On Symmetry XIII

"Every equation and every explanation used in physics must be compatible with the symmetry of time. Thus we can no longer regard effect as subsequent to cause. If we think of the present as pushed into existence by the past, we must in precisely the same sense think of it pulled into existence by the future." (Gilbert N Lewis, "The Symmetry of Time in Physics", Science, 1930) 

"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"A symmetric game is often the convenient representative of a collection of related games, most of which are asymmetric. [...] The symmetric games are used so often, especially in introductions to game theory, that it is easy to forget they represent a very special case. For each strategy of the row player in a symmetric game, there must be an equivalent strategy for the column player." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"How often even ordinal symmetry appears in the real world is an open question. A great many situations are approximately symmetric, and symmetric payoffs provide a useful starting point for analysis. On the other hand, symmetric games present a problem that human beings are equipped to evade. Symmetry theoretically erases distinctions between players, but real people are capable of exploiting very subtle distinctions." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Order graphs allow us to describe games easily, and our indexing system lets us lay out the games in a systematic and revealing way. Symmetries in the order graph repre-sentation shed some light on the nature of symmetric and reflected games, and on the structure of the space of the 2 × 2 games. They are not directly useful for analysing behaviour." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Symmetry has long been important in the study of physical systems. Connections between the geometric symmetry of crystalline systems and their x-ray diffraction spectra were found to be crucial to the interpretation of the diffraction patterns and the extraction therefrom of information locating the atoms in the crystal. The geometric symmetries of molecules determine which vibrational modes will be active in absorbing or emitting  radiation; the symmetries of periodic systems have implications as to their energy bands, their ability to conduct electricity, and even their superconductivity." (George B Arfken et al, "Mathematical Methods for Physicists: A comprehensive guide", 2013)

"The invariance of physical laws with respect to position or orientation (i.e., the symmetry of space) gives rise to conservation laws for linear and angular momentum. Sometimes the implications of symmetry invariance are far more complicated or sophisticated than might at first be supposed; the invariance of the forces predicted by electromagnetic theory when measurements are made in observation frames moving uniformly at different speeds (inertial frames) was an important clue leading Einstein to the discovery of special relativity. With the advent of quantum mechanics, considerations of angular momentum and spin introduced new symmetry concepts into physics. These ideas have since catalyzed the modern development of particle theory." (George B Arfken et al, "Mathematical Methods for Physicists: A comprehensive guide", 2013)

On Symmetry: On the Symmetry of Time

"Every equation and every explanation used in physics must be compatible with the symmetry of time. Thus we can no longer regard effect as subsequent to cause. If we think of the present as pushed into existence by the past, we must in precisely the same sense think of it pulled into existence by the future." (Gilbert N Lewis, "The Symmetry of Time in Physics", Science, 1930)

"Everywhere […] in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since […] these laws are time symmetric- they permit […] time-reverse. […] The initial conditions play a decisive role in endowing the world with its sense of temporal direction. […] some prescription for initial conditions is crucial if we are to understand […]"  (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)

"A single snapshot of the pattern. A fixed spiral is elegant in form, but it does not possess symmetry in the mathematical sense. However, a rotating spiral does possess a dynamic symmetry - as does any continuously rotating object. It is a symmetry that takes place in both space and time. Start with the spiral, let time pass. It rotates to a new position. Now rotate it backwards through the appropriate angle, and it looks exactly as it did to begin with. This is a space-time symmetry: translate time and rotate space. Indeed, the rotating spiral has lots of space-time symmetries, because every period of time corresponds to some spatial rotation." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Dynamics involves a crucial element: time. Symmetry is about shapes, which are things in space. How can symmetries apply to time? [...] We can even put both together, and look at patterns in space and time combined. Turning space into time is a typical mathematician's trick, a kind of 'technology transfer'. However, the 'technology' involved is conceptual technology - ideas - rather than gadgetry. No mathematician would ever waste a good idea by restricting it to just one incarnation. Many physically different systems possess an underlying mathematical identity: they obey rules which, in the abstract, are exactly the same - and must therefore lead to corresponding conclusions." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"More generally, thermodynamics shows that there is an irreversible flow of time. Rather than there being time symmetry and indeed a reversibility of time as postulated in classical physics, a clear distinction is drawn between the past and future. An arrow of time results within open systems in the loss of organization and an increase in randomness or disorder over time. This accumulation of disorder or positive entropy results from the Second Law of Thermodynamics." (John Urry, "Global Complexity", 2003)

"Eternal constancy of the laws of physics is a symmetry. What we see as we look back in time, or we peer through telescopes out into space, or we look through our powerful microscopes (particle accelerators), is the same system of laws of physics governing the whole universe at all times and at all places. These are the basic symmetries of the structure of our universe and its contents and, at a deeper level, the symmetries of the laws that govern the universe themselves. Indeed, the symmetries we uncover are the basic principles that define our laws of nature and the laws of physics, hence those that control our universe." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The space and time of the universe that we humans inhabit contain symmetries. These are almost obvious yet subtle, even mysterious. Space and time form the stage upon which the dynamics - that is, the motion and interactions of the physical systems, atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of space and time control the dynamics of the physical interactions of matter." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The concept of symmetry is used widely in physics. If the laws that determine relations between physical magnitudes and a change of these magnitudes in the course of time do not vary at the definite operations (transformations), they say, that these laws have symmetry (or they are invariant) with respect to the given transformations. For example, the law of gravitation is valid for any points of space, that is, this law is invariant with respect to the system of coordinates." (Alexey Stakhov et al, "The Mathematics of Harmony", 2009)

"Many statistical procedures perform more effectively on data that are normally distributed, or at least are symmetric and not excessively kurtotic (fat-tailed), and where the mean and variance are approximately constant. Observed time series frequently require some form of transformation before they exhibit these distributional properties, for in their 'raw' form they are often asymmetric." (Terence C Mills, "Applied Time Series Analysis: A practical guide to modeling and forecasting", 2019)

On Perfection: Perfect Symmetry

"The fact is that the beautiful, humanly speaking, is merely form considered in its simplest aspect, in its most perfect symmetry, in its most entire harmony with our make-up." (Victor Hugo, "Cromwell", 1909)

"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection." (Herman Weyl, "Symmetry", 1938)

"[...] nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop." (Chen-Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)

"Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible." (Ian Stewart & Martin Golubitsky,"Fearful Symmetry: Is God a Geometer?", 1992)

"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & ‎K Nishijima ,"Broken Symmetry: Selected Papers Of Y Nambu", 1995)

"The possibility of translating uncertainties into risks is much more restricted in the propensity view. Propensities are properties of an object, such as the physical symmetry of a die. If a die is constructed to be perfectly symmetrical, then the probability of rolling a six is 1 in 6. The reference to a physical design, mechanism, or trait that determines the risk of an event is the essence of the propensity interpretation of probability. Note how propensity differs from the subjective interpretation: It is not sufficient that someone’s subjective probabilities about the outcomes of a die roll are coherent, that is, that they satisfy the laws of probability. What matters is the die’s design. If the design is not known, there are no probabilities." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy,"Symmetry: A Journey into the Patterns of Nature", 2008)

"Mathematical symmetry is an idealized model. However, slightly imperfect symmetry requires explanation; it’s not enough just to say ‘it’s asymmetric’." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

Avner Ash - Collected Quotes

"A group is a set along with a rule that tells how to combine any two elements in the set to get another element in the set. We usually use the word composition to describe the act of combining two elements of the group to get a third." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"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)

"Although it is not difficult to count the holes in a real pretzel in your hand, prior to eating it, when a surface pops out of an abstract mathematical construction it can be very difficult to figure out its properties, such as how many holes it has. The cohomology groups can help us to do so." (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)

"Many people believe that all of mathematics has already been discovered and codified. Mathematicians (they think) do nothing except rearrange the material in different ways for different types of students. This seems to be the result of the cut-and-dried method of teaching mathematics in many high schools and universities. The facts are laid out in the cleanest logical order. Little attempt is made to show how someone once had to invent it all, at first in a confused way, and that only later was it possible to give it this neat form." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematicians are just lucky that elliptic curves are fairly simple - they only involve two variables and no powers higher than the cube - and yet are so rich." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematicians often get bored by a problem after they have fully understood it and have given proofs of their conjectures. Sometimes they even forget the precise details of what they have done after the lapse of years, having refocused their interest in another area. The common notion of the mathematician contemplating timeless truths, thinking over the same proof again and again - Euclid looking on beauty bare - is rarely true in any static sense." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Mathematics is like a game. It has rules, and to enjoy playing or watching it, you have to know and understand the rules. Mathematicians make up the rules as they go along." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"One thing mathematicians do is connect concepts that occur in different trains of thought." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Some number patterns, like even and odd numbers, lie on the surface. But the more you learn about numbers, both experimentally and theoretically, the more you discover patterns that are not so obvious. […] After a hidden pattern is exposed, it can be used to find more hidden patterns. At the end of a long chain of patterned reasoning, you can get to very difficult theorems, exploring facts about numbers that you otherwise would not know were true." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Still, in the end, we find ourselves drawn to the beauty of the patterns themselves, and the amazing fact that we humans are smart enough to prove even a feeble fraction of all possible theorems about them. Often, greater than the contemplation of this beauty for the active mathematician is the excitement of the chase. Trying to discover first what patterns actually do or do not occur, then finding the correct statement of a conjecture, and finally proving it - these things are exhilarating when accomplished successfully. Like all risk-takers, mathematicians labor months or years for these moments of success." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"The set of complex numbers is another example of a field. It is handy because every polynomial in one variable with integer coefficients can be factored into linear factors if we use complex numbers. Equivalently, every such polynomial has a complex root. This gives us a standard place to keep track of the solutions to polynomial equations." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"The word conjecture means 'guess'. The way it is used in mathematics is 'educated guess'." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"There is a big debate as to whether logic is part of mathematics or mathematics is part of logic. We use logic to think. We notice that our thinking, when it is valid, goes in certain patterns. These patterns can be studied mathematically. Thus, logic is a part of mathematics, called 'mathematical logic'." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006) 

"What is a group? It is just a pattern that certain things can exhibit when you have a composition law for always getting a third thing by combining any two others." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"For centuries, some people tried to prove that the parallel postulate could be derived from the other axioms and postulates. In the nineteenth century, this was shown to be impossible. How? Mathematicians constructed various models in which all the other axioms and postulates held but the parallel postulate failed." (Avner Ash & Robert Gross, "Summing It Up: From one plus one to modern number theory", 2016)

"Often a yes/no problem in mathematics becomes more interesting when it turns into a counting problem. One reason for this is that counting may introduce more structure, or a finer structure, into the data or the conceptual framework of the problem. Similarly, sometimes the counting problem becomes more interesting if it can be turned into a group theory problem. Sometimes, the yes/no problem has an obvious answer but the counting problem still yields a fascinating and beautiful theory. Another reason to solve counting problems is that we are more naturally interested in a more precise and more quantitative question than the simple binary puzzle that we began with. Of course, only with success in achieving an interesting answer will we be satisfied with the more complex approach." (Avner Ash & Robert Gross, "Summing It Up: From one plus one to modern number theory", 2016)

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)

On Symmetry XI

"The investigation of the symmetries of a given mathematical structure has always yielded the most powerful results. Symmetries are maps which preserve certain properties." (Emil Artin, "Geometric Algebra", 1957)

"Whereas the continuous symmetries always lead to conservation laws in classical mechanics, a discrete symmetry does not. With the introduction of quantum mechanics, however, this difference between the discrete and continuous symmetries disappears. The law of right-left symmetry then leads also to a conservation law: the conservation of parity." (Chen-Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)

"More generally, thermodynamics shows that there is an irreversible flow of time. Rather than there being time symmetry and indeed a reversibility of time as postulated in classical physics, a clear distinction is drawn between the past and future. An arrow of time results within open systems in the loss of organization and an increase in randomness or disorder over time. This accumulation of disorder or positive entropy results from the Second Law of Thermodynamics." (John Urry, "Global Complexity", 2003)

"A physical system is said to possess a symmetry if one can make a change in the system such that, after the change, the system is exactly the same as it was before. We call the change we are making to the system a symmetry operation or a symmetry transformation. If a system stays the same when we do a transformation to it, we say that the system is invariant under the transformation." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"So, a scientist's definition of symmetry would be something like this: symmetry is an invariance of an object or system to a transformation. The invariance is the sameness or constancy of the system in form, appearance, composition, arrangement, and so on, and a transformation is the abstract action we apply to the system that takes it from one state into another, equivalent, one. There are often numerous transformations we can apply on a given system that take it into an equivalent state." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"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 of some mathematical structure is a transformation of that structure, of a specified kind, that leaves specified properties of the structure unchanged." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"A system governed by a deterministic theory can only evolve along a single trajectory - namely, that dictated by its laws and initial conditions; all other trajectories are excluded. Symmetry principles, on the other hand, fit the freedom-inducing model. Rather than distinguishing what is excluded from what is bound to happen, these principles distinguish what is excluded from what is possible. In other words, although they place restrictions on what is possible, they do not usually determine a single trajectory." (Yemima Ben-Menahem, "Causation in Science", 2018)

"Symmetries are transformations that keep certain parameters (properties, equations, and so on) invariant, that is, the parameters they refer to are conserved under these transformations. It is to be expected, therefore, that the identification of conserved quantities is inseparable from the identification of fundamental symmetries in the laws of nature. Symmetries single out 'privileged' operations, conservation laws single out 'privileged' quantities or properties that correspond to these operations. Yet the specific connections between a particular symmetry and the invariance it entails are far from obvious. For instance, the isotropy of space (the indistinguishability of its directions) is intuitive enough, but the conservation of angular momentum based on that symmetry, and indeed, the concept of angular momentum, are far less intuitive." (Yemima Ben-Menahem, "Causation in Science", 2018)

Leon M Lederman - Collected Quotes

"A physical system is said to possess a symmetry if one can make a change in the system such that, after the change, the system is exactly the same as it was before. We call the change we are making to the system a symmetry operation or a symmetry transformation. If a system stays the same when we do a transformation to it, we say that the system is invariant under the transformation." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Although nature suggests a pathway to a mathematical description of everything, it has thus far eluded a final or complete grand mathematical synthesis. […] Mathematics is therefore inspired by nature. But it does not have to conduct experimental observations to proceed. The worlds of mathematics and theoretical physics are therefore distinct - they have different 'mission statements'. Whereas theoretical physics maps the properties of the nature we experience, mathematics builds a map of all possible 'natures' that logic permits to exist." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Art imitates nature through the incorporation of discrete symmetries, and indeed, reflection symmetry can be found throughout nature." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Eternal constancy of the laws of physics is a symmetry. What we see as we look back in time, or we peer through telescopes out into space, or we look through our powerful microscopes (particle accelerators), is the same system of laws of physics governing the whole universe at all times and at all places. These are the basic symmetries of the structure of our universe and its contents and, at a deeper level, the symmetries of the laws that govern the universe themselves. Indeed, the symmetries we uncover are the basic principles that define our laws of nature and the laws of physics, hence those that control our universe." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"In the physics of complex systems we can introduce a statistical concept, a measure of randomness, called entropy. In a quiet equilibrium, like hot onion soup sitting in a thermos bottle with no escaping heat, the entropy remains constant in time. However, in violent nonequilibrium processes, like shattering glass or explosions, the entropy always increases. Essentially, entropy, as a measure of randomness, will always increase when a very ordered initial condition leads to a very disordered final state through the normal laws of physics. The fact that entropy at best stays the same in equilibrium, or increases in all other processes, is called the second law of thermodynamics." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004) 

"Mathematicians have evolved a systematic way of thinking about symmetries that is fairly easy to grasp at the outset and a lot of fun to play with. This almost magical subject is known as group theory. […] Group theory is the mathematical language of symmetry, and it is so important that it seems to play a fundamental role in the very structure of nature. It governs the forces we see and is believed to be the organizing principle underlying all of the dynamics of elementary particles. Indeed, in modem physics the concept of symmetry serves as perhaps the most crucial concept of all. Symmetry principles are now known to dictate the basic laws of physics, to control the structure and dynamics of matter, and to define the fundamental forces in nature. Nature, at its most fundamental level, is defined by symmetry." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Quantum-mechanical effects appear in physical systems that are exceedingly small. A small system means very tiny objects with very tiny amounts of energy, moving around over very short time intervals. Quantum effects show up dramatically once we arrive at length scales the size of the atom, about one ten-thousandth of a millionth of a meter. In fact, we simply cannot understand an atom without quantum mechanics. This is not to say that nature itself suddenly 'switches off'' classical mechanics and 'switches on' quantum mechanics when we enter this new submicroscopic realm. Quantum mechanics is always valid and always holds true at all scales of nature. Rather, quantum effects gradually become more and more pronounced as we descend into the world of atoms. Quantum mechanics is the ultimate set of rules, as far as we know, that governs how nature works" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"So, a scientist's definition of symmetry would be something like this: symmetry is an invariance of an object or system to a transformation. The invariance is the sameness or constancy of the system in form, appearance, composition, arrangement, and so on, and a transformation is the abstract action we apply to the system that takes it from one state into another, equivalent, one. There are often numerous transformations we can apply on a given system that take it into an equivalent state." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Suppose that while flipping a coin, a small black hole passed by and ate the coin. As long as we got to see the coin, the probabilities of heads and tails would add to one, but the possibility of a coin disappearing altogether into a black hole would have to be included. Once the coin crosses the event horizon of the black hole, it simply does not meaningfully exist in our universe anymore. Can we simply adjust our probabilistic interpretation to accommodate this outcome? Will we ever encounter negative probabilities?" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"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 space and time of the universe that we humans inhabit contain symmetries. These are almost obvious yet subtle, even mysterious. Space and time form the stage upon which the dynamics - that is, the motion and interactions of the physical systems, atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of space and time control the dynamics of the physical interactions of matter." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The subject of mathematics has its own identity, on the other hand, and, in contrast to physicists, mathematicians attempt to create a roadmap of all possible logical systems that could consistently exist, whether they ultimately have anything to do with nature or not. Yet nature provides the basis of abstraction that gives birth to mathematics. " (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)

"Theoretical physics borrows from mathematics (or, if there's none to borrow, they invent new mathematics) in order to create a mathematical roadmap of things that can happen in the real world, in nature. It strives to explain all of the many different phenomena observed in the universe, perhaps ultimately seeking one elegant and economical logical system. However, physicists usually settle for lesser triumphs, in which many physical systems with common and comprehensible behaviors are successfully described. This description is always created in the abstract language of mathematics." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"We have come, in our time, to systematize our understanding of the rules of nature. We say that these rules are the laws of physics. The language of the laws of nature is mathematics. We acknowledge that our understanding of the laws is still incomplete, yet we know how to proceed to enlarge our understanding by means of the 'scientific method' - a logical process of observation and reason that distills the empirically true statements we can make about nature." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"We often wonder why the more complex systems seem to indicate a preferred direction of time, or an arrow of time, whereas their elementary counterparts do not. […] This has to do with the if-then nature of physics questions. Anything we observe involves laws of motion but also particular initial conditions. […] The initial conditions are what make a situation look peculiar when we time reverse it." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

On Randomness XIX (Chaos II)

"The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results. Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical find randomness of the natural environment." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"Chaos demonstrates that deterministic causes can have random effects […] There's a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. […] This paradox, that symmetry can get lost between cause and effect, is called symmetry-breaking. […] From the smallest scales to the largest, many of nature's patterns are a result of broken symmetry; […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"Often, we use the word random loosely to describe something that is disordered, irregular, patternless, or unpredictable. We link it with chance, probability, luck, and coincidence. However, when we examine what we mean by random in various contexts, ambiguities and uncertainties inevitably arise. Tackling the subtleties of randomness allows us to go to the root of what we can understand of the universe we inhabit and helps us to define the limits of what we can know with certainty." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Randomness, chaos, uncertainty, and chance are all a part of our lives. They reside at the ill-defined boundaries between what we know, what we can know, and what is beyond our knowing. They make life interesting." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"There are only patterns, patterns on top of patterns, patterns that affect other patterns. Patterns hidden by patterns. Patterns within patterns. If you watch close, history does nothing but repeat itself. What we call chaos is just patterns we haven't recognized. What we call random is just patterns we can't decipher. what we can't understand we call nonsense. What we can't read we call gibberish. There is no free will. There are no variables." (Chuck Palahniuk, "Survivor", 1999)

"Heat is the energy of random chaotic motion, and entropy is the amount of hidden microscopic information." (Leonard Susskind, "The Black Hole War", 2008)

"Chaos is impatient. It's random. And above all it's selfish. It tears down everything just for the sake of change, feeding on itself in constant hunger. But Chaos can also be appealing. It tempts you to believe that nothing matters except what you want." (Rick Riordan, "The Throne of Fire", 2011)

"A system in which a few things interacting produce tremendously divergent behavior; deterministic chaos; it looks random but its not." (Chris Langton)

Statistical Tools II: Dices

"God's dice always have a lucky roll." (Sophocles, 5th century BC)

"[...] to repeat the same throw ten thousand times with the dice would be impossible, whereas to make it once or twice is comparatively easy." (Aristotle, "On the Heavens", cca. 350 BC)

"Four dice are cast and a Venus throw results-that is chance; but do you think it would be chance, too, if in one hundred casts you made one hundred Venus throws? It is possible for paints flung at rando mon a canvas to form the outline of a face; but do you imagine that an accidental scattering of pigments could produce the beautiful portrait of Venus of Cos? Suppose that a hog should form a letter 'A' on the ground with its snout; is that a reason for believing that it could write out Ennius's poem The Andromche?" (Marcus Tullius Cicero, cca. 44 BC)

"Is it possible, then, for any man to apprehend in advance occurrences for which no cause or reason can be assigned? What do we mean when we employ such terms as luck, fortune, accident, turn of the die, except that we are seeking to describe events which happened and came to pass in such a way that they either might not have happened and come to pass at all or might have happened and come to pass under quite different circumstances? How then can an event be anticipated and predicted which occurs fortuitously and as a result of blind chance and of the spinning of Fortune's wheel?" (Marcus Tullius Cicero, "De Divinatione", 44 BC)

"The law of large numbers is noted in events which are attributed to pure chance since we do not know their causes or because they are too complicated. Thus, games, in which the circumstances determining the occurrence of a certain card or certain number of points on a die infinitely vary, can not be subjected to any calculus. If the series of trials is continued for a long time, the different outcomes nevertheless appear in constant ratios. Then, if calculations according to the rules of a game are possible, the respective probabilities of eventual outcomes conform to the known Jakob Bernoulli theorem. However, in most problems of contingency a prior determination of chances of the various events is impossible and, on the contrary, they are calculated from the observed result." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"A perfect equity adjusts its balance in all parts of life. The dice of God are always loaded. The world looks like a multiplication-table, or a mathematical equation, which, turn it how you will, balances itself." (Ralph W Emerson, "Compensation", 1841)

"Without doubt, matter is unlimited in extent, and, in this sense, infinite; and the forces of Nature mould it into an innumerable number of worlds. Would it be at all astonishing if, from the universal dice-box, out of an innumberable number of throws, there should be thrown out one world infinitely perfect? Nay, does not the calculus of probabilities prove to us that one such world out of an infinite number, must be produced of necessity?" (Philippe Buchez & William B Greene, "Remarks on the Science of History: Followed by an a priori autobiography", 1849)

"As an instrument for selecting at random, I have found nothing superior to dice. It is most tedious to shuffle cards thoroughly be- tween each successive draw, and the method of mixing and stirring up marked balls in a bag is more tedious still. A teetotum or some form of roulette is preferable to these, but dice are better than all. When they are shaken and tossed in a basket, they hurtle so variously against one another and against the ribs of the basket-work that they tumble wildly about, and their positions at the outset afford no perceptible clue to what they will be after even a single good shake and toss." (Francis Galton, Nature vol. 42, 1890) 

"A throw of the dice will never abolish chance." (Stéphane Mallarmé, 1897)

"If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force—and every other representation remains indefinite and therefore useless—it follows that, in the great dice game of existence, it must pass through calculable number of combinations. In infinite time, every possible combination would at some time or another be realized; more: it would be realized an infinite number of times. And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combination conditions of the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demonstrated: the world as a circular movement that has already repeated itself infinitely often and plays its game in infinitum. This conception is not simply a mechanistic conception; for if it were that, it would not condition an infinite recurrence of identical cases, but a final state. Because the world has not reached this, mechanistic theory must be considered an imperfect and merely provisional hypothesis." (Friedrich Nietzsche, "The Will to Power", [notes written 1883-1888] 1901)

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one." I, at any rate, am convinced that He does not throw dice." (Albert Einstein, [Letter to Max Born] 1926)

"It seems hard to sneak a look at God's cards. But that He plays dice and uses 'telepathic' methods [...] is something that I cannot believe for a single moment." (Albert Einstein, [Letter to Cornel Lanczos] 1942)

"You believe in the God who plays dice, and I in complete law and order in a world that objectively exists, and which I, in a wildly speculative way, am trying to capture. [...] Even the great initial success of the quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility. No doubt the day will come when we will see whose instinctive attitude was the correct one." (Albert Einstein, [Letter to Max Born] 1944)

"If God has made the world a perfect mechanism, He has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success." (Max Born, "Albert Einstein: Philosopher-Scientist", 1949)

"In the game of heads and tails, if head comes up a hundred times in a row then this appears to us extraordinary, because after dividing the nearly infinite number of combinations that can arise in a hundred throws into regular sequences, or those in which we observe a rule that is easy to grasp, and into irregular sequences, the latter are incomparably more numerous." (Pierre-Simon Laplace, "A Philosophical Essay on Probability Theories", 1951)

"The picture of scientific method drafted by modern philosophy is very different from traditional conceptions. Gone is the ideal of a universe whose course follows strict rules, a predetermined cosmos that unwinds itself like an unwinding clock. Gone is the ideal of the scientist who knows the absolute truth. The happenings of nature are like rolling dice rather than like revolving stars; they are controlled by probability laws, not by causality, and the scientist resembles a gambler more than a prophet. He can tell you only his best posits - he never knows beforehand whether they will come true. He is a better gambler, though, than the man at the green table, because his statistical methods are superior. And his goal is staked higher - the goal of foretelling the rolling dice of the cosmos. If he is asked why he follows his methods, with what title he makes his predictions, he cannot answer that he has an irrefutable knowledge of the future; he can only lay his best bets. But he can prove that they are best bets, that making them is the best he can do - and if a man does his best, what else can you ask of him?" (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"We must emphasize that such terms as 'select at random', 'choose at random', and the like, always mean that some mechanical device, such as coins, cards, dice, or tables of random numbers, is used." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)

"Consideration of particle emission from black holes would seem to suggest that God not only plays dice, but also sometimes throws them where they cannot be seen." (Stephen Hawking, "The Quantum Mechanics of Black Holes", Scientific American, 1977)

"Not only does God play dice with the world - He does not let us see what He has rolled." (Stanisław Lem, "Imaginary Magnitude", 1981)

"Specialists [...] are slowly coming to the realization that the universe is biased and leans to the left. [...] Many scientists have come to believe that this odd state of affairs has somethittg to do with the weak nuclear force. It seems that the weak force tends to impart a left-handed spin to electrons, and this effect may bias some kinds of molecular synthesis to the left. [...] But scientific speculation of this ilk leads to a deeper question. Was it purely a matter of chance that left-handedness became the preferred direction in our universe, or is there some reason behind it? Did the sinister bent of existence that scientists have observed stem from a roll of the dice, or is God a semiambidextrous southpaw?" (Malcolm W Browne, 1986)

"[In statistics] you have the fact that the concepts are not very clean. The idea of probability, of randomness, is not a clean mathematical idea. You cannot produce random numbers mathematically. They can only be produced by things like tossing dice or spinning a roulette wheel. With a formula, any formula, the number you get would be predictable and therefore not random. So as a statistician you have to rely on some conception of a world where things happen in some way at random, a conception which mathematicians don’t have." (Lucien LeCam, [interview] 1988)

"Combinatorics, a sort of glorified dice-throwing." (Robert Kanigel, "The Man Who Knew Infinity: A Life of the Genius Ramanujan", 1991)

"Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"It is true that every aspect of the roll of dice may be suspect: the dice themselves, the form and texture of the surface, the person throwing them. If we push the analysis to its extreme, we may even wonder what chance has to do with it at all. Neither the course of the dice nor their rebounds rely on chance; they are governed by the strict determinism of rational mechanics. Billiards is based on the same principles, and it has never been considered a game of chance. So in the final analysis, chance lies in the clumsiness, the inexperience, or the naiveté of the thrower - or in the eye of the observer." (Ivar Ekeland, "The Broken Dice, and Other Mathematical Tales of Chance", 1993)

"Whether we shuffle cards or roll dice, chance is only a result of our human lack of deftness: we don't have enough control to immobilize a die at will or to individually direct the cards in a deck. The comparison is an important one nonetheless, and highlights the limits of this method of creating chance - it doesn't matter who rolls the dice, but we wouldn't let just anyone shuffle the cards." (Ivar Ekeland, "The Broken Dice, and Other Mathematical Tales of Chance", 1993)

"So Einstein was wrong when he said, 'God does not play dice'. Consideration of black holes suggests, not only that God does play dice, but that he sometimes confuses us by throwing them where they can't be seen." (Stephen Hawking, 1994)

"Yet, Einstein's theories are also not the last word: quantum theory and relativity are inconsistent, and Einstein himself, proclaiming that 'God does not play dice!', rejected the basic reliance of quantum theory on chance events, and looked forward to a theory which would be deterministic. Recent experiments suggest that this view of Einstein's conflicts with his other deeply held beliefs about the nature of the physical universe. Certain it is that somewhere, beyond physicists' current horizons, are even more powerful theories of how the world is." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"In systems such as contemporary society, evolution is always a promise and devolution is always a threat. No system comes with a guarantee of ongoing evolution. The challenge is real. To ignore it is to play dice with all we have. To accept it is not to play God - it is to become an instrument of whatever divine purpose infuses the universe." (Ervin László, "The systems view of the world", 1996)

"[...] an apparently random universe could be obeying every whim of a deterministic deity who chooses how the dice roll; a universe that has obeyed perfect mathematical laws for the last ten billion years could suddenly start to play truly random dice. So the distinction is about how we model the system, and what point of view seems most useful, rather than about any inherent feature of the system itself." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Chaos teaches us that anybody, God or cat, can play dice deterministically, while the naïve onlooker imagines that something random is going on." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Indeed a deterministic die behaves very much as if it has six attractors, the steady states corresponding to its six faces, all of whose basins are intertwined. For technical reasons that can't quite be true, but it is true that deterministic systems with intertwined basins are wonderful substitutes for dice; in fact they're super-dice, behaving even more ‘randomly’ - apparently - than ordinary dice. Super-dice are so chaotic that they are uncomputable. Even if you know the equations for the system perfectly, then given an initial state, you cannot calculate which attractor it will end up on. The tiniest error of approximation – and there will always be such an error - will change the answer completely." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Perhaps God can play dice, and create a universe of complete law and order, in the same breath." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Simple laws may not produce simple behaviour. Deterministic laws can produce behaviour that appears random. Order can breed its own kind of chaos. The question is not so much whether God plays dice, but how God plays dice.", 1997)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing - seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"From the moment we first roll a die in a children’s board game, or pick a card (any card), we start to learn what probability is. But even as adults, it is not easy to tell what it is, in the general way." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"[...] the chance of a head (or a double six) is just a chance. The whole point of probability is to discuss uncertain eventualities before they occur. After this event, things are completely different. As the simplest illustration of this, note that even though we agree that if we ¯ip a coin and roll two dice then the chance of a head is greater than the chance of a double six, nevertheless it may turn out that the coin shows a tail when the dice show a double six." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"[…] we would like to observe that the butterfly effect lies at the root of many events which we call random. The final result of throwing a dice depends on the position of the hand throwing it, on the air resistance, on the base that the die falls on, and on many other factors. The result appears random because we are not able to take into account all of these factors with sufficient accuracy. Even the tiniest bump on the table and the most imperceptible move of the wrist affect the position in which the die finally lands. It would be reasonable to assume that chaos lies at the root of all random phenomena." (Iwo Bialynicki-Birula & Iwona Bialynicka-Birula, "Modeling Reality: How Computers Mirror Life", 2004)

"Random number generators do not always need to be symmetrical. This misconception of assuming equal likelihood for each outcome is fostered in a restricted learning environment, where learners see only such situations (that is, dice, coins and spinners). It is therefore very important for learners to be aware of situations where the different outcomes are not equally likely (as with the drawing-pins example)." (Alan Graham, "Developing Thinking in Statistics", 2006)

"There is no such thing as randomness. No one who could detect every force operating on a pair of dice would ever play dice games, because there would never be any doubt about the outcome. The randomness, such as it is, applies to our ignorance of the possible outcomes. It doesn’t apply to the outcomes themselves. They are 100% determined and are not random in the slightest. Scientists have become so confused by this that they now imagine that things really do happen randomly, i.e. for no reason at all." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"God may not play dice with the universe, but something strange is going on with the prime numbers." (Paul Erdos)

"God plays dice with the universe, but they’re loaded dice. And the main objective of physics now is to find out by what rules were they loaded and how can we use them for our own ends." (Joseph Ford)

"I shall never believe that God plays dice with the world." (Albert Einstein)

"The perfect die does not lose its usefulness or justification by the fact that real dice fail to live up to it." (William Feller)

Catastrophe Theory II

"What I am offering, is not a scientific theory, but a method; the first step in the construction of a model is to describe the dynamical models compatible with an empirically given morphology, and this is also the first step in understanding the phenomena under consideration. [...] We may hope that theoreticians will develop a quantitative model [for specific processes described by catastrophe theory ...] But this is only a hope." (René Thom, "Structural Stability and Morphogenesis", 1972)

"The catastrophe model is at the same time much less and much more than a scientific theory; one should consider it as a language, a method, which permits classification and systematization of given empirical data [...] In fact, any phenomenon at all can be explained by a suitable model from catastrophe theory." (René F Thom, 1973)

"First, nature's line patterns are not all of the same sort; the triple junctions generic in mud cracks cannot occur with caustics. Second, the geometrical optics of cylindrically symmetric artifacts such as telescopes, where departures from the ideal point focus are treated as 'aberrations', is very different from the geometrical optics of nature, where the generic forms of caustic surfaces are governed by the mathematics of catastrophe theory." Michael V Berry & John F Nye, "Fine Structure in Caustic Junctions", Nature Vol. 267 (3606), 1977)

"the claims made for the theory are greatly exaggerated and its accomplishments, at least in the biological and social sciences, are insignificant. [...] Catastrophe theory is one of many attempts that have been made to deduce the world by thought alone [...] an appealing dream for mathematicians, but a dream that cannot come true."  (Héctor J Sussmann & Raphael S Zahler, Nature, 1977)

"A catastrophe, in the very broad sense [René] Thom gives to the word, is any discontinuous transition that occurs when a system can have more than one stable state, or can follow more than one stable pathway of change. The catastrophe is the 'jump' from one state or pathway to another." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"It is not enough to know the critical stress, that is, the quantitative breaking point of a complex design; one should also know as much as possible of the qualitative geometry of its failure modes, because what will happen beyond the critical stress level can be very different from one case to the next, depending on just which path the buckling takes. And here catastrophe theory, joined with bifurcation theory, can be very helpful by indicating how new failure modes appear." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"The unfoldings are called catastrophes because each of them has regions where a dynamic system can jump suddenly from one state to another, although the factors controlling the process change continuously. Each of the seven catastrophes represents a pattern of behavior determined only by the number of control factors, not by their nature or by the interior mechanisms that connect them to the system's behavior. Therefore, the elementary catastrophes can be models for a wide variety of processes, even those in which we know little about the quantitative laws involved." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Two assumptions are needed to apply catastrophe theory as it now stands: first, that the system described be governed by a potential, and second, that its behavior depend on a limited number of control factors. Without these assumptions, the classification of the elementary catastrophes is impossible." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"It is more a philosophy than mathematics, and even as a philosophy it doesn't explain the real world [...] as mathematics, it brings together two of the most basic ideas in modern math: the study of dynamic systems and the study of the singularities of maps. Together, they cover a very wide area - but catastrophe theory brings them together in an arbitrary and constrained way." (Steven Smale)

"While it must be granted that a number of immoderate claims in the form of 'catastrophe theory can do everything' have been made in the literature, on the basis of too little experience, it doesn't seem that the proper response is an equally immoderate claim that 'catastrophe theory can do nothing' on the basis of that same body of experience." (Robert Rosen)

Edward Witten - Collected Quotes

"Discovery of supersymmetry would be one of the real milestones in physics, made even more exciting by its close links to still more ambitious theoretical ideas. Indeed, supersymmetry is one of the basic requirements of 'string theory', which is the framework in which theoretical physicists have had some success in unifying gravity with the rest of the elementary particle forces. Discovery of supersymmetry would would certainly give string theory an enormous boost." (Edward Witten, [preface to (Gordon Kane, "Supersymmetry: Unveiling the Ultimate Laws of Nature", 2000) 1999)

"In string theory, to understand the nature of the Big Bang, or the quantum fate of a black hole, or the nature of the vacuum state that determines the properties of the elementary particles, requires information beyond perturbation theory [...] Perturbation theory is not everything. It is just the way the [string] theory was discovered." (Edward Witten, "The Past and Future of String Theory", 2003)

"Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. [...] 2. It greatly reduces the number of possible theories. [...] 3. It gives the first hint that string theory will change our notions of spacetime." (Edward Witten, "The Past and Future of String Theory", 2003)

"To put it differently, global symmetry is a property of a system, but gauge symmetry in general is a property of a description of a system. What we really learn from the centrality of gauge symmetry in modern physics is that physics is described by subtle laws that are 'geometrical'. 
This concept is hard to define, but what it means in practice is that the laws of Nature are subtle in a way that defies efforts to make them explicit without making choices. The difficulty of making these laws explicit in a natural and non-redundant way is the reason for 'gauge symmetry'." (Edward Witten, "Symmetry and Emergence", 2018)

"We can see the relation between gauge symmetry and global symmetry in another way if we imagine whether physics as we know it could one day be derived from something much deeper – maybe unimaginably deeper than we now have. Maybe the spacetime we experience and the particles and fields in it are all 'emergent' from something much deeper." (Edward Witten, "Symmetry and Emergence", 2018)

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

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

"Even though it is, properly speaking, a postprediction, in the sense that the experiment was made before the theory, the fact that gravity is a consequence of string theory, to me, is one of the greatest theoretical insights ever." (Edward Witten)

"I would expect that a proper elucidation of what string theory really is all about would involve a revolution in our concepts of the basic laws of physics - similar in scope to any that occurred in the past. (Edward Witten [interview])

♾️On Complex Numbers XVI

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

 "√-1 is take for granted today. No serious mathematician would deny that it is a number. Yet it took centuries for √-1 to be officially admitted to the pantheon of numbers. For almost three centuries, it was controversial; mathematicians didn't know what to make of it; many of them worked with it successfully without admitting its existence. […] Primarily for cognitive reasons. Mathematicians simply could not make it fit their idea of what a number was supposed to be. A number was supposed to be a magnitude. √-1 is not a magnitude comparable to the magnitudes of real numbers. No tree can be √-1 units high. You cannot owe someone √-1 dollars. Numbers were supposed to be linearly ordered. √-1 is not linearly ordered with respect to other numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"From a formal perspective, much about complex numbers and arithmetic seems arbitrary. From a purely algebraic point of view, i arises as a solution to the equation x^2+1=0. There is nothing geometric about this - no complex plane at all. Yet in the complex plane, the i-axis is 90° from the x-axis. Why? Complex numbers in the complex plane add like vectors. Why? Complex numbers have a weird rule of multiplication […]" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"[…] i is not a real number-not ordered anywhere relative to the real numbers! In other words, it does not even have the central property of ‘numbers’, indicating a magnitude that can be linearly compared to all other magnitudes. You can see why i has been called imaginary. It has almost none of the properties of the small natural numbers-not subitizability, not groupability, and not even relative magnitude. If i is to be a number, it is a number only by virtue of closure and the laws of arithmetic." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The complex plane is just the 90° rotation plane-the rotation plane with the structure imposed by the 90° Rotation metaphor added to it. Multiplication by i is 'just' rotation by 90°. This is not arbitrary; it makes sense. Multiplication by-1 is rotation by 180°. A rotation of 180° is the result of two 90° rotations. Since i times i is -1, it makes sense that multiplication by i should be a rotation by 90°, since two of them yield a rotation by 180°, which is multiplication by -1." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Negative numbers posed some of the same quandaries that the imaginary numbers did to Renaissance mathematicians - they didn’t seem to correspond to quantities associated with physical objects or geometrical figures. But they proved less conceptually challenging than the imaginaries. For instance, negative numbers can be thought of as monetary debts, providing a readily grasped way to make sense of them." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the 'i times π' power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The fact that multiplying positive 4i times positive 4i yields negative 16 seems like saying that the friend of my friend is my enemy. Which in turn suggests that bad things would happen if i and its offspring were granted citizenship in the number world. Unlike real numbers, which always feel friendly toward the friends of their friends, the i-things would plainly be subject to insane fits of jealousy, causing them to treat numbers that cozy up to their friends as threats. That might cause a general breakdown of numerical civility." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Basis real and imaginary numbers have eternal and necessary reality. Complex numbers do not. They are temporal and contingent in the sense that for complex numbers to exist, we first have to carry out an operation: adding basis real and imaginary numbers together. Complex numbers therefore do not exist in their own right. They are constructed. They are derived. Symmetry breaking is exactly where constructed numbers come into existence. The very act of adding a sine wave to a cosine wave is the sufficient condition to create a broken symmetry: a complex number. The 'Big Bang', mathematically, is simply where a perfect array of basis sine and cosine waves start entering into linear combinations, creating a chain reaction, an 'explosion', of complex numbers - which corresponds to the 'physical' universe." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

String Theory I

"The odd thing about string theory was very odd indeed. It required that the universe have at least ten dimensions. As we live in a universe of only four dimensions, the theory postulated that the other dimensions [...] had collapsed into structures so tiny that we do not notice them. (Timothy Ferris, "Coming of Age in the Milky Way", 1988)

"Theoretical physicists are accustomed to living in a world which is removed from tangible objects by two levels of abstraction. From tangible atoms we move by one level of abstraction to invisible fields and particles. A second level of abstraction takes us from fields and particles to the symmetry-groups by which fields and particles are related. The superstring theory takes us beyond symmetry-groups to two further levels of abstraction. The third level of abstraction is the interpretation of symmetry-groups in terms of states in ten-dimensional space-time. The fourth level is the world of the superstrings by whose dynamical behavior the states are defined." (Freeman J Dyson, "Infinite in All Directions", 1988)

"No other theory known to science [other than superstring theory] uses such powerful mathematics at such a fundamental level. […] because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory. […] The new mathematics, which is responsible for the merger of these two theories, is topology, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity." (Michio Kaku, "Hyperspace", 1995)

"Discovery of supersymmetry would be one of the real milestones in physics, made even more exciting by its close links to still more ambitious theoretical ideas. Indeed, supersymmetry is one of the basic requirements of 'string theory', which is the framework in which theoretical physicists have had some success in unifying gravity with the rest of the elementary particle forces. Discovery of supersymmetry would would certainly give string theory an enormous boost." (Edward Witten, [preface to (Gordon Kane, "Supersymmetry: Unveiling the Ultimate Laws of Nature", 2000) 1999)

"For string theory to make sense, the universe should have nine spatial dimensions and one time dimension, for a total of ten dimensions." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"If string theory is right, the microscopic fabric of our universe is a richly intertwined multidimensional labyrinth within which the strings of the universe endlessly twist and vibrate, rhythmically beating out the laws of the cosmos." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"To appreciate string theory, one must understand what it is that is being generalized." (Edward Witten, "Physical Law and the Quest for Mathematical Understanding", Bulletin of the American Mathematical Society Vol. 40 (1), 2002)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory." (Edward Witten)

"[The discovery of string theory is like] wandering around the desert and then stumbling on a tiny pebble. But when we examine it carefully, we find that it is the tip of a gigantic pyramid." (Michio Kaku)

Complex Systems III

"Complexity must be grown from simple systems that already work." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Even though these complex systems differ in detail, the question of coherence under change is the central enigma for each." (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)

"By irreducibly complex I mean a single system composed of several well-matched, interacting parts that contribute to the basic function, wherein the removal of any one of the parts causes the system to effectively cease functioning. An irreducibly complex system cannot be produced directly (that is, by continuously improving the initial function, which continues to work by the same mechanism) by slight, successive modification of a precursor, system, because any precursors to an irreducibly complex system that is missing a part is by definition nonfunctional." (Michael Behe, "Darwin’s Black Box", 1996)

"A dictionary definition of the word ‘complex’ is: ‘consisting of interconnected or interwoven parts’ […] Loosely speaking, the complexity of a system is the amount of information needed in order to describe it. The complexity depends on the level of detail required in the description. A more formal definition can be understood in a simple way. If we have a system that could have many possible states, but we would like to specify which state it is actually in, then the number of binary digits (bits) we need to specify this particular state is related to the number of states that are possible." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"When the behavior of the system depends on the behavior of the parts, the complexity of the whole must involve a description of the parts, thus it is large. The smaller the parts that must be described to describe the behavior of the whole, the larger the complexity of the entire system. […] A complex system is a system formed out of many components whose behavior is emergent, that is, the behavior of the system cannot be simply inferred from the behavior of its components." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"Complex systems operate under conditions far from equilibrium. Complex systems need a constant flow of energy to change, evolve and survive as complex entities. Equilibrium, symmetry and complete stability mean death. Just as the flow, of energy is necessary to fight entropy and maintain the complex structure of the system, society can only survive as a process. It is defined not by its origins or its goals, but by what it is doing." (Paul Cilliers,"Complexity and Postmodernism: Understanding Complex Systems", 1998)

"There is no over-arching theory of complexity that allows us to ignore the contingent aspects of complex systems. If something really is complex, it cannot by adequately described by means of a simple theory. Engaging with complexity entails engaging with specific complex systems. Despite this we can, at a very basic level, make general remarks concerning the conditions for complex behaviour and the dynamics of complex systems. Furthermore, I suggest that complex systems can be modelled." (Paul Cilliers," Complexity and Postmodernism", 1998)

"The self-similarity of fractal structures implies that there is some redundancy because of the repetition of details at all scales. Even though some of these structures may appear to teeter on the edge of randomness, they actually represent complex systems at the interface of order and disorder."  (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

Marcus du Sautoy - Collected Quotes

"Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations." (Marcus du Sautoy, "The Music of the Primes", 1998)

"[…] despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge." (Marcus du Sautoy, "The Music of the Primes", 2003)

"It seems paradoxical that the fundamental objects on which we build our order-filled world of mathematics should behave so wildly and unpredictably." (Marcus du Sautoy, "The Music of the Primes", 2003) 

"Despite the unworldly nature of mathematics, mathematicians still have egos that need massaging. Nothing acts as a better drive to the creative process than the thought of the immortality bestowed by having your name attached to a theorem." (Marcus du Sautoy, "The Music of the Primes", 2003)

"For centuries, mathematicians had been listening to the primes and hearing only disorganised noise. These numbers were like random notes wildly dotted on a mathematical stave with no discernible tune. Now Riemann had found new ears with which to listen to these mysterious tones. The sine-like waves that Riemann had created from the zeros in his zeta landscape revealed some hidden harmonic structure." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)

"A proof is like a piece of theatre or music, with moments of high drama where some major shift takes the audience into a new realm." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"As mathematicians had gradually got to grips with what symmetry actually meant, they seemed to be gazing upon an endless world filled with a chaotic and infinitely varied range of symmetrical objects." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"For both primes and symmetries, zeta functions act as black boxes. They are built from a formula which binds together the numbers you are trying to understand. The hope is that the zeta function will reveal new insights into the numbers of symmetries. It provides a way of getting from part of the mathematical world where chaos seems to reign to a completely different region where one can start to pick out patterns." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"For the mathematician, the pattern searcher, understanding symmetry is one of the principal themes in the quest to chart the mathematical world." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Science is about discovery, but it is also about communication. An idea can hardly be said to exist if you do not awaken that same idea in someone else." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Symmetry continues to inform the way we craft words in songs and poetry. From the first cave paintings to modern art, from primitive drumbeats to contemporary music, artists have continually pushed symmetry to the extremes."(Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008) 

"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Why, though, is symmetry so pervasive in nature? It is not just a matter of aesthetics. Just as it is for me and mathematics, symmetry in nature is about language. It provides a way for animals and plants to convey a multitude of messages, from genetic superiority to nutritional information. Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008) 

"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth ‘listening to’." (Marcus du Sautoy, "Listen by Numbers: Music and Maths", 2011)

"Mathematics has beauty and romance. It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there." (Marcus Du Sautoy)

"The reason why we do maths is because it's like poetry. It's about patterns, and that really turned me on. It made me feel that maths was in tune with the other things I liked doing." (Marcus du Sautoy)