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Equation Quotes (93 quotes)

Natura non facit saltum or, Nature does not make leaps… If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).
From Benoit B. Mandelbrot and Richard Hudson, The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward (2004,2010), 85-86.
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Thomasina: Every week I plot your equations dot for dot, x’s against y’s in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?
Septimus: We do.
Thomasina: Then why do your shapes describe only the shapes of manufacture?
Septimus: I do not know.
Thomasina: Armed thus, God could only make a cabinet.
In the play, Acadia (1993), Scene 3, 37.
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Toutes les fois que dans une équation finale on trouve deux quantités inconnues, on a un lieu, l'extrémité de l'une d’elles décrivant une ligne droite ou courbe. La ligne droite est simple et unique dans son genre; les espèces des courbes sont en nombre indéfini, cercle, parabole, hyperbole, ellipse, etc.
Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many,—circle, parabola, hyperbola, ellipse, etc.
Introduction aux Lieux Plans et Solides (1679) collected in OEuvres de Fermat (1896), Vol. 3, 85. Introduction to Plane and Solid Loci, as translated by Joseph Seidlin in David E. Smith(ed.)A Source Book in Mathematics (1959), 389. Alternate translation using Google Translate: “Whenever in a final equation there are two unknown quantities, there is a locus, the end of one of them describing a straight line or curve. The line is simple and unique in its kind, species curves are indefinite in number,—circle, parabola, hyperbola, ellipse, etc.”
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A good deal of my research in physics has consisted in not setting out to solve some particular problem, but simply examining mathematical quantities of a kind that physicists use and trying to fit them together in an interesting way, regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later to have an application. Then one has good luck. At age 78.
International Journal of Theoretical Physics (1982), 21, 603. In A. Pais, 'Playing With Equations, the Dirac Way'. Behram N. Kursunoglu (Ed.) and Eugene Paul Wigner (Ed.), Paul Adrien Maurice Dirac: Reminiscences about a Great Physicist (1990), 110.
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A great deal of my work is just playing with equations and seeing what they give.
Quoted in Frank Wilczek, ',The Dirac Equation'. Proceedings of the Dirac Centennial Symposium (2003), 45.
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A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
In Time Enough for Love: The Lives of Lazarus Long (1973), 265.
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A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain—the medial orbito-frontal cortex—like looking at a great painting or listening to a piece of music. … Neuroscience can’t tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved; you can find beauty in anything.
As quoted in James Gallagher, 'Mathematics: Why The Brain Sees Maths As Beauty,' BBC News (13 Feb 2014), on bbc.co.uk web site.
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Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.
J. C. Maxwell on Louis Poinsôt (1777-1859) in 'On a Dynamical Top' (1857). In W. D. Niven (ed.), The Scientific Papers of James Clerk Maxwell (1890), Vol. 1, 248.
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An equation means nothing to me unless it expresses a thought of God.
Quoted in Clifford A. Pickover, A Passion for Mathematics (2005), 1; but with no footnote to primary source.
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Edward [Teller] isn’t the cloistered kind of scientist. He gets his ideas in conversation and develops them by trying them out on people. We were coming back from Europe on the Ile de France and I was standing in the ship’s nightclub when he came up and said, 'Freddie, I think I have an idea.’ It was something he’d just thought of about magnetohydrodynamics. I was a bachelor then and I’d located several good-looking girls on the ship, but I knew what I had to do, so I disappeared and started working on the calculations. I’d get something finished and start prowling on the deck again when Edward would turn up out of the night and we’d walk the deck together while he talked and I was the brick wall he was bouncing these things off of. By the end of the trip we had a paper. He’d had the ideas, and I’d done some solving of equations. But he insisted that we sign in alphabetical order, which put my name first.
As quoted in Robert Coughlan, 'Dr. Edward Teller’s Magnificent Obsession', Life (6 Sep 1954), 61-62.
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Engineers think that equations approximate the real world.
Physicists think that the real world approximates equations.
Mathematicians are unable to make the connection.
Anonymous
In Jon Fripp, Michael Fripp and Deborah Fripp, Speaking of Science (2000), 41.
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Equations are Expressions of Arithmetical Computation, and properly have no place in Geometry, except as far as Quantities truly Geometrical (that is, Lines, Surfaces, Solids, and Proportions) may be said to be some equal to others. Multiplications, Divisions, and such sort of Computations, are newly received into Geometry, and that unwarily, and contrary to the first Design of this Science. For whosoever considers the Construction of a Problem by a right Line and a Circle, found out by the first Geometricians, will easily perceive that Geometry was invented that we might expeditiously avoid, by drawing Lines, the Tediousness of Computation. Therefore these two Sciences ought not to be confounded. The Ancients did so industriously distinguish them from one another, that they never introduced Arithmetical Terms into Geometry. And the Moderns, by confounding both, have lost the Simplicity in which all the Elegance of Geometry consists. Wherefore that is Arithmetically more simple which is determined by the more simple Equation, but that is Geometrically more simple which is determined by the more simple drawing of Lines; and in Geometry, that ought to be reckoned best which is geometrically most simple.
In 'On the Linear Construction of Equations', Universal Arithmetic (1769), Vol. 2, 470.
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Equations are more important to me, because politics is for the present, but an equation is something for eternity.
Quoted in Carl Seelig (ed.), Helle Zeit, Dunkle Zeit: In Memoriam Albert Einstein (1956), 71.
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Equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge.
The Great Equations: Breakthroughs in Science: from Pythagoras to Heisenberg (2009), 17.
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Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 38. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 112-113.
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Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?
A Brief History of Time (1998), 190.
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For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations—even a calculating machine can often do so—but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols—subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes. Was it Eddington who said that we once thought if we understood 1 we understood 2, for 1 and 1 are 2, but we have since found we must learn a good deal more about “and”?
In 'The Biological Basis of Imagination', American Thought: 1947 (1947), 81.
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G=A – W
Glück gleich Arbeit weniger Widerstand.
Happiness is equal to work minus resistance.
Quoted by E. P. Hillpern (previously an assistant to Ostwald), 'Some Personal Qualities of Wilhehn Ostwald Recalled by a Former Student', Chymia (1949), 2, 59. (Hillpern had been an assistant to Ostwald)
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Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it is the truth.
In a lecture, after establishing the relation eπ/2 = i-i in a lecture, “which evidently had a strong hold on his imagination. He dropped his chalk and rubber, put his hands in his pockets, and after contemplating the formula a few minutes turned to his class and said [this quote] very slowly and impressively.” As quoted in W. E. Byerly (writing as a Professor Emeritus at Harvard University, but a former student of Peirce), 'Benjamin Peirce: II. Reminiscences', The American Mathematical Monthly (Jan 1925), 32, No. 1, 6.
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How did Biot arrive at the partial differential equation? [the heat conduction equation] … Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals.
The Tragicomical History of Thermodynamics, 1822-1854 (1980), 51.
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I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.
Quoted in F Wilczek, B Devine, Longing for the Harmonies.
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I do not personally want to believe that we already know the equations that determine the evolution and fate of the universe; it would make life too dull for me as a scientist. … I hope, and believe, that the Space Telescope might make the Big Bang cosmology appear incorrect to future generations, perhaps somewhat analogous to the way that Galileo’s telescope showed that the earth-centered, Ptolemaic system was inadequate.
From 'The Space Telescope (the Hubble Space Telescope): Out Where the Stars Do Not Twinkle', in NASA Authorization for Fiscal Year 1978: Hearings before the Committee on Commerce, Science and Transportation, United States Senate, 95th Congress, first session on S.365 (1977), 124. This was testimony to support of authorization for NASA beginning the construction of the Space Telescope, which later became known as the Hubble Space Telescope.
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I have done a terrible thing: I have postulated a particle that cannot be detected.
After postulating the existence of the neutrino, a particle with no mass and no electric charge, in order to balance an equation. In an article in Mercury, 29, 29, it is stated, without citation, that “the day after making the proposal Pauli told his colleague Walter Baade.” Webmaster has looked, but not yet found a primary source. Can you help?
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I think equation guessing might be the best method to proceed to obtain the laws for the part of physics which is presently unknown.
In his Nobel Prize Lecture (11 Dec 1965), 'The Development of the Space-Time View of Quantum Electrodynamics'. Collected in Stig Lundqvist, Nobel Lectures: Physics, 1963-1970 (1998), 177.
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I think it is a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.
At age 60.
"Interview with T. Kuhn (7 May 1963), Niels Bohr Library, American Intitute of Physics, New York. In A. Pais, 'Playing With Equations, the Dirac Way'. Behram N. Kursunoglu (Ed.) and Eugene Paul Wigner (Ed.), Paul Adrien Maurice Dirac: Reminiscences about a Great Physicist (1990), 109.
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If an angel were to tell us about his philosophy, I believe many of his statements might well sound like '2 x 2= 13'.
Lichtenberg: Aphorisms & Letters (1969), 31.
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If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways. We consider, however—this is the most essential point of the whole calculation—E to be composed of a well-defined number of equal parts and use thereto the constant of nature h = 6.55 ×10-27 erg sec. This constant multiplied by the common frequency ν of the resonators gives us the energy element ε in erg, and dividing E by ε we get the number P of energy elements which must be divided over the N resonators.
[Planck’s constant, as introduced in 1900; subsequently written e = hν.]
In 'On the theory of the energy distribution law of the normal spectrum', in D. ter Haar and Stephen G. Brush, trans., Planck’s Original Papers in Quantum Physics (1972), 40.
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If ever an equation has come into its own it is Einstein’s e = me2. Everyone can rattle it off now, from the highest to the lowest.
In Asimov on Physics (1976), 96. Also in Isaac Asimov’s Book of Science and Nature Quotations (1988), 212.
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If time is treated in modern physics as a dimension on a par with the dimensions of space, why should we a priori exclude the possibility that we are pulled as well as pushed along its axis? The future has, after all, as much or as little reality as the past, and there is nothing logically inconceivable in introducing, as a working hypothesis, an element of finality, supplementary to the element of causality, into our equations. It betrays a great lack of imagination to believe that the concept of “purpose” must necessarily be associated with some anthropomorphic deity.
In 'Epilogue', The Sleepwalkers: A History of Man’s Changing Vision of the Universe (1959, 1968), 537.
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In my youth I regarded the universe as an open book, printed in the language of physical equations, whereas now it appears to me as a text written in invisible ink, of which in our rare moments of grace we are able to decipher a small fragment.
From Epilogue in Bricks to Babel (1980).
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In the social equation, the value of a single life is nil; in the cosmic equation, it is infinite… Not only communism, but any political movement which implicitly relies on purely utilitarian ethics, must become a victim to the same fatal error. It is a fallacy as naïve as a mathematical teaser, and yet its consequences lead straight to Goya’s Disasters, to the reign of the guillotine, the torture chambers of the Inquisition, or the cellars of the Lubianka.
In 'The Invisible Writing', Arrow in the Blue: An Autobiography (1952), Vol. 2, 357.
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It is not only a decided preference for synthesis and a complete denial of general methods which characterizes the ancient mathematics as against our newer Science [modern mathematics]: besides this extemal formal difference there is another real, more deeply seated, contrast, which arises from the different attitudes which the two assumed relative to the use of the concept of variability. For while the ancients, on account of considerations which had been transmitted to them from the Philosophie school of the Eleatics, never employed the concept of motion, the spatial expression for variability, in their rigorous system, and made incidental use of it only in the treatment of phonoromically generated curves, modern geometry dates from the instant that Descartes left the purely algebraic treatment of equations and proceeded to investigate the variations which an algebraic expression undergoes when one of its variables assumes a continuous succession of values.
In 'Untersuchungen über die unendlich oft oszillierenden und unstetigen Functionen', Ostwald’s Klassiker der exacten Wissenschaften (1905), No. 153, 44-45. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 115. From the original German, “Nicht allein entschiedene Vorliebe für die Synthese und gänzliche Verleugnung allgemeiner Methoden charakterisiert die antike Mathematik gegenüber unserer neueren Wissenschaft; es gibt neben diesem mehr äußeren, formalen, noch einen tiefliegenden realen Gegensatz, welcher aus der verschiedenen Stellung entspringt, in welche sich beide zu der wissenschaftlichen Verwendung des Begriffes der Veränderlichkeit gesetzt haben. Denn während die Alten den Begriff der Bewegung, des räumlichen Ausdruckes der Veränderlichkeit, aus Bedenken, die aus der philosophischen Schule der Eleaten auf sie übergegangen waren, in ihrem strengen Systeme niemals und auch in der Behandlung phoronomisch erzeugter Kurven nur vorübergehend verwenden, so datiert die neuere Mathematik von dem Augenblicke, als Descartes von der rein algebraischen Behandlung der Gleichungen dazu fortschritt, die Größenveränderungen zu untersuchen, welche ein algebraischer Ausdruck erleidet, indem eine in ihm allgemein bezeichnete Größe eine stetige Folge von Werten durchläuft.”
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It must be admitted that science has its castes. The man whose chief apparatus is the differential equation looks down upon one who uses a galvanometer, and he in turn upon those who putter about with sticky and smelly things in test tubes. But all of these, and most biologists too, join together in their contempt for the pariah who, not through a glass darkly, but with keen unaided vision, observes the massing of a thundercloud on the horizon, the petal as it unfolds, or the swarming of a hive of bees. And yet sometimes I think that our laboratories are but little earthworks which men build about themselves, and whose puny tops too often conceal from view the Olympian heights; that we who work in these laboratories are but skilled artisans compared with the man who is able to observe, and to draw accurate deductions from the world about him.
The Anatomy of Science (1926), 170- 1.
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It required unusual inquisitiveness to pursue the development of scientific curiosities such as charged pith balls, the voltaic cell, and the electrostatic machine. Without such endeavors and the evolution of associated instrumentation, initially of purely scientific interest, most of the investigations that lead to the basic equations of electromagnetism would have been missed. … We would have been deprived of electromagnetic machinery as well as knowledge of electromagnetic waves.
From The Science Matrix: The Journey, Travails, Triumphs (1992, 1998), 14.
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It was basic research in the photoelectric field—in the photoelectric effect that would one day lead to solar panels. It was basic research in physics that would eventually produce the CAT scan. The calculations of today's GPS satellites are based on the equations that Einstein put to paper more than a century ago.
Speech to the National Academy of Sciences Annual Meeting (27 Apr 2009).
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It would be difficult and perhaps foolhardy to analyze the chances of further progress in almost every part of mathematics one is stopped by unsurmountable difficulties, improvements in the details seem to be the only possibilities which are left… All these difficulties seem to announce that the power of our analysis is almost exhausted, even as the power of ordinary algebra with regard to transcendental geometry in the time of Leibniz and Newton, and that there is a need of combinations opening a new field to the calculation of transcendental quantities and to the solution of the equations including them.
From Rapport historique sur les progrès des sciences mathématiques depuis 1789, et sur leur état actuel (1810), 131. As translated in George Sarton, The Study of the History of Mathematics (1936), 13. In the original French: “Il seroit difficile et peut-être téméraire d’analyser les chances que l’avenir offre à l’avancement des mathématiques: dans presque toutes les parties, on est arrêté par des difficultés insurmontables; des perfectionnements de détail semblent la seule chose qui reste à faire… Toutes ces difficultés semblent annoncer que la puissance de notre analyse est à-peu-près épuisée, comme celle de l’algèbre ordinaire l’étoit par rapport à la géométrie transcendante au temps de Leibnitz et de Newton, et qu’il faut des combinaisons qui ouvrent un nouveau champ au calcul des transcendantes et à la résolution des équations qui les contiennent.” Sarton states this comes from “the report on mathematical progress prepared for the French Academy of Sciences at Napoleon’s request”.
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It would seem that more than function itself, simplicity is the deciding factor in the aesthetic equation. One might call the process beauty through function and simplification.
As quoted in Christian Science Monitor (7 May 1952).
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I’ve been very involved in science literacy because it’s critically important in our world today. … As a public, we’re asked to vote on issues, we’re asked to accept explanations, we’re asked to figure out what to do with our own health care, and you can’t do that unless you have some level of science literacy. Science literacy isn’t about figuring out how to solve equations like E=MC². Rather, it’s about being able to read an article in the newspaper about the environment, about health care and figuring out how to vote on it. It’s about being able to prepare nutritious meals. It’s about being able to think your way through the day.
As quoted in 'Then & Now: Dr. Mae Jemison' (19 Jun 2005) on CNN web site.
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John [H.] Van Vleck, who was a leading young theoretical physicist when I was also a leading young theoretical physicist, said to me one day, “I never have made a contribution to physics that I didn’t get by fiddling with the equations,” and I said, “I’ve never made a contribution that I didn’t get by just having a new idea. Then I would fiddle with the equations to help support the new idea.” Van Vleck was essentially a mathematical physicist, you might say, and I was essentially a person of ideas. I don’t think I’m primarily mathematical. … I have a great curiosity about the nature of the world as a whole, and most of my ideas are qualitative rather than quantitative.
Interview with George B. Kauffman and Laurie M. Kauffman, in 'Linus Pauling: Reflections', American Scientist (Nov-Dec 1994), 82, No. 6, 523.
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Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future ... If someone can hit on the right lines along which to make this development, it m may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply the ... My own belief is that this is a more likely line of progress than trying to guess at physical pictures.
'The Evolution of the Physicist's Picture of Nature', Scientific American, May 1963, 208, 47. In Steve Adams, Frontiers (2000), 57.
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Mathematicians may flatter themselves that they possess new ideas which mere human language is as yet unable to express. Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but, we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations into their minds.
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Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence”.
In 'A Plea for the Mathematician', Nature, 1, 238 in Collected Mathematical Papers, Vol. 2, 655-56.
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Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius.
In 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences,' Communications in Pure and Applied Mathematics (Feb 1960), 13, No. 1 (February 1960). Collected in Eugene Paul Wigner, A.S. Wightman (ed.), Jagdish Mehra (ed.), The Collected Works of Eugene Paul Wigner (1955), Vol. 6, 537.
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One has to divide one’s time between politics and our equations. But our equations are much more important to me, because politics is for the present, while such an equation is for eternity.
Remark to his assistant, Ernst Straus,(late 1940s) while at Princeton University Institute of Advanced Study. As quoted in Albrecht Fösling and Ewald Osers (trans.), Albert Einstein: A Biography (1997), 725. At the time Einstein was one of the Emergency Committee of Atomic Scientists, concerned with informing the public on the atomic bomb and its effects. (He was its Chairman from May 1946.)
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People were pretty well spellbound by what Bohr said… While I was very much impressed by [him], his arguments were mainly of a qualitative nature, and I was not able to really pinpoint the facts behind them. What I wanted was statements which could be expressed in terms of equations, and Bohr's work very seldom provided such statements. I am really not sure how much later my work was influenced by these lectures of Bohr's... He certainly did not have a direct influence because he did not stimulate one to think of new equations.
Recalling the occasion in May 1925 (a year before receiving his Ph.D.) when he met Niels Bohr who was in Cambridge to give a talk on the fundamental difficulties of the quantum theory.
In History of Twentieth Century Physics (1977), 109. In A. Pais, 'Playing With Equations, the Dirac Way'. Behram N. Kursunoglu (Ed.) and Eugene Paul Wigner (Ed.), Paul Adrien Maurice Dirac: Reminiscences about a Great Physicist (1990), 94.
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Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really reflects the interconnectedness of Nature's laws at different levels.
Nobel Banquet Speech (10 Dec 1969), in Wilhelm Odelberg (ed.),Les Prix Nobel en 1969 (1970).
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Science is beautiful when it makes simple explanations of phenomena or connections between different observations. Examples include the double helix in biology, and the fundamental equations of physics.
[Answer to question: What are the things you find most beautiful in science?]
'Stephen Hawking: "There is no heaven; it's a fairy story"', interview in newspaper The Guardian (15 May 2011).
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Someone told me that each equation I included in the book would halve the sales.
In A Brief History of Time: From the Big Bang to Black Holes (1988, 1990), vi.
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Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
Getting Even (1978), 44.
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Suppose physics soon succeeds, as Stephen Hawking and a few other physicists hope and believe, in reducing physics to a single equation or a small set of equations that will “explain” all of nature’s fundamental laws. We can then ask the unanswerable question, "Why this set of equations?”
In Introduction, The Night Is Large: Collected Essays 1938-1995 (1996), xvii.
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That small word “Force,” they make a barber's block,
Ready to put on
Meanings most strange and various, fit to shock
Pupils of Newton....
The phrases of last century in this
Linger to play tricks—
Vis viva and Vis Mortua and Vis Acceleratrix:
Those long-nebbed words that to our text books still
Cling by their titles,
And from them creep, as entozoa will,
Into our vitals.
But see! Tait writes in lucid symbols clear
One small equation;
And Force becomes of Energy a mere
Space-variation.
'Report on Tait's Lecture on Force:— B.A., 1876', reproduced in Bruce Clarke, Energy Forms: Allegory and Science in the Era of Classical Thermodynamics (2001), 19. Maxwell's verse was inspired by a paper delivered at the British Association (B.A.. He was satirizing a “considerable cofusion of nomenclature” at the time, and supported his friend Tait's desire to establish a redefinition of energy on a thermnodynamic basis.
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The aim of research is the discovery of the equations which subsist between the elements of phenomena.
In Popular Scientific Lectures (1910), 205.
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The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. That such is the case a few indications will suffice to show. The mathematical concepts of constant and variable are represented familiarly in life by the notions of fixedness and change. The concept of equation or that of an equational system, imposing restriction upon variability, is matched in life by the concept of natural and spiritual law, giving order to what were else chaotic change and providing partial freedom in lieu of none at all. What is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal, some excellence high-dwelling among the rocks, an “ever flying perfect” as Emerson calls it, unto which we may approximate nearer and nearer, but which we can never quite attain, save in aspiration. The supreme concept of functionality finds its correlate in life in the all-pervasive sense of interdependence and mutual determination among the elements of the world. What is known in mathematics as transformation—that is, lawful transfer of attention, serving to match in orderly fashion the things of one system with those of another—is conceived in life as a process of transmutation by which, in the flux of the world, the content of the present has come out of the past and in its turn, in ceasing to be, gives birth to its successor, as the boy is father to the man and as things, in general, become what they are not. The mathematical concept of invariance and that of infinitude, especially the imposing doctrines that explain their meanings and bear their names—What are they but mathematicizations of that which has ever been the chief of life’s hopes and dreams, of that which has ever been the object of its deepest passion and of its dominant enterprise, I mean the finding of the worth that abides, the finding of permanence in the midst of change, and the discovery of a presence, in what has seemed to be a finite world, of being that is infinite? It is needless further to multiply examples of a correlation that is so abounding and complete as indeed to suggest a doubt whether it be juster to view mathematics as the abstract idealization of life than to regard life as the concrete realization of mathematics.
In 'The Humanization of Teaching of Mathematics', Science, New Series, 35, 645-46.
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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.
In Positive Philosophy, Bk. 1, chap. 2.
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The dollar is the final term in almost every equation which arises in the practice of engineering in any or all of its branches, except qualifiedly as to military and naval engineering, where in some cases cost may be ignored.
From Address on 'Industrial Engineering' at Purdue University (24 Feb 1905). Reprinted by Yale & Towne Mfg Co of New York and Stamford, Conn. for the use of students in its works.
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The equation eπi = -1 has been called the eutectic point of mathematics, for no matter how you boil down and explain this equation, which relates four of the most remarkable numbers of mathematics, it still has a certain mystery about it that cannot be explained away.
Anonymous
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The equation of animal and vegetable life is too complicated a problem for human intelligence to solve, and we can never know how wide a circle of disturbance we produce in the harmonies of nature when we throw the smallest pebble into the ocean of organic life.
Man and Nature, (1864), 103.
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The equations of dynamics completely express the laws of the historical method as applied to matter, but the application of these equations implies a perfect knowledge of all the data. But the smallest portion of matter which we can subject to experiment consists of millions of molecules, not one of which ever becomes individually sensible to us. We cannot, therefore, ascertain the actual motion of anyone of these molecules; so that we are obliged to abandon the strict historical method, and to adopt the statistical method of dealing with large groups of molecules … Thus molecular science teaches us that our experiments can never give us anything more than statistical information, and that no law derived from them can pretend to absolute precision. But when we pass from the contemplation of our experiments to that of the molecules themselves, we leave a world of chance and change, and enter a region where everything is certain and immutable.
'Molecules' (1873). In W. D. Niven (ed.), The Scientific Papers of James Clerk Maxwell (1890), Vol. 2, 374.
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The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up.
The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else’s Problem field.
The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the check [bill], the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.)
Life, the Universe and Everything (1982, 1995), 47-48.
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The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.
'Quantum Mechanics of Many-Electron Systems', Proceedings of the Royal Society (1929), A, 123, 714-733. Quoted in Steven M. Bachrach, Computational Organic Chemistry, Preface, xiii.
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The great object of all knowledge is to enlarge and purify the soul, to fill the mind with noble contemplations, to furnish a refined pleasure, and to lead our feeble reason from the works of nature up to its great Author and Sustainer. Considering this as the ultimate end of science, no branch of it can surely claim precedence of Astronomy. No other science furnishes such a palpable embodiment of the abstractions which lie at the foundation of our intellectual system; the great ideas of time, and space, and extension, and magnitude, and number, and motion, and power. How grand the conception of the ages on ages required for several of the secular equations of the solar system; of distances from which the light of a fixed star would not reach us in twenty millions of years, of magnitudes compared with which the earth is but a foot-ball; of starry hosts—suns like our own—numberless as the sands on the shore; of worlds and systems shooting through the infinite spaces.
Oration at Inauguration of the Dudley Astronomical Observatory, Albany (28 Jul 1856). Text published as The Uses of Astronomy (1856), 36.
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The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon… when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms.
Théorie Analytique de la Chaleur (1822), Art. 428, trans. Ivor Grattan-Guinness.
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The mathematician can afford to leave to his clients, the engineers, or perhaps the popular philosophers, the emotion of belief: for himself he keeps the lyrical pleasure of metre and of evolving equations: and it is a pleasant surprise to him and an added problem if he finds that the arts can use his calculations, or that the senses can verify them, much as if a composer found that sailors could heave better when singing his songs.
In 'Revolution in Science', Some Turns of Thought in Modern Philosophy (1933), 81.
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The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.
Opening of Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree. The paper was originally published (1824) in French, as a pamphlet, in Oslo. Collected in Œuvres Complètes (1881), Vol. 1, 28. Translation by W.H. Langdon collected in David Eugene Smith, A Source Book in Mathematics (2012), 261. In this work, he showed why—despite two centuries of efforts by mathematicians—solving equations of the fifth degree would remain futile. The insights from this paper led to the modern theory of equations.
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The quantum hypothesis will eventually find its exact expression in certain equations which will be a more exact formulation of the law of causality.
In Max Planck and James Vincent Murphy (trans.), Where is Science Going?, (1932), 143.
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The reason Dick's [Richard Feynman] physics was so hard for ordinary people to grasp was that he did not use equations. The usual theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations. This was the way Hans [Bethe] and Oppy [Oppenheimer] and Julian Schwinger did physics. Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the picture gave him the solutions directly with a minimum of calculation. It was no wonder that people who had spent their lives solving equations were baffled by him. Their minds were analytical; his was pictorial.
Quoted in Michio Kaku and Jennifer Trainer Thompson, Beyond Einstein: the Cosmic Quest for the Theory of the Universe (1987, 1999), 56-57, citing Freeman Dyson, Disturbing the Universe (1979, 1981), 55-56.
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The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity... the rigid electron is no working hypothesis, but a working hindrance. Approaching Maxwell's equations with the concept of the rigid electron seems to me the same thing as going to a concert with your ears stopped up with cotton wool. We must admire the courage and the power of the school of the rigid electron which leaps across the widest mathematical hurdles with fabulous hypotheses, with the hope to land safely over there on experimental-physical ground.
In Arthur I. Miller, Albert Einstein's Special Theory of Relativity (1981), 350.
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The science of calculation … is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be indulged in by one who is to have a profession to follow for his subsistence.
In Letter (18 Jun 1799) to William G. Munford. On founders.archives.gov website.
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The successes of the differential equation paradigm were impressive and extensive. Many problems, including basic and important ones, led to equations that could be solved. A process of self-selection set in, whereby equations that could not be solved were automatically of less interest than those that could.
In Does God Play Dice? The Mathematics of Chaos (1989, 1997), 33.
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The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject. … The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One should be influenced very much in this choice by considerations of mathematical beauty. It would probably be a good thing also to give a preference to those branches of mathematics that have an interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations are of more fundamental importance than equations.
From Lecture delivered on presentation of the James Scott prize, (6 Feb 1939), 'The Relation Between Mathematics And Physics', printed in Proceedings of the Royal Society of Edinburgh (1938-1939), 59, Part 2, 122.
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The universe is governed by science. But science tells us that we can’t solve the equations, directly in the abstract. We need to use the effective theory of Darwinian natural selection of those societies most likely to survive. We assign them higher value.
[Answer to question: What is the value in knowing “Why are we here?”]
'Stephen Hawking: "There is no heaven; it’s a fairy story"', interview in newspaper The Guardian (15 May 2011).
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The whole art of making experiments in chemistry is founded on the principle: we must always suppose an exact equality or equation between the principles of the body examined and those of the products of its analysis.
From Traité Élémentaire de Chimie (1789), 140. Translation as given in James Riddick Partington, A Short History of Chemistry (1960), 124. This is an alternate translation of part of the same passage on this page that begins “We may lay it down as an incontestible axiom…”
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The world looks like a multiplication-table, or a mathematical equation, which, turn it how you will, balances itself.
From 'Compensation', collected in The Complete Works of Ralph Waldo Emerson (1903), 102.
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There is a strange disparity between the sciences of inert matter and those of life. Astronomy, mechanics, and physics are based on concepts which can be expressed, tersely and elegantly, in mathematical language. They have built up a universe as harmonious as the monuments of ancient Greece. They weave about it a magnificent texture of calculations and hypotheses. They search for reality beyond the realm of common thought up to unutterable abstractions consisting only of equations of symbols. Such is not the position of biological sciences. Those who investigate the phenomena of life are as if lost in an inextricable jungle, in the midst of a magic forest, whose countless trees unceasingly change their place and their shape. They are crushed under a mass of facts, which they can describe but are incapable of defining in algebraic equations.
Man the Unknown (1935), 1.
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There may only be a small number of laws, which are self-consistent and which lead to complicated beings like ourselves. … And even if there is only one unique set of possible laws, it is only a set of equations. What is it that breathes fire into the equations and makes a universe for them to govern? Is the ultimate unified theory so compelling that it brings about its own existence?
Lecture (1987), 'The Origin of the Universe', collected in Black Holes And Baby Universes And Other Essays (1993), 99.
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Think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be imagined without it.
In The Golden Compass (1995, 2001), 372-373.
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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.
In The First Three Minutes: A Modern View of the Origin of the Universe (1977, Rev. ed. 1993), 131-132.
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This is one of the greatest advantages of modern geometry over the ancient, to be able, through the consideration of positive and negative quantities, to include in a single enunciation the several cases which the same theorem may present by a change in the relative position of the different parts of a figure. Thus in our day the nine principal problems and the numerous particular cases, which form the object of eighty-three theorems in the two books De sectione determinata of Appolonius constitute only one problem which is resolved by a single equation.
In Histoire de la Géométrie, chap. 1, sect. 35.
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Those who think 'Science is Measurement' should search Darwin's works for numbers and equations.
'David H. Hubel', in Larry R. Squire (ed.), The History of Neuroscience in Autobiography (1996), Vol. 1, 313.
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Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.
In Populär-wissenschafliche Vorlesungen (1903), 224-225.
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Throughout the 1960s and 1970s devoted Beckett readers greeted each successively shorter volume from the master with a mixture of awe and apprehensiveness; it was like watching a great mathematician wielding an infinitesimal calculus, his equations approaching nearer and still nearer to the null point.
Quoted in a review of Samuel Beckett’s Nohow On: Ill Seen Ill Said, Worstward Ho, in 'The Last Word', The New York Review of Books (13 Aug 1992).
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To emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations, ... let me take one more instance, by choosing a subject in which the purely mathematical interest is deemed supreme, the theory of functions of a complex variable. That at least is a theory in pure mathematics, initiated in that region, and developed in that region; it is built up in scores of papers, and its plan certainly has not been, and is not now, dominated or guided by considerations of applicability to natural phenomena. Yet what has turned out to be its relation to practical issues? The investigations of Lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation; which is merely the general geometrical method of regarding functional relations in that theory. Again, the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, made in the last twenty years, can all be, and now are all, I believe, deduced from similar considerations by interpreting functional relations between complex variables. In the dynamics of a rotating heavy body, the only substantial extension of our knowledge since the time of Lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion. Further, under the title of conjugate functions, the theory has been applied to various questions in electrostatics, particularly in connection with condensers and electrometers. And, lastly, in the domain of physical astronomy, some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas, the principles, the methods, and the results of the theory of functions. … the refined and extremely difficult work of Poincare and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects, developments which were made without a thought of such applications.
In Presidential Address British Association for the Advancement of Science, Section A, (1897), Nature, 56, 377.
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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.
http://web.archive.org/web/20070109161311/http://www.knowprose.com/node/12961
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Two kinds of symbol must surely be distinguished. The algebraic symbol comes naked into the world of mathematics and is clothed with value by its masters. A poetic symbol—like the Rose, for Love, in Guillaume de Lorris—comes trailing clouds of glory from the real world, clouds whose shape and colour largely determine and explain its poetic use. In an equation, x and y will do as well as a and b; but the Romance of the Rose could not, without loss, be re-written as the Romance of the Onion, and if a man did not see why, we could only send him back to the real world to study roses, onions, and love, all of them still untouched by poetry, still raw.
C.S. Lewis and E.M. Tillyard, The Personal Heresy: A Controversy (1936), 97.
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Well, in the first place, it leads to great anxiety as to whether it’s going to be correct or not … I expect that’s the dominating feeling. It gets to be rather a fever…
At age 60, when asked about his feelings on discovering the Dirac equation.
"Interview with T. Kuhn (7 May 1963), Niels Bohr Library, American Intitute of Physics, New York. In A. Pais, 'Playing With Equations, the Dirac Way'. Behram N. Kursunoglu (Ed.) and Eugene Paul Wigner (Ed.), Paul Adrien Maurice Dirac: Reminiscences about a Great Physicist (1990), 110.
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What appear to be the most valuable aspects of the theoretical physics we have are the mathematical descriptions which enable us to predict events. These equations are, we would argue, the only realities we can be certain of in physics; any other ways we have of thinking about the situation are visual aids or mnemonics which make it easier for beings with our sort of macroscopic experience to use and remember the equations.
In The Lost Cause: Causation and the Mind-body Problem (2003).
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Whereas, to borrow an illustration from mathematics, life was formerly an equation of, say, 100 unknown quantities, it is now one of 99 only, inasmuch as memory and heredity have been shown to be one and the same thing.
Samuel Butler, Henry Festing Jones (ed.), The Note-Books of Samuel Butler (1917), 57.
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Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever.
…...
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You know the formula m over naught equals infinity, m being any positive number? [m/0 = ∞]. Well, why not reduce the equation to a simpler form by multiplying both sides by naught? In which case you have m equals infinity times naught [m = ∞ × 0]. That is to say, a positive number is the product of zero and infinity. Doesn't that demonstrate the creation of the Universe by an infinite power out of nothing? Doesn't it?
In Point Counter Point (1928), 162.
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[My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who] devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis.
Quoted in Max Davidson, 'Bill Bryson: Have faith, science can solve our problems', Daily Telegraph (26 Sep 2010)
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[Richard Feynman] would be standing in front of the hall smiling at us all as we came in, his fingers tapping out a complicated rhythm on the black top of the demonstration bench that crossed the front of the lecture hall. As latecomers took their seats, he picked up the chalk and began spinning it rapidly through his fingers in a manner of a professional gambler playing with a poker chip, still smiling happily as if at some secret joke. And then—still smiling—he talked to us about physics, his diagrams and equations helping us to share his understanding. It was no secret joke that brought the smile and the sparkle in his eye, it was physics. The joy of physics!
Describing his experience as a student attending Feynman lectures, in Introduction to Richard P. Feynman Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character (1986, 2010), 9-10.
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[Student:} I only use my math book on special equations.
From movie Rock 'n' Roll High School (1979). Writers, Richard Whitley, Russ Dvonch and Joseph McBride. In Larry Langman and Paul Gold, Comedy Quotes from the Movies (2001), 359.
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[S]ome physicists describe gravity in terms of ten dimensions all curled up. But those aren't real words—just placeholders, used to refer to parts of abstract equations.
In God's Debris: A Thought Experiment (2004), 20-21.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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