Mental Models LVI (Conceptual Models III)

"Mere deductive logic, whether you clothe it in mathematical symbols and phraseology or whether you enlarge its scope into a more general symbolic technique, can never take the place of clear relevant initial concepts of the meaning of your symbols, and among symbols I include words. If you are dealing with nature, your meanings must directly relate to the immediate facts of observation. We have to analyse first the most general characteristics of things observed, and then the more casual contingent occurrences. There can be no true physical science which looks first to mathematics for the provision of a conceptual model. Such a procedure is to repeat the errors of the logicians of the middle-ages." (Alfred N Whitehead, "Principle of Relativity", 1922)

"The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other ; it is rather a word denoting a kind of conceptual construction, as, e.g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas." (Moritz Schlick, "Allgemeine Erkenntnislehre", 1925)

"The rule is derived inductively from experience, therefore does not have any inner necessity, is always valid only for special cases and can anytime be refuted by opposite facts. On the contrary, the law is a logical relation between conceptual constructions; it is therefore deductible from upper laws and enables the derivation of lower laws; it has as such a logical necessity in concordance with its upper premises; it is not a mere statement of probability, but has a compelling, apodictic logical value once its premises are accepted."(Ludwig von Bertalanffy, "Kritische Theorie der Formbildung", 1928)

"As perceivers we select from all the stimuli falling on our senses only those which interest us, and our interests are governed by a pattern-making tendency, sometimes called a schema." (Mary Douglas, "Purity and Danger", 1966)

"Whether or not a given conceptual model or representation of a physical system happens to be picturable, is irrelevant to the semantics of the theory to which it eventually becomes attached. Picturability is a fortunate psychological occurrence, not a scientific necessity. Few of the models that pass for visual representations are picturable anyhow. For one thing, the model may be and usually is constituted by imperceptible items such as unextended particles and invisible fields. True, a model can be given a graphic representation - but so can any idea as long as symbolic or conventional diagrams are allowed. Diagrams, whether representational or symbolic, are meaningless unless attached to some body of theory. On the other hand theories are in no need of diagrams save for psychological purposes. Let us then keep theoretical models apart from visual analogues."  (Mario Bunge, "Philosophy of Physics", 1973)

"The understanding of a thing begins and ends with some conceptual model of it. The model is the better, the more accurate, and inclusive. But even rough models can be used to guide - or misguide - research." (Bunge A Mario, "Philosophy in Crisis: The Need for Reconstruction", 2001)

"A conceptual model is a mental image of a system, its components, its interactions. It lays the foundation for more elaborate models, such as physical or numerical models. A conceptual model provides a framework in which to think about the workings of a system or about problem solving in general. An ensuing operational model can be no better than its underlying conceptualization." (Henry N Pollack, "Uncertain Science … Uncertain World", 2005)

"[...] a single thing may elicit several appearances, various conceptual models of it, or several plans of action for it, depending on the subject’s abilities and interests." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Although fiction is not fact, paradoxically we need some fictions, particularly mathematical ideas and highly idealized models, to describe, explain, and predict facts.  This is not because the universe is mathematical, but because our brains invent or use refined and law-abiding fictions, not only for intellectual pleasure but also to construct conceptual models of reality." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"At all events, our world pictures may have components of all three kinds: perceptual, conceptual, and praxiological (action-theoretical).  This is because there are three gates to the outer world: perception, conception, and action. However, ordinarily only one or two of them need be opened: combinations of all three, as in building a house according to a blueprint, are the exception.  We may contemplate a landscape without forming either a conceptual model of it or a plan to act upon it.  And we may build a theoretical model of an imperceptible thing, such as an invisible extrasolar planet, on which we cannot act." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

On Mental Models XLV

"A symbol, therefore, may have no effect and indeed ordinarily will have no effect on the image of the immediate future around one.   It does produce an effect, however, of what might be called the image of the image, on the image of the future, on the image of the past, on the image of the potential or even of the image of the possible." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"I have suggested that one of the basic theorems of the theory of the image is that it is the image which in fact determines what might be called the current behavior of any organism […] The image acts as a field. The behavior consists in gravitating toward the most highly valued part of the field." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"Within the confines of my abstraction, for instance, it is clear that the problem of truth and validity cannot be solved completely, if what we mean by the truth of an image is its correspondence with some reality in the world outside it.  The difficulty with any correspondence theory of truth is that images can only be compared with images.  They can never be compared with any outside reality.  The difficulty with the coherence theory of truth, on the other hand, is that the coherence or consistency of the image is simply not what we mean by its truth." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"As perceivers we select from all the stimuli falling on our senses only those which interest us, and our interests are governed by a pattern-making tendency, sometimes called a schema. In a chaos of shifting impressions each of us constructs a stable world in which objects have recognisable shapes, are located in depth and have permanence." (Mary Douglas, "Purity and Danger", 1966)

"It [system dynamics] focuses on building system dynamics models with teams in order to enhance team learning, to foster consensus and to create commitment with a resulting decision […] System dynamics can be helpful to elicit and integrate mental models into a more holistic view of the problem and to explore the dynamics of this holistic view […] It must be understood that the ultimate goal of the intervention is not to build a system dynamics model. The system dynamics model is a means to achieve other ends […] putting people in a position to learn about a messy problem … create a shared social reality […] a shared understanding of the problem and potential solutions … to foster consensus within the team [..]" (Jac A M Vennix, "Group Model Building: Facilitating Team Learning Using System Dynamics", 1996)

"A vision is a clear mental picture of a desired future outcome. If you have ever put together a large 1,000-piece jigsaw puzzle, the chances are you used the picture on the top of the puzzle box to guide the placement of the pieces. That picture on the top of the box is the end result or the vision of what you are trying to turn into a reality. It is much more difficult - if not impossible - to put the jigsaw puzzle together without ever looking at the picture." (Jane Flaherty & Peter B Stark, "The Manager's Pocket Guide to Leadership Skills", 1999)

"Deep change in mental models, or double-loop learning, arises when evidence not only alters our decisions within the context of existing frames, but also feeds back to alter our mental models. As our mental models change, we change the structure of our systems, creating different decision rules and new strategies. The same information, interpreted by a different model, now yields a different decision. Systems thinking is an iterative learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view, reinventing our policies and institutions accordingly." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"Eliciting and mapping the participant's mental models, while necessary, is far from sufficient [...] the result of the elicitation and mapping process is never more than a set of causal attributions, initial hypotheses about the structure of a system, which must then be tested. Simulation is the only practical way to test these models. The complexity of the cognitive maps produced in an elicitation workshop vastly exceeds our capacity to understand their implications. Qualitative maps are simply too ambiguous and too difficult to simulate mentally to provide much useful information on the adequacy of the model structure or guidance about the future development of the system or the effects of policies." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"To form a mental picture of the event, the knowledge developer attempts to integrate his or her perception of the situation with the expert’s perception. That mental picture is then recorded. What happens is a continuous shuttle process; the knowledge developer mentally moves back and forth from the initial impression of the event to the later evaluation of the event. What is finally recorded is the evaluation made during this retrospective period. Because a time lapse can make details of a situation less clear, the information is not always valid." (Elias M Awad, "Knowledge Management", 2003)

"Images are generally resistant to change and ignore messages that do not conform to their internal settings. Sometimes, however, they do react and can alter in an incremental or even revolutionary manner. Humans can talk about and share their images and, in the symbolic universe they create, reflect upon what is and what might be." (Michael C Jackson, "Critical Systems Thinking and the Management of Complexity", 2019)

On Cybernetics (1940-1949)

"As soon as we are convinced that all technical and non-technical feedback systems are closely related, these relationships must not be distinguished by their specific designs in anatomy or technology; on the contrary their only common characterisation is the analogy of signal flows and the dynamics of control." (Hermann Schmidt, "Regelungstechnik - die technische Aufgabe und ihre wissenschaftliche, sozialpolitische und kulturpolitische Auswirkung", Verein Deutscher Ingenieure, Zeitschrift Vol. 85 (4), 1941)

"[…] the physiologist is forced to use simplified schemes or models for the description of organic regulation processes. He finds such simplified schemes in regulation technology in a great variety of forms, as if ready made for his use. [...] The state can also be viewed in some of its activities as a regulator of free forces […]" (Hermann Schmidt, "Regelungstechnik - die technische Aufgabe und ihre wissenschaftliche, sozialpolitische und kulturpolitische Auswirkung", Verein Deutscher Ingenieure, Zeitschrift Vol. 85 (4), 1941)

"Given any object, relatively abstracted from its surroundings for study, the behavioristic approach consists in the examination of the output of the object and of the relations of this output to the input. By output is meant any change produced in the surroundings by the object. By input, conversely, is meant any event external to the object that modifies this object in any manner." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Teleology", Philosophy of Science 10, 1943)

"In classifying behavior the term teleology was used as synonymous with purpose controlled by feed-back. Teleology has been interpreted in the past to imply purpose and the vague concept of a final cause has been often added. This concept of final causes has led to the opposition of teleology to determinism. […] teleology is not opposed to determinism, but to non-teleology. Both teleological and nonteleological systems are deterministic when the behavior considered belongs to the realm where determinism applies. The concept of teleology shares only one thing with the concept of causality: a time axis. But causality implies a one-way, relatively irreversible functional relationship, whereas teleology is concerned with behavior, not with functional relationships." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Technology", Philosophy of Science Vol. 10 (1), 1943)

"Since we consider purposefulness a concept necessary for the understanding of certain modes of behavior we suggest that a teleological study is useful if it avoids problems of causality and concerns itself merely with an investigation of purpose." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Technology", Philosophy of Science Vol. 10 (1), 1943)

"Purposeful active behavior may be subdivided into two classes: feed-back (or teleological) and non-feed-back (or non-teleological)." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Technology", Philosophy of Science Vol. 10 (1), 1943)

"Besides electrical engineering theory of the transmission of messages, there is a larger field [cybernetics] which includes not only the study of language but the study of messages as a means of controlling machinery and society, the development of computing machines and other such automata, certain reflections upon psychology and the nervous system, and a tentative new theory of scientific method." (Norbert Wiener, "Cybernetics", 1948)

"Cybernetics is a word invented to define a new field in science. It combines under one heading the study of what in a human context is sometimes loosely described as thinking and in engineering is known as control and communication. In other words, cybernetics attempts to find the common elements in the functioning of automatic machines and of the human nervous system, and to develop a theory which will cover the entire field of control and communication in machines and in living organisms." (Norbert Wiener, "Cybernetics", 1948) 

"The concept of teleological mechanisms however it be expressed in many terms, may be viewed as an attempt to escape from these older mechanistic formulations that now appear inadequate, and to provide new and more fruitful conceptions and more effective methodologies for studying self-regulating processes, self-orienting systems and organisms, and self-directing personalities. Thus, the terms feedback, servomechanisms, circular systems, and circular processes may be viewed as different but equivalent expressions of much the same basic conception." (Lawrence K Frank, 1948)

"We have decided to call the entire field of control and communication theory, whether in the machine or in the animal, by the name Cybernetics, which we form from the Greek [...] for steersman. In choosing this term, we wish to recognize that the first significant paper on feedback mechanisms is an article on governors, which was published by Clerk Maxwell in 1868, and that governor is derived from a Latin corruption [...] We also wish to refer to the fact that the steering engines of a ship are indeed one of the earliest and best-developed forms of feedback mechanisms." (Norbert Wiener, "Cybernetics", 1948)

On Metaphors VIII

"The world is emblematic. Parts of speech are metaphors, because the whole of nature is a metaphor of the human mind." (Ralph W Emerson, "Nature", 1836)

"Everything which distinguishes man from the animals depends upon this ability to volatilize perceptual metaphors in a schema, and thus to dissolve an image into a concept. For something is possible in the realm of these schemata which could never be achieved with the vivid first impressions: the construction of a pyramidal order according to castes and degrees, the creation of a new world of laws, privileges, subordinations, and clearly marked boundaries - a new world, one which now confronts that other vivid world of first impressions as more solid, more universal, better known, and more human than the immediately perceived world, and thus as the regulative and imperative world." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"The various languages placed side by side show that with words it is never a question of truth, never a question of adequate expression; otherwise, there would not be so many languages. The ‘thing in itself’ (which is precisely what the pure truth, apart from any of its consequences, would be) is likewise something quite incomprehensible to the creator of language and something not in the least worth striving for. This creator only designates the relations of things to men, and for expressing these relations he lays hold of the boldest metaphors. To begin with, a nerve stimulus is transferred into an image: first metaphor. The image, in turn, is imitated in a sound: second metaphor. And each time there is a complete overleaping of one sphere, right into the middle of an entirely new and different one." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"One should employ a metaphor in science only when there is good evidence that an important similarity or analogy exists between its primary and secondary subjects. One should seek to discover more about the relevant similarities or analogies, always considering the possibility that there are no important similarities or analogies, or alternatively, that there are quite distinct similarities for which distinct terminology should be introduced. One should try to discover what the 'essential' features of the similarities or analogies are, and one should try to assimilate one’s account of them to other theoretical work in the same subject area - that is, one should attempt to explicate the metaphor." (Richard Boyd, "Metaphor and Theory Change: What Is ‘Metaphor’ a Metaphor For?", 1979)

"Myth is the system of basic metaphors, images, and stories that in-forms the perceptions, memories, and aspirations of a people; provides the rationale for its institutions, rituals and power structure; and gives a map of the purpose and stages of life." (Sam Keen, "The Passionate Life", 1983)

"Organizations are complex and paradoxical phenomena that can be understood in many different ways. Many of our taken-for-granted ideas about organizations are metaphorical, even though we may not recognize them as such. For example, we frequently talk about organizations as if they were machines designed to achieve predetermined goals and objectives, and which should operate smoothly and efficiently. And as a result of this kind of thinking, we often attempt to organize and manage them in a mechanistic way, forcing their human qualities into a background role. By using different metaphors to understand the complex and paradoxical character of organizational life, we are able to manage and design organizations in ways that we may not have thought possible before." (Gareth Morgan, "Images of Organization", 1986)

"We consider the notion of ‘system’ as an organising concept, before going on to look in detail at various systemic metaphors that may be used as a basis for structuring thinking about organisations and problem situations.“ (Mike Jackson, Creative Problem Solving: Total Systems Intervention, 1991)

"Perhaps our ultimate understanding of scientific topics is measured in terms of our ability to generate metaphoric pictures of what is going on. Maybe understanding is coming up with metaphoric pictures." (Per Bak, "How Nature Works: the science of self-organized criticality", 1996)

"We are a people captivated by the power and romance of metaphor, forever seeking the invisible through the image of the visible." (Lewis H Lapham, "Waiting For The Barbarians", 1997)

"Metaphor lives a secret life all around us. We utter about six metaphors a minute. Metaphorical thinking is essential to how we understand ourselves and others, how we communicate and learn, discover and invent." (James Geary, "I Is an Other: The Secret Life of Metaphor and How it Shapes the Way We See the World", 2011)

Mental Models XLIII

"We both are, and know that we are, and delight in our being, and our knowledge of it. Moreover, in these three things no true-seeming illusion disturbs us; for we do not come into contact with these by some bodily sense, as we perceive the things outside of us of all which sensible objects it is the images resembling them, but not themselves which we perceive in the mind and hold in the memory, and which excite us to desire the objects. But, without any delusive representation of images or phantasms, I am most certain that I am, and that I know and delight in this." (Aurelius Augustinus, "The City of God", early 400s)

"The imagination is an eye where images remain forever." (Joseph Joubert, [Letter to Revd. Dr. Trusler] 1799) 

"There is a kind, I might almost say, of artistic satisfaction, when we are able to survey the enormous wealth of Nature as a regularly ordered whole - a kosmos, an image of the logical thought of our own mind." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1881) 

"I call the combination of a concept and a sound-image a sign, but in current usage the term generally designates only a sound-image, a word, for example (arbor, etc.). One tends to forget that arbor is called a sign only because it carries the concept ‘tree’, with the result that the idea of the sensory part implies the idea of the whole." (Ferdinand de Saussure, "Course in General Linguistics", 1915) 

"Thought is prior to language and consists in the simultaneous presentation to the mind of two different images." (Thomas E Hulme, "Notes on Language and Style", 1929) 

"In this way things, external objects, are assimilated to more or less ordered motor schemas, and in this continuous assimilation of objects the child's own activity is the starting point of play. Not only this, but when to pure movement are added language and imagination, the assimilation is strengthened, and wherever the mind feels no actual need for accommodating itself to reality, its natural tendency will be to distort the objects that surround it in accordance with its desires or its fantasy, in short to use them for its satisfaction. Such is the intellectual egocentrism that characterizes the earliest form of child thought." (Jean Piaget, "The Moral Judgment of the Child", 1932)

"What now is the answer to the question as to the bridge between the perception of the senses and the concepts, which is now reduced to the question as to the bridge between the outer perceptions and those inner image-like representations. It seems to me one has to postulate a cosmic order of nature - outside of our arbitrariness- to which the outer material objects are subjected as are the inner images […] The organizing and regulating has to be posited beyond the differentiation of physical and psychical […] I am all for it to call this ‘organizing and regulating’ ‘archetypes’. It would then be inadmissible to define these as psychic contents. Rather, the above-mentioned inner pictures (dominants of the collective unconscious, see Jung) are the psychic manifestations of the archetypes, but which would have to produce and condition all nature laws belonging to the world of matter. The nature laws of matter would then be the physical manifestation of the archetypes." (Wolfgang Pauli, [Letter to Markus Fierz], 1948) 

"My suggestion is that at each state the proper order of operation of the mind requires an overall grasp of what is generally known, not only in formal logical, mathematical terms, but also intuitively, in images, feelings, poetic usage of language, etc." (David Bohm,"Wholeness and the Implicate Order Wholeness and the Implicate Order", 1980) 

"Visual thinking is often driven more strongly by the conceptual knowledge we use to organize our images than by the contents of the images themselves. Chess masters are known for their remarkable memory for the pieces on a chessboard. But it's not because people with photographic memories become chess masters. The masters are no better than beginners when remembering a board of randomly arranged pieces. Their memory captures meaningful relations among the pieces, such as threats and defenses, not just their distribution in space."(Steven Pinker, "How the Mind Works", 1997)

On Classification I: Science

"No occupation is more worthy of an intelligent and enlightened mind, than the study of Nature and natural objects; and whether we labour to investigate the structure and function of the human system, whether we direct our attention to the classification and habits of the animal kingdom, or prosecute our researches in the more pleasing and varied field of vegetable life, we shall constantly find some new object to attract our attention, some fresh beauties to excite our imagination, and some previously undiscovered source of gratification and delight." (Sir Joseph Paxton, "A Practical Treatise on the Cultivation of the Dahlia", 1838)

"Are our systems the inventions of naturalists, or only their reading of the Book of Nature? and can that book have more than one reading? If these classifications are not mere inventions, if they are not an attempt to classify for our own convenience the objects we study, then they are thoughts which, whether we detect them or not, are expressed in Nature, - then Nature is the work of thought, the production of intelligence carried out according to plan, therefore premeditated, - and in our study of natural objects we are approaching the thoughts of the Creator, reading His conceptions, interpreting a system that is His and not ours." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"Science is the systematic classification of experience." (George H Lewes, "The Physical Basis of Mind", 1877)

"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)

"The sole purpose of physical theory is to provide a representation and classification of experimental laws; the only test permitting us to judge a physical theory and pronounce it good or bad is the comparison between the consequences of this theory and the experimental laws it has to represent and classify."  (Pierre-Maurice-Marie Duhem, “The Aim and Structure of Physical Theory”, 1908)

"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)

"A […] difference between most system-building in the social sciences and systems of thought and classification of the natural sciences is to be seen in their evolution. In the natural sciences both theories and descriptive systems grow by adaptation to the increasing knowledge and experience of the scientists. In the social sciences, systems often issue fully formed from the mind of one man. Then they may be much discussed if they attract attention, but progressive adaptive modification as a result of the concerted efforts of great numbers of men is rare." (Lawrence J Henderson, "The Study of Man", 1941)

"The purpose of a classification scheme is to arrange information, in documents on shelves or on cards in indexes, in a sequence that will be helpful to the user." (Douglas J Foskett, Classification and indexing in the social sciences, 1963)

"The classification of a system as complex or simple will depend upon the observer of the system and upon the purpose he has for considering the system." (Mike Jackson, "Towards a System of Systems Methodologies", 1984)

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

On Abstraction (1960-1969)

"It is of our very nature to see the universe as a place that we can talk about. In particular, you will remember, the brain tends to compute by organizing all of its input into certain general patterns. It is natural for us, therefore, to try to make these grand abstractions, to seek for one formula, one model, one God, around which we can organize all our communication and the whole business of living." (John Z Young, "Doubt and Certainty in Science: A Biologist’s Reflections on the Brain", 1960)

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

"Scientists, it should already be clear, never learn concepts, laws, and theories in the abstract and by themselves. Instead, these intellectual tools are from the start encountered in a historically and pedagogically prior unit that displays them with and through their applications." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractions, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range for its applications." (Aleksandr D Aleksandrov, 1963)

"A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense..." (Hermann Bondi. "Relativity and Common Sense", 1964)

"If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details." (John G Kemeny, "Random Essays on Mathematics, Education, and Computers", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"Learning is any change in a system that produces a more or less permanent change in its capacity for adapting to its environment. Understanding systems, especially systems capable of understanding problems in new task domains, are learning systems." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The more we are willing to abstract from the detail of a set of phenomena, the easier it becomes to simulate the phenomena. Moreover we do not have to know, or guess at, all the internal structure of the system but only that part of it that is crucial to the abstraction." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

On Abstraction (1930-1939)

"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"The fundamental concepts of physical science, it is now understood, are abstractions, framed by our mind, so as to bring order to an apparent chaos of phenomena." (Sir William C Dampier, "A History of Science and its Relations with Philosophy & Religion", 1931)

"It is the function of notions in science to be useful, to be interesting, to be verifiable and to acquire value from anyone of these qualities. Scientific notions have little to gain as science from being forced into relation with that formidable abstraction, ‘general truth’." (Wilfred Trotter, [paper delivered before the Royal College of Surgeons of England] 1932)

"We love to discover in the cosmos the geometrical forms that exist in the depths of our consciousness. The exactitude of the proportions of our monuments and the precision of our machines express a fundamental character of our mind. Geometry does not exist in the earthly world. It has originated in ourselves. The methods of nature are never so precise as those of man. We do not find in the universe the clearness and accuracy of our thought. We attempt, therefore, to abstract from the complexity of phenomena some simple systems whose components bear to one another certain relations susceptible of being described mathematically." (Alexis Carrel, "Man the Unknown", 1935)

„[...] the abstract mathematical theory has an independent, if lonely existence of its own. But when a sufficient number of its terms are given physical definitions it becomes a part of a vital organism concerning itself at every instant with matters full of human significance. Every theorem can be given the form ‘if you do so and so, such and such will happen'." (Oswald Veblen, "Remarks on the Foundation of Geometry", Bulletin of the American Mathematical Society, Vol. 35, 1935)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T. Sanders, "Mathematics", National Mathematics Magazine, 1937)

"Sooner or later the cold plunge into pure abstraction must be taken if one is to learn to swim in mathematics and to reason as rational, thinking human beings do." (Eric T Bell, "The Handmaiden of the Sciences", 1937)

"The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes." (Eric T Bell, "Men of Mathematics", 1937)

"Matter-of-fact is an abstraction, arrived at by confining thought to purely formal relations which then masquerade as the final reality. This is why science, in its perfection, relapses into the study of differential equations. The concrete world has slipped through the meshes of the scientific net." (Alfred N Whitehead, "Modes of Thought", 1938)

Mental Models XXXI

"When I have before my eyes a pair of stereoscopic drawings which are hard to combine, it is difficult to bring the lines and points that correspond, to cover each other, and with every little motion of the eyes they glide apart. But if I chance to gain a lively mental image (Anschauungsbild) of the represented solid form (a thing that often occurs by lucky chance), I then move my two eyes with perfect certainty over the figure without the picture separating again." (Hermann von Helmholtz, "Tonempfindungen" ["Sensations of Tone"], 1863)

"When a mental image, as such, is the object of my apprehension, there is no meaning in seeking to distinguish its existence in my consciousness (in me) from its existence out of my consciousness (in itself) ; for the object apprehended is, in this case, one which does not even exist, as the objects of external perception do, in itself outside of my consciousness. It exists only within me." (Friedrich Ueberweg, “System of Logic and History of Logical Doctrines”, 1871)

"Ideas are substitutions which require a secondary process when what is symbolized by them is translated into the images and experiences it replaces; and this secondary process is frequently not performed at all, generally only performed to a very small extent. Let anyone closely examine what has passed in his mind when he has constructed a chain of reasoning, and he will be surprised at the fewness and faintness of the images which have accompanied the ideas." (George H Lewes "Problems of Life and Mind", 1873)

"The leading characteristic of algebra is that of operation on relations. This also is the leading characteristic of Thought. Algebra cannot exist without values, nor Thought without Feelings. The operations are so many blank forms till the values are assigned. Words are vacant sounds, ideas are blank forms, unless they symbolize images and sensations which are their values. Nevertheless it is rigorously true, and of the greatest importance, that analysts carry on very extensive operations with blank forms, never pausing to supply the symbols with values until the calculation is completed; and ordinary men, no less than philosophers, carry on long trains of thought without pausing to translate their ideas (words) into images." (George H Lewes "Problems of Life and Mind", 1873)

"In this sense the fundamental ideas of mechanics, together with the principles connecting them, represent the simplest image which physics can produce of things in the sensible world and the processes which occur in it. By varying the choice of the propositions which we take as fundamental, we can give various representations of the principles of mechanics. Hence we can thus obtain various images of things; and these images we can test and compare with each other in respect of permissibility, correctness, and appropriateness." (Heinrich Hertz, “The Principles of Mechanics Presented in a New Form”, 1894)

"The images which we may form of things are not determined without ambiguity by the requirement that the consequents of the images must be the images of the consequents. Various images of the same objects are possible, and these images may differ in various respects. We should at once denote as inadmissible all images which implicitly contradict the laws of our thought. Hence we postulate in the first place that all our images shall be logically permissible or, briefly, that they shall be permissible. We shall denote as incorrect any permissible images, if their essential relations contradict the relations of external things, i.e. if they do not satisfy our first fundamental requirement. Hence we postulate in the second place that our images shall be correct." (Heinrich Hertz, “The Principles of Mechanics Presented in a New Form”, 1894)

"Thinking in pictures is, therefore, only a very incomplete form of becoming conscious. In some way, too, it stands nearer to unconscious processes than does thinking in words, and it is unquestionably older than the latter both ontogenetically and phylogenetically." (Sigmund Freud, "The Ego And The Id", 1923)

"Common sense […] may be thought of as a series of concepts and conceptual schemes which have proved highly satisfactory for the practical uses of mankind. Some of those concepts and conceptual schemes were carried over into science with only a little pruning and whittling and for a long time proved useful. As the recent revolutions in physics indicate, however, many errors can be made by failure to examine carefully just how common sense ideas should be defined in terms of what the experimenter plans to do." (James B Conant, "Science and Common Sense", 1951)

"We look upon economic theory as a sequence of conceptual models that seek to express in simplified form different aspects of an always more complicated reality." (Tjalling Koopmans, "Three Essays", 1957)

On Concepts III

"With the synthesis of every new concept in the aggregation of coordinate characteristics the extensive or complex distinctness is increased; with the further analysis of concepts in the series of subordinate characteristics the intensive or deep distinctness is increased. The latter kind of distinctness, as it necessarily serves the thoroughness and conclusiveness of cognition, is therefore mainly the business of philosophy and is carried farthest especially in metaphysical investigations." (Immanuel Kant, "Logic", 1800)

"In speaking here of ‘comprehensibility’, the expression is used in its most modest sense. It implies: the production being produced by the creation of general concepts, relations between these concepts and sense experience. It is in this sense that the world of our sense experiences is comprehensible. The fact that it is comprehensible is a miracle." (Albert Einstein, "Out of My Later Years", 1950)

"What in fact is the schema of the object? In one essential respect it is a schema belonging to intelligence. To have the concept of an object is to attribute the perceived figure to a substantial basis, so that the figure and the substance that it thus indicates continue to exist outside the perceptual field. The permanence of the object seen from this viewpoint is not only a product of intelligence, but constitutes the very first of those fundamental ideas of conservation which we shall see developing within the thought process." (Jean Piaget, "The Psychology of Intelligence", 1950)

"A conceptual scheme is never discarded merely because of a few stubborn facts with which it cannot be reconciled; a conceptual scheme is either modified or replaced by a better one, never abandoned with nothing left to take its place." (James B Conant, "Science and Common Sense", 1951)

"[…] the link between observation and formulation is one of the most difficult and crucial in the scientific enterprise. It is the process of interpreting our theory or, as some say, of ‘operationalizing our concepts’. Our creations in the world of possibility must be fitted in the world of probability; in Kant’s epigram, ‘Concepts without precepts are empty’. It is also the process of relating our observations to theory; to finish the epigram, ‘Precepts without concepts are blind’." (Scott Greer, "The Logic of Social Inquiry", 1969)

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

"The basic idea is that schemata are data structures for representing the generic concepts stored in memory. There are schemata for generalized concepts underlying objects, situations, events, sequences of events, actions, and sequences of actions. Roughly, schemata are like models of the outside world. To process information with the use of a schema is to determine which model best fits the incoming information. Ultimately, consistent configurations of schemata are discovered which, in concert, offer the best account for the input. This configuration of schemata together constitutes the interpretation of the input. (David E Rumelhart, Paul Smolensky, James L McClelland & Geoffrey E Hinton, "Schemata and sequential thought processes in PDP models", 1986)

"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)

"[…] all mathematical cognition has this pecularity: that it must first exhibit its concept in intuitional form. […] Without this, mathematics cannot take a single step. Its judgements are therefore always intuitional, whereas philosophy must make do with discursive judgements from mere concepts. It may illustrate its judgements by means of a visual form, but it can never derive them from such a form." (Immanuel Kant)

"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." (Paul Halmos)

Mental Models XXX

"Everything which distinguishes man from the animals depends upon this ability to volatilize perceptual metaphors in a schema, and thus to dissolve an image into a concept. For something is possible in the realm of these schemata which could never be achieved with the vivid first impressions: the construction of a pyramidal order according to castes and degrees, the creation of a new world of laws, privileges, subordinations, and clearly marked boundaries - a new world, one which now confronts that other vivid world of first impressions as more solid, more universal, better known, and more human than the immediately perceived world, and thus as the regulative and imperative world." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"That immense framework and planking of concepts to which the needy man clings his whole life long in order to preserve himself is nothing but a scaffolding and toy for the most audacious feats of the liberated intellect. And when it smashes this framework to pieces, throws it into confusion, and puts it back together in an ironic fashion, pairing the most alien things and separating the closest, it is demonstrating that it has no need of these makeshifts of indigence and that it will now be guided by intuitions rather than by concepts. There is no regular path which leads from these intuitions into the land of ghostly schemata, the land of abstractions. There exists no word for these intuitions; when man sees them he grows dumb, or else he speaks only in forbidden metaphors and in unheard - of combinations of concepts. He does this so that by shattering and mocking the old conceptual barriers he may at least correspond creatively to the impression of the powerful present intuition." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"This is the reason why mechanical explanations are better understood than stories, even though they are more difficult to reproduce. The exposition, even if it is faulty, excites analogous schemas already existing in the listener’s mind; so that what takes place is not genuine understanding, but a convergence of acquired schemas of thought. In the case of stories, this convergence is not possible, and the schemas brought into play are usually divergent." (Jean Piaget, "The Language and Thought of the Child", 1926)

"At first sight the assimilative tendency shown by thought seems sufficient to secure stability in judgments. To assimilate, in psychology as in biology, is to reproduce oneself by means of the external world; it is to transform perceptions until they are identical with one’s own thought, i.e. with previous schemas. Assimilation is therefore preservation and, in a certain sense, identification." (Jean Piaget, "Judgement and Reasoning in the Child", 1928)

"In the presence of certain objects of thought or of certain affirmations the child, in virtue of previous experiences, adopts a certain way of reacting and thinking which is always the same, and which might be called a schema of reasoning. Such schemas are the functional equivalents of general propositions, but since the child is not conscious of these schemas before discussion and a desire for proof have laid them bare and at the same time changed their character, they cannot be said to constitute implicit general propositions. They simply constitute certain unconscious tendencies which live their own life but are submitted to no general systematization and consequently lead to no logical exactitude. To put it in another way, they form a logic of action but not yet a logic of thought." (Jean Piaget, "Judgement and Reasoning in the Child", 1928)

"The more the schemata are differentiated, the smaller the gap between the new and the familiar becomes, so that novelty, instead of constituting an annoyance avoided by the subject, becomes a problem and invites searching." (Jean Piaget, "The Construction Of Reality In The Child", 1950)

"Accommodation of mental structures to reality implies the existence of assimilatory schemata apart from which any structure would be impossible." (Jean Piaget, "The Construction Of Reality In The Child", 1950)

"What in fact is the schema of the object? In one essential respect it is a schema belonging to intelligence. To have the concept of an object is to attribute the perceived figure to a substantial basis, so that the figure and the substance that it thus indicates continue to exist outside the perceptual field. The permanence of the object seen from this viewpoint is not only a product of intelligence, but constitutes the very first of those fundamental ideas of conservation which we shall see developing within the thought process." (Jean Piaget, "The Psychology of Intelligence", 1950)

"[…] there is perhaps a difference between the ideas which are associated in the sense of their patterns being tired to the original one and available in connexion with it, and being actually associated or aroused. Our mental modelling of the outer world may imitate it and its sequences from moment to moment, but only that which is fairly frequent, or fits into other patterns, will remain for long, and of that only a portion will arise in response to other ideas. " (Kenneth J W Craik, “The Nature of Psychology”, 1966)

"[Schemata are] knowledge structures that represent objects or events and provide default assumptions about their characteristics, relationships, and entailments under conditions of incomplete information." (Paul J DiMaggio, "Culture and Cognition", Annual Review of Sociology No. 23, 1997)

On Discovery (1975-1999)

"It is one of our most exciting discoveries that local discovery leads to a complex of further discoveries. Corollary to this we find that we no sooner get a problem solved than we are overwhelmed with a multiplicity of additional problems in a most beautiful payoff of heretofore unknown, previously unrecognized, and as-yet unsolved problems." (Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)

"You cannot learn, through common sense, how things are you can only discover where they fit into the existing scheme of things." (Stuart Hall, 1977)

"Science is not a heartless pursuit of objective information. It is a creative human activity, its geniuses acting more as artists than information processors. Changes in theory are not simply the derivative results of the new discoveries but the work of creative imagination influenced by contemporary social and political forces. " (Stephen J Gould, "Ever Since Darwin: Reflections in Natural History", 1977)

"Every discovery, every enlargement of the understanding, begins as an imaginative preconception of what the truth might be. The imaginative preconception - a ‘hypothesis’ - arises by a process as easy or as difficult to understand as any other creative act of mind; it is a brainwave, an inspired guess, a product of a blaze of insight. It comes anyway from within and cannot be achieved by the exercise of any known calculus of discovery. " (Sir Peter B Medawar, "Advice to a Young Scientist", 1979)

"It is hard for us today to assimilate all the new ideas that are being suggested in response to the new information we have. We must remember that our picture of the universe is based not only on our scientific knowledge but also on our culture and our philosophy. What new discoveries lie ahead no one can say. There may well be civilizations in other parts of our galaxy or in other galaxies that have already accomplished much of what lies ahead for mankind. Others may just be beginning. The universe clearly presents an unending challenge." (Necia H Apfel & J Allen Hynek, "Architecture of the Universe", 1979)

"One cannot ‘invent’ the structure of an object. The most we can do is to patiently bring it to the light of day, with humility - in making it known, it is ‘discovered’. If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we are in no sense ‘making’ or ’building’ these ‘structures’. They have not waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we have been detecting or discovering, the reticent structure which we are trying to grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to ‘invent’ the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to ‘construct’ with the help of this language, bit by bit, those ‘theories’ which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them by a language in constant state improvement [...].The sole thing that constitutes the true inventiveness and imagination of the researcher is the quality of his attention as he listens to the voices of things." (Alexander Grothendieck, "Récoltes et semailles –Rélexions et témoignage sur un passé de mathématicien", 1985) 

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

"[…] there is an external world which can in principle be exhaustively described in scientific language. The scientist, as both observer and language-user, can capture the external facts of the world in propositions that are true if they correspond to the facts and false if they do not. Science is ideally a linguistic system in which true propositions are in one-to-one relation to facts, including facts that are not directly observed because they involve hidden entities or properties, or past events or far distant events. These hidden events are described in theories, and theories can be inferred from observation, that is, the hidden explanatory mechanism of the world can be discovered from what is open to observation. Man as scientist is regarded as standing apart from the world and able to experiment and theorize about it objectively and dispassionately." (Mary B Hesse, "Revolutions and Reconstructions in the Philosophy of Science", 1980)

"One cannot ‘invent’ the structure of an object. The most we can do is to patiently bring it to the light of day, with humility - in making it known, it is ‘discovered’. If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we are in no sense ‘making’ or ’building’ these ‘structures’. They have not waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we have been detecting or discovering, the reticent structure which we are trying to grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to ‘invent’ the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to ‘construct’ with the help of this language, bit by bit, those ‘theories’ which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them by a language in constant state improvement [...].The sole thing that constitutes the true inventiveness and imagination of the researcher is the quality of his attention as he listens to the voices of things." (Alexander Grothendieck, "Récoltes et semailles –Rélexions et témoignage sur un passé de mathématicien", 1985)

"Ultimately, discovery and invention are both problems of classification, and classification is fundamentally a problem of finding sameness. When we classify, we seek to group things that have a common structure or exhibit a common behavior." (Grady Booch, "Object-oriented design: With Applications", 1991)

"In brief, the way we do mathematics is human, very much so. But mathematicians have no doubt that there is a mathematical reality beyond our puny existence. We discover mathematical truth, we do not create it. We ask ourselves what seems to be a natural question and start working on it, and not uncommonly we find the solution (or someone else does). And we know that the answer could not have been different." (David Ruelle, "Chance and Chaos", 1991)

"As a result, surprisingly enough, scientific advance rarely comes solely through the accumulation of new facts. It comes most often through the construction of new theoretical frameworks. [..] To understand scientific development, it is not enough merely to chronicle new discoveries and inventions. We must also trace the succession of worldviews" (Nancy R Pearcey & Charles B. Thaxton, "The Soul of Science: Christian Faith and Natural Philosophy", 1994)

"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)

On Discovery (2000-2019)

"It is through proof that human mathematicians transcend the limitations of their humanity. Proofs link human mathematicians to truths of the universe. In the romance, proofs are discoveries of those truths." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)

"We tackle a multifaceted universe one face at a time, tailoring our models and equations to fit the facts at hand. Whatever mechanical conception proves appropriate, that is the one to use. Discovering worlds within worlds, a practical observer will deal with each realm on its own terms. It is the only sensible approach to take." (Michael Munowitz, "Knowing: The Nature of Physical Law", 2005)

"Equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge." (Robert P Crease, "The Great Equations", 2009)

"The urge to discover, to invent, to know the unknown, seems so deeply human that we cannot imagine our history without it." (Alan Lightman, "The Discoveries: Great Breakthroughs in 20th-Century Science, Including the Original Papers", 2009)

"Metaphor lives a secret life all around us. We utter about six metaphors a minute. Metaphorical thinking is essential to how we understand ourselves and others, how we communicate and learn, discover and invent." (James Geary, "I Is an Other: The Secret Life of Metaphor and How it Shapes the Way We See the World", 2011)

"Science would be better understood if we called theories ‘misconceptions’ from the outset, instead of only after we have discovered their successors." (David Deutsch, "Beginning of Infinity", 2011)

"Models do not only describe reality, they are also instruments for exploring reality. They are not only involved in the integration of known data, but also in the discovery of new data.” (Andreas Bartels, "The Standard Model of Cosmology as a Tool for Interpretation and Discovery”, 2013)

"Stories are how we think. They are how we make meaning of life. Call them schemas, scripts, mental maps, ideas, metaphors, or narratives. Stories are how we inspire and motivate human beings. Great stories help us to understand our place in the world, create our identity, discover our purpose, form our character and define and teach human values." (Jeroninio Almeida, "Karma Kurry for the Mind, Body, Heart & Soul", 2013)

Mental Models XXIII

"We must make a threefold distinction and think of that which becomes, that in which it becomes, and the model which it resembles” (Plato, “Timaeus”, 360 BC)

"An image is, after all, a reminder; it is to the illiterate what a book is to the literate, and what the word is to the hearing, the image is to sight. All this is the approach through the senses: but it is with the mind that we lay hold on the image." (John of Damascus, cca. 8th century) 

“An image (in the most strict signification of the word) is the Resemblance of some thing visible […] (Thomas Hobbes, “Leviathan”, 1651)

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

"This formal and pure condition of sensibility to which the employment of the concept of understanding is restricted, we shall entitle the schema of the concept. The procedure of understanding in these schemata we shall entitle the schematism of pure understanding.
 The schema is in itself always a product of imagination. Since, however, the synthesis of imagination aims at no special intuition, but only at unity in the determination of sensibility, the schema has to be distinguished from the image." (Immanuel Kant," Critique of Pure Reason", 1781)

"The scientific worker has elected primarily to know, not do. He does not directly seek, like the practical man, to realize the ideal of exploiting nature and controlling life – though he makes this more possible; he seeks rather to idealize – to conceptualize – the real, or at least those aspects of reality that are available in his experience. He thinks more of lucidity and formulae than of loaves and fishes. He is more concerned with knowing Nature than with enjoying her. His main intention is to describe the sequences in Nature in the simplest possible formulae, to make a working thought-model of the known world. He would make the world translucent, not that emotion may catch the glimmer of the indefinable light that shines through, but for other reasons – because of his inborn inquisitiveness, because of his dislike of obscurities, because of his craving for a system – an intellectual system in which phenomena are at least provisionally unified." (Sir John A Thomson," Introduction to Science", 1911)

"Once we give serious consideration to the hypothesis of the unconscious, it follows that our view of the world can be but a provisional one; for if we effect so radical an alteration in the subject of perception and cognition as this dual focus implies, the result must be a world view very different from any known before." (Carl Gustav Jung, "The Structure And Dynamics Of The Psyche", 1960) 

“Cultural archetypes are the unconscious models that help us make sense of the world: they are the myths, narratives, images, symbols, and files into which we organize the data of our life experience” (Clotaire Rapaille, “Cultural Imprints”, Executive Excellence Vol. 16 (10), 1999)

"Mental models represent possibilities, and the theory of mental models postulates three systems of mental processes underlying inference: (0) the construction of an intensional representation of a premise’s meaning – a process guided by a parser; (1) the building of an initial mental model from the intension, and the drawing of a conclusion based on heuristics and the model; and (2) on some occasions, the search for alternative models, such as a counterexample in which the conclusion is false. System 0 is linguistic, and it may be autonomous. System 1 is rapid and prone to systematic errors, because it makes no use of a working memory for intermediate results. System 2 has access to working memory, and so it can carry out recursive processes, such as the construction of alternative models." (Sangeet Khemlania & P.N. Johnson-Laird, "The processes of inference", Argument and Computation, 2012)

“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) 

Mental Models XXI

"You may read any quantity of books, and you may be almost as ignorant as you were at starting, if you don’t have, at the back of your minds, the change for words in definite images which can only be acquired through the operation of your observing faculties on the phenomena of nature." (Thomas H Huxley, "Science and Education", 1877)

"[…] we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind of that which is, though it is invisible to us."  (Arabella B Buckley, "The Fairy-Land of Science", 1899)

"We wish to obtain a representation of phenomena and form an image of them in our minds. Till now, we have always attempted to form these images by means of the ordinary notions of time and space. These notions are perhaps innate; in any case they have been developed by our daily observations. For me, these notions are clear, and I confess that I am unable to gain any idea of physics without them. […] I would like to retain this ideal of other days and describe everything that occurs in this world in terms of clear pictures." (Hendrik A Lorentz, [Fifth Solvay Conference] 1927)

"Just as the eye sees details that are not there if they fit in with the sense of the picture, or overlooks them if they make no sense, so also very little inherent certainty will suffice to secure the highest scientific value to an alleged fact, if only it fits in with a great scientific generalization, while the most stubborn facts will be set aside if there is no place for them in the established framework of science." (Michael Polanyi, "Personal Knowledge", 1962)

"If we are to have meaningful, connected experiences; ones that we can comprehend and reason about; we must be able to discern patterns to our actions, perceptions, and conceptions. Underlying our vast network of interrelated literal meanings (all of those words about objects and actions) are those imaginative structures of understanding such as schema and metaphor, such as the mental imagery that allows us to extrapolate a path, or zoom in on one part of the whole, or zoom out until the trees merge into a forest." (William H Calvin, "The Cerebral Code", 1996)

"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)

“We start from vague pictures or ideas […] which we encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialog between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs” (E Brian Davies, “Science in the Looking Glass”, 2003)

“What is consciousness? Our brain simulates reality. So, our everyday experiences are a form of dreaming, which is to say, they are mental models, simulations, not the things they appear to be.” (Stephen LaBerge, “Losi in Lucidity”, 2014)

“A mathematician possesses a mental model of the mathematical entity she works on. This internal mental model is accessible to her direct observation and manipulation. At the same time, it is socially and culturally controlled, to conform to the mathematics community's collective model of the entity in question. The mathematician observes a property of her own internal model of that mathematical entity. Then she must find a recipe, a set of instructions, that enables other competent, qualified mathematicians to observe the corresponding property of their corresponding mental model. That recipe is the proof. It establishes that property of the mathematical entity.” (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)

“Men of broader intellect know that there is no clear distinction between the real and the unreal, that things appear as they seem only by virtue of the delicate physical and mental instruments through which we perceive them.” (Howard P Lovecraft)

Mental Models XIX

"'Schema' refers to an active organisation of past reactions, or of past experiences, which must always be supposed to be operating in any well-adapted organic response. That is, whenever there is any order or regularity of behavior, a particular response is possible only because it is related to other similar responses which have been serially organised, yet which operate, not simply as individual members coming one after another, but as a unitary mass. Determination by schemata is the most fundamental of all the ways in which we can be influenced by reactions and experiences which occurred some time in the past. All incoming impulses of a certain kind, or mode, go together to build up an active, organised setting: visual, auditory, various types of cutaneous impulses and the like, at a relatively low level; all the experiences connected by a common interest: in sport, in literature, history, art, science, philosophy, and so on, on a higher level." (Frederic C Bartlett, "Remembering: A study in experimental and social psychology", 1932)

"[T]he sudden inventions characteristic of the sixth stage [of infant development] are in reality the product of a long evolution of schemata and not only of an internal maturation of perceptive structures. [..] This is revealed by the existence of a fifth stage, characterized by experimental groping. […] What does this mean if not that the practice of actual experience is necessary in order to acquire the practice of mental experience and that invention does not arise entirely preformed despite appearances? (Jean Piaget, "The origin of intelligence in children" 1936)

"My hypothesis then is that thought models, or parallels, reality - that its essential feature is not ‘the mind’, ‘the self’, ‘sense-data’, nor propositions but symbolism, and that this symbolism is largely of the same kind as that which is familiar to us in mechanical devices which aid thought and calculation." (Kenneth Craik, "The Nature of Explanation", 1943)

"A person is changed by the contingencies of reinforcement under which he behaves; he does not store the contingencies. In particular, he does not store copies of the stimuli which have played a part in the contingencies. There are no 'iconic representations' in his mind; there are no 'data structures stored in his memory'; he has no 'cognitive map' of the world in which he has lived. He has simply been changed in such a way that stimuli now control particular kinds of perceptual behavior." (Burrhus F Skinner, "About behaviorism", 1974)

"Imagining is not perceiving, but images are indeed derivatives of perceptual activity. In particular, they are the anticipatory phases of that activity, schemata that the perceiver has detached from the perceptual cycle for other purposes. […] The experience of having an image is just the inner aspect of a readiness to perceive the imagined object. (Ulrich Neisser, "Cognition and Reality" 1976)

"[I]t seems (to many) that we cannot account for perception unless we suppose it provides us with an internal image (or model or map) of the external world, and yet what good would that image do us unless we have an inner eye to perceive it, and how are we to explain its capacity for perception? It also seems (to many) that understanding a heard sentence must be somehow translating it into some internal message, but how will this message be understood: by translating it into something else? The problem is an old one, and let’s call it Hume’s Problem, for while he did not state it explicitly, he appreciated its force and strove mightily to escape its clutches. (Daniel Dennett, "Brainstorms: Philosophical essays on mind and psychology", 1978)

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

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

"Since mental models can take many forms and serve many purposes, their contents are very varied. They can contain nothing but tokens that represent individuals and identities between them, as in the sorts of models that are required for syllogistic reasoning. They can represent spatial relations between entities, and the temporal or causal relations between events. A rich imaginary model of the world can be used to compute the projective relations required for an image. Models have a content and form that fits them to their purpose, whether it be to explain, to predict, or to control." (Philip Johnson-Laird, "Mental models: Toward a cognitive science of language, inference, and consciousness", 1983)

"The basic idea is that schemata are data structures for representing the generic concepts stored in memory. There are schemata for generalized concepts underlying objects, situations, events, sequences of events, actions, and sequences of actions. Roughly, schemata are like models of the outside world. To process information with the use of a schema is to determine which model best fits the incoming information. Ultimately, consistent configurations of schemata are discovered which, in concert, offer the best account for the input. This configuration of schemata together constitutes the interpretation of the input. (David E Rumelhart, Paul Smolensky, James L McClelland & Geoffrey E Hinton, "Schemata and sequential thought processes in PDP models", 1986)

Mental Models XVII

"As infinite kinds of almost identical images arise continually from the innumerable atoms and flow out to us from the gods, so we should take the keenest pleasure in turning and bending our mind and reason to grasp these images, in order to understand the nature of these blessed and eternal beings." (Marcus TulliusCicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC)

"The imagination is one of the highest prerogatives of man. By this faculty he unites, independently of the will, former images and ideas, and thus creates brilliant and novel results […] The value of the products of our imagination depends of course on the number, accuracy, and clearness of our impressions; on our judgment and taste in selecting or rejecting the involuntary combinations, and to a certain extent on our power of voluntarily combining them." (Charles Darwin, "The Descent of Man", 1874)

"That faculty which perceives and recognizes the noble proportions in what is given to the senses, and in other things situated outside itself, must be ascribed to the soul. It lies very close to the faculty which supplies formal schemata to the senses, or deeper still, and thus adjacent to the purely vital power of the soul, which does not think discursively […] Now it might be asked how this faculty of the soul, which does not engage in conceptual thinking, and can therefore have no proper knowledge of harmonic relations, should be capable of recognizing what is given in the outside world. For to recognize is to compare the sense perception outside with the original pictures inside, and to judge that it conforms to them.” (Johannes Kepler, “Harmonices Mundi” [“Harmony of the World”, 1619)

"The entire method consists in the order and arrangement of the things to which the mind's eye must turn so that we can discover some truth." (René Descartes, "Rules for the Direction of the Mind", 1628)

“[…] inner images are rather psychic manifestations of the archetypes which, however, would also have to put forth, create, condition anything lawlike in the behavior of the corporeal world. The laws of this world would then be the physical manifestations of the archetypes. […] Each law of nature should then have an inner correspondence and vice versa, even though this is not always directly visible today.” (Wolfgang Pauli, [letter to Markus Fierz] 1948)

“The process of understanding in nature, together with the joy that man feels in understanding, i.e., in becoming acquainted with new knowledge, seems therefore to rest upon a correspondence, a coming into congruence of preexistent internal images of the human psyche with external objects and their behavior. […] the place of clear concepts is taken by images of strongly emotional content, which are not thought but  are seen pictorially, as it were, before the minds eye.” (Wolfgang Pauli, “Der Einfluss archetypischer Vorstellungen auf die Bildung  naturwissenschaftlicher Theorien bei Kepler”, 1952)

“You cannot learn, through common sense, how things are you can only discover where they fit into the existing scheme of things.”  (Stuart Hall, 1977)

"Imagination is our means of interpreting the world, and it also is our means of forming images in the mind. The images themselves are not separate from our interpretations of the world; they are our way of thinking of the objects in the world. We see the forms in our mind’s eye and we see these very forms in the world. We could not do one of these things if we could not do the other" (Mary Warnock, "Imagination", 1978)

“[…] the human brain must work in models. The trick is to have your brain work better than the other person’s brain because it understands the most fundamental models: ones that will do most work per unit. If you get into the mental habit of relating what you’re reading to the basic structure of the underlying ideas being demonstrated, you gradually accumulate some wisdom."  (Charles T Munger, “Poor Charlie’s Almanack”, 2005)

“We know the world by a process of constantly transforming it into ourselves." (Alan Watts) 

Mental Models XV

“We invoke the imagination and the intervals that it furnishes, since the form itself is without motion or genesis, indivisible and free of all underlying matter, though the elements latent in the form are produced distinctly and individually on the screen of imagination. What projects the images is the understanding; the source of what is projected is the form in the understanding; and what they are projected in is this 'passive nous' that unfolds in revolution about the partlessness of genuine Nous.” (Proclus Lycaeus, “A Commentary on the First Book of Euclid’s Elements”, cca 5th century) 

“Know that the figures employed by prophets are of two kinds: first, where every word which occurs in the simile represents a certain idea: and secondly, where the simile, as a whole, represents a general idea, but has a great many points which have no reference whatever to that idea: they are simply required to give to the simile its proper form and order, or better to conceal the idea: the simile is therefore continued as far as necessary, according to its literal sense.” (Moses Maimonides, “The Guide for the Perplexed”, 1190) 

“With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses.” (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

“Mental schemas are not rigid. By lasting and laborious processes the mental schemas accommodate themselves, in the long run, to the features of real situations and become progressively more fit to manage them and to solve the problems with which we are faced. Each period of mental development is characterized by a system. of basic mental schemas which determine the capacity of the child to learn, to interpret, and to use the information he gets.” (Efraim Fischbein, "Intuition and Proof", For The Leaning of Mathematics 3 (2), 1982)

“People’s views of the world, of themselves, of their own capabilities, and of the tasks that they are asked to perform, or topics they are asked to learn, depend heavily on the conceptualizations that they bring to the task. In interacting with the environment, with others, and with the artifacts of technology, people form internal, mental models of themselves and of the things with which they are interacting. These models provide predictive and explanatory power for understanding the interaction.” (Donald A Norman, “Some observations on Mental Models”, 1983)

"You must know the big ideas in the big disciplines and use them routinely – all of them, not just a few.  Most people are trained in one model – economics, for example – and try to solve all problems in one way. This is a dumb way of handling problems. […] What you need is a latticework of mental models in your head.  And, with that system, things gradually get to fit together in a way that enhances cognition.” (Charles T Munger, “Poor Charlie’s Almanack”, 2005)

“Art is constructivist in nature, aimed at the deliberate refinement and elaboration of mental models and worldviews. These are the natural products of cognition itself, the outcome of the brain’s tendency to strive for the integration of perceptual and conceptual material over time. […] human culture is essentially a distributed cognitive system within which worldviews and mental models are constructed and shared by the members of a society. Artists are traditionally at the forefront of that process, and have a large influence on our worldviews and mental models.” (Mark Turner, “The Artful Mind : cognitive science and the riddle of human creativity”, 2006)

“It is hard to navigate across one’s environment without having some ideas, however coarse, about it. Indeed, to face any situation we must know whether it is real or imaginary, profane or sacred, sensitive or insensitive to our actions, and so on. This is why even lowly organisms develop, if not worldviews, at least rough sensory maps of their immediate environment – as noted by ethologists from the start. But it is generally assumed that only humans can build conceptual models of their environments. And, except for some philosophers, humans distinguish maps from the territories they represent.” (Mario Bunge, “Matter and Mind: A Philosophical Inquiry”, 2010)

“Also known as worldview, mental model, or mind-set, our perspective of the world is based on the sum total of our knowledge and experiences. It defines us, shaping our thoughts and actions because it represents the way we see ourselves and situations, how we judge the relative importance of things, and how we establish a meaningful relationship with everything around us.” (Navi Radjou, Prasad Kaipa, “From Smart to Wise: Acting and Leading with Wisdom”, 2013) 

Mental Models XIV

"A human being, what is a human being? Everything and nothing. Through the power of thought it can mirror everything it experiences. Through memory and knowledge it becomes a microcosm, carrying the world within itself. A mirror of things, a mirror of facts. Each human being becomes a little universe within the universe!” (Guy de Maupassant, [in “The Journal of a Madman”] 1851)

“Every definite image in the mind is steeped and dyed in the free water that flows around it. With it goes the sense of its relations, near and remote, the dying echo of whence it came to us, the dawning sense of whither it is to lead. The significance, the value, of the image is all in this halo or penumbra that surrounds and escorts it, - or rather that is fused into one with it and has become bone of its bone and flesh of its flesh; leaving it, it is true, an image of the same thing it was before, but making it an image of that thing newly taken and freshly understood." (William James, “The Principles of Psychology”, 1890)

"Great thinkers have vast premonitory glimpses of schemes of relations between terms, which hardly even as verbal images enter the mind, so rapid is the whole process. We all of us have this permanent consciousness of whither our thought is going." (William James, “The Principles of Psychology”, 1890)

“It is a common and necessary feature of human intelligence that we can neither conceive of things nor define them conceptually without adding attributes to them that simply do not exist. This applies not only to every thought and imagination of ordinary life, even the sciences do not proceed otherwise. Only philosophy seeks and finds the difference between things that exist and things that we perceive, and also sees the necessity of this difference. […] What we add are therefore not incorrect conceptions but the conditions for such conceptions in general. We cannot simply remove them and replace them with better ones; either we must add them, or we must abstain from all conceptions of this kind.” (Heinrich Hertz, “Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt”, 1894)

“The words of the language, as they are written or spoken, do not seem to play any role in any mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced or combined. […]  But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought - before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will.” (Albert Einstein, [letter to Hadamard, in (Jacques Hadamard, “The Psychology of Invention in the Mathematical Field,1945)])

“Visual thinking calls, more broadly, for the ability to see visual shapes as images of the patterns of forces that underlie our existence - the functioning of minds, of bodies or machines, the structure of societies or ideas.” (Rudolf Arnheim, “Visual Thinking”, 1969)

“People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like something else (association); and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. Personally, I put a lot of effort into ‘listening’ to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations.” (William P Thurston, “On proof and progress in mathematics”, Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"[...] images are probably the main content of our thoughts, regardless of the sensory modality in which they are generated and regardless of whether they are about a thing or a process involving things; or about words or other symbols, in a given language, which correspond to a thing or process. Hidden behind those images, never or rarely knowable by us, there are indeed numerous processes that guide the generation and deployment of those images in space and time. Those processes utilize rules and strategies embodied in dispositional representations. They are essential for our thinking but are not a content of our thoughts.” (Antonio R Damasio, “Descartes' Error. Emotion, Reason, and the Human Brain”, 1994)

“Intuition isn't direct perception of something external. It's the effect in the mind/brain of manipulating concrete objects - at a later stage, of making marks on paper, and still later, manipulating mental images. This experience leaves a trace, an effect, in your mind/brain.” (Reuben Hersh, “What Is Mathematics, Really?”, 1998)

“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)