On Heuristics II

"Models of bounded rationality describe how a judgement or decision is reached (that is, the heuristic processes or proximal mechanisms) rather than merely the outcome of the decision, and they describe the class of environments in which these heuristics will succeed or fail." (Gerd Gigerenzer & Reinhard Selten [Eds., "Bounded Rationality: The Adaptive Toolbox", 2001)

"A second class of metaphors - mathematical algorithms, heuristics, and models - brings us closer to the world of computer science programs, simulations, and approximations of the brain and its cognitive processes." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"In particular, the accurate intuitions of experts are better explained by the effects of prolonged practice than by heuristics. We can now draw a richer and more balanced picture, in which skill and heuristics are alternative sources of intuitive judgments and choices." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"This is the essence of intuitive heuristics: when faced with a difficult question, we often answer an easier one instead, usually without noticing the substitution." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)

"Heuristics are an evolutionary solution to an ongoing problem: we have limited mental resources. As such, they have a very long and thoroughly time-tested history of helping us - on average - make better decisions." (Peter H Diamandis, "Abundance: The Future is Better Than You Think", 2012)

"Heuristics are simplified rules of thumb that make things simple and easy to implement. But their main advantage is that the user knows that they are not perfect, just expedient, and is therefore less fooled by their powers. They become dangerous when we forget that." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"The art of reasoned persuasion is an iterative, recursive heuristic, meaning that we must go back and forth between the facts and the rules until we have a good fit. We cannot see the facts properly until we know what framework to place them into, and we cannot determine what framework to place them into until we see the basic contours of the facts." (Joel P Trachtman, "The Tools of Argument", 2013)

"Heuristic decision making is fast and frugal and is often based on the evaluation of one or two salient bits of information." (Amitav Chakravarti, "Why People (Don’t) Buy: The Go and Stop Signals", 2015)

"A heuristic is a strategy we derive from previous experience with a similar problem." (Darius Foroux, "Think Straight", 2017)

"The social world that humans have made for themselves is so complex that the mind simplifies the world by using heuristics, customs, and habits, and by making models or assumptions about how things generally work (the ‘causal structure of the world’). And because people rely upon (and are invested in) these mental models, they usually prefer that they remain uncontested." (Dr James Brennan, "Psychological  Adjustment to Illness and Injury", West of England Medical Journal Vol. 117 (2), 2018)

On Generalization (1800-1849)

"An idea is always a generalization, and generalization is a property of thinking. To generalize means to think." (Georg W F Hegel, "The Philosophy of Right", 1820)

"To minds of a certain cast there is nothing so captivating as simplification and generalization." (Thomas R Malthus, "Principles of Political Economy", 1820)

"General assertions, like general truths, are not always applicable to individual cases; though Fortune's wheel is generally on the turn, sometimes when it gets into the mud, it sticks there." (Letitia E Landon, "Romance and Reality", 1831)

"It is not easy to anatomize the constitution and the operations of a mind which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)

"Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "System of Logic", 1843)

"The mere accumulation of unconnected observations of details, devoid of generalization of ideas, may doubtlessly have tended to create and foster the deeply rooted prejudice, that the study of the exact sciences must necessarily chill the feelings, and diminish the nobler enjoyments attendant upon a contemplation of nature." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe", 1845)

On Symbols (2000-2009)

"Precision is greatly enhanced by the human capacity to symbolize. Symbols can be devised to stand for mathematical ideas, entities, operations, and relations. Symbols also permit precise and repeatable calculation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"A symbol is a mental representation regarding the internal reality referring to its object by a convention and produced by the conscious interpretation of a sign. In contrast to signals, symbols may be used every time if the receiver has the corresponding representation. Symbols also relate to feelings and thus give access not only to information but also to the communicator’s motivational and emotional state. The use of symbols makes it possible for the organism using it to evoke in the receiver the same response it evokes in himself. To communicate with symbols is to use a language." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"In the definition of meaning, it is assumed that both the source and receiver have previously coded (and stored) signals of the same or similar referents, such that the messages may have meaning and relate to behaviour. That is, the used symbols must have the same signification for both sender and receiver. If not, the receiver will create a different mental picture than intended by the transmitter. Meaning is generated by individuals in a process of social interaction with a more or less common environment. It is a relation subsisting within a field of experience and appears as an emergent property of a symbolic representation when used in culturally accepted interaction. The relation between the symbolic representation and its meaning is random. Of this, however, the mathematical theory has nothing to say. If human links in the chain of communication are missing, of course no questions of meaning will arise." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"A person thinking in the nonverbal mode is actually thinking with the meaning of the language in the form of mental pictures of the concepts and ideas it contains. Nonverbal thought doesn't require literacy. An illiterate person can communicate without knowing what the symbols look like. [...] Literacy, then, is established as the person learns how the symbols look and becomes able to recognize them as representing certain things or concepts." (Ronald D Davis & Eldon M Braun, "The Gift of Learning", 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. […] [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"I often told the fanatics of realism that there is no such thing as realism in art: it only exists in the mind of the observer. Art is a symbol, a thing conjuring up reality in our mental image. That is why I don't see any contradiction between abstract and figurative art either." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)

"A symbol is an object, act, or event that conveys meaning to others. Symbols can be considered a rich, non-verbal language that vibrantly conveys the organization’s important values concerning how people relate to one another and interact with the environment" (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"But because of the way in which depictions represent, there is a correspondence between parts and spatial relations of the representation and those of the object; this structural mapping, which confers a type of resemblance, underlies the way images convey specific content. In this respect images are like pictures. Unlike words and symbols, depictions are not arbitrarily paired with what they represent." (Stephen Kosslyn et al," The Case for Mental Imagery", 2006)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"[...] the scientific models of concrete things are symbolic rather than iconic: they are systems of propositions, not pictures. Besides, such models are seldom if ever completely accurate, if only because they involve more or less brutal simplifications, such as pretending that a metallic surface is smooth, a crystal has no impurities, a biopopulation has a single predator, or a market is in equilibrium.  These are all fictions. However, they are stylizations rather than wild fantasies. Hence, introducing and using them to account for real existents does not commit us to fictionism, just as defending the role of experience need not make us empiricists, nor is admitting the role of intuition enough to qualify as intuitionist." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"But notice, a subatomic particle is itself a holon [hole/parts]. And so is a cell. And so is a symbol, and an image, and a concept. What all of those entities are, before they are anything else, is a holon. So the world is not composed of atoms or symbols or cells or concepts. It is composed of holons." (Ken Wilber, "A Brief History of Everything", 2007)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre O Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

Mental Models LVIII

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"A mental model is conceived here 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", [in "The Universe of General Relativity"] 2000)

"In the language of mental models, such past experience provided the default assumptions necessary to fill the gaps in the emerging and necessarily incomplete framework of a relativistic theory of gravitation. It was precisely the nature of these default assumptions that allowed them to be discarded again in the light of novel information - provided, for instance, by the further elaboration of the mathematical formalism - without, however, having to abandon the underlying mental models which could thus continue to function as heuristic orientations." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", [in "The Universe of General Relativity"] 2000)

"A mental model represents a possibility, or, to be precise, the structure and content of the model capture what is common to the different ways in which the possibility could occur [...]" (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"According to mental model theory, human reasoning relies on the construction of integrated mental representations of the information that is given in the reasoning problem's premises. These integrated representations are the mental models. A mental model is a mental representation that captures what is common to all the different ways in which the premises can be interpreted. It represents in "small scale" how "reality" could be— according to what is stated in the premises of a reasoning problem. Mental models, though, must not be confused with images. A mental model often forms the basis of one or more visual images, but some of them represent situations that cannot be visualized. Instead, mental models are often likened to diagrams since, as with diagrams, their structure is analogous to the structure of the states of affairs they represent." (Carsten Held et al, "Mental Models and the Mind", 2006)

"Mental models are mental representations of a certain type. The main problem in the philosophy of mental representation is to characterize the relation between a mental representation and the represented object. Naively speaking, a mental representation is an entity that 'stands for' another—the represented object - , but here 'stands for' is just a metaphoric place-holder for 'represents', thus requires further explanation." (Carsten Held et al, "Mental Models and the Mind", 2006)

"Prom the processing view, the model theory distinguishes between three different operations. In the construction phase, reasoners construct the mental model that reflects the information from the premises. In the inspection phase, this model is inspected to find new information that is not explicitly given in the premises. In most variants of the model theory, the inspection process is conceptualized as a spatial focus that scans the model to find new information not given in the premises.. In the variation phase, reasoners try to construct alternative models from the premises that refute the putative conclusion. If no such model is found, the putative conclusion is considered true." (Carsten Held et al, "Mental Models and the Mind", 2006)

"The model theory postulates that mental models are parsimonious. They represent what is possible, but not what is impossible, according to assertions. This principle of parsimony minimizes the load on working memory, and so it applies unless something exceptional occurs to overrule it." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"Just as physicists have created models of the atom based on observed data and intuitive synthesis of the patterns in their data, so must designers create models of users based on observed behaviors and intuitive synthesis of the patterns in the data. Only after we formalize such patterns can we hope to systematically construct patterns of interaction that smoothly match the behavior patterns, mental models, and goals of users. Personas provide this formalization." (Alan Cooper et al, "About Face 3: The Essentials of Interaction Design", 2007)

"We tend to form mental models that are simpler than reality; so if we create represented models that are simpler than the actual implementation model, we help the user achieve a better understanding. […] Understanding how software actually works always helps someone to use it, but this understanding usually comes at a significant cost. One of the most significant ways in which computers can assist human beings is by putting a simple face on complex processes and situations. As a result, user interfaces that are consistent with users’ mental models are vastly superior to those that are merely reflections of the implementation model." (Alan Cooper et al,  "About Face 3: The Essentials of Interaction Design", 2007)

Mental Models LV

"To imagine - to form an image - we must have the numerous relations of things present to the mind, and see the objects in their actual order. In this we are of course greatly aided by the mass of organised experience, which allows us rapidly to estimate the relations of gravity or affinity just as we remember that fire burns and that heated bodies expand. But be the aid great or small, and the result victorious or disastrous, the imaginative process is always the same." (George H Lewes, "The Principles of Success in Literature", 1865)

"The degree in which each mind habitually substitutes signs for images will be, CETERIS PARIBUS [with other conditions remaining the same], the degree in which it is liable to error. This is not contradicted by the fact that mathematical, astronomical, and physical reasonings may, when complex, be carried on more successfully by the employment of signs; because in these cases the signs themselves accurately represent the abstractness of the relations. Such sciences deal only with relations, and not with objects; hence greater simplification ensures greater accuracy. But no sooner do we quit this sphere of abstractions to enter that of concrete things, than the use of symbols becomes a source of weakness. Vigorous and effective minds habitually deal with concrete images." (George H Lewes, "The Principles of Success in Literature", 1865)

"The steps to scientific as well as other knowledge consist in a series of logical fictions which are as legitimate as they are indispensable in the operations of thought, but whose relations to the phenomena whereof they are the partial and not unfrequently merely symbolical representations must never be lost sight of." (John Stallo, "The Concepts and Theories of Modern Physics", 1884)

"Myths and science fulfil a similar function: they both provide human beings with a representation of the world and of the forces that are supposed to govern it. They both fix the limits of what is considered as possible." (François Jacob, "The Possible and the Actual", 1982)

"Many people would accept that we do not really have knowledge of the world; we have knowledge only of our representations of the world. Yet we seem condemned by our consitution to treat these representations as if they were the world, for our everyday experience feels as if it were of a given and immediate world." (Francisco Varela, "The Embodied Mind", 1991)

"The seemingly stable scene you normally see is really a mental model that you construct - the eyes are actually darting all around, producing a retinal image as jerky as an amateur video, and some of what you thought you saw was instead filled in from memory." (William H Calvin, "How Brains Think", 1996) 

"[…] the 'reality' that we perceive is based on mental models in which things don't usually change their shapes or disappear, despite their changing appearances. We mainly react to what we expect - and tend to represent the things that we see as though they remain the same as we move atomic." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)

"We all construct mental models that describe our various mental states, bodies of knowledge about our abilities, depictions of our acquaintances, and collections of stories about our pasts. Then, whenever we use our models of ourselves, we tend to use terms like conscious - when those reflections lead to choices we make, and we use unconscious or unintentional to describe those activities that we regard as beyond our control." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)

"We solve easy problems in routine ways, scarcely thinking about how we accomplish these - but when our usual methods don't work, we start to 'reflect' on what went wrong and find ourselves to be switching around in a network of 'models', each of which purports to represent some facet or aspect of ourselves, so that we end representing ourselves with a loosely connected collection of images, models, and anecdotes." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006) 

"Why must those models be simplifications? Each model must help us to focus on only those aspects that matter in some particular context; that's what makes a map more useful to us than seeing the entire landscape that it depicts. The same applies to what we store in our minds. Consider how messy our minds would become if we filled them up with descriptions of things whose details had too little significance. So instead, we spend large parts of our lives at trying to tidy up our minds - selecting the portions we want to keep, suppressing others we'd like to forget, and refining the ones we're dissatisfied with." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)

On Equilibrium (1950-1959)

"Now a system is said to be at equilibrium when it has no further tendency to change its properties." (Walter J Moore, "Physical chemistry", 1950)

"Physical irreversibility manifests itself in the fact that, whenever the system is in a state far removed from equilibrium, it is much more likely to move toward equilibrium, than in the opposite direction." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

"Equilibrium requires that the whole of the structure, the form of its elements, and the means of interconnection be so combined that at the supports there will automatically be produced passive forces or reactions that are able to balance the forces acting upon the structures, including the force of its own weight."  (Eduardo Torroja, "Philosophy of Structure", 1951) 

"[…] there are three different but interconnected conceptions to be considered in every structure, and in every structural element involved: equilibrium, resistance, and stability." (Eduardo Torroja, "Philosophy of Structure" , 1951) 

"Business-cycle theorists concerned themselves with why the economy naturally generated fluctuations in employment and output, [while the rest of the profession] continued to operate on the assumption that full employment was the natural, equilibrium position for the economy." (Robert A Gordon, "Business Fluctuations", 1952)

"Biological communities are systems of interacting components and thus display characteristic properties of systems, such as mutual interdependence, self-regulation, adaptation to disturbances, approach to states of equilibrium, etc." (Ludwig von Bertalanffy, "Problems of Life", 1952)

"Every stable system has the property that if displaced from a state of equilibrium and released, the subsequent movement is so matched to the initial displacement that the system is brought back to the state of equilibrium. A variety of disturbances will therefore evoke a variety of matched reactions." (W Ross Ashby, "Design for a Brain: The Origin of Adaptive Behavior", 1952)

"The primary fact is that all isolated state-determined dynamic systems are selective: from whatever state they have initially, they go towards states of equilibrium. These states of equilibrium are always characterised, in their relation to the change-inducing laws of the system, by being exceptionally resistant." (W Ross Ashby, "Design for a Brain: The Origin of Adaptive Behavior", 1952)

"Multiple equilibria are not necessarily useless, but from the standpoint of any exact science the existence of a uniquely determined equilibrium is, of course, of the utmost importance, even if proof has to be purchased at the price of very restrictive assumptions; without any possibility of proving the existence of (a) uniquely determined equilibrium - or at all events, of a small number of possible equilibria - at however high a level of abstraction, a field of phenomena is really a chaos that is not under analytical control." (Joseph A Schumpeter, "History of Economic Analysis", 1954)

"Perhaps as important is the relation between the existence of solutions to a competitive equilibrium and the problems of normative or welfare economics." (Kenneth J Arrow & Gerard Debreu. "Existence of an equilibrium for a competitive economy", Econometrica: Journal of the Econometric Society, 1954)

"As shorthand, when the phenomena are suitably simple, words such as equilibrium and stability are of great value and convenience. Nevertheless, it should be always borne in mind that they are mere shorthand, and that the phenomena will not always have the simplicity that these words presuppose." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Reversible processes are not, in fact, processes at all, they are sequences of states of equilibrium. The processes which we encounter in real life are always irreversible processes." (Arnold Sommerfeld, "Thermodynamics and Statistical Mechanics", Lectures on Theoretical - Physics Vol. V, 1956)

"Thus, if the whole is at a state of equilibrium, each part must be in a state of equilibrium in the conditions provided by the other. [...] the whole is at a state of equilibrium if and only if each part is at a state of equilibrium in the conditions provided by the other part. [...] No state (of the whole) can be a state of equilibrium unless it is acceptable to every one of the component parts, each acting in the conditions given by the others." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"It is clear to all that the animal organism is a highly complex system consisting of an almost infinite series of parts connected both with one another and, as a total complex, with the surrounding world, with which it is in a state of equilibrium. (Ivan P Pavlov, "Experimental psychology, and other essays", 1957)

"[Equilibrium] is a notion which can be employed usefully in varying degrees of looseness. It is an absolutely indispensable part of the toolbag of the economist and one which he can often contribute usefully to other sciences which are occasionally apt to get lost in the trackless exfoliations of purely dynamic systems." (Kenneth Boulding, The Skills of the Economist", Journal of Political Economy 67 (1), 1959)

"One of the most important skills of the economist, therefore, is that of simplification of the model." (Kenneth Boulding, "The Skills of the Economist", Journal of Political Economy 67 (1), 1959)

"The ability to work with systems of general equilibrium is perhaps one of the most important skills of the economist - a skill which he shares with many other scientists, but in which he has perhaps a certain comparative advantage." (Kenneth Boulding, "The Skills of the Economist", Journal of Political Economy 67 (1), 1959)

On Equilibrium (1960-1969)

"The static stability of a system is defined by the initial tendency to return to equilibrium conditions following some disturbance from equilibrium. […] If the object has a tendency to continue in the direction of disturbance, negative static stability or static instability exists. […] If the object subject to disturbance has neither the tendency to return nor the tendency to continue in the displacement direction, neutral static stability exists." (Hugh H Hurt, "Aerodynamics for Naval Aviators", 1960)

"While static stability is concerned with the tendency of a displaced body to return to equilibrium, dynamic stability is concerned with the resulting motion with time. If an object is disturbed from equilibrium, the time history of the resulting motion indicates the dynamic stability of the system. In general, the system will demonstrate positive dynamic stability if the amplitude of the motion decreases with time." (Hugh H Hurt, "Aerodynamics for Naval Aviators", 1960)

An economy may be in equilibrium from a short-period point of view and yet contain within itself incompatibilities that are soon going to knock it out of equilibrium." (Joan Robinson, "Essays in the Theory of Economic Growth", 1965)

"[...] in a state of dynamic equilibrium with their environments. If they do not maintain this equilibrium they die; if they do maintain it they show a degree of spontaneity, variability, and purposiveness of response unknown in the non-living world. This is what is meant by ‘adaptation to environment’ […] [Its] essential feature […] is stability - that is, the ability to withstand disturbances." (Kenneth Craik, 'Living organisms', “The Nature of Psychology”, 1966)

"Higher, directed forms of energy (e.g., mechanical, electric, chemical) are dissipated, that is, progressively converted into the lowest form of energy, i.e., undirected heat movement of molecules; chemical systems tend toward equilibria with maximum entropy; machines wear out owing to friction; in communication channels, information can only be lost by conversion of messages into noise but not vice versa, and so forth." (Ludwig von Bertalanffy, "Robots, Men and Minds", 1967)

"[The equilibrium model describes systems] which, in moving to an equilibrium point, typically lose organization, and then tend to hold that minimum level within relatively narrow conditions of disturbance." (Walter F Buckley, "Sociology and modern systems theory", 1967)

"A more viable model, one much more faithful to the kind of system that society is more and more recognized to be, is in process of developing out of, or is in keeping with, the modern systems perspective (which we use loosely here to refer to general systems research, cybernetics, information and communication theory, and related fields). Society, or the sociocultural system, is not, then, principally an equilibrium system or a homeostatic system, but what we shall simply refer to as a complex adaptive system." (Walter F Buckley, "Society as a complex adaptive system", 1968)

"Biologically, life is not maintenance or restoration of equilibrium but is essentially maintenance of disequilibria, as the doctrine of the organism as open system reveals. Reaching equilibrium means death and consequent decay. Psychologically, behaviour not only tends to release tensions but also builds up tensions; if this stops, the patient is a decaying mental corpse in the same way a living organism becomes a body in decay when tensions and forces keeping it from equilibrium have stopped." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Conventional physics deals only with closed systems, i.e. systems which are considered to be isolated from their environment. [...] However, we find systems which by their very nature and definition are not closed systems. Every living organism is essentially an open system. It maintains itself in a continuous inflow and outflow, a building up and breaking down of components, never being, so long as it is alive, in a state of chemical and thermodynamic equilibrium but maintained in a so-called steady state which is distinct from the latter." (Ludwig von Bertalanffy, "General System Theory", 1968)

"We have argued at some length in another place that the mechanical equilibrium model and the organismic homeostasis models of society that have underlain most modern sociological theory have outlived their usefulness." (Walter F Buckley, "Society as a complex adaptive system", 1968)

"Peace is an unstable equilibrium, which can be preserved only by acknowledged supremacy or equal power." (Will Durant, The Lessons of History, 1968)

On Observation (1970-1979)

"Science consists simply of the formulation and testing of hypotheses based on observational evidence; experiments are important where applicable, but their function is merely to simplify observation by imposing controlled conditions." (Henry L Batten, "Evolution of the Earth", 1971)

"A hypothesis is empirical or scientific only if it can be tested by experience. […] A hypothesis or theory which cannot be, at least in principle, falsified by empirical observations and experiments does not belong to the realm of science." (Francisco J Ayala, "Biological Evolution: Natural Selection or Random Walk", American Scientist, 1974)

"Science is systematic organisation of knowledge about the universe on the basis of explanatory hypotheses which are genuinely testable. Science advances by developing gradually more comprehensive theories; that is, by formulating theories of greater generality which can account for observational statements and hypotheses which appear as prima facie unrelated." (Francisco J Ayala, "Studies in the Philosophy of Biology: Reduction and Related Problems", 1974)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)

"The essential function of a hypothesis consists in the guidance it affords to new observations and experiments, by which our conjecture is either confirmed or refuted." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1976)

"[…] a body of practices widely regarded by outsiders as well organized, logical, and coherent, in fact consists of a disordered array of observations with which scientists struggle to produce order." (Bruno Latour & S Woolgar, Laboratory Life: The Social Construction of Scientific Facts, 1979)

On Concepts IV

"While science is pursuing a steady onward movement, it is convenient from time to time to cast a glance back on the route already traversed, and especially to consider the new conceptions which aim at discovering the general meaning of the stock of facts accumulated from day to day in our laboratories." (Dmitry Mendeleyev, "The Periodic Law of the Chemical Elements", Journal of the Chemical Society Vol. 55, 1889)

"The aim of ‘science’ is to attain conceptions so adequate and exact that we shall never need to change them." (William James, "The Principles of Psychology", 1890)

"Science like life feeds on its own decay. New facts burst old rules; then newly developed concepts bind old and new together into a reconciling law." (William James, "The Will to Believe and Other Essays in Popular Philosophy", 1896)

"Science works by the slow method of the classification of data, arranging the detail patiently in a periodic system into groups of facts, in series like the strata of the rocks. For each series there must be a vocabulary of special words which do not always make good sense when used in another series. But the laws of periodicity seem to hold throughout, among the elements and in every sphere of thought, and we must learn to co-ordinate the whole through our new conception of the reign of relativity." (William H Pallister, "Poems of Science", 1931)

"The distinguishing feature of modern scientific thought lies in the fact that it begins by discarding all a priori conceptions about the nature of reality - or about the ultimate nature of the universe - such as had characterized practically all Greek philosophy and all medieval thinking as well, and takes instead, as its starting point, well-authenticated, carefully tested experimental facts, no matter whether these facts seen at the moment to fit into any general philosophical scheme or not - that is, no matter whether they seem at the moment to be reasonable or not." (Robert A Millikan, "Professor Einstein at the California Institute of Technology", Science Vol. 73 (1893), 1931)

"Physics too deals with mathematical concepts; however, these concepts attain physical content only by the clear determination of their relation to the objects of experience." (Albert Einstein, "Out of My Later Years", 1950)

"Science is still the versatile, unpredictable hero of the play, creating endless new situations, opening romantic vistas and challenging accepted concepts." (René J Dubos, "Louis Pasteur: Free Lance of Science", 1950)

"The important point for us to observe is that all these constructions and the laws connecting them can be arrived at by the principle of looking for the mathematically simplest concepts and the link between them. In the limited number of mathematically existent simple field types, and the simple equations possible between them, lies the theorist’s hope of grasping the real in all its depth." (Albert Einstein, "Ideas and Opinions", 1954)

"For Science in its totality, the ultimate goal is the creation of a monistic system in which - on the symbolic level and in terms of the inferred components of invisibility and intangibly fine structure - the world’s enormous multiplicity is reduced to something like unity, and the endless successions of unique events of a great many different kinds get tidied and simplified into a single rational order. Whether this goal will ever be reached remains to be seen. Meanwhile we have the various sciences, each with its own system coordinating concepts, its own criterion of explanation." (Aldous Huxley, "Literature and Science", 1963)

“All the efforts of the researcher to find other models, conceptions, different mathematical forms, better linguistic modes of expression, to do justice to newly discovered layers of being mean self-transformation. The researcher in his place is the human being in self-transformation to more profound insight into what is given.” (John Dessauer)

On Proofs (1850-1899)

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

"We must never assume that which is incapable of proof.”  (George H Lewes, “The Physiology of Common Life” Vol. 2, 1860)

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"The mathematician starts with a few propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature: authority and tradition furnish the data, and the mental operations are deductive." (Thomas H Huxley, 1869)

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

“’Divide et impera’ is as true in algebra as in statecraft; but no less true and even more fertile is the maxim ‘auge et impera’. The more to do or to prove, the easier the doing or the proof.” (James J Sylvester, “Proof of the Fundamental Theorem of Invariants”, Philosophic Magazine, 1878)

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

“That which is provable, ought not to be believed in science without proof” (Richard Dedekind, “Was sind und was sollen die Zahlen?”, 1888)

“Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.” (Christian H Dillmann, „Die Mathematik die Fackelträgerin einer neuen Zeit“, 1889)

“If men of science owe anything to us, we may learn much from them that is essential. For they can show how to test proof, how to secure fulness and soundness in induction, how to restrain and to employ with safety hypothesis and analogy.” (Lord John Acton, [Lecture] “The Study of History”, 1895)

“The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us.” (Paul Valéry, 1895)

"Just give me the insights. I can always come up with the proofs!" (Bernhard Riemann)

"Analogy cannot serve as proof." (Louis Pasteur)

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

Mental Models VIII (Limitations II)

“This is the greatest degree of impoverishment; the [mental] image, deprived little by little of its own characteristics, is nothing more than a shadow. […] Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others.” (Théodule-Armand Ribot, “Essay on the Creative Imagination”, 1900)

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

“Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities.” (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures”, 1956)

“Mental models are fuzzy, incomplete, and imprecisely stated. Furthermore, within a single individual, mental models change with time, even during the flow of a single conversation. The human mind assembles a few relationships to fit the context of a discussion. As debate shifts, so do the mental models. Even when only a single topic is being discussed, each participant in a conversation employs a different mental model to interpret the subject. Fundamental assumptions differ but are never brought into the open. […] A mental model may be correct in structure and assumptions but, even so, the human mind - either individually or as a group consensus - is apt to draw the wrong implications for the future.” (Jay W Forrester, “Counterintuitive Behaviour of Social Systems”, Technology Review, 1971)

“At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs.” (Philip Johnson-Laird, “Mental Models” 1983)

"Almost every aspect of our lives is shaped in some way by how we make sense of the world. Our thinking and our actions are affected by the mental models we hold. These models define our limits or open our opportunities. Despite their power and pervasiveness, these models are usually virtually invisible to us. We don't realize they are there at all.” (Robert Gunther et al, “The Power of Impossible Thinking: Transform the Business of Your Life and the Life of Your Business”, 2004)

“Humans have difficulty perceiving variables accurately […]. However, in general, they tend to have inaccurate perceptions of system states, including past, current, and future states. This is due, in part, to limited ‘mental models’ of the phenomena of interest in terms of both how things work and how to influence things. Consequently, people have difficulty determining the full implications of what is known, as well as considering future contingencies for potential systems states and the long-term value of addressing these contingencies. ” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007)

“[…] we cannot accurately assess both what a mental model is and what it is becoming because the act of assessment affects the model.” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007) [see Werner Heisenberg’s principle]

“Mental models are problematic because they typically operate unconsciously. This means that they influence behavior and structure thinking in ways that individuals do not recognize and therefore cannot easily articulate. This makes certain kinds of exploratory conversations difficult or impossible, and causes even reasonable propositions to be rejected out of hand.” (Kim Erwin, “Communicating The New: Methods to Shape and Accelerate Innovation”, 2013)

“A mental model is not necessarily founded on facts or complete understanding of reality. Let's be honest, most of our mental models are flawed in many ways, and that's perfectly normal. They work because they are fast and simple and not because they are a complete representation of the reality. […] The most important thing about a person's mental model is that it's simplified and very limited compared to what it models.” (Peter W Szabo, “User Experience Mapping”, 2017)

See also:
Mental Models VIII – More on their Limits
Mental Models I, II, III, V, VI, VII

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)

On Randomness II (Trivia II)

“From a purely operational point of viewpoint […] the concept of randomness is so elusive as to cease to be viable." (Mark Kac, 1983)

“The popular image of mathematics as a collection of precise facts, linked together by well-defined logical paths, is revealed to be false. There is randomness and hence uncertainty in mathematics, just as there is in physics.” (Paul Davis, “The Mind of God”, 1992)

“Randomness, chaos, uncertainty, and chance are all a part of our lives. They reside at the ill-defined boundaries between what we know, what we can know, and what is beyond our knowing. They make life interesting.” (Ivars Peterson, “The Jungles of Randomness: A Mathematical Safari”, 1998)

“[…] we underestimate the share of randomness in about everything […]  The degree of resistance to randomness in one’s life is an abstract idea, part of its logic counterintuitive, and, to confuse matters, its realizations nonobservable.” (Nassim N Taleb, “Fooled by Randomness”, 2001)

“Mathematics, far from being stymied by this situation, finds enormous value in it. The fecundity of ‘randomness’ is astounding; it is an inexhaustible source of scientific riches. Could ‘randomness’ be such a rich notion because of the inner contradiction that it contains, not despite it? The depth we sense in ‘randomness’ comes from something that lies behind any specific mathematical definition.” (William Byers, “How Mathematicians Think”, 2007)

“A Black Swan is a highly improbable event with three principal characteristics: It is unpredictable; it carries a massive impact; and, after the fact, we concoct an explanation that makes it appear less random, and more predictable, than it was. […] The Black Swan idea is based on the structure of randomness in empirical reality. [...] the Black Swan is what we leave out of simplification.” (Nassim N Taleb, “The Black Swan”, 2007)

“The key to understanding randomness and all of mathematics is not being able to intuit the answer to every problem immediately but merely having the tools to figure out the answer.” (Leonard Mlodinow, “The Drunkard’s Walk: How Randomness Rules Our Lives”, 2008)

"The randomness which lies at the very foundations of pure mathematics of necessity permeates every human description of nature" (Joseph Ford)

“Our concept of randomness is merely an attempt to characterize and distinguish the sort of series which bamboozles the most people. […] It is thus irrelevant whether a series has been made up by a penny, a calculating machine, a Geiger counter or a practical joker. What matters is its effect on those who see it, not how it was produced.” (Spencer Brown)

On Proofs II

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

"It is by logic that we prove, but by intuition that we discover." (Henri Poincaré, “Science and Method”, 1908)

"Symbols, formulae and proofs have another hypnotic effect. Because they are not immediately understood, they, like certain jokes, are suspected of holding in some sort of magic embrace the secret of the universe, or at least some of its more hidden parts." (Scott Buchanan, “Poetry and Mathematics”, 1975)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya,  "How to Solve It", 1973)

"A proof only becomes a proof after the social act of ‘accepting it as a proof’." (Yu I Manin, "A Course in Mathematical Logic", 1977)

"A proof in science does more than eliminate doubt. It eliminates inconsistencies and provides the underlying logical basis of the statement." (Edward Teller, “The Pursuit of Simplicity”, 1981)

"A proof only becomes a proof after the social act of 'accepting it as a proof'" (Yuri I Manin, "Provable and Unprovable", 1982)

"A math lecture without a proof is like a movie without a love scene." (Hendrik Lenstra, 2002)

“The more powerful the mathematical tools used to prove a result, the shorter that proof might be expected to be […]” (Julian Havil, “Nonplussed!”, 2007

More on Theories

"It seems to be one of the fundamental features of nature the fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. " (Paul A M Dirac , “The Evolution of the Physicist’s Picture of Nature" , Scientific American, 1963)

“It [a theory] ought to furnish a compass which, if followed, will lead the observer further and further into previously unexplored regions. Whether these regions will be barren or fertile experience alone will decide; but, at any rate, one who is guided in this way will travel onward in a definite direction, and will not wander aimlessly to and fro.” (Sir Joseph J Thomson, “The Corpuscular Theory of Matter”, 1907)

”As soon as we inquire into the reasons for the phenomena, we enter the domain of theory, which connects the observed phenomena and traces them back to a single ‘pure’ phenomena, thus bringing about a logical arrangement of an enormous amount of observational material.” (Georg Joos, “Theoretical Physics”, 1968)

"[...] we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete." (Stephen Hawking, “Gödel and the End of the Universe” )

“The scientist who discovers a theory is usually guided to his discovery by guesses; he cannot name a method by means of which he found the theory and can only say that it appeared plausible to him, that he had the right hunch or that he saw intuitively which assumption would fit the facts.” (Hans Reichenbach, “The Rise of Scientific Philosophy”, 1951)

“It is in the nature of theoretical science that there can be no such thing as certainty. A theory is only ‘true’ for as long as the majority of the scientific community maintain the view that the theory is the one best able to explain the observations.” (Jim Baggott, “The Meaning of Quantum Theory”, 1992)

"Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about  reverence, not mastery." (Richard Power, “Gold Bug Variations”, 1993)

”Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulas, are the beginning of every physical theory.” (Leopold Infeld, “The Evolution of Physics”, 1961)

”A discovery in science, or a new theory, even where it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalyzed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow: it takes a vast world unchallenged and for granted.” (James R Oppenheimer, “Atom and Void”, 1989)

"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)