Knowledge Representation: On Decision-Making (Quotes)

 "It [probability] is the very guide of life, and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of probabilities." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Great decisions in the realms of thought and momentous discoveries and solutions of problems are only possible to an individual working in solitude." (Sigmund Freud, "Group Psychology and the Analysis of the Ego", 1921)

“The fundamental gospel of statistics is to push back the domain of ignorance, prejudice, rule-of-thumb, arbitrary or premature decisions, tradition, and dogmatism and to increase the domain in which decisions are made and principles are formulated on the basis of analyzed quantitative facts.” (Robert W Burgess, “The Whole Duty of the Statistical Forecaster”, Journal of the American Statistical Association , Vol. 32, No. 200, 1937)

"Years ago a statistician might have claimed that statistics deals with the processing of data [...]today’s statistician will be more likely to say that statistics is concerned with decision making in the face of uncertainty." (Herman Chernoff & Lincoln E Moses, "Elementary Decision Theory", 1959) 

"[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not." (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)E

"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Each of us uses models constantly. Every person in his private life and in his business life instinctively uses models for decision making. The mental image of the world around you which you carry in your head is a model. […] A mental image is a model. All our decisions are taken on the basis of models." (Jay W Forrester, "Counter-Intuitive Behaviour of Social Systems", Technological Review 73, 1971)

"A somewhat deeper analysis suggests that a decision is a deliberate act of selection, by the mind, of an alternative from a set of competing alternatives in the hope, expectation, or belief that the actions envisioned in carrying out the selected alternative will accomplish certain goals. Decision is the selection of a mental state: it is a commitment to certain actions or inactions. Other people may observe our actions but they do not directly observe our decisions." (Peter C Fishburn, "Personalistic Decision Theory: Exposition and Critique", 1972)

"The human condition can almost be summed up in the observation that, whereas all experiences are of the past, all decisions are about the future. It is the great task of human knowledge to bridge this gap and to find those patterns in the past which can be projected into the future as realistic images." (Kenneth E Boulding, [foreword] 1972)

"Decision-making problems (hypothesis testing) involve situations where it is desired to make a choice among various alternative decisions (hypotheses). Such problems can be viewed as generalized state estimation problems where the definition of state has simply been expanded." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"Statistics is a body of methods and theory applied to numerical evidence in making decisions in the face of uncertainty." (Lawrence Lapin, "Statistics for Modern Business Decisions", 1973)

"The science of statistics may be described as exploring, analyzing and summarizing data; designing or choosing appropriate ways of collecting data and extracting information from them; and communicating that information. Statistics also involves constructing and testing models for describing chance phenomena. These models can be used as a basis for making inferences and drawing conclusions and, finally, perhaps for making decisions." (Fergus Daly et al, "Elements of Statistics", 1995)

"The major fault in this process - and thus, in the way we were making decisions - is that it lacks an organizing framework. In pursuing a variety of goals and objectives, in whatever situation we manage, we often fail to see that some of them are in conflict and that the achievement of one might come at the expense of achieving another. In weighing up the actions we might take to reach our goals and objectives, we have no way to account for nature's complexity and only rarely factor it in." (Allan Savory & Jody Butterfield, "Holistic Management: A new framework for decision making", 1988)

"A holistic perspective is essential in management. If we base management decisions on any other perspective, we are likely to experience results different from those intended because only the whole is reality." (Allan Savory & Jody Butterfield, "Holistic Management: A new framework for decision making", 1988)

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Just as dynamics arise from feedback, so too all learning depends on feedback. We make decisions that alter the real world; we gather information feedback about the real world, and using the new information we revise our understanding of the world and the decisions we make to bring our perception of the state of the system closer to our goals." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

"No plea about inadequacy of our understanding of the decision-making processes can excuse us from estimating decision making criteria. To omit a decision point is to deny its presence - a mistake of far greater magnitude than any errors in our best estimate of the process." (Jay W Forrester, "Perspectives on the modelling process", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Diversity and independence are important because the best collective decisions are the product of disagreement and contest, not consensus or compromise." (James Surowiecki, "The Wisdom of Crowds", 2005)

"The fact that cognitive diversity matters does not mean that if you assemble a group of diverse but thoroughly uninformed people, their collective wisdom will be smarter than an expert's. But if you can assemble a diverse group of people who possess varying degrees of knowledge and insight, you're better off entrusting it with major decisions rather than leaving them in the hands of one or two people, no matter how smart those people are." (James Surowiecki, "The Wisdom of Crowds", 2005)

"[...] under the right circumstances, groups are remarkably intelligent, and are often smarter than the smartest people in them. Groups do not need to be dominated by exceptionally intelligent people in order to be smart. Even if most of the people within a group are not especially well-informed or rational, it can still reach a collectively wise decision." (James Surowiecki, "The Wisdom of Crowds", 2005)

"[...] a model is a tool for taking decisions and any decision taken is the result of a process of reasoning that takes place within the limits of the human mind. So, models have eventually to be understood in such a way that at least some layer of the process of simulation is comprehensible by the human mind. Otherwise, we may find ourselves acting on the basis of models that we don’t understand, or no model at all.” (Ugo Bardi, “The Limits to Growth Revisited”, 2011)

"In general, when building statistical models, we must not forget that the aim is to understand something about the real world. Or predict, choose an action, make a decision, summarize evidence, and so on, but always about the real world, not an abstract mathematical world: our models are not the reality - a point well made by George Box in his oft-cited remark that "all models are wrong, but some are useful". (David Hand, "Wonderful examples, but let's not close our eyes", Statistical Science 29, 2014)

"So everyone has and uses mental representations. What sets expert performers apart from everyone else is the quality and quantity of their mental representations. Through years of practice, they develop highly complex and sophisticated representations of the various situations they are likely to encounter in their fields - such as the vast number of arrangements of chess pieces that can appear during games. These representations allow them to make faster, more accurate decisions and respond more quickly and effectively in a given situation. This, more than anything else, explains the difference in performance between novices and experts." (Anders Ericsson & Robert Pool, "Peak: Secrets from  the  New  Science  of  Expertise" , 2016)

Knowledge Representation: On Knowledge (Quotes)

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular." (Proclus Lycaeus, cca 5th century)

"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)

"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)

"There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured […] Of these sciences the gate and key is mathematics […] He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world." (Roger Bacon, "Opus Majus", 1267)

"There are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth unless the mind discovers it by the path of experience." (Roger Bacon, "Opus Majus", 1267)

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"[…] the highest probability amounts not to certainty, without which there can be no true knowledge." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Mathematical knowledge adds vigor to the mind, frees it from prejudice, credulity, and superstition." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)

"It is your opinion, the ideas we perceive by our senses are not real things, but images, or copies of them. Our knowledge therefore is no farther real, than as our ideas are the true representations of those originals. But as these supposed originals are in themselves unknown, it is impossible to know how far our ideas resemble them; or whether they resemble them at all. We cannot therefore be sure we have any real knowledge." (George Berkeley, "Three Dialogues", 1713)

"There is nothing more pleasant for man than the certainty of knowledge; whoever has once tasted of it is repelled by everything in which he perceives nothing but uncertainty. This is why the mathematicians who always deal with certain knowledge have been repelled by philosophy and other things, and have found nothing more pleasant than to spend their time with lines and letters." (Christian Wolff, 1741)

"He that would make a real progress in knowledge must dedicate his age as well as first fruits - the latter growth as well as the first-fruits - at the altar of truth." (Bishop George Berkeley, "Siris", 1744)

"Those who have not imbibed the prejudices of philosophers, are easily convinced that natural knowledge is to be founded on experiment and observation." (Colin Maclaurin, "An Account of Sir Isaac Newton’s Philosophical Discoveries", 1748)

"Our knowledge springs from two fundamental sources of the mind; the first is the capacity of receiving representations (receptivity for impressions), the second is the power of knowing an object through these representations (spontaneity [in the production] of concepts)." (Immanuel Kant, "Critique of Pure Reason", 1781)

"Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts." (Immanuel Kant, "Critique of Pure Reason", 1781)

"Thoughts without content are empty, intuitions without concepts are blind. The understanding can intuit nothing, the senses can think nothing. Only through their unison can knowledge arise." (Immanuel Kant, "Critique of Pure Reason", 1781)

"The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"On completing one discovery we never fail to get an imperfect knowledge of others of which you could have no idea before […]" (Joseph Priestley, 1786)

"As there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves, by their own exertions, to distinction and eminence. […] Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"The power of Reason […] is unquestionably the most important by far of those which are comprehended under the general title of Intellectual. It is on the right use of this power that our success in the pursuit of both knowledge and of  happiness depends; and it is by the exclusive possession of it that man is distinguished, in the most essential respects, from the lower animals. It is, indeed, from their subserviency to its operations, that the other faculties […] derive their chief value." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"Each item of knowledge involves a second, a third step, and so on ad infinitum. If we pursue the life of the tree in its roots, or in its branches and twigs, one thing always follows from another. And the more vitally any concern of knowledge takes hold of us, the more we find ourselves driven to pursue it in its ramifications, both up and down." (Johann Wolfgang von Goethe, "Annals", 1807)

"Knowledge is only real and can only be set forth fully in the form of science, in the form of system." (G W Friedrich Hegel, "The Phenomenology of Mind", 1807)

"One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth - induction and analogy - are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities", 1814) 

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

"The highest knowledge can be nothing more than the shortest and clearest road to truth; all the rest is pretension, not performance, mere verbiage and grandiloquence, from which we can learn nothing." (Charles C Colton, "Lacon", 1820)

"We [...] are profiting not only by the knowledge, but also by the ignorance, not only by the discoveries, but also by the errors of our forefathers; for the march of science, like that of time, has been progressing in the darkness, no less than in the light." (Charles C Colton, "Lacon", 1820)

"The first steps in the path of discovery, and the first approximate measures, are those which add most to the existing knowledge of mankind." (Charles Babbage, "Reflections on the Decline of Science in England", 1830)

"Our knowledge of circumstances has increased, but our uncertainty, instead of having diminished, has only increased. The reason of this is, that we do not gain all our experience at once, but by degrees; so our determinations continue to be assailed incessantly by fresh experience; and the mind, if we may use the expression, must always be under arms." (Carl von Clausewitz, "On War", 1832)

"Truth in itself is rarely sufficient to make men act. Hence the step is always long from cognition to volition, from knowledge to ability. The most powerful springs of action in men lie in his emotions." (Carl von Clausewitz, "On War", 1832)

"Science and knowledge are subject, in their extension and increase, to laws quite opposite to those which regulate the material world. Unlike the forces of molecular attraction, which cease at sensible distances; or that of gravity, which decreases rapidly with the increasing distance from the point of its origin; the farther we advance from the origin of our knowledge, the larger it becomes, and the greater power it bestows upon its cultivators, to add new fields to its dominions." (Charles Babbage, "On the Economy of Machinery and Manufactures", 1832)

"The peculiar character of mathematical truth is that it is necessarily and inevitably true; and one of the most important lessons which we learn from our mathematical studies is a knowledge that there are such truths." (William Whewell, "Principles of English University Education", 1838)

"[…] in order that the facts obtained by observation and experiment may be capable of being used in furtherance of our exact and solid knowledge, they must be apprehended and analysed according to some Conception which, applied for this purpose, gives distinct and definite results, such as can be steadily taken hold of and reasoned from […]" (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon their History" Vol. 2, 1840)

"[…] there do exist among us doctrines of solid and acknowledged certainty, and truths of which the discovery has been received with universal applause. These constitute what we commonly term Sciences; and of these bodies of exact and enduring knowledge, we have within our reach so large and varied a collection, that we may examine them, and the history of their formation, with good prospect of deriving from the study such instruction as we seek." (William Whewell, "The Philosophy of the Inductive Sciences Founded upon Their History" Vol. 1, 1847)

"Remember that accumulated knowledge, like accumulated capital, increases at compound interest: but it differs from the accumulation of capital in this; that the increase of knowledge produces a more rapid rate of progress, whilst the accumulation of capital leads to a lower rate of interest. Capital thus checks its own accumulation: knowledge thus accelerates its own advance. Each generation, therefore, to deserve comparison with its predecessor, is bound to add much more largely to the common stock than that which it immediately succeeds." (Charles Babbage, "The Exposition of 1851: Or the Views of Industry, Science and Government of England", 1851)

"All knowledge is profitable; profitable in its ennobling effect on the character, in the pleasure it imparts in its acquisition, as well as in the power it gives over the operations of mind and of matter. All knowledge is useful; every part of this complex system of nature is connected with every other. Nothing is isolated. The discovery of to-day, which appears unconnected with any useful process, may, in the course of a few years, become the fruitful source of a thousand inventions." (Joseph Henry, "Report of the Secretary" [Sixth Annual Report of the Board of Regents of the Smithsonian Institution for 1851], 1852)

"SYSTEM (σύστημα, σύν ἵστημιavu, to place together) - is a full and connected view of all the truths of some department of knowledge. An organized body of truth, or truths arranged under one and the same idea, which idea is as the life or soul which assimilates all those truths. No truth is altogether isolated. Every truth has relation to some other. And we should try to unite the facts of our knowledge so as to see them in their several bearings. This we do when we frame them into a system. To do so legitimately we must begin by analysis and end with synthesis. But system applies not only to our knowledge, but to the objects of our knowledge. Thus we speak of the planetary system, the muscular system, the nervous system. We believe that the order to which we would reduce our ideas has a foundation in the nature of things. And it is this belief that encourages us to reduce our knowledge of things into systematic order. The doing so is attended with many advantages. At the same time a spirit of systematizing may be carried too far. It is only in so far as it is in accordance with the order of nature that it can be useful or sound." (William Fleming, "Vocabulary of philosophy, mental, moral, and metaphysical; with quotations and references; for the use of students", 1857)

"Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained - a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas." (Chauncey Wright, "The Philosophy of Herbert Spencer", North American Review, 1865)

"One of the greatest obstacles to the free and universal movement of human knowledge is the tendency that leads different kinds of knowledge to separate into systems." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Isolated facts and experiments have in themselves no value, however great their number may be. They only become valuable in a theoretical or practical point of view when they make us acquainted with the law of a series of uniformly recurring phenomena, or, it may be, only give a negative result showing an incompleteness in our knowledge of such a law, till then held to be perfect." (Hermann von Helmholtz, "The Aim and Progress of Physical Science", 1869)

"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress." (William T Kelvin, "Elements of Natural Philosophy", 1873)

"The whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects; and it is only so far, therefore, as we can discover and register resemblances that we can turn our observations to account." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Science is the observation of things possible, whether present or past; prescience is the knowledge of things which may come to pass, though but slowly." (Leonardo da Vinci, "The Notebooks of Leonardo da Vinci", 1883)

"[…] when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of science." (William T Kelvin, "Electrical Units of Measurement", 1883)

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

"The smallest group of facts, if properly classified and logically dealt with, will form a stone which has its proper place in the great building of knowledge, wholly independent of the individual workman who has shaped it." (Karl Pearson, "The Grammar of Science", 1892)

"The true aim of the teacher must be to impart an appreciation of method and not a knowledge of facts." (Karl Pearson, "The Grammar of Science", 1892)

"There is no short cut to truth, no way to gain a knowledge of the universe except through the gateway of scientific method." (Karl Pearson, "The Grammar of Science", 1892)

"Mature knowledge regards logical clearness as of prime importance: only logically clear images does it test as to correctness; only correct images does it compare as to appropriateness. By pressure of circumstances the process is often reversed. Images are found to be suitable for a certain purpose; are next tested as to their correctness ; and only in the last place purged of implied contradictions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"Without a theory all our knowledge of nature would be reduced to a mere inventory of the results of observation. Every scientific theory must be regarded as an effort of the human mind to grasp the truth, and as long as it is consistent with the facts, it forms a chain by which they are linked together and woven into harmony." (Thomas Preston, "The Theory of Heat", 1894)

"A system is not so important as a method. A system is of significance because it brings order and clearness into our knowledge, but he who hopes by its help to reach something more, he who thinks to extend his knowledge by means of a system is self-deceived." (Harald Høffding, "A history of modern philosophy", 1900)

"Man's determination not to be deceived is precisely the origin of the problem of knowledge. The question is always and only this: to learn to know and to grasp reality in the midst of a thousand causes of error which tend to vitiate our observation." (Federigo Enriques, "Problems of Science", 1906)

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

"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)

"The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another." (Henri Poincaré, 1913)

"By intuition is frequently understood perception, or the knowledge of actual reality, the apprehension of something as real. […] Intuition is the undifferentiated unity of the perception of the real and of the simple image of the possible. " (Benedetto Croce, "The Essence of Æsthetic", 1921)

"Observed facts must be built up, woven together, ordered, arranged, systematized into conclusions and theories by reflection and reason, if they are to have full bearing on life and the universe. Knowledge is the accumulation of facts. Wisdom is the establishment of relations. And just because the latter process is delicate and perilous, it is all the more delightful." (Gamaliel Bradford, "Darwin", 1926)

"With fuller knowledge we should sweep away the references to probability and substitute the exact facts." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"Scientific discovery and scientific knowledge have been achieved only by those who have gone in pursuit of them without any practical purpose whatsoever in view." (Max Planck, "Where is Science Going?", 1932)

"As facts and knowledge accumulate, the claim of the scientist to an understanding of the world in a certain sense diminishes." (Werner K Heisenberg, "Zur Geschichte der physikalischen Naturerklärung", 1933)

"The urge to knowledge is so deeply rooted in man that it can scarcely be omitted from a list of life's important needs." (Hans Reichenbach, "Atom and Cosmos: The World of Modern Physics", 1933)

"The laws of science are the permanent contributions to knowledge - the individual pieces that are fitted together in an attempt to form a picture of the physical universe in action. As the pieces fall into place, we often catch glimpses of emerging patterns, called theories; they set us searching for the missing pieces that will fill in the gaps and complete the patterns. These theories, these provisional interpretations of the data in hand, are mere working hypotheses, and they are treated with scant respect until they can be tested by new pieces of the puzzle." (Edwin P Whipple, "Experiment and Experience", [Commencement Address, California Institute of Technology] 1938)

"We have discovered that it is actually an aid in the search for knowledge to understand the nature of the knowledge we seek." (Arthur S Eddington, "The Philosophy of Physical Science", 1938)

"It is by abstraction that one can separate movements, knowledge, and affectivity. And the analysis is, here, so far from being a real dismemberment that it is given only as probable. One can never effectively reduce an [mental] image to its elements, for the reason that an image, like all other psychic syntheses, is something more and different from the sum of its elements. […] We will always go from image to image. Comprehension is a movement which is never-ending, it is the reaction of the mind to an image by another image, to this one by another image and so on, in principle to infinity. "(Jean-Paul Sartre, "The Imaginary: A phenomenological psychology of the imagination", 1940)

"In perception, a knowledge forms itself slowly; in the [mental] image the knowledge is immediate. We see now that the image is a synthetic act which unites a concrete, nonimagined, knowledge to elements which are more actually representative. The image teaches nothing: it is organized exactly like the objects which do produce knowledge, but it is complete at the very moment of its appearance." (Jean-Paul Sartre, "The Psychology of Imagination", 1940)

"Science, in the broadest sense, is the entire body of the most accurately tested, critically established, systematized knowledge available about that part of the universe which has come under human observation. For the most part this knowledge concerns the forces impinging upon human beings in the serious business of living and thus affecting man’s adjustment to and of the physical and the social world. […] Pure science is more interested in understanding, and applied science is more interested in control […]" (Austin L Porterfield, "Creative Factors in Scientific Research", 1941)

"Our theory has some bleaker consequences. [...] What is knowledge, if we are but a part of the mechanical system of the world we seek to know? What becomes of our ceaseless effort to explain the universe we live in, if explanation is but a part of the mechanical process?" (Kenneth Craik, "The Nature of Explanation", 1943)

"Whenever a man increases the content of his mind he gains new knowledge, and this occurs each time a new relation is established between the worlds on the two sides of the sense-organs - the world of ideas in an individual mind, and the world of objects existing outside individual minds which is common to us all." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"Science usually advances by a succession of small steps, through a fog in which even the most keen-sighted explorer can seldom see more than a few paces ahead. Occasionally the fog lifts, an eminence is gained, and a wider stretch of territory can be surveyed - sometimes with startling results. A whole science may then seem to undergo a kaleidoscopic ‘rearrangement’, fragments of knowledge being found to fit together in a hitherto unsuspected manner. Sometimes the shock of readjustment may spread to other sciences; sometimes it may divert the whole current of human thought." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"Every bit of knowledge we gain and every conclusion we draw about the universe or about any part or feature of it depends finally upon some observation or measurement. Mankind has had again and again the humiliating experience of trusting to intuitive, apparently logical conclusions without observations, and has seen Nature sail by in her radiant chariot of gold in an entirely different direction." (Oliver J Lee, "Measuring Our Universe: From the Inner Atom to Outer Space", 1950)

"The essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Knowledge rests on knowledge; what is new is meaningful because it departs slightly from what was known before; this is a world of frontiers, where even the liveliest of actors or observers will be absent most of the time from most of them." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

Science cannot be based on dogma or authority of any kind, nor on any institution or revelation, unless indeed it be of the Book of Nature that lies open before our eyes. We need not dwell on the processes of acquiring knowledge by observation, experiment, and inductive and deductive reasoning. The study of scientific method both in theory and practice is of great importance. It is inherent in the philosophy that the record may be imperfect and the conceptions erroneous; the potential fallibility of our science is not only acknowledged but also insisted upon." (Sir Robert Robinson, "Science and the Scientist", Nature Vol. 176 (4479), 1955)

"There comes a point where the mind takes a leap - call it intuition or what you will - and comes out upon a higher plane of knowledge, but can never prove how it got there. All great discoveries have involved such a leap." (Albert Einstein, [interview in Life, "Death of a Genius"] 1955)

"Knowledge is not something which exists and grows in the abstract. It is a function of human organisms and of social organization. Knowledge, that is to say, is always what somebody knows: the most perfect transcript of knowledge in writing is not knowledge if nobody knows it. Knowledge however grows by the receipt of meaningful information - that is, by the intake of messages by a knower which are capable of reorganising his knowledge." (Kenneth E Boulding, "General Systems Theory - The Skeleton of Science", Management Science Vol. 2 (3), 1956)

"The mathematical formulas indeed no longer portray nature, but rather our knowledge of nature." (Werner K Heisenberg, "The Representation of Nature in Contemporary Physics", Daedalus Vol. 87 (3), 1958)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The New Science", 1959)

"Incomplete knowledge must be considered as perfectly normal in probability theory; we might even say that, if we knew all the circumstances of a phenomenon, there would be no place for probability, and we would know the outcome with certainty." (Félix E Borel, Probability and Certainty", 1963)

"In its efforts to learn as much as possible about nature, modem physics has found that certain things can never be ‘known’ with certainty. Much of our knowledge must always remain uncertain. The most we can know is in terms of probabilities." (Richard P Feynman, "The Feynman Lectures on Physics", 1964)

"Knowing reality means constructing systems of transformations that correspond, more or less adequately, to reality. They are more or less isomorphic to transformations of reality. The transformational structures of which knowledge consists are not copies of the transformations in reality; they are simply possible isomorphic models among which experience can enable us to choose. Knowledge, then, is a system of transformations that become progressively adequate." (Jean Piaget, "Genetic Epistemology", 1968)

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

"The idea of knowledge as an improbable structure is still a good place to start. Knowledge, however, has a dimension which goes beyond that of mere information or improbability. This is a dimension of significance which is very hard to reduce to quantitative form. Two knowledge structures might be equally improbable but one might be much more significant than the other." (Kenneth E Boulding, "Beyond Economics: Essays on Society", 1968)

"Models constitute a framework or a skeleton and the flesh and blood will have to be added by a lot of common sense and knowledge of details."(Jan Tinbergen, "The Use of Models: Experience," 1969)

"Inductive inference is the only process known to us by which essential new knowledge comes into the world." (Sir Ronald A Fisher, "The Design of Experiments", 1971)

"A discovery must be, by definition, at variance with existing knowledge." (Albert Szent-Gyorgyi, "Dionysians and Apollonians", Science 176, 1972)

"The human condition can almost be summed up in the observation that, whereas all experiences are of the past, all decisions are about the future. It is the great task of human knowledge to bridge this gap and to find those patterns in the past which can be projected into the future as realistic images." (Kenneth E Boulding, [foreword] 1972)

"Human knowledge is personal and responsible, an unending adventure at the edge of uncertainty." (Jacob Bronowski, "The Ascent of Man", 1973)

"Discoveries are made by pursuing possibilities suggested by existing knowledge." (Michael Polanyi, "Meaning", 1975)

"Knowledge is not a series of self-consistent theories that converges toward an ideal view; it is rather an ever increasing ocean of mutually incompatible (and perhaps even incommensurable) alternatives, each single theory, each fairy tale, each myth that is part of the collection forcing the others into greater articulation and all of them contributing, via this process of competition, to the development of our consciousness." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

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

"Certainty, simplicity, vividness originate in popular knowledge. That is where the expert obtains his faith in this triad as the ideal of knowledge. Therein lies the general epistemological significance of popular science." (Ludwik Fleck, "Genesis and Development of a Scientific Fact", 1979)

"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)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"Knowledge is the appropriate collection of information, such that it's intent is to be useful. Knowledge is a deterministic process. When someone 'memorizes' information (as less-aspiring test-bound students often do), then they have amassed knowledge. This knowledge has useful meaning to them, but it does not provide for, in and of itself, an integration such as would infer further knowledge." (Russell L Ackoff, "Towards a Systems Theory of Organization", 1985)

"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend, "Farewell to Reason", 1987)

"We admit knowledge whenever we observe an effective (or adequate) behavior in a given context, i.e., in a realm or domain which we define by a question (explicit or implicit)." (Humberto Maturana & Francisco J Varela, "The Tree of Knowledge", 1987)

"Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance." (John A Wheeler, Scientific American Vol. 267, 1992)

"Indeed, knowledge that one will be judged on some criterion of ‘creativeness’ or ‘originality’ tends to narrow the scope of what one can produce (leading to products that are then judged as relatively conventional); in contrast, the absence of an evaluations seems to liberate creativity." (Howard Gardner,  "Creating Minds", 1993)

"Knowledge is theory. We should be thankful if action of management is based on theory. Knowledge has temporal spread. Information is not knowledge. The world is drowning in information but is slow in acquisition of knowledge. There is no substitute for knowledge." (William E Deming, "The New Economics for Industry, Government, Education", 1993) 

"Discourses are ways of referring to or constructing knowledge about a particular topic of practice: a cluster (or formation) of ideas, images and practices, which provide ways of talking about, forms of knowledge and conduct associated with, a particular topic, social activity or institutional site in society. These discursive formations, as they are known, define what is and is not appropriate in our formulation of, and our practices in relation to, a particular subject or site of social activity." (Stuart Hall, "Representation: Cultural Representations and Signifying Practices", 1997)

"The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge."  (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)

"All human knowledge - including statistics - is created  through people's actions; everything we know is shaped by our language, culture, and society. Sociologists call this the social construction of knowledge. Saying that knowledge is socially constructed does not mean that all we know is somehow fanciful, arbitrary, flawed, or wrong. For example, scientific knowledge can be remarkably accurate, so accurate that we may forget the people and social processes that produced it." (Joel Best, Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists, 2001)

"Knowledge is factual when evidence supports it and we have great confidence in its accuracy. What we call 'hard fact' is information supported by  strong, convincing evidence; this means evidence that, so far as we know, we cannot deny, however we examine or test it. Facts always can be questioned, but they hold up under questioning. How did people come by this information? How did they interpret it? Are other interpretations possible? The more satisfactory the answers to such questions, the 'harder' the facts."(Joel Best, Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists, 2001)

"It is the nature of an uncertainty that it is not known and can never be known, whether the best estimate is greater or less than the true value." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"All knowledge that is not the real product of observation, or of consequences deduced from observation, is entirely groundless and illusory." (Jean-Baptiste Lamarck)

"By observation, facts are distinctly and minutely impressed in the mind; by analogy, similar facts are connected ; by experiment, new facts are discovered ; and, in the progression of knowledge, observation, guided by analogy, leads to experiment, and analogy, confirmed by experiment, becomes scientific truth." (Sir Humphry Davy)

"In imaginative thought there is no real knowledge of anything but similarities (ultimately identities): knowledge of differences is merely a transition to a new knowledge of similarities."  (Northrop Frye)

"Real wisdom is not the knowledge of everything, but the knowledge of which things in life are necessary, which are less necessary, and which are completely unnecessary to know." (Lev N Tolstoy)

"The goal of education is not to increase the amount of knowledge but to create the possibilities for a child to invent and discover, to create men who are capable of doing new things." (Jean Piaget)

"We call it 'explanation', but it is 'description' which distinguishes us from earlier stages of knowledge and science. We describe better - we explain just as little who came before us [...] We operate with nothing but things which do not exist, with lines, planes, bodies, atoms, divisible time, divisible space - how should explanation even be possible when we first make everything into an image, into our image!" (Friedrich W Nietzsche)

You know how the divine Simplicity enfolds all things. Mind is the image of this enfolding Simplicity. If, then, you called this divine Simplicity infinite Mind, it will be the exemplar of our mind. If you called the divine mind the totality of the truth of things, you will call our mind the totality of the assimilation of things, so that it may be a totality of ideas. In the divine Mind conception is the production of things; in our mind conception is the knowledge of things. If the divine Mind is absolute Being, then its conception is the creation of beings; and conception in the human mind is the assimilation of beings." (Nicholas of Cusa)

Knowledge Representation: On Definitions (Quotes)

"The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning." (Thomas Hobbes, "The Moral and Political Works of Thomas Hobbes of Malmesbury", 1750)

"Definitions might be good if we did not employ words in making them." (Jean-Jacques Rousseau, "Emile, or, Treatise on Education", 1762)

"A definition is nothing else but an explication of the meaning of a word, by words whose meaning is already known. Hence it is evident that every word cannot be defined; for the definition must consist of words; and there could be no definition, if there were not words previously understood without definition." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"There is nothing more difficult than a good definition, for it is scarcely possible to express, in a few words, the abstracted view of an infinite variety of facts." (Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher" , 1830)

"It is the essence of a scientific definition to be causative, not by introduction of imaginary somewhats, natural or supernatural, under the name of causes, but by announcing the law of action in the particular case, in subordination to the common law of which all the phenomena are modifications or results." (Samuel T Coleridge, "Hints Towards the Formation of a More Comprehensive Theory of Life, The Nature of Life", 1847)

"The dimmed outlines of phenomenal things all merge into one another unless we put on the focusing-glass of theory, and screw it up sometimes to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world." (James C Maxwell, "Are There Real Analogies in Nature?", 1856)

"Questions of Definition are of the very highest importance in Philosophy, and they need to be watched accordingly." (George Campbell, "A Fourth State of Matter, Nature", 1880)

"The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language. (Anton Chekhov, [letter to A.S. Suvorin] 1892)

"A definition is the enclosing a wilderness of idea within a wall of words." (Samuel Butler, "The Note-Books of Samuel Butler", 1912)

"In every new and growing science there are many working hypotheses that never attain to any sort of reality. On the other hand, in the old and abstract sciences of mathematics, where it is hard to tell how much is mere definition or convention, the problem of reality is not so much doubtful as it is meaningless." (Gilbert N Lewis, "The Anatomy of Science", 1926)

"Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought. (Frank P Ramsey, 'Last Papers: Philosophy', 1929)

"A Weltanschauung [worldview] is an intellectual construction which solves all the problems of our existence uniformly on the basis of one overriding hypothesis, which, accordingly, leaves no question unanswered and in which everything that interests us finds its fixed place [...] the worldview of science already departs noticeably from our definition. It is true that it too assumes the uniformity of the explanation of the universe; but it does so only as a programme, the fulfillment of which is relegated to the future." Sigmund Freud, "New introductory lectures on psycho-analysis", 1932)

"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1937)

"The view is often defended that sciences should be built up on clear and sharply defined basal concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them." (Sigmund Freud, "Collected Papers", 1950)

"Being built on concepts, hypotheses, and experiments, laws are no more accurate or trustworthy than the wording of the definitions and the accuracy and extent of the supporting experiments." (Gerald Holton, "Introduction to Concepts and Theories in Physical Science", 1952)

"The word 'definition' has come to have a dangerously reassuring sound, owing no doubt to its frequent occurrence in logical and mathematical writings." (Willard van Orman Quine, From a Logical Point of View, 1953)

"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)

"The view is often defended that sciences should be built up on clear and sharply defined basal concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them." (Sigmund Freud, "General Psychological Theory", 1963)

"This other world is the so-called physical world image; it is merely an intellectual structure. To a certain extent it is arbitrary. It is a kind of model or idealization created in order to avoid the inaccuracy inherent in every measurement and to facilitate exact definition." (Max Planck, "The Philosophy of Physics", 1963)

"Definitions are the guardians of rationality, the first line of defense against the chaos of mental disintegration. (Ayn Rand, The Romantic Manifesto, 1969)

"Concepts form the basis for any science. These are ideas, usually somewhat vague (especially when first encountered), which often defy really adequate definition. The meaning of a new concept can seldom be grasped from reading a one-paragraph discussion. There must be time to become accustomed to the concept, to investigate it with prior knowledge, and to associate it with personal experience. Inability to work with details of a new subject can often be traced to inadequate understanding of its basic concepts." (William C Reynolds & Harry C Perkins, "Engineering Thermodynamics", 1977)

"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)

"When terms [...] evolve and change definition with time; and when the social reality which terms are intended to organize and render intelligible is also seen to be in flux, capturing the truth in a net of words becomes a matter of intuition and style more than of any scientific method that can be replicated by others and made to achieve the same result every time someone asks the same question, or undertakes the same operations." (William H McNeill, "Discrepancies among the Social Sciences", 1981)

"Definitions are temporary verbalizations of concepts, and concepts - particularly difficult concepts - are usually revised repeatedly as our knowledge and understanding grows." (Ernst Mayr, "The Growth of Biological Thought", 1982) 

"A full definition of an object must include the whole of human experience, both as a criterion of truth and a practical indicator of its connection with human wants." (Vladimir Lenin) 

"Fundamental definitions do not arise at the start but at the end of the exploration, because in order to define a thing you must know what it is and what it is good for." (Hans Freudenthal)

"We begin to reason from sensible objects, and definition is the end and epilogue of science. It is not the beginning of our knowing, but only of our teaching." (Tommaso Campanella)

Knowledge Representation: On Patterns (Quotes)

"Most surprising and far-reaching analogies revealed themselves between apparently quite disparate natural processes. It seemed that nature had built the most various things on exactly the same pattern; or, in the dry words of the analyst, the same differential equations hold for the most various phenomena." (Ludwig Boltzmann, "On the methods of theoretical physics", 1892)

"It [knowledge] is clearly related to information, which we can now measure; and an economist especially is tempted to regard knowledge as a kind of capital structure, corresponding to information as an income flow. Knowledge, that is to say, is some kind of improbable structure or stock made up essentially of patterns - that is, improbable arrangements, and the more improbable the arrangements, we might suppose, the more knowledge there is." (Kenneth Boulding, "Beyond Economics: Essays on Society", 1968)

"The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos. […] This is the task of natural science: to show that the wonderful is not incomprehensible, to show how it can be comprehended - but not to destroy wonder. For when we have explained the wonderful, unmasked the hidden pattern, a new wonder arises at how complexity was woven out of simplicity. The aesthetics of natural science and mathematics is at one with the aesthetics of music and painting - both inhere in the discovery of a partially concealed pattern." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Faced with information overload, we have no alternative but pattern-recognition."(Marshall McLuhan, "Counterblast", 1969) 

"Without the hard little bits of marble which are called 'facts' or 'data' one cannot compose a mosaic; what matters, however, are not so much the individual bits, but the successive patterns into which you arrange them, then break them up and rearrange them." (Arthur Koestler, "The Act of Creation", 1970) 

"To do science is to search for repeated patterns, not simply to accumulate facts […]" (Robert H. MacArthur, "Geographical Ecology", 1972)

"A pattern has an integrity independent of the medium by virtue of which you have received the information that it exists. Each of the chemical elements is a pattern integrity. Each individual is a pattern integrity. The pattern integrity of the human individual is evolutionary and not static." (Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975) 

"In everyday language, the words 'pattern' and 'symmetry' are used almost interchangeably, to indicate a property possessed by a regular arrangement of more-or-less identical units […]” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"Complexity is looking at interacting elements and asking how they form patterns and how the patterns unfold. It’s important to point out that the patterns may never be finished. They’re open-ended. In standard science this hit some things that most scientists have a negative reaction to. Science doesn’t like perpetual novelty." (W Brian Arthur, 1999)

"Knowledge is encoded in models. Models are synthetic sets of rules, and pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Didier Sornette, "Why Stock Markets Crash - Critical Events in Complex Systems", 2003) 

"[...] when data is presented in certain ways, the patterns can be readily perceived. If we can understand how perception works, our knowledge can be translated into rules for displaying information. Following perception‐based rules, we can present our data in such a way that the important and informative patterns stand out. If we disobey the rules, our data will be incomprehensible or misleading." (Colin Ware, "Information Visualization: Perception for Design" 2nd Ed., 2004)

"Perceiving the world as well designed and thus the product of a designer, and even seeing divine providence in the daily affairs of life, may be the product of a brain adapted to finding patterns in nature. We are pattern seeking and pattern-finding animals." (Michael Shermer, "Why Darwin Matters: The Case Against Intelligent Design", 2007) 

"It is the consistency of the information that matters for a good story, not its completeness. Indeed, you will often find that knowing little makes it easier to fit everything you know into a coherent pattern." (Daniel Kahneman, "Thinking, Fast and Slow", 2011) 

"Randomness might be defined in terms of order - its absence, that is. […] Everything we care about lies somewhere in the middle, where pattern and randomness interlace." (James Gleick, "The Information: A History, a Theory, a Flood", 2011)

"Finding patterns is easy in any kind of data-rich environment […] The key is in determining whether the patterns represent signal or noise." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don’t", 2012)

"Another way to secure statistical significance is to use the data to discover a theory. Statistical tests assume that the researcher starts with a theory, collects data to test the theory, and reports the results - whether statistically significant or not. Many people work in the other direction, scrutinizing the data until they find a pattern and then making up a theory that fits the pattern." (Gary Smith, "Standard Deviations", 2014)

"If somebody ransacks data to find a pattern, we still need a theory that makes sense. On the other hand, a theory is just a theory until it is tested with persuasive data." (Gary Smith, "Standard Deviations", 2014)

"[…] regard it in fact as the great advantage of the mathematical technique that it allows us to describe, by means of algebraic equations, the general character of a pattern even where we are ignorant of the numerical values which will determine its particular manifestation." (Friedrich A von Hayek, "The Market and Other Orders", 2014)

"Remember that even random coin flips can yield striking, even stunning, patterns that mean nothing at all. When someone shows you a pattern, no matter how impressive the person’s credentials, consider the possibility that the pattern is just a coincidence. Ask why, not what. No matter what the pattern, the question is: Why should we expect to find this pattern?" (Gary Smith, "Standard Deviations", 2014)

"We are genetically predisposed to look for patterns and to believe that the patterns we observe are meaningful. […] Don’t be fooled into thinking that a pattern is proof. We need a logical, persuasive explanation and we need to test the explanation with fresh data." (Gary Smith, "Standard Deviations", 2014)

"We are hardwired to make sense of the world around us - to notice patterns and invent theories to explain these patterns. We underestimate how easily pat - terns can be created by inexplicable random events - by good luck and bad luck." (Gary Smith, "Standard Deviations", 2014)

"We are seduced by patterns and we want explanations for these patterns. When we see a string of successes, we think that a hot hand has made success more likely. If we see a string of failures, we think a cold hand has made failure more likely. It is easy to dismiss such theories when they involve coin flips, but it is not so easy with humans. We surely have emotions and ailments that can cause our abilities to go up and down. The question is whether these fluctuations are important or trivial." (Gary Smith, "Standard Deviations", 2014)

"A pattern is a design or model that helps grasp something. Patterns help connect things that may not appear to be connected. Patterns help cut through complexity and reveal simpler understandable trends. […] Patterns can be temporal, which is something that regularly occurs over time. Patterns can also be spatial, such as things being organized in a certain way. Patterns can be functional, in that doing certain things leads to certain effects. Good patterns are often symmetric. They echo basic structures and patterns that we are already aware of." (Anil K. Maheshwari, "Business Intelligence and Data Mining", 2015)

Knowledge Representation: On Truth (Quotes)

“As being is to become, so is truth to belief” (Plato, “Timaeus”, cca. 360 BC)

“The first duty of man is the seeking after and the investigation of truth.” (Marcus Tullius Cicero, “De Officiis”, [“On Duties”], cca. 44 BC)

“The exact kind of language we employ in philosophical analyses of abstract truth is one thing, and the language used in attempts to popularize the subject is another.” (Marcus Tullius Cicero, “De officiis” [“On Duties”], cca.44 BC)

“Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.” (Marcus Aurelius, "Meditations", cca. 2nd century)

“We are meant to take them [the words ‘increase and multiply’] in a figurative sense. […] It is only in the case of signs outwardly given that we find increase and multiplication in the sense that a single truth can be expressed by several different means […] that a single expression can be interpreted in several different ways.” (St. Augustine, “Confessions”, 397- 400)

"Truth is sought for itself, but the truths are immersed in uncertainties." (Abu Ali al-Hasan ibn al-Haytham [Alhazen], "Aporias against Ptolemy", 1025-1028)

“If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” (Roger Bacon, “Opus Majus” Book 1, 1267)

"Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience."(Roger Bacon, "Opus Majus", cca. 1267) 

“The truth of voice perishes with the sound; truth latent in the mind is hidden wisdom and invisible treasure; but the truth which illuminates books desires to manifest itself to every disciplinable sense. Let us consider how great a commodity of doctrine exists in books, - how easily, how secretly, how safely, they expose the nakedness of human ignorance without putting it to shame. These are the masters that instruct us without rods and ferules, without hard words and anger, without clothes or money. If you approach them, they are not asleep; if, investigating, you interrogate them, they conceal nothing; if you mistake them, they never grumble; if you are ignorant, they cannot laugh at you.” (Richard de Burry, “Philobiblon”, 1344)

"Man's mind is so formed that it is far more susceptible to falsehood than to truth." (Desiderius Erasmus, "Praise of Folly", 1509)

“[…] no pleasure is comparable to the standing upon the vantage ground of truth […]” (Sir Francis Bacon, “Essays”, 1597)

“The first and most ancient inquirers into truth were wont to throw their knowledge into aphorisms, or short, scattered, unmethodical sentences.” (Lord Bacon, “Novum Organum”, 1620)

"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)

“[…] thus each truth discovered was a rule available in the discovery of subsequent ones.” (René Descartes, “Discourse on Method”, 1637)

 “Every man is not a proper champion for truth, nor fit to take up the gauntlet in the cause of verity: many from the ignorance of these maxims, and an inconsiderate zeal for truth, have too rashly charged the troops of error, and remain as trophies unto the enemies of truth. A man may be in as just possession of truth as of a city, and yet be forced to surrender: ’tis therefore far better to enjoy her with peace than to hazard her on a battle: if therefore there rise any doubts in my way, I do forget them, or at least defer them, till my better settled judgment and more manly reason be able to resolve them.” (Sir Thomas Browne, ”Religio Medici”, 1643)

“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.” (René Descartes, “Principles of Philosophy”, 1644)

“Knowledge is made by oblivion, and to purchase a clear and warrantable body of truth, we must forget and part with much we know.” (Sir Thomas Browne, “Pseudodoxia Epidemica”, 1646)

“Men are apt to prefer a prosperous error before an afflicted truth.” (Jeremy Taylor, “The Rule and Exercises of Holy Living”, 1650)

 “All things being double-handed, and having the appearances both of truth and falsehood, where our affections have engaged us we attend only to the former.” (Joseph Glanvill, “Scepsis”, 1665)

“Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” (Blaise Pascal, “Pensées”, 1670)

“We see neither justice nor injustice which does not change its nature with change in climate. Three degrees of latitude reverse all jurisprudence: a meridian decides the truth.” (Blaise Pascal, “Pensées”, 1670)

“In logic, they teach that contraries laid together more evidently appear: it follows, then, that all controversy being permitted, falsehood will appear more false, and truth the more true; which must needs conduce much to the general confirmation of an implicit truth.” (John Milton, “True Religion, Heresy, Schism, Toleration, and what best means may be used against the Growth of Popery”, 1673)

“In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow.” (Baruch Spinoza, [letter to Hugo Boxel], 1674)

“Truth is always consistent with itself, and needs nothing to help it out; it is always near at hand, and sits upon our lips, and is ready to drop out before we are aware; whereas a lie is troublesome, and sets a man’s invention upon the rack, and one trick needs a great many more to make it good.” (John Tillotson, “Sermons”, 1682)

"And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Two things are identical if one can be substituted for the other without affecting the truth." (Gottfried W Leibniz, "Table de definitions", 1704)

“Not only the investigation of truth, but the communication of it also, is often practised in such a method as neither agrees precisely to synthetic or analytic.”  (Isaac Watts, “Logic, or The right use of reason, in the inquiry after truth”, 1725)

“He that would make a real progress in knowledge must dedicate his age as well as first fruits - the latter growth as well as the first-fruits - at the altar of truth.” (Bishop George Berkeley, “Siris”, 1744)

“If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty.” (Edmund Burke, “Essay on the Sublime and Beautiful”, 1756)

"There are truths which are not for all men, nor for all times." (Voltaire, [Letter to François-Joachim de Pierre] 1764)

"Ignorance is preferable to error; and he is less remote from the truth who believes nothing, than he who believes what is wrong." (Thomas Jefferson, "Notes on the State of Virginia", 1781)

"General abstract truth is the most precious of all blessings; without it, man is blind; it is the eye of reason." (Jean-Jacques Rousseau, "The Confessions of J. J. Rousseau", 1783)

“The discovery of truth by slow, progressive meditation is talent. Intuition of the truth, not preceded by perceptible meditation, is genius.” (Johann K Lavater, 1787) 

“Everything possible to be believed is an image of truth.” (William Blake, “The Marriage of Heaven and Hell”, 1790)

“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world.” (Immanuel Kant, “Critique of Judgment”, 1790)

"We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoine Lavoisier, "Elements of Chemistry", 1790)

"It is an acknowledged truth in philosophy that a just theory will always be confirmed by experiment." (Thomas R Malthus, "An Essay on The Principle of Population", 1798)

“Forgetting that the only eternal part for man to act is man, and that the only immutable greatness is truth.” (Alphonse Lamartine, “The History of the Restoration of Monarchy in France”, 1851)

"Accuracy of language is one of the bulwarks of truth." (Anna B Jameson, "A Commonplace Book of Thoughts, Memories, and Fancies", 1854)

"We must therefore discover some method of investigation which allows the mind at every step to lay hold of a clear physical conception, without being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis." (James C Maxwell, "On Faraday’s lines of force", 1855)

"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)

“The excellence of every art is its intensity, capable of making all disagreeables evaporate from their being in close relationship with beauty and truth.” (John Keats. [letter to George and Thomas Keats] 1817)

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

"We are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it." (Thomas Jefferson, [Letter to William Roscoe] 1820)

"Mathematics, like dialectics, is an instrument of the inner higher sense, while in practice it is an art like rhetoric. For both of these, nothing has value but form; content is immaterial. Whether mathematics is adding up pennies or guineas, whether rhetoric is defending truth or falsehood, makes no difference to either.” (Johann Wolfgang von Goethe, "Wilhelm Meisters Wanderjahre" ["Reflections in the Spirit of the Wanderers"], 1821)

“Facts are the mere dross of history. It is from the abstract truth which interpenetrates them, and lies latent among them, like gold in the ore, that the mass derives its whole value: and the precious particles are generally combined with the baser in such a manner that the separation is a task of the utmost difficulty.” (Thomas B Macaulay, “History”, 1828)

"Truth in itself is rarely sufficient to make men act. Hence the step is always long from cognition to volition, from knowledge to ability. The most powerful springs of action in men lie in his emotions." (Carl von Clausewitz, "On War", 1832)

“It is difficult to discriminate the voice of truth from amid the clamour raised by heated partisans.” (Friedrich Schiller, “Schillers Sammtliche Werke”, 1834)

“The most important and lasting truths are the most obvious ones. Nature cheats us with her mysteries, one after another, like a juggler with his tricks; but shews us her plain honest face, without our paying for it.” (William Hazlitt, “Characteristics: In the Manner of Rochefoucault's Maxims”, 1837)

“In truth, ideas and principles are independent of men; the application of them and their illustration is man's duty and merit.” (Edward Forbes, 1847)

“The peculiarity of the evidence of mathematical truths is that all the argument is on one side.” (John Stuart Mill, “On Liberty”, 1859)

[…] the besetting danger is not so much of embracing falsehood for truth, as of mistaking a part of the truth for the whole.” (John Stuart Mill, “Dissertations and Discussions: Political, Philosophical, and Historical”, 1864) 

"No departure from the truth of nature shall be discovered by the closest scrutiny." (Henry P Robinson, "Pictorial Effect in Photography", 1869)

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

“Pure truth cannot be assimilated by the crowd; it must be communicated by contagion.” (Henri-Frédéric Amiel, [journal entry] 1875)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)

“Convictions are more dangerous enemies of truth than lies.” (Friedrich Nietzsche, “Human, All Too Human: A book for Free Spirits”, 1878) 

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

“How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?” (Sir Arthur Conan Doyle, “The Sign of Four”, 1890)

“Accuracy of statement is one of the first elements of truth; inaccuracy is a near kin to falsehood.” (Tyron Edwards, “A Dictionary of Thoughts”, 1891)

"There is no short cut to truth, no way to gain a knowledge of the universe except through the gateway of scientific method." (Karl Pearson, “The Grammar of Science”, 1892)

"It is they who hold the secret of the mysterious property of the mind by which error ministers to truth, and truth slowly but irrevocably prevails. Theirs is the logic of discovery, the demonstration of the advance of knowledge and the development of ideas, which as the earthly wants and passions of men remain almost unchanged, are the charter of progress, and the vital spark in history." (Lord John Acton, "The Study of History", [lecture delivered at Cambridge] 1895)

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

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notations, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. (William S Jevons, "The Principles of Science" 2nd Ed., 1900)

"The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe." (Karl Pearson, “The Grammar of Science”, 1900)

“Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty.” (Henri Poincaré, “Science and Hypothesis”, 1902) 

“Logic, then, is not necessarily an instrument for finding truth; on the contrary, truth is necessarily an instrument for using logic - for using it, that is, for the discovery of further truth and for the profit of humanity. Briefly, you can only find truth with logic if you have already found truth without it.” (Gilbert K Chesterton, Daily News, 1905)

"The forceps of our minds are clumsy forceps, and crush the truth a little in taking hold of it." (Herbert G Wells, "Scepticism of the Instrument: A Modern Utopia", 1905)

“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)

“The truth of an idea is not a stagnant property inherent in it. Truth happens to an idea. It becomes true, is made true by events. Its verity is in fact an event, a process: the process namely of its verifying itself, its verification. Its validity is the process of its validation.” (William James, “Pragmatism: A New Name for Some Old Ways of Thinking”, 1907)

"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré," Science and Method", 1908)

"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned, these, and such like, are the surest groimds of its title of human regard, and would remain unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.” (James J Sylvester, "A Plea for the Mathematician", Nature, 1908)

“Truth is on a curve whose asymptote our spirit follows eternally.” (Léo Errera, “Recueil d'Œuvres de Léo Errera: Botanique Générale”, 1908) 

“Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts.” (Paul Carus, “Philosophy as a Science”, 1909)

“The pursuit of truth is chimerical. […] There is no permanent absolute unchangeable truth;  what we should pursue is the most convenient arrangement of our ideas.” (Samuel Butler, “Notebooks”, 1912)

“Only in men’s imagination does every truth find an effective and undeniable existence.” (Joseph Conrad, “Some Reminiscences”, 1912)

“The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics.” (J Craig, “A Course of Study for the Preparation of Rural School Teachers”, 1912)

"The conception of logical laws must be the decisive factor in the treatment of logic, and that conception depends upon what we understand by the word ‘true’. It is generally admitted at the very beginning that logical laws must be rules of conduct to guide thought to truth […]" (Gottlob Frege," Grundgesetze", The Monist, 1915) 

“As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies. “ (Albert Einstein: “Relativity: The Special and General Theory”, 1916)

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

“[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination.” (James B Shaw, “Lectures on the Philosophy of Mathematics”, 1918)

“Ignorance may find a truth on its doorstep that erudition vainly seeks in the stars.” (George Iles, “Canadian Stories”, 1918)

"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)

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

“It is not the possession of truth, but the success which attends the seeking after it, that enriches the seeker and brings happiness to him.” (Max Planck, “Where is Science Going?”, 1932) 

“Apart from blunt truth, our lives sink decadently amid the perfume of hints and suggestions.” (Alfred N Whitehead, “Adventures of Ideas”, 1933)

”[…] the merit of mathematics, in all its forms, consists in its truth; truth conveyed to the understanding, not directly by words but by symbols which serve as the world’s only universal written language.” (David Eugene Smith, “The Poetry of Mathematics and Other Essays”,  1934)

“Mathematics is the science of number and space. It starts from a group of self-evident truths and by infallible deduction arrives at incontestable conclusions […] the facts of mathematics are absolute, unalterable, and eternal truths.” (E Russell Stabler, “An Interpretation and Comparison of Three Schools of Thought in the Foundations of Mathematics”, The Mathematics Teacher Vol 26, 1935)

"Science makes no pretension to eternal truth or absolute truth; some of its rivals do. That science is in some respects inhuman may be the secret of its success in alleviating human misery and mitigating human stupidity." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1938)

"Even if all parts of a problem seem to fit together like the pieces of a jigsaw puzzle, one has to remember that the probable need not necessarily be the truth and the truth not always probable." (Sigmund Freud, "Moses and Monotheism", 1939)

“When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic.” (Jean Rostand, “Pensées d'un Biologiste”, 1939)

“A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling.” (Robert Musil, “Man Without Qualities”, 1943)

"Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans," Physics and Philosophy" 3rd Ed., 1943) 

"Thus we do not try to prove the existence of the external world – we discover it, because the fundamental power of words or other symbols to represent events [...] permits us to put forward hypotheses and test their truth by reference to experience." (Kenneth Craik, "The Nature of Explanation", 1943)

“When two hypotheses are possible, we provisionally choose that which our minds adjudge to the simpler on the supposition that this Is the more likely to lead in the direction of the truth.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)

"After all, the ultimate goal of all research is not objectivity, but truth." (Helene Deutsch, "The Psychology of Women", 1944)

"The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists." (Erwin Schrödinger, „What is Life?", 1944)

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

"Science condemns itself to failure when, yielding to the infatuation of the serious, it aspires to attain being, to contain it, and to possess it; but it finds its truth if it considers itself as a free engagement of thought in the given, aiming, at each discovery, not at fusion with the thing, but at the possibility of new discoveries; what the mind then projects is the concrete accomplishment of its freedom." (Simone de Beauvoir, "The Ethics of Ambiguity", 1947)

"Any useful logic must concern itself with Ideas with a fringe of vagueness and a Truth that is a matter of degree.” (Norbert Wiener, “Cybernetics”, 1948)

"A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." (Max Planck, "A Scientific Autobiography", 1949)

“Representation of the world, like the world itself, is the work of men; they describe it from their own point of view, which they confuse with absolute truth.” (Simone de Beauvoir, “The Second Sex”, 1949)

"Science makes no pretension to eternal truth or absolute truth; some of its rivals do." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"It is true that the grasping of truth is not possible without empirical basis. However, the deeper we penetrate and the more extensive and embracing our theories become the less empirical knowledge is needed to determine those theories." (Albert Einstein, 1952)

“Logic and truth are two very different things, but they often look the same to the mind that’s performing the logic. " (Theodore Sturgeon, “More Than Human”, 1953)

"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)

"There are no whole truths; all truths are half-truths. It is trying to treat them as whole truths that plays the devil.” (Alfred North Whitehead, “Dialogues”, 1954) 

"The history of science is rich in the example of the fruitfulness of bringing two sets of techniques, two sets of ideas, developed in separate contexts for the pursuit of new truth, into touch with one another." (J. Robert Oppenheimer, "Science and the common understanding", 1954)

"Science is the creation of concepts and their exploration in the facts. It has no other test of the concept than its empirical truth to fact." (Jacob Bronowski, "Science and Human Values", 1956)

“Starting from statistical observations and applying to them a clear and precise concept of probability it is possible to arrive at conclusions which are just as reliable and ‘truth-full’ and quite as practically useful as those obtained in any other exact science.” (Richard von Mises, “Probability, Statistics, and Truth”2nd Ed., 1957)

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

"We speak in terms of ‘acceptance’, ‘confidence’, and ‘probability’, not ‘proof’. If by proof it is meant the establishment of eternal and absolute truth, open to no possible exception or modification, then proof has no place in the natural sciences." (George G Simpson, “Life: An Introduction to Biology”, 1957)

“We can never achieve absolute truth but we can live hopefully by a system of calculated probabilities. The law of probability gives to natural and human sciences - to human experience as a whole - the unity of life we seek.” (Agnes E Meyer, “Education for a New Morality”, 1957)

"It will never be possible by pure reason to arrive at some absolute truth." (Werner K Heisenberg, "Physics and Philosophy: The revolution in modern science", 1958)

"Scientific method is the way to truth, but it affords, even in principle, no unique definition of truth. Any so-called pragmatic definition of truth is doomed to failure equally." (Willard v O Quine, "Word and Object", 1960) 

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)

"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)

"When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic." (Jean Rostand, "The substance of man", 1962)

“Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected.” (Richard Feynman, “The Feynman Lectures on Physics” Vol. 1,1964)

“[…] in the statistical world you can multiply ignorance by a constant and get truth.” (Raymond F Jones, “The Non-Statistical Man”, 1964)

"The belief that there is only one truth and that oneself is in possession of it, seems to me the deepest root of all that is evil in the world." (Max Born, "Natural Philosophy of Cause and Chance", 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 views are only probable, and a doctrine of probability which is not bound to a truth dissolves into thin air. In order to describe the probable, you must have a firm hold on the true. Therefore, before there can be any truth whatsoever, there must be absolute truth.” (Jean-Paul Sartre, “The Philosophy of Existentialism”, 1965)

“Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth.” (Salomon Bochner, “Why Mathematics Grows”, Journal of the History of Ideas, 1965)

“[…] truth is the intersection of independent lies.” (Richard Levins, “The Strategy of Model Building in Population Biology”, 1966)

"Primary scientific papers are not meant to be final statement of indisputable truths; each is merely a tiny tentative step forward, through the jungle of ignorance." (Erwin Schrödinger, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)

"At root what is needed for scientific inquiry is just receptivity to data, skill in reasoning, and yearning for truth. Admittedly, ingenuity can help too." (Willard v O Quine, "The Web of Belief", 1970)

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn," The Structure of Scientific Revolutions", 1970)

“Probability is truth in some degree […]” (Errol E Harris, “Hypothesis and Perception: The Roots of Scientific Method”, 1970)

“In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.” (Martin Gardner, “Mathematical Games”, Scientific American, 1973)

“No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect.” (Arthur C Clarke, “Profiles of the Future”, 1973)

“No matter how much reverence is paid to anything purporting to be ‘statistics’, the term has no meaning unless the source, relevance, and truth are all checked.” (Tom Burnam, “The Dictionary of Misinformation”, 1975)

“It seems that truth

Is progressive approximation

In which the relative fraction

Of our spontaneously tolerated residual error

Constantly diminishes.” (R Buckminster Fuller, “And It Came to Pass - Not to Stay”, 1976)

„[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)

"Truth cannot be defined or tested by agreement with 'the world'; for not only do truths differ for different worlds but the nature of „[…] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)

“Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous.” (Nelson Goodman, “Ways of Worldmaking”, 1978)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural.” (Stephen J Gould, “The Mismeasure of Man”, 1980)

"[…] the truth or likeness to truth that much of science pursues is of a rather special kind – we might call it 'physically necessary truth'" (L Jonathan Cohen, "What has science to do with truth?", Synthese 45, 1980)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, “The Role of Mathematics in the Rise of Science”, 1981)

"True, the initial ideas are in general those of an individual, but the establishment of the reality and truth is in general the work of more than one person." (Willard Libby, "Talking to people", 1981)

“In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it.” (Rudy Rucker, “Infinity and the Mind: The science and philosophy of the infinite”, 1982)

"Scientific theories must tell us both what is true in nature, and how we are to explain it. […] Scientific theories are thought to explain by dint of the descriptions they give of reality." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite, The Mathematical Intelligencer, Vol. 5, No. 4, 1983)

"The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me - both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension." (Paul R Halmos, “I Want to Be a Mathematician”, 1985)

"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend, “Farewell to Reason”, 1987)

“Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it.” (Isaac Asimov, [Interview by Bill Moyers] 1988)

“[…] 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”, 1990)

"It is often the scientist’s experience that he senses the nearness of truth when such connections are envisioned. A connection is a step toward simplification, unification. Simplicity is indeed often the sign of truth and a criterion of beauty.” (Mahlon B Hoagland, “Toward the Habit of Truth”, 1990)

"It is not merely the truth of science that makes it beautiful, but its simplicity.” (Walker Percy, “Signposts in a Strange Land”, 1991)

„[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less ‚beautiful‘. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula.“ (Jean Dieudonné, „Mathematics - The Music of Reason“, 1992)

“Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth.” (Clifford Truesdell and Walter Noll, “The Non-Linear Field Theories of Mechanics” 2nd Ed., 1992)

"Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God." (Malba Tahan & Patricia R Baquero, “The Man Who Counted”, 1993)

“The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth’.” (Richard Feynman, “Six Easy Pieces”, 1994)

“[…] equations are like poetry: They speak truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend.” (Michael Guillen, “Five Equations That Changed the World”, 1995)

“In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world.” (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

“Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith.” (K C Cole, “The Universe and the Teacup: The Mathematics of Truth and Beauty”, 1997)

"Mathematical logic deals not with the truth but only with the game of truth.” (Gian-Carlo Rota, “Indiscrete Thoughts”, 1997)

“Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics.” (Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty”, 1997)

“Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net.” (John Polkinghorne, “Belief in God in an Age of Science”, 1998)

“Mathematics has no privileged road to the truth.”(Donald C Benson, “The Moment of Proof: Mathematical Epiphanies”, 1999)

“Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!” (Gregory J Chaitin, “A century of controversy over the foundations of mathematics”, 2000)

"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)

“Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley.” (Yōko Ogawa, "The Housekeeper and the Professor", 2003)

“A model is a simplification or approximation of reality and hence will not reflect all of reality. […] Box noted that ‘all models are wrong, but some are useful’. While a model can never be ‘truth’, a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless.” (Kenneth P Burnham & David R Anderson, “Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach” 2nd Ed., 2005)

"It is art which invents the lies that raise falsehood to its highest affirmative power, that turns the will to deceive into something which is affirmed in the power of falsehood. For the artist, appearance no longer means the negation of the real in this world but this kind of selection, correction, redoubling and affirmation. Then truth perhaps takes on a new sense. Truth is appearance." (Gilles Deleuze, "Nietzsche as Philosopher", 2005)

“Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception.“ (David Ruelle, “The Mathematician's Brain”, 2007)

"It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject.” (Steven G Krantz, "The History and Concept of Mathematical", 2007)

“Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty.” (William Byers, “How Mathematicians Think”, 2007)

“Geometrical truth is (as we now speak) synthetic: it states facts about the world. Such truths are not ordinary truths but essential truths, giving the reality of the empirical world in which they are imperfect embodied.” (Fred Wilson, “The External World and Our Knowledge of It”, 2008)

"The concept of symmetry (invariance) with its rigorous mathematical formulation and generalization has guided us to know the most fundamental of physical laws. Symmetry as a concept has helped mankind not only to define ‘beauty’ but also to express the ‘truth’. Physical laws tries to quantify the truth that appears to be ‘transient’ at the level of phenomena but symmetry promotes that truth to the level of ‘eternity’.” (Vladimir G Ivancevic & Tijana T Ivancevic, “Quantum Leap”, 2008)

"Mathematicians, like priests, seek ‘ideal’, immutable truths and then often try to apply these truths to the real world." (Clifford A Pickover, "The Loom of God: Tapestries of Mathematics and Mysticism", 2009)

"Philosophers have sometimes made a distinction between analytic and synthetic truths. Analytic truths are not verified by observation; true analytic statements are tautologies and are true by virtue of the definitions of their terms and their logical structure. Synthetic truths relate to the material world; the truth of synthetic statements depends on their correspondence to how physical reality works. Mathematics, according to this distinction, deals exclusively with analytic truths. Its statements are all tautologies and are (analytically) true by virtue of their adherence to formal rules of construction." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"There is an absolute nature to truth in mathematics, which is unmatched in any other branch of knowledge. A theorem, once proven, requires independent checking but not repetition or independent derivation to be accepted as correct. […] Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity” (James Glimm, "Reflections and Prospectives", 2009)

“A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms.” (Raymond S Nickerson, “Mathematical Reasoning”, 2010)

“What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?” (Raymond S Nickerson, “Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs”, 2010)

“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri  & Jan von Plato, “Proof Analysis”, 2011)

“[…] statistics is a method of pursuing truth. At a minimum, statistics can tell you the likelihood that your hunch is true in this time and place and with these sorts of people. This type of pursuit of truth, especially in the form of an event’s future likelihood, is the essence of psychology, of science, and of human evolution.” (Arthur Aron et al, "Statistics for Psychology" 6th Ed., 2012)

“Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.” (Edward Frenkel, “Love and Math”, 2014)

"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)

“Scientists generally agree that no theory is 100 percent correct. Thus, the real test of knowledge is not truth, but utility.” (Yuval N Harari, “Sapiens: A brief history of humankind”, 2017)