🅿Edgar E Peters - Collected Quotes

"Because we hate living with uncertainty, we often try to make complex systems, such as the economy, more predictable and less uncertain. However, making a complex system more predictable also makes it less resilient to shocks, and less creative. Lowering uncertainty reduces complexity, often with disastrous effects." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"Complex systems, then, have local uncertainty and global certainty. They generate change, and they are resilient to unexpected shocks. They turn uncertainty into order, and they reverse order back into uncertainty. They evolve and change through time, and they do so without a central planner. Complex systems are everywhere. In fact, real life is one huge complex system. How is such behavior possible? First, we need to understand the general class of complex systems. By understanding their nature, we will see the important role uncertainty plays in maintaining stability. When we understand natural systems, we will understand the role of uncertainty in a free society." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

In a free-market economy, then, uncertainty is a necessary element. Only when the economy is in a state of uncertainty can the participants efficiently search for solutions to problems and find creative answers. In addition, only a system that depends on uncertainty can survive unexpected shocks. A complex process can take multiple paths to an optimal solution. It does not require 'ideal' conditions; in fact, shocks often force it to find a better solution, a higher hill in the fitness landscape. The 'creative destruction' identified by the Austrian school suggests that a free-market economy is not only resilient to shocks, but is also creative and capable of generating innovation. It can only do so while in a high state of uncertainty." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"It seems obvious that uncertainty reigns during times of crisis. However, crisis itself can be a positive development. It is only negative in that it specifies that change is coming. Most people are uncomfortable with change and equate it with hard times. Mainstream economics tends to take the same view, calling such events 'shocks'. They are even referred to as 'exogenous' - outside of the system. If it were not for change, according to the mainstream school, everything would continue along in perfect balance, a 'circular flow'. Change is like an alien invasion. The mainstream view ignores the fact that change is necessary." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"Uncertainty is not necessarily bad or synonymous with risk. Complex systems use uncertainty to their advantage as they adapt to changes in their environment and learn to be resilient to unexpected shocks. Uncertainty then, rather than being the source of so many problems, becomes a necessary element if a market and a society are to remain free." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999) 

📒🅰Brian W Aldiss - Collected Quotes

"If a machine had broken down, it would have been quickly replaced. But who can replace a man?" (Brian W Aldiss, "Who Can Replace a Man?", 1958)

"Science fiction is no more written for scientists than ghost stories are written for ghosts." (Brian Aldiss, Penguin Science Fiction, 1961)

"What was that epigram that he had trotted forth too often, about civilization being the distance man placed between himself and his excreta? But it was nearer the truth to say that civilization was the distance man had placed between himself and everything else." (Brian W Aldiss, "The Dark Light Years", 1964)

"The waking brain is perpetually lapped by the unconscious." (Brian W Aldiss, "Man in His Time", 1966)

"The silence seemed to carry as much weight as that deep mass of foliage which covered all the land on the day side of the planet. It was a silence built of millions upon millions of years, intensifying as the sun overhead poured forth more and more energy in the first stages of its decline. Not that the silence signified lack of life. Life was everywhere, life on a formidable scale. But the increased solar radiation that had brought the extinction of most of the animal kingdom had spelt the triumph of plant life. Everywhere, in a thousand forms and guises, the plants ruled. And vegetables have no voices." (Brian W Aldiss," Nomansland", 1961)

"Science fiction is the search for a definition of man and his status in the universe which will stand in our advanced but confused state of knowledge (science), and is characteristically cast in the Gothic or post-Gothic mould." (Brian W Aldiss, "Billion Year Spree: The True History of Science Fiction", 1973)

"Over most of the universe, God was spread in fossil radiation, too old, too thin." (Brian W. Aldiss, "Non-Isotropic", 1978)

"We take foul medicines to improve our health; so we must entertain foul thoughts on occasion, to strengthen wisdom." (Brian W Aldiss, "The Small Stones of Tu Fu", 1978)

📒🆉George Zebrowski - Collected Quotes

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"Apocalypse is the eye of a needle, through which we pass into a different world." (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"Part of human life is the need to reassure ourselves about the future that we may never live to see, rather than fool ourselves, as many did in the last century, that there won’t be any future and they might as well lie down and die." (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"Picture this: a mobile space colony, supporting more than a million people. No, not a colony, but an organism which can move and grow as long as it can obtain resources and maintain a food supply within its ecology. It’s a living organism because it can respond to stimuli through its optical and sensory nervous system. It thinks with the intellects of its human and cybernetic intelligences. And it can reproduce, which is what we expect from a living organism." (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"Progress is a tension between the notion of perfection and the notion that striving, not finding, is important." (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife", 1979)

"She reached out across the cathedral of space-time to those hopelessly distant candle-furnaces, where all the material elements had been forged again and again inside the generation of suns, where alien sunspaces were certain to contain other humanities, however different, and she wondered if someone there might be her friend." (George Zebrowski, "Macrolife", 1979)

"In a perfectly rational universe, infinities turn back on themselves [...]" (George Zebrowski, "Is Science Rational?", OMNI Magazine, 1994)

"The rationality of our universe is best suggested by the fact that we can discover more about it from any starting point, as if it were a fabric that will unravel from any thread." (George Zebrowski, "Is Science Rational?", OMNI Magazine, 1994)

"Time is a relationship that we have with the rest of the universe; or more accurately, we are one of the clocks, measuring one kind of time." (George Zebrowski, OMNI Magazine, 1994)

"Science, when it runs up against infinities, seeks to eliminate them, because a proliferation of entities is the enemy of explanation." (George Zebrowski, "Time Is Nothing But A Clock" , OMNI Magazine Vol. 17 (1), 1994)

📒🆂Clifford D Simak - Collected Quotes

"The chain of life runs smoothly from one generation to the next and none of the links stand out except here and there a link one sees by accident." (Clifford D Simak, "Time and Again", 1951)

"To cover up actual lack of knowledge, the tale develops an explanation which amounts to divine intervention. It is an easy and, to the primitive mind, a plausible and satisfactory way to explain something of which nothing at all is known." (Clifford D Simak, "City", 1952)

"We thought all the time that we were passing through time when we really weren’t, when we never have. We’ve just been moving along with time. We said, there’s another second gone, there’s another minute and another hour and another day, when, as a matter of fact the second or the minute or the hour was never gone. It was the same one all the time. It had just moved along and we had moved with it." (Clifford D Simak, "City", 1952)

"And time itself? Time was a never-ending medium that stretched into the future and the past - except there was no future and no past, but an infinite number of brackets, extending either way, each bracket enclosing its single phase of the Universe." (Clifford D Simak,"Ring Around the Sun", 1954)

"Once again the universe was spread far out before him and it was a different and in some ways a better universe, a more diagrammatic universe, and in time, he knew, if there were such a thing as time, he'd gain some completer understanding and acceptance of it." (Clifford D Simak, "All the Traps of Earth", 1960)

"It was a place without a single feature of the space-time matrix that he knew. It was a place where nothing yet had happened - an utter emptiness. There was neither light nor dark: there was nothing here but emptiness. There had never been anything in this place, nor was anything ever intended to occupy this place [...]" (Clifford D Simak, "Time is the Simplest Thing", 1961)

"There was so much knowledge in the galaxy and he knew so little of it, understood so little of the little that he knew. [...] Out among the stars lay a massive body of knowledge, some of it an extension of what mankind knew, some of it concerning matters which Man had not yet suspected, and used in ways and for purposes that Man had not as yet imagined. And never might imagine, if left on his own." (Clifford D Simak, "Way Station", 1963)

"It's not the machine itself that does the trick. The machine merely acts as an intermediary between the sensitive and the spiritual force. It is an extension of the sensitive. It magnifies the capability of the sensitive and acts as a link of some sort. It enables the sensitive to perform his function." (Clifford D Simak, "Way Station", 1963)

"There was a comfort in the thought, a strange sort of personal comfort in being able to believe that some intelligence might have solved the riddle of that mysterious equation of the universe. And how, perhaps, that mysterious equation might tie in with the spiritual force that was idealistic brother to time and space and all those other elemental factors that held the universe together." (Clifford D Simak, "Way Station", 1963)

"The robots are not technologically minded. They were not built to be. They were built to bolster human vanity and pride, to meet a strange longing that seems to be built into the human ego - the need to have other humans (or a reasonable facsimile of other humans) to minister to our wants and needs, human slaves to be dominated, human beings over which a man or woman (or a child) can assert authority, thus building up a false feeling of superiority." (Clifford D Simak, "A Choice of Gods", 1972)

"These kind never change. The machine does something to a man. It brutalizes him. It serves as a buffer between himself and his environment and he is the worst for it. It arouses an opportunistic instinct and makes possible a greed that makes a man inhuman." (Clifford D Simak, "A Choice of Gods", 1972)

"If mankind were to continue in other than the present barbarism, a new path must be found, a new civilization based on some other method than technology. Space is an illusion, and time as well. There is no such factor as either time or space. We have been blinded by our own cleverness, blinded by false perceptions of those qualities that we term eternity and infinity. There is another factor that explains it all, and once this universal factor is recognized, everything grows simple. There is no longer any mystery, no longer any wonder, no longer any doubt; for the simplicity of it all lies before us [...]" (Clifford D Simak,"A Heritage of Stars", 1977)

📒🅰Poul Anderson - Collected Quotes

"He was well on the way to inventing differential calculus when his mother called him down to breakfast." (Poul Anderson, "Brain Wave", 1954)

"No one person can change a whole culture." (Poul Anderson, "Ghetto", 1954)

"Memory is not a passive filing cabinet, but a continuous process beneath the level of consciousness; in a way, you are always reliving your entire past." (Poul Anderson, "Journeys End", 1957)

"Time is the bridge that always burns behind us." (Poul Anderson, "The Burning Bridge", 1960)

"We live with our archetypes, but can we live in them?" (Poul Anderson, "The Fatal Fulfillment", 1970

"Homo can truly be called sapiens when he practices his specialty of being unspecialized. His repeated attempts to freeze himself into an all-answering pattern or culture or ideology, or whatever he has named it, have repeatedly brought ruin. Give him the pragmatic business of making his living, and he will usually do rather well. He adapts, within broad limits." (Poul Anderson, "The Queen of Air and Darkness", 1971)

"The best foundation that a decision is ever allowed is our fallible assessment of the probabilities." (Poul Anderson, "The People of the Wind", 1973)

"The fact is that thresholds exist throughout reality, and that things on their far sides are altogether different from things on their hither sides." (Poul Anderson, "The Saturn Game", 1981)

"Absolute proof of absolute knowledge is impossible." (Poul Anderson, "The Boat of a Million Years" 1989)

"I don’t pretend to understand what the physicists mean by time, but for people, it isn’t so-and-so many measured units; it’s events, experiences. A man who crowds his life and dies young has lived longer than one who got old sitting in tame sameness." (Poul Anderson, "The Boat of a Million Years" 1989)

"Before the spirit could seek into it, the mind must. She studied the tensor equations as once she studied the sutras, she meditated upon the koans of science, and at last she began to feel her oneness with all that was, and in the vision find peace." (Poul Anderson, "The Boat of a Million Years" 1989)

🅵Pierre de Fermat - Collected Quotes

"And this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long." (Pierre de Fermat, [letter] 1640) 

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." (Pierre de Fermat, [Letter to Frénicle] 1657)

"And perhaps, posterity will thank me for having shown that the ancients did not know everything." (Pierre de Fermat, 'Relation of New Discoveries in the Science of Numbers', in [Letter to Pierre de Carcavi] Aug 1659) 

"The result of my work has been the most extraordinary, the most unforeseen, and the happiest, that ever was; for, after having performed all the equations, multiplications, antitheses, and other operations of my method, and having finally finished the problem, I have found that my principle gives exactly and precisely the same proportion for the refractions which Monsieur Descartes has established." (Pierre de Fermat, 1662)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"A prime number, which exceeds a multiple of four by unity, is only once the hypotenuse of a right triangle." (Pierre de Fermat)

"I have discovered a truly remarkable proof of this theorem which this margin is too small to contain." (Pierre de Fermat) [Note written on the margins of his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus] 

"[...] light travels between two given points along the path of shortest time [...]" (Pierre de Fermat)

"The essential quality of a proof is to compel belief." (Pierre de Fermat)

"We found a beautiful and most general proposition, namely, that every integer is either a square, or the sum of two, three or at most four squares. This theorem depends on some of the most recondite mysteries of numbers, and it is not possible to present its proof on the margin of this page." (Pierre de Fermat)

🅲Péter Csermely - Collected Quotes

"A mathematical fractal is generated by an infinitely recursive process, in which the final level of detail is never reached, and never can be reached by increasing the scale at which observations are made. In reality, fractals are generated by finite processes, and exhibit no visible change in detail after a certain resolution limit. This behavior of natural fractal objects is similar to the exponential cutoff, which can be observed in many degree distributions of real networks." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"As opposed to engineered systems, evolutionary networks are integrated and their parts cannot be optimized separately." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Creative elements add random elements to network behavior, inducing an increase in noise. This is highly beneficial to a certain extent as we saw in the previous box, but becomes intolerable if it exceeds a certain threshold. This threshold is high if the hosting network lives an individual life and often meets unexpected situations. However, the same threshold becomes low if the hosting network is part of a higher level organization which provides a stable environment." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Democracy is a highly robust, but at the same time very fragile, complex system, which requires constant change to maintain its self-organized state. Democracy is by definition not in equilibrium. The random graph pattern may pose a system of boring and very low-complexity equilibrium. If all of us ever find ourselves with plenty of resources, that will be THE END of our history." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Fractals are self-similar objects. However, not every self-similar object is a fractal, with a scale-free form distribution. If we put identical cubes on top of each other, we get a self-similar object. However, this object will not have scale-free statistics: since it has only one measure of rectangular forms, it is single-scaled. We need a growing number of smaller and smaller self-similar objects to satisfy the scale-free distribution." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"If a network has violently changing properties, it is most probably not very stable. How can we measure stability, if a network remains unchanged? The assessment of stability often requires a test, and this test comes in the form of a perturbation to the network. A stable network should try to restore its original status after a perturbation. However, this is not easy. Most networks are open systems and therefore undergo a continuous series of perturbations." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Network stability may be a key element in the development of multilevel, nested networks. The formation of nested networks obviously requires at least a few contacts between the bottom networks. However, evolutionary selection requires the independence and at least temporary isolation of the bottom networks themselves. Weak links are probably the only tools for solving this apparent paradox." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Noise is bad for the network, if high and continuous noise levels disturb all network functions. So far, the take-home message is that we have to stop noise in order to survive. This assumption is wrong. Reducing the noise to zero would mean no interaction of the network with the environment. Isolation is clearly a bad strategy, since such an isolated network will die. However, zero noise is bad for another reason too. Noise can be helpful in many ways. The first documented observations of good noise were sailors’ reports on the peculiar phenomenon that  disordered raindrops falling on the ocean can calm roughseas. Another example of the optimal level of noise is opinion formation. A low noise is not enough for modulation of opinion formation, while strong fluctuations prevent the formation of a definitive collective opinion." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Perturbations are often regarded as noise. What is the difference? Noise is usually understood from the point of the experimenter. If we measure it from the outside, noise is the fluctuation of the value we measure. However, from the point of view of the network, noise is a series ofperturbations changing its original status. Network perturbations can be called either signals or noise." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Practically every complex system can be imagined as a network. Atoms form a network making macromolecules. Proteins form a network making cells. Cells form a network making organs and bodies. We form a network making our societies, and so on. Most of these networks are a result of self-organization. In fact, self-organization seems to be an inherent property of matter in our Universe. The resulting networks have a lot of common features, from their topology to their dynamism." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Scale-free degree distribution is not always the optimal solution to the requirement of cost efficiency. As mentioned before, in smallworld networks, building and maintaining links between network elements requires energy. Therefore, network topology is a result of an optimization process. It is optimized with respect to the available resources to ensure optimal communication between different network parts. If the network enjoys unlimited resources, it will have a random distribution with plenty of links. Scale-free degree distribution occurs as a result of optimization in systems with finite resources. If the network experiences even more limited resources, the degree distribution will be steeper than in the case of a scale-free network, and a transition will therefore occur towards a star network. These phenomena are called topological phase transitions [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Scale-free topologies enable more sensitive responses to various changes than those allowed by random networks (Bar-Yam and Epstein, 2004). This can be a very important property for explaining why scale-free networks have been selected and maintained in many systems." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Self-organizing networks suffer various types of random damage. Therefore, if the network remained static, it would soon become dysfunctional. Some networks have developed highly specific screening systems which recognize and repair random damage. On the one hand, this process requires energy, which arrives in the form of perturbations or noise. On the other hand, noise-triggered network restructuring will repeat a few steps of the original self-organization and therefore constitutes a much cheaper way of providing a continuous repair function, with the additional advantage that it is always adaptive with respect to the actual environment of the network." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"The scale-free distribution pattern has been most studied on the degree distribution of networks. What is a degree? The degree of a network. element is the number of connections it has. A scale-free degree distribution means that the network has a large number of elements with very few neighbors, but it has a non-zero number of elements with an extraordinarily large number of neighbors. These connection-rich elements are called hubs. If an element has just a few connections, it is often called a node." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"The scale-free system is a ‘borderline’ case between the random graph and the star phase. It is very fragile and transient, but in spite of this, it is very robust. We call it democracy. This democracy net always keeps a delicate balance betweenanarchy (random net) and dictatorship (star net). Fortunately, in democratic systems, the society is not segmented and weak links flourish. Consequently, the fragile system becomes robust. Democratic systems show the greatest complexity of all. However, weak links and their buffering may grow too great. A democratic society remains flexible for smaller challenges, but occasionally may become overcomplicated and unable to make a fast response to a life-threatening danger." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"To understand, how noise is related to scale-freeness, we have to do some mathematics again. Noise is usually characterized by a mathematical trick. The seemingly random fluctuation of the signal is regarded as a sum of sinusoidal waves. The components of the million waves giving the final noise structure are characterized by their frequency. To describe noise, we plot the contribution (called spectral density) of the various waves we use to model the noise as a function of their frequency. This transformation is called a Fourier transformation [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Weak links are both elusive and overwhelming. Science has grown used, to examining strong links. Strong links are always there. Strong links are reproducible. Strong links are few in number and hence comprehensible. Strong links are already known. Strong links are scientific. Strong links are exciting. In short, strong links are like friends to us. In contrast, weak links are transient. Weak links are undetectable. Weak links are overwhelmingly numerous. Weak links are unknown. Weak links are unscientific. Weak links are hopeless. In short, weak links are like foes to us."(Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Weak links are links between network elements, which connect them with a low intensity. Weak links may also connect network elements with a higher intensity, but in this case they are only transient. [...] A link is defined as weak when its addition or removal does not change the mean value of a target measure in a statistically discernible way." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"[...] weak links seem to be necessary for the development of small-worldness, are a consequence of scale-freeness, and make a key contribution to the formation of nestedness. Weak links are also general and important elements of networks, forming most of their contacts. When we talk about the reason why we like networks, we have to talk about weak-linkedness." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

🅵Daniel Fleisch - Collected Quotes

"Among the differences that will always be with you are the small overshoots and oscillations just before and after the vertical jumps in the square waves. This is called 'Gibbs ripple' and it will cause an overshoot of about 9% at the discontinuities of the square wave no matter how many terms of the series you add. But [...] adding more terms increases the frequency of the Gibbs ripple and reduces its horizontal extent in the vicinity of the jumps." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"An understanding of complex numbers can make the study of waves consid erably less mysterious, and you probably already have an idea that complex numbers have real and imaginary parts. Unfortunately, the term 'imaginary' often leads to confusion about the nature and usefulness of complex numbers." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"As the mechanical wave source moves through the medium, it pushes on a nearby segment of the material, and that segment moves away from the source and is compressed (that is, the same amount of mass is squeezed into a smaller volume, so the density of the segment increases). That segment of increased density exerts pressure on adjacent segments, and in this way a pulse (if the source gives a single push) or a harmonic wave (if the source oscillates back and forth) is generated by the source and propagates through the material." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"Before considering the wave equation for mechanical waves, you should understand the difference between the motion of individual particles and the motion of the wave itself. Although the medium is disturbed as a wave goes by, which means that the particles of the medium are displaced from their equilibrium positions, those particles don’t travel very far from their undisturbed positions. The particles oscillate about their equilibrium positions, but the wave does not carry the particles along – a wave is not like a steady breeze or an ocean current which transports material in bulk from one location to another. For mechanical waves, the net displacement of material produced by the wave over one cycle, or over one million cycles, is zero. So, if the particles aren’t being carried along with the wave, what actually moves at the speed of the wave? […] the answer is energy." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"But the presence of √−1 (the rotation operator between the two perpendicular numbe rlines in the complex plane) in the exponent causes the expression e^ix to move from the real to the imaginary number line. As it does so, its real and imaginary parts oscillate in a sinusoidal fashion […] So the real and imaginary parts of the expression e^ix oscillate in exactly the same way as the real and imaginary components of the rotating phasor […]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"So a very useful way to think about i (√−1) is as an operator that produces a 90◦ rotation of any vector to which it is applied. Thus the two perpendicular number lines form the basis of what we know today as the complex plane. Unfortunately, since multiplication by √−1 is needed to get from the horizontal to the vertical number line, the numbers along the vertical number line are called 'imaginary'. We say 'unfortunately' because these numbers are every bit as real as the numbers along the horizontal number line. But the terminology is pervasive, so when you first learned about complex numbers, you probably learned that they consist of a 'real' and an 'imaginary' part." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"That’s where boundary conditions come in. A boundary condition 'ties down' a function or its derivative to a specified value at a specified location in space or time. By constraining the solution of a differential equation top satisfy the boundary condition(s), you may be able to determine the value of the function or its derivatives at other locations. We say 'may' because boundary conditions that are not well-posed may provide insufficient or contradictory information." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"The 'disturbance' of such waves involves three things: the longitudinal displacement of material, changes in the density of the material, and variation of the pressure within the material. So pressure waves could also be called 'density waves' or even 'longitudinal displacement waves', and when you see graphs of the wave disturbance in physics and engineering textbooks, you should make sure you understand which of these quantities is being plotted as the 'displacement' of the wave." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"The easiest way to think about the shape of a wave is to imagine taking a snapshot of the wave at some instant of time. To keep the notation simple, you can call the time at which the snapshot is taken t = 0; snapshots taken later will be timed relative to this first one. At the time of that first snapshot […] can be written as y = f(x, 0) […] Many waves maintain the same shape over time – the wave moves in the direction of propagation, but all peaks and troughs move in unison, so the shape does not change as the wave moves." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"This equation is considered by some mathematicians and physicists to be the most important equation ever devised. In Euler’s relation, both sides of the equation are expressions for a complex number on the unit circle. The left side emphasizes the magnitude (the 1 multiplying e^iθ ) and direction in the complex plane (θ), while the right side emphasizes the real (cos θ) and imaginary (sin θ) components. Another approach to demonstrating the equivalence of the two sides of Euler’s relation is to write out the power-series representation of each side; [...]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"Why are boundary conditions important in wave theory? One reason is this: Differential equations, by their very nature, tell you about the change in a function (or, if the equation involves second derivatives, about the change in the change of the function). Knowing how a function changes is very useful, and may be all you need in certain problems. But in many problems you wish to know not only how the function changes, but also what value the function takes on at certain locations or times." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

🅳George B Dyson - Collected Quotes

"An Internet search engine is a finite-state, deterministic machine, except at those junctures where people, individually and collectively, make a nondeterministic choice as to which results are selected as meaningful and given a click. These clicks are then immediately incorporated into the state of the deterministic machine, which grows ever so incrementally more knowledgeable with every click. This is what Turing defined as an oracle machine."  (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"If life, by some chance, happens to have originated, and survived, elsewhere in the universe, it will have had time to explore an unfathomable diversity of forms. Those best able to survive the passage of time, adapt to changing environments, and migrate across interstellar distances will become the most widespread. A life form that assumes digital representation, for all or part of its life cycle, will be able to travel at the speed of light." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"In our universe, we measure time with clocks, and computers have a 'clock speed', but the clocks that govern the digital universe are very different from the clocks that govern ours. In the digital universe, clocks exist to synchronize the translation between bits that are stored in memory (as structures in space) and bits that are communicated by code (as sequences in time). They are clocks more in the sense of regulating escapement than in the sense of measuring time."(George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"It is characteristic of objects of low complexity that it is easier to talk about the object than produce it and easier to predict its properties than to build it." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Sixty-some years ago, biochemical organisms began to assemble digital computers. Now digital computers are beginning to assemble biochemical organisms. Viewed from a distance, this looks like part of a life cycle. But which part? Are biochemical organisms the larval phase of digital computers? Or are digital computers the larval phase of biochemical organisms?" (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Life evolved, so far, by making use of the viral cloud as a source of backup copies and a way to rapidly exchange genetic code. Life may be better adapted to the digital universe than we think." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity [...] (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Monte Carlo is able to discover practical solutions to otherwise intractable problems because the most efficient search of an unmapped territory takes the form of a random walk. Today’s search engines, long descended from their ENIAC-era ancestors, still bear the imprint of their Monte Carlo origins: random search paths being accounted for, statistically, to accumulate increasingly accurate results. The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Music allows us to assemble temporal sequences into mental scaffolding that transcends the thinness of time in which we live." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"One of the facets of extreme originality is not to regard as obvious the things that lesser minds call obvious," (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Over long distances, it is expensive to transport structures, and inexpensive to transmit sequences. Turing machines, which by definition are structures that can be encoded as sequences, are already propagating themselves, locally, at the speed of light. The notion that one particular computer resides in one particular location at one time is obsolete. (George Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012) 

"Random search can be more efficient than nonrandom search - something that Good and Turing had discovered at Bletchley Park. A random network, whether of neurons, computers, words, or ideas, contains solutions, waiting to be discovered, to problems that need not be explicitly defined." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The brain is a statistical, probabilistic system, with logic and mathematics running as higher-level processes. The computer is a logical, mathematical system, upon which higher-level statistical, probabilistic systems, such as human language and intelligence, could possibly be built." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The good news is that, as Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the noncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance."  (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The fundamental, indivisible unit of information is the bit. The fundamental, indivisible unit of digital computation is the transformation of a bit between its two possible forms of existence: as structure (memory) or as sequence (code). This is what a Turing Machine does when reading a mark (or the absence of a mark) on a square of tape, changing its state of mind accordingly, and making (or erasing) a mark somewhere else." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The paradox of artificial intelligence is that any system simple enough to be understandable is not complicated enough to behave intelligently, and any system complicated enough to behave intelligently is not simple enough to understand. The path to artificial intelligence, suggested Turing, is to construct a machine with the curiosity of a child, and let intelligence evolve." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The question of whether something is feasible in a type belongs to a higher logical type. It is characteristic of objects of low complexity that it is easier to talk about the object than produce it and easier to predict its properties than to build it. But in the complicated parts of formal logic it is always one order of magnitude harder to tell what an object can do than to produce the object." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"There are two kinds of creation myths: those where life arises out of the mud, and those where life falls from the sky. In this creation myth, computers arose from the mud, and code fell from the sky." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Where does meaning come in? If everything is assigned a number, does this diminish the meaning in the world? What Gödel (and Turing) proved is that formal systems will, sooner or later, produce meaningful statements whose truth can be proved only outside the system itself. This limitation does not confine us to a world with any less meaning. It proves, on the contrary, that we live in a world where higher meaning exists." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Nature uses digital computing for generation-to-generation information storage, combinatorics, and error correction but relies on analog computing for real-time intelligence and control." (George B Dyson, Analogia: The Emergence of Technology Beyond Programmable Control", 2020)