🅳Avinash K Dixit - Collected Quotes

"A game is a situation of strategic interdependence: the outcome of your choices (strategies) depends upon the choices of one or more other persons acting purposely. The decision makers involved in a game are called players, and their choices are called moves. The interests of the players in a game may be in strict conflict; one person’s gain is always another’s loss. Such games are called zero-sum. More typically, there are zones of commonality of interests as well as of conflict and so, there can be combinations of mutually gainful or mutually harmful strategies. Nevertheless, we usually refer to the other players in a game as one’s rivals." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Chess experts have been successful at characterizing optimal strategies near the end of the game. Once the chessboard has only a small number of pieces on it, experts are able to look ahead to the end of the game and determine by backward reasoning whether one side has a guaranteed win or whether the other side can obtain a draw. But the middle of the game, when several pieces remain on the board, is far harder. Looking ahead five pairs of moves, which is about as much as can be done by experts in a reasonable amount of time, is not going to simplify the situation to a point where the endgame can be solved completely from there on." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Chess strategy illustrates another important practical feature of looking forward and reasoning backward: you have to play the game from the perspective of both players. While it is hard to calculate your best move in a complicated tree, it is even harder to predict what the other side will do." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"John Nash’s beautiful equilibrium was designed as a theoretical way to square just such circles of thinking about thinking about other people’s choices in games of strategy. The idea is to look for an outcome where each player in the game chooses the strategy that best serves his or her own interest, in response to the other’s strategy. If such a configuration of strategies arises, neither player has any reason to change his choice unilaterally. Therefore, this is a potentially stable outcome of a game where the players make individual and simultaneous choices of strategies." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Many mathematical game theorists dislike the dependence of an outcome on historical, cultural, or linguistic aspects of the game or on purely arbitrary devices like round numbers; they would prefer the solution be determined purely by the abstract mathematical facts about the game - the number of players, the strategies available to each, and the payoffs to each in relation to the strategy choices of all. We disagree. We think it entirely appropriate that the outcome of a game played by humans interacting in a society should depend on the social and psychological aspects of the game." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Science and art, by their very nature, differ in that science can be learned in a systematic and logical way, whereas expertise in art has to be acquired by example, experience, and practice." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Strategic thinking starts with your basic skills and considers how best to use them. Knowing the law, you must decide the strategy for defending your client. Knowing how well your football team can pass or run and how well the other team can defend against each choice, your decision as the coach is whether to pass or to run. Sometimes, as in the case of nuclear brinkmanship, strategic thinking also means knowing when not to play." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"The essence of a game of strategy is the interdependence of the players’ decisions. These interactions arise in two ways. The first is sequential [...] The players make alternating moves. [...] The second kind of interaction is simultaneous, as in the prisoners’ dilemma [...] The players act at the same time, in ignorance of the others’ current actions. However, each must be aware that there are other active players, who in turn are similarly aware, and so on. Therefore each must figuratively put himself in the shoes of all and try to calculate the outcome. His own best action is an integral part of this overall calculation. When you find yourself playing a strategic game, you must determine whether the interaction is simultaneous or sequential." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"When playing mixed or random strategies, you can’t fool the opposition every time. The best you can hope for is to keep them guessing and fool them some of the time. You can know the likelihood of your success but cannot say in advance whether you will succeed on any particular occasion. In this regard, when you know that you are talking to a person who wants to mislead you, it may be best to ignore any statements he makes rather than accept them at face value or to infer that exactly the opposite must be the truth." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

🏛️On Art IX: On Metaphysics

"Metaphysics may be, after all, only the art of being sure of something that is not so and logic only the art of going wrong with confidence." (Joseph W Krutch, "The Modern Temper", 1929)

"As to the role of emotions in art and the subconscious mechanism that serves as the integrating factor both in artistic creation and in man's response to art, they involve a psychological phenomenon which we call a sense of life. A sense of life is a pre-conceptual equivalent of metaphysics, an emotional, subconsciously integrated appraisal of man and of existence." (Ayn Rand, "The Romantic Manifesto: A Philosophy of Literature", 1969)

"For some years now the activity of the artist in our society has been trending more toward the function of the ecologist: one who deals with environmental relationships. Ecology is defined as the totality or pattern of relations between organisms and their environment. Thus the act of creation for the new artist is not so much the invention of new objects as the revelation of previously unrecognized relation- ships between existing phenomena, both physical and metaphysical. So we find that ecology is art in the most fundamental and pragmatic sense, expanding our apprehension of reality." (Gene Youngblood, "Expanded Cinema", 1970)

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

 "Science, as usually taught to liberal arts students, emphasizes results rather than method, and tries to teach technique rather than to give insight into and understanding of the scientific habit of thought. What is needed, however, is not a dose of metaphysics but a truly humanistic teaching of science." (Harry D Gideonse)

🏛️On Art VIII: On Physics

"It is impossible to follow the march of one of the greatest theories of physics, to see it unroll majestically its regular deductions starting from initial hypotheses, to see its consequences represent a multitude of experimental laws down to the smallest detail, without being charmed by the beauty of such a construction, without feeling keenly that such a creation of the human mind is truly a work of art." (Pierre-Maurice-Marie DuhemDuhem, "The Aim and Structure of Physical Theory", 1908)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Part of the art and skill of the engineer and of the experimental physicist is to create conditions in which certain events are sure to occur." (Eugene P Wigner, "Symmetries and Reflections", 1979)

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

🅲Subrahmanyan Chandrasekhar - Collected Quotes

"All of us are sensitive to nature's beauty. It is not unreasonable that some aspects of this beauty are shared by the natural sciences." (Subrahmanyan Chandrasekhar, "Beauty and the Quest for Beauty in Science", Physics Today Vol. 32 (7), [lecture] 1979)

"It is, indeed an incredible fact that what the human mind, at its deepest and most profound, perceives as beautiful finds its realization in external nature.[...] What is intelligible is also beautiful." (Subrahmanyan Chandrasekhar, "Beauty and the Quest for Beauty in Science", Physics Today Vol. 32 (7), [lecture] 1979)

"The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time." (Subrahmanyan Chandrasekhar, "The Mathematical Theory of Black Holes", 1983)

"A black hole partitions the three-dimensional space into two regions: an inner region which is bounded by a smooth two-dimensional surface called the event horizon; and an outer region, external to the event horizon, which is asymptotically flat; and it is required (as a part of the definition) that no point in the inner region can communicate with any point of the outer region. This incommunicability is guaranteed by the impossibility of any light signal, originating in the inner region, crossing the event horizon. The requirement of asymptotic flatness of the outer region is equivalent to the requirement that the black hole is isolated in space and that far from the event horizon the space-time approaches the customary space-time of terrestrial physics." (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability", [Nobel lecture] 1983)

"The mathematical theory of black holes is a subject of immense complexity; but its study has convinced me of the basic truth of the ancient mottoes 'The simple is the seal of the true' and 'beauty is the splendor of truth.'" (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability", [Nobel lecture] 1983)

"Turning to the physical properties of the black holes, we can study them best by examining their reaction to external perturbations such as the incidence of waves of different sorts. Such studies reveal an analytic richness of the Kerr space-time which one could hardly have expected. This is not the occasion to elaborate on these technical matters. Let it suffice to say that contrary to every prior expectation, all the standard equations of mathematical physics can be solved exactly in the Kerr space-time. And the solutions predict a variety and range of physical phenomena which black holes must exhibit in their interaction with the world outside." (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability", [Nobel lecture] 1983)

"In some strange way, any new fact or insight that I may have found has not seemed to me as a 'discovery' of mine, but rather something that had always been there and that I had chanced to pick up." (Subrahmanyan Chandrasekhar, "Truth and Beauty: Aesthetics and Motivations in Science", 1987)

"The pursuit of science has often been compared to the scaling of mountains, high and not so high. But who amongst us can hope, even in imagination, to scale the Everest and reach its summit when the sky is blue and the air is still, and in the stillness of the air survey the entire Himalayan range in the dazzling white of the snow stretching to infinity? None of us can hope for a comparable vision of nature and of the universe around us. But there is nothing mean or lowly in standing in the valley below and awaiting the sun to rise over Kinchinjunga." (Subrahmanyan Chandrasekhar, "Truth and Beauty: Aesthetics and Motivations in Science, 1987)

"When a supremely great creative mind is kindled, it leaves a blazing trail that remains a beacon for centuries." (Subrahmanyan Chandrasekhar, "Newton and Michelangelo", Current Science Vol. 67 (7), 1994)

"This 'shuddering before the beautiful', this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound." (Subrahmanyan Chandrasekhar)

On Figurative Figures IV: On Circumference

"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"A system of nature proceeding from subjects of the most simple organization to such as are more perfect, or from the circumference to the centre, is called a Mathematical System." (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"Philosophical systems do not depend upon individual productions which are subject to continual variation, but upon eternal and unchangeable ideas. These always proceed from the centre to the circumference, or from the most perfect productions to those of a lower order." (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"The world can no more have two summits than a circumference can have two centres." (Pierre T de Chardin, "The Divine Milieu", 1960)

"An observer of our biological sciences today sees dark figures moving over a bridge of glass. We are faced with an ever expanding universe of light and darkness. The greater the circle of understanding becomes, the greater is the circumference of surrounding ignorance." (Erwin Chargaff, "Essays on Nucleic Acids", 1963)

"The universe has no circumference, for if it had a center and a circumference there would be some and some thing beyond the world, suppositions which are wholly lacking in truth. Since, therefore, it is impossible that the universe should be enclosed within a corporeal center and corporeal boundary, it is not within our power to understand the universe, whose center and circumference are God. And though the universe." (Nicholas of Cusa[Nicolaus Cusanus])

On Geometrical Figures: On Circumference

"Now discourse is necessarily limited by its point of departure and its point of arrival, and since these are in mutual opposition we speak of contradiction. For the discursive reason these terms are opposed and distinct. In the realm of the reason, therefore, there is a necessary disjunction between extremes, as, for example, in the rational definition of the circle where the lines from the center to the circumference are equal and where the center cannot coincide with the circumference." (Nicholas of Cusa, "Apologia Doctae ignorantiae" ["The Defense of Learned Ignorance"], 1449)

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"The circumference of any circle being given, if that circumference be brought into the form of a square, the area of that square is equal to the area of another circle, the circumscribed square of which is equal to the area of the circle whose circumference is first given." (John A Parker, "The Quadrature of the Circle", 1874)

"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"To square a circle means to find a square whose area is equal to the area of a given circle. In its first form this problem asked for a rectangle whose dimensions have the same ratio as that of the circumference of a circle to its radius. The proof of the impossibility of solving this by use of ruler and compasses alone followed immediately from the proof, in very recent times, that π cannot be the root of a polynomial equation with rational coefficients." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The digits of pi beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places are sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere" (Petr Beckmann, "A History of Pi", 1976)

"There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as the 'Squaring of the Circle'. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it." (Robert Lawlor, "Sacred Geometry", 1982)

"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"In the case of circle squaring, since the problem requires pinpointing the ratio between a circle’s diameter and circumference, the irrational number the investigator bumps into is pi (π). Perhaps because of its extreme (in fact, total) difficulty - similar to the alchemist’s hope of turning lead into gold - circle squaring offered its pursuers the dream of international fame in the discovery of an unknown quantity seemingly woven into the fabric of the universe." (Daniel J Cohen, "Equations from God: Pure Mathematics and Victorian Faith", 2007)

"Is it possible to construct a square, using only a compass and a straightedge, that is exactly equal in area to the area of a given circle? If π could be expressed as a rational fraction or as the root of a first- or second-degree equation, then it would be possible, with compass and straightedge, to construct a straight line exactly equal to the circumference of a circle. The squaring of the circle would quickly follow. We have only to construct a rectangle with one side equal to the circle’s radius and the other equal to half the circumference. This rectangle has an area equal to that of the circle, and there are simple procedures for converting the rectangle to a square of the same area. Conversely, if the circle could be squared, a means would exist for constructing a line segment exactly equal to π. However, there are ironclad proofs that π is transcendental and that no straight line of transcendental length can be constructed with compass and straightedge. " (Martin Gartner, "Sphere Packing, Lewis Carroll, and Reversi", 2009)

"The engineer and the mathematician have a completely different understanding of the number pi. In the eyes of an engineer, pi is simply a value of measurement between three and four, albeit fiddlier than either of these whole numbers. [...] Mathematicians know the number pi differently, more intimately. What is pi to them? It is the length of a circle’s round line (its circumference) divided by the straight length (its diameter) that splits the circle into perfect halves. It is an essential response to the question, ‘What is a circle?’ But this response – when expressed in digits – is infinite: the number has no last digit, and therefore no last-but-one digit, no antepenultimate digit, no third-from-last digit, and so on." (Daniel Tammet, "Thinking in Numbers" , 2012)

"It just so happens that π can be characterised precisely without any reference to decimals, because it is simply the ratio of any circle’s circumference to its diameter. Likewise can be characterised as the positive number which squares to 2. However, most irrational numbers can’t be characterised in this way." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center. It’s a minimal form of change, a way to change and curve in the slightest way possible. And, of course, circles are symmetrical. If you rotate a circle about its center, it looks unchanged. That rotational symmetry may be why circles are so ubiquitous. Whenever some aspect of nature doesn’t care about direction, circles are bound to appear. Consider what happens when a raindrop hits a puddle: tiny ripples expand outward from the point of impact. Because they spread equally fast in all directions and because they started at a single point, the ripples have to be circles. Symmetry demands it." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process. But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker stepping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter. That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan) [attributed]

"Ten decimal places of pi are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope." (Simon Newcomb)

"The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems - the determination of the diagonal of a square and that of the circumference of a circle - revealed the existence of new mathematical beings for which no place could be found within the rational domain." (Tobias Dantzig)

🅷William Herschel - Collected Quotes

"We may strongly suspect that there is not, in strictness of speaking, one fixed star in the heavens, and reasons which I shall adduce will render this so obvious that there can hardly remain a doubt of the general motion of all the starry systems, and, consequently, of the solar one among the rest." (William Herschel, "On the Proper Motion of the Sun and Solar System", Philosophical Transactions of the Royal Society of London Vol. 73, 1783)

"If we indulge a fanciful imagination and build worlds of our own, we must not wonder at our going wide from the path of truth and nature; but these will vanish like the Cartesian vortices, that soon gave way when better theories were offered." (William Herschel, "The Construction of the Heavens", Philosophical Transactions of the Royal Society of London Vol. LXXV, 1785) 

"The subject of the Construction of the Heavens is of so extensive and important a nature, that we cannot exert too much attention in our endeavors to throw all possible light upon it." (William Herschel, "The Construction of the Heavens", Philosophical Transactions of the Royal Society of London Vol. LXXV, 1785) 

"The phenomena of nature, especially those that fall under the inspection of the astronomer, are to be viewed, not only with the usual attention to facts as they occur, but with the eye of reason and experience." (William Herschel, "An Account of Three Volcanoes in the Moon", Philosophical Transactions of the Royal Society of London Vol. 67, 1787)

"It is sometimes of great use in natural philosophy to doubt of things that are commonly taken for granted; especially as the means of resolving any doubt, when once it is entertained, are often within our reach." (William Herschel, "Investigation of the Powers the Prismatic Colours to Heat and Illuminate Objects", Philosophical Transactions of the Royal Society of London Vol. 90, 1800)

"Familiar objects and events are far from presenting themselves to our senses in that aspect and with those connections under which science requires them to be viewed, and which constitute their rational explanation." (William Herschel, "Outlines of Astronomy", 1849)

"I have looked farther into space than ever human being did before me. I have observed stars of which the light, it can be proved, must take two millions of years to reach this earth!" (William Herschel)

"There are two kinds of happiness or contentment for which we mortals are adapted; the first we experience in thinking and the other in feeling. The first is the purest and most unmixed. Let a man once know what sort of a being he is; how great the being which brought him into existence, how utterly transitory is everything in the material world, and let him realize this without passion in a quiet philosophical temper, and I maintain that then he is happy; as happy indeed as it is possible for him to be." (William Herschel [letter to Jacob Herschel)

"When an object is once discovered by a superior power, an inferior one will suffice to see if afterwards." (William Herschel)

On Simplicity: Simple Rules

"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)

"Simple rules can have complex consequences. This simple rule has such a wealth of implications that it is worth examining in detail. It is the far from self-evident guiding principle of reductionism and of most modern investigations into cosmic complexity. Reductionism will not be truly successful until physicists and cosmologists demonstrate that the large-scale phenomena of the world arise from fundamental physics alone. This lofty goal is still out of reach. There is uncertainty not only in how physics generates the structures of our world but also in what the truly fundamental rules of physics are. (William Poundstone, "The Recursive Universe", 1985)

"As glimpsed by physicists, Nature's rules are simple, but also intricate: Different rules are subtly related to each other. The intricate relations between the rules produce interesting effects in many physical situations. [...] Nature's design is not only simple, but minimally so, in the sense that were the design any simpler, the universe would be a much duller place." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"All reality is a game. Physics at its most fundamental, the very fabric of our universe, results directly from the interaction of certain fairly simple rules, and chance; the same description may be applied to the best, most elegant and both intellectually and aesthetically satisfying games. By being unknowable, by resulting from events which, at the sub-atomic level, cannot be fully predicted, the future remains malleable, and retains the possibility of change, the hope of coming to prevail; victory, to use an unfashionable word. In this, the future is a game; time is one of its rules." (Iain Banks, "The Player of Games", 1988)

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"With the growing interest in complex adaptive systems, artificial life, swarms and simulated societies, the concept of 'collective intelligence' is coming more and more to the fore. The basic idea is that a group of individuals (e. g. people, insects, robots, or software agents) can be smart in a way that none of its members is. Complex, apparently intelligent behavior may emerge from the synergy created by simple interactions between individuals that follow simple rules." (Francis Heylighen, "Collective Intelligence and its Implementation on the Web", 1999)

"Through self-organization, the behavior of the group emerges from the collective interactions of all the individuals. In fact, a major recurring theme in swarm intelligence (and of complexity science in general) is that even if individuals follow simple rules, the resulting group behavior can be surprisingly complex - and remarkably effective. And, to a large extent, flexibility and robustness result from self-organization." (Eric Bonabeau & Christopher Meyer, "Swarm Intelligence: A Whole New Way to Think About Business", Harvard Business Review, 2001)

"Chaos theory revealed that simple nonlinear systems could behave in extremely complicated ways, and showed us how to understand them with pictures instead of equations. Complexity theory taught us that many simple units interacting according to simple rules could generate unexpected order. But where complexity theory has largely failed is in explaining where the order comes from, in a deep mathematical sense, and in tying the theory to real phenomena in a convincing way. For these reasons, it has had little impact on the thinking of most mathematicians and scientists." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"[a complex system is] a system in which large networks of components with no central control and simple rules of operation give rise to complex collective behavior, sophisticated information processing, and adaptation via learning or evolution." (Melanie Mitchell, "Complexity: A Guided Tour", 2009)

"[...] the Game of Life, in which a few simple rules executed repeatedly can generate a surprising degree of complexity. Recall that the game treats squares, or pixels, as simply on or off (filled or blank) and the update rules are given in terms of the state of the nearest neighbours. The theory of networks is closely analogous. An electrical network, for example, consists of a collection of switches with wires connecting them. Switches can be on or off, and simple rules determine whether a given switch is flipped, according to the signals coming down the wires from the neighbouring switches. The whole network, which is easy to model on a computer, can be put in a specific starting state and then updated step by step, just like a cellular automaton. The ensuing patterns of activity depend both on the wiring diagram (the topology of the network) and the starting state. The theory of networks can be developed quite generally as a mathematical exercise: the switches are called ‘nodes’ and the wires are called ‘edges’. From very simple network rules, rich and complex activity can follow." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

On Symmetry XII

"The beauty that Nature has revealed to physicists in Her laws is a beauty of design, a beauty that recalls, to some extent, the beauty of classical architecture, with its emphasis on geometry and symmetry. The system of aesthetics used by physicists in judging Nature also draws its inspiration from the austere finality of geometry." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"To detect a symmetry in the fundamental design, one would have to check the covariance of each of the many equations of motion in the differential formulation. With the action formulation, on the other hand, one has the considerably easier task of checking the invariance of the action." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The impossibility of defining absolute motion can be seen as the manifestation of a symmetry known as relativistic invariance. In the same way that parity invariance tells us that we cannot distinguish the mirror-image world from our world, relativistic invariance tells us that it is impossible to decide whether we are at rest or moving steadily." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The precise mathematical definition of symmetry involves the notion of invariance. A geometrical figure is said to be symmetric under certain operations if those operations leave it unchanged." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Unlike an architect, Nature does not go around expounding on the wondrous symmetries of Her design. Instead, theoretical physicists must deduce them. Some symmetries, such as parity and rotational invariances, are intuitively obvious. We expect Nature to possess these symmetries, and we are shocked if She does not. Other symmetries, such as Lorentz invariance and general covariance, are more subtle and not grounded in our everyday perceptions. But, in any case, in order to find out if Nature employs a certain symmetry, we must compare the implications of the symmetry with observation." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Symmetry is ubiquitous. Symmetry has myriad incarnations in the innumerable patterns designed by nature. It is a key element, often the central or defining theme, in art, music, dance, poetry, or architecture. Symmetry permeates all of science, occupying a prominent place in chemistry, biology, physiology, and astronomy. Symmetry pervades the inner world of the structure of matter, the outer world of the cosmos, and the abstract world of mathematics itself. The basic laws of physics, the most fundamental statements we can make about nature, are founded upon symmetry." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The symmetries that we sense and observe in the world around us affirm the notion of the existence of a perfect order and harmony underlying everything in the universe. Through symmetry we sense an apparent logic at work in the universe, external to, yet resonant with, our own minds. [...] Symmetry gives wings to our creativity. It provides organizing principles for our artistic impulses and our thinking, and it is a source of hypotheses that we can make to understand the physical world." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"A symmetry is a function that preserves what we feel is important about an object." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Lie groups turn up when we study a geometric object with a lot of symmetry, such as a sphere, a circle, or flat spacetime. Because there is so much symmetry, there are many functions from the object to itself that preserve the geometry, and these functions become the elements of the group." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center. It’s a minimal form of change, a way to change and curve in the slightest way possible. And, of course, circles are symmetrical. If you rotate a circle about its center, it looks unchanged. That rotational symmetry may be why circles are so ubiquitous. Whenever some aspect of nature doesn’t care about direction, circles are bound to appear. Consider what happens when a raindrop hits a puddle: tiny ripples expand outward from the point of impact. Because they spread equally fast in all directions and because they started at a single point, the ripples have to be circles. Symmetry demands it." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

🆉Anthony Zee - Collected Quotes

"As glimpsed by physicists, Nature's rules are simple, but also intricate: Different rules are subtly related to each other. The intricate relations between the rules produce interesting effects in many physical situations. [...] Nature's design is not only simple, but minimally so, in the sense that were the design any simpler, the universe would be a much duller place." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"In science, one tries to say what no one else has ever said before. In poetry, one tries to say what everyone else has already said, but better. This explains, in essence, why good poetry is as rare as good science." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"In the path-integral formulation, the essence of quantum physics may be summarized with two fundamental rules: (1). The classical action determines the probability amplitude for a specific chain of events to occur, and (2) the probability that either one or the other chain of events occurs is determined by the probability amplitudes corresponding to the two chains of events. Finding these rules represents a stunning achievement by the founders of quantum physics." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Physicists dream of a unified description of Nature. Symmetry, in its power to tie together apparently unrelated aspects of physics, is linked closely to the notion of unity." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Physics is the most reductionistic of sciences. [...] Contemporary physics rests on the cornerstone of reductionism. As we delve deeper, Nature appears ever simpler. That this is so is, in fact, astonishing. We have no a priori reason to expect the universe, with its fantastic wealth of bewilderingly complex phenomena, to be governed ultimately by a few simple rules." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The beauty that Nature has revealed to physicists in Her laws is a beauty of design, a beauty that recalls, to some extent, the beauty of classical architecture, with its emphasis on geometry and symmetry. The system of aesthetics used by physicists in judging Nature also draws its inspiration from the austere finality of geometry." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The impossibility of defining absolute motion can be seen as the manifestation of a symmetry known as relativistic invariance. In the same way that parity invariance tells us that we cannot distinguish the mirror-image world from our world, relativistic invariance tells us that it is impossible to decide whether we are at rest or moving steadily." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The power and glory of symmetry allow us to bypass completely the construction of strong interaction theories of dubious utility. We are able to contain and isolate our ignorance. [...] Symmetry tells us that states in the same multiplet must have the same energy, but it cannot tell us what that energy is." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The precise mathematical definition of symmetry involves the notion of invariance. A geometrical figure is said to be symmetric under certain operations if those operations leave it unchanged." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The search for fundamental symmetries boils down to the study of transformations that do not change fundamental physical action - such transformations as reflection, rotation, the Lorentz transformation, and the like." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"To detect a symmetry in the fundamental design, one would have to check the covariance of each of the many equations of motion in the differential formulation. With the action formulation, on the other hand, one has the considerably easier task of checking the invariance of the action." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Toward the end of the last century, many physicists felt that the mathematical description of physics was getting ever more complicated. Instead, the mathematics involved has become ever more abstract, rather than more complicated. The mind of God appears to be abstract but not complicated. He also appears to like group theory." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Unlike an architect, Nature does not go around expounding on the wondrous symmetries of Her design. Instead, theoretical physicists must deduce them. Some symmetries, such as parity and rotational invariances, are intuitively obvious. We expect Nature to possess these symmetries, and we are shocked if She does not. Other symmetries, such as Lorentz invariance and general covariance, are more subtle and not grounded in our everyday perceptions. But, in any case, in order to find out if Nature employs a certain symmetry, we must compare the implications of the symmetry with observation." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"We intuitively know space to be a smooth continuum, an arena in which the fundamental particles move and interact. This assumption underpins our physical theories, and no experimental evidence has ever contradicted it. However, the possibility that space may not be smooth cannot be excluded." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Welcome to the strange world of the quantum, where one cannot determine how a particle gets from here to there. Physicists are reduced to bookies, posting odds on the various possibilities." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)