On Insight (1960-1969)

 "The attempt to characterize exactly models of an empirical theory almost inevitably yields a more precise and clearer understanding of the exact character of a theory. The emptiness and shallowness of many classical theories in the social sciences is well brought out by the attempt to formulate in any exact fashion what constitutes a model of the theory. The kind of theory which mainly consists of insightful remarks and heuristic slogans will not be amenable to this treatment. The effort to make it exact will at the same time reveal the weakness of the theory." (Patrick Suppes," A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences", Synthese  Vol. 12 (2/3), 1960)

"Model-making, the imaginative and logical steps which precede the experiment, may be judged the most valuable part of scientific method because skill and insight in these matters are rare. Without them we do not know what experiment to do. But it is the experiment which provides the raw material for scientific theory. Scientific theory cannot be built directly from the conclusions of conceptual models." (Herbert G Andrewartha, "Introduction to the Study of Animal Population", 1961)

"Since we are assured that the all-wise Creator has observed the most exact proportions of number, weight and measure in the make of all things, the most likely way therefore to get any insight into the nature of those parts of the Creation which come within our observation must in all reason be to number, weigh and measure." (Stephen Hales, "Vegetable Staticks", 1961)

"A point of view can be a dangerous luxury when substituted for insight and understanding." (Marshall McLuhan, "The Gutenberg Galaxy", 1962)

"The purpose of computing is insight, not numbers […] sometimes […] the purpose of computing numbers is not yet in sight." (Richard Hamming, [Motto for the book] "Numerical Methods for Scientists and Engineers", 1962)

"[…] it is more important to have beauty in one's equations that to have them fit experiment. […] It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." (Paul A M Dirac, "The Evolution of the Physicist’s Picture of Nature ", Scientific American, 1963)

"The original insight is most likely to come when elements stored in different compartments of the mind drift into the open, jostle one another, and now and then form new combinations." (Eric Hoffer, "The Ordeal of Change", 1963)

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practices to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuition of humans. Here and there an intuition is scooped out and replaced by a firmly built pillar of thought; however, this pillar is based on some deeper, perhaps less clearly defined, intuition. Though the process of replacing intuitions with precise thoughts does not change the nature of the ground on which mathematics ultimately rests, it does add strength and height to the structure." (Morris Kline, "Mathematics in Western Culture", 1964)

"The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links." (Arthur Koestler, "The Act of Creation", 1964)

"All great insights and discoveries are not only usually thought by several people at the same time, they must also be re-thought in that unique effort to truly say the same thing about the same thing." (Martin Heidegger, "What Is A Thing", 1967)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"More than a burial ground for unacceptable ideas and wishes, the unconscious is the spawning ground of intuition and insight, the source of humor, of poetic imagery, and of scientific analogy." (Judith Groch, "The Right to Create", 1969)

On Insight (1990-1999)

 "Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through some process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics." (William P Thurston, "Mathematical education", Notices AMS 37, 1990)

"[…] new insights fail to get put into practice because they conflict with deeply held internal images of how the world works [...] images that limit us to familiar ways of thinking and acting. That is why the discipline of managing mental models - surfacing, testing, and improving our internal pictures of how the world works - promises to be a major breakthrough for learning organizations." (Peter Senge, "The Fifth Discipline: The Art and Practice of the Learning Organization", 1990)

"Science is (or should be) a precise art. Precise, because data may be taken or theories formulated with a certain amount of accuracy; an art, because putting the information into the most useful form for investigation or for presentation requires a certain amount of creativity and insight." (Patricia H Reiff, "The Use and Misuse of Statistics in Space Physics", Journal of Geomagnetism and Geoelectricity 42, 1990)

"Insight is 'mental vision,' one of the ways in which the mind escapes the limits of the obvious or the familiar." (Jennifer James, "Thinking In The Future Tense", 1991) 

"It is not surprising to find many mathematical ideas interconnected or linked. The expansion of mathematics depends on previously developed ideas. The formation of any mathematical system begins with some undefined terms and axioms (assumptions) and proceeds from there to definitions, theorems, more axioms and so on. But history points out this is not necessarily the route that creativity" (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"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 scope of Theories of Everything is infinite but bounded; they are necessary parts of a full understanding of things but they are far from sufficient to reveal everything about a Universe like ours. In the pages of this book, we have seen something of what a Theory of Everything might hope to teach us about the unity of the Universe and the way in which it may contain elements that transcend our present compartmentalized view of Nature's ingredients. But we have also learnt that there is more to Everything than meets the eye. Unlike many others that we can imagine, our world contains prospective elements. Theories of Everything can make no impression upon predicting these prospective attributes of reality; yet, strangely, many of these qualities will themselves be employed in the human selection and approval of an aesthetically acceptable Theory of Everything. There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all." (John D Barrow, "New Theories of Everything", 1991)

"Fitting is essential to visualizing hypervariate data. The structure of data in many dimensions can be exceedingly complex. The visualization of a fit to hypervariate data, by reducing the amount of noise, can often lead to more insight. The fit is a hypervariate surface, a function of three or more variables. As with bivariate and trivariate data, our fitting tools are loess and parametric fitting by least-squares. And each tool can employ bisquare iterations to produce robust estimates when outliers or other forms of leptokurtosis are present." (William S Cleveland, "Visualizing Data", 1993)

"Puzzle composers share another feature with mathematicians. They know that, generally speaking, the simpler a puzzle is to express, the more attractive it is likely to be found: similarly, simplicity is for both a desirable feature of the solution. Especially satisfying solutions are often described as 'elegant', a word that - no surprise here - is also used by scientists, engineers and designers, indeed by anyone with a problem to solve. However, simplicity is by no means the only reward of success. Far from it! Mathematicians (and scientists and others) can reasonably expect two further returns: they are (in no particular order) firstly the power to do things, and secondly the perception of connections which were never before suspected, leading in turn to the insight and illumination that mathematicians expect from their best arguments." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"[...] two related deficiencies have prevented real progress in understanding insight and its role in problem solving. First, we do not yet have a system of classifying problems into those in which insight occurs versus those in which it does not. However, only if we can isolate problems in which insight occurs will we be able to set on a firm base our theories of the mechanisms underlying insight. Second, formulation of such a taxonomic system requires that we agree on a definition of insight." (Robert W Weisberg, "Prolegomena to theories of insight in problem solving: a taxonomy of problems", 1995)

"Ideas about organization are always based on implicit images or metaphors that persuade us to see, understand, and manage situations in a particular way. Metaphors create insight. But they also distort. They have strengths. But they also have limitations. In creating ways of seeing, they create ways of not seeing. There can be no single theory or metaphor that gives an all-purpose point of view, and there can be no simple 'correct theory' for structuring everything we do." (Gareth Morgan, "Imaginization", 1997)

"Our searches for numerical order lead as often to terminal nuttiness as to profound insight." (Stephen J Gould, "Questioning the Millennium: A Rationalist's Guide to a Precisely Arbitrary Countdown", 1997)

"Cleaning up old proofs is an important part of the mathematical enterprise that often yields new insights that can be used to solve new problems and build more beautiful and encompassing theories." (Bruce Schecter, "My Brain is Open", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world."  (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"When scientists need to explain difficult points of theory, illustration by hypothetical example - rather than by total abstraction - works well (perhaps indispensably) as a rhetorical device. Such cases do not function as speculations in the pejorative sense - as silly stories that provide insight into complex mechanisms - but rather as idealized illustrations to exemplify a difficult point of theory." (Stephen Jay Gould, "Leonardo's Mountain of Clams and the Diet of Worms", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world."  (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation." (Kenneth E Iverson, "Math for the Layman", 1999)

On Insight (1980-1989)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"Instead of insight, maybe all a man gets is strength to wander for a while. Maybe the only gift is a chance to inquire, to know nothing for certain. An inheritance of wonder and nothing more." (William Least Heat-Moon, "Blue Highways", 1982)

"The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases." (Paul Halmos, "Selecta: Expository writing", 1983)

"All the efforts of the researcher to find other models, conceptions, different mathematical forms, better linguistic modes of expression, to do justice to newly discovered layers of being mean self-transformation. The researcher in his place is the human being in self-transformation to more profound insight into what is given." (John H. Dessauer, Universitas: A Quarterly German Review of the Arts and Sciences Vol. 26 (4), 1984)

"When one combines the new insights gained from studying far-from-equilibrium states and nonlinear processes, along with these complicated feedback systems, a whole new approach is opened that makes it possible to relate the so-called hard sciences to the softer sciences of life - and perhaps even to social processes as well. […] It is these panoramic vistas that are opened to us by Order Out of Chaos." (Ilya Prigogine, "Order Out of Chaos: Man's New Dialogue with Nature", 1984)

"Mathematics is good if it enriches the subject, if it opens up new vistas, if it solves old problems, if it fills gaps, fitting snugly and satisfyingly into what is already known, or if it forges new links between previously unconnected parts of the subject It is bad if it is trivial, overelaborate, or lacks any definable mathematical purpose or direction It is pure if its methods are pure - that is, if it doesn't cheat and tackle one problem while pretending to tackle another, and if there are no gaping holes in its logic It is applied if it leads to useful insights outside mathematics. By these criteria, today's mathematics contains as high a proportion of good work as at any other period, and as any other area, and much of it manages to be both pure and applied at the same time." (Ian Stewart, "The Problems of Mathematics", 1987)

"One of the features that distinguishes applied mathematics is its interest in framing important questions about the observed world in a mathematical way. This process of translation into a mathematical form can give a better handle for certain problems than would be otherwise possible. We call this the modeling process. It combines formal reasoning with intuitive insights. Understanding the models devised by others is a first step in learning some of the skills involved, and that is how we proceed in this text, which is an informal introduction to the mathematics of dynamical systems." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"Although discrete mathematics and statistics provide necessary foundations for computer engineering and social sciences, calculus remains the archetype of higher mathematics. It is a powerful and elegant example of the mathematical method, leading both to major applications and to major theories. The language of calculus has spread to all scientific fields; the insight it conveys about the nature of change is something that no educated person can afford to be without." (Mathematical Sciences Education Board, "Everybody Counts: A Report to the Nation on the Future of Mathematics Education", 1989)

On Insight (-1899)

"[…] it is from long experience chiefly that we are to expect the most certain rules of practice, yet it is withal to be remembered, that observations, and to put us upon the most probable means of improving any art, is to get the best insight we can into the nature and properties of those things which we are desirous to cultivate and improve." (Stephen Hales, "Vegetable Staticks", 1727)

"Reason in a creature is a faculty of widening the rules and purposes of the use of all its powers far beyond natural instinct; it acknowledges no limits to its projects. Reason itself does not work instinctively, but requires trial, practice, and instruction in order gradually to progress from one level of insight to another." (Immanuel Kant, "Idea for a Universal History from a Cosmopolitan Point of View", 1784)

"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)

"[...] it should be noted that the seeds of wisdom that are to bear fruit in the intellect are sown less by critical studies and learned monographs than by insights, broad impressions, and flashes of intuition." (Carl von Clausewitz, "On War", 1832)

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"The desire to explain what is simple by what is complex, what is easy by what is difficult, is a calamity affecting the whole body of science, known, it is true, to men of insight, but not generally admitted." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"The mysterious complexity of our life is not to be embraced by maxims... to lace ourselves up in formulas of that sort is to repress all the divine promptings and inspirations that spring from growing insight and sympathy." (George Eliot, "The Mill on the Floss", 1860)

"The sciences are said, and they are truly said, to have a mutual connection, that any one of them may be the better understood, for an insight into the rest." (Samuel Horsley, "Sermons", 1860)

"It often happens that the pursuit of the beautiful and appropriate, or, as it may be otherwise expressed, the endeavor after the perfect, is rewarded with a new insight into the true." (James J Sylvester, "Separation of the Roots of an Algebraical Equation", Philosophical Magazine, 1866)

"A law of nature, however, is not a mere logical conception that we have adopted as a kind of memoria technical to enable us to more readily remember facts. We of the present day have already sufficient insight to know that the laws of nature are not things which we can evolve by any speculative method. On the contrary, we have to discover them in the facts; we have to test them by repeated observation or experiment, in constantly new cases, under ever-varying circumstances; and in proportion only as they hold good under a constantly increasing change of conditions, in a constantly increasing number of cases with greater delicacy in the means of observation, does our confidence in their trustworthiness rise." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

"Science corrects the old creeds, sweeps away, with every new perception, our infantile catechisms, and necessitates a faith commensurate with the grander orbits and universal laws which it discloses yet it does not surprise the moral sentiment that was older and awaited expectant these larger insights." (Ralph Waldo Emerson, "Letters and Social Aims", 1876)

"Our power of scientific insight is but feeble when compared with the profundity of nature, because deep truths require deep thought to enable us to understand and value them." (George Gore, 'The Art of Scientific Discovery", 1878)

"The aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of one truth upon another. After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely." (Gottlob Frege, "The Foundations of Arithmetic", 1884)

"By thus thinking [by analogy] you will get a more real grasp of the subject and insight into the actual processes occurring in Nature - unknown though these may still strictly be - than if you employed the old ideas of action at a distance, or contented yourselves with no theory at all on which to link the facts. You will have made a step in the direction of the truth, but I must beg you to understand that it is only a step; that what modifications and additions will have to be made to it before it becomes a complete theory of electricity I am unable fully to tell you." (Oliver J Lodge, "Modern Views of Electricity", 1889)

"The history of science shows that great discoveries are made by means of imaginative insight, but it also teaches that mere imagination without dependence upon known facts is frequently a source of much mischief." (James E Creighton, "An Introductory Logic", 1898)

On Insight (2010-2019)

"The idea of natural computation has grown into a new scientific paradigm and has proved to be a rich source of new insights about nature. Many processes in nature exhibit key characteristics of computation, especially discrete units or steps and repetition according to a fixed set of rules. Although the processes may be highly complex, their regularity makes them highly amenable to simulation [...]." (David G Green & Tania Leishman, "Computing and Complexity: Networks, Nature and Virtual Worlds, Philosophy of Complex Systems, 2011)

"A genuine experience of the unexpected, in maths as much as in magic, demands of its performer at once originality of insight and a lightness of touch. Even a single step too many in a method renders ugly and clumsy the theorem or the trick." (Daniel Tammet, "Thinking in Numbers", 2012)

"Regression analysis, like all forms of statistical inference, is designed to offer us insights into the world around us. We seek patterns that will hold true for the larger population. However, our results are valid only for a population that is similar to the sample on which the analysis has been done." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"Understanding chaos requires much less advanced mathematics than other current areas of physics research such as general relativity or particle physics. Observing chaos and fractals requires no specialized equipment; chaos is seen in scores of everyday phenomena - a boiling pot of water, a dripping faucet, shifting weather patterns. And fractals are almost ubiquitous in the natural world. Thus, it is possible to teach the central ideas and insights of chaos in a rigorous, genuine, and relevant way to students with relatively little mathematics background." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"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)

"Mathematical modeling is the modern version of both applied mathematics and theoretical physics. In earlier times, one proposed not a model but a theory. By talking today of a model rather than a theory, one acknowledges that the way one studies the phenomenon is not unique; it could also be studied other ways. One's model need not claim to be unique or final. It merits consideration if it provides an insight that isn't better provided by some other model." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"Sometimes mathematical advances happen by just looking at something in a slightly different way, which doesn’t mean building something new or going somewhere different, it just means changing your perspective and opening up huge new possibilities as a result. This particular insight leads to calculus and hence the understanding of anything curved, anything in motion, anything fluid or continuously changing." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Artificial intelligence is defined as the branch of science and technology that is concerned with the study of software and hardware to provide machines the ability to learn insights from data and the environment, and the ability to adapt in changing situations with high precision, accuracy and speed." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"The goal of data science is to improve decision making by basing decisions on insights extracted from large data sets. As a field of activity, data science encompasses a set of principles, problem definitions, algorithms, and processes for extracting nonobvious and useful patterns from large data sets. It is closely related to the fields of data mining and machine learning, but it is broader in scope." (John D Kelleher & Brendan Tierney, "Data Science", 2018)

"The patterns that we extract using data science are useful only if they give us insight into the problem that enables us to do something to help solve the problem." (John D Kelleher & Brendan Tierney, "Data Science", 2018)

On Insight (2000-2009)

"Mathematics has given us dazzling insights into the power of exponential growth and how the same patterns recur in numbers, regardless of the phenomena being observed." (Richar Koch, "The Power Laws", 2000)

"Somehow mathematicians seem to long for more than just results from their proofs; they want insight." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"When we have difficulties solving a problem, insight into its solution can come about by restructuring the problem." (S Ian Robertson, "Problem Solving", 2001)

"Natural frequencies facilitate inferences made on the basis of numerical information. The representation does part of the reasoning, taking care of the multiplication the mind would have to perform if given probabilities. In this sense, insight can come from outside the mind." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Why does representing information in terms of natural frequencies rather than probabilities or percentages foster insight? For two reasons. First, computational simplicity: The representation does part of the computation. And second, evolutionary and developmental primacy: Our minds are adapted to natural frequencies." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Becoming a real researcher has been the ultimate humbling experience for me. Nature is the examiner from hell; if you find new things at all, you always find them the hard way, with sweat and tears. Only then do you notice that there was a really easy way to find them. But this insight rarely arrives before you have been utterly humiliated and reduced to despair." (João Magueijo, "Faster Than The Speed Of Light: The Story of a Scientific Speculation", 2003)

"Proofs should be as short, transparent, elegant, and insightful as possible." (Burkard Polster,"Q.E.D.: Beauty in Mathematical Proof", 2004)

"Central tendency is the formal expression for the notion of where data is centered, best understood by most readers as 'average'. There is no one way of measuring where data are centered, and different measures provide different insights." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Insight is not a lightbulb that goes off inside our heads. It is a flickering candle that can easily be snuffed out." (Malcolm Gladwell, "Blink: The Power of Thinking Without Thinking", 2005)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Insight (1970-1979)

"Insight doesn't happen often in the click of the moment, like a lucky snapshot, but it comes in its own time and more slowly and from nowhere but within." (Eudora Welty, "One Time, One Place", 1971)

"Early scientific thinking was holistic, but speculative - the modern scientific temper reacted by being empirical, but atomistic. Neither is free from error, the former because it replaces factual inquiry with faith and insight, and the latter because it sacrifices coherence at the altar of facticity. We witness today another shift in ways of thinking: the shift toward rigorous but holistic theories. This means thinking in terms of facts and events in the context of wholes, forming integrated sets with their own properties and relationships." (Ervin László, "Introduction to Systems Philosophy", 1972)

"Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework." (Melvin Schwartz, "Principles of Electrodynamics", 1972)

"Science cannot develop unless it is pursued for the sake of pure knowledge and insight. It will not survive unless it is used intensely and wisely for the betterment of humanity and not as an instrument of domination by one group over another." (Victor F Weisskopf, "Physics in the Twentieth Century: Selected Essays", 1972)

"Though we can say that mathematics is not art, some mathematicians think of themselves as artists of pure form. It seems clear, however, that their elegant and near aesthetic forms fail as art, because they are secondary visual ideas, the product of an intellectual set of restraints, rather than the cause of a felt insight realized in and through visual form." (Robert E Mueller, "Idols of Computer Art", 1972)

"[...] it is rather more difficult to recapture directness and simplicity than to advance in the direction of ever more sophistication and complexity. Any third-rate engineer or researcher can increase complexity; but it takes a certain flair of real insight to make things simple again." (Ernst F Schumacher, "Small Is Beautiful", 1973)

"The advantage of this way of proceeding is evident: insights and skills obtained on the model-side can be - certain transference criteria satisfied - transferred to the original, [in this way] the model-builder obtains a new knowledge about the modeled original […]" (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)

"All of us must cross the line between ignorance and insight many times before we truly understand." (David Hawkins, "The Informed Vision: Essays on Learning and Human Nature", 1974)

"The history of science is full of revolutionary advances that required small insights that anyone might have had, but that, in fact, only one person did." (Isaac Asimov, "The Three Numbers", Ellery Queen's Mystery Magazine, 1974)

"A metaphor is a word used in an unfamiliar context to give us a new insight; a good metaphor moves us to see our ordinary world in an extraordinary way." (Sallie McFague, "Speaking in Parables", 1975)

"To gauge the understanding and insight that metaphysics provides is to ask whether, in the final analysis, it helps us to cope with our world and harmonize our existence with nature, humanity, and ourselves, and leads to greater freedom and self-realization. Metaphysics is only the beginning. The end is human progress." (Rudolph Rummel, "Understanding Conflict and War: The dynamic psychological field", 1975)

"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics." (Martin Gardner, "Aha! Insight", 1978)

"The word ‘induction’ has two essentially different meanings. Scientific induction is a process by which scientists make observations of particular cases, such as noticing that some crows are black, then leap to the universal conclusion that all crows are black. The conclusion is never certain. There is always the possibility that at least one unobserved crow is not black." (Martin Gardner, "Aha! Insight", 1978)

"Every discovery, every enlargement of the understanding, begins as an imaginative preconception of what the truth might be. The imaginative preconception - a ‘hypothesis’ - arises by a process as easy or as difficult to understand as any other creative act of mind; it is a brainwave, an inspired guess, a product of a blaze of insight. It comes anyway from within and cannot be achieved by the exercise of any known calculus of discovery. " (Sir Peter B Medawar, "Advice to a Young Scientist", 1979)

"The truth is not in nature waiting to declare itself, and we cannot know a priori which observations are relevant and which are not; every discovery, every enlargement of the understanding begins as an imaginative preconception of what the truth might be. This imaginative preconception - a 'hypothesis' - arises by a process as easy or as difficult to understand as any other creative act of mind; it is a brainwave, an inspired guess, the product of a blaze of insight. It comes, anyway, from within and cannot be arrived at by the exercise of any known calculus of discovery." (Sir Peter B Medawar, "Advice to a Young Scientist", 1979)

On Insight (1950-1959)

"Insight is not the same as scientific deduction, but even at that it may be more reliable than statistics." (Anthony Standen, "Science Is a Sacred Cow", 1950)

"Another approach for a given problem is to try to restate it in just as many different forms as you can. Change the words. Change the viewpoint. Look at it from every possible angle. After you’ve done that, you can try to look at it from several angles at the same time and perhaps you can get an insight into the real basic issues of the problem, so that you can correlate the important factors and come out with the solution." (Claude E Shannon, "Creative Thinking", 1952)

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practice to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuitions of man." (Morris Kline, "Mathematics in Western Culture", 1953)

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas." (Saunders MacLane, "Of Course and Courses"The American Mathematical Monthly, Vol 61, No 3, 1954)

"By far the most important consequence of the conceptual revolution brought about in physics by relativity and quantum theory lies not in such details as that meter sticks shorten when they move or that simultaneous position and momentum have no meaning, but in the insight that we had not been using our minds properly and that it is important to find out how to do so." (Percy W Bridgman, "Quo Vadis", 1958)

On Insight (1900-1949)

"Not only virtue, but also insight, not only sanctity but also wisdom, are the duties and tasks of mankind." (Otto Weininger, "Sex and Character", 1903)

"The introduction into geometrical work of conceptions such as the infinite, the imaginary, and the relations of hyperspace, none of which can be directly imagined, has a psychological significance well worthy of examination. It gives a deep insight into the resources and working of the human mind. We arrive at the borderland of mathematics and psychology." (John Theodore Merz, "History of European Thought in the Nineteenth Century", 1903)

"The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect, - the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae." (John T Merz, "History of European Thought in the 19th Century" Vol. 1, 1904)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor,"A Theory of Motives, Ideals and Values in Education" 1907)

"In fact, the opposition of instinct and reason is mainly illusory. Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realms, it is insight that first arrives at what is new." (Bertrand Russell, "Our Knowledge of the External World", 1914)

"No student ought to complete a course in mathematics without the feeling that there must be something in it, without catching a glimpse, however fleeting, of its possibilities, without at least a few moments of pleasure in achievement and insight." (Helen A Merrill, 'Why Students Fail in Mathematics", The Mathematics Teacher, 1918) 

"Mathematics is the foe of nature and of insight into nature. A human being who possesses mathematical knowledge, the lingo of formulas, and with it approaches nature, must be like a woman who, with soaped hands, wants to grab a fish: surely, she will not seize it. Yet it is an unprecedented arrogance of mathematicians to pose in front of the world and of nature, and to say that they only had eyes for things." (Alfred Döblin, "Die abscheuliche Relativitätslehre" ["The Abominable Relativity Theory"], Berliner Tageblatt, 1923)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; "Die logischen Grundlagen der Mathematik." Mathematische Annalen 88 (1), 1923)

"The primary purposes of the teaching of mathematics should be to develop those powers of understanding und analyzing relations of quantity and of space which are necessary to an insight into and a control over our environment and to an appreciation of the progress of civilization its various aspects, and to develop those habits of thought and of action which will make those powers in effective in the life of the individual." (J W Young [Ed] The Reorganization of Mathematics in Secondary Education, 1927)

"It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper, "The Open Society and Its Enemies", 1945)

"The theory [of categories] also emphasizes that, whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these new objects as an integral part of their definition. The pursuit of this program entails a simultaneous consideration of objects and their mappings (in our terminology, this means the consideration not of individual objects but of categories). This emphasis on the specification of the type of mappings employed gives more insight onto the degree of invariance of the various concepts involved." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

On Problem Solving XIX: Insight

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner, The New Yorker, 1963)

"[...] two related deficiencies have prevented real progress in understanding insight and its role in problem solving. First, we do not yet have a system of classifying problems into those in which insight occurs versus those in which it does not. However, only if we can isolate problems in which insight occurs will we be able to set on a firm base our theories of the mechanisms underlying insight. Second, formulation of such a taxonomic system requires that we agree on a definition of insight." (Robert W Weisberg, "Prolegomena to theories of insight in problem solving: a taxonomy of problems", 1995)

"Therefore, although the notion of insight as a distinct process has a long history in the psychological study of problem solving, it might be useful at this point to refrain from using analytic and insight as theoretical terms applied a priori to problems." (Jason M Chein et al, "Working memory and insight in the nine-dot problem", Memory & Cognition 38, 2010)

"A feeling of insight is a kind of 'Aha!' characterized by suddenness and obviousness (and often relief!) - like a revelation. You are relatively confident that your solution is correct without having to check it. In contrast, you experienced no Aha! if the solution occurs to you slowly and stepwise. As an example, imagine a light bulb that is switched on all at once in contrast to slowly dimming it up." (Amory H Danek et al, "Working wonders? Investigating insight with magic tricks", Cognition 130, 2014)

"If insight involves an abrupt change from a state of not knowing how to solve the problem to a state of knowing [...], with no conscious awareness of what caused the change, the person experiencing the insight can still be expected to be able to report that the change occurred." (Edward Bowden & Kristin Grunewald, "Whose insight is it anyway?", [in "Insight], 2018)

"In the problem-solving literature, the term 'insight' is used to designate the clear and sudden understanding of how to solve a problem. Insight is thought to arise as the result of a solver breaking free of unwarranted assumptions, or forming novel, task-related connections between existing concepts or skills. Although it would be preferable to have a single, clear, universally accepted definition of insight, this may not be possible – and almost certainly will not be possible until we start to assess on a regular basis the single feature that seems to be widely acknowledged as distinguishing insight solutions from other solutions: the 'Aha!'" (Edward Bowden & Kristin Grunewald, "Whose insight is it anyway?", [in "Insight], 2018)

"If insight involves an abrupt change from a state of not knowing how to solve the problem to a state of knowing how to solve the problem (or, in some cases, knowing the solution), with no conscious awareness of what caused the change, the person experiencing the insight can still be expected to be able to report that the change occurred." (Edward Bowden & Kristin Grunewald, "Whose insight is it anyway?", [in "Insight], 2018)

Previous <<||>> Next

On Continuity XI (Thought II)

"The function of man’s highest faculty, his reason, consists precisely of the continuous limitation of infinity, the breaking up of infinity into convenient, easily digestible portions - differentials. This is precisely what lends my field, mathematics, its divine beauty." (Yevgeny Zamiatin, "We", 1924)

"Rationality consists [of] the continuous adaptation of our language to our continually expanding world, and metaphor is one of the chief means by which this is accomplished." (Mary B Hesse, "Models and Analogies in Science", 1966)

"Truth is a totality, the sum of many overlapping partial images. History, on the other hand, sacrifices totality in the interest of continuity." (Edmund Leach, "Brain-Twister", 1967)

"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent 'forward' movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission." (Carl Eckart, "Our Modern Idol: Mathematical Science", 1984)

"To form a mental picture of the event, the knowledge developer attempts to integrate his or her perception of the situation with the expert’s perception. That mental picture is then recorded. What happens is a continuous shuttle process; the knowledge developer mentally moves back and forth from the initial impression of the event to the later evaluation of the event. What is finally recorded is the evaluation made during this retrospective period. Because a time lapse can make details of a situation less clear, the information is not always valid." (Elias M Awad, "Knowledge Management", 2003)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Diagrams (1975-1999)

"Pencil and paper for construction of distributions, scatter diagrams, and run-charts to compare small groups and to detect trends are more efficient methods of estimation than statistical inference that depends on variances and standard errors, as the simple techniques preserve the information in the original data." (W Edwards Deming, "On Probability as Basis for Action", American Statistician, Volume 29, Number 4, November 1975)

"The formalist makes a distinction between geometry as a deductive structure and geometry as a descriptive science. Only the first is regarded as mathematical. The use of pictures or diagrams, or even mental imagery, all are non- mathematical. In principle, they should be unnecessary. Consequently. he regards them as inappropriate in a mathematics text, perhaps even in a mathematics class." (Philip J Davis & Reuben Hersh, "The Mathematical Experience", 1981)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"[The diagram] is only an heuristic to prompt certain trains of inference; [...] it is dispensable as a proof-theoretic device; indeed, [...] it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array." (Neil Tennant, "The withering away of formal semantics", Mind and Language Vol. 1 (4), 1986)

"We distinguish diagrammatic from sentential paper-and-pencil representations of information by developing alternative models of information-processing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, like the propositions in a text. Diagrammatic representations are indexed by location in a plane. Diagrammatic representations also typically display information that is only implicit in sentential representations and that therefore has to be computed, sometimes at great cost, to make it explicit for use. We then contrast the computational efficiency of these representations for solving several. illustrative problems in mathematics and physics." (Herbert A Simon, "Why a diagram is (sometimes) worth ten thousand words", 1987) 

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

"The value of diagram techniques even at this rudimentary level should be clear by now: it is easier to visualize where simplifications may be found in a complicated network by searching for a reducible linkage than by examining a complicated algebraic expression."(Geoffrey E Stedman, "Diagram Techniques in Group Theory", 1990)

"Diagrams are physical situations. They must be, since we can see them. As such, they obey their own set of constraints. […] By choosing a representational scheme appropriately, so that the constraints on the diagrams have a good match with the constraints on the described situation, the diagram can generate a lot of information that the user never need infer. Rather, the user can simply read off facts from the diagram as needed." (Jon Barwise & John Etchemendy, "Visual information and valid reasoning", [in "Visualization in Teaching and Learning Mathematics"], 1991)

"It has been said that the art of geometry is to reason well from false diagrams." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. We will use diagrams to depict mental models for a variety of concepts, and it is important to keep in mind that any diagram, or even a non-diagrammatic representation that represents the same essential relations would be equally effective." (Lyn D English & Graeme S Halford," "Mathematics Education: Models and Processes", 1995) 

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 1995)

"Given particular tasks of reasoning, different types of diagrams show different degrees of suitedness. For example, Euler diagrams are superior in handling certain problems concerning inclusion and membership among classes and individuals, but they cannot be generally applied to such problems without special provisos. Diagrams make many proofs in geometry shorter and more intuitive, while they take certain precautions of the reasoner's to be used validly. […] Mathematicians experience that coming up with the 'right' sorts of diagrams is more than half-way to the solution of most complicated problems." (Atsushi Shimojima, "Operational Constraints in Diagrammatic Reasoning" , [in "Logical Reasoning with Diagrams"], 1996)

"Making a good choice of representational conventions is always important in solving a problem, but especially true of charts. This sensitivity of type of chart to the particularities of the task at hand makes a very general logic of charts useless." (Jon Barwise & Eric Hammer, "Diagrams and the Concept of Logical System", [in "Logical Reasoning with Diagrams"], 1996)

"Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic.  In short, pictures can prove theorems." (James R Brown, "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures", 1999)

Mental Models XLIV (Spiritual & Religious Writings)

"When a soul has advanced so far on the spiritual road as to be lost to all the natural methods of communing with God; when it seeks Him no longer by meditation, images, impressions, nor by any other created ways, or representations of sense, but only by rising above them all, in the joyful communion with Him by faith and love, then it may be said to have found God of a truth, because it has truly lost itself as to all that is not God, and also as to its own self." (John of the Cross," Spiritual Canticle of The Soul and The Bridegroom", 1578) 

"Every man, as the Stoics used to say, is first and principally recommended to his own care; and every man is certainly, in every respect, fitter and abler to take care of himself than of any other person. Every man feels his own pleasures and his own pains more sensibly than those of other people. The former are the original sensations; the latter the reflected or sympathetic images of those sensations. The former may be said to be the substance; the latter the shadow. (Adam Smith, "The Theory of Moral Sentiments", 1759)

"Even Truth is of many types, like – Imaginative Truth, Practical Truth and Philosophical Truth. That which is in three times, that is called Truth and God itself is the first and the last truth.  But in practical life, truth takes many forms and as the practical truth I understand the sensible world's hard comprehensible truth. The one attained by the research of intellect, I call philosophical truth and imaginative that which illustrates through the subtle pictures of the mind." (Laxmi Prasad Devkota, "Art and Life", 1945)

"The eye projects and focuses the inner image (idea) onto the physical world in the same manner that a motion picture camera transfers an image onto a screen. The mouth creates words. The ears create sound. The difficulty in understanding this principle is due to the fact that we’ve taken it for granted that the image and sound already exist for the senses to interpret. Actually the senses are the channels of creation by which idea is projected into material expression." (Jane Roberts,"The Seth Material", 1970)

"[…] the intellect is incapable of knowing the supreme Truth; it can only range about seeking Truth, and catching fragmentary representations of it, not the thing itself, and trying to piece them together." (Ghose Aurobindo, "The Riddle of the World", 1973)

"Meditation is the emptying of the mind of the known. It cannot be done by thought or by the hidden prompting of thought, nor by desire in the form of prayer, nor through the self-effacing hypnotism of words, images, hopes, and vanities. All these have to come to an end, easily, without effort and choice, in the flame of awareness." (Jiddu Krishnamurti, "Meditations", 1979) 

"It is astonishingly beautiful and interesting, how thought is absent when you have an insight. Thought cannot have an insight. It is only when the mind is not operating mechanically in the structure of thought that you have an insight. Having had an insight, thought draws a conclusion from that insight. And then thought acts and thought is mechanical. So I have to find out whether having an insight into myself, which means into the world, and not drawing a conclusion from it is possible. If I draw a conclusion, I act on an idea, on an image, on a symbol, which is the structure of thought, and so I am constantly preventing myself from having insight, from understanding things as they are." (Jiddu Krishnamurti," On Mind and Thought", 1993)

"To say that a thing is imaginary is not to dispose of it in the realm of mind, for the imagination, or the image making faculty, is a very important part of our mental functioning. An image formed by the imagination is a reality from the point of view of psychology; it is quite true that it has no physical existence, but are we going to limit reality to that which is material? We shall be far out of our reckoning if we do, for mental images are potent things, and although they do not actually exist on the physical plane, they influence it far more than most people suspect." (Dion Fortune," Spiritualism and Occultism", 2000)

On Analogy (1960-1969)

"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure." (André Weil, "De la Métaphysique aux Mathématiques", 1960)

"Analogy is even slipperier than logic." (Robert A Heinlein, "Stranger in a Strange Land", 1961)

“Science fiction is, very strictly and literally, analogous to science facts. It is a convenient analog system for thinking about new scientific, social, and economic ideas - and for re-examining old ideas.” (John W Campbell Jr., “Prologue to Analog”, 1962)

"When a science approaches the frontiers of its knowledge, it seeks refuge in allegory or in analogy." (Erwin Chargaff, "Essays on Nucleic Acids", 1963)

"Analogy serves to provoke certain types of questions which can, on investigation, lead to the recognition of more comprehensive ranges of order in the archaeological data." (Lewis R Binford, "Smudge Pits and Hide Smoking: The Use of Analogy in Archaeological Reasoning, American Antiquity Vol. 32 (1), 1967)

"Everything lives by movement, everything is maintained by equilibrium, and harmony results from the analogy of contraries; this law is the form of forms." (Eliphas Levi, "Transcendental Magic", 1968)

"It is probably no exaggeration to say that all of theoretical physics proceeds by analogy." (Jeremy Bernstein, "Elementary Particles and Their Currents", 1968)

"More than a burial ground for unacceptable ideas and wishes, the unconscious is the spawning ground of intuition and insight, the source of humor, of poetic imagery, and of scientific analogy." (Judith Groch, "The Right to Create", 1969)

Theoni Pappas - Collected Quotes

"To experience the joy of mathematics is to realize mathematics is not some isolated subject that has little relationship to the things around us other than to frustrate us with unbalanced check books and complicated computations. Few grasp the true nature of mathematics - so entwined in our environment and in our lives." (Theoni Pappas, "The Joy of Mathematics: Discovering Mathematics All Around You", 1986)

"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"Perhaps mathematicians' fascination with pi over the centuries can be likened to the drive that motivates mountain climbers to attempt an ascent." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"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)

"Statistics is a very powerful and persuasive mathematical tool. People put a lot of faith in printed numbers. It seems when a situation is described by assigning it a numerical value, the validity of the report increases in the mind of the viewer. It is the statistician's obligation to be aware that data in the eyes of the uninformed or poor data in the eyes of the naive viewer can be as deceptive as any falsehoods." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"It is not surprising to find many mathematical ideas interconnected or linked. The expansion of mathematics depends on previously developed ideas. The formation of any mathematical system begins with some undefined terms and axioms (assumptions) and proceeds from there to definitions, theorems, more axioms and so on. But history points out this is not necessarily the route that creativity" (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"Throughout the evolution of mathematics, problems have acted as catalysts in the discovery and development of mathematical ideas. In fact, the history of mathematics can probably be traced by studying the problems that mathematicians have tried to solve over the centuries. It is almost disheartening when an old problem is finally solved, for it will no longer be around to challenge and stimulate mathematical thought." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"When looking at the end result of any statistical analysis, one must be very cautious not to over interpret the data. Care must be taken to know the size of the sample, and to be certain the method forg athering information is consistent with other samples gathered. […] No one should ever base conclusions without knowing the size of the sample and how random a sample it was. But all too often such data is not mentioned when the statistics are given - perhaps it is overlooked or even intentionally omitted." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"Just as mathematical objects do not precisely describe things in our world, so traditional logic cannot be perfectly applied to the real world and real-world situations." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)

"Objects in nature have provided and do provide models for stimulating mathematical discoveries. Nature has a way of achieving an equilibrium and an exquisite balance in its creations. The key to understanding the workings of nature is with mathematics and the sciences. [.] Mathematical tools provide a means by which we try to understand, explain, and copy natural phenomena. One discovery leads to the next." (Theoni Pappas, "The Magic of Mathematics: Discovering the spell of mathematics", 1994)

On Proofs (1990-1999)

"[...] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.“ (Ivars Peterson, „Islands of Truth: A Mathematical Mystery Cruise“, 1990)

"A distinctive feature of mathematics, that feature in virtue of which it stands as a paradigmatically rational discipline, is that assertions are not accepted without proof. […] By proof is meant a deductively valid, rationally compelling argument which shows why this must be so, given what it is to be a triangle. But arguments always have premises so that if there are to be any proofs there must also be starting points, premises which are agreed to be necessarily true, self-evident, neither capable of, nor standing in need of, further justification. The conception of mathematics as a discipline in which proofs are required must therefore also be a conception of a discipline in which a systematic and hierarchical order is imposed on its various branches. Some propositions appear as first principles, accepted without proof, and others are ordered on the basis of how directly they can be proved from these first principle. Basic theorems, once proved, are then used to prove further results, and so on. Thus there is a sense in which, so long as mathematicians demand and provide proofs, they must necessarily organize their discipline along lines approximating to the pattern to be found in Euclid's Elements." (Mary Tiles,"Mathematics and the Image of Reason" , 1991) 

"Notice also that scientists generally avoid the use of the word proof. Evidence can support a hypothesis or a theory, but it cannot prove a theory to be true. It is always possible that in the future a new idea will provide a better explanation of the evidence." (James E McLaren, “Heath Biology”, 1991)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 1994)

“Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode.” (William P Thurston, “On Proof and Progress in Mathematics”, 1994)

"Mathematics is about theorems: how to find them; how to prove them; how to generalize them; how to use them; how to understand them. […] But great theorems do not stand in isolation; they lead to great theories. […] And great theories in mathematics are like great poems, great paintings, or great literature: it takes time for them to mature and be recognized as being 'great'." (John L Casti, "Five Golden Rules", 1995)

"The ingredient that knits this landscape together is proof. Proof determines the route from one fact to another. To professional mathematicians, no statement is considered valid unless it is proved beyond any possibility of logical error. But there are limits to what can be proved, and how it can be proved. A great deal of work in philosophy and the foundations of mathematics has established that you can't prove everything, because you have to start somewhere; and even when you've decided where to start, some statements may be neither provable nor disprovable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

“What's so awful about using intuition or using inductive arguments? […] without them we would have virtually no mathematics at all; for, until the last few centuries, mathematics was advanced almost solely by intuition, inductive observation, and arguments designed to compel belief, not by laboured proofs, and certainly not through proofs of the ghastliness required by today's academic journals” (Jon MacKeman, “What's the point of proof?”, Mathematics Teaching 155, 1996)

"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 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)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"In practice, proofs are simply whatever it takes to convince colleagues that a mathematical idea is true." (Claudia Henrion, "Women in Mathematics", 1997)

"Cleaning up old proofs is an important part of the mathematical enterprise that often yields new insights that can be used to solve new problems and build more beautiful and encompassing theories." (Bruce Schecter, "My Brain is Open", 1998)

See also:
Proofs I, II, III, IV, V, VI, VIII, IX
Theorems I, II, III, IV, V, VI, VII, VIII, IX, X

On Intuition (1960-1969)

"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one’s craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful." (Jerome S Bruner, "On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)

"The first [principle], is that a mathematical theory can only he developed axiomatically in a fruitful way when the student has already acquired some familiarity with the corresponding material - a familiarity gained by working long enough with it on a kind of experimental, or semiexperimental basis, i.e. with constant appeal to intuition. The other principle [...]  is that when logical inference is introduced in some mathematical question, it should always he presented with absolute honesty - that is, without trying to hide gaps or flaws in the argument; any other way, in my opinion, is worse than giving no proof at all." (Jean Dieudonné, "Thinking in School Mathematics", 1961)

"The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error." (R Willem van Bemmelen, "The Scientific Character of Geology", The Journal of Geology Vol 69 (4), 1961)

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it." (George Polya, "Mathematical Discovery", 1962)

"Intuition is the collection of odds and ends where we place all the intellectual mechanisms which we do not know how to analyze or even name with precision, or which we are not interested in analyzing or naming." (Mario Bunge, "Intuition and Science", 1962)

"Living mathematics rests on the fluctuation between the antithesis powers of intuition and logic, the individuality of 'grounded' problems and the generality of far-reaching abstractions. We ourselves must prevent the development being forced to only one pole of the life-giving antithesis." (Richard Courant, 1962)

"The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links." (Arthur Koestler, "The Act of Creation", 1964)

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practices to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuition of humans. Here and there an intuition is scooped out and replaced by a firmly built pillar of thought; however, this pillar is based on some deeper, perhaps less clearly defined, intuition. Though the process of replacing intuitions with precise thoughts does not change the nature of the ground on which mathematics ultimately rests, it does add strength and height to the structure." (Morris Kline, "Mathematics in Western Culture ", 1964)

"Mathematical intuition is more often conservative than revolutionary, more often hampering than liberating." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American Vol,. 211 (3), 1964)

 "People may come along and argue philosophically that they like one better than another; but we have learned from much experience that all philosophical intuitions about what nature is going to do fail." (Richard Feynman, "The Character of Physical Law", 1965)

"The most natural way to give an independence proof is to establish a model with the required properties. This is not the only way to proceed since one can attempt to deal directly and analyze the structure of proofs. However, such an approach to set theoretic questions is unnatural since all our intuition come from our belief in the natural, almost physical model of the mathematical universe." (Paul J Cohen, "Set Theory and the Continuum Hypothesis", 1966)

"Scientific research was much like prospecting: you went out and you hunted, armed with your maps and instruments, but in the end your preparations did not matter, or even your intuition. You needed your luck, and whatever benefits accrued to the diligent, through sheer, grinding hard work." (Michael Crichton, "The Andromeda Strain", 1969)

On Truth (1920-1929)

"The terrible thing about the quest for truth is that you find it." (Rémy de Gourmont, "Philosophic Nights in Paris", 1920)

"It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability." (John M Keynes, "A Treatise on Probability", 1921)

"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)

“We all know that Art is not truth. Art is a lie that makes us realize truth.” (Pablo Picasso, “The Arts”, 1923)

“Science does not aim at establishing immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved.” (Bertrand Russell, “The ABC of Relativity”, 1925)

"Progress in truth - truth of science and truth of religion - is mainly a progress in the framing of concepts, in discarding artificial abstractions or partial metaphors, and in evolving notions which strike more deeply into the root of reality." (Alfred N Whitehead, "Religion in the Making", 1926)

"The scientist is a lover of truth for the very love of truth itself, wherever it may lead." (Luther Burbank, "Why I Am An Infidel", 1926)

“If our so-called facts are changing shadows, they are shadows cast by the light of constant truth.” (Sir Arthur S Eddington, “Science and the Unseen World”, 1929) 

“Try to be conspicuously accurate in everything, pictures as well as text. Truth is not only stranger than fiction, it is more interesting.” (William R Hearst, “Letter of Instruction to Hearst Publishers”, 1929)

On Theorems (1950-1969)

“On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory.” (Kurt Gödel, 1951)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, “Mathematics in Western Culture”, 1953)

"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem." (Milton Friedman, "Essays in Positive Economics", 1953)

“You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing” (George Polya, “Mathematics and plausible reasoning” Vol. 1, 1954)

“Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas.” (Saunders MacLane, “Of Course and Courses”, The American Mathematical Monthly, Vol. 61, No. 3, March, 1954)

”Mathematics is a creation of the mind. To begin with, there is a collection of things, which exist only in the mind, assumed to be distinguishable from one another; and there is a collection of statements about these things, which are taken for granted. Starting with the assumed statements concerning these invented or imagined things, the mathematician discovers other statements, called theorems, and proves them as necessary consequences. This, in brief, is the pattern of mathematics. The mathematician is an artist whose medium is the mind and whose creations are ideas.” (Hubert Stanley Wall, “Creative Mathematics”, 1963)

“So the first thing we have to accept is that even in mathematics you can start in different places. If all these various theorems are interconnected by reasoning there is no real way to say ‘These are the most fundamental axioms’, because if you were told something different instead you could also run the reasoning the other way. It is like a bridge with lots of members, and it is over-connected; if pieces have dropped out you can reconnect it another way.” (Richard Feynman, “The Character of Physical Law”, 1965)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

“A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective.” (George Spencer-Brown, “Laws of Form”, 1969)

See also:
Theorems I, II, III, IV, V, VII, VIII, IX, X
Proofs I, II, III, IV, V,. VI, VII, VIII, IX

On Proofs (1975 - 1989)

“The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed.” (Michael Dummett, “The philosophical basis of intuitionistic logic”, 1975)

“No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions.”  (Paul K Feyerabend, “Against Method: Outline of an Anarchistic Theory of Knowledge”, 1975)

“There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it.” (Imre Lakatos, “Proofs and Refutations”, 1976)

“On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists.” (Imre Lakatos, “Mathematics, Science and Epistemology” Vol. 2, 1978)

"Mathematics is a way of finding out, step by step, facts which are inherent in the statement of the problem but which are not immediately obvious. Usually, in applying mathematics one must first hit on the facts and then verify them by proof. Here we come upon a knotty problem, for the proofs which satisfied mathematicians of an earlier day do not satisfy modem mathematicians." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

"Mathematicians start out with certain assumptions and definitions, and then by means of mathematical arguments or proofs they are able to show that certain statements or theorems are true." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

“When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively.” (Morris Kline, “Mathematics: The Loss of Certainty”, 1980)

“No proof is final. New counterexamples undermine old proofs. The proofs are then revised and mistakenly considered proven for all time. But history tells us that this merely means that the time has not yet come for a critical examination of the proof” (Morris Kline, “Mathematics: The Loss of Certainty”, 1980)

“If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book […] it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.” (George Pólya, “Mathematical Discovery: on Understanding, Learning, and Teaching Problem Solving”, 1981)

“A proof transmits conviction from its premises down to its conclusion, so it must start with premises […] for which there already is conviction; otherwise, there will be nothing to transmit.” (Robert Nozick, Philosophical Explanations, 1981)

“We often hear that mathematics consists mainly in ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.” (Gian-Carlo Rota, “Complicating Mathematics” in “Discrete Thoughts”, 1981)

“People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it. Yet the conception of a new argument is hardly likely actually to proceed in this way. There is a globality and seemingly vague conceptual content that is necessary in the construction of a mathematical argument; and this can bear little relation to the time that it would seem to take in order fully to appreciate a serially presented proof” (Roger Penrose, “The Emperor’s New Mind”, 1989)

See also:

Proofs I, II, III, IV, V, VI, VIII, IX
Theorems I, II, III, IV, V, VI, VII, VIII, IX, X