🪷On Mind: Mental Models I

“We form ourselves images or symbols of external objects; and the form which we give them is such that the necessary consequents of the images in thought are always the images of the necessary consequents in nature of the things pictured." (Heinrich Hertz, 1894)

“All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly." (Ludwig Boltzmann, 1899)

“The logical picture of the facts is the thought. […] A picture is a model of reality. In a picture objects have the elements of the picture corresponding to them. The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way.” (Ludwig Wittgenstein, “Tractatus Logico-Philosophicus”, 1922)

“If the organism carries a ‘small-scale model’ of external reality and of its possible actions within its head, it is able to try out various alternatives, conclude which is the best of them, react to future simulations before they arise, utilize the knowledge of past events in dealing with the present and the future, and in every way to react in a much fuller, safer, and more competent manner to the emergencies which face it.” (Kenneth Craik, “The Nature of Explanation”, 1943)

“While the stuff from which our world picture is build is yielded exclusively from the sense organs as organs of the mind, so that every man's world picture is and always remains a construct of his mind and cannot be proved to have any other existence […]” (Erwin Schrodinger, “What is Life?”, 1944)

“As our mental eye penetrates into smaller and smaller distances and shorter and shorter times, we find nature behaving so entirely differently from what we observe in visible and palpable bodies of our surroundings that no model shaped after our large-scale experiences can ever be ‘true’. A complete satisfactory model of this type is not only practically inaccessible, but not even thinkable. Or, to be precise, we can, of course, think of it, but however we think it, it is wrong; not perhaps quite as meaningless as a ‘triangular circle’, but more so than a ‘winged lion’.” (Erwin Schrödinger, “Science and Humanism”, 1952)
 
“This other world is the so-called physical world image; it is merely an intellectual structure. To a certain extent it is arbitrary. It is a kind of model or idealization created in order to avoid the inaccuracy inherent in every measurement and to facilitate exact definition.” (Max Planck, “The Philosophy of Physics”, 1963)

“They [archetypes] are, at the same time, both images and emotions. One can speak of an archetype only when these two aspects are simultaneous. When there is merely the image, then there is simply a word picture of little consequence. But by being charged with emotion, the image gains numinosity (or psychic energy); it becomes dynamic, and consequences of some kind must flow from it.” (Carl G Jung, “Man and His Symbols”, 1964)

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

See also:
Mental Models II, III, IV, V, VI, VII, VIII

On (Scientific) Bias II

“The human mind can hardly remain entirely free from bias, and decisive opinions are often formed before a thorough examination of a subject from all its aspects has been made.” (Helena P. Blavatsky, “The Secret Doctrine”, 1888)

“The classification of facts, the recognition of their sequence and relative significance is the function of science, and the habit of forming a judgment upon these facts unbiased by personal feeling is characteristic of what may be termed the scientific frame of mind.” (Karl Pearson, “The Grammar of Science”, 1892)

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

"Scientific discovery, or the formulation of scientific theory, starts in with the unvarnished and unembroidered evidence of the senses. It starts with simple observation - simple, unbiased, unprejudiced, naive, or innocent observation - and out of this sensory evidence, embodied in the form of simple propositions or declarations of fact, generalizations will grow up and take shape, almost as if some process of crystallization or condensation were taking place. Out of a disorderly array of facts, an orderly theory, an orderly general statement, will somehow emerge." (Sir Peter B Medawar, "Is the Scientific Paper Fraudulent?", The Saturday Review, 1964)

“Numbers have undoubted powers to beguile and benumb, but critics must probe behind numbers to the character of arguments and the biases that motivate them.” (Stephen Jay Gould, “An Urchin in the Storm: Essays About Books and Ideas”, 1987)

“But our ways of learning about the world are strongly influenced by the social preconceptions and biased modes of thinking that each scientist must apply to any problem. The stereotype of a fully rational and objective ‘scientific method’, with individual scientists as logical (and interchangeable) robots, is self-serving mythology.” (Stephen Jay Gould, “This View of Life: In the Mind of the Beholder”, “Natural History”, Vol. 103, No. 2, 1994)

“The human brain always concocts biases to aid in the construction of a coherent mental life, exclusively suitable for an individual’s personal needs.” (Abhijit Naskar, “We Are All Black: A Treatise on Racism”, 2017)

“Science is the search for truth, that is the effort to understand the world: it involves the rejection of bias, of dogma, of revelation, but not the rejection of morality.” (Linus Pauling)

“Facts and values are entangled in science. It's not because scientists are biased, not because they are partial or influenced by other kinds of interests, but because of a commitment to reason, consistency, coherence, plausibility and replicability. These are value commitments.” (Alva Noë)

“A scientist has to be neutral in his search for the truth, but he cannot be neutral as to the use of that truth when found. If you know more than other people, you have more responsibility, rather than less.” (Charles P Snow)

Early Glimpses of Bias

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

“Men willingly believe what they wish to be true.” (Julius Caesar, “De Bello Gallico”, Book III, 58–49 BC)

“You can have no greater or lesser dominion than the one over yourself. The greatest deception men suffer is from their own opinions.” (Leonardo da Vinci)

“Man prefers to believe what he prefers to be true.” (Francis Bacon, “Novum Organum”, 1620)

“The human brain is a complex organ with the wonderful power of enabling man to find reasons for continuing to believe whatever it is that he wants to believe.” (Voltaire)

“Men judge things according to the disposition of their minds, and had rather imagine things than understand them.” (Baruch Spinoza, “Ethics”, Book I, 1677)

“Reasoning will never make a Man correct an ill Opinion, which by Reasoning he never acquired.” ( Jonathan Swift, “A Letter to a Young Gentleman, Lately Enter’d Into Holy Orders by a Person of Quality”, 1721)

“It is hard to prevent oneself from believing what one so keenly desires, and who can doubt that the interest we have in admitting or denying the reality of the Judgement to come determines the faith of most men in accordance with their hopes and fears.” (Jean-Jacques Rousseau, “Reveries of the Solitary Walker”, 1782)

“Men are not to be reasoned out of an opinion that they have not reasoned themselves into.” (Fisher Ames, 1786)

“How little ground there can be to hope that men may be reasoned out of their errors, when in fact they were never reasoned into them." (Lyman Beecher, 1823)

📊On Statistics: Looking into the Crystal Ball of Statistics

“The aim of every science is foresight. For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past.” (Auguste Compte, "Plan des travaux scientifiques nécessaires pour réorganiser la société", 1822) 

“No matter how solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not contradict it, if we undertake to verify it . […] It is far better to foresee even without certainty than not to foresee at all.” (Henri Poincaré, “The Foundations of Science”, 1913)

“[…] the statistical prediction of the future from the past cannot be generally valid, because whatever is future to any given past, is in tum past for some future. That is, whoever continually revises his judgment of the probability of a statistical generalization by its successively observed verifications and failures, cannot fail to make more successful predictions than if he should disregard the past in his anticipation of the future. This might be called the ‘Principle of statistical accumulation’.” (Clarence I Lewis, “Mind and the World-Order: Outline of a Theory of Knowledge”, 1929)

“The only useful function of a statistician is to make predictions, and thus to provide a basis for action.” (William E Deming)

“To say that observations of the past are certain, whereas predictions are merely probable, is not the ultimate answer to the question of induction; it is only a sort of intermediate answer, which is incomplete unless a theory of probability is developed that explains what we should mean by ‘probable’ and on what ground we can assert probabilities.” (Hans Reichenbach, “The Rise of Scientific Philosophy”, 1951)

“It is never possible to predict a physical occurrence with unlimited precision.” (Max Planck, “The Meaning of Causality in Physics”, 1953)

“Predictions, prophecies, and perhaps even guidance - those who suggested this title to me must have hoped for such-even though occasional indulgences in such actions by statisticians has undoubtedly contributed to the characterization of a statistician as a man who draws straight lines from insufficient data to foregone conclusions!” (John W Tukey, “Where do We Go From Here?”, Journal of the American Statistical Association, Vol. 55, No. 289, 1960)

“Can there be laws of chance? The answer, it would seem should be negative, since chance is in fact defined as the characteristic of the phenomena which follow no law, phenomena whose causes are too complex to permit prediction.” (Félix E Borel, “Probabilities and Life”, 1962)

“All predictions are statistical, but some predictions have such a high probability that one tends to regard them as certain.” (Marshall J Walker, “The Nature of Scientific Thought”, 1963)

“The moment you forecast you know you’re going to be wrong, you just don’t know when and in which direction.” (Edgar R Fiedler, “Across the Board”, 1977)

🎲On Probability Theory: On Probability (Trivia)

"Man is so complicated a machine that it is impossible to get a clear idea of the machine beforehand, and hence impossible to define it. For this reason, all the investigations have been vain, which the greatest philosophers have made à priori, that is to say, in so far as they use, as it were, the wings of the spirit. Thus it is only à posteriori or by trying to disentangle the soul from the organs of the body, so to speak, that one can reach the highest probability concerning man's own nature, even though one can not discover with certainty what his nature is." (Julien Offray de La Mettrie, "Man a Machine", 1747)

"Coincidences, in general, are great stumbling blocks in the way of that class of thinkers who have been educated to know nothing of the theory of probabilities - that theory to which the most glorious objects of human research are indebted for the most glorious of illustrations." (Edgar Allen Poe, "The Murders in the Rue Morgue", 1841)

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

"Life is a gamble at terrible odds; if it was a bet you wouldn't take it." (Tom Stoppard, "Rosencrantz and Guildenstern are Dead", 1955)

"As is known, the question of the objectivity or the subjectivity of probability has divided the world of science into two camps. Some maintain that there exist two types of probability, as above, others, that only the subjective exists, because regardless of what is supposed to take place, we cannot have full knowledge of it. Therefore, some lay the uncertainty of future events at the door of our knowledge of them, whereas others place it within the realm of the events themselves." (Stanisław Lem, " A Perfect Vacuum", 1971)

"People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual coincidence is far more unusual than any coincidence could possibly be." (Isaac Asimov, "The Planet That Wasn't", 1976)

"Luck was not probability, but it acted through probability. It was, so to speak, quantities of probability, a quantitative average throughout the universe. And like any other fixed quantity, it could only be concentrated or increased at the cost of a diminution elsewhere." (Barrington J Bayley, "The Grand Wheel", 1977)

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

"Probability does pervade the universe, and in this sense, the old chestnut about baseball imitating life really has validity. The statistics of streaks and slumps, properly understood, do teach an important lesson about epistemology, and life in general. The history of a species, or any natural phenomenon, that requires unbroken continuity in a world of trouble, works like a batting streak. All are games of a gambler playing with a limited stake against a house with infinite resources. The gambler must eventually go bust. His aim can only be to stick around as long as possible, to have some fun while he's at it, and, if he happens to be a moral agent as well, to worry about staying the course with honor!" (Stephen J Gould, 1991)

"Misunderstanding of probability may be the greatest of all impediments to scientific literacy." (Stephen J Gould, "Dinosaur in a haystack: reflections in natural history", 1995)

🎲On Probability Theory: On Probability (1950-1974)

"It is never possible to predict a physical occurrence with unlimited precision." (Max Planck, "The Meaning of Causality in Physics", 1953)

"The epistemological value of probability theory is based on the fact that chance phenomena, considered collectively and on a grand scale, create non-random regularity." (Andrey Kolmogorov, "Limit Distributions for Sums of Independent Random Variables", 1954)

"Just as in applied statistics the crux of a problem is often the devising of some method of sampling that avoids bias, our problem is that of finding a probability assignment which avoids bias, while agreeing with whatever information is given. The great advance provided by information theory lies in the discovery that there is a unique, unambiguous criterion for the 'amount of uncertainty' represented by a discrete probability distribution, which agrees with our intuitive notions that a broad distribution represents more uncertainty than does a sharply peaked one, and satisfies all other conditions which make it reasonable." (Edwin T Jaynes, "Information Theory and Statistical Mechanics" I, 1956)

"[...] the whole course of events is determined by the laws of probability; to a state in space there corresponds a definite probability, which is given by the de Brogile wave associated with the state." (Max Born, "Atomic Physics", 1957)

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

"Probability is a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation." (William Feller, "An Introduction to Probability Theory and Its Applications", 1957)

"Everybody has some idea of the meaning of the term 'probability' but there is no agreement among scientists on a precise definition of the term for the purpose of scientific methodology. It is sufficient for our purpose, however, if the concept is interpreted in terms of relative frequency, or more simply, how many times a particular event is likely to occur in a large population." (Alfred R Ilersic, "Statistics", 1959)

"The mathematician, the statistician, and the philosopher do different things with a theory of probability. The mathematician develops its formal consequences, the statistician applies the work of the mathematician and the philosopher describes in general terms what this application consists in. The mathematician develops symbolic tools without worrying overmuch what the tools are for; the statistician uses them; the philosopher talks about them. Each does his job better if he knows something about the work of the other two." (Irvin J Good, "Kinds of Probability", Science Vol. 129, 1959)

"The practical power of a statistical test is the product of its’ statistical power and the probability of use." (John W Tukey, A Quick, "Compact, Two Sample Test to Duckworth’s Specifications", 1959)

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

"The probability concept used in probability theory has exactly the same structure as have the fundamental concepts in any field in which mathematical analysis is applied to describe and represent reality." (Richard von Mises, "Mathematical Theory of Probability and Statistics", 1964)

"After all, without the experiment - either a real one or a mathematical model - there would be no reason for a theory of probability." (Thornton C Fry, "Probability and Its Engineering Uses", 1965)

"The statistician has no magic touch by which he may come in at the stage of tabulation and make something of nothing. Neither will his advice, however wise in the early stages of a study, ensure successful execution and conclusion. Many a study, launched on the ways of elegant statistical design, later boggled in execution, ends up with results to which the theory of probability can contribute little." (W Edwards Deming, "Principles of Professional Statistical Practice", Annals of Mathematical Statistics, 36(6), 1965)

"[...] in probability theory we are faced with situations in which our intuition or some physical experiments we have carried out suggest certain results. Intuition and experience lead us to an assignment of probabilities to events. As far as the mathematics is concerned, any assignment of probabilities will do, subject to the rules of mathematical consistency." (Robert Ash, "Basic probability theory", 1970)

"Probability theory, for us, is not so much a part of mathematics as a part of logic, inductive logic, really. It provides a consistent framework for reasoning about statements whose correctness or incorrectness cannot be deduced from the hypothesis. The information available is sufficient only to make the inferences 'plausible' to a greater or lesser extent." (Ralph Baierlein, "Atoms and Information Theory: An Introduction to Statistical Mechanics", 1971)

"[...] we will adopt the broad view and will take 'probability', to be a quantitative relation, between a hypothesis and an inference, corresponding to the degree of rational belief in the correctness of the inference, given the hypothesis. The hypothesis is the information we possess, or assume for the sake of argument. The inference is a statement that, to a greater or lesser extent, is justified by the hypothesis. Thus 'the probability' of an inference, given a hypothesis, is the degree of rational belief in the correctness of the inference, given the hypothesis." (Ralph Baierlein, "Atoms and Information Theory: An Introduction to Statistical Mechanics", 1971)

"The relevant question is not whether ANOVA assumptions are met exactly, but rather whether the plausible violations of the assumptions have serious consequences on the validity of probability statements based on the standard assumptions." (Gene V Glass et al, "Consequences of Failure to Meet Assumptions Underlying the Fixed Effects Analyses of Variance and Covariance", Review of Educational Research Vol. 42 (3), 1972)

"Since small differences in probability cannot be appreciated by the human mind, there seems little point in being excessively precise about uncertainty." (George E P Box & G C Tiao, "Bayesian inference in statistical analysis", 1973)

🎲On Probability Theory: On Probability (1900-1924)

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

"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)

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"Nature prefers the more probable states to the less probable because in nature processes take place in the direction of greater probability. Heat goes from a body at higher temperature to a body at lower temperature because the state of equal temperature distribution is more probable than a state of unequal temperature distribution." (Max Planck, "The Atomic Theory of Matter", 1909

"One can hardly give a satisfactory definition of probability." (Henri Poincaré, "Calcul des Probabilités", 1912)

"Sometimes the probability in favor of a generalization is enormous, but the infinite probability of certainty is never reached." (William Dampier-Whetham, "Science and the Human Mind", 1912)

"It is difficult to find an intelligible account of the meaning of ‘probability’, or of how we are ever to determine the probability of any particular proposition; and yet treatises on the subject profess to arrive at complicated results of the greatest precision and the most profound practical importance." (John M Keynes, "A Treatise on Probability", 1921)

"We know that the probability of well-established induction is great, but, when we are asked to name its degree we cannot. Common sense tells us that some inductive arguments are stronger than others, and that some are very strong. But how much stronger or how strong we cannot express." (John M Keynes, "A Treatise on Probability", 1921)

"The second law of thermodynamics appears solely as a law of probability, entropy as a measure of the probability, and the increase of entropy is equivalent to a statement that more probable events follow less probable ones." (Max Planck, "A Survey of Physics", 1923)

🎲On Probability Theory: On Probability (1800-1824)

"The function just found cannot, it is true, express rigorously the probabilities of the errors: for since the possible errors are in all cases confined within certain limits, the probability of errors exceeding those limits ought always to be zero, while our formula always gives some value. However, this defect, which every analytical function must, from its nature, labor under, is of no importance in practice, because the value of our function decreases so rapidly […] that it can safely be considered as vanishing. Besides, the nature of the subject never admits of assigning with absolute rigor the limits of error." (Carl F Gauss, "Theoria motus corporum coelestium in sectionibus conicis solem ambientum", 1809)

"Probability has reference partly to our ignorance, partly to our knowledge [..] The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all cases possible is the measure of this probability, which is thus simply a fraction whose number is the number of favorable cases and whose denominator is the number of all cases possible." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities", 1814)

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

"One of the great advantages of the calculus of probabilities is to teach us to distrust first opinions." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities", 1814)

"In all speculations on the origin, or agents that have produced the changes on this globe, it is probable that we ought to keep within the boundaries of the probable effects resulting from the regular operations of the great laws of nature which our experience and observation have brought within the sphere of our knowledge. When we overleap those limits, and suppose a total change in nature's laws, we embark on the sea of uncertainty, where one conjecture is perhaps as probable as another; for none of them can have any support, or derive any authority from the practical facts wherewith our experience has brought us acquainted." (William Maclure, "Observations on the Geology of the United States of America", 1817)

"From the foregoing we see that the two justifications each leave something to be desired. The first depends entirely on the hypothetical form of the probability of the error; as soon as that form is rejected, the values of the unknowns produced by the method of least squares are no more the most probable values than is the arithmetic mean in the simplest case mentioned above. The second justification leaves us entirely in the dark about what to do when the number of observations is not large. In this case the method of least squares no longer has the status of a law ordained by the probability calculus but has only the simplicity of the operations it entails to recommend it." (Carl Friedrich Gauss, "Anzeige: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior", Göttingische gelehrte Anzeigen, 1821)

🎲On Probability Theory: On Probability (1700-1749)

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"We define the art of conjecture, or stochastic art, as the art of evaluating as exactly as possible the probabilities of things, so that in our judgments and actions we can always base ourselves on what has been found to be the best, the most appropriate, the most certain, the best advised; this is the only object of the wisdom of the philosopher and the prudence of the statesman." (Jacob Bernoulli, "Ars Conjectandi", 1713)

"Probability is a degree of certainty and it differs from certainty as a part from a whole." (Jacob Bernoulli, "Ars Conjectandi", 1713)

"The probability of an Event is greater, or less, according to the number of Chances by which it may Happen, compar’d with the number of all the Chances, by which it may either Happen or Fail. […] Therefore, if the Probability of Happening and Failing are added together, the Sum will always be equal to Unit." (Abraham De Moivre, "The Doctrine of Chances", 1718)

"Probable evidence, in its very nature, affords but an imperfect kind of information, and is to be considered as relative only to beings of limited capacities. For nothing which is the possible object of knowledge, whether past, present, or future, can be probable to an infinite Intelligence; […] to us, probability is the very guide of life." (Joseph Butler, "The Analogy of Religion, Natural and Revealed, to the Constitution and Course of Nature", 1736)

"Do not expect to arrive at certainty in every subject which you pursue. There are a hundred things wherein we mortals […] must be content with probability, where our best light and reasoning will reach no farther." (Isaac Watts," The Improvement of the Mind", 1741)

🎲On Probability Theory: On Probability (1600-1699)

"Thus, joining the rigor of demonstrations in mathematics with the uncertainty of chance, and conciliating these apparently contradictory matters, it can, taking its name from both of them, with justice arrogate the stupefying name: The Mathematics of Chance." (Blaise Pascal, [Address to the Academie Parisienne de Mathematiques] 1654)

"As a Foundation to the following Proposition, I shall take Leave to lay down this Self-evident Truth: That any one Chance or Expectation to win any thing is worth just such a Sum, as wou’d procure in the same Chance and Expectation at a fair Lay." (Christiaan Huygens, "De ratiociniis in ludo aleae", 1657)

"The good or evil of an event should be considered in view of the event's likelihood of occurrence." (Antoine Amauld & Pierre Nicole, "The Art of Thinking: Port-Royal Logic", 1662)

"Take away probability, and you can no longer please the world; give probability, and you can no longer displease it." (Blaise Pascal, "Thoughts", 1670)

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

"Probability is a degree of possibility." (Gottfried W Leibniz, "On estimating the uncertain", 1676)

"Probability, however, is not something absolute, [it is] drawn from certain information which, although it does not suffice to resolve the problem, nevertheless ensures that we judge correctly which of the two opposites is the easiest (facilius) given the conditions known to us." (Gottfried W Leibniz, "Forethoughts for an encyclopaedia or universal science", cca. 1679)

"Consider however (imitating Mathematicians) certainty or truth to be like the whole; and probabilities [to be like] parts, such that probabilities would be to truths what an acute angle [is] to a right [angle]." (Gottfried W Leibniz, [ to Vincent Placcius] 1687)

"The probable is something which lies midway between truth and error" (Christian Thomasius, "Institutes of Divine Jurisprudence", 1688)

"Probability is the appearance of agreement upon fallible proofs. As demonstration is the showing the agreement or disagreement of two ideas by the intervention of one or more proofs, which have a constant, immutable, and visible connexion one with another; so probability is nothing but the appearance of such an agreement or disagreement by the intervention of proofs, whose connexion is not constant and immutable, or at least is not perceived to be so, but is, or appears for the most part to be so, and is enough to induce the mind to judge the proposition to be true or false, rather than the contrary." (John Locke, "An Essay Concerning Human Understanding", Book IV, 1689)

"Probability is likeliness to be true, the very notation of the word signifying such a proposition, for which there be arguments or proofs to make it pass, or be received for true. […] The grounds of probability are two: conformity with our own experience, or the testimony of others' experience. Probability then, being to supply the defect of our knowledge and to guide us where that fails, is always conversant about propositions whereof we have no certainty, but only some inducements to receive them for true."  (John Locke, "An Essay Concerning Human Understanding", Book IV, 1689)

"It is impossible for a Die, with such determin’d force and direction, not to fall on such a determin’d side, only I don’t know the force and direction which makes it fall on such a determin’d side, and therefore I call that Chance, which is nothing but want of Art…" (John Arbuthnot, "Of the Laws of Chance", 1692)