On Impossibility (1800-1849)

"The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"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)

"The application of imaginary quantities to the theory of equations, has perhaps been made more extensively than to any other part of analysis. To consider the propriety of this application on the grounds of perspicuity and conciseness, a long discussion would be necessary. I may, however, be here permited merely to state my opinion, that impossible quantities must be employed in the theory of equations, in order to obtain general rules and compendious methods." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"But by far the greatest obstacle to the progress of science and to the undertaking of new tasks and provinces therein is found in this: that men despair and think things impossible." (Sophie Germain, 1813)

"The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations." (Niels H Abel,"Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree", 1824)

"§ 6. It is impossible for the human mind, itself a finite creation, to regard nature, whether her powers or her productions are considered, in the light of the whole manifestation of an infinite power, but only as parts or fragments of such manifestation. But to comprehend these as one whole, that is, as an eternal and immutable yet ever varying body, or, as innumerable forms of one highest whole, is the end. of all disquisition, the sum of which we call a System." (John Lindley, Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us." (Auguste Comte, "Course of Positive Philosophy", 1830)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan,"On the Study and Difficulties of Mathematics", 1830)

"The nature of mathematical demonstration is totally different from all other, and the difference consists in this - that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us." (Augustus De Morgan, "On the Study and Difficulties of Mathematics", 1830)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." (Carl F Gauss,"Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)

"Our general arithmetic, so far surpassing in extent the geometry of the ancients, is entirely the creation of modern times. Starting originally from the notion of absolute integers, it has gradually enlarged its domain. To integers have been added fractions, to rational quantities the irrational, to positive the negative .and to the real the imaginary. This advance, however, has always been made at first with timorous and hesitating step. The early algebraists called the negative roots of equations false roots, and these are indeed so when the problem to which they relate has been stated in such a form that the character of the quantity sought allows of no opposite. But just as in general arithmetic no one would hesitate to admit fractions, although there are so many countable things where a fraction has no meaning, so we ought not to deny to, negative numbers the rights accorded to positive simply because innumerable things allow no opposite. The reality of negative numbers is sufficiently justified since in innumerable other cases they find an adequate substratum. This has long been admitted, but the imaginary quantities - formerly and occasionally now, though improperly, called impossible-as opposed to real quantities are still rather tolerated than fully naturalized, and appear more like an empty play upon symbols to which a thinkable substratum is denied unhesitatingly by those who would not depreciate the rich contribution which this play upon symbols has made to the treasure of the relations of real quantities." (Carl F Gauss, "Theoria residuorum biquadraticorum, Commentatio secunda", Göttingische gelehrte Anzeigen, 1831)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." (Carl F Gauss, "Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)

"Every mathematical method has its inverse, as truly, and for the same reason, as it is impossible to make a road from one town to another, without at the same time making one from the second to the first. The combinatorial analysis is analysis by means of combinations; the calculus of generating functions is combination by means of analysis." (Augustus de Morgan, "The Differential and Integral Calculus", 1836)

"[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us." (Auguste Comte,"Cours de Philosophie Positive", 1830-1842)

"A definite Integral always presupposes numeric values; consequently equations in which definite integrals occur are seldom or never correct as general (formal) equations, but can only be admitted as numeric equations; consequently the convergence of any infinite series which may occur in them is an indispensable condition, whereas the condition of convergence with respect to a general series in general investigations, such as must be necessarily first established as the foundation of the possibility of any calculation, is quite as absurd [...]" (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)

"Those who can, in common algebra, find a square root of -1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √-1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

On Impossibility (1850-1874)

"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore, the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind." (James C Maxwell, 1850)

"They say that Understanding ought to work by the rules of right reason. These rules are, or ought to he, contained in Logic; but the actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind." (James C Maxwell, "Quetlet on Probabilities", Edinburgh Review 92, 1850)

"It is impossible by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects. [Footnote: ] If this axiom be denied for all temperatures, it would have to be admitted that a self-acting machine might be set to work and produce mechanical effect by cooling the sea or earth, with no limit but the total loss of heat from the earth and sea, or in reality, from the whole material world." (William Thomson, "On the Dynamical Theory of Heat with Numerical Results Deduced from Mr Joule's Equivalent of a Thermal Unit and M. Regnault's Observations on Steam", Transactions of the Royal Society of Edinburgh, 1851)

"The chemists who uphold dualism are far from being agreed among themselves; nevertheless, all of them in maintaining their opinion, rely upon the phenomena of chemical reactions. For a long time the uncertainty of this method has been pointed out: it has been shown repeatedly, that the atoms put into movement during a reaction take at that time a new arrangement, and that it is impossible to deduce the old arrangement from the new one. It is as if, in the middle of a game of chess, after the disarrangement of all the pieces, one of the players should wish, from the inspection of the new place occupied by each piece, to determine that which it originally occupied." (Auguste Laurent, "Chemical Method", 1855)

"The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language." (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)

"[…] in the pursuit of physical science, the imagination should be taught to present the subject investigated in all possible and even impossible views […] (Michael Faraday, "Experimental Researches in Chemistry and Physics", 1859)

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"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)

"When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition." (Gottlob Frege,"Methods of Calculation based on an Extension of the Concept of Quantity", 1874)

"I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them on the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge, - knowledge mingled with ignorance, producing doubt. (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition." (Gottlob Frege, "Methods of Calculation based on an Extension of the Concept of Quantity", 1874)

On Impossibility (1875-1899)

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

"But I thoroughly believe myself, and hope to prove to you, that science is full of beautiful pictures, of real poetry, and of wonder-working fairies; and what is more […] though they themselves will always remain invisible, yet you will see their wonderful power at work everywhere around you. […] There is only one gift we must have before we can learn to know them - we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind, of that which is, though it is invisible to us." (Arabella Buckley, Fairyland, 1879)

"Thought often leads us far beyond the imaginable without thereby depriving us of the basis for our conclusions. Even if, as it appears, thought without mental pictures is impossible for us men, still their connection with the object of thought can be wholly superficial, arbitrary, and conventional." (Gottlob Frege,"The Foundations of Arithmetic", 1884) 

"Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state, that is a perpetual motion machine." (Ludwig E Boltzmann, "The Second Law of Thermodynamics", [Address to a Formal meeting of the Imperial Academy of Science], 1886)

"I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them on the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge, - knowledge mingled with ignorance, producing doubt." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, American Journal of Physics 12, 1890)

"[employment of] exact or mathematical methods […] unfortunately is impossible in most branches of science (particularly in biology), because the empirical foundations are much too imperfect and the present problems much too complicated. Mathematical treatment of these does more harm than good because it gives a deceptive semblance of certainty which is not actually attainable. Part of physiology also involves problems which are difficult or impossible to resolve exactly, and these include the chorology and ecology of plankton." (Ernst Häckel,"Plantonic studies", 1891)

"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction." (Giuseppe Peano, "Notations de Logique Mathématique", 1894)

"The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, .is, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language." (Thomas Hill, North American Review Vol. 85, 1898)

"[…] we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind of that which is, though it is invisible to us." (Arabella B Buckley, "The Fairy-Land of Science", 1899)

📓On Literature: On Impossibility (From Fiction to Science Fiction)

"[Sherlock Holmes:] How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?" (Sir Arthur C Doyle, "Sign of the Four", 1890)

"Except under controlled conditions, or in circumstances where it is possible to ignore individuals and consider only large numbers and the law of averages, any kind of accurate foresight is impossible." (Aldous Huxley, "Time Must Have a Stop", 1944)

"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)

"It is really quite amazing by what margins competent but conservative scientists and engineers can miss the mark, when they start with the preconceived idea that what they are investigating is impossible." (Arthur C Clarke, "Profiles of the Future", 1962)

"It is said that science fiction and fantasy are two different things. Science fiction is the improbable made possible, and fantasy is the impossible made probable." (Rod Serling, The Twilight Zone, "The Fugitive", 1962)

"The only way of discovering the limits of the possible is to venture a little way past them into the impossible." (Arthur C Clarke, "Profiles of the Future", 1962)

"When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong." (Arthur C Clarke, "Profiles of the Future", 1962)

"It is impossible to import things into an infinite area, there being no outside to import things in from." (Douglas N Adams, "The Original Hitchhiker Radio Script, Fit the Fifth", 1978)

"Fantasy is the impossible made probable. Science Fiction is the improbable made possible." (Rod Sterling)

On Impossibility (2010-2019)

"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)

"The problem of complexity is at the heart of mankind’s inability to predict future events with any accuracy. Complexity science has demonstrated that the more factors found within a complex system, the more chances of unpredictable behavior. And without predictability, any meaningful control is nearly impossible. Obviously, this means that you cannot control what you cannot predict. The ability ever to predict long-term events is a pipedream. Mankind has little to do with changing climate; complexity does." (Lawrence K Samuels, "The Real Science Behind Changing Climate", 2014)

"To understand the precise point when the possible becomes the impossible, you have to appreciate and understand the laws of physics." (Michio Kaku, "The Future of the Mind: The Scientific Quest to Understand, Enhance, and Empower the Mind", 2014)

"An act of creativity is the result of an insight that arises discontinuously. Of course the insight must be preceded by something that is deeply problematic; it is so deeply problematic that a resolution may well seem impossible. This is why the resolution does not arise through systematic means but only occurs when all systematic approaches have been exhausted to no effect, that is, if you want to be creative you must sometimes be prepared to fly blind. This is not easy to do. Creativity involves living for protracted periods with the kind of tension that arises in situations of cognitive dissonance." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Data contain descriptions. Some are true, some are not. Some are useful, most are not. Skillful use of data requires that we learn to pick out the pieces that are true and useful. [...] To find signals in data, we must learn to reduce the noise - not just the noise that resides in the data, but also the noise that resides in us. It is nearly impossible for noisy minds to perceive anything but noise in data. […] Signals always point to something. In this sense, a signal is not a thing but a relationship. Data becomes useful knowledge of something that matters when it builds a bridge between a question and an answer. This connection is the signal." (Stephen Few, "Signal: Understanding What Matters in a World of Noise", 2015)

"When a culture is founded on the principle of immediacy of experience, there is no need for numeracy. It is impossible to consume more than one thing at a time, so differentiating between 'a small amount', 'a larger amount' and 'many' is enough for survival." (The Open University, "Understanding the environment: learning and communication", 2016)

"Parameter estimation is a basic aspect of model construction and historically it has been assumed that data are sufficient to estimate the parameters, for instance, correlations that are part of the model; however, when the number of parameters is too large for the amount of data, accurate parameter estimation becomes impossible. The result is model uncertainty." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"Since it’s impossible to express an irrational number such as π as a fraction, the quest for a fraction equal to π could never be successful. Ancient mathematicians didn’t know that, however. As noted above, it wasn’t until the eighteenth century that the irrationality of π was demonstrated. Their labors weren’t in vain, though. While enthusiastically pursuing their fundamentally doomed enterprise, they developed a lot of interesting mathematics as well as impressively accurate approximations of π." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The most accurate but least interpretable form of data presentation is to make a table, showing every single value. But it is difficult or impossible for most people to detect patterns and trends in such data, and so we rely on graphs and charts. Graphs come in two broad types: Either they represent every data point visually (as in a scatter plot) or they implement a form of data reduction in which we summarize the data, looking, for example, only at means or medians." (Daniel J Levitin, "Weaponized Lies", 2017)

"The laws of the universe cannot be different from the universe. The laws of the universe must actually be the universe. Otherwise you create an impossible Cartesian dualism of laws and things operated on by laws. How can a law operate on a non-law? That’s a category error. The only way to make the laws and 'stuff' of the universe the same is via mathematics." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"Statistical metrics can show us facts and trends that would be impossible to see in any other way, but often they’re used as a substitute for relevant experience, by managers or politicians without specific expertise or a close-up view. " (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

On Impossibility (2000-2009)

 "Statisticians can calculate the probability that such random samples represent the population; this is usually expressed in terms of sampling error [...]. The real problem is that few samples are random. Even when researchers know the nature of the population, it can be time-consuming and expensive to draw a random sample; all too often, it is impossible to draw a true random sample because the population cannot be defined. This is particularly true for studies of social problems. [...] The best samples are those that come as close as possible to being random." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"[...] a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration." Yu-Chi Ho & David L Pepyne, "Simple explanation of the no-free-lunch theorem and its implications", Journal of Optimization Theory and Applications 115, 2002)

"The diversity of networks in business and the economy is mindboggling. There are policy networks, ownership networks, collaboration networks, organizational networks, network marketing-you name it. It would be impossible to integrate these diverse interactions into a single all-encompassing web. Yet no matter what organizational level we look at, the same robust and universal laws that govern nature's webs seem to greet us. The challenge is for economic and network research alike to put these laws into practice." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"At the basis of the impossibility of making reliable predictions for systems such as the atmosphere, there is a phenomenon known today as the butterfly effect. This deals with the progressive limitless magnification of the slightest imprecision (error) present in the measurement of the initial data (the incomplete knowledge of the current state of each molecule of air), which, although in principle negligible, will increasingly expand during the course of the model’s evolution, until it renders any prediction on future states (atmospheric weather conditions when the forecast refers to more than a few days ahead) completely insignificant, as these states appear completely different from the calculated ones." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems", 2003)

"How is it that -1 can have a square root? The square of a positive number is always positive, and the square of a negative number is again positive (and the square of 0 is just 0 again, so that is hardly of use to us here). It seems impossible that we can find a number whose square is actually negative." (Sir Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", 2004)

"What was impossible, inconceivable, and incoherent based on literal vocabulary becomes possible, conceivable, and coherent through metaphoric redescription. Combinations of terms that were incoherent, in relation to the conventional rules of meaning, become meaningful. Metaphoric description arises from a momentary suspension of the rules for literal vocabulary. The semantics of a metaphor convey an alternative realm of conceptual possibilities, through a new set of possible attributes. Of course, not all scientific language is metaphoric. But when unexpected empirical findings raise serious doubts about a familiar scientific theory, a satisfactory resolution occur through the use of metaphoric vocabulary." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

 "Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"It is impossible for √-1 to be a real number, since its square is negative. This implies that √-1 is neither greater nor less than zero, so we cannot see √-1 on the real line. However, √-1 behaves like a number with respect to + and x. This prompts us to look elsewhere for it, and indeed we find it on another line (the imaginary axis) perpendicular to the real line." (John Stillwell, "Yearning for the impossible: the surprising truths of mathematics", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Once you understand a problem, many aspects of it suddenly become much simpler. As mathematicians the world over say, everything is either impossible or trivial." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"The model theory postulates that mental models are parsimonious. They represent what is possible, but not what is impossible, according to assertions. This principle of parsimony minimizes the load on working memory, and so it applies unless something exceptional occurs to overrule it." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"People don’t need to know all the details of how a complex mechanism actually works in order to use it, so they create a cognitive shorthand for explaining it, one that is powerful enough to cover their interactions with it, but that doesn’t necessarily reflect its actual inner mechanics. […] In the digital world, however, the differences between a user’s mental model and the implementation model are often quite distinct. The discrepancy between implementation and mental models is particularly stark in the case of software applications, where the complexity of implementation can make it nearly impossible for the user to see the mechanistic connections between his actions and the program’s reactions." (Alan Cooper et al, "About Face 3: The Essentials of Interaction Design", 2007)

"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)

"Algebraic symbols carry a universality of interpretation that allows them to be manipulated in a way that words cannot. Indeed, this was the key breakthrough that allowed mathematics to flourish in a way that was not possible until the advent of algebra. All higher mathematics relies on constant use of algebraic manipulation and would be impossible without it." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"It is impossible to construct a model that provides an entirely accurate picture of network behavior. Statistical models are almost always based on idealized assumptions, such as independent and identically distributed (i.i.d.) interarrival times, and it is often difficult to capture features such as machine breakdowns, disconnected links, scheduled repairs, or uncertainty in processing rates." (Sean Meyn, "Control Techniques for Complex Networks", 2008)

"Prior to the discovery of the butterfly effect it was generally believed that small differences averaged out and were of no real significance. The butterfly effect showed that small things do matter. This has major implications for our notions of predictability, as over time these small differences can lead to quite unpredictable outcomes. For example, first of all, can we be sure that we are aware of all the small things that affect any given system or situation? Second, how do we know how these will affect the long-term outcome of the system or situation under study? The butterfly effect demonstrates the near impossibility of determining with any real degree of accuracy the long term outcomes of a series of events." (Elizabeth McMillan, Complexity, "Management and the Dynamics of Change: Challenges for practice", 2008)

"Topology allows the possibility of making qualitative predictions when quantitative ones are impossible." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)

"Perhaps the simplest way to explain symmetry is to follow the operational approach used by mathematicians: a symmetry is a motion. That is, suppose you have an object and pick it up, move it around, and set it down. If it is impossible to distinguish between the object in its original and final positions, we say that it has a symmetry." (Michael Field & Martin Golubitsky, "Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature" 2nd Ed, 2009)

"With Kurt Gödel, we find in the twentieth century the idea that formal systems are incomplete, a concept that is perhaps important to chess theory. If undecidable statements exist in chess, then it is impossible to solve them completely with a computer chess program." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

On Impossibility (1975-1999)

"A mature science, with respect to the matter of errors in variables, is not one that measures its variables without error, for this is impossible. It is, rather, a science which properly manages its errors, controlling their magnitudes and correctly calculating their implications for substantive conclusions." (Otis D Duncan, "Introduction to Structural Equation Models", 1975)

"Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)

"[Human consciousness] depends wholly on our seeing the outside world in such categories. And the problems of consciousness arise from putting reconstitution beside internalization, from our also being able to see ourselves as if we were objects in the outside world. That is in the very nature of language; it is impossible to have a symbolic system without it." (Jacob Bronowski, "The origins of knowledge and imagination", 1978)

"The most complex system imaginable is the mind - by definition, since the mind must be at least one degree more complex than whatever it imagines. Catastrophe theory proposes that qualitative stability is a necessary attribute of thought; without it, recognition and memory would be impossible." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

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

"The world's present industrial civilization is handicapped by the coexistence of two universal, overlapping, and incompatible intellectual systems: the accumulated knowledge of the last four centuries of the properties and interrelationships of matter and energy; and the associated monetary culture which has evolved from folkways of prehistoric origin. […] Despite their inherent incompatibilities, these two systems during the last two centuries have had one fundamental characteristic in common, namely exponential growth, which has made a reasonably stable coexistence possible. But, for various reasons, it is impossible for the matter-energy system to sustain exponential growth for more than a few tens of doublings, and this phase is by now almost over. The monetary system has no such constraints, and according to one of its most fundamental rules, it must continue to grow by compound interest." (Marion K Hubbert, "Two Intellectual Systems: Matter-energy and the Monetary Culture", [seminar] 1981)

"It is actually impossible in theory to determine exactly what the hidden mechanism is without opening the box, since there are always many different mechanisms with identical behavior. Quite apart from this, analysis is more difficult than invention in the sense in which, generally, induction takes more time to perform than deduction: in induction one has to search for the way, whereas in deduction one follows a straightforward path." (Valentino Braitenberg, "Vehicles: Experiments in Synthetic Psychology", 1984)

"If we want to solve problems effectively [...] we must keep in mind not only many features but also the influences among them. Complexity is the label we will give to the existence of many interdependent variables in a given system. The more variables and the greater their interdependence, the greater the system's complexity. Great complexity places high demands on a planner's capacity to gather information, integrate findings, and design effective actions. The links between the variables oblige us to attend to a great many features simultaneously, and that, concomitantly, makes it impossible for us to undertake only one action in a complex system." (Dietrich Dorner, "The Logic of Failure: Recognizing and Avoiding Error in Complex Situations", 1989)

"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)

"In practice, it may be impossible to purge a real system of its actual randomness and observe the consequences, but often we can guess what these would be by turning to theory. Most theoretical studies of real phenomena are studies of approximations." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Why is it so important to find primes, or to show that a certain integer is one? A very practical application in cryptography rests on the fact that since it is extremely hard to factor very large numbers, a two-hundred-digit number that was the product of two primes could govern text encoding: It would be virtually impossible to guess what the two numbers were if you didn't know them in advance, and out of the question (save perhaps on a state-of-the-art supercomputer) to go at it by trial and error." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"The impossibility of constructing a complete, accurate quantitative description of a complex system forces observers to pick which aspects of the system they most wish to understand." (Thomas Levenson, "Measure for Measure: A musical history of science", 1994)

"It is impossible to understand the true meaning of an equation, or to appreciate its beauty, unless it is read in the delightfully quirky language in which it was penned." (Michael Guillen, "Five Equations That Changed the World", 1995)

"How surprising it is that the laws of nature and the initial conditions of the universe should allow for the existence of beings who could observe it. Life as we know it would be impossible if any one of several physical quantities had slightly different values." (Steven Weinberg, Life in the Quantum Universe", Scientific American, 1995)

"The Law of Entropy Nonconservation required that life be lived forward, from birth to death. […] To wish for the reverse was to wish for the entropy of the universe to diminish with time, which was impossible. One might as well wish for autumn leaves to assemble themselves in neat stacks just as soon as they had fallen from trees or for water to freeze whenever it was heated." (Michael Guillen," Five Equations That Changed the World", 1995)

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

"Small changes in the initial conditions in a chaotic system produce dramatically different evolutionary histories. It is because of this sensitivity to initial conditions that chaotic systems are inherently unpredictable. To predict a future state of a system, one has to be able to rely on numerical calculations and initial measurements of the state variables. Yet slight errors in measurement combined with extremely small computational errors (from roundoff or truncation) make prediction impossible from a practical perspective. Moreover, small initial errors in prediction grow exponentially in chaotic systems as the trajectories evolve. Thus, theoretically, prediction may be possible with some chaotic processes if one is interested only in the movement between two relatively close points on a trajectory. When longer time intervals are involved, the situation becomes hopeless." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"There are no hard problems, only problems that are hard to a certain level of intelligence. Move the smallest bit upwards, and some problems will suddenly move from 'impossible' to 'obvious'. Move a substantial degree upwards, and all of them will become obvious. Move a huge distance upwards [...]" (Eliezer S Yudkowsky, "Staring into the Singularity", 1996)

"All things which are proved to be impossible must obviously rest on some assumptions, and when one or more of these assumptions are not true then the impossibility proof fails - but the expert seldom remembers to carefully inspect the assumptions before making their 'impossible' statements." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)

"Another limit imposed by reality is its sheer complexity, which makes it impossible to predict some ordinary things (like weather) at the same time that it’s possible to predict truly extraordinary things (like the fate of the universe)." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", 1997)   

"Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

On Impossibility (1950-1974)

"Every organism represents a system, by which term we mean a complex of elements in mutual interaction. From this obvious statement the limitations of the analytical and summative conceptions must follow. First, it is impossible to resolve the phenomena of life completely into elementary units; for each individual part and each individual event depends not only on conditions within itself, but also to a greater or lesser extent on the conditions within the whole, or within superordinate units of which it is a part. Hence the behavior of an isolated part is, in general, different from its behavior within the context of the whole. [...] Secondly, the actual whole shows properties that are absent from its isolated parts." (Ludwig von Bertalanffy, "Problems of Life", 1952)

"The history of mathematics shows that the introduction of better and better symbolism and operations has made a commonplace of processes that would have been impossible with the unimproved techniques." (Morris Kline,"Mathematics in Western culture", 1953)

"There comes a stage, however, as the system becomes larger and larger, when the reception of all the information is impossible by reason of its sheer bulk. Either the recording channels cannot carry all the information, or the observer, presented with it all, is overwhelmed. When this occurs, what is he to do? The answer is clear: he must give up any ambition to know the whole system. His aim must be to achieve a partial knowledge that, though partial over the whole, is none the less complete within itself, and is sufficient for his ultimate practical purpose." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Uncertainty is introduced, however, by the impossibility of making generalizations, most of the time, which happens to all members of a class. Even scientific truth is a matter of probability and the degree of probability stops somewhere short of certainty." (Wayne C Minnick,"The Art of Persuasion", 1957)

"The outstanding feature of behavior is that it is often quite easy to recognize but extremely difficult or impossible to describe with precision." (Anatol Rapoport. "An Essay on Mind", General Systems, 1962)

"A theory in its scientific context is not a static museum piece, but is always being extended and modified to account for new phenomena. […] Moreover, without a model, it will be impossible to use a theory for one of the essential purposes we demand of it, namely to make predictions in new domains of phenomena." (Mary B Hesse," Models and Analogies in Science", 1963)

"I shall not attempt to prove that mathematics is useful. I will admit it and so save myself the trouble that here is a great and respected discipline where all is impossible yet much is useful. The usefulness largely flows from the impossibility. Mathematical concepts have been simplified and generalized until they describe an imaginative world no part of which could possibly exist outside men’s minds." (Billy E Goetz, "The Usefulness of the Impossible", 1963)

"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem." (Howard W Eves, "A Survey of Geometry", 1963)

"It is not impossible that our own Model will die a violent death, ruthlessly smashed by an unprovoked assault of new facts […]." (Clive S Lewis, "The Discarded Image: An Introduction to Medieval and Renaissance Literature", 1964) 

"In all of natural philosophy, the most deeply and repeatedly studied part, next to pure geometry, is mechanics. […] The picture of nature as a whole given us by mechanics may be compared to a black-and-white photograph: It neglects a great deal, but within its limitations, it can be highly precise. Developing sharper and more flexible black-and-white photography has not attained pictures in color or three-dimensional casts, but it serves in cases where color and thickness are irrelevant, presently impossible to get in the required precision, or distractive from the true content." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"In order to define a concept we have to indicate first of all that it is a special case of a more general concept. This is impossible for the concept of set, since this concept is already as broad as possible and is thus not a special case of any other concept." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"It is impossible, and it has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form." (Jacob Klein, "Greek Mathematical Thought and the Origin of Algebra", 1968)

"Scientific knowledge is not created solely by the piecemeal mining of discrete facts by uniformly accurate and reliable individual scientific investigations. The process of criticism and evaluation, of analysis and synthesis, are essential to the whole system. It is impossible for each one of us to be continually aware of all that is going on around us, so that we can immediately decide the significance of every new paper that is published. The job of making such judgments must therefore be delegated to the best and wisest among us, who speak, not with their own personal voices, but on behalf of the whole community of Science. […] It is impossible for the consensus - public knowledge - to be voiced at all, unless it is channeled through the minds of selected persons, and restated in their words for all to hear." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

"The choice of model is often the most critical aspect of a design and development engineering job, but it is impossible to give explicit rules or techniques.' (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"It is impossible to devise a scientific experiment to describe the creation process, or even to ascertain whether such a process can take place. The Creator does not create at the whim of a scientist." (Henry M. Morris, "Scientific Creationism", 1974) 

"The field of probability and statistics is then transformed into a Tower of Babel, in which only the most naive amateur claims to understand what he says and hears, and this because, in a language devoid of convention, the fundamental distinctions between what is certain and what is not, and between what is impossible and what is not, are abolished. Certainty and impossibility then become confused with high or low degrees of a subjective probability, which is itself denied precisely by this falsification of the language. On the contrary, the preservation of a clear, terse distinction between certainty and uncertainty, impossibility and possibility, is the unique and essential precondition for making meaningful statements (which could be either right or wrong), whereas the alternative transforms every sentence into a nonsense." (Bruno de Finetti, "Theory of Probability", 1974)

On Impossibility (1925-1949)

"I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment." (Tobias Dantzig, "The Two Realities", 1930)

"The validity of demonstrably wrong law cannot conceivably be justified. However, any answer to the question of the purpose of law other than by enumerating the manifold partisan views about it has proved impossible - and it is precisely on that impossibility of any natural law, and on that alone, that the validity of positive law may be founded. At this point relativism, so far only the method of our approach, enters our system as a structural element." (Gustav Radbruch, "Rechtsphilosophie", 1932)

"While it is true that theory often sets difficult, if not impossible tasks for the experiment, it does, on the other hand, often lighten the work of the experimenter by disclosing cogent relationships which make possible the indirect determination of inaccessible quantities and thus render difficult measurements unnecessary." (Georg Joos, "Theoretical Physics", 1934)

"It is impossible to make a clear cut between science, religion, and art. The whole is never equal simply to the sum of its various parts." (Max Planck, "The Philosophy of Physics", 1936)

"Statements about impossibility in mathematics are of a wholly different character. A problem in mathematics which may not be solved for centuries to come is not always impossible. 'Impossible' in mathematics means theoretically impossible, and has nothing to do with the present state of our knowledge." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"When the number of factors coming into play in a phenomenological complex is too large, scientific method in most cases fails us. One need only think of the weather, in which case prediction even for a few days ahead is impossible. Nevertheless no one doubts that we are confronted with a causal connection whose causal components are in the main known to us. Occurrences in this domain are beyond the reach of exact prediction because of the variety of factors in operation, not because of any lack of order in nature." (Albert Einstein, "Science and Religion", 1941)

"Events with a sufficiently small probability never occur, or at least we must act, in all circumstances, as if they were impossible." (Félix E Borel, "Probabilities and Life", 1943)

"We have now to enquire how the neural mechanism, in producing numerical measurement and calculation, has managed to function in a way so much more universal and flexible than any other. Our question, to emphasize it once again, is not to ask what kind of thing a number is, but to think what kind of mechanism could represent so many physically possible or impossible, and yet self-consistent, processes as number does." (Kenneth Craik, "The Nature of Explanation", 1943)

"It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge." (George Pólya, "How to solve it", 1945)

"The rules of algebra show that the square of any number, whether positive or negative, is a positive number: therefore, to speak of the square root of a negative number is mere absurdity. Now, Cardan deliberately commits that absurdity and begins to calculate on such 'imaginary' quantities.     One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament - which, of course, was, in the nineteenth century, established on solid and rigorous bases. It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)

On Impossibility (1900-1924)

"To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary about it. In the choice of this convention we can be guided only by the principle of sufficient reason. Unfortunately, this principle is very vague and very elastic, and in the cursory examination we have just made we have seen it assume different forms. The form under which we meet it most often is the belief in continuity, a belief which it would be difficult to justify by apodeictic reasoning, but without which all science would be impossible. Finally, the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothesis made at the outset, provided only that this hypothesis satisfies the condition of continuity. (Henri Poincaré, "Science and Hypothesis", 1901)

 "With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science." (David Hilbert,"Mathematical Problems", Bulletin of the American Mathematical Society Vol. 8, 1902)

"The most ordinary things are to philosophy a source of insoluble puzzles. In order to explain our perceptions it constructs the concept of matter and then finds matter quite useless either for itself having or for causing perceptions in a mind. With infinite ingenuity it constructs a concept of space or time and then finds it absolutely impossible that there be objects in this space or that processes occur during this time [...] The source of this kind of logic lies in excessive confidence in the so-called laws of thought." (Ludwig E Boltzmann, "On Statistical Mechanics", 1904)

"Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt, a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility." (Henri Poincaré, "The Value of Science", 1905)

"An exceedingly small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation 'approximately'. If that enabled us to predict the succeeding situation with 'the same approximation', that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon." (Jules H Poincaré, "Science and Method", 1908)

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

"I do not say that the notion of infinity should be banished; I only call attention to its exceptional nature, and this so far as I can see, is due to the part which zero plays in it, and we must never forget that like the irrational it represents a function which possesses a definite character but can never be executed to the finish If we bear in mind the imaginary nature of these functions, their oddities will not disturb us, but if we misunderstand their origin and significance we are confronted by impossibilities." (Paul Carus,"The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"And here is what makes this analysis situs interesting to us; it is that geometric intuition really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning." (Henri Poincaré, "Dernières pensées", 1913)

"The great difference between induction and hypothesis is that the former infers the existence of phenomena such as we have observed in cases which are similar, while hypothesis supposes something of a different kind from what we have directly observed, and frequently something which it would be impossible for us to observe directly." (Charles S Peirce, "Chance, Love and Logic: Philosophical Essays, Deduction, Induction, Hypothesis", 1914)

 "The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist." (Godfrey H Hardy, 1915)

"It may be impossible for human intelligence to comprehend absolute truth, but it is possible to observe Nature with an unbiased mind and to bear truthful testimony of things seen." (Sir Richard A Gregory,"Discovery, Or, The Spirit and Service of Science", 1916)

"The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s ‘blank form of a universe’ without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition." (Lawrence J Henderson, "The Order of Nature: An Essay", 1917)

"The laws of nature cannot be intelligently applied until they are understood, and in order to understand them, many experiments bearing upon the ultimate nature of things must be made, in order that all may be combined in a far-reaching generalization impossible without the detailed knowledge upon which it rests." (Theodore W Richards, "The Problem of Radioactive Lead", 1918)

"In Continuity, it is impossible to distinguish phenomena at their merging-points, so we look for them at their extremes." (Charles Fort,"The Book of the Damned", 1919)