28 April 2022

Évariste Galois - Collected Quotes

"Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author never does more damage to his readers than when he hides a difficulty." (Évariste Galois, "Deux mémoires d'Analyse pure", 1831)

"Concerning the theory of equations, I have tried to find out under what circumstances equations are solvable by radicals, which gave me the opportunity of investigating thoroughly, and describing, all transformations possible on an equation, even if it is the case that it is not solvable by radicals." (Évariste Galois, [letter to Auguste Chevalier] 1832)

"From the beginning of this century, computational procedures have become so complicated that any progress by those means has become impossible." (Évariste Galois)

"Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work." (Évariste Galois)

"It seems there is no fruit to be drawn from the solution we offer." (Évariste Galois)

"[Mathematics] is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it." (Évariste Galois)

"Science progresses by a series of combinations in which chance plays not the least role. Its life is rough and resembles that of minerals which grow by juxtaposition [accretion]. This applies not only to science such as it emerges [results] from the work of a series of scientists, but also to the particular research of each one of them. In vain would analysts dissimulate: (however abstract it may be, analysis is no more our power than that of others); they do not deduce, they combine, they compare: (it must be sought out, sounded out, solicited.) When they arrive at the truth it is by cannoning from one side to another that they come across it." (Évariste Galois)

"The analysts try in vain to conceal the fact that they do not deduce: they combine, they compose [...] when they do arrive at the truth they stumble over it after groping their way along." (Évariste Galois)

26 April 2022

Nikolai I Lobachevsky - Collected Quotes

"Up to now, one as has not succeeded to find a rigourous proof of that truth [Euclid’s axiom on parallels]. Those which were given may be named only explanations, but do not deserve to be considered, in the full sense, mathematical proofs. " (Nikolai I Lobachevsky, 1823)

"[…] with Newton's and Descartes' time, the whole Mathematics, becoming Analytic, walked so rapid steps forward that they left far behind themselves this study without which they already could do and which had ceased to draw on itself that attention which it deserved before." (Nikolai I Lobachevsky, 1829)

"The theory of parallels explained by us supposes that the lines and the angles are in a dependence which, as we shall see later, nobody is in a position to prove whether it is found in nature or not." (Nikolai I Lobachevsky, "Elements of Geometry", 1830)

"In the nature we properly know only the movement, without which the sensory impressions are impossible. So, every other concepts, for instance, the Geometrical ones, are artificially produced by our intellect, taken from the properties of movement; and that is why the space in itself, separately, for us does not exist. In consequence there cannot be any contradiction in our intellect when we admit that certain forces in nature follow one Geometry and others another particular one of their own." (Nikolai I Lobachevsky, "New Elements of Geometry with a complete theory of Parallels", cca 1835-1838)

"In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it came to us from Euclid.
As belonging to these imperfections, I consider the obscurity in the fundamental concepts of the geometrical magnitudes and in the manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory of parallels, to fill which all efforts of mathematicians have so far been in vain." (Nikolai I Lobachevsky," Geometric researches on the theory of parallels", 1840)

"The new (non-Euclidean) geometry, whose foundation has been laid here, even if it does not exist in nature, nevertheless may exist in our imagination; and, while remaining unused for actual measurements, it opens a new, extensive field for the mutual application of geometry and analysis." (Nikolai I Lobachevsky)

"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." (Nikolai I Lobachevsky)

24 April 2022

On Beliefs: From Fiction to Science-Fiction

"Freethinkers are those who are willing to use their minds without prejudice and without fearing to understand things that clash with their own customs, privileges, or beliefs. This state of mind is not common, but it is essential for right thinking." (Leo Tolstoy, "War and Peace", 1865-1869)

"The world is devoted to physical science, because it believes theses discoveries will increase its capacity of luxury and self-indulgence. But the pursuit of science only leads to the insoluble." (Benjamin Disraeli, "Lothair", 1870)

"The facts of life do not penetrate to the sphere in which our beliefs are cherished; as it was not they that engendered those beliefs, so they are powerless to destroy them." (Marcel Proust, "In Search of Lost Time", 1913)

"So the universe is not quite as you thought it was. You'd better rearrange your beliefs, then. Because you certainly can't rearrange the universe." (Isaac Asimov, "Nightfall", 1941)

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

"The point is that we are all capable of believing things which we know to be untrue, and then, when we are finally proved wrong, impudently twisting the facts so as to show that we were right. Intellectually, it is possible to carry on this process for an indefinite time: the only check on it is that sooner or later a false belief bumps up against solid reality, usually on a battlefield." (George Orwell, "In Front of Your Nose", Tribune, 1946)

"Doublethink means the power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them." (George Orwell, "Nineteen Eighty-Four", 1949)

"There is no belief, however foolish, that will not gather its faithful adherents who will defend it to the death." (Isaac Asimov, "The Stars in Their Courses", 1971)

"Reality is that which when you stop believing in it, it doesn’t go away." (Philip K Dick, "Valis", 1981)

"Dangers lurk in all systems. Systems incorporate the unexamined beliefs of their creators. Adopt a system, accept its beliefs, and you help strengthen the resistance to change." (Frank Herbert, "God Emperor of Dune", 1984)

"Where is the world whose people don't prefer a comfortable, warm, and well-worn belief, however illogical, to the chilly winds of uncertainty?" (Isaac Asimov, "Foundation and Earth", 1986)

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

"Chaos is impatient. It's random. And above all it's selfish. It tears down everything just for the sake of change, feeding on itself in constant hunger. But Chaos can also be appealing. It tempts you to believe that nothing matters except what you want." (Rick Riordan, "The Throne of Fire", 2011)

On Beliefs (1925-1949)

"Rule 2. Any summary of a distribution of numbers in terms of symmetric functions should not give an objective degree of belief in any one of the inferences or predictions to be made therefrom that would cause human action significantly different from what this action would be if the original distributions had been taken as evidence." (Walter A Shewhart, "Economic Control of Quality of Manufactured Product", 1931)

"There is scarcely a scientific axiom that is not nowadays denied by somebody. And at the same time almost any nonsensical theory that may be put forward in the name of science would be almost sure to find believers and disciples." (Max Planck, "Where is Science Going?", 1932)

"A scientist commonly professes to base his beliefs on observations, not theories [...] have never come across anyone who carries this profession into practice. [...] Observation is not sufficient [...]  theory has an important share in determining belief." (Arthur S Eddington, "The Expanding Universe", 1933)

"As words are not the things we speak about, and structure is the only link between them, structure becomes the only content of knowledge. If we gamble on verbal structures that have no observable empirical structures, such gambling can never give us any structural information about the world. Therefore such verbal structures are structurally obsolete, and if we believe in them, they induce delusions or other semantic disturbances." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1933)

"The most important application of the theory of probability is to what we may call 'chance-like' or 'random' events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe - after many unsuccessful attempts - that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"Every new theory as it arises believes in the flush of youth that it has the long sought goal; it sees no limits to its applicability, and believes that at last it is the fortunate theory to achieve the 'right' answer." (Percy W Bridgman, "The Nature of Physical Theory", 1936)

"Both religion and natural science require a belief in God for their activities, to the former He is the starting point, and to the latter the goal of every thought process. To the former He is the foundation, to the latter, the crown of the edifice of every generalized world view." (Max Planck, "Religion and Natural Science", 1937)

"It happens at times that a person believes that he has a world-view, but that there is yet one particular phenomenon that is of such a nature that it baffles the understanding, and that he explains differently and attempts to ignore in order not to harbor the thought that this phenomenon might overthrow the whole view, or that his reflection does not possess enough courage and resolution to penetrate the phenomenon with his world-view." (Søren Kierkegaard, 1938)

"In every writer on philosophy there is a concealed metaphysic, usually unconscious; even if his subject is metaphysics, he is almost certain to have an uncritically believed system which underlies his specific arguments." (Bertrand Russell, "Dewey’s New Logic" [in"The Philosophy of John Dewey", ed. by Paul A Schilpp & Lewis E Hahn, 1939])

"Classes and concepts may, however, also be conceived as real objects, namely classes as 'pluralities of things' or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions." (Kurt Gödel, "The Philosophy of Bertrand Russell", 1944)

"To a scientist a theory is something to be tested. He seeks not to defend his beliefs, but to improve them. He is, above everything else, an expert at ‘changing his mind’." (Wendell Johnson, 1946)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

"Belief has its structures, and its symbols change. Its tradition changes. All the relationships within these forms are inter-dependent. We look at the symbols, we hope to read them, we hope for sharing and communication." (Muriel Rukeyser, "The Life of Poetry", 1949)

On Beliefs (Unsourced)

"A man protesting against error is on the way towards uniting himself with all men that believe in truth." (Thomas Carlyle)

"A theory is a supposition which we hope to be true, a hypothesis is a supposition which we expect to be useful; fictions belong to the realm of art; if made to intrude elsewhere, they become either make-believes or mistakes." (G Johnstone Stoney)

"General impressions are never to be trusted. Unfortunately when they are of long standing they become fixed rules of life and assume a prescriptive right not to be questioned. Consequently those who are not accustomed to original inquiry entertain a hatred and horror of statistics. They cannot endure the idea of submitting sacred impressions to cold-blooded verification. But it is the triumph of scientific men to rise superior to such superstitions, to desire tests by which the value of beliefs may be ascertained, and to feel sufficiently masters of themselves to discard contemptuously whatever may be found untrue." (Sir Francis Galton)

"He who searches for truth must not appease his urge by giving himself up to the narcotic of belief." (Hans Reichenbach)

"[…] if one really understood the central point and its necessity in the construction of the world, one ought to be able to state it in one clear, simple sentence. Until we see the quantum principle with this simplicity we can well believe that we do not know the first thing about the universe, about ourselves, and about our place in the universe." (John A Wheeler)

"Logic makes us reject certain arguments but it cannot make us believe any argument." (Henri Lebesgue)

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." (Leonhard Euler)

"Much of the best mathematical inspiration comes from experience and that it is hardly possible to believe in the existence of an absolute, immutable concept of mathematical rigor, dissociated from all human experience." (John von Neumann)

"On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols’ […]. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient." (Jean Dieudonné)

"Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas." (Albert Einstein)

"Science is the art of creating suitable illusions which the fool believes or argues against, but the wise man enjoys for their beauty or their ingenuity, without being blind to the fact that they are human veils and curtains concealing the abysmal darkness of the unknowable." (Carl G Jung)

"[…] there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as ‘pure mathematicians’ (though ‘purist’ would be more adequate) while the second is, with equal inadequacy, referred to as ‘applied’." (Mark Kac)

"The essential quality of a proof is to compel belief." (Pierre de Fermat)

 "[…] the mathematician learns early to accept no fact, to believe no statement, however apparently reasonable or obvious or trivial, until it has been proved, rigorously and totally by a series of steps proceeding from universally accepted first principles." (Alfred Adler)

"There exists, among mathematicians, a deep-seated and strong belief which sustains them in their abstract studies, namely that none of their problems can remain without any answer." (Gheorghe Ţiţeica)

"We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye, each believing that it can see better with one eye that with two." (Augustus De Morgan)

On Beliefs (-1799)

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

"Nothing is easier than self-deceit. For what each man wishes, that he also believes to be true." (Demosthenes, "Olynthiac", 349 BC)

"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)

"That is probable which for the most part usually comes to pass, or which is a part of the ordinary beliefs of mankind, or which contains in itself some resemblance to these qualities, whether such resemblance be true or false." (Marcus Tullius Cicero, "De Inventione", cca 50 BC)

"To believe only in possibilities, is not faith, but mere Philosophy." (Sir Thomas Browne," Religio Medici", 1643)

"Grant a mathematician but one minute principle, he immediately draws a consequence from it, to which you must necessarily assent; and from this consequence another, till he leads you so far (whether you will or no) that you have much ado to believe all he has proved, and what you have already assented to." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)

"People believe they see Things, which better Observations will convince them could not possibly be seen: Therefore assert nothing till after repeated Experiments and Examinations in all Lights and in all Positions." (Henry Baker, "The Microscope Made Easy", 1742)

"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)

"Conjectures in philosophy are termed hypotheses or theories; and the investigation of an hypothesis founded on some slight probability, which accounts for many appearances in nature, has too often been considered as the highest attainment of a philosopher. If the hypothesis (sic) hangs well together, is embellished with a lively imagination, and serves to account for common appearances - it is considered by many, as having all the qualities that should recommend it to our belief, and all that ought to be required in a philosophical system." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

On Belief: Belief and Mathematics

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality." (Cassius J Keyser,"The Humanization of the Teaching of Mathematics", 1912)

"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35 (904), 1912)

"It is only when one looks not towards the outside, at their utility, but within mathematics itself at the relationship among the unused parts that one sees the other, real face of this science. It is not goal-oriented but uneconomical and passionate… (The mathematician) believes that what he is doing will probably eventually lead to some practical cash value, but this is not what spurs him on; he serves the truth, which is to say his destiny, not its purpose. The result may be economical a thousand times over; what is immanent is a total surrender and a passionate devotion." (Robert Musil, "The Mathematical Man", 1913)

"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)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1950)

"The trouble seems to lie chiefly in the assumption that mathematics is by nature something absolute, unchanging with time and place, and therefore capable of being identified once the genius with the eye sharp enough to perceive and characterize it appears on the human scene. And, since mathematics is nothing of the sort (although the layman will probably go on for centuries hence believing that it is), only failure can ensue from the attempt so to characterize it." (Raymond L Wilder,"Introduction to the Foundations of Mathematics", 1952)

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

"Up until now most economists have concerned themselves with linear systems, not because of any belief that the facts were so simple, but rather because of the mathematical difficulties involved in nonlinear systems [... Linear systems are] mathematically simple, and exact solutions are known. But a high price is paid for this simplicity in terms of special assumptions which must be made." (Paul A Samuelson, "Foundations of Economic Analysis", 1966)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Mathematical truth ultimately depends on an irreducible set of assumptions, which are adopted without demonstration. But to qualify as true knowledge, the assumptions require a warrant for their assertion. There is no valid warrant for mathematical knowledge other than demonstration or proof. Therefore the assumptions are beliefs, not knowledge, and remain open to challenge, and thus to doubt." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. [...] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)

"The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. So deep does this belief run that a branch of science is considered not to be properly understood until it can be cast in mathematics." (Paul C W Davies, "The Mind of God: The Scientific Basis for a Rational World", 1992)

"A fundamental difference between religious and scientific thought is that the received beliefs in religion are ultimately based on revelations or pronouncements, usually by some long dead prophet or priest.[...] Dogma is interpreted by a caste of priests and is accepted by the multitude on faith or under duress. In contrast, the statements of science are derived from the data of observations and experiment, and from the manipulation of these data according to logical and often mathematical procedures." (John A Moore, "Science as a Way of Knowing: The Foundations of Modern Biology", 1993)

"Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations." (Anna Sierpinska,"Understanding in Mathematics", 1994)

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

"To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not ‘more’ beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics." (Calvin C Clawson, "Mathematical Mysteries", 1996)   

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

"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently." (Michael F Atiyah, 2004)

"Contrary to popular belief, mathematics is not a universal language. Rather, mathematics is based on a strict set of definitions and rules that have been instated and to which meaning has been given." (Christopher Tremblay, "Mathematics for Game Developers", 2004)

"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Our inner weighing of evidence is not a careful mathematical calculation resulting in a probabilistic estimate of truth, but more like a whirlpool blending of the objective and the personal. The result is a set of beliefs - both conscious and unconscious - that guide us in interpreting all the events of our lives." (Leonard Mlodinow, "War of the Worldviews: Where Science and Spirituality Meet - and Do Not", 2011)

"There are thousands of apparent mathematical truths out there that we humans have discovered and believe to be true but have so far been unable to prove. They are called conjectures. A conjecture is simply a statement about mathematical reality that you believe to be true [..]" (Paul Lockhart, "Measurement", 2012)

"We tend to think of maths as being an 'exact' discipline, where answers are right or wrong. And it's true that there is a huge part of maths that is about exactness. But in everyday life, numerical answers are sometimes just the start of the debate. If we are trained to believe that every numerical question has a definite, 'right' answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest." (Rob Eastaway, "Maths on the Back of an Envelope", 2019)

"There exists, among mathematicians, a deep-seated and strong belief which sustains them in their abstract studies, namely that none of their problems can remain without any answer." (Gheorghe Ţiţeica)

On Belief: Belief and Science

"The processes of nature lie so deep, that, after all the pains we can take, much, perhaps, will remain undiscovered beyond the reach of human art or skill. But this is no reason why we should give ourselves up to the belief of fictions, be they ever so ingenious, instead of hearkening to the unerring voice of nature […]" (Colin Maclaurin, "An Account of Sir Isaac Newton’s Philosophical Discoveries", 1748)

"Man must cling to the belief that the incomprehensible is comprehensible. Else he would give up investigating. " (Johann Wolfgang von Goethe, "Maxims and Reflections", 1829)

"By degree of probability we really mean, or ought to mean, degree of belief [...] Probability then, refers to and implies belief, more or less, and belief is but another name for imperfect knowledge, or it may be, expresses the mind in a state of imperfect knowledge." (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

"Science for the past is a description, for the future a belief." (Karl Pearson, "The Grammar of Science", 1892)

"For the truly scientific man, the hypothesis is destined solely to enable him to get the facts of nature in some definite order, an order which shall make apparent their connection with the great order and harmony which is believed to be present in the universe." (James M Baldwin, "The Processes of Life Revealed by the Microscope: A Plea for Physiological Histology", Science N.S. Vol. 2 (34), 1895)

"Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us." (Henri Poincaré, "Science and Hypothesis", 1901)

"It [science] involves an intelligent and persistent endeavor to revise current beliefs so as to weed out what is erroneous, to add to their accuracy, and, above all, to give them such shape that the dependencies of the various facts upon one another may be as obvious as possible." (John Dewey, "Democracy and Education", 1916)

"Science herself consults her heart when she lays it down that the infinite ascertainment of fact and correction of false belief are the supreme goods for man." (William James, "Selected Papers on Philosophy", 1918)

"The question whether any branch of science can ever become purely deductive is easily answered. It cannot. If science deals with the external world, as we believe it does, and not merely with the relations of propositions then no branch of science can ever be purely deductive. Deductive reasoning by itself can never tell us about facts. The use of deduction in science is to serve as a calculus to make our observations go further, not to take the place of observation." (Arthur D Ritchie, "Scientific Method: An Inquiry into the Character and Validity of Natural Laws", 1923)

"There is scarcely a scientific axiom that is not nowadays denied by somebody. And at the same time almost any nonsensical theory that may be put forward in the name of science would be almost sure to find believers and disciples." (Max Planck, "Where is Science Going?", 1932)

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The important distinction between science and those other systematizations [i.e., art, philosophy, and theology] is that science is self-testing and self-correcting. Here the essential point of science is respect for objective fact. What is correctly observed must be believed [...] the competent scientist does quite the opposite of the popular stereotype of setting out to prove a theory; he seeks to disprove it. (George G Simpson, "Notes on the Nature of Science", 1962)

"Science, indeed, tells us a very great deal less about the universe than we have been accustomed to suppose, and there is no reason to believe that all we can ever know must be couched in terms of its thin and largely arbitrary abstractions." (John W N Sullivan, "Art and Reality", 1964)

"[…] the social scientist who lacks a mathematical mind and regards a mathematical formula as a magic recipe, rather than as the formulation of a supposition, does not hold forth much promise. A mathematical formula is never more than a precise statement. It must not be made into a Procrustean bed - and that is what one is driven to by the desire to quantify at any cost. It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can, would once have believed in witchcraft. The chief merit of mathematicization is that it compels us to become conscious of what we are assuming." (Bertrand de Jouvenel, "The Art of Conjecture", 1967)

"In science we always know much less than we believe we do." (Erwin Chargaff, "Uncertainties Great, Is the Gain Worth the Risk?", Chemical and Engineering News, 1977)

"The so-called exact sciences often are not as exact as is commonly believed. How often they infer the existence of a hat from the emergence of a rabbit!" (Erwin Chargaff, "Voices in the Labyrinth: Nature, Man and Science", 1977)

"I find it more difficult, but also much more fun, to get the right answer by indirect reasoning and before all the evidence is in. It’s what a theoretician does in science. But the conclusions drawn in this way are obviously more risky than those drawn by direct measurement, and most scientists withhold judgment until there is more direct evidence available. The principal function of such detective work - apart from entertaining the theoretician - is probably to so annoy and enrage the observationalists that they are forced, in a fury of disbelief, to perform the critical measurements." (Carl Sagan, "The Cosmic Connection: An Extraterrestrial Perspective", 1975)

"Many people believe that reasoning, and therefore science, is a different activity from imagining. But this is a fallacy […] Reasoning is constructed with movable images just as certainly as poetry is." (Jacob Bronowski, "Visionary Eye", 1978)

"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true - a preconception that always, and necessarily, goes a little way (sometimes a long way) beyond anything which we have logical or factual authority to believe in. It is the invention of a possible world, or of a tiny fraction of that world. The conjecture is then exposed to criticism to find out whether or not that imagined world is anything like the real one. Scientific reasoning is therefore at all levels an interaction between two episodes of thought - a dialogue between two voices, the one imaginative and the other critical; a dialogue, as I have put it, between the possible and the actual, between proposal and disposal, conjecture and criticism, between what might be true and what is in fact the case. (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The scientific method is a potentiation of common sense, exercised with a specially firm determination not to persist in error if any exertion of hand or mind can deliver us from it. Like other exploratory processes, it can be resolved into a dialogue between fact and fancy, the actual and the possible; between what could be true and what is in fact the case. The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticise and modify as we go along, so that it ends by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The degree of confirmation assigned to any given hypothesis is sensitive to properties of the entire belief system [...] simplicity, plausibility, and conservatism are properties that theories have in virtue of their relation to the whole structure of scientific beliefs taken collectively. A measure of conservatism or simplicity would be a metric over global properties of belief systems." (Jerry Fodor, "Modularity of Mind", 1983)

"There are those who try to generalize, synthesize, and build models, and there are those who believe nothing and constantly call for more data. The tension between these two groups is a healthy one; science develops mainly because of the model builders, yet they need the second group to keep them honest." (Andrew Miall,"Principles of Sedimentary Basin Analysis", 1984)

"To a considerable degree science consists in originating the maximum amount of information with the minimum expenditure of energy. Beauty is the cleanness of line in such formulations along with symmetry, surprise, and congruence with other prevailing beliefs." (Edward O Wilson, "Biophilia", 1984)

"Modern philosophy of science has gone far beyond the naive belief that science reveals the truth. Even if it could, we would have no means of proving it. Certainty seems unattainable. All scientific statements remain open to doubt. […] We cannot reach the absolute at least as far as science is concerned; we have to content ourselves with the relative." (Rolf Sattler, "Biophilosophy", 1986)

"Competent scientists do not believe their own models or theories, but rather treat them as convenient fictions. […] The issue to a scientist is not whether a model is true, but rather whether there is another whose predictive power is enough better to justify movement from today's fiction to a new one." (Steve Vardeman, " Comment", Journal of the American Statistical Association 82, 1987)

"Cybernetics is simultaneously the most important science of the age and the least recognized and understood. It is neither robotics nor freezing dead people. It is not limited to computer applications and it has as much to say about human interactions as it does about machine intelligence. Today’s cybernetics is at the root of major revolutions in biology, artificial intelligence, neural modeling, psychology, education, and mathematics. At last there is a unifying framework that suspends long-held differences between science and art, and between external reality and internal belief." (Paul Pangaro, "New Order From Old: The Rise of Second-Order Cybernetics and Its Implications for Machine Intelligence", 1988)

"The use of Occam’s razor, along with the related critical, skeptical view toward any speculations about the unknown, is perhaps the most misunderstood aspect of the scientific method. People confuse doubt with denial. Science doesn’t deny anything, but it doubts everything not required by the data. Note, however, that doubt does not necessarily mean rejection, just an attitude of disbelief that can be changed when the facts require it." (Victor J Stenger, "Physics and Psychics: The Search for a World Beyond the Senses", 1990)

"On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting - the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings." (John D Barrow, New Theories of Everything", 1991)

"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. [...] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)

"The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. So deep does this belief run that a branch of science is considered not to be properly understood until it can be cast in mathematics." (Paul C W Davies, "The Mind of God: The Scientific Basis for a Rational World", 1992)

"The most natural way to give an independence proof is to establish a model with the required properties. This is not the only way to proceed since one can attempt to deal directly and analyze the structure of proofs. However, such an approach to set theoretic questions is unnatural since all our intuition come from our belief in the natural, almost physical model of the mathematical universe." (Paul J Cohen, Set Theory and the Continuum Hypothesis", 1966) instead of with words, sounds or watercolors." (John Casti, "Reality Rules", 1992)

"A fundamental difference between religious and scientific thought is that the received beliefs in religion are ultimately based on revelations or pronouncements, usually by some long dead prophet or priest.[...] Dogma is interpreted by a caste of priests and is accepted by the multitude on faith or under duress. In contrast, the statements of science are derived from the data of observations and experiment, and from the manipulation of these data according to logical and often mathematical procedures." (John A Moore, "Science as a Way of Knowing: The Foundations of Modern Biology", 1993)

"Our beliefs about ourselves in relation to the world around us are the roots of our values, and our values determine not only our immediate actions, but also, over the course of time, the form of our society. Our beliefs are increasingly determined by science." (Henry P Stapp, "Mind, Matter and Quantum Mechanics", 1993)

"Science demands a tolerance for ambiguity. Where we are ignorant, we withhold belief. Whatever annoyance the uncertainty engenders serves a higher purpose: It drives us to accumulate better data. This attitude is the difference between science and so much else. Science offers little in the way of cheap thrills. The standards of evidence are strict. But when followed they allow us to see far, illuminating even a great darkness." (Carl Sagan, "Pale Blue Dot: A Vision of the Human Future in Space", 1994)

"Having a scientific outlook means being willing to divest yourself of a pet hypothesis, whether it relates to easy self-help improvements, homeopathy, graphology, spontaneous generation, or any other concept, when the data produced by a carefully designed experiment contradict that hypothesis. Retaining a belief in a hypothesis that cannot be supported by data is the hallmark of both the pseudoscientist and the fanatic. Often the more deeply held the hypothesis, the more reactionary is the response to nonsupportive data." (Michael Zimmerman, "Science, Nonscience, and Nonsense: Approaching Environmental Literacy", 1995)

"In science, for a theory to be believed, it must make a prediction - different from those made by previous theories - for an experiment not yet done. For the experiment to be meaningful, we must be able to get an answer that disagrees with that prediction. When this is the case, we say that a theory is falsifiable - vulnerable to being shown false. The theory also has to be confirmable, it must be possible to verify a new prediction that only this theory makes. Only when a theory has been tested and the results agree with the theory do we advance the statement to the rank of a true scientific theory." (Lee Smolin, "The Trouble with Physics", 2006)

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition." (Thomas S Kuhn, "The Essential Tension: Selected Studies in Scientific Tradition and Change", 2011)

"In the classical deterministic scenario, a model consists of a few variables and physical constants. The relational structure of the model is conceptualized by the scientist via intuition gained from thinking about the physical world. Intuition means that the scientist has some mental construct regarding the interactions beyond positing a skeletal mathematical system he believes is sufficiently rich to capture the interactions and then depending upon data to infer the relational structure and estimate a large number of parameters." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

On Belief: Belief and Probability

"That is probable which for the most part usually comes to pass, or which is a part of the ordinary beliefs of mankind, or which contains in itself some resemblance to these qualities, whether such resemblance be true or false." (Marcus Tullius Cicero, "De Inventione", cca 50 BC)

"Every probability - and most of our common, working beliefs are probabilities - is provided with buffers at both ends, which break the force of opposite opinions clashing against it; but scientific certainty has no spring in it, no courtesy, no possibility of yielding. All this must react on the minds which handle these forms of truth." (Paul-Henri T d'Holbach [Baron d'Holbach], "The System of Nature, Or, Laws of the Moral and Physical World", 1770)

"Conjectures in philosophy are termed hypotheses or theories; and the investigation of an hypothesis founded on some slight probability, which accounts for many appearances in nature, has too often been considered as the highest attainment of a philosopher. If the hypothesis (sic) hangs well together, is embellished with a lively imagination, and serves to account for common appearances - it is considered by many, as having all the qualities that should recommend it to our belief, and all that ought to be required in a philosophical system." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

"The probability of an event is the reason we have to believe that it has taken place, or that it will take place." (Siméon-Denis Poisson,"'Règles générales des probabilités", 1837)

"By degree of probability we really mean, or ought to mean, degree of belief [...] Probability then, refers to and implies belief, more or less, and belief is but another name for imperfect knowledge, or it may be, expresses the mind in a state of imperfect knowledge." (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

"Every probability - and most of our common, working beliefs are probabilities - is provided with buffers at both ends, which break the force of opposite opinions clashing against it; but scientific certainty has no spring in it, no courtesy, no possibility of yielding. All this must react on the minds which handle these forms of truth. (Oliver W Holmes, "The Autocrat of the Breakfast-Table", 1891)

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

"We must have a logical intuition of the probable relations between propositions. Once the existence of this relation between evidence and conclusion, the latter becomes the subject of the degree of belief." (John M Keynes, "Treatise on Probability", 1921)

"The most important application of the theory of probability is to what we may call 'chance-like' or 'random' events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe - after many unsuccessful attempts - that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"To a scientist a theory is something to be tested. He seeks not to defend his beliefs, but to improve them. He is, above everything else, an expert at ‘changing his mind’." (Wendell Johnson, 1946)

"As a set of cognitive beliefs, religion is a speculative hypothesis of an extremely low order of probability." (Sidney Hook, The Partisan Review, 1950)

"Meanwhile, for those who are not aware of it, it is necessary to mention that in the conception we follow and sustain here only subjective probabilities exist - i.e. the degree of belief in the occurrence of an event attributed by a given person at a given instant and with a given set of in information. This is in contrast to other conceptions which limit themselves to special types of cases in which they attribute meaning to 'objective probabilities' (for instance, cases of symmetry as for dice etc., 'statistical' cases of 'repeatable' events, etc.)." (Bruno de Finetti, "Theory of Probability", 1974)

"Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs." (Bruno de Finetti, "Theory of Probability", 1974)

"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Burton G Malkiel, "A Random Walk Down Wall Street", 1989)

"By a variable we will mean an attribute, measurement or inquiry that may take on one of several possible outcomes, or values, from a specified domain. If we have beliefs (i.e., probabilities) attached to the possible values that a variable may attain, we will call that variable a random variable." (Judea Pearl, "Causality: Models, Reasoning, and Inference", 2000)

"Probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive." (Nassim N Taleb, "Fooled by Randomness", 2001)

"The objectivist view is that probabilities are real aspects of the universe - propensities of objects to behave in certain ways - rather than being just descriptions of an observer’s degree of belief. For example, the fact that a fair coin comes up heads with probability 0.5 is a propensity of the coin itself. In this view, frequentist measurements are attempts to observe these propensities. Most physicists agree that quantum phenomena are objectively probabilistic, but uncertainty at the macroscopic scale - e.g., in coin tossing - usually arises from ignorance of initial conditions and does not seem consistent with the propensity view. (Stuart J. Russell & Peter Norvig, "Artificial Intelligence: A Modern Approach", 2010)

"We are far too willing to reject the belief that much of what we see in life is random." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Our inner weighing of evidence is not a careful mathematical calculation resulting in a probabilistic estimate of truth, but more like a whirlpool blending of the objective and the personal. The result is a set of beliefs - both conscious and unconscious - that guide us in interpreting all the events of our lives." (Leonard Mlodinow, "War of the Worldviews: Where Science and Spirituality Meet - and Do Not", 2011)

"One kind of probability - classic probability - is based on the idea of symmetry and equal likelihood […] In the classic case, we know the parameters of the system and thus can calculate the probabilities for the events each system will generate. […] A second kind of probability arises because in daily life we often want to know something about the likelihood of other events occurring […]. In this second case, we need to estimate the parameters of the system because we don’t know what those parameters are. […] A third kind of probability differs from these first two because it’s not obtained from an experiment or a replicable event - rather, it expresses an opinion or degree of belief about how likely a particular event is to occur. This is called subjective probability […]." (Daniel J Levitin, "Weaponized Lies", 2017)

On Beliefs (2010-2019)

"Each person has a different mental model and, therefore, potentially a different interpretation of the Facts. The danger comes when we start to assume that our interpretation of the Facts is the only interpretation and we believe that what we see and think is the Truth, and that there is only one Truth." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

"The objectivist view is that probabilities are real aspects of the universe - propensities of objects to behave in certain ways - rather than being just descriptions of an observer’s degree of belief. For example, the fact that a fair coin comes up heads with probability 0.5 is a propensity of the coin itself. In this view, frequentist measurements are attempts to observe these propensities. Most physicists agree that quantum phenomena are objectively probabilistic, but uncertainty at the macroscopic scale - e.g., in coin tossing - usually arises from ignorance of initial conditions and does not seem consistent with the propensity view. (Stuart J. Russell and Peter Norvig, "Artificial Intelligence: A Modern Approach", 2010)

"A general limitation of the human mind is its imperfect ability to reconstruct past states of knowledge, or beliefs that have changed. Once you adopt a new view of the world (or any part of it), you immediately lose much of your ability to recall what you used to believe before your mind changed." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Chaos is impatient. It's random. And above all it's selfish. It tears down everything just for the sake of change, feeding on itself in constant hunger. But Chaos can also be appealing. It tempts you to believe that nothing matters except what you want." (Rick Riordan, "The Throne of Fire", 2011)

"Mental models and rules are intimately intertwined, since the models often suggest clear rules for societies to follow. Religions are more than theories; they are prescriptive moral codes that seek to enforce rules on their followers. They, like the rules they enjoin, are invested with considerable emotional meaning and therefore are believed for intrinsic reasons and not simply because they are accurate or useful. While religious beliefs cannot be verified, they are also difficult to falsify. All of this reinforces the fundamental conservatism of human societies, because mental models of reality once adopted are hard to change in the light of new evidence that they are not working." (Francis Fukuyama, "The Origins of Political Order", 2011)

"Our inner weighing of evidence is not a careful mathematical calculation resulting in a probabilistic estimate of truth, but more like a whirlpool blending of the objective and the personal. The result is a set of beliefs - both conscious and unconscious - that guide us in interpreting all the events of our lives." (Leonard Mlodinow, "War of the Worldviews: Where Science and Spirituality Meet - and Do Not", 2011)

"The confidence that individuals have in their beliefs depends mostly on the quality of the story they can tell about what they see, even if they see little." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition." (Thomas S Kuhn, "The Essential Tension: Selected Studies in Scientific Tradition and Change", 2011)

"When people believe a conclusion is true, they are also very likely to believe arguments that appear to support it, even when these arguments are unsound." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Image theory is an attempt to describe decision making as it actually occurs. […] The concept of images is central to the theory. They represent visions held by individuals and organisations that constitute how they believe the world should exist. When considering individuals, the theory refers to these images as the value image, trajectory image and strategic image. The value image is based on an individual’s ethics, morals and beliefs. The trajectory images encompass the decision maker’s goals and aspirations. Finally, for each trajectory image, a decision maker may have one or more strategic images that contain their plans, tactics and forecasts for their goal. […] In an organisational decision-making setting, these images are referred to as culture, vision and strategy." (Christopher B Stephenson, "What causes top management teams to make poor strategic decisions?", 2012)

"There are thousands of apparent mathematical truths out there that we humans have discovered and believe to be true but have so far been unable to prove. They are called conjectures. A conjecture is simply a statement about mathematical reality that you believe to be true [..]" (Paul Lockhart, "Measurement", 2012)

"System dynamics models have little impact unless they change the way people perceive a situation. A model must help to organize information in a more understandable way. A model should link the past to the present by showing how present conditions arose, and extend the present into persuasive alternative futures under a variety of scenarios determined by policy alternatives. In other words, a system dynamics model, if it is to be effective, must communicate with and modify the prior mental models. Only people's beliefs - that is, their mental models - will determine action. Computer models must relate to and improve mental models if the computer models are to fill an effective role." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"A very different - and very incorrect - argument is that successes must be balanced by failures (and failures by successes) so that things average out. Every coin flip that lands heads makes tails more likely. Every red at roulette makes black more likely. […] These beliefs are all incorrect. Good luck will certainly not continue indefinitely, but do not assume that good luck makes bad luck more likely, or vice versa." (Gary Smith, "Standard Deviations", 2014)

"The search for better numbers, like the quest for new technologies to improve our lives, is certainly worthwhile. But the belief that a few simple numbers, a few basic averages, can capture the multifaceted nature of national and global economic systems is a myth. Rather than seeking new simple numbers to replace our old simple numbers, we need to tap into both the power of our information age and our ability to construct our own maps of the world to answer the questions we need answering." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 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)

"Learning is a dynamic event and so the belief that learning is primarily about the acquisition of facts is fundamentally flawed - the acquisition and manipulation of data is at best a prerequisite to learning. Real learning involves acquiring knowledge and understanding." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"A worldview consists of observations of the individual and other people with respect to the self, time and space, the natural and the supernatural and the sacred and profane. […] Beliefs about the world do not reside in the human mind in chaotic disorder; rather they form a latent system. A worldview cannot, however, be viewed as a well-organised network of cognitive models or a static collection of values; instead it should be regarded as the product of a process shaped by historical, cultural and social perspectives and contexts." (Helena Helve, "A longitudinal perspective on worldviews, values and identities", 2016)

"In the classical deterministic scenario, a model consists of a few variables and physical constants. The relational structure of the model is conceptualized by the scientist via intuition gained from thinking about the physical world. Intuition means that the scientist has some mental construct regarding the interactions beyond positing a skeletal mathematical system he believes is sufficiently rich to capture the interactions and then depending upon data to infer the relational structure and estimate a large number of parameters." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"One kind of probability - classic probability - is based on the idea of symmetry and equal likelihood […] In the classic case, we know the parameters of the system and thus can calculate the probabilities for the events each system will generate. […] A second kind of probability arises because in daily life we often want to know something about the likelihood of other events occurring […]. In this second case, we need to estimate the parameters of the system because we don’t know what those parameters are. […] A third kind of probability differs from these first two because it’s not obtained from an experiment or a replicable event - rather, it expresses an opinion or degree of belief about how likely a particular event is to occur. This is called subjective probability […]." (Daniel J Levitin, "Weaponized Lies", 2017)

"We tend to think of maths as being an 'exact' discipline, where answers are right or wrong. And it's true that there is a huge part of maths that is about exactness. But in everyday life, numerical answers are sometimes just the start of the debate. If we are trained to believe that every numerical question has a definite, 'right' answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest." (Rob Eastaway, "Maths on the Back of an Envelope", 2019)

On Beliefs (2000-2009)

"By a variable we will mean an attribute, measurement or inquiry that may take on one of several possible outcomes, or values, from a specified domain. If we have beliefs (i.e., probabilities) attached to the possible values that a variable may attain, we will call that variable a random variable." (Judea Pearl, "Causality: Models, Reasoning, and Inference", 2000)

"Of course, the information systems governing the feedback we receive can change as we learn. They are part of the feedback structure of our systems. Through our mental models we define constructs such as GDP or scientific research, create metrics for these ideas, and design information systems to evaluate and report them. These then condition the perceptions we form. Changes in our mental models are constrained by what we previously chose to define, measure, and attend to. Seeing is believing and believing is seeing. They feed back on one another." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Agent subroutines may pass information back and forth, but subroutines are not changed as a result of the interaction, as people are. In real social interaction, information is exchanged, but also something else, perhaps more important: individuals exchange rules, tips, beliefs about how to process the information. Thus a social interaction typically results in a change in the thinking processes - not just the contents - of the participants." (James F Kennedy et al, "Swarm Intelligence", 2001)

"Probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive." (Nassim N Taleb, "Fooled by Randomness", 2001)

"The danger arises when a culture takes its own story as the absolute truth, and seeks to impose this truth on others as the yardstick of all knowledge and belief." (F David Peat, "From Certainty to Uncertainty", 2002)

"There are endless examples of elaborate structures and apparently complex processes being generated through simple repetitive rules, all of which can be easily simulated on a computer. It is therefore tempting to believe that, because many complex patterns can be generated out of a simple algorithmic rule, all complexity is created in this way." (F David Peat, "From Certainty to Uncertainty", 2002)

"If 'memory' is a misleading metaphor for recording devices, so is the epithet 'problem solver' for our computing machines. Of course, they are no problem solvers, because they do not have any problems in the first place. It is our problems they help us solve like any other useful tool, say, a hammer which may be dubbed a 'problem solver' for driving nails into a board. The danger in this subtle semantic twist by which the responsibility for action is shifted from man to a machine lies in making us lose sight of the problem of cognition. By making us believe that the issue is how to find solutions to some well defined problems, we may forget to ask first what constitutes a 'problem', what is its 'solution', and - when a problem is identified - what makes us want to solve it." (Heinz von Foerster, "Understanding Understanding: Essays on Cybernetics and Cognition", 2003)

"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Didier Sornette, "Why Stock Markets Crash: Critical events in complex financial systems", 2003)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently." (Michael F Atiyah, 2004)

"Contrary to popular belief, mathematics is not a universal language. Rather, mathematics is based on a strict set of definitions and rules that have been instated and to which meaning has been given." (Christopher Tremblay,"Mathematics for Game Developers", 2004)

"The Bayesian approach is based on the following postulates: (B1) Probability describes degree of belief, not limiting frequency. As such, we can make probability statements about lots of things, not just data which are subject to random variation. […] (B2) We can make probability statements about parameters, even though they are fixed constants. (B3) We make inferences about a parameter θ by producing a probability distribution for θ. Inferences, such as point estimates and interval estimates, may then be extracted from this distribution. (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"The important thing is to understand that frequentist and Bayesian methods are answering different questions. To combine prior beliefs with data in a principled way, use Bayesian inference. To construct procedures with guaranteed long run performance, such as confidence intervals, use frequentist methods. Generally, Bayesian methods run into problems when the parameter space is high dimensional." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"We must begin by distinguishing between visual mental imagery and visual perception: Visual perception occurs while a stimulus is being viewed, and includes functions such as visual recognition (i. e., registering that a stimulus is familiar) and identification (i. e., recalling the name, context, or other information associated with the object). Two types of mechanisms are used in visual perception: ‘bottom-up’ mechanisms are driven by the input from the eyes; in contrast, ‘top-down’ mechanisms make use of stored information (such as knowledge, belief, expectations, and goals). Visual mental imagery is a set of representations that gives rise to the experience of viewing a stimulus in the absence of appropriate sensory input. In this case, information in memory underlies the internal events that produce the experience. Unlike afterimages, mental images are relatively prolonged." (Stephen M Kosslyn, "Mental images and the brain", Cognitive Neuropsychology 22, 2005)

"Context is not as simple as being in a different space [...] context includes elements like our emotions, recent experiences, beliefs, and the surrounding environment - each element possesses attributes, that when considered in a certain light, informs what is possible in the discussion." (George Siemens, "Knowing Knowledge", 2006)

"In science, for a theory to be believed, it must make a prediction - different from those made by previous theories - for an experiment not yet done. For the experiment to be meaningful, we must be able to get an answer that disagrees with that prediction. When this is the case, we say that a theory is falsifiable - vulnerable to being shown false. The theory also has to be confirmable, it must be possible to verify a new prediction that only this theory makes. Only when a theory has been tested and the results agree with the theory do we advance the statement to the rank of a true scientific theory." (Lee Smolin, "The Trouble with Physics", 2006)

"Often our mental models serve a very useful and practical purpose by helping us to make very quick sense of our experiences and interactions. However, the danger for us is that our mental models do not always reflect the truth, i.e. the way things really are. Often they reflect what we believe to be true, and sometimes we get it wrong." (John Middleton, "Upgrade Your Brain: 52 brilliant ideas for everyday genius", 2006)

"Our generational perspective contributes to the mental models we hold about ourselves, the world, and the way things ‘should’ be. These beliefs create blind spots that can become our undoing as we pursue our values and seek to accomplish our goals. Likewise, they can have a powerful effect on our culture." (Deborah Gilburg,"Empowering Multigenerational Collaboration in the Workplace", The Systems Thinker Vol. 18 No. 4, 2007)

"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Mental models reflect the beliefs, values, and assumptions that we personally hold, and they underlie our reasons for doing things the way we do." (Kambiz E Maani & Robert Y Cavana,"Systems Methodology", The Systems Thinker Vol. 18 No. 8, 2007)

"Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities." (Tom G Palmer, "Realizing Freedom: Libertarian Theory, History, and Practice", 2009)

"Gaining awareness about how the system is built up and how it works can also help us to avoid solutions that only treat the symptoms of an underlying problem without curing the problem itself. System thinking is powerful because it helps us to see our own mental models and how these models color our perception of the world. In many cases, it is difficult for us to alter our mental models. There are always some beliefs or viewpoints that we are not willing to change, no matter what evidence is presented against it. This causes a certain resistance to new concepts. Problems can occur, however, when a rigid mental model stands in the way of a solution that might solve a problem. In such situations, adherence to mental models can be dangerous to the health of the organization." (Akhilesh Bajaj & Stanisław Wrycza, "Systems Analysis and Design for Advanced Modeling Methods: Best Practices", 2009)

"Knowledge about things beyond our immediate environment may be acquired through deduction, if the initial premises are believed to be correct." (Nayef Al-Rodhan, "Sustainable History and the Dignity of Man", 2009)

On Beliefs (1990-1999)

"It is always hard when reality intrudes on belief." (Alan D Foster, "Cyber Way", 1990)

"The use of Occam’s razor, along with the related critical, skeptical view toward any speculations about the unknown, is perhaps the most misunderstood aspect of the scientific method. People confuse doubt with denial. Science doesn’t deny anything, but it doubts everything not required by the data. Note, however, that doubt does not necessarily mean rejection, just an attitude of disbelief that can be changed when the facts require it." (Victor J Stenger, "Physics and Psychics: The Search for a World Beyond the Senses", 1990)

"A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective." (Edward Teller, "Conversations on the Dark Secrets of Physics", 1991)

"A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective." (Edward Teller, "Conversations on the Dark Secrets of Physics", 1991)

"A model can not be proved to be correct; at best it can only be found to be reasonably consistant and not to contradict some of our beliefs of what reality is." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Mathematical truth ultimately depends on an irreducible set of assumptions, which are adopted without demonstration. But to qualify as true knowledge, the assumptions require a warrant for their assertion. There is no valid warrant for mathematical knowledge other than demonstration or proof. Therefore the assumptions are beliefs, not knowledge, and remain open to challenge, and thus to doubt." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting - the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings." (John D Barrow, New Theories of Everything", 1991)

"The most persuasive positive argument for mental images as objects is [that] whenever one thinks one is seeing something there must be something one is seeing. It might be an object directly, or it might be a mental picture. [This] point is so plausible that it is deniable only at the peril of becoming arbitrary. One should concede that the question whether mental images are entities of some sort is not resolvable by logical or linguistic analysis, and believe what makes sense of experience." (Eva T H Brann, "The World of Imagination" , 1991)

"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. [...] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)

"The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. So deep does this belief run that a branch of science is considered not to be properly understood until it can be cast in mathematics." (Paul C W Davies, "The Mind of God: The Scientific Basis for a Rational World", 1992)

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

"A fundamental difference between religious and scientific thought is that the received beliefs in religion are ultimately based on revelations or pronouncements, usually by some long dead prophet or priest.[...] Dogma is interpreted by a caste of priests and is accepted by the multitude on faith or under duress. In contrast, the statements of science are derived from the data of observations and experiment, and from the manipulation of these data according to logical and often mathematical procedures." (John A Moore, "Science as a Way of Knowing: The Foundations of Modern Biology", 1993)

"Our beliefs about ourselves in relation to the world around us are the roots of our values, and our values determine not only our immediate actions, but also, over the course of time, the form of our society. Our beliefs are increasingly determined by science." (Henry P Stapp, "Mind, Matter and Quantum Mechanics", 1993)

"Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations." (Anna Sierpinska, "Understanding in Mathematics", 1994)

"Science demands a tolerance for ambiguity. Where we are ignorant, we withhold belief. Whatever annoyance the uncertainty engenders serves a higher purpose: It drives us to accumulate better data. This attitude is the difference between science and so much else. Science offers little in the way of cheap thrills. The standards of evidence are strict. But when followed they allow us to see far, illuminating even a great darkness." (Carl Sagan, "Pale Blue Dot: A Vision of the Human Future in Space", 1994)

"Having a scientific outlook means being willing to divest yourself of a pet hypothesis, whether it relates to easy self-help improvements, homeopathy, graphology, spontaneous generation, or any other concept, when the data produced by a carefully designed experiment contradict that hypothesis. Retaining a belief in a hypothesis that cannot be supported by data is the hallmark of both the pseudoscientist and the fanatic. Often the more deeply held the hypothesis, the more reactionary is the response to nonsupportive data." (Michael Zimmerman, "Science, Nonscience, and Nonsense: Approaching Environmental Literacy", 1995)

"The leading edge of growth of intelligence is at the cultural and societal level. It is like a mind that is struggling to wake up. This is necessary because the most difficult problems we face are now collective ones. They are caused by complex global interactions and are beyond the scope of individuals to understand and solve. Individual mind, with its isolated viewpoints and narrow interests, is no longer enough." (Jeff Wright, "Basic Beliefs", [email] 1995)

"The mental model, in turn, can be considered as a syntactic language of thought whose semantic interpretation is provided by the actual world. In this sense, a person's beliefs are true to the extent that they correspond to the world." (William J Rapaport, "Understanding Understanding: Syntactic Semantics and Computational Cognition", Philosophical Perspectives Vol. 9, 1995)

"To talk about sensemaking is to talk about reality as an ongoing accomplishment that takes form when people make retrospective sense of the situations in which they find themselves and their creations. There is a strong reflexive quality to this process. People make sense of things by seeing a world on which they already imposed what they believe. In other words, people discover their own inventions. This is why sensemaking can be understood as invention and interpretations understood as discovery. These are complementary ideas. If sensemaking is viewed as an act of invention, then it is also possible to argue that the artifacts it produces include language games and texts." (Karl E Weick, "Sensemaking in Organizations", 1995)

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

"The more we know about our universe, the more difficult it becomes to believe in determinism." (Ilya Prigogine, "The End of Certainty", 1997)

"To begin with, we must understand that any mindset consists of mental models, or concepts, that influence our interpretation of situations and predispose us to certain responses. These models, which are replete with beliefs and assumptions, thus strongly determine the way we understand the world and act in it. The irony is, they become so ingrained in us, as tendencies and predispositions, that we seldom pay attention to them." (Stephen G Haines, "The Manager's Pocket Guide to Strategic and Business Planning", 1998)

"To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not ‘more’ beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics." (Calvin C Clawson, "Mathematical Mysteries", 1996)   

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

"Beliefs are those ideas we take as true and use to guide our actions. We all have beliefs about what sort of people we are and what we are capable of. These beliefs act as permissions for or limitations on what we do. When we believe something is possible, we will try it; if we believe it impossible, we will not." (Joseph O’Connor, "Leading With NLP: Essential Leadership Skills for Influencing and Managing People", 1998)

"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)

"People respond to their experience, not to reality itself. We do not know what reality is. Our senses, beliefs, and past experience give us a map of the world from which to operate." (Joseph O’Connor, "Leading With NLP: Essential Leadership Skills for Influencing and Managing People", 1998)

On Beliefs (1980-1989)

"A schema, then is a data structure for representing the generic concepts stored in memory. There are schemata representing our knowledge about all concepts; those underlying objects, situations, events, sequences of events, actions and sequences of actions. A schema contains, as part of its specification, the network of interrelations that is believed to normally hold among the constituents of the concept in question. A schema theory embodies a prototype theory of meaning. That is, inasmuch as a schema underlying a concept stored in memory corresponds to the meaning of that concept, meanings are encoded in terms of the typical or normal situations or events that instantiate that concept." (David E Rumelhart, "Schemata: The building blocks of cognition", 1980)

"Once we have accepted a configuration of schemata, the schemata themselves provide a richness that goes far beyond our observations. […] In fact, once we have determined that a particular schema accounts for some event, we may not be able to determine which aspects of our beliefs are based on direct sensory information and which are merely consequences of our interpretation." (David E Rumelhart, "Schemata: The building blocks of cognition", 1980)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Reality is that which when you stop believing in it, it doesn’t go away." (Philip K Dick, "Valis", 1981)

"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true - a preconception that always, and necessarily, goes a little way (sometimes a long way) beyond anything which we have logical or factual authority to believe in. It is the invention of a possible world, or of a tiny fraction of that world. The conjecture is then exposed to criticism to find out whether or not that imagined world is anything like the real one. Scientific reasoning is therefore at all levels an interaction between two episodes of thought - a dialogue between two voices, the one imaginative and the other critical; a dialogue, as I have put it, between the possible and the actual, between proposal and disposal, conjecture and criticism, between what might be true and what is in fact the case. (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The scientific method is a potentiation of common sense, exercised with a specially firm determination not to persist in error if any exertion of hand or mind can deliver us from it. Like other exploratory processes, it can be resolved into a dialogue between fact and fancy, the actual and the possible; between what could be true and what is in fact the case. The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticise and modify as we go along, so that it ends by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"The connection between cause and effect takes place in time. This temporary relation may be defined in various ways. Some people believe that cause always precedes effect, that there is a certain interval between the time when the cause begins to act (for example, the interaction of two systems) and the time the effect appears. For a certain time cause and effect coexist, then the cause dies out and the consequence ultimately becomes the cause of something else. And so on to infinity." (Alexander Spirkin, "Dialectical Materialism", 1983)

"The degree of confirmation assigned to any given hypothesis is sensitive to properties of the entire belief system [...] simplicity, plausibility, and conservatism are properties that theories have in virtue of their relation to the whole structure of scientific beliefs taken collectively. A measure of conservatism or simplicity would be a metric over global properties of belief systems." (Jerry Fodor, "Modularity of Mind", 1983)

"Dangers lurk in all systems. Systems incorporate the unexamined beliefs of their creators. Adopt a system, accept its beliefs, and you help strengthen the resistance to change." (Frank Herbert, "God Emperor of Dune", 1984)

"There are those who try to generalize, synthesize, and build models, and there are those who believe nothing and constantly call for more data. The tension between these two groups is a healthy one; science develops mainly because of the model builders, yet they need the second group to keep them honest." (Andrew Miall, "Principles of Sedimentary Basin Analysis", 1984)

"To a considerable degree science consists in originating the maximum amount of information with the minimum expenditure of energy. Beauty is the cleanness of line in such formulations along with symmetry, surprise, and congruence with other prevailing beliefs." (Edward O Wilson, "Biophilia", 1984)

"Modern philosophy of science has gone far beyond the naive belief that science reveals the truth. Even if it could, we would have no means of proving it. Certainty seems unattainable. All scientific statements remain open to doubt. […] We cannot reach the absolute at least as far as science is concerned; we have to content ourselves with the relative." (Rolf Sattler, "Biophilosophy", 1986)

"The assumption of rationality has a favored position in economics. It is accorded all the methodological privileges of a self-evident truth, a reasonable idealization, a tautology, and a null hypothesis. Each of these interpretations either puts the hypothesis of rational action beyond question or places the burden of proof squarely on any alternative analysis of belief and choice. The advantage of the rational model is compounded because no other theory of judgment and decision can ever match it in scope, power, and simplicity." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"Competent scientists do not believe their own models or theories, but rather treat them as convenient fictions. […] The issue to a scientist is not whether a model is true, but rather whether there is another whose predictive power is enough better to justify movement from today's fiction to a new one." (Steve Vardeman, " Comment", Journal of the American Statistical Association 82, 1987)

"Cybernetics is simultaneously the most important science of the age and the least recognized and understood. It is neither robotics nor freezing dead people. It is not limited to computer applications and it has as much to say about human interactions as it does about machine intelligence. Today’s cybernetics is at the root of major revolutions in biology, artificial intelligence, neural modeling, psychology, education, and mathematics. At last there is a unifying framework that suspends long-held differences between science and art, and between external reality and internal belief." (Paul Pangaro, "New Order From Old: The Rise of Second-Order Cybernetics and Its Implications for Machine Intelligence", 1988)

"A culture may be conceived as a network of beliefs and purposes in which any string in the net pulls and is pulled by the others, thus perpetually changing the configuration of the whole. If the cultural element called morals takes on a new shape, we must ask what other strings have pulled it out of line. It cannot be one solitary string, nor even the strings nearby, for the network is three-dimensional at least." (Jacques Barzun, "The Culture We Deserve", 1989)

"A model is generally more believable if it can predict what will happen, rather than 'explain' something that has already occurred. […] Model building is not so much the safe and cozy codification of what we are confident about as it is a means of orderly speculation." (James R Thompson, "Empirical Model Building", 1989)

"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Burton G Malkiel, "A Random Walk Down Wall Street", 1989)

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