Out of Context: On Analogy (Definitions)

"Analogy, although it is not infallible, is yet that telescope of the mind by which it is marvellously assisted in the discovery of both physical and moral truth." (Charles C Colton, "Lacon", 1820)

"On the whole, Analogy is to be regarded as a step towards satisfactory proof, much in advance of first presumptions, if skillfully applied; [...]" (Francis W Newman, "Lectures on Logic", 1838)

"Analogies are figures intended to serve as fatal weapons if they succeed, and as innocent toys if they fail." (Henry Adams, "Mont Saint Michel and Chartres", 1904)

“It must be gently but firmly pointed out that analogy is the very corner-stone of scientific method.” (Archie E Heath, “On Analogy”, The Cambridge Magazine, 1918)

"[…] analogies are not ‘aids’ to the establishment of theories; they are an utterly essential part of theories, without which theories would be completely valueless and unworthy of the name." (Norman R Campbell, "Physics: The Elements", 1920)

"An analogy is not a reason […]" (Hardy Cross, "Engineers and Ivory Towers", 1952)

"Analogy is even slipperier than logic." (Robert A Heinlein, "Stranger in a Strange Land", 1961)

"An analogy is a relationship between two entities, processes, or what you will, which allows inferences to be made about one of the things, usually that about which we know least, on the basis of what we know about the other." (Rom Harré," The Philosophies of Science" , 1972)

"Analogies, metaphors, and emblems are the threads by which the mind holds on to the world even when, absentmindedly, it has lost direct contact with it, and they guarantee the unity of human experience." (Hannah Arendt, "The Life of the Mind", 1981)

"Analogy, in its most general sense, is this ability to think about relational patterns.” (Keith Holyoak et al, “Introduction: The Place of Analogy in Cognition”, 2001)

On Conjecture (1800-1899)

"In order to supply the defects of experience, we will have recourse to the probable conjectures of analogy, conclusions which we will bequeath to our posterity to be ascertained by new observations, which, if we augur rightly, will serve to establish our theory and to carry it gradually nearer to absolute certainty." (Johann H Lambert, "The System of the World", 1800)

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

"The science of the mathematics performs more than it promises, but the science of metaphysics promises more than it performs. The study of the mathematics, like the Nile, begins in minuteness but ends in magnificence; but the study of metaphysics begins with a torrent of tropes, and a copious current of words, yet loses itself at last in obscurity and conjecture, like the Niger in his barren deserts of sand." (Charles C Colton, "Lacon", 1820)

"We know the effects of many things, but the causes of few; experience, therefore, is a surer guide than imagination, and inquiry than conjecture." (Charles C Colton, "Lacon", 1820)

"Let me be permitted to recall that the object of mathematics is not to investigate the causes that one can assign to natural phenomena. This science would lose both its character and credit if, renouncing the support of general well-confirmed facts, it sought within the realm of nebulous conjectures, a realm which has always been a fertile source of error for ways of satisfying the thirst fo rexplanation." (Sophie Germain, "Examen des principes qui peuvent conduire a la connaissance des lois de requilibre et du mouvement des solides elastiques", Annales de Chimie 38, 1828)

"Life is not the object of Science: we see a little, very little; And what is beyond we can only conjecture." (Samuel Johnson, "Causes Which Produce Diversity of Opinion", 1840)

"The entire annals of Observation probably do not elsewhere exhibit so extraordinary a verification of any theoretical conjecture adventured on by the human spirit!" (John P Nichol, "The Planet Neptune: An Exposition and History", 1848)

"The philosophical study of nature rises above the requirements of mere delineation, and does not consist in the sterile accumulation of isolated facts. The active and inquiring spirit of man may therefore be occasionally permitted to escape from the present into the domain of the past, to conjecture that which cannot yet be clearly determined, and thus to revel amid the ancient and ever-recurring myths of geology." (Alexander von Humboldt, "Views of Nature: Or Contemplation of the Sublime Phenomena of Creation", 1850)

"The rules of scientific investigation always require us, when we enter the domains of conjecture, to adopt that hypothesis by which the greatest number of known facts and phenomena may be reconciled." (Matthew F Maury, "The Physical Geography of the Sea", 1855)

"There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact." (Samuel L Clemens [Mark Twain], "Life on the Mississippi", 1883)

"I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain. [...] But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors." (Hermann von Helmholtz, 1891)

On Conjecture (1750-1799)

"One of the most intimate of all associations in the human mind is that of cause and effect. They suggest one another with the utmost readiness upon all occasions; so that it is almost impossible to contemplate the one, without having some idea of, or forming some conjecture about the other." (Joseph Priestley, "The History and Present State of Electricity", 1767)

"It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed with in the field of experience, and the truth and sufficiency of which are, at the same time, insured by experience. With these principles it rises, in obedience to the laws of its own nature, to ever higher and more remote conditions. But it quickly discovers that, in this way, its labours must remain ever incomplete, because new questions never cease to present themselves; and thus it finds itself compelled to have recourse to principles which transcend the region of experience, while they are regarded by common sense without distrust. It thus falls into confusion and contradictions, from which it conjectures the presence of latent errors, which, however, it is unable to discover, because the principles it employs, transcending the limits of experience, cannot be tested by that criterion. The arena of these endless contests is called Metaphysic." (Immanuel Kant, "The Critique of Pure Reason", 1781)

"[...] the lofty aspirations of humanity and not delusions; they are realities. They link us to a purer order of existence, which makes us heirs of immortality. We repose order a confident and unwavering assurance that, in God’s own time, these earth-mists will be dispersed, and the dim twilight of conjecture will yield to the glorious, unclouded noonday of knowledge." (John LeConte, "The Nebular Hypothesis", The Popular Science Monthly Vol. 2, 1873)

"On the other hand, if we add observation to observation, without attempting to draw no only certain conclusions, but also conjectural views from them, we offend against the very end for which only observations ought to be made." (Friedrich W Herschel, "On the Construction of the Heavens", Philosophical Transactions of the Royal Society of London Vol. LXXV, 1785)

"The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"Conjecture may lead you to form opinions, but it cannot produce knowledge. Natural philosophy must be built upon the phenomena of nature discovered by observation and experiment." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

"Conjectures in philosophy are termed hypotheses or theories; and the investigation of an hypothesis founded on some slight probability, which accounts for many appearances in nature, has too often been considered as the highest attainment of a philosopher. If the hypothesis (sic) hangs well together, is embellished with a lively imagination, and serves to account for common appearances - it is considered by many, as having all the qualities that should recommend it to our belief, and all that ought to be required in a philosophical system." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

On Creativity (Mathematics I)

"Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing." (Robert C Buck, "Teaching Machines and Mathematics Programs", American Mathematical Monthly 69, 1962)

"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory." (Mark Kac, "Enigmas Of Chance", 1985)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful!" (James Tanton, "Thinking Mathematics")

"Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there." (Meir Shalev)

"No discovery has been made in mathematics, or anywhere else for that matter, by an effort of deductive logic; it results from the work of creative imagination which builds what seems to be truth, guided sometimes by analogies, sometimes by an esthetic ideal, but which does not hold at all on solid logical bases. Once a discovery is made, logic intervenes to act as a control; it is logic that ultimately decides whether the discovery is really true or is illusory; its role therefore, though considerable, is only secondary." (Henri Lebesgue)

"The essential feature of mathematical creativity is the exploration, under the pressure of powerful implosive forces, of difficult problems for whose validity and importance the explorer is eventually held accountable by reality." (Alfred Adler)

On Induction (-1849)

"The only possible way to conceive universal is by induction, since we come to know abstractions by induction. But unless we have sense experience, we cannot make inductions. Even though sense perception relates to particular things, scientific knowledge concerning such can only be constructed by the successive steps of sense perception, induction, and formulation of universals." (Aristotle, "Posterior Analytics", cca. 350 BC)

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, "The New Organon", 1620)

"In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions." (Isaac Newton, "The Principia: Mathematical Principles of Natural Philosophy", 1687)

"As in Mathematics, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions but such as are taken from Experiments, or other certain Truths." (Sir Isaac Newton, "Opticks", 1704)

"It is often in our Power to obtain an Analogy where we cannot have an Induction." (David Hartley, "Observations on Man, His Frame, His Duty, and His Expectations", 1749)

"Especially when we investigate the general laws of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law; although, whether they can be made to agree in this way better than in any Other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law." (Roger J Boscovich, "De Lege Continuitatis" ["On the law of continuity"], 1754)

"A discovery in mathematics, or a successful induction of facts, when once completed, cannot be too soon given to the world. But […] an hypothesis is a work of fancy, useless in science, and fit only for the amusement of a vacant hour." (Henry Brougham, Edinburgh Review 1, 1803)

"The most important questions of life are, for the most part, really only problems of probability. Strictly speaking one may even say that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, induction and analogy, the principal means for discovering truth, are based on probabilities, so that the entire system of human knowledge is connected with this theory." (Pierre-Simon Laplace, "Theorie Analytique des Probabilités", 1812)

"Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities”, 1814)

"Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1814)

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

"It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations." (Carl Friedrich Gauss, 1817)

"Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many." (Sir John Herschel, "A Preliminary Discourse on the Study of Natural Philosophy" , 1830)

"We have here spoken of the prediction of facts of the same kind as those from which our rule was collected. But the evidence in favour of our induction is of a much higher and more forcible character when it enables us to explain and determine cases of a kind different from those which were contemplated in the formation of our hypothesis. The instances in which this has occurred, indeed, impress us with a conviction that the truth of our hypothesis is certain. No accident could give rise to such an extraordinary coincidence. No false supposition could, after being adjusted to one class of phenomena, so exactly represent a different class, when the agreement was unforeseen and contemplated. That rules springing from remote and unconnected quarters should thus leap to the same point, can only arise from that being where truth resides." (William Whewell, "The Philosophy of the Inductive Sciences" Vol. 2, 1840)

"There is in every step of an arithmetical or algebraical calculation a real induction, a real inference from facts to facts, and what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of its language." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)

📁 On DIKW: On Facts (1800-1824)

"Isolated facts, those that can only be obtained by rough estimate and that require development, can only be presented in memoires; but those that can be presented in a body, with details, and on whose accuracy one can rely, may be expounded in tables." (Emmanuel Duvillard, "Memoire sur le travail du Bureau de statistique", 1806)

"The foundations of chemical philosophy are observation, experiment, and analogy. By observation, facts are distinctly and minutely impressed on the mind. By analogy, similar facts are connected. By experiment, new facts are discovered; and, in the progression of knowledge, observation, guided by analogy, lends to experiment, and analogy confirmed by experiment, becomes scientific truth." (Sir Humphry Davy, "Elements of Chemical Philosophy" Vol. 4, 1812)

"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)

"Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1814)

"The substitution of analogy for fact is the bane of chemical philosophy; the legitimate use of analogy is to connect facts together and to guide to new experiments." (Sir Humphry Davy, "Journal of Science and the Arts", 1816)

"Force is not a fact at all, but an idea embodying what is approximately the fact." (William K Clifford, "The Common Sense of the Exact Sciences", 1823)

Catastrophe Theory I

"[...] if the behavior points for the entire control surface are plotted and then connected, they form a smooth surface: the behavior surface. The surface has an overall slope from high values where rage predominates to low values in the region where fear is the prevailing state of mind, but the slope is not its most distinctive feature. Catastrophe theory reveals that in the middle of the surface there must be a smooth double fold, creating a pleat without creases, which grows narrower from the front of the surface to the back and eventually disappears in a singular point where the three sheets of the pleat come together. It is the pleat that gives the model its most interesting characteristics. All the points on the behavior surface represent the most probable behavior [...], with the exception of those on the middle sheet, which represent least probable behavior. Through catastrophe theory we can deduce the shape of the entire surface from the fact that the behavior is bimodal for some control points." (E Cristopher Zeeman, "Catastrophe Theory", Scientific American, 1976)

"Catastrophe Theory is - quite likely - the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations." (René F Thom," La Théorie des catastrophes: État présent et perspective", 1977)

"'Catastrophe theory' denotes both a purely mathematical discipline describing certain singularities of smooth maps, as well as the concerted effort to apply these theorems to a wide variety of problems in fields ranging from linguistics and psychology to embryology, evolution, physics, and engineering." (Héctor J Sussmann & Raphael S Zahler, "Catastrophe Theory as Applied to the Social and Biological Sciences: A Critique" Synthese Vol. 37 (2), 1978)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"But there is another kind of change, too, change that is less suited to mathematical analysis: the abrupt bursting of a bubble, the discontinuous transition from ice at its melting point to water at its freezing point, the qualitative shift in our minds when we 'get' a pun or a play on words. Catastrophe theory is a mathematical language created to describe and classify this second type of change. It challenges scientists to change the way they think about processes and events in many fields." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Catastrophe theory is a controversial new way of thinking about change - change in a course of events, change in an object's shape, change in a system's behavior, change in ideas themselves. Its name suggests disaster, and indeed the theory can be applied to literal catastrophes such as the collapse of a bridge or the downfall of an empire. But it also deals with changes as quiet as the dancing of sunlight on the bottom of a pool of water and as subtle as the transition from waking to sleep." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Catastrophes are often stimulated by the failure to feel the emergence of a domain, and so what cannot be felt in the imagination is experienced as embodied sensation in the catastrophe. (William I Thompson, "Gaia, a Way of Knowing: Political Implications of the New Biology", 1987)

"A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. [...] Catastrophe theory is concerned with the mathematical modeling of sudden changes - so called 'catastrophes' - in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

"Catastrophe theory can be thought of as a link between classical analysis, dynamical systems, differential topology (including singularity theory), modern bifurcation theory and the theory of complex systems. [...] The name ‘catastrophe theory’ is used for a combination of singularity theory and its applications. [...] From the didactical point of view, there are two main positions for courses in catastrophe theory at university level: Trying to teach the theory as a perfect axiomatic system consisting of exact definitions, theorems and proofs or trying to teach mathematics as it can be developed from historical or from natural problems." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

"Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

🪷On Mind: Mental Models XLIX

"For imagination is different from either perceiving or discursive thinking, though it is not found without sensation, or judgement without it. That this activity is not the same kind of thinking as judgement is obvious. For imagining lies within our own power whenever we wish (e.g. we can call up a picture, as in the practice of mnemonics by the use of mental images), but in forming opinions we are not free: we cannot escape the alternative of falsehood or truth." (Aristotle, "De Anima", cca. 350 BC)

"Since it seems that there is nothing outside and separate in existence from sensible spatial magnitudes, the objects of thought are in the sensible forms, viz. both the abstract objects and all the states and affections of sensible things. Hence no one can learn or understand anything in the absence of sense, and when the mind is actively aware of anything it is necessarily aware of it along with an image; for images are like sensuous contents except in that they contain no matter. Imagination is different from assertion and denial; for what is true or false involves a synthesis of thoughts. In what will the primary thoughts differ from images? Must we not say that neither these nor even our other thoughts are images, though they necessarily involve them?" (Aristotle, "De Anima", cca. 350 BC)

"Thinking is different from perceiving and is held to be in part imagination, in part judgement: we must therefore first mark off the sphere of imagination and then speak of judgement. If then imagination is that in virtue of which an image arises for us, excluding metaphorical uses of the term, is it a single faculty or disposition relative to images, in virtue of which we discriminate and are either in error or not? The faculties in virtue of which we do this are sense, opinion, knowledge, thought." (Aristotle, "De Anima", cca. 350 BC)

"As regards the question, therefore, what memory or remembering is, it has now been shown that it is the having of an image, related as a likeness to that of which it is an image; and as to the question of which of the faculties within us memory is a function, it has been shown that it is a function of the primary faculty of sense-perception, i.e. of that faculty whereby we perceive time."  (Aristotle," De Memoria et Reminiscentia" ["On Memory and Recollection"], 4th century BC)

"But since we have, in our work on the soul, treated of imagination, and the faculty of imagination is identical with that of sense-perception, though the being of a faculty of imagination is different from that of a faculty of sense-perception; and since imagination is the movement set up by a sensory faculty when actually discharging its function, while a dream appears to be an image (for which occurs in sleep - whether simply or in some particular way - is what we call a dream): it manifestly follows that dreaming is an activity of the faculty of sense-perception, but belongs to this faculty qua imaginative." (Aristotle, On Dreams, 4th century BC)

"For according to the arguments from the existence of the sciences there will be Forms of all things of which there are sciences, and according to the argument that there is one attribute common to many things there will be Forms even of negations, and according to the argument that there is an object for thought even when the thing has perished, there will be Forms of perishable things; for we can have an image of these." (Aristotle, "Metaphysics", 4th century BC)

"All that is required between cognizer and cognized is a likeness in terms of representation, not a likeness in terms of an agreement in nature. For it's plain that the form of a stone in the soul is of a far higher nature than the form of a stone in matter. But that form, insofar as it represents the stone, is to that extent the principle leading to its cognition." (Thomas Aquinas, "Quaestiones disputatae de veritate", cca. 1256-1259) 

"Imagery is not past but present. It rests with what we call our mental processes to place these images in a temporal order." (George H Mead, 1929)

"It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience. Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme - the quantum theory - which seems entirely adequate for the treatment of atomic processes; for visualisation, however, we must content ourselves with two incomplete analogies - the wave picture and the corpuscular picture." (Werner Heisenberg, "On Quantum Physics", 1930)

"The solution of the difficulty is that the two mental pictures which experiment lead us to form - the one of the particles, the other of the waves - are both incomplete and have only the validity of analogies which are accurate only in limiting cases." (Werner Heisenberg,"On Quantum Mechanics", 1930)

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)

From Parts to Wholes (1800-1849)

"In every moment of her duration Nature is one connected whole; in every moment each individual part must be what it is, because all the others are what they are; and you could not remove a single grain of sand from its place, without thereby, although perhaps imperceptibly to you, altering something throughout all parts of the immeasurable whole." (Johann G Fichte, "The Vocation of Man”, 1800)

"It is the destiny of our race to become united into one great body, thoroughly connected in all its parts, and possessed of similar culture. Nature, and even the passions and vices of Man, have from the beginning tended towards this end. A great part of the way towards it is already passed, and we may surely calculate that it will in time be reached." (Johann G Fichte, "The Vocation of Man", 1800)

"It is probable that what we call thought is not an actual being, but no more than the relation between certain parts of that infinitely varied mass, of which the rest of the universe is composed, and which ceases to exist as soon as those parts change their position with regard to each other." (Percy B Shelley, "On a Future State", 1815)

"Each of the parts of philosophy is a philosophical whole, a circle rounded and complete in itself. In each of these parts, however, the philosophical Idea is found in a particular specificality or medium. The single circle, because it is a real totality, bursts through the limits imposed by its special medium, and gives rise to a wider circle. The whole of philosophy in this way resembles a circle of circles. The Idea appears in each single circle, but, at the same time, the whole Idea is constituted by the system of these peculiar phases, and each is a necessary member of the organisation." (Georg W F Hegel, "Encyclopedia of the Philosophical Sciences", 1816)

"When the whole and the parts are seen at once, as mutually producing and explaining each other, as unity in multeity, there results shapeliness." (Samuel T Coleridge, "Letters", 1836)

"Science is nothing but the finding of analogy, identity, in the most remote parts." (Ralph W Emerson, 1837)

"So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically." (Antoine A Cournot, "Researches into the Mathematical Principles of the Theory of Wealth", 1838)

"The component parts of a vegetable or animal substance do not lose their mechanical and chemical properties as separate agents, when, by a peculiar mode of juxtaposition, they, as an aggregate whole, acquire physiological or vital properties in addition. Those bodies continue, as before, to obey mechanical and chemical laws, in so far as the operation of those laws is not counteracted by the new laws which govern them as organized beings. […] Though there are laws which, like those of chemistry and physiology, owe their existence to a breach of the principle of Composition of Causes, it does not follow that these peculiar, or, as they might be termed, heteropathic laws, are not capable of composition with one another." (John S Mill, "A System of Logic: Ratioconative and Inductive", 1843) [the heteropathic laws is synonymous with emergence]

On Equilbrium (1850-1874)

"Everything lives by movement, everything is maintained by equilibrium, and harmony results from the analogy of contraries; this law is the form of forms. (Eliphas Levi, Transcendental Magic, 1854)

"Living things have no inertia, and tend to no equilibrium." (Thomas H Huxley, "On the Educational Value of the Natural History Sciences", 1854)

"[...] the small is great, the great is small; all is in equilibrium in necessity […]" (Victor Hugo, Les Misérables, 1862

"Nature! […] She creates needs because she loves action. Wondrous! that she produces all this action so easily. Every need is a benefit, swiftly satisfied, swiftly renewed. - Every fresh want is a new source of pleasure, but she soon reaches an equilibrium." (Johann Wolfgang von Goethe, Nature, 1870)

"We cannot prevent equilibrium from producing its effects. We may brave human laws, but we cannot resist natural ones." (Jules Verne, "Twenty Thousand Leagues Under the Sea", 1870)

🪷On Mind: Mental Models XXXIX

"A spirit is one simple, undivided, active being: as it perceives ideas, it is called the understanding, and as it produces or otherwise operates about them, it is called the will. Hence there can be no idea formed of a soul or spirit: for all ideas whatever, being passive and inert, vide, they cannot represent unto us, by way of image or likeness, that which acts." (George Berkeley, "Principles of Human Knowledge", 1710)

"The ideas imprinted on the senses by the Author of Nature are called real things: and those excited in the imagination being less regular, vivid and constant, are more properly termed ideas, or images of things, which they copy and represent. But then our sensations, be they never so vivid and distinct, are nevertheless ideas, that is, they exist in the mind, or are perceived by it, as truly as the ideas of its own framing. The ideas of sense are allowed to have more reality in them, that is, to be more strong, orderly, and coherent than the creatures of the mind; but this is no argument that they exist without the mind. They are also less dependent on the spirit, or thinking substance which perceives them, in that they are excited by the will of another and more powerful spirit: yet still they are ideas, and certainly no idea, whether faint or strong, can exist otherwise than in a mind perceiving it." (George Berkeley, "Principles of Human Knowledge", 1710)

"It is your opinion, the ideas we perceive by our senses are not real things, but images, or copies of them. Our knowledge therefore is no farther real, than as our ideas are the true representations of those originals. But as these supposed originals are in themselves unknown, it is impossible to know how far our ideas resemble them; or whether they resemble them at all. We cannot therefore be sure we have any real knowledge." (George Berkeley, "Three Dialogues", 1713)

"To speak the truth, Philonous, I think there are two kinds of objects, the one perceived immediately, which are likewise called ideas, the other are real things or external objects perceived by the mediation of ideas, which are their images and representations. Now I own, ideas do not exist without the mind; but the latter sort of objects do." (George Berkeley, "Three Dialogues", 1713)

"[...] things which do not now exist in the mind itself, can only be perceived, remembered, or imagined, by means of ideas or images of them in the mind, which are the immediate objects of perception, remembrance, and imagination. This doctrine appears evidently to be borrowed from the old system; which taught, that external things make impressions upon the mind, like the impressions of a seal upon wax; that it is by means of those impressions that we perceive, remember) or imagine them; and that those impressions must resemble the things from which they are taken. When we form our notions of the operations of the mind by analogy, this way of conceiving them seems to be very natural, and offers itself to our thoughts: for as every thing which is felt must make some impression upon the body, we are apt to think, that every thing which is understood must make some impression upon the mind." (Thomas Reid, "An Inquiry into the Human Mind", 1734)

"It seems evident, that men are carried, by a natural instinct or prepossession, to repose faith in their senses; and that, without any reasoning, or even almost before the use of reason, we always suppose an external universe, which depends not on our perception, but would exist, though we and every sensible creature were absent or annihilated. […] It seems also evident, that, when men follow this blind and powerful instinct of nature, they always suppose the very images, presented by the senses, to be the external objects." (David Hume, "An Enquiry Concerning Human Understanding", 1748)

"Like pictures, images seem to depict information about interval spatial extents. The scanning experiments support the claim that portions of images depict corresponding portions of the represented objects, and that the spatial relations between portions of the image index the spatial relations between the corresponding portions of the imaged objects." (Stephen Kosslyn," Image and Mind", 1980) 

"The crucial problem is that of describing what is ‘seen in the mind’s eye’ and what is ‘heard in one’s head’. What are spoken of as ‘visual images’, ‘mental pictures’, ‘auditory images’ and, in one use, ‘ideas’ are commonly taken to be entities which are genuinely found existing and found existing elsewhere than in the external world. So minds are nominated for their theatres." (Gilbert Ryle, "The Concept of Mind", 1949)

„Like pictures, images seem to depict information about interval spatial extents. The scanning experiments support the claim that portions of images depict corresponding portions of the represented objects, and that the spatial relations between portions of the image index the spatial relations between the corresponding portions of the imaged objects.“ (Stephen Kosslyn, “Image and Mind”. 1980) 

"An image of thought called philosophy has been formed historically and it effectively stops people from thinking." (Gilles Deleuze,"Dialogues II", 1987) 

On Abstraction (2000-2009)

"Abstraction is itself an abstract word and has no single meaning. […] Every word in our language is abstract, because it represents something else." (Eric Maisel, "The Creativity Book: A Year's Worth of Inspiration and Guidance", 2000)

"What cognitive capabilities underlie our fundamental human achievements? Although a complete answer remains elusive, one basic component is a special kind of symbolic activity - the ability to pick out patterns, to identify recurrences of these patterns despite variation in the elements that compose them, to form concepts that abstract and reify these patterns, and to express these concepts in language. Analogy, in its most general sense, is this ability to think about relational patterns." (Keith Holyoak et al, "Introduction: The Place of Analogy in Cognition", 2001)

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

"A model isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)

"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Do not be afraid of the word 'theory'. Yes, it can sound dauntingly abstract at times, and in the hands of some writers can appear to have precious little to do with the actual, visual world around us. Good theory however, is an awesome thing. [...] But unless we actually use it, it borders on the metaphysical and might as well not be used at all." (Richard Howells,  Visual Culture, 2003)

"Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verifiable predictions about the behavior of atoms, molecules and solids." (Arthur M Lesk, "Introduction to Symmetry and Group Theory for Chemists", 2004)

"Mathematics is not about abstract entities alone but is about relation of abstract entities with real entities. […] Adequacy relations between abstract and real entities provide space or opportunity where mathematical and logical thought operates parsimoniously."  (Navjyoti Singh, "Classical Indian Mathematical Thought", 2005)

"Calculating with letters instead of numbers is a big step forward in everyone’s education. It is rightly appreciated as a step from the concrete to the abstract, from the particular to the general, from arithmetic to algebra; but it is not always recognized as a step from confusion to clarity. To appreciate the clarity of algebra, ask yourself: what are the rules for calculating with numbers?" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 2007)

"Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities." (Tom G Palmer, Realizing Freedom: Libertarian Theory, History, and Practice, 2009)

"In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions." (Bruce Hannon and Matthias Ruth, "Dynamic Modeling of Diseases and Pests", 2009)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Simplicity IX

"By the word simplicity, is not always meant folly or ignorance; but often, pure and upright Nature, free from artifice, craft or deceitful ornament." (Benjamin Franklin,  Poor Richard's Almanack, 1733-1758)

"Simplicity and precision ought to be the characteristics of a scientific nomenclature: words should signify things, or the analogies of things, and not opinions." (Sir Humphry Davy, Elements of Chemical Philosophy", 1812)

"Discoveries are not generally made in the order of their scientific arrangement: their connexions and relations are made out gradually; and it is only when the fermentation of invention has subsided that the whole clears into simplicity and order. " (William Whewell, "An Elementary Treatise on Mechanics" Vol. I, 1819)

"Simplicity is naturally agreeable to a mind of limited powers, but to an infinite mind all things are simple." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, [Paris International Congress] 1900)

"Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof." (David Hilbert, "Mathematical Problems", Bulletin of the American Mathematical Society Vol. 8, 1902)

"There is one, but only one branch which may claim certainty; in which experts have not seriously disagreed, - Mathematics. […] Mathematics possesses also the second desideratum, - Simplicity. It begins, as is well known, with few and uncomplicated definitions and postulates, and proceeds, step by step, to quite elaborate cases. […] Mathematics possesses also the third desideratum, - Applicability of the skill acquired." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability." (Albert Einstein, "Autobiographical Notes", 1949)

"Whether mathematical simplicity is God’s affair or our, the fact remains that this feature more than any other remains the mainspring of progress in the physical sciences." (Paul C W Davies, "The Edge of Infinity", 1981)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

On Emergence I

"This, however, is very speculative; the point of interest for our present enquiry is that physical reality is built up, apparently, from a few fundamental types of units whose properties determine many of the properties of the most complicated phenomena, and this seems to afford a sufficient explanation of the emergence of analogies between mechanisms and similarities of relation-structure among these combinations without the necessity of any theory of objective universals." (Kenneth Craik, "The Nature of Explanation", 1943)

"The moment of truth, the sudden emergence of a new insight, is an act of intuition. Such intuitions give the appearance of miraculous flushes, or short-circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links. (Arthur Koestler, "The Act of Creation", 1964)

"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)

"Catastrophes are often stimulated by the failure to feel the emergence of a domain, and so what cannot be felt in the imagination is experienced as embodied sensation in the catastrophe. (William I Thompson, "Gaia, a Way of Knowing: Political Implications of the New Biology", 1987)

"The phenomenon of emergence takes place at critical points of instability that arise from fluctuations in the environment, amplified by feedback loops." (Fritjof Capra, "The Hidden Connections", 2002)

"This spontaneous emergence of order at critical points of instability is one of the most important concepts of the new understanding of life. It is technically known as self-organization and is often referred to simply as ‘emergence’. It has been recognized as the dynamic origin of development, learning and evolution. In other words, creativity-the generation of new forms-is a key property of all living systems. And since emergence is an integral part of the dynamics of open systems, we reach the important conclusion that open systems develop and evolve. Life constantly reaches out into novelty." (Fritjof  Capra, "The Hidden Connections", 2002)

"Emergence is the phenomenon of properties, capabilities and behaviours evident in the whole system that are not exclusively ascribable to any of its parts." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"The beauty of nature insists on taking its time. Everything is prepared. Nothing is rushed. The rhythm of emergence is a gradual, slow beat; always inching its way forward, change remains faithful to itself until the new unfolds in the full confidence of true arrival. Because nothing is abrupt, the beginning of spring nearly always catches us unawares. It is there before we see it; and then we can look nowhere without seeing it. (John O'Donohue, "To Bless the Space Between Us: A Book of Blessings", 2008)

"System theorists know that it's easy to couple simple-to-understand systems into a ‘super system’ that's capable of displaying behavioral modes that cannot be seen in any of its constituent parts. This is the process called ‘emergence’." (John L Casti, [interview with Austin Allen], 2012)

"This spontaneous emergence of order at critical points of instability, which is often referred to simply as 'emergence', is one of the hallmarks of life. It has been recognized as the dynamic origin of development, learning, and evolution. In other words, creativity-the generation of new forms-is a key property of all living systems." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

🪷On Mind: Mental Models XXXII

"The imagination […] that reconciling and mediatory power, which incorporating the reason in images of the sense and organizing (as it were) the flux of the senses by the permanence and self-circling energies of the reason, gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors." (Samuel T Coleridge, "The Statesman's Manual", 1816)

"It seems to be like taking the pieces of a dissected map out of its box. We first look at one part, and then at another, then join and dove-tail them; and when the successive acts of attention have been completed, there is a retrogressive effort of mind to behold it as a whole. The poet should paint to the imagination, not to the fancy; and I know no happier case to exemplify the distinction between these two faculties." (Samuel T Coleridge," Biographia Literaria", 1817) 

"The mechanism of thought consists in combinations, separations, and recombinations of representative images or symbols […] the object of thought is adaptation to environment." (Paul Carus, “Le probeme de la conscience du moi", 1893)

"It is now known that as the physical basis of any word, be it noun or verb, there is a series of mental images acquired through different senses, located in different regions of the gray cortex of the brain, and joined together in a unit by a series of association-tracts which pass in the white matter under the cortex. The word ‘concept’ long used by psychologists to denote congeries of mental images making up an idea conveyed by a single word may be adopted by the pathologist to indicate this collection of mental images. To be complete, such a concept must have all its parts intact and the connections between those parts also intact."  (Anon, "Aphasia", Psychological Review Vol. I (1), 1894)

"[…] the image is an act which envisions an absent or non-existent object as a body, by means of a physical or mental content which is present only as an 'analogical representative' of the object envisioned." (Jean-Paul Sartre, "The Psychology of Imagination", 1940)

"The crucial problem is that of describing what is ‘seen in the mind’s eye’ and what is ‘heard in one’s head’. What are spoken of as ‘visual images’, ‘mental pictures’ […] are commonly taken to be entities which are genuinely found existing and found existing elsewhere than in the external world. So minds are nominated for their theaters." (Gilbert Ryle, "The Concept of Mind" , 1949)

"We may not speak of the image as a thing, like a canvas only in our heads. But we may say that in thinking with images we are thinking analogically, or by means of representations, just as we are when we look at somebody’s portrait rather than at himself. […] The image is our attempt to reach the non-existent or absent object in our thoughts as we concentrate on this or that aspect of it, its visible appearance, its sound, its smell. […] The images themselves are not separate from our interpretations of the world; they are our way of thinking of objects in the world." (Mary Warnock, "Imagination", 1978) 

"Hard though the scientists of mental imagery try, they cannot get around the fact that the representations they deal with are like pictures. […] The methods have to assume, and the experiments continually corroborate, that having imagery is somehow like perceptual seeing, and that it is somehow like seeing pictures. […] The minimal reason for this assumption is that people do naturally talk of seeing pictures before their mind’s eye." (Eva T H Brann,"The World of Imagination", 1991)

"The most persuasive positive argument for mental images as objects is [that] whenever one thinks one is seeing something there must be something one is seeing. It might be an object directly, or it might be a mental picture. [This] point is so plausible that it is deniable only at the peril of becoming arbitrary. One should concede that the question whether mental images are entities of some sort is not resolvable by logical or linguistic analysis, and believe what makes sense of experience." (Eva T H Brann,"The World of Imagination" , 1991)

On Metaphors VII

"The greatest thing by far is to be a master of metaphor. It is the one thing that cannot be learnt from others; it is also a sign of genius, since a good metaphor implies an intuitive perception of the similarity in dissimilars." (Aristotle, "Poetics", cca. 335 BC)

"[Metaphor] is a loose word, at best, and we must beware of attributing to it stricter rules of usage than are usually found in practice." (Max Black, “Models and Metaphors”, 1962)

"Rationality consists [of] the continuous adaptation of our language to our continually expanding world, and metaphor is one of the chief means by which this is accomplished." (Mary B Hesse, "Models and Analogies in Science", 1966)

"The very nature of science is such that scientists need the metaphor as a bridge between old and new theories." (Earl R MacCormac, "Metaphor and Myth in Science and Religion", 1976)

“[…] the use of analogies, particularly with metaphor, adds richness and dimension to arguments and descriptions not possible with ordinary discourse or with propositional reasoning.” (Jeanette M Gallagher, “The Future of Formal Thought Research: The Study of Analogy and Metaphor”, 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)

"Metaphor is for most people a device of the poetic imagination and the rhetorical flourish - a matter of extraordinary rather than ordinary language. Moreover, metaphor is typieully viewed as characteristic of language alone, a matter of words rather than thought or action. For this reason, most people think they can get along perfectly well without metaphor. We have found, on the contrary, that metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature." (George Lakoff & Mark Johnson, "Metaphors we Live by", 1980)

"Metaphor plays an essential role in establishing a link between scientific language and the world. Those links are not, however, given once and for all. Theory change, in particular, is accompanied by a change in some of the relevant metaphors and in the corresponding parts of the network of similarities through which terms attach to nature." (Thomas S Kuhn, "Metaphor in science", 1993)

“Metaphorizing is a  manner of thinking, not a  property of thinking. It is a  capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects.” (Ivan Mladenov, “Conceptualizing Metaphors: On Charles Peirce’s marginalia”, 2006)

On Metaphors V

"Metaphor consists in giving the thing a name that belongs to something else; the transference being either from genus to species, or from species to genus, or from species to species, or on grounds of analogy." (Aristotle, "Poetics", cca. 335 BC)
 
"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)

"[…] 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)

"[…] theoretical science is essentially disciplined exploitation of metaphor." (Anatol Rapoport, "Operational Philosophy", 1953)

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

"The symbol and the metaphor are as necessary to science as to poetry." (Jacob Bronowski, "Science and Human Values", 1956) 

"The model is only a suggestive metaphor, a fiction about the messy and unwieldy observations of the real world. In order for it to be persuasive, to convey a sense of credibility, it is important that it not be too complicated and that the assumptions that are made be clearly in evidence. In short, the model must be simple, transparent, and verifiable." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)
 
"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)

"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)

On Analogy (2000-)

“What cognitive capabilities underlie our fundamental human achievements? Although a complete answer remains elusive, one basic component is a special kind of symbolic activity - the ability to pick out patterns, to identify recurrences of these patterns despite variation in the elements that compose them, to form concepts that abstract and reify these patterns, and to express these concepts in language. Analogy, in its most general sense, is this ability to think about relational patterns.” (Keith Holyoak et al, “Introduction: The Place of Analogy in Cognition”, 2001)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Education is a set of analogies to a genuinely human existence, of which the arts are the model. Merely human life is of course a demonic analogy or parody of genuinely human life." (Northrop Frye, "Notebooks and Lectures on the Bible and Other Religious Texts", 2003)

“It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct.” (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"What could mathematics and poetry share, except that the mention of either one is sometimes enough to bring an uneasy chill into a conversation? [...] Both fields use analogies - comparisons of all sorts - to explain things, to express unknown or unknowable concepts, and to teach." (Marcia Birken & Anne C Coon, “Discovering Patterns in Mathematics and Poetry”, 2008)

"By bringing together what we know and what we don't know through analogy, metaphorical thinking strikes the spark that ignites discovery." (James Geary, [TED talk] 2009)

"The human mind delights in finding pattern - so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 2010)

"This is always the case in analogical reasoning: Relations between two dissimilar domains never map completely to one another. In fact, it is often the salient similarities between the base and target domains that provoke thought and increase the usefulness of an analogy as a problem-solving tool." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

On Analogy (1925-1949)

"Analogies prove nothing, that is quite true, but they can make one feel more at home." (Sigmund Freud, Neue Folge der Vorlesungen zur Einführung in die Psychoanalyse" ["New Introductory Lectures on Psychoanalysis"], 1933)

"If the scientist has, during the whole of his life, observed carefully, trained himself to be on the look-out for analogy and possessed himself of relevant knowledge, then the ‘instrument of feeling’ […] will become a powerful divining rod leading the scientist to discover order in the midst of chaos by providing him with a clue, a hint, or an hypothesis upon which to base his experiments.' (Rosamund E M Harding, "An Anatomy of Inspiration", 1940)

"The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof." (Richard Courant & Herbert Robbins, "What Is Mathematics?: An Elementary Approach to Ideas and Methods" , 1941) 

"Analogies are useful for analysis in unexplored fields. By means of analogies an unfamiliar system may be compared with one that is better known. The relations and actions are more easily visualized, the mathematics more readily applied, and the analytical solutions more readily obtained in the familiar system." (Harry F Olson, "Dynamical Analogies", 1943)

"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)

"This, however, is very speculative; the point of interest for our present enquiry is that physical reality is built up, apparently, from a few fundamental types of units whose properties determine many of the properties of the most complicated phenomena, and this seems to afford a sufficient explanation of the emergence of analogies between mechanisms and similarities of relation-structure among these combinations without the necessity of any theory of objective universals." (Kenneth Craik, "The Nature of Explanation", 1943)

"Analogy is a sort of similarity. Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts." (George Pólya, "How to solve it", 1945)

"[…] analogy [is] an important source of conjectures. In mathematics, as in the natural and physical sciences, discovery often starts from observation, analogy, and induction. These means, tastefully used in framing a plausible heuristic argument, appeal particularly to the physicist and the engineer." (George Pólya, "How to solve it", 1945) 

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements." (George Pólya, "How to Solve It", 1945)

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements. Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision. All sorts of analogy may play a role in the discovery of the solution and so we should not neglect any sort." (George Pólya, "How to solve it", 1945)

"Induction is the process of discovering general laws by the observation and combination of particular instances. […] Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further special cases." (George Pólya, "How to solve it", 1945)

"Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning." (George Pólya, "How to Solve It", 1945)

"[…] the number of available analogies is a determining factor in the growth and progress of science." (Morris R Cohen, "The Meaning of Human History", 1947)

"A man desiring to understand the world looks about for a clue to its comprehension. He pitches upon some area of commonsense fact and tries to understand other areas in terms of this one. The original area becomes his basic analogy or root metaphor." (Stephen Pepper, "World Hypotheses: A Study in Evidence", 1948)

"The solution of the difficulty is that the two mental pictures which experiment lead us to form - the one of the particles, the other of the waves - are both incomplete and have only the validity of analogies which are accurate only in limiting cases." (Werner Heisenberg, "The Physical Principles of the Quantum Theory", 1949)