On Paradoxes (Unsourced)

"Failures of perspective in decision-making can be due to aspects of the social utility paradox, but more often result from simple mistakes caused by inadequate thought." (Herman Kahn) 

"How wonderful that we have met with a paradox. Now we have some hope of making progress." (Niels H D Bohr)

"It is a paradox in mathematics and physics that we have no good model for the teaching of models." (Hartley Rogers Jr)

"Scientific truth is always paradox, if judged by everyday experience, which catches only the delusive appearance of things." (Karl Marx) 

"There is no sharply drawn line between those contradictions which occur in the daily work of every mathematician, beginner or master of his craft, as a result of more or less easily detected mistakes, and the major paradoxes which provide food for logical thought for decades and sometimes centuries." (Nicholas Bourbaki) 

"We have a paradox in the method of science. The research man may often think and work like an artist, but he has to talk like a bookkeeper in terms of facts, fi gures and logical sequence of thought." (H D Smyth) 

"We no longer pretend to be able to grasp reality in a physical theory; we see in it rather an analytic or geometric mold useful and fertile for a tentative representation of phenomena, no longer believing that the agreement of a theory with experience demonstrates that the theory expresses the reality of things. Such statements have sometimes seemed discouraging; we ought rather to marvel that, with representations of things more or less distant and discolored, the human spirit has been able to fi nd its way through the chaos of so many phenomena and to derive from scientific knowledge the ideas of beauty and harmony. It is no paradox to say that science puts order, at least tentative order, into nature." (Charles E Picard)

📓On Literature: On Paradoxes (From Fiction to Science-Fiction)

"Yet a man may love a paradox, without losing either his wit or his honesty." (Ralph W Emerson, "The Dial, XII", 1841)

"He played with the idea, and grew willful; tossed it into the air and transformed it; let it escape and recaptured it; made it iridescent with fancy, and winged it with paradox."  (Oscar Wilde, "The Picture of Dorian Gray", 1890) 

"The way of paradoxes is the way of truth. To test Reality we must see it on the tight-rope. When the Verities become acrobats we can judge them." (Oscar Wilde, "The Picture of Dorian Gray", 1890)

"Paradox though it may seem - and paradoxes are always dangerous things - it is none the less true that Life imitates art far more than Art imitates life." (Oscar Wilde, "The Decay of Lying", 1891)

"Paradox is the poisonous flower of quietism, the iridescent surface of the rotting mind, the greatest depravity of all." (Thomas Mann,"The Magic Mountain", 1924)

"There is in life an element of elfin coincidence which people reckoning on the prosaic may perpetually miss. As it has been well expressed in the paradox of Poe, wisdom should reason on the unforeseen." (Gilbert K Chesterton, "The Father Brown omnibus", 1951)

"Speculative fantasy, as I prefer to call the more serious fringe of science fiction, is an especially potent method of using one’s imagination to construct a paradoxical universe where dream and reality become fused together, each retaining its own distinctive quality and yet in some way assuming the role of its opposite, and where by an undeniable logic black simultaneously becomes white." (James G Ballard, "Time, Memory, and Inner Space’", 1963)

"Irony is a form of paradox. Paradox is what is good and great at the same time." (K W Friedrich Schlegel, "Aphorism 48", "Dialogue on Poetry and Literary Aphorisms", 1968)

"With sufficient knowledge, any possible so-called paradox can be resolved." (Edward E Smith, "Masters of the Vortex", 1968) 

"Her dance spoke of nothing more and nothing less than the tragedy of being alive, and being human. It spoke, most eloquently, of pain. It spoke, most knowingly, of despair. It spoke of the cruel humor of limitless ambition yoked to limited ability, of eternal hope invested in an ephemeral lifetime, of the driving need to try and create an inexorably predetermined future. It spoke of fear, and of hunger, and, most clearly, of the basic loneliness and alienation of the human animal. It described the universe through the eyes of man: a hostile environment, the embodiment of entropy, into which we are all thrown alone, forbidden by our nature to touch another mind save secondhand, by proxy. It spoke of the blind perversity which forces man to strive hugely for a peace which, once attained, becomes boredom. And it spoke of folly, of the terrible paradox by which man is simultaneously capable of reason and unreason, forever unable to cooperate even with himself." (Spider Robinson & Jeanne Robinson, "Stardance", 1977)

"Technology is both a tool for helping humans and for destroying them. This is the paradox of our times which we're compelled to face." (Frank Herbert, ["Conversations in Port Townsend," interview with Tim O'Reilly] 1983) 

"It is a paradox of Life that all species breed past mere replacement. Any paradise of plenty soon fills to become paradise no more." (David Brin, "Brightness Reef", 1995)

"The old knowledge had been difficult but not distressing. It had been all paradox and myth, and it had made sense. The new knowledge was all fact and reason, and it made no sense." (Ursula K Le Guin," "A Man of the People", 1995)

On Paradoxes (2000-2009)

"One distinguishing feature of contemporary culture in America is that information is consumed for diversion rather than for consequential purposes. Paradoxically, in other words, information of all kinds is processed as low-level cognitive distraction, a form of mass entertainment. The collapsing of news, data, broadcast, and leisure media has resulted in an entertainment culture where the act of consuming information is as diversionary as watching a variety show or a situation comedy. [...] The simple term 'information' may no longer be adequate to describe the spectrum of input available to human beings in their capacities as data processors." (Joseph Urgo, "In the Age of Distraction", 2000)

"That is the paradox of the epidemic: that in order to create one contagious movement, you often have to create many small movements first." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)

"Emergent self-organization in multi-agent systems appears to contradict the second law of thermodynamics. This paradox has been explained in terms of a coupling between the macro level that hosts self-organization (and an apparent reduction in entropy), and the micro level (where random processes greatly increase entropy). Metaphorically, the micro level serves as an entropy 'sink', permitting overall system entropy to increase while sequestering this increase from the interactions where self-organization is desired." (H Van Dyke Parunak & Sven Brueckner, "Entropy and Self-Organization in Multi-Agent Systems", Proceedings of the International Conference on Autonomous Agents, 2001)

"The best reaction to a paradox is to invent a genuinely new and deep idea." (Ian Hacking, "An Introduction to Probability and Inductive Logic", 2001)

"As the least conscious layer of the user experience, the conceptual model has the paradoxical quality of also having the most impact on usability. If an appropriate conceptual model is faithfully represented throughout the interface, after users recognize and internalize the model, they will have a fundamental understanding of what the application does and how to operate it." (Bob Baxley, "Making the Web Work: Designing Effective Web Applications", 2002)

"A sudden change in the evolutive dynamics of a system (a ‘surprise’) can emerge, apparently violating a symmetrical law that was formulated by making a reduction on some (or many) finite sequences of numerical data. This is the crucial point. As we have said on a number of occasions, complexity emerges as a breakdown of symmetry (a system that, by evolving with continuity, suddenly passes from one attractor to another) in laws which, expressed in mathematical form, are symmetrical. Nonetheless, this breakdown happens. It is the surprise, the paradox, a sort of butterfly effect that can highlight small differences between numbers that are very close to one another in the continuum of real numbers; differences that may evade the experimental interpretation of data, but that may increasingly amplify in the system’s dynamics." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems", 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) 

"Paradox is the sharpest scalpel in the satchel of science. Nothing concentrates the mind as effectively, regardless of whether it pits two competing theories against each other, or theory against observation, or a compelling mathematical deduction against ordinary common sense." (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"An apparent paradox is that chaos is deterministic, generated by fixed rules which do not themselves involve any elements of change. We even speak of deterministic chaos. In principle, the future is completely determined by the past; but in practice small uncertainties, much like minute errors of measurement which enter into calculations, are amplified, with the effect that even though the behavior is predictable in the short term, it is unpredictable over the long term." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"Nature does weird things. It lives on the edge. But it is careful to bob and weave from the fatal punch of logical paradox." (Brian Greene, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", 2004)

"Although fiction is not fact, paradoxically we need some fictions, particularly mathematical ideas and highly idealized models, to describe, explain, and predict facts.  This is not because the universe is mathematical, but because our brains invent or use refined and law-abiding fictions, not only for intellectual pleasure but also to construct conceptual models of reality." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"One of the apparent paradoxes in probability is that, while the outcome of the next roll of a die or toss of a coin may be unpredictable, there are nevertheless underlying patterns in the outcomes overall. Specifically, when a fair die is rolled many times, there is a 'settling down' effect as the proportion of each outcome (1, 2, 3, …, 6) gradually approaches 1/6. In the limiting case, as the number of rolls reaches infinity, the shape of the probability distribution becomes uniform." (Alan Graham, "Developing Thinking in Statistics", 2006)

"The existing general descriptions of quantum theory emphasize puzzles and paradoxes in a way that tend to make non-physicists leery of using in any significant away the profound changes in our understanding of both man and nature wrought by the quantum revolution. Yet in the final analysis quantum mechanics is more understandable than classical mechanics because it is more deeply in line with our common sense ideas about our role in nature than the ‘automaton’ notion promulgated by classical physics." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

"Simpson’s Paradox can occur whenever data are aggregated. If data are collapsed across a subclassification (such as grades, race, or age), the overall difference observed may not represent what is going on. Standardization can help correct this, but nothing short of random assignment of individuals to groups will prevent the possibility of yet another subclassificatiion, as yet unidentified, changing things around again. But I believe that knowing of the possibility helps us, so that we can contain the enthusiasm of our impulsive first inferences." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"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)

On Paradoxes (1980-1989)

"But […] we recognize that the wave-particle duality does not arise because of anything paradoxical about the behavior of elementary particles, but simply from the fact that we have asked the wrong question. If we had asked 'How does an elementary particle behave?' instead of asking 'Does it behave like a particle or a wave?', we would have been able to give a perfectly sensible answer. An elementary particle is not a particle in the sense that a bullet is, and it is not a wave like the surf. It exhibits some properties that we normally associate with each of these kinds of things, but it is an entirely new kind of phenomenon." (James Trefil," From Atoms to Quarks: An Introduction to the Strange World of Particle Physics, 1980) 

"A paradox is truth standing on its head to attract attention." (Nicholas Falletta, "The Paradoxicon", 1983) 

"If there are paradoxes in mathematics, think of how many paradoxes there must be in ordinary speech. The existence of paradoxes involves logic. This led naturally to the question of local logics. It seems to me that it is no more natural to expect that a universal logic will hold than that a universal geometry holds." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Organizations are complex and paradoxical phenomena that can be understood in many different ways. Many of our taken-for-granted ideas about organizations are metaphorical, even though we may not recognize them as such. For example, we frequently talk about organizations as if they were machines designed to achieve predetermined goals and objectives, and which should operate smoothly and efficiently. And as a result of this kind of thinking, we often attempt to organize and manage them in a mechanistic way, forcing their human qualities into a background role. By using different metaphors to understand the complex and paradoxical character of organizational life, we are able to manage and design organizations in ways that we may not have thought possible before." (Gareth Morgan, "Images of Organization", 1986)

"Central to the development of the calculus were the concepts of convergence and limit, and with these concepts at hand it became at last possible to resolve the ancient paradoxes of infinity which had so much intrigued Zeno." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

 "The world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)

"A discovery in science, or a new theory, even where it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalyzed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow: it takes a vast world unchallenged and for granted." (James R Oppenheimer, "Atom and Void", 1989)

On Paradoxes (1990-1999)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, 'true on all scales'. Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"There is a seeming paradox: the more the indeterminism, the more the control. This is obvious in the physical sciences. Quantum physics takes for granted that nature is at bottom irreducibly stochastic. Precisely that discovery has immeasurably enhanced our ability to interfere with and alter the course of nature." (Ian Hacking, "The Taming of Chance", 1990)

"It would seem that there may be a possible synthesis that can de-bifurcate bifurcation theory and give a semblance of order to the House of Discontinuity. Reasonable continuity and stability can exist for many processes and structures at certain scales of perception and analysis while at other scales quantum chaotic discontinuity reigns. It may be God or it may be the Law of Large Numbers which accounts for this seemingly paradoxical coexistence." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"Probability is too important to be left to the experts. […] The experts, by their very expert training and practice, often miss the obvious and distort reality seriously. [...] The desire of the experts to publish and gain credit in the eyes of their peers has distorted the development of probability theory from the needs of the average user. The comparatively late rise of the theory of probability shows how hard it is to grasp, and the many paradoxes show clearly that we, as humans, lack a well-grounded intuition in the matter. Neither the intuition of the man in the street, nor the sophisticated results of the experts provides a safe basis for important actions in the world we live in." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Quantum mechanics, like other physical theories, consists of a mathematical part, and an operational part that tells you how a certain piece of physical reality is described by the mathematics. Both the mathematical and the operational aspects of quantum mechanics are straightforward and involve no logical paradoxes. Furthermore, the agreement between theory and experiment is as good as one can hope for. Nevertheless, the new mechanics has given rise to many controversies, which involve its probabilistic aspect, the relation of its operational concepts with those of classical mechanics […]" (David Ruelle, "Chance and Chaos", 1991)

"This remarkable state of affairs [overuse of significance testing] is analogous to engineers’ teaching (and believing) that light consists only of waves while ignoring its particle characteristics - and losing in the process, of course, any motivation to pursue the most interesting puzzles and paradoxes in the field." (Geoffrey R Loftus, "On the tyranny of hypothesis testing in the social sciences", Contemporary Psychology 36, 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)

"Even distinguished philosophers of science [...] recognize the failure of philosophy to help understand the nature of science. They have not discovered a scientific method that provides a formula or prescriptions for how to make discoveries. But many famous scientists have given advice: try many things; do what makes your heart leap; think big; dare to explore where there is no light; challenge expectation; cherchez le paradox; be sloppy so that something unexpected happens, but not so sloppy that you can’t tell what happened; turn it on its head; never try to solve a problem until you can guess the answer; precision encourages the imagination; seek simplicity; seek beauty. [...] One could do no better than to try them all." (Lewis Wolpert, "The Unnatural Nature of Science", 1992)

"Perhaps the greatest strength of graph theory is the abundance of natural and beautiful problems waiting to be solved. […] Paradoxically, much of what is wrong with graph theory is due to this richness of problems. It is all too easy to find new problems based on no theory whatsoever, and to solve the first few cases by straightforward methods. Unfortunately, in some instances the problems are unlikely to lead anywhere […]" (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)

"Arm chair reflections on the concept of causation [are] not going to yield new insights. The grandfather paradox is simply a way of pointing to the fact that if the usual laws of physics are supposed to hold true in a chronology violating spacetime, then consistency constraints emerge. [To understand these constraints] involves solving problems in physics, not armchair philosophical reflections." (John Earman,"Recent Work on Time Travel", 1995)

"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematician should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Not only is it easy to lie with maps, it's essential. To portray meaningful relationships for a complex, three-dimensional world on a flat sheet of paper or a video screen, a map must distort reality. As a scale model, the map must use symbols that almost always are proportionally much bigger or thicker than the features they represent. To avoid hiding critical information in a fog of detail, the map must offer a selective, incomplete view of reality. There's no escape from the cartographic paradox: to present a useful and truthful picture, an accurate map must tell white lies." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Books on paradoxes in statistics are similar to mystery books. They have a faithful readership, and they follow a rigorous sequence in their presentation, like Greek tragedies." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997) 

On Paradoxes (1970-1979)

"On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: 'Mathematics is just a combination of meaningless symbols…'" (Jean Dieudonné, The Works of Nicholas Bourbaki", The American Mathematical Monthly, Vol. 77 (2), 1970) 

"Adaptability is an asset for biological survival, but paradoxically, the greatest threat to the quality of human life is that the human species is so immensely adaptable that it can survive even under the most objectionable conditions." (René J Dubos, "Reason Awake", 1970)

"Paradox is the technique for seizing the conflicting aspects of any problem. Paradox coalesces or telescopes various facets of a complex process in a single instant." (Marshall McLuhan, "Take Today: The Executive as Dropout", 1972)

"[...] mathematics and poetry move together between two extremes of mysticism, the mysticism of the commonplace where ideas illuminate and create facts, and the mysticism of the extraordinary where God, the Infinite, the Real, poses the riddles of desire and disappointment, sin and salvation, effort and failure, question and paradoxical answer [...]" (Scott Buchanan, "Poetry and Mathematics", 1975)

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Whenever the Eastern mystics express their knowledge in words - be it with the help of myths, symbols, poetic images or paradoxical statements-they are well aware of the limitations imposed by language and 'linear' thinking. Modern physics has come to take exactly the same attitude with regard to its verbal models and theories. They, too, are only approximate and necessarily inaccurate. They are the counterparts of the Eastern myths, symbols and poetic images, and it is at this level that I shall draw the parallels. The same idea about matter is conveyed, for example, to the Hindu by the cosmic dance of the god Shiva as to the physicist by certain aspects of quantum field theory. Both the dancing god and the physical theory are creations of the mind: models to describe their authors' intuition of reality." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"The paradox of reality is that no image is as compelling as the one which exists only in the mind's eye." (Shana Alexander, "Talking Woman", 1976)

"It is a bizarre paradox we are facing, for we find that experimental scientists (who are supposed to be fair) at times make the Spanish Inquisition a model of fair hearings and unbiased judgment." (Lorraine L Cudmore,"The Center of Life: A Natural History of the Cell", 1977) 

On Paradoxes (1960-1969)

"It is a well-known statistical paradox that the average age of women over forty is under forty." (Morris J Slonim, "Sampling in a Nutshell", 1960)

"Our conscious appreciation of the fact that one event follows another is of a different kind from our awareness of either event separately. If two events are to be represented as occurring in succession, then - paradoxically - they must also be thought of simultaneously." (Gerald J Whitrow, "The Natural Philosophy of Time", 1961)

"Mathematics then is a formidable and bold bridge between ourselves and the external world. Though it is a purely human creation, the access it has given us to some domains of nature enable us to progress far beyond all expectations. Indeed it is paradoxical that abstractions so remote from reality should achieve so much. Artificial the mathematical account may be, a fairy tale perhaps, but one with a moral." (Morris Kline,"Mathematics: A Cultural Approach", 1962)

"In consequence, we are caught up in a paradox, one which might be called the paradox of conceptualization. The proper concepts are needed to formulate a good theory, but we need a good theory to arrive at the proper concepts." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories." (Philip J Davis, Number", Scientific American, No 211 (3), 1964)

"We are caught up in a paradox, one which might be called the paradox of conceptualization. The proper concepts are needed to formulate a good theory, but we need a good theory to arrive at the proper concepts." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis,"The Mathematics of Matrices", 1965)

"To an important extent, indeed, scientific research has become the secular religion of materialistic society; and it is somewhat paradoxical that a country whose constitution enforces the strict separation of church and state should have contributed so much public money to the establishment and propagation of scientific pessimism." (Harry G Johnson, [in "Basic Research and National Goals: A Report to the Committee on Science and Astronautics Federal Support of Basic Research - Some Economic Issues], 1965) 

"Paradoxes have played a dramatic part in intellectual history, often foreshadowing revolutionary developments in science, mathematics, and logic. Whenever, in any discipline, we discover a problem that cannot be solved within the conceptual framework that supposedly should apply, we experience an intellectual shock. The shock may compel us to discard the old framework and adopt a new one. It is to this process of intellectual molting that we owe the birth of many of the major ideas in mathematics and science." (Anatol Rapoport, "Escape from Paradox", Scientific American Vol. 217 (1), 1967) 

"But, really, mathematics is not religion; it cannot be founded on faith. And what was most important, the methods yielding such remarkable results in the hands of the great masters began to lead to errors and paradoxes when employed by their less talented students. The masters were kept from error by their perfect mathematical intuition, that subconscious feeling that often leads to the right answer more quickly than lengthy logical reasoning. But the students did not possess this intuition […]" (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"I am here not only to evade for a while the clamor and filth and confusion of the cultural apparatus but also to confront, immediately and directly, if it’s possible, the bare bones of existence, the elemental and fundamental, the bedrock which sustains us. I want to be able to look at and into a juniper tree, a piece of quartz, a vulture, a spider, and see it as it is in itself, devoid of all humanly ascribed qualities, anti-Kantian, even the categories of scientific description. To meet God or Medusa face to face, even if it means risking everything human in myself. I dream of a hard and brutal mysticism in which the naked self merges with a nonhuman world and yet somehow survives still intact, individual, separate. Paradox and bedrock." (Edward Abbey, "Desert Solitaire, 1968) 

"[…] mathematicians and philosophers have always been interested in the concept of infinity. This interest arose at the very moment when it became clear that each natural number has a successor, i.e., that the number sequence is infinite. However, even the first attempts to cope with infinity lead to numerous paradoxes." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"There are living systems; there is no living 'matter'. No substance, no single molecule, extracted and isolated from a living being possess, of its own, the aforementioned paradoxical properties. They are present in living systems only; that is to say, nowhere below the level of the cell." (Jacques Monod, "From Biology to Ethics", 1969)

On Paradoxes (1925-1949)

"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it is the truth." (Benjamin Peirce [in William E Byerly, "Benjamin Peirce: II. Reminiscences", The American Mathematical Monthly 32 (1), 1925)

"The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact." (Alfred N Whitehead, "Science and the Modern World" 1925)

"Roughly it amounts to this: mathematical analysis as it works today must make use of irrational numbers (such as the square root of two); the sense if any in which such numbers exist is hazy. Their reputed mathematical existence implies the disputed theories of the infi nite. The paradoxes remain. Without a satisfactory theory of irrational numbers, among other things, Achilles does not catch up with the tortoise, and the earth cannot turn on its axis. But as Galileo remarked, it does. It would seem to follow that something is wrong with our attempts to compass the infinite." (Eric T Bell, "Debunking Science", 1930) 

"Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man." (Bertrand Russell, "The Scientific Outlook", 1931)

"The every-day language reeks with philosophies. […] It shatters at every touch of advancing knowledge. At its heart lies paradox. The language of mathematics, on the contrary, stands and grows in firmness. It gives service to men beyond all other language." (Arthur Bentley, "Linguistic Analysis of Mathematics", 1932)

"Mathematics is an activity governed by the same rules imposed upon the symphonies of Beethoven, the paintings of DaVinci, and the poetry of Homer. Just as scales, as the laws of perspective, as the rules of metre seem to lack fire, the formal rules of mathematics may appear to be without lustre. Yet ultimately, mathematics reaches pinnacles as high as those attained by the imagination in its most daring reconnoiters. And this conceals, perhaps, the ultimate paradox of science. For in their prosaic plodding both logic and mathematics often outstrip their advance guard and show that the world of pure reason is stranger than the world of pure fancy." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes." (James R Newman, "Mathematics and the Imagination", 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"Paradoxical as it may seem, a Latin prose or a geometry problem, even though they are done wrong, may be of a great service one day, provided we devote the right kind of effort to them. Should the occasion arise, they can one day make us better able to give someone in affliction exactly the help required to save him, at the supreme moment of his need." (Simone Weil, "Reflections on the Right Use of School Studies with a View to the Love of God", 1942)

"What a curious picture it is to find man, Homo sapiens, of divine origin, we are told, seriously considering going underground to escape the consequences of his own folly. With a little wisdom and foresight, surely it is not yet necessary to forsake life in the fresh air and in the warmth of the sunlight. What a paradox if our own cleverness in science should force us to live underground with the moles." (James W Fulbright, "The Effect of the Atomic Bomb on American Foreign Policy", Congressional Record Vol. 91, 1945) 

"Every process, event, happening – call it what you will; in a word, everything that is going on in Nature means an increase of the entropy of the part of the world where it is going on. Thus a living organism continually increases its entropy – or, as you may say, produces positive entropy – and thus tends to approach the dangerous state of maximum entropy, which is death. It can only keep aloof from it, i.e. alive, by continually drawing from its environment negative entropy – which is something very positive as we shall immediately see. What an organism feeds upon is negative entropy. Or, to put it less paradoxically, the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive." (Erwin Schrödinger, "What is Life?", 1944)

"[In quantum mechanics] we have the paradoxical situation that observable events obey laws of chance, but that the probability for these events itself spreads according to laws which are in all essential features causal laws." (Max Born, Natural Philosophy of Cause and Chance, 1949)

On Paradoxes (1850-1899)

"Even with the examples of the infinite considered so far it could not escape our notice that not all infinite multitudes are to be regarded as equal to one another in respect of their plurality, but that some of them are greater (or smaller) than others, i.e. another multitude is contained as a part in one multitude (or on the contrary one multitude occurs in another as a mere part). This also is a claim which sounds to many paradoxical." (Bernard Bolzano, "Paradoxes of the Infinite", 1851)

"Disinterestedness is as great a puzzle and paradox as ever. Indeed, strictly speaking, it is a species of irrationality, or insanity, as regards the individual's self; a contradiction of the most essential nature of a sentient being, which is to move to pleasure and from pain." (Alexander Bain, "On the Study of Character, including an estimate of phrenology", 1861) 

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

"The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself."  (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)

"As lightning clears the air of impalpable vapours, so an incisive paradox frees the human intelligence from the lethargic influence of latent and unsuspected assumptions. Paradox is the slayer of Prejudice." (James J Sylvester, "The Collected Mathematical Papers of James Joseph Sylvester", 1875)

"It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors." (Thomas H Huxley, "The Progress of Science", 1887)

"Paradoxes have no place in science. Their removal is the substitution of true for false statements and thoughts." (William T Kelvin, "On Sun’s Heat", 1887) 

"However we define error, the idea of calculating its extent may appear paradoxical. A science of errors seems a contradiction in terms." (Francis Y Edgeworth, "The Element of Chance in Competitive Examinations", Journal of the Royal Statistical Society Vol. 53, 1890) 

"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)

On Paradoxes (1900-1924)

"It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)

"[...] operations of an almost mysterious character, which run counter to ordinary procedure in a more or less paradoxical way. They are methods which give an onlooker the impression of magic if he be not himself initiated or equally skilled in the mechanism.  (Hans Vaihinger, "The Philosophy of 'As if': A System of the Theoretical, Practical and Religious Fictions of Mankind", 1911)

"The very name calculus of probabilities is a paradox. Probability opposed to certainty is what we do not know, and how can we calculate what we do not know?" (Henri Poincaré, "The Foundations of Science", 1913)

"We know not to what are due the accidental errors, and precisely because we do not know, we are aware they obey the law of Gauss. Such is the paradox." (Henri Poincaré, "The Foundations of Science", 1913)

"The scientific paradox is only an exception to some familiar but too inclusive generalization. It, therefore, has both the appeal of the riddle and the charm of surprise - the surprise, the instant the truth is seen, of a sudden and unexpected discovery." (William J Humphries, "A Bundle of Meteorological Paradoxes", Annual Report of the Board of Regents of the Smithsonian Institution, 1919)

"We live in a world full of contradiction and paradox, a fact of which perhaps the most fundamental illustration is this: that the existence of a problem of knowledge depends on the future being different than the past, while the possibility of the solution of the problem depends on the future being like the past." (Frank H Knight, "Risk, Uncertainty, and Profit", 1921)

🦋On Chaos Theory: On Randomness XIX

"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)

🅵Bas C van Fraassen - Collected Quotes

"A map is a graphical representation of geographical or astronomical features, but this may range from a sketch of a subway system, to an interactive, zoomable, or animated map on a computer which constantly changes in front of the eyes."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A map is designed to help one get around in the landscape it depicts. [...]"  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A model often contains much that does not correspond to any observable feature in the domain. Then, from an empiricist point of view, the model’s structure must be taken to reveal structure in the observable phenomena, while the rest of the model must be serving that purpose indirectly. It may be practically as well as theoretically useful to think of the phenomena as embedded in a larger - and largely unobservable - structure."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A scale model represents, and yields information about what it is a model of, by selective resemblance. [...] Scale models can be produced for the sheer aesthetic pleasure of it, but more typically they serve in studies meant to design the very things of which they are meant to be the scaled down versions."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A science presents us with representations of the phenomena through artifacts, both abstract, such as theories and mathematical models, and concrete such as graphs, tables, charts, and ‘table-top’ models. These representations do not form a haphazard compilation though any unity, in the range of representations made available, is visible mainly at the more abstract levels."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"An algebra is a mathematical structure consisting of a set of elements and a collection of operators on those elements - though the term is variously defined in different contexts in mathematics, so as to narrow the meaning (e.g. an algebra is a vector space with a bilinear multiplication operation)."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"[...] construction of a data model is precisely the selective relevant depiction of the phenomena by the user of the theory required for the possibility of representation of the phenomenon."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"For unqualified adequacy of the theory, what is required is that the surface models of phenomena fit properly with or into the theoretical models. The surface models will provide probability functions for events that are classified as outcomes in situations classified as measurements of given observables."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"In maps we have scale models of terrain, but projected onto a plane, thus producing occlusion of a sort not inherent to three-dimensional imaging. Maps do not usually have an obvious perspective; but we see perspectivity when, for example, the curvature of the earth makes marginal distortion inevitable as a result of this projection that lowers the dimensionality. A map too is the product of a measuring procedure, but they bring to light a much more important point about ‘point of view’, essentially independent of these limitations in cartography. The point extends to all varieties of modeling, but is made salient by the sense in which use enters the concept of ‘map’ from the beginning. A map is not only an object used to represent features of a terrain, it is an object for the use of the industrial designer, the navigator, and most of all the traveler, to plan and direct action. This brings us to an aspect of scientific representation not touched on so far, though implicit in the discussion of perspective, crucial to its overall understanding: its indexicality."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The general concept of a map is not so different from that of a model, though the one is extrapolated from a graph with spatial similarity to certain features of a landscape, and the other from a table-top contraption."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

'Model' is a metaphor, whose base is the simply constructed table-top model."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Scale modeling displays the characteristics of picturing, by relying on selective resemblance to achieve its aim, but in a way that is subject to inevitable occlusion or distortion."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Scientific theories represent how things are, doing so mainly by presenting a range of models as candidate representations of the phenomena. [...] A theory provides, in essence, a set of models. The 'in essence' signals much that must be delicately expanded and qualified; [...] These models - the theoretical models - are provided in the first instance to fit observed and observable phenomena. Since the description of these phenomena is in practice already by means of models - the ‘data models’ or ‘surface models’, we can put the requirement as follows: the data or surface models must ideally be isomorphically embeddable in theoretical models."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The activity of representation is successful in that case only if the recipients are able to receive that information through their ‘viewing’ of the representation. [...] In science the original creation of a model may have been a purely theoretical activity, but eventually it provides input for an application, where conditional predictions made on the basis of that model feed into planning and action."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The observable phenomenon makes its appearance to us first of all in the outcome of a specific measurement, or large set of such measurements - or at slight remove, in a data model constructed from these individual outcomes, or at a slightly further remove yet, in the surface model constructed by extrapolating the patterns in the data model to something finer than our instruments can register."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The physical sciences give us representations of nature, and scientific representation is in general three-faceted. From a purely foundational point of view, the theoretical models that depict the ‘underlying reality’ are the main thing. But some elements or substructures of those models are meant to represent the observable phenomena - the empirical substructures."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Theories represent the phenomena just in case their models, in some sense, 'share the same structure' with those phenomena - that, in slogan form, is what is called the semantic view of theories. [...] Embedding, that means displaying an isomorphism to selected parts of those models. Here is the argument to present the first challenge. For a phenomenon to be embeddable in a model, that means that it is isomorphic to a part of that model. So the two, the phenomenon and the relevant model part must have the same structure. Therefore, the phenomenon must have a structure, and this shared structure is obviously not itself a physical, concrete individual - so what is implied here is something of the order of realism about universals."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"[...] when a theoretical model is said to represent certain phenomena, there is indeed reference to a matching, namely between parts of the theoretical models and the relevant data models - both of them abstract entities. Note now, the crucial word in this sentence: the punch comes in the word 'relevant'. There is nothing in an abstract structure itself that can determine that it is the relevant data model, to be matched by the theory. A particular data model is relevant because it was constructed on the basis of results gathered in a certain way, selected by specific criteria of relevance, on certain occasions, in a practical experimental or observational setting, designed for that purpose."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

🅱Bernard Bolzano - Collected Quotes

"At the present time there is still lacking a precise definition of the most important concepts: line, surface, solid." (Bernard Bolzano, 1810

"Every word in language serves to designate an idea and some of them even complete propositions. Therefore, it is only natural to suppose that each idea is composed of at least as many parts as there are words in its expression." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"The most plausible way to [conceive of the relation between grounding and causality] is that somehow those truths that state the existence of the properties of a cause be considered as the ground, and those that concern the existence and the properties of the effect be considered as the consequence." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"] ,1837)

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"Already within the domain of those things which do not have any pretension of reality, but only of possibility, there indisputably are sets that are infinite. The set of propositions and truths in themselves is infinite, as one can easily see." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"But rather they are able, in spite of that relationship between them that is the same for both of them, to have a relationship of inequality in their pluralities, so that one of them can be presented as a whole, of which the other is a part. An equality of these pluralities may only be concluded if some other reason is added, such as that both multitudes have exactly the same determining grounds, e.g. they have exactly the same way of being formed." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Even in the realm of things which do not claim actuality, and do not even claim possibility, there exist beyond dispute sets which are infinite. The set of all ‘absolute propositions and truths' is easily seen to be infinite." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Even with the examples of the infinite considered so far it could not escape our notice that not all infinite multitudes are to be regarded as equal to one another in respect of their plurality, but that some of them are greater (or smaller) than others, i.e. another multitude is contained as a part in one multitude (or on the contrary one multitude occurs in another as a mere part).This also is a claim which sounds to many paradoxical." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"If they [mathematicians] find a quantity greater than any finite number of the assumed units, they call it infinitely great; if they find one so small that its every finite multiple is smaller than the unit, they call it infinitely small; nor do they recognise any other kind of infinitude than these two, together with the quantities derived from them as being infinite to a higher order of greatness or smallness, and thus based after all on the same idea." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"[...] from that circumstance alone we are not allowed to conclude that both sets, if they are infinite, are equal to each other with respect to the multiplicity of their parts (that is, if we abstract from all differences between them); [...] Equality of those multiplicities can only be inferred when some other reason is added, for instance that both sets have absolutely equal grounds of determination, i.e., that their mode of formation is absolutely equal." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Space and time - which again do not belong to the domain of the actual, though they can be determinations of the actual - form a very important category of infinitely great quantities." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"The existence of infinite sets, at least with non-actual members, is something which I now regard as sufficiently proved and defended; as also, that the set of all absolute truths is an infinite set." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"The very word in finite shows that we put the infinite into contrast with the merely finite. Again, the derivation of the former name from the latter betrays the additional fact that we consider the idea of the infinite to arise from the idea of the finite by, and only by, the adjunction of a new element; for such in fact is the abstract idea of negation." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Therefore both multitudes have one and the same plurality, as one can also say, equal plurality. Obviously this conclusion becomes void as soon as the multitude of things in A is an infinite multitude, for now not only do we never reach, by counting, the last thing in A, but rather, by virtue of the definition of an infinite multitude, in itself there is no last thing in A, i.e. however many have already been designated, there are always others to designate." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"By a 'Satz an sich' [senternce in itself] I mean any statement whatever to the effect that something is or is not, irrespective of whether the statement be true or false, irrespective of whether any person ever formulated it in words, and even irrespective of whether it ever entered into any mind as a thought."  (Bernard Bolzano)

"My special pleasure in mathematics rested particularly on its purely speculative part." (Bernard Bolzano)

🦋On Chaos Theory I

"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)

"The term chaos is also used in a general sense to describe the body of chaos theory, the complete sequence of behaviours generated by feed-back rules, the properties of those rules and that behaviour." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"Chaos theory reconciles our intuitive sense of free will with the deterministic laws of nature. However, it has an even deeper philosophical ramification. Not only do we have freedom to control our actions, but also the sensitivity to initial conditions implies that even our smallest act can drastically alter the course of history, for better or for worse. Like the butterfly flapping its wings, the results of our behavior are amplified with each day that passes, eventually producing a completely different world than would have existed in our absence!" (Julien C Sprott, "Strange Attractors: Creating Patterns in Chaos", 2000)

"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds", 2004)

"This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment." (Greg Rae, "Chaos Theory: A Brief Introduction", 2006)

"Yet, with the discovery of the butterfly effect in chaos theory, it is now understood that there is some emergent order over time even in weather occurrence, so that weather prediction is not next to being impossible as was once thought, although the science of meteorology is far from the state of perfection." (Peter Baofu, "The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos", 2007)

"[chaos theory] presents a universe that is at once deterministic and obeys the fundamental physical laws, but is capable of disorder, complexity, and unpredictability. It shows that predictability is a rare phenomenon operating only within the constraints that science has filtered out from the rich diversity of our complex world." (Ziauddin Sardar & Iwona Abrams, "Introducing Chaos: A Graphic Guide", 2008)

"Complexity theory can be defined broadly as the study of how order, structure, pattern, and novelty arise from extremely complicated, apparently chaotic systems and conversely, how complex behavior and structure emerges from simple underlying rules. As such, it includes those other areas of study that are collectively known as chaos theory, and nonlinear dynamical theory." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)

"An ending is an artificial device; we like endings, they are satisfying, convenient, and a point has been made. But time does does not end, and stories march in step with time. Equally, chaos theory does not assume an ending; the ripple effect goes on, and on." (Penelope Lively, "How It All Began", 2011)

"The things that really change the world, according to Chaos theory, are the tiny things. A butterfly flaps its wings in the Amazonian jungle, and subsequently a storm ravages half of Europe." (Neil Gaiman, "Good Omens", 2011)

On Paradoxes (1950-1959)

"There is in life an element of elfin coincidence which people reckoning on the prosaic may perpetually miss. As it has been well expressed in the paradox of Poe, wisdom should reason on the unforeseen." (Gilbert K Chesterton, "The Father Brown omnibus", 1951)

"The sweeping development of mathematics during the last two centuries is due in large part to the introduction of complex numbers; paradoxically, this is based on the seemingly absurd notion that there are numbers whose squares are negative." (Emile Borel, 1952)

"It is true that the field of mathematical activity proper, both in analysis and in geometry, is not directly affected by the antinomies. They appear chiefly in a region of extreme generalization, beyond the domain in which the concepts of these disciplines are actually used. It is in general not difficult to take precautionary measures in order to avoid the dangerous region. This is the main reason why many mathematicians recoiled so quickly from the initial shock caused by the appearance of the antinomies. The very fact that one continued to speak of paradoxes, or antinomies, rather than of contradictions serves as an indication that deep in their heart most modern mathematicians did not want to be expelled from the paradise into which Cantor’s discoveries had led them." (Abraham Fraenkel et al, "Foundations of Set Theory" 2nd Ed., 1953)

"A discovery in science, or a new theory, even when it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalysed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow; it takes a vast world unchallenged and for granted. This is one reason why, however great the novelty or scope of new discovery, we neither can, nor need, rebuild the house of the mind very rapidly. This is one reason why science, for all its revolutions, is conservative. This is why we will have to accept the fact that no one of us really will ever know very much. This is why we shall have to find comfort in the fact that, taken together, we know more and more." (J. Robert Oppenheimer, Science and the Common Understanding, 1954)

"Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications." (Alfred N Whitehead, "An Introduction to Mathematics", 1958)

On Paradoxes (-1849)

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this […] is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another." (Galileo Galilei, "Two New Sciences", 1638)

"Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found." (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632

"It will seem not a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful. (Leonhard Euler, "Specimen de usu observationum in mathesi pura" , Novi Commentarii academiae scientiarum Petropolitanae 6, 1756/57) 

"It is to be desired, that the charges of paradox and mystery, said to be introduced into algebra by negative and impossible quantities, should be proposed distinctly, in a precise form, fit to be apprehended and made the subject of discussion." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"Mathematical science has been at times embarrassed with contradictions and paradoxes; yet they are not to be imputed to imaginary symbols, rather than to any other symbols invented for the purpose of rendering demonstration compendious, and expeditious. It may; however, be justly remarked, that mathematicians, neglecting to exercise mental superintendance, are too prone to trust to mechanical dexterity; and that some, instead of establishing the truth of conclusions on antecedent reasons, have endeavoured to prop it by imperfect analogies or mere algebraic forms. On the other hand, there are mathematicians, whose zeal for just reasoning has been alarmed at a verbal absurdity and, from a name improperly applied, or a definition incautiously given, l have been hurried to the precipitate conclusion, that operations with symbols of which the mind can form no idea, must necessarily be doubtful and unintelligible." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"[...] all knowledge, and especially the weightiest knowledge of the truth, to which only a brief triumph is allotted between the two long periods in which it is condemned as paradoxical or disparaged as trivial." (Arthur Schopenhauer, "The World as Will and Representation", 1819)

"To concentrate on a craft is the best procedure. For the person of inferior gifts it will always remain a craft. The more gifted person will raise it to an art. And as for the man of highest endowment, in doing one thing he does all things; or, to put it less paradoxically, in the one thing that he does properly, he sees a symbol of all things that are done right. " (Johann Wolfgang von Goethe, 1829)

"There is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear." (Augustus De Morgan, "On the Study and Difficulties of Mathematics", 1830)

"Divergent series are in general very mischievous affairs, and it is shameful that any one should have founded a demonstration upon them. You can demonstrate anything you please by employing them, and it is they who have caused so much misfortune, and given birth to so many paradoxes." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)

"One should not think slightly of the paradoxical; for the paradox is the source of the thinker's passion, and the thinker without a paradox is like a lover without feeling: a paltry mediocrity." (Søren Kierkegaard, "Philosophical Fragments: Or, A Fragment of Philosophy" 1844)

"It is the duty of the human understanding to understand that there are things which it cannot understand, and what those things are. Human understanding has vulgarly occupied itself with nothing but understanding, but if it would only take the trouble to understand itself at the same time it would simply have to posit the paradox." (Søren Kierkegaard, [The Journals of Søren Kierkegaard] 1847)

"Take away the paradox from a thinker and you have a professor." (Søren Kierkegaard, 

[journal entry] 1849)

🚀On Invention (2000-2019)

"The passion and beauty and joy of science is that we humans have invented a process to understand the universe in a way that is true for everyone. We are finding universal truths." (Bill Nye, 2000) 

"The best reaction to a paradox is to invent a genuinely new and deep idea. (Ian Hacking, "An Introduction to Probability and Inductive Logic", 2001)

"The brain highlights what it imagines as patterns; it disregards contradictory information. Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it." (Benoît Mandelbrot, "The (Mis)Behavior of Markets", 2004)

"Metaphorizing is a manner of thinking, not a property of thinking. It is a capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects." (Ivan Mladenov, “Conceptualizing Metaphors: On Charles Peirce’s marginalia”, 2006)

"The strength of a theory is not what it allows, but what it prohibits; if you can invent an equally persuasive explanation for any outcome, you have zero knowledge." (Eliezer Yudkowsky, "An Alien God", 2007)

"We didn't invent nature. Nature invented us. Nature bats last, the saying goes, but, even more importantly, it's her playing field. We would be wise to learn the ground rules and play by them." (Kenny Ausubel, [speech at Bioneers Conference, 2003)

"Scientists often invent words to fill the holes in their understanding. These words are meant as conveniences until real understanding can be found. […] Words such as dimension and field and infinity […] are not descriptions of reality, yet we accept them as such because everyone is sure someone else knows what the words mean." (Scott Adams, "God's Debris: A Thought Experiment", 2004)

"The urge to discover, to invent, to know the unknown, seems so deeply human that we cannot imagine our history without it." (Alan Lightman, "The Discoveries: Great Breakthroughs in 20th-Century Science, Including the Original Papers", 2009)

On Abstraction (1940-1949)

"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics." (Eric T Bell, "The Development of Mathematics", 1940)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"This abstracting of common experience is one of the principal sources of the utility of mathematics and the secret of its scientific power. The world that impinges on the senses of all but introverted solipsists is too intricate for any exact description yet imagined by human beings. By abstracting and simplifying the evidence of the senses, mathematics brings the worlds of science and daily life into focus with our myopic comprehension, and makes possible a rational description of our experiences which accords remarkably well with observation." (Eric T Bell, "The Development of Mathematics", 1940)

"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for […] The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"A mathematical point is an idea - an abstraction - something we can think about. It cannot be defined logically. Fortunately, our common experience with physical objects makes it relatively certain that each of us has the same idea in mind when he uses the word point, and so the lack of a logical definition presents no particular difficulty." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"Mathematics being a very abstract science should be presented very concretely." (George Pólya, "How to Solve It", 1945)

"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)

"I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas." (John von Neumann,  "The Mathematician", The Works of the Mind Vol. I (1), 1947)

🅲Benjamin N Cardozo - Collected Quotes

"[…] law, like other branches of social science, must be satisfied to test the validity of its conclusions by the logic of probabilities rather than the logic of certainty." (Benjamin N Cardozo, "The Growth of the Law", 1924)

"Methods, when classified and separated, acquire their true bearing and perspective as a means to an end, not as ends in themselves." (Benjamin N Cardozo, "The Growth of the Law", 1924)

"The theorist has a hard time to make his way in an ungrateful world. He is supposed to be indifferent to realities; yet his life is spent in the exposure of realities, which, till illuminated by his searchlight, were hidden and unknown." (Benjamin N Cardozo, "The Growth of the Law", 1924)

"Where the line is to be drawn the important and the trivial cannot be settled by a formula." (Benjamin N Cardozo, 1921)

"Often a liberal antidote of experience supplies a sovereign cure for a paralyzing abstraction built upon a theory." (Benjamin N Cardozo, "The Paradoxes of Legal Science", 1928)

"Method is much, technique is much, but inspiration is even more." (Benjamin N Cardozo, "The Game of the Law", 1931)

Discovery in Physics (1950-1999)

"All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things." (Max Born, "Physical Reality", Philosophical Quarterly, Vol. 3, No. 11,1953)

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"We know many laws of nature and we hope and expect to discover more. Nobody can foresee the next such law that will be discovered. Nevertheless, there is a structure in laws of nature which we call the laws of invariance. This structure is so far-reaching in some cases that laws of nature were guessed on the basis of the postulate that they fit into the invariance structure." (Eugene P Wigner, "The Role of Invariance Principles in Natural Philosophy", 1963)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"The world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)

"As we explore physics at higher and higher energy, revealing its structure at shorter and shorter distances, we discover more and more symmetry." (David J Gross, "The Role of Symmetry in Fundamental Physics", 1996)