🅱Johann I Bernoulli - Collected Quotes

"A quantity diminished or enlarged by an infinitely smaller quantity is neither diminished nor enlarged." (Johann Bernoulli, cca 1691-92)

"But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not." (Johann I Bernoulli, cca. 1691–1692

"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)

"Here, we call function of a variable magnitude, a quantity formed in whatever manner with that variable magnitude and constants." (Johann I Bernoulli, 1718)

"If nature could pass from one extremity to another, for example, from rest to movement, from movement to rest, or from a movement in one direction to a movement in the opposite direction, without passing through all the imperceptible movements that lead from the one to the other; the first state must be destroyed, without nature knowing to which new state it must become; for in the end by what reason should one be chosen for preference, and of which one could not ask why this one and not that one? since having no necessary connection between these two states, no passage from movement to rest, from rest to movement, or from a movement [in one direction] to a movement in an opposite direction; no reason at all will determine producing one thing rather than any other." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"In fact, a similar principle of hardness cannot exist; it is a chimera which offends that general law which nature constantly observes in all its operations; I speak of that immutable and perpetual order, established since the creation of the Universe, that can be called the LAW OF CONTINUITY, by virtue of which everything that takes place, takes place by infinitely small degrees. It seems that common sense dictates that no change can take place at a jump; natura non operatur per saltion; nothing can pass from one extreme to the other without passing through all the degrees in between." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

On Research (-1899)

"Nothing proves more clearly that the mind seeks truth, and nothing reflects more glory upon it, than the delight it takes, sometimes in spite of itself, in the driest and thorniest researches of algebra." (Bernard de Fontenelle, "Histoire du Renouvellement de l'Académie des Sciences", 1708)

"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"'Natural system' - a contradiction in terms. Nature has no system; she has, she is life and its progress from an unknown center toward an unknowable goal. Scientific research is therefore endless, whether one proceed analytically into minutiae or follow the trail as a whole, in all its breadth and height." (Johann Wolfgang von Goethe, 1823)

"But a thousand unconnected observations have no more value, as a demonstrative proof, than a single one. If we do not succeed in discovering causes by our researches, we have no right to create them by the imagination; we must not allow mere fancy to proceed beyond the bounds of our knowledge." (Justus von Liebig, "The Lancet", 1844)

"The aim of research is the discovery of the equations which subsist between the elements of phenomena." (Ernst Mach, 1898)

"Physical research by experimental methods is both a broadening and a narrowing field. There are many gaps yet to be filled, data to be accumulated, measurements to be made with great precision, but the limits within which we must work are becoming, at the same time, more and more defined." (Elihu Thomson, "Annual Report of the Board of Regents of the Smithsonian Institution", 1899)

On Research (2000-2019)

"Statisticians can calculate the probability that such random samples represent the population; this is usually expressed in terms of sampling error [...]. The real problem is that few samples are random. Even when researchers know the nature of the population, it can be time-consuming and expensive to draw a random sample; all too often, it is impossible to draw a true random sample because the population cannot be defined. This is particularly true for studies of social problems. [...] The best samples are those that come as close as possible to being random." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"[...] it is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illumi￾nated, and some of the meaning hidden in the nature of things is finally revealed." (David Ruelle, "The Mathematician's Brain", 2007)

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

"Meta-analytic thinking is the consideration of any result in relation to previous results on the same or similar questions, and awareness that combination with future results is likely to be valuable. Meta-analytic thinking is the application of estimation thinking to more than a single study. It prompts us to seek meta-analysis of previous related studies at the planning stage of research, then to report our results in a way that makes it easy to include them in future meta-analyses. Meta-analytic thinking is a type of estimation thinking, because it, too, focuses on estimates and uncertainty." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Statistical cognition is concerned with obtaining cognitive evidence about various statistical techniques and ways to present data. It’s certainly important to choose an appropriate statistical model, use the correct formulas, and carry out accurate calculations. It’s also important, however, to focus on understanding, and to consider statistics as communication between researchers and readers." (Geoff Cumming, "Understanding the New Statistics", 2012)

"Another way to secure statistical significance is to use the data to discover a theory. Statistical tests assume that the researcher starts with a theory, collects data to test the theory, and reports the results - whether statistically significant or not. Many people work in the other direction, scrutinizing the data until they find a pattern and then making up a theory that fits the pattern." (Gary Smith, "Standard Deviations", 2014)

"How can we tell the difference between a good theory and quackery? There are two effective antidotes: common sense and fresh data. If it is a ridiculous theory, we shouldn’t be persuaded by anything less than overwhelming evidence, and even then be skeptical. Extraordinary claims require extraordinary evidence. Unfortunately, common sense is an uncommon commodity these days, and many silly theories have been seriously promoted by honest researchers." (Gary Smith, "Standard Deviations", 2014)

"These practices - selective reporting and data pillaging - are known as data grubbing. The discovery of statistical significance by data grubbing shows little other than the researcher’s endurance. We cannot tell whether a data grubbing marathon demonstrates the validity of a useful theory or the perseverance of a determined researcher until independent tests confirm or refute the finding. But more often than not, the tests stop there. After all, you won’t become a star by confirming other people’s research, so why not spend your time discovering new theories? The data-grubbed theory consequently sits out there, untested and unchallenged." (Gary Smith, "Standard Deviations", 2014)

"A conceptual model is a framework that is initially used in research to outline the possible courses of action or to present an idea or thought. When a conceptual model is developed in a logical manner, it will provide a rigor to the research process." (N Elangovan & R Rajendran, "Conceptual Model: A Framework for Institutionalizing the Vigor in Business Research", 2015)

"Even properly done statistics can’t be trusted. The plethora of available statistical techniques and analyses grants researchers an enormous amount of freedom when analyzing their data, and it is trivially easy to ‘torture the data until it confesses’." (Alex Reinhart, "Statistics Done Wrong: The Woefully Complete Guide", 2015)

"The correlational technique known as multiple regression is used frequently in medical and social science research. This technique essentially correlates many independent (or predictor) variables simultaneously with a given dependent variable (outcome or output). It asks, 'Net of the effects of all the other variables, what is the effect of variable A on the dependent variable?' Despite its popularity, the technique is inherently weak and often yields misleading results. The problem is due to self-selection. If we don’t assign cases to a particular treatment, the cases may differ in any number of ways that could be causing them to differ along some dimension related to the dependent variable. We can know that the answer given by a multiple regression analysis is wrong because randomized control experiments, frequently referred to as the gold standard of research techniques, may give answers that are quite different from those obtained by multiple regression analysis." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Collecting data through sampling therefore becomes a never-ending battle to avoid sources of bias. [...] While trying to obtain a random sample, researchers sometimes make errors in judgment about whether every person or thing is equally likely to be sampled." (Daniel J Levitin, "Weaponized Lies", 2017)

"Samples give us estimates of something, and they will almost always deviate from the true number by some amount, large or small, and that is the margin of error. […] The margin of error does not address underlying flaws in the research, only the degree of error in the sampling procedure. But ignoring those deeper possible flaws for the moment, there is another measurement or statistic that accompanies any rigorously defined sample: the confidence interval." (Daniel J Levitin, "Weaponized Lies", 2017)

"The job of the statistician is to formulate an inventory of all those things that matter in order to obtain a representative sample. Researchers have to avoid the tendency to capture variables that are easy to identify or collect data on - sometimes the things that matter are not obvious or are difficult to measure." (Daniel J Levitin, "Weaponized Lies", 2017)

On Research (1950-1999)

"The hypothesis is the principal intellectual instrument in research. Its function is to indicate new experiments and observations and it therefore sometimes leads to discoveries even when not correct itself. We must resist the temptation to become too attached to our hypothesis, and strive to judge it objectively and modify it or discard it as soon as contrary evidence is brought to light. Vigilance is needed to prevent our observations and interpretations being biased in favor of the hypothesis. Suppositions can be used without being believed." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"Mathematical models for empirical phenomena aid the development of a science when a sufficient body of quantitative information has been accumulated. This accumulation can be used to point the direction in which models should be constructed and to test the adequacy of such models in their interim states. Models, in turn, frequently are useful in organizing and interpreting experimental data and in suggesting new directions for experimental research." (Robert R. Bush & Frederick Mosteller, "A Mathematical Model for Simple Learning", Psychological Review 58, 1951)

"Statistics is the fundamental and most important part of inductive logic. It is both an art and a science, and it deals with the collection, the tabulation, the analysis and interpretation of quantitative and qualitative measurements. It is concerned with the classifying and determining of actual attributes as well as the making of estimates and the testing of various hypotheses by which probable, or expected, values are obtained. It is one of the means of carrying on scientific research in order to ascertain the laws of behavior of things - be they animate or inanimate. Statistics is the technique of the Scientific Method." (Bruce D Greenschields & Frank M Weida, "Statistics with Applications to Highway Traffic Analyses", 1952)

"In a general way it may be said that to think in terms of systems seems the most appropriate conceptual response so far available when the phenomena under study - at any level and in any domain - display the character of being organized, and when understanding the nature of the interdependencies constitutes the research task. In the behavioral sciences, the first steps in building a systems theory were taken in connection with the analysis of internal processes in organisms, or organizations, when the parts had to be related to the whole." (Fred Emery, "The Causal Texture of Organizational Environments", 1963)

"Most scientific theories, however, are ephemeral. Exceptions will likely be found that invalidate a theory in one or more of its tenets. These can then stimulate a new round of research leading either to a more comprehensive theory or perhaps to a more restrictive (i.e., more precisely defined) theory. Nothing is ever completely finished in science; the search for better theories is endless. The interpretation of a scientific experiment should not be extended beyond the limits of the available data. In the building of theories, however, scientists propose general principles by extrapolation beyond available data. When former theories have been shown to be inadequate, scientists should be prepared to relinquish the old and embrace the new in their never-ending search for better solutions. It is unscientific, therefore, to claim to have 'proof of the truth' when all that scientific methodology can provide is evidence in support of a theory." (William D Stansfield, "The Science of Evolution", 1977)

"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"Mathematical research should be as broad and as original as possible, with very long range-goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be discovered." (Arthur Jaffe, "Ordering the universe: the role of mathematics", SIAM Review Vol 26. No 4, 1984)

"Basic research is like shooting an arrow into the air and, where it lands, painting a target." (Homer B Adkins, 1984)

"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)

"Thus my advice to model builders in the social sciences is to think in terms of social processes that might require algebraic structures that could yield catastrophe potential. Build models from an intimate knowledge of these processes while remaining aware of the algebraic requirements for catastrophes. The art of nonlinear model building is a delicate dance with two partners, algebraic forms that produce known effects and a substantive understanding of the complexities of social phenomena. Coordinating the two by mixing structure to match complexity is the job of the theorist, and it is the single greatest creative challenge of any researcher." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"If the null hypothesis is not rejected, [Sir Ronald] Fisher's position was that nothing could be concluded. But researchers find it hard to go to all the trouble of conducting a study only to conclude that nothing can be concluded." (Frank L Schmidt, "Statistical Significance Testing and Cumulative Knowledge", "Psychology: Implications for Training of Researchers, Psychological Methods" Vol. 1 (2), 1996)

On Research (1900-1949)

"Scientific facts are of little value in themselves. Their significance is their bearing upon other facts, enabling us to generalize and so to discover principles, just as the accurate measurement of the position of a star may be without value in itself, but in relation to other similar measurement of other stars may become the means of discovering their proper motions. We refine our instruments; we render more trustworthy our means of observation we extend our range of experimental inquiry, and thus lay the foundation for the future work, with the full knowledge that, although our researches can not extend beyond certain limits, the field itself is, even within those limits, inexhaustible." (Elihu Thompson, "The Field of Experimental Research", 1901)

"[…] scientific research is somewhat like unraveling complicated tangles of strings, in which luck is almost as vital as skill and accurate observation." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1905)

"At bottom each 'exact' science is, and must be speculative, and its chief tool of research, too rarely used with both courage and judgement, is the regulated imagination." (Reginald A Daly, "Igneous Rocks and their Origin", 1914)

"Physics has progressed because, in the first place, she accepted the uniformity of nature; because, in the next place, she early discovered the value of exact measurements; because, in the third place, she concentrated her attention on the regularities that underlie the complexities of phenomena as they appear to us; and lastly, and not the least significant, because she emphasized the importance of the experimental method of research. An ideal or crucial experiment is a study of an event, controlled so as to give a definite and measurable answer to a question - an answer in terms of specific theoretical ideas, or better still an answer in terms of better understood relations." (Thomas H Morgan, "The Relation of Biology to Physics", Science Vol. LXV (1679),  1927)

"Research is fundamentally a state of mind involving continual re­examination of doctrines and axioms upon which current thought and action are based. It is, therefore, critical of existing practices." (Theobald Smith, "The Influence of Research in Bringing into Closer Relationship the Practice of Medicine and Public Health Activities", American Journal of Medical Sciences, 1929)

"In every important advance the physicist finds that the fundamental laws are simplified more and more as experimental research advances. He is astonished to notice how sublime order emerges from what appeared to be chaos. And this cannot be traced back to the workings of his own mind but is due to a quality that is inherent in the world of perception." (Albert Einstein, 1932)

"Statistics is a scientific discipline concerned with collection, analysis, and interpretation of data obtained from observation or experiment. The subject has a coherent structure based on the theory of Probability and includes many different procedures which contribute to research and development throughout the whole of Science and Technology." (Egon Pearson, 1936)

"The researcher worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty. […] It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society , Volume LIX, 1939)

"While practical benefits often result from pure academic research […], such benefits are not guaranteed and cannot be predicted; nor need they be seen as the ultimate goal." (Abraham Flexner, 1939)

"[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts. " (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"Mathematical research can lend its organisational characteristics to poetry, whereby disjointed metaphors take on a universal sense. Similarly, the axiomatic foundations of group theory can be assimilated into a larger moral concept of a unified universe. Without this, mathematics would be a laborious Barbary." (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"Unlike art, science is genuinely progressive. Achievement in the fields of research and technology is cumulative; each generation begins at the point where its predecessor left off." (Aldous Huxley, "Science, Liberty and Peace", 1946)

"A successful hypothesis is not necessarily a permanent hypothesis, but it is one which stimulates additional research, opens up new fields, or explains and coordinates previously unrelated facts." (Farrington Daniels, "Outlines of Physical Chemistry", 1948)

On Structure: Structure in Mathematics I

"[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts." (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects ‘elegance’ in its ‘architectural’ structural makeup." (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I (1), 1947]) 

"The constructions of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow-mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity." (Hermann Weyl, "A Half-Century of Mathematics", The American Mathematical Monthly, 1951)

"Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas." (Saunders MacLane, "Of Course and Courses"The American Mathematical Monthly, Vol 61, No 3, 1954)

"The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1962)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"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 bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design." (Freeman J Dyson, [interview] 1994)

"In abstract mathematics, special attention is given to particular properties of numbers. Then those properties are taken in a very pure (and primitive) form. Those properties in pure form are then assigned to a given set. Therefore, by studying in details the internal mathematical structure of a set, we should be able to clarify the meaning of original properties of the objects. Likewise, in set theory, numbers disappear and only the concept of sets and characteristic properties of sets remain." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not." (Michael Atiyah, 2009)

On Structure: Structure in Knowledge (2000-2009)

"[...] information feedback about the real world not only alters our decisions within the context of existing frames and decision rules 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, processed and interpreted by a different decision rule, now yields a different decision. Altering the structure of our systems then alters their patterns of behavior. The development of systems thinking is a double-loop learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view and then redesign our policies and institutions accordingly." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

"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)

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

"Representations of real or imaginary structure in the human mind enabling orientation as well as goal orientated actions and movements" (Ralf Wagner, "Customizing Multimedia with Multi-Trees" [in "Encyclopedia of Multimedia Technology and Networking" 2nd Ed.], 2009)

On Structure: Structure in Knowledge (1990-1999)

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

"The essential idea of semantic networks is that the graph-theoretic structure of relations and abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The great organizing principle of thought is abstraction. By assigning particular things to abstract categories we are able to dispense with irrelevant detail and yet instantly draw copious conclusions about a thing due to its membership in various categories. Semantic networks specify the structure of interrelated abstract categories and use this structure to draw conclusions." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)

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

"All systems evolve, although the rates of evolution may vary over time both between and within systems. The rate of evolution is a function of both the inherent stability of the system and changing environmental circumstances. But no system can be stabilized forever. For the universe as a whole, an isolated system, time’s arrow points toward greater and greater breakdown, leading to complete molecular chaos, maximum entropy, and heat death. For open systems, including the living systems that are of major interest to us and that interchange matter and energy with their external environments, time’s arrow points to evolution toward greater and greater complexity. Thus, the universe consists of islands of increasing order in a sea of decreasing order. Open systems evolve and maintain structure by exporting entropy to their external environments." (L Douglas Kiel, "Chaos Theory in the Social Sciences: Foundations and Applications", 1996)

"Ideas about organization are always based on implicit images or metaphors that persuade us to see, understand, and manage situations in a particular way. Metaphors create insight. But they also distort. They have strengths. But they also have limitations. In creating ways of seeing, they create ways of not seeing. There can be no single theory or metaphor that gives an all-purpose point of view, and there can be no simple 'correct theory' for structuring everything we do." (Gareth Morgan, "Imaginization", 1997)

"[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)

"[A mental model] is a relatively enduring and accessible, but limited, internal conceptual representation of an external system (historical, existing, or projected) [italics in original] whose structure is analogous to the perceived structure of that system." (James K Doyle & David N Ford, "Mental models concepts revisited: Some clarifications and a reply to Lane", System Dynamics Review 15 (4), 1999)

"[…] philosophical theories are structured by conceptual metaphors that constrain which inferences can be drawn within that philosophical theory. The (typically unconscious) conceptual metaphors that are constitutive of a philosophical theory have the causal effect of constraining how you can reason within that philosophical framework." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought", 1999)

"What it means for a mental model to be a structural analog is that it embodies a representation of the spatial and temporal relations among, and the causal structures connecting the events and entities depicted and whatever other information that is relevant to the problem-solving talks. […] The essential points are that a mental model can be nonlinguistic in form and the mental mechanisms are such that they can satisfy the model-building and simulative constraints necessary for the activity of mental modeling." (Nancy J Nersessian, "Model-based reasoning in conceptual change", 1999)

On Structure: Structure in Knowledge (1980-1989)

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

"These organizational processes result in our perceptions being structured into units corresponding to objects and properties of objects. It is these larger units that may be stored and later assembled into images that are experienced as quasi-pictorial, spatial entities resembling those evoked during perception itself [...] It is erroneous to equate image representations with mental photographs, since this would overlook the fact that images are composed from highly processed perceptual encodings." (Stephen Kosslyn, "Image and Mind", 1980)

"Facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world's data. Theories are structures of ideas that explain and interpret facts. Facts do not go away while scientists debate rival theories for explaining them." (Stephen J Gould "Evolution as Fact and Theory", 1981)

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

"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)

"A mental model is a cognitive construct that describes a person's understanding of a particular content domain in the world." (John Sown, "Conceptual Structures: Information Processing in Mind and Machine", 1984)

"Curiously, the unexpected complexity that has been discovered in nature has not led to a slowdown in the progress of science, but on the contrary to the emergence of new conceptual structures that now appear as essential to our understanding of the physical world - the world that includes us. (Isabelle Stengers, "Order Out of Chaos", 1984)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to motor mechanisms' of the knowledge." (John F Sowa, "Conceptual Structures", 1984)

"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 mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding." (Mark Johnson, "The Body in the Mind", 1987)

"The mapping from linguistic inputs to mental models is not a one-one mapping. So semantic properties of sentences may not be recoverable from a mental model. Reading or listening is typically for content not for form. People want to know what is being said to them, not how it is being said. [...] A mental model is a representation of the content of a text that need bear no resemblance to any of the text's linguistic representations. Its structure is similar to the situation described by the text." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"[…] a mental model is a mapping from a domain into a mental representation which contains the main characteristics of the domain; a model can be ‘run’ to generate explanations and expectations with respect to potential states. Mental models have been proposed in particular as the kind of knowledge structures that people use to understand a specific domain […]" (Helmut Jungermann, Holger Schütz & Manfred Thuering, "Mental models in risk assessment: Informing people about drugs", Risk Analysis 8 (1), 1988)

"Model is used as a theory. It becomes theory when the purpose of building a model is to understand the mechanisms involved in the developmental process. Hence as theory, model does not carve up or change the world, but it explains how change takes place and in what way or manner. This leads to build change in the structures." (Laxmi K Patnaik, "Model Building in Political Science", The Indian Journal of Political Science Vol. 50 (2), 1989)

On Structure: Structure in Knowledge (1970-1979)

"In general, one might define a complex of semantic components connected by logical constants as a concept. The dictionary of a language is then a system of concepts in which a phonological form and certain syntactic and morphological characteristics are assigned to each concept. This system of concepts is structured by several types of relations. It is supplemented, furthermore, by redundancy or implicational rules […] representing general properties of the whole system of concepts. […] At least a relevant part of these general rules is not bound to particular languages, but represents presumably universal structures of natural languages. They are not learned, but are rather a part of the human ability to acquire an arbitrary natural language." (Manfred Bierwisch, "Semantics", 1970)

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

"The essential functions of the mind consist in understanding and in inventing, in other words, in building up structures by structuring reality." (Jean Piaget, 1971)

"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)

"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful' . In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed." (Michael Dummett, "The philosophical basis of intuitionistic logic", 1975)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"Owing to his lack of knowledge, the ordinary man cannot attempt to resolve conflicting theories of conflicting advice into a single organized structure. He is likely to assume the information available to him is on the order of what we might think of as a few pieces of an enormous jigsaw puzzle. If a given piece fails to fit, it is not because it is fraudulent; more likely the contradictions and inconsistencies within his information are due to his lack of understanding and to the fact that he possesses only a few pieces of the puzzle. Differing statements about the nature of things […] are to be collected eagerly and be made a part of the individual's collection of puzzle pieces. Ultimately, after many lifetimes, the pieces will fit together and the individual will attain clear and certain knowledge." (Alan R Beals, "Strategies of Resort to Curers in South India" [contributed in Charles M. Leslie (ed.), "Asian Medical Systems: A Comparative Study", 1976]) 

"The cognitive map is not a picture or image which 'looks like' what it represents; rather, it is an information structure from which map-like images can be reconstructed and from which behaviour dependent upon place information can be generated." (John O'Keefe & Lynn Nadel, "The Hippocampus as a Cognitive Map", 1978)

"The evolutionary vision is agnostic in regard to systems in the universe of greater complexity than those of which human beings have clear knowledge. It recognizes aesthetic, moral, and religious ideas and experiences as a species, in this case of mental structures or of images, which clearly interacts with other species in the world's great' ecosystem." (Kenneth Boulding," Ecodynamics: A New Theory of Societal Evolution", 1978)

"The use of metaphor is one of many devices available to the scientific community to accomplish the task of accommodation of language to the causal structure of the world." (Richard Boyd, "Metaphor and theory change: what is ‘metaphor’ a metaphor for?", 1979)