profound theories." (Pierre-Simon Laplace)

"Mathematics is pure language - the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms. (In a spoken language, there exist words, like "happiness", that defy definition.) It is also an art - the most intellectual and classical of the arts." (Alfred Adler)

”Mathematical language, precise and adequate, nay, absolutely convertible with mathematical thought, can afford us no example of those fallacies which so easily arise from the ambiguities of ordinary language; its study cannot, therefore, it is evident, supply us with any means of obviating those illusions from which it is itself exempt. The contrast of mathematics and philosophy, in this respect, is an interesting object of speculation; tut, as imitation is impossible, one of no practical result.” (William Hamilton)

“What is the inner secret of mathematical power? Briefly stated, it is that mathematics discloses the skeletal outlines of all closely articulated relational systems. For this purpose mathematics uses the language of pure logic with its score or so of symbolic words, which, in its important forms of expression, enables the mind to comprehend systems of relations otherwise completely beyond its power. These forms are creative discoveries which, once made, remain permanently at our disposal. By means of them the scientific imagination is enabled to penetrate ever more deeply into the rationale of the universe about us.” (George D Birkhoff)

“[…] mathematics has liberated itself from language; and one who knows the tremendous labor put into this process and its ever-recurring surprising success, cannot help feeling that mathematics nowadays is more efficient in it particular sphere of the intellectual world than, say the modern languages in their deplorable condition of decay or even music are on their fronts.” (Andreas Speiser)

“[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words.” (Charles Nordmann)

“There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction.” (Thomas Merton)

Mathematicians vs Theorems

“A mathematician, then, will be defined in what follows as someone who has published the proof of at least one non-trivial theorem.” (Jean Dieudonné)

“Everybody knows that mathematics is about Miracles, only mathematicians have a name for them: Theorems.” (Roger Howe, 1998)

“A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought.” (Paul Carus, “Friedrich Schiller”, 1905)

“[…] the mathematician is always walking upon the brink of a precipice, for, no matter how many theorems he deduces, he cannot tell that some contradiction will not await him in the infinity of consequences.” (Richard A Arms, “The Notion of Number and the Notion of Class”, 1917)

"The mathematician is still regarded as the hermit who knows little of the ways of life outside his cell, who spends his time compounding incredible and incomprehensible theorems in a strange, clipped, unintelligible jargon." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

“A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem.” (Steven Krantz, “Conformal Mappings”, 1999)

“The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done." (Godfrey H Hardy, “A Mathematician’s Apology”, 1940)

“Good mathematicians see analogies between theorems or theories, the very best ones see analogies between analogies.” (Stefan Banach)

“No mathematician nowadays sets any store on the discovery of isolated theorems, except as affording hints of an unsuspected new sphere of thought, like meteorites detached from some undiscovered planetary orb of speculation.” (James J Sylvester)

“So if you could be the Devil and offer a mathematician to sell his soul for the proof of one theorem - what theorem would most mathematicians ask for?” (H Montgomery)

On Theorems I (Unsourced)

“To state a theorem and then to show examples of it is literally to teach backwards." (E. Kim Nebeuts)

"Young men should prove theorems, old men should write books." (Godfrey H Hardy)

“We often hear that mathematics consists mainly of proving theorems. Is a writer's job mainly that of writing sentences?" (Gian-Carlo Rota)

“The product of mathematics is clarity and understanding. Not theorems, by themselves.” (Bill Thurston)

"Old theorems never die; they turn into definitions." (Edwin Hewitt)

“[…] a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences." (Godfrey H Hardy)

“I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.” (Friedrich L G Frege)

“Ah, there’s no excitement to beat the excitement of proving a theorem! Until you find out the next day that it’s wrong.” (Cathleen S Morawetz)

“It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all.” (James J Sylvester)

“I think that science may be styled the knowledge of universals, or abstract wisdom; and art is science reduced to practice - or science is reason, and art the mechanism of it - and may be called practical science. Science, in fine, is the theorem, and art the problem.” (Laurence Sterne)

Nature and Mathematics VI (Mathematics and the Laws of Nature)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George Ogden Abell)

 "Natural Philosophy consists in discovering the frame and operations of Nature, and reducing them, as far as may be, to general Rules or Laws - establishing these rules by observations and experiments, and thence deducing the causes and effects of things.” (Isaac Newton)

"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them."(Albert Einstein, “Principles of Research”, 1918)

"The simplicities of natural laws arise through the complexities of the language we use for their expression.” (Eugene Wigner)

“It is impossible to transcend the laws of nature. You can only determine that your understanding of nature has changed." (Nick Powers)

“The secret of nature is symmetry. When searching for new and more fundamental laws of nature, we should search for new symmetries.” (David Gross)

“The laws of Nature are written in the language of mathematics […]” (Galileo Galilei)

“We know many laws of nature and we hope and expect to discover more. Nobody can foresee the next such law that will be discovered. Nevertheless, there is a structure in laws of nature which we call the laws of invariance. This structure is so far-reaching in some cases that laws of nature were guessed on the basis of the postulate that they fit into the invariance structure.” (Eugene P Wigner)

Mathematics and Its Laws

"The laws of mathematics are not merely human inventions or creations. They simply ‘are’ they exist quite independently of the human intellect. The most that any man with a keen intellect can do is to find out that they are there and to take cognizance of them." (M C Escher)

“In mathematics there is no understanding. In mathematics there are only necessities, laws of existence, invariant relationships. Thus any mathematico-mechanistic outlook must, in the last analysis, waive all understanding. For, we only understand when we know the motives; where there are no motives, all understanding ceases.” (Friedrich Nietzsche)

“People think of axioms as laws you have to follow, or true things you have to assume, and I think neither of these perspectives is correct. It's more accurate to think of axioms as a way to agree that we're talking about the same thing." (Qiaochu Yuan)

“All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true." (Georg C Lichtenberg)

"All the mathematical sciences are founded on the relations between physical laws and laws of numbers.” (James C Maxwell)

“Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.” (Melvin Schwartz, “Principles of Electrodynamics”, 1972)

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." (Eugene Wigner)

"Mathematics is a way of expressing natural laws, it is the easiest and best way to describe a general law or the flow of a phenomenon, it is the most perfect language in which one can narrate a natural phenomenon." (Gheorghe Ţiţeica)

Reality and Mathematics

"How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality." (Albert Einstein)

"We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori." (Karl Friedrich Gauss, 1830)

"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite)

"Mathematics is not only real, but it is the only reality. [The] entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." (Martin Gardner)

“I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply the notes of our observations." (Godfrey H Hardy, “A Mathematician's Apology”, 1941)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, “The Role of Mathematics in the Rise of Science”, 1981)

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw Ulam, “Adventures of a Mathematician”, 1976)

“On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols’ […]. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.” (Jean Dieudonné)

“Human thought, flying on the trapezes of the star-filled universe, with mathematics stretched beneath, was like an acrobat working with a net but suddenly noticing that in reality there is no net.” (Vladimir Nabokov)

“A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.” (Henri Poincaré)

“Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.” (Edward Frenkel, “Love and Math”, 2014)

On Topology I

"Topology is geometry without distance or angle. The geometrical objects of study, not rigid but rather made of rubber or elastic, are especially stretchy." (Stephen Huggett & David Jordan, “A Topological Aperitif”, 2009)

“In geometry, topology is the study of properties of shapes that are independent of size or shape and are not changed by stretching, bending, knotting, or twisting.” (M C Escher, 1971)

“Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything.” (Edward Witten)

"Topology is the science of fundamental pattern and structural relationships of event constellations." (R Buckminster Fuller)

“Topology is an elastic version of geometry that retains the idea of continuity but relaxes rigid metric notions of distance.” (Samuel Eilenberg)

"Topology is precisely that mathematical discipline which allows a passage from the local to the global." (René Thom)

"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl)

“The geometry of Algebraic Topology is so pretty, it would seem a pity to slight it and miss all the intuition which it provides. At deeper levels, algebra becomes increasingly important, so for the sake of balance it seems only fair to emphasize geometry at the beginning.” (Allen Hatcher)

Imagination in Mathematics

"One factor that has remained constant through all the twists and turns of the history of physics is the decisive importance of the mathematical imagination." (Freeman J Dyson)

"The moving power of mathematical invention is not reasoning but imagination." (Augustus De Morgan)

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." (John W N Sullivan)

"The mere formulation of a problem is often far more essential than its solution. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advances in science." (Albert Einstein)

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity.” (Howard W Eves, “In Mathematical Circles”, 1969)

“Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there." (Meir Shalev)

"One of the most fundamental notions in mathematics is that of number. Although the idea of number is basic, the numbers themselves possess both nuance and complexity that spark the imagination." (Edward B Burger, “Exploring the Number Jungle”, 2000)

"The mathematician's best work is art […] a high and perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch each other." (Gustav Mittag-Leffler)

"The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning." (Alfred North Whitehead, “A Treatise on Universal Algebra”, 1898)

“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”, 1962)

On Complex Numbers I

“[…] neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^3 - 6xx + 13x - 10 = 0 as having three roots, yet there is just one real root, which is 2, and the other two [2+i and 2-i]], however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.” (René Descartes, “Gemetry”, 1637)

“We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible.” (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)

“Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being.” (Gottfried Leibniz, 1702)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -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 the question." (Carl Friedrich Gauss)

“All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.” (Euler, Algebra, 1770)

“In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just as thoroughly existent as the finite positive integers.” (Georg Cantor)

“One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?” (Stephen W Hawking, “The Universe in a Nutshell”, 2001).

“The shortest path between two truths in the real domain passes through the complex domain." (Jacque Hadamard [misquoted])

“The origin and immediate purpose of the introduction of complex magnitudes into mathematics lie in the theory of simple laws of dependence between variable magnitudes expressed by means of operations on magnitudes. If we enlarge the scope of applications of these laws by assigning to the variables they involve complex values, then there appears an otherwise hidden harmony and regularity.” (Heinz-Dieter Ebbinghaus et al., “Numbers”, 1983)

“The more science I studied, the more I saw that physics becomes metaphysics and numbers become imaginary numbers. The farther you go into science, the mushier the ground gets. You start to say, 'Oh, there is an order and a spiritual aspect to science.’” (Dan Brown)

"Adam and Eve are like imaginary number, like the square root of minus one… If you include it in your equation, you can calculate all manners of things, which cannot be imagined without it." (Philip Pullman)

See also:
5 Books 10 Quotes: Complex Numbers
More Quotes on Complex Numbers III
More Quotes on Complex Numbers II
More Quotes on Complex Numbers I 

Algebra

"[…] algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world." (Alfred North Whitehead, “The Organization of Thought”, 1974)

 "The symbolism of algebra is its glory. But it also is its curse." (William Betz) "Algebra begins with the unknown and ends with the unknowable." (Anonymous)

“Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable.” (Alfred North Whitehead)

“We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.”  (William K Clifford, “Common Sense of the Exact Sciences”, 1885)

“The word is of Arabic origin. "Al" is the Arabic article the, and "gebar" is the verb to set, to restitute.” (Tobias Dantzig & Joseph Mazur, “Number: The Language of Science”, 1930)

”Nothing proves more clearly that the mind seeks truth, and nothing reflects more glory upon it, than the delight it takes, sometimes in spite of itself, in the driest and thorniest researches of algebra.” (Bernard de Fontenelle, “Histoire du Renouvellement de l'Académie des Sciences”, 1708)

”The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process.” (Augustus de Morgan, “On the Study and Difficulties of Mathematics”, 1830)

“The science of algebra, independently of any of its uses, has all the advantages which belong to mathematics in general as an object of study, and which it is not necessary to enumerate. Viewed either as a science of quantity, or as a language of symbols, it may be made of the greatest service to those who are sufficiently acquainted with arithmetic, and who have sufficient power of comprehension to enter fairly upon its difficulties.” (Augustus de Morgan, “Elements of Algebra”, 1837)

“They that are ignorant of Algebra cannot imagine the wonders in this kind are to be done by it: and what further improvements and helps advantageous to other parts of knowledge the sagacious mind of man may yet find out, it is not easy to determine. This at least I believe, that the ideas of quantity are not those alone that are capable of demonstration and knowledge; and that other, and perhaps more useful, parts of contemplation, would afford us certainty, if vices, passions, and domineering interest did not oppose and menace such endeavors.” (John Locke, “An Essay Concerning Human Understanding”, 1689)

”Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra.” (Susanne K Langer, “Philosophy in a New Key”, 1957)

“Every moment of time dictated and determined the following moment, and was itself dictated and determined by the moment that came before it. Everything was calculable: everything happened because it must; the commandments were erased from the tables of the law; and in their place came the cosmic algebra: the equations of the mathematicians.” (George B Shaw, “Too True to Be Good”, 1934)

“Algebra is applied to the clouds, the irradiation of the planet benefits the rose, and no thinker would dare to say that the perfume of the hawthorn is useless to the constellation.” (Victor Hugo, “Les Miserables”, 1938)

“In mathematics itself abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics  and that of a research subject with a highly active life of its own.” (Israel N Herstein, ”Abstract Algebra”, 1986)

On Truth (Unsourced)

"Truth in science can be defined as the working hypothesis best suited to open the way to the next better one.” (Konrad Lorenz)

“Truth is what stands the test of experience.” (Albert Einstein)

“The truth is that which works.” (John Dewey)

"Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.” (Isaac Newton)

“The opposite of a correct statement is a false statement. The opposite of a profound truth may well be another profound truth.” (Niels Bohr)

“Every truth is true only up to a point. Beyond that, by way of counter-point, it becomes untruth.” (Søren Kierkegaard)

“All truths are easy to understand once they are discovered; the point is to discover them.” (Galileo Galilei)

“Truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not gold.” (Lev Nikolaevich Tolstoy)

“It is the mark of an instructed mind to rest assured with that degree of precision that the nature of the subject admits, and not to seek exactness when only an approximation of the truth is possible.” (Aristotle)

“We can never achieve absolute truth, but we can live hopefully by a system of calculated probabilities. The law of probability gives to natural and human sciences - to human experience as a whole - the unity of life we seek.” (Agnes Meyer)

On Truth: Geometrical Truth

"The connected course of reasoning by which any Geometrical truth is established is called a demonstration." (Robert Potts, "Euclid's Elements of Geometry", 1845)

"In all cases, however, it must be kept in view that every geometrical truth is deduced by a comparison between two others, which agree, one in one particular part, and the other in another, with the conclusion so deduced." (?,"Miscallanea Mathematica", American Railroad Journal, No. X, 1846)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"Geometrical truth is a product of reason; that makes it superior to empirical truth, which is found through generalization of a great number of instances." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1954)

"Conventionalism as geometrical and mathematical truths are created by our choices, not dictated by or imposed on us by scientific theory. The idea that geometrical truth is truth we create by the understanding of certain conventions in the discovery of non-Euclidean geometries." (Clifford Singer, "Engineering a Visual Field", 1955)

"To enter a temple constructed wholly of invariable geometric proportions is to enter an abode of eternal truth." (Robert Lawlor, "Sacred Geometry", 1982)

"Geometrical truth is (as we now speak) synthetic: it states facts about the world. Such truths are not ordinary truths but essential truths, giving the reality of the empirical world in which they are imperfect embodied." (Fred Wilson, "The External World and Our Knowledge of It", 2008)

Truth and Error (Unsourced)

"It is much easier to meet with error than to find truth; error is on the surface, and can be more easily met with; truth is hid in great depths, the way to seek does not appear to all the world." (Johann Wolfgang von Goethe)

"Error is just as important a condition of life as truth." (Carl Gustav Jung)

"Give me fruitful error any time, full of seeds, bursting with its own corrections. You can keep your sterile truth for yourself." (Vilfredo Pareto)

"All errors spring up in the neighborhood of some truth; they grow round about it, and, for the most part, derive their strength from such contiguity." (Thomas Binney)

"In all science, error precedes the truth, and it is better it should go first than last." (Hugh Walpole)

"All extremes are error. The reverse of error is not truth, but error still. Truth lies between these extremes." (Lord David Cecil)

"An error is the more dangerous in proportion to the degree of truth which it contains." (Henri-Frédéric Amiel)

"Truth emerges more readily from error than from confusion." (Francis Bacon)

"A few observation and much reasoning lead to error; many observations and a little reasoning to truth." (Alexis Carrel)

 "An error does not become truth by reason of multiplied propagation, nor does truth become error because nobody sees it." (Mahatma Gandhi)

"To rise from error to truth is rare and beautiful." (Victor Hugo)

On Truth: Mathematical Truth (Unsourced)

"A mathematical truth is neither simple nor complicated in itself, it is." (Émile Lemoine)

"Beauty in mathematics is seeing the truth without effort." (George Pólya)

"Either one or the other [analysis or synthesis] may be direct or indirect. The direct procedure is when the point of departure is known-direct synthesis in the elements of geometry. By combining at random simple truths with each other, more complicated ones are deduced from them. This is the method of discovery, the special method of inventions, contrary to popular opinion." (André-Marie Ampère)

"Geometrical truths are in a way asymptotes to physical truths, that is to say, the latter approach the former indefinitely near without ever reaching them exactly." (Jean le Rond D’Alembert)

"[It used to be that] geometry must, like logic, rely on formal reasoning in order to rebut the quibblers. But the tables have turned. All reasoning concerned with what common sense knows in advance, serves only to conceal the truth and to weary the reader and is today disregarded." (Alexis C Clairaut)

"Math is the only place where truth and beauty mean the same thing." (Danica McKellar)

"Mathematics connect themselves on the one side with common life and the physical sciences; on the other side with philosophy, in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics, and the foundation of our knowledge of them." (Arthur Cayley)

"Mathematics is a form of poetry which transcends poetry in that it proclaims the truth, a form of reasoning which transcends reasoning in that it wants to accomplish the truth it proclaims, a form of action, ritual behavior, which finds fulfillment in the act, but must proclaim and elaborate a poetic form of truth." (Salomon Bochner)

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz)

"Show all these fanatics a little geometry, and they learn it quite easily. But, strangely enough, their minds are not thereby rectified. They perceive the truths of geometry, but it does not teach them to weigh probabilities. Their minds have set hard. They will reason in a topsy-turvy wall all their lives, and I am sorry for it." (Voltaire)

"The character of necessity ascribed to the truths of mathematics and even the peculiar certainty attributed to them is an illusion." (John S Mill)

"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite) [in The Mathematical Intelligencer, Vol. 5, No. 4, 1983)

Intuition and Mathematics

"The two operations of our understanding, intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge." (René Descartes)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. […] The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasonings." (Alan Turing)

"[…] all mathematical cognition has this pecularity: that it must first exhibit its concept in intuitional form. […] Without this, mathematics cannot take a single step. Its judgements are therefore always intuitional, whereas philosophy must make do with discursive judgements from mere concepts. It may illustrate its judgements by means of a visual form, but it can never derive them from such a form.” (Immanuel Kant)

"The object of mathematical rigor is to sanction and legimize the conquests of intuition, and there never was any other object for it." (Jacques S Hadamard)

“Logic merely sanctions the conquests of the intuition.” (Jacques S Hadamard)

"It is by logic we prove; it is by intuition we discover." (Henri Poincaré)

"For, compared with the immense expanse of modern mathematics, what would the wretched remnants mean, the few isolated results incomplete and unrelated, that the intuitionists have obtained." (David Hilbert)

“Living mathematics rests on the fluctuation between the antithesis powers of intuition and logic, the individuality of 'grounded' problems and the generality of far-reaching abstractions. We ourselves must prevent the development being forced to only one pole of the life-giving antithesis.” (Richard Courant, 1962)

“Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool.” (Charles P Steinmetz, “Transactions of the American Institute of Electrical Engineers”, 1909)

 "Mathematics as an expression of the human mind reflects the active will, the contemplative reason. and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science." (Richard Courant ‎& Herbert Robbins, “What is Mathematics?”, 1941)

“All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things.” (Max Born, “Physical Reality”, Philosophical Quarterly, Vol. 3, No. 11,1953)

 “Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practice to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuitions of man.” (Morris Kline, “Mathematics in Western Culture”, 1953)

“Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor.” (Ian Stewart, “Concepts of Modern Mathematics”, 1995)

"Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.” (Paul Lockhart, “A Mathematician's Lament”, 2009)

What is Intuition?

"Intuition is the source of scientific knowledge." (Aristotle)

"Knowledge has three degrees - opinion, science, illumination. The means or instrument of the first is sense; of the second, dialectic; of the third, intuition." (Plotinus)

"Intuition is the conception of an attentive mind, so clear, so distinct, and so effortless that we cannot doubt what we have so conceived.” (René Descartes)

“The disclosure of a new fact, the leap forward, the conquest over yesterday’s ignorance, is an act not of reason but of imagination, of intuition.” (Charles Nicolle)

"The supreme task is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can lead to them." (Albert Einstein)

“Intuition is perception via the unconscious that brings forth ideas, images, new possibilities and ways out of blocked situations.” (Carl Jung)

“Intuition is the art, peculiar to the human mind, of working out the correct answer from data that is, in itself, incomplete or even, perhaps, misleading.” (Isaac Asimov, “Forward the Foundation”, 1993)

“Intuition, mind’s originary act of ‘seeing’ what is given to him […]” (Hermann Weyl, “Mind and Nature”, 2009)

 “Intuition is the supra-logic that cuts out all routine processes of thought and leaps straight from the problem to the answer.” (Robert Graves)

"This covert mechanism would be the source of what we call intuition, the mysterious mechanism by which we arrive at the solution of a problem without reasoning toward it." (Antonio Damasio, “Descartes' Error”, 2005)

"Patterns experienced again and again become intuitions. […] Intuitive judgments are made by our use of imagery; intuition is the result of mental model building. […] The mental model used and the form of the intuition is dependent upon the question being answered." (Roger Frantz, “Two Minds”, 2005)

"An intuition is neither caprice nor a sixth sense but a form of unconscious intelligence." (Gerd Gigerenzer, “Risk Savvy”, 2015)