Mental Models IV (Limitations I)

“However, and conversely, our models fall far short of representing the world fully. That is why we make mistakes and why we are regularly surprised. In our heads, we can keep track of only a few variables at one time. We often draw illogical conclusions from accurate assumptions, or logical conclusions from inaccurate assumptions. Most of us, for instance, are surprised by the amount of growth an exponential process can generate. Few of us can intuit how to damp oscillations in a complex system.” (Donella H Meadows, “Limits to Growth”, 1972)

"The problem with mental models lie not in whether they are right or wrong - by definition, all models are simplifications. The problem with mental models arise when they become implicit - when they exist below the level of our awareness. “[…] models, if unexamined, limit an organization's range of actions to what is familiar and comfortable. [...] Each person's mental model focuses on different parts of the system. Each emphasizes different cause-effect chains. This makes it virtually impossible for a shared picture of the system as a whole to emerge in normal conversation." (Peter Senge, “The Fifth Discipline”, 1990)

“Mental models are the images, assumptions, and stories which we carry in our minds of ourselves, other people, institutions, and every aspect of the world. Like a pane of glass framing and subtly distorting our vision, mental models determine what we see. Human beings cannot navigate through the complex environments of our world without cognitive ‘mental maps’; and all of these mental maps, by definition, are flawed in some way.” (Peter M Senge et al, “The Fifth Discipline Fieldbook: Strategies and Tools for Building a Learning Organization”, 1994)

“What are the models? Well, the first rule is that you’ve got to have multiple models - because if you just have one or two that you’re using, the nature of human psychology is such that you’ll torture reality so that it fits your models, or at least you’ll think it does.” (Charles Munger, 1994)

"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)

“[…] our mental models fail to take into account the complications of the real world - at least those ways that one can see from a systems perspective. It is a warning list. Here is where hidden snags lie. You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long-term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays. You are likely to mistreat, misdesign, or misread systems if you don’t respect their properties of resilience, self-organization, and hierarchy.” (Donella H Meadows, “Thinking in Systems: A Primer”, 2008)

“The discrepancy between our mental models and the real world may be a major problem of our times; especially in view of the difficulty of collecting, analyzing, and making sense of the unbelievable amount of data to which we have access today.” (Ugo Bardi, “The Limits to Growth Revisited”, 2011)

Mental Models III

"Theories usually result from the precipitate reasoning of an impatient mind which would like to be rid of phenomena and replace them with images, concepts, indeed often with mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

“Everything we think we know about the world is a model. Every word and every language is a model. All maps and statistics, books and databases, equations and computer programs are models. So are the ways I picture the world in my head - my mental models. None of these is or ever will be the real world. […] Our models usually have a strong congruence with the world. That is why we are such a successful species in the biosphere. Especially complex and sophisticated are the mental models we develop from direct, intimate experience of nature, people, and organizations immediately around us.” (Donella Meadows, “Limits to Growth”, 1972)

“Concepts are inventions of the human mind used to construct a model of the world. They package reality into discrete units for further processing, they support powerful mechanisms for doing logic, and they are indispensable for precise, extended chains of reasoning. […] A mental model is a cognitive construct that describes a person's understanding of a particular content domain in the world.” (John Sown, “Conceptual Structures: Information Processing in Mind and Machine”, 1984)

“We construct mental models that provide us with situations in which we can interact with mental objects that represent objects, properties and relations and that behave in ways that simulate the objects, properties and relations that our models represent. […] The concepts and principles that a person understands, in this sense, are embedded in the kinds of objects that he or she includes in mental models and in the ways in which those objects behave, including how they combine and separate to form other objects.” (James G Greeno, “Number sense as situated knowing in a conceptual domain”, Journal for Research on Mathematics Education Vol. 22 No. 3, 1991)

“A mental model is conceived […] as a knowledge structure possessing slots that can be filled not only with empirically gained information but also with ‘default assumptions’ resulting from prior experience. These default assumptions can be substituted by updated information so that inferences based on the model can be corrected without abandoning the model as a whole. Information is assimilated to the slots of a mental model in the form of ‘frames’ which are understood here as ‘chunks’ of knowledge with a well-defined meaning anchored in a given body of shared knowledge.” (Jürgen Renn, “Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy”, “The Universe of General Relativity” Ed. by A.J. Kox & Jean Eisenstaedt, 2005)

“Mental models can be literal representations of the external world (as they often are with visual imagery) or arbitrary representations (as they are with propositional, mathematical, or verbal models). In either case, they are explanatory or descriptive representations of the external world.” (Gregory J Feist, “The Psychology of Science and the Origins of the Scientific Mind”, 2006)

“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)

“All models (whether mental or those turned into computer maps/models) are developed using a particular lens of what we value - what we think is important to understand, or what performance we wish to develop or improve. Although organizations can build forum models focusing on the performance measure du jour, they would be well advised to use a systemic or integral framework for what to include.” (Peggy Holman et al, “The Change Handbook”, 2007)

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

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

Mathematical Models II

"Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models." (Ludwig Boltzmann, "Lectures on Gas Theory", 1896)

"It is characteristic of modern physics to represent all processes in terms of mathematical equations. But the close connection between the two sciences must not blur their essential difference." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

“Formulation of a mathematical model is the first step in the process of analyzing the behaviour of any real system. However, to produce a useful model, one must first adopt a set of simplifying assumptions which have to be relevant in relation to the physical features of the system to be modelled and to the specific information one is interested in. Thus, the aim of modelling is to produce an idealized description of reality, which is both expressible in a tractable mathematical form and sufficiently close to reality as far as the physical mechanisms of interest are concerned.” (Francois Axisa, “Discrete Systems” Vol. I, 2001)

“A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, “Process Modelling and Model Analysis”, 2001)

“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)

“Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem.” (John A Adam, “Mathematics in Nature”, 2003)

“Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it.” (Sandip Banerjee, “Mathematical Modeling: Models, Analysis and Applications”, 2014)

“A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. [...] A mathematical model is never a completely accurate representation of a physical situation - it is an idealization." (James Stewart, “Calculus: Early Transcedentals” 8th Ed., 2016)

"Different models serve different purposes. Setting up a model involves focusing on features of the phenomenon that are compatible with the methodology being proposed, and neglecting features that are not compatible with it. A mathematical model in applied science explicitly refrains from attempting to be a complete picture of the phenomenon being modeled." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)

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

See also:
Mathematical Models I
Models in Physics
On Models
Good Models
Models vs. Facts

On Models: Are All Models Wrong?

“[…] no models are [true] = not even the Newtonian laws. When you construct a model you leave out all the details which you, with the knowledge at your disposal, consider inessential. […] Models should not be true, but it is important that they are applicable, and whether they are applicable for any given purpose must of course be investigated. This also means that a model is never accepted finally, only on trial.” (Georg Rasch, “Probabilistic Models for Some Intelligence and Attainment Tests”, 1960)

“Celestial navigation is based on the premise that the Earth is the center of the universe. The premise is wrong, but the navigation works. An incorrect model can be a useful tool.” (R A J Phillips, “A Day in the Life of Kelvin Throop”, Analog Science Fiction and Science Fact, Vol. 73 No. 5, 1964)

“Since all models are wrong the scientist cannot obtain a ‘correct’ one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity.” (George Box, “Science and Statistics", Journal of the American Statistical Association 71, 1976)

“A model of the universe does not require faith, but a telescope. If it is wrong, it is wrong.” (Paul C W Davies, “Space and Time in the Modern Universe”, 1977)

"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)

“The fact that [the model] is an approximation does not necessarily detract from its usefulness because models are approximations. All models are wrong, but some are useful.” (George Box, 1987)

“[…] it does not seem helpful just to say that all models are wrong. The very word model implies simplification and idealization. The idea that complex physical, biological or sociological systems can be exactly described by a few formulae is patently absurd. The construction of idealized representations that capture important stable aspects of such systems is, however, a vital part of general scientific analysis and statistical models, especially substantive ones, do not seem essentially different from other kinds of model.” (Sir David Cox, "Comment on ‘Model uncertainty, data mining and statistical inference’", Journal of the Royal Statistical Society, Series A 158, 1995)

“I do not know that my view is more correct; I do not even think that ‘right’ and ‘wrong’ are good categories for assessing complex mental models of external reality - for models in science are judged [as] useful or detrimental, not as true or false.” (Stephen Jay Gould, “Dinosaur in a Haystack: Reflections in Natural History”, 1995)

“No matter how beautiful the whole model may be, no matter how naturally it all seems to hang together now, if it disagrees with experiment, then it is wrong.” (John Gribbin, “Almost Everyone’s Guide to Science”, 1999)

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

"First, we affirm that all models are wrong, some of them are useful. Since a model is an abstraction of reality, and that too only from a particular perspective, they are fundamentally wrong because they are not reality. That gives no license to models that are wrongly built - after all, two wrongs don’t make a right. So usefulness, or purpose, is what determines a model’s role, given that it is correctly formed. Models therefore have teleological value even though they are ontologically erroneous." (John Boardman & Brian Sauser, "Systems Thinking: Coping with 21st Century Problems", 2008)

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

Models with a Twist

"Every theory of the course of events in nature is necessarily based on some process of simplification and is to some extent, therefore, a fairy tale." (Sir Napier Shaw, “Manual of Meteorology”, 1932)

“Models - in contrast to those who sat for Renoir - improve with age.” (Erwin Chargaff, “Heraclitean Fire”, 1978)

“What is a model? A model is like an Austrian timetable. Austrian trains are always late. A Prussian visitor asks the Austrian conductor why they bother to print timetables. The conductor replies ‘If we did not, how would we know how late the trains are?’” (Victor F Weisskopf)

“A model is a work of fiction.” (Nancy Cartwright, 1983) “In science, as in life, it is extremely dangerous to fall in love with beautiful models.” (Vijay Pande)

“It is a paradox in mathematics and physics that we have no good model for the teaching of models.” (Hartley Rogers Jr)

“Old models never die; they just fade away.” (Robert M Solow, “How Did Economics Get That Way and What Way Did It Get?”, Daedalus, Vol. 126, No. 1, 1997) [Link]

"There are no surprising facts, only models that are surprised by facts; and if a model is surprised by the facts, it is no credit to that model." (Eliezer S Yudkowsky, "Quantum Explanations", 2008)

“Classical models tell us more than we at first can know.” (Karl Popper)

“A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.” (Manfred Eigen, 'The Origin of Biological Information' [in “The Physicists's Conception of Nature”, Ed. by Jagdish Mehra, 1973)

“The best model of a cat is a cat. Preferably the same cat.” (Arturo Rosenblueth, “Philosophy of Science”, 1945) [also attributed to Norbert Wiener]

Models in Physics

“Physics is the attempt at the conceptual construction of a model of the real world and its lawful structure.” (Albert Einstein, [letter to Moritz Schlick] 1931)

“The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts.” (Ernst Mach, “The Science of Mechanics” 5th Ed, 1942)

“All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things.” (Max Born, “Physical Reality”, Philosophical Quarterly, Vol. 3, 1953)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from 'ultimate' or 'fundamental' truth. It is as ridiculous to deride continuum physics because it is not obtained from nuclear physics as it would be to reproach it with lack of foundation in the Bible." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics", 1965)

“The pre-eminence of astronomy rests on the peculiarity that it can be treated mathematically; and the progress of physics, and most recently biology, has hinged equally on finding formulations of their laws that can be displayed as mathematical models.” (Jacob Bronowski, “The Ascent of Man”, 1973)

“Here is one way to look at physics: the physicists are men looking for new interpretations of the Book of Nature. After each pedestrian period of normal science, they dream up a new model, a new picture, a new vocabulary, and then they announce that the true meaning of the Book has been discovered.” (Richard Rorty, “Philosophy as a Kind of Writing”, 1978)

“[…] the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that.” (Richard P Feynman, “QED: The Strange Theory of Light and Matter”, 1985)

“Physicists are all too apt to look for the wrong sorts of generalizations, to concoct theoretical models that are too neat, too powerful, and too clean. Not surprisingly, these seldom fit well with data. To produce a really good biological theory, one must try to see through the clutter produced by evolution to the basic mechanisms. What seems to physicists to be a hopelessly complicated process may have been what nature found simplest, because nature could build on what was already there.” (Francis H C Crick, “What Mad Pursuit?: A Personal View of Scientific Discovery”, 1988)

“Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?” (Stephen W Hawking, “A Brief History of Time: From the Big Bang to Black Holes”, 1988)

“Physicists’ models are like maps: never final, never complete until they grow as large and complex as the reality they represent.” (James Gleick, “Genius”, 1992)

On Metaphors II

“The drive toward the formation of metaphors is the fundamental human drive, which one cannot for a single instant dispense with in thought, for one would thereby dispense with man himself.” (Friedrich Nietzsche, “On Truth and Lies in a Nonmoral Sense”, 1873)

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

"[...] one cannot describe reality; only give metaphors that indicate it. All human modes of description (photographic, mathematical, and literary) are metaphorical. Even the most precise scientific description of an object or movement is a tissue of metaphors." (John Fowles, “'Notes on an Unfinished Novel”, 1969) 

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

"The essence of metaphor is understanding and experiencing one kind of thing in terms of another. […] Metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature.” (George Lakoff and Mark Johnson, Metaphors We Live By, 1980)

“Metaphors can have profound significance because, as images or figures, they allow the mind to grasp or discover unsuspected ideal and material relationships between objects.” (Giuseppe Del Re, “Cosmic Dance”, 1999)

“[metaphors] are always open to more than one interpretation. But far from being a defect this essential openness is the reason why a number of those metaphors have had a very long life and have been able to survive great changes both in science and in the social background against which they first appeared.” (Olaf Pedersen, “The Book of Nature”, 1992)

“A metaphor is not an ornament. It is an organ of perception. Through metaphors, we see the world as one thing or another.” (Neil Postman, “The End of Education: Redefining the Value of School”, 1996)

“Metaphor is evidence of the human ability to visualize the universe as a coherent organism. Proof of our capacity, not just to see one thing in another but to change the very nature of things. When a metaphor is accepted as fact, it enters groupthink, taking on an existence in the real world. [...] Metaphor is the default form of thought, providing many angles from which to literally 'see' the world." (Marcel Danesi, "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture", 2004)

"Metaphor is a primary cognitive tool by which we make sense of the world." (Terry Marks-Tarlow, "Psyche's Veil: Psychotherapy, Fractals and Complexity", 2008)

Mental Models II

"’Mental models’ are deeply ingrained assumptions, generalizations, or even pictures or images that influence how we understand the world and how we take action. Very often, we are not consciously aware of our mental models or the effects they have on our behavior. […] Mental models focus on the openness needed to unearth shortcomings in our present ways of seeing the world. [...] Mental models are deeply held internal images of how the world works, images that limit us to familiar ways of thinking and acting. Very often, we are not consciously aware of our mental models or the effects they have on our behavior.” (Peter Senge, “The Fifth Discipline”, 1990)

“A mental model is a knowledge structure that incorporates both declarative knowledge (e.g., device models) and procedural knowledge (e.g., procedures for determining distributions of voltages within a circuit), and a control structure that determines how the procedural and declarative knowledge are used in solving problems (e.g., mentally simulating the behavior of a circuit).” (Barbara Y White & John R Frederiksen, “Causal Model Progressions as a Foundation for Intelligent Learning Environments”, Artificial Intelligence 42, 1990)

"We all depend on models to interpret our everyday experiences. We interpret what we see in terms of mental models constructed on past experience and education. They are constructs that we use to understand the pattern of our experiences." (David Bartholomew, “What is Statistics?”, 1995)

“I do not know that my view is more correct; I do not even think that ‘right’ and ‘wrong’ are good categories for assessing complex mental models of external reality - for models in science are judged [as] useful or detrimental, not as true or false.” (Stephen Jay Gould, “Dinosaur in a Haystack: Reflections in Natural History”, 1995)

“The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states.” (Robert Hafner & Jim Stewart, “Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’”, Science Education 79 (2), 1995)

“Our mental model of the way the world works must shift from images of a clockwork, machinelike universe that is fixed and determined, to the model of a universe that is open, dynamic, interconnected, and full of living qualities.” (Joseph Jaworski, “Synchronicity: The Inner Path of Leadership”, 1996)

“Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music.” (Northrop Frye, “The Educated Imagination”, 2002)

“Each generation builds a mental picture that reflects their own understanding of this world. They construct mental tools that penetrate more and more deeply into it, so that they can explore aspects of it that were previously hidden.” (Alain Connes, “The Princeton Companion to Mathematics”, Ed. by Timothy Gowers et al, 2008)

"When a particular image appears in the mind's eye often enough it begins to connect apparently unrelated ideas leading to models and theories. […] Patterns experienced again and again become intuitions. […] Intuitive judgments are made by our use of imagery; intuition is the result of mental model building. […] The mental model used and the form of the intuition is dependent upon the question being answered." (Roger Frantz, “Two Minds”, 2005)

“We all have mental models: the lens through which we see the world that drive our responses to everything we experience. Being aware of your mental models is key to being objective.” (Elizabeth Thornton, “Learn to Be an Objective Leader without Losing Everything”, 2015)

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

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)

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 (Trivia)

“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 (1950-1999)

"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)

"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)

"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)

"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)

"[I]n 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)

"Of course, we know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, "The Coming of Age of the Cell", Science, 1975)

"The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient." (Benoit Mandelbrot, "The Fractal Geometry of Nature", 1977)

"In decision theory, mathematical analysis shows that once the sampling distribution, loss function, and sample are specified, the only remaining basis for a choice among different admissible decisions lies in the prior probabilities. Therefore, the logical foundations of decision theory cannot be put in fully satisfactory form until the old problem of arbitrariness (sometimes called 'subjectiveness') in assigning prior probabilities is resolved." (Edwin T Jaynes, "Prior Probabilities", 1978)

"Events may appear to us to be random, but this could be attributed to human ignorance about the details of the processes involved." (Brain S Everitt, "Chance Rules", 1999)

On Probability (1900-1949)

"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)

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

"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)

"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 rational concept of probability, which is the only basis of probability calculus, applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time. Using the language of physics, we may say that in order to apply the theory of probability we must have a practically unlimited sequence of uniform observations." (Richard von Mises, "Probability, Statistics and Truth", 1928)

"There can be no unique probability attached to any event or behaviour: we can only speak of ‘probability in the light of certain given information’, and the probability alters according to the extent of the information." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means." (Bertrand Russell, 1929)

"When an observation is made on any atomic system that has been prepared in a given way and is thus in a given state, the result will not in general be determinate, i.e. if the experiment is repeated several times under identical conditions several different results may be obtained. If the experiment is repeated a large number of times it will be found that each particular result will be obtained a definite fraction of the total number of times, so that one can say there is a definite probability of its being obtained any time that the experiment is performed. This probability the theory enables one to calculate." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)

"The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra." (Andrey Kolmogorov, "Foundations of the Theory of Probability", 1933)

"Starting from statistical observations, it is possible to arrive at conclusions which not less reliable or useful than those obtained in any other exact science. It is only necessary to apply a clear and precise concept of probability to such observations. " (Richard von Mises, "Probability, Statistics, and Truth", 1939)

"Probabilities must be regarded as analogous to the measurement of physical magnitudes; that is to say, they can never be known exactly, but only within certain approximation." (Emile Borel, "Probabilities and Life", 1943)

"The conception of chance enters in the very first steps of scientific activity in virtue of the fact that no observation is absolutely correct. I think chance is a more fundamental conception that causality; for whether in a concrete case, a cause-effect relation holds or not can only be judged by applying the laws of chance to the observation." (Max Born, 1949)

On Probability (1800-1899)

"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)

“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)

“The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen).” (Siméon-Denis Poisson, “'Règles générales des probabilités”, 1837)

“I consider the world probability as meaning the state of mind with respect to an assertion, a coming event, or any other matter on which absolute knowledge does not exist.” (Augustus De Morgan, “Essay on Probability”, 1838)

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

“[…] probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.” (George Boole, “The Laws of Thought”, 1854)

"There is no more remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with, and justify, the conclusions to which mankind have been led, not by reasoning, but by instinct and experience, both of the individual and of the race. At the same time it has corrected, extended, and invested them with a definiteness and precision of which these crude, though sound, appreciations of common sense were till then devoid." (Morgan W Crofton, "Probability", Encyclopaedia Britannica 9th Ed,, 1885)

"The scientific imagination always restrains itself within the limits of probability." (Thomas H Huxley, "Science and Christian Tradition", 1893)

On Probability (1700-1799)

“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)

"Events are independent when the happening of any one of them does neither increase nor abate the probability of the rest." (Thomas Bayes, "An Essay towards solving a Problem in the Doctrine of Chances", 1763)

"[...] the probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon it's happening." (Thomas Bayes, "An Essay towards solving a Problem in the Doctrine of Chances", 1763)

“If an event can be produced by a number n of different causes, the probabilities of the existence of these causes, given the event (prises de l'événement), are to each other as the probabilities of the event, given the causes: and the probability of each cause is equal to the probability of the event, given that cause, divided by the sum of all the probabilities of the event, given each of the causes.” (Pierre-Simon Laplace, "Mémoire sur la Probabilité des Causes par les Événements", 1774)

“The word ‘chance’ then expresses only our ignorance of the causes of the phenomena that we observe to occur and to succeed one another in no apparent order. Probability is relative in part to this ignorance, and in part to our knowledge.” (Pierre-Simon Laplace, "Mémoire sur les Approximations des Formules qui sont Fonctions de Très Grands Nombres", 1783)

“[…] determine the probability of a future or unknown event not on the basis of the number of possible combinations resulting in this event or in its complementary event, but only on the basis of the knowledge of order of familiar previous events of this kind” (Marquis de Condorcet, “Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix”, 1785)

“The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs. This kind of calculation is more complicated and more difficult than it is commonly thought to be […]” (Antoine-Laurent Lavoisier, cca. 1790)

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)