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: On Nature (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)