On Literature: On Analysis (From Fiction to Science-Fiction)

"The simplicity of nature is not that which may easily be read, but is inexhaustible. The last analysis can no wise be made." (Ralph W Emerson, "Essays, First Series", 1841)

"It requires a very unusual mind to undertake the analysis of the obvious." (Alfred N Whitehead,"Science in the Modern World", 1925)

"The double analysis kills the single analysis, and the treble kills the double, until at last a sufficiency of statistics comes very near to common sense." (Hilaire Belloc," The Silence of the Sea", 1940) 

"The machine is only a tool after all, which can help humanity progress faster by taking some of the burdens of calculations and interpretations off its back. The task of the human brain remains what it has always been; that of discovering new data to be analyzed, and of devising new concepts to be tested." (Isaac Asimov, "I, Robot", 1950)

"The evil are always foolish in the final analysis." (Gene Wolfe, "The Island of Doctor Death and Other Stories", 1970)

"A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects." (Robert A Heinlein, "Time Enough for Love", 1973)

"So together they left the office and walked into the uncertainty of the rest of their lives. That, in the final analysis, is the great adventure in which each of us takes part; what more courageous thing is there, after all, than facing the unknown we all share, the danger and joy that awaits us in the unread pages of the Book of the Future [...]" (George Alec Effinger," The World of Pez Pavilion: Preliminary to the Groundbreaking Ceremony", 1983)

"The complexities of cause and effect defy analysis." (Douglas Adams, "Dirk Gently's Holistic Detective Agency", 1987)

"Computers bootstrap their own offspring, grow so wise and incomprehensible that their communiqués assume the hallmarks of dementia: unfocused and irrelevant to the barely-intelligent creatures left behind. And when your surpassing creations find the answers you asked for, you can't understand their analysis and you can't verify their answers. You have to take their word on faith." (Peter Watts, "Blindsight", 2006)

On Literature: On Imagination (From Fiction to Science-Fiction)

"The limit of man's knowledge in any subject possesses a high interest which is perhaps increased by its close neighbourhood to the realms of imagination." (Charles Darwin, "Journal of Researches Into the Geology and Natural History of the Various Countries Visited by H.M.S. Beagle: Under the Command of Captain FitzRoy, R. N., from 1832-6", 1836) 

"There is a fifth dimension beyond that which is known to Man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ... The Twilight Zone." (Rod Serling, "The Twilight Zone" [TV series] 1959)

"You’re traveling through another dimension, a dimension not only of sight and sound but of mind; a journey into a wondrous land whose boundaries are that of imagination. That’s the signpost up ahead - your next stop, The Twilight Zone." (Rod Serling, "The Twilight Zone", [opening narration] 1961)

"That perfected machines may one day succeed us is, I remember, an extremely commonplace notion on Earth. It prevails not only among poets and romantics but in all classes of society. Perhaps it is because it is so widespread, born spontaneously in popular imagination, that it irritates scientific minds. Perhaps it is also for this very reason that it contains a germ of truth. Only a germ: Machines will always be machines; the most perfected robot, always a robot." (Pierre Boulle, "Planet of the Apes", 1963)

"You unlock this door with the key of imagination. Beyond it is another dimension. A dimension of sound. A dimension of sight. A dimension of mind. You’re moving into a land of both shadow and substance, of things and ideas. You’ve just crossed over into The Twilight Zone." (Rod Serling, "The Twilight Zone", [opening narration] 1963)

"We cannot predict the new forces, powers, and discoveries that will be disclosed to us when we reach the other planets and set up new laboratories in space. They are as much beyond our vision today as fire or electricity would be beyond the imagination of a fish." (Arthur C Clarke, "Space and the Spirit of Man", 1965)

"The simple truth is that interstellar distances will not fit into the human imagination." (Douglas N Adams, "The Hitchhiker's Guide to the Galaxy", 1979)

"The dreams of people are in the machines, a planet network of active imaginations hooked into their made-up, make-believe worlds. Artificial reality is taking over; it has its own children." (Storm Constantine, "Immaculate", 1991)

On Literature: On Knowledge (From Fiction to Science-Fiction)

"The limit of man's knowledge in any subject possesses a high interest which is perhaps increased by its close neighbourhood to the realms of imagination." (Charles Darwin, "Journal of Researches Into the Geology and Natural History of the Various Countries Visited by H.M.S. Beagle: Under the Command of Captain FitzRoy, R. N., from 1832-6", 1836

"My desire for knowledge is intermittent; but my desire to bathe my head in atmospheres unknown to my feet is perennial and constant. The highest that we can attain to is not Knowledge, but Sympathy with Intelligence. I do not know that this higher knowledge amounts to anything more definite than a novel and grand surprise on a sudden revelation of the insufficiency of all that we called Knowledge before - a discovery that there are more things in heaven and earth than are dreamed of in our philosophy. It is the lighting up of the mist by the sun." (Henry D Thoreau, "Walking", 1851)

"If the fresh facts which come to our knowledge all fit themselves into the scheme, then our hypothesis may gradually become a solution." (Arthur C Doyle, "The Adventure of Wisteria Lodge", 1908)

"Knowledge is the distilled essence of our intuitions, corroborated by experience." (Elbert Hubbard, "A Thousand & One Epigrams, 1911)

"As the traveller who has been once from home is wiser than he who has never left his own door step, so a knowledge of one other culture should sharpen our ability to scrutinise more steadily, to appreciate more lovingly, our own." (Margaret Mead, "Coming of Age in Samoa", 1928)

"A civilization is a heritage of beliefs, customs, and knowledge slowly accumulated in the course of centuries, elements difficult at times to justify by logic, but justifying themselves as paths when they lead somewhere, since they open up for man his inner distance." (Antoine de Saint-Exupéry, "Flight to Arras", 1942)

"Can any of us fix anything? No. None of us can do that. We're specialized. Each one of us has his own line, his own work. I understand my work, you understand yours. The tendency in evolution is toward greater and greater specialization. Man's society is an ecology that forces adaptation to it. Continued complexity makes it impossible for us to know anything outside our own personal field - I can't follow the work of the man sitting at the next desk over from me. Too much knowledge has piled up in each field. And there are too many fields." (Philip K. Dick, The Variable Man", 1952)

"There is a fifth dimension beyond that which is known to Man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ... The Twilight Zone." (Rod Serling, "The Twilight Zone" [TV series] 1959)

"The meeting between ignorance and knowledge, between brutality and culture - it begins in the dignity with which we treat our dead." (Frank Herbert, "Dune", 1965)

"These dwarfs amass knowledge as others do treasure; for this reason they are called Hoarders of the Absolute. Their wisdom lies in the fact that they collect knowledge but never use it." (Stanislaw Lem, "How Erg the Self-Inducing Slew a Paleface", 1965)

"But you must not change one thing, one pebble, one grain of sand, until you know what good and evil will follow on that act. The world is in balance, in Equilibrium. […] It is dangerous, that power. [...] It must follow knowledge, and serve need." (Ursula K Le Guin, "A Wizard of Earthsea", 1968)

"Every judgment teeters on the brink of error. To claim absolute knowledge is to become monstrous. Knowledge is an unending adventure at the edge of uncertainty." (Frank Herbert, "Children of Dune", 1976)

"Their minds sang with the ecstatic knowledge that either what they were doing was completely and utterly and totally impossible or that physics had a lot of catching up to do." (Douglas Adams, "So Long, and Thanks for All the Fish", 1985)

"What is all your studying worth, all your learning, all your knowledge, if it doesn't lead to wisdom? And what's wisdom but knowing what is right, and what is the right thing to do?" (Iain Banks, "Use of Weapons", 1990)

"All knowledge is local, all truth is partial. [...] No truth can make another truth untrue. All knowledge is part of the whole knowledge. A true line, a true color. Once you have seen the larger pattern, you cannot get back to seeing the part as the whole." (Ursula K Le Guin, "A Man of the People", 1995)

On Literature: On Waves (From Fiction to Science-Fiction)

"A single thought is that which it is from other thoughts as a wave of the sea takes its form and shape from the waves which precede and follow it." (Samuel T Coleridge, "Letters", 1836)

"The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

"How beautifully simple is Wessel’s idea. Multiplying by √-1 is, geometrically, simply a rotation by 90 degrees in the counter clockwise sense [...] Because of this property √-1 is often said to be the rotation operator, in addition to being an imaginary number. As one historian of mathematics has observed, the elegance and sheer wonderful simplicity of this interpretation suggests 'that there is no occasion for anyone to muddle himself into a state of mystic wonderment over the grossly misnamed ‘imaginaries'. This is not to say, however, that this geometric interpretation wasn’t a huge leap forward in human understanding. Indeed, it is only the start of a tidal wave of elegant calculations." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)

"God created two acts of folly. First, He created the Universe in a Big Bang. Second, He was negligent enough to leave behind evidence for this act, in the form of the microwave radiation." (Paul Erdős)

"Reality is a wave function traveling both backward and forward in time." (John L Casti)

On Literature: On God (From Fiction to Science-Fiction)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"Sometimes the gods themselves forget the answers to their own riddles." (Edwin L. Arnold, "Lieut. Gullivar Jones: His Vacation", 1905)

"People believe in God because they’ve been conditioned to believe in God." (Aldous Huxley, "Brave New World", 1932)

"All perfection comes from within, and the perfection that is imposed from without is as frivolous and stupid as the trimmings on gingercake. The free man may be bad, but only the free man can be good. And all the kingdom and the power and the glory—call it of God, call it of Cosmos - must arise from the free will of man." (Anthony Boucher, "The Barrier", 1942)

"The gods do not speak the language of men, any more than men can speak the language of the gods." (Miriam Allen deFord, "The Apotheosis of Ki", 1956)

"Only the man who has had to face despair is really convinced that he needs mercy. Those who do not want mercy never seek it. It is better to find God on the threshold of despair than to risk our lives in a complacency that has never felt the need of forgiveness. A life that is without problems may literally be more hopeless than one that always verges on despair." (Thomas Merton, "No Man Is an Island", 1955)

"Man does not create gods, in spite of appearances. The times, the age, impose them on him." (Stanislaw Lem, "Solaris", 1961)

"If there are any gods whose chief concern is man, they cannot be very important gods." (Arthur C Clarke, "Rocket to the Renaissance", [revised version] 1962)

"To be an atheist is to maintain God." (Ursula K Le Guin, "The Left Hand of Darkness", 1969)

"I have seen God creating the cosmos, watching its growth, and finally destroying it." (Olaf Stapledon, "Nebula Maker", 1976)

"Man and the true God are identical—as the Logos and the true God are - but a lunatic blind creator and his screwed-up world separate man from God. That the blind creator sincerely imagines that he is the true God only reveals the extent of his occlusion." (Philip K Dick, "Valis", 1981)

"All the universe is just a dream in God's mind, and as long as he's asleep, he believes in it, and things stay real." (Orson Scott Card, "The Tales of Alvin Maker: Seventh Son", 1987)

"People needed to believe in gods, if only because it was so hard to believe in people." (Terry Pratchett, "Pyramids", 1989)

"God created two acts of folly. First, He created the Universe in a Big Bang. Second, He was negligent enough to leave behind evidence for this act, in the form of the microwave radiation." (Paul Erdős)

On Algebra: On Lie Algebra

"During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. [...] The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments." (Henri P Cartan & Samuel Eilenberg, "Homological Algebra", 1956)

"Algebraic deformation represents a method for the quantization of Lie groups and Lie algebras. Quantum groups are not groups, but Hopf algebras. [3]

"Lie groups describe finite symmetries or symmetries which smoothly depend on a finite number of real parameters. Lie algebras are the linearization of Lie groups at the unit element. The passage from Lie groups to Lie algebras simplifies considerably the approach. Lie algebras are frequently called infinitesimal symmetries." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Representations of symmetries with the aid of linear operators (e.g., matrices) play a crucial role in modern physics. In particular, this concerns the linear representations of groups, Lie algebras, and quantum groups (Hopf algebras)" (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Solvable Lie algebras are close to both upper triangular matrices and commutative Lie algebras. In contrast to this, semisimple Lie algebras are as far as possible from being commutative. By Levi’s decomposition theorem, any Lie algebra is built out of a solvable and a semisimple one. The nontrivial prototype of a solvable Lie algebra is the Heisenberg algebra." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"The fundamental Levi decomposition of a Lie algebra is the prototype of a semidirect product of Lie algebras." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"There is a fundamental relationship between Lie groupoids and Lie algebroids which is similar to the relationship between Lie groups and Lie algebras. There are, however, significant differences in the conventions involved, and it important to be aware of these." (Mike Crampin & David Saunders, "Cartan Geometries and their Symmetries: A Lie Algebroid Approach", 2016)

On Edward Beltrami - Historical Perspectives

"Early mathematicians had no difficulty accepting the first four of Euclid’s axioms. But the fifth was thought not to be so obvious as the other four, and mathematicians tried into the 19th century to derive the fifth postulate from the first four. All such attempts were doomed to failure, but in making the effort, investigators developed a great deal of modern mathematics. It was finally discovered (in the 19th century by Bolyai, Gauss, Lobachevsky, Beltrami, and others) that the fifth postulate is in fact independent of the first four and that one gets a perfectly legitimate geometry by discarding Euclid’s parallel axiom and replacing it with a different one: 'The hyperbolic parallel axiom: Given a line l and a point P not on l, there are infinitely many lines through P that are parallel to l.'" (Bruce Crauder et al, "Functions and Change: A Modeling Approach to College Algebra and Trigonometry", 2008)

"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"Beltrami started this train of thought in 1865, by asking which surfaces can be mapped onto the plane in such a way that their geodesics go to straight lines. He found that the answer is precisely the surfaces of constant Gaussian curvature. For example, great circles on the sphere can be mapped to lines on the plane, and the map that does the trick (to be precise, for the hemisphere) is central projection [...]. Rays from the center O to any great circle form a plane, which of course meets any other plane in a line. Thus projection from O to a plane sends great circles to lines, though only half of the sphere is mapped." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful [...]. What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone. This was noticed by the third great contributor to non-Euclidean geometry after Beltrami and Klein - the French mathematician Henri Poincaré [...]" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"[...] quaternions give a nice approach to symmetric objects in three-dimensional space: the regular polyhedra. But this leads in turn to the regular polytopes, a family of four-dimensional symmetrical objects as remarkable as the regular polyhedra. One then becomes convinced that four-dimensional space is not just a set of quadruples; it is a world of genuine geometry." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

On Literature: On Probability (From Fiction to Science-Fiction)

"That is probable which for the most part usually comes to pass, or which is a part of the ordinary beliefs of mankind, or which contains in itself some resemblance to these qualities, whether such resemblance be true or false." (Marcus T Cicero, "De Inventione", cca. 86–84 BC)

"Take away probability, and you can no longer please the world; give probability, and you can no longer displease it." (Blaise Pascal, "Thoughts", 1670)

"Ignorance gives one a large range of probabilities." (George Eliot, "Daniel Deronda", 1876)

"When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth." (Arthur C Doyle, "The Sign of Four", 1890

"Every probability - and most of our common, working beliefs are probabilities - is provided with buffers at both ends, which break the force of opposite opinions clashing against it […]" (Oliver W Holmes, "The Autocrat of the Breakfast-Table", 1891) 

"It is more than possible; it is probable." (Arthur C Doyle, "The Memoirs of Sherlock Holmes", 1893)

"If everything, everything were known, statistical estimates would be unnecessary. The science of probability gives mathematical expression to our ignorance, not to our wisdom." (Samuel R Delany, "Time Considered as a Helix of Semi-Precious Stones", 1969)

"People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual coincidence is far more unusual than any coincidence could possibly be." (Isaac Asimov, "The Planet That Wasn't", 1976)

"In the real world irrational things happened, impossible coincidences happened, because probability required that coincidences rarely, but not never, occur." (Orson Scott Card, "Ender’s Game", 1985)

"One of the elementary rules of nature is that, in the absence of a law prohibiting an event or phenomenon, it is bound to occur with some degree of probability. To put it simply and crudely: Anything that can happen does happen." (Kenneth W Ford)

On Systems: On Paths

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow recti-linear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"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 pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)

"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 is, however, fair to say that very few applications of swarm intelligence have been developed. One of the main reasons for this relative lack of success resides in the fact that swarm-intelligent systems are hard to 'program', because the paths to problem solving are not predefined but emergent in these systems and result from interactions among individuals and between individuals and their environment as much as from the behaviors of the individuals themselves. Therefore, using a swarm-intelligent system to solve a problem requires a thorough knowledge not only of what individual behaviors must be implemented but also of what interactions are needed to produce such or such global behavior." (Eric Bonabeau et al, "Swarm Intelligence: From Natural to Artificial Systems", 1999)

"Complexity theory shows that great changes can emerge from small actions. Change involves a belief in the possible, even the 'impossible'. Moreover, social innovators don’t follow a linear pathway of change; there are ups and downs, roller-coaster rides along cascades of dynamic interactions, unexpected and unanticipated divergences, tipping points and critical mass momentum shifts. Indeed, things often get worse before they get better as systems change creates resistance to and pushback against the new. Traditional evaluation approaches are not well suited for such turbulence. Traditional evaluation aims to control and predict, to bring order to chaos. Developmental evaluation accepts such turbulence as the way the world of social innovation unfolds in the face of complexity. Developmental evaluation adapts to the realities of complex nonlinear dynamics rather than trying to impose order and certainty on a disorderly and uncertain world." (Michael Q Patton, "Developmental Evaluation", 2010)

"In the 'computation' that is the economy, large and small probabilistic events at particular non-repeatable moments determine the attractors fallen into, the temporal structures that form and die away, the technologies that are brought to life, the economic structures and institutions that result from these, the technologies and structures that in turn build upon these; indeed the future shape of the economy - the future path taken. The economy at all levels and at all times is path dependent. History again becomes important. And time reappears." (W Brian Arthur, "Complexity and the Economy", 2015)

"Feedback systems are closed loop systems, and the inputs are changed on the basis of output. A feedback system has a closed loop structure that brings back the results of the past action to control the future action. In a closed system, the problem is perceived, action is taken and the result influences the further action. Thus, the distinguishing feature of a closed loop system is a feedback path of information, decision and action connecting the output to input." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

On Paths: On Shortest Path

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900)

"Imagine the forehead of a bull, with the protuberances from which the horns and ears start, and with the collars hollowed out between these protuberances; but elongate these horns and ears without limit so that they extend to infinity; then you will have one of the surfaces we wish to study. On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears. Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear. [...] If, therefore, a material point is thrown on the surface studied starting from a geometrically given position with a geometrically given velocity, mathematical deduction can determine the trajectory of this point and tell whether this path goes to infinity or not. But, for the physicist, this deduction is forever useless. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)