Mental Models XL

"When the Soul wants to experience something she throws out an image in front of her and then steps into it." (Eckhart von Hochheim [aka Meister Eckhart] cca. 14th century)

"Materialism is the recognition of 'objects in themselves' , or outside the mind; ideas and sensations are copies of images of those objects." (Vladimir Lenin, "Materialism and Empirio-Criticism", 1908) 

"Images apparently occupy a curious position somewhere between the statements of language, which are intended to convey a meaning, and the things of nature, to which we only can give a meaning." (Ernst Gombrich, "Symbolic Images", 1972) 

 "When one analyzes the pre-conscious step to concepts, one always finds ideas which consist of 'symbolic images'. The first step to thinking is a painted vision of these inner pictures whose origin cannot be reduced only and firstly to the sensual perception but which are produced by an 'instinct to imagining' and which are re-produced by different individuals independently, i. e. collectively [...] But the archaic image is also the necessary predisposition and the source of a scientific attitude. To a total recognition belong also those images out of which have grown the rational concepts." (Wolfgang Pauli, [Letter to Markus Fierz] 1948) 

" The symbol is the tool which gives man his power, and it is the same tool whether the symbols are images or words, mathematical signs or mesons." (Jacob Bronowski, "The Reach of Imagination", 1967) 

"To imagine means to make images and to move them about inside one's head in new arrangements." (Jacob Bronowski, "The Reach of Imagination", 1967)

"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) 

"Vision is a process that produces from images of the external world a description that is useful to the viewer and not cluttered with irrelevant information." (David Marr, "Vision", 1982)

"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) 

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

Mental Models XXXVII

"This world is finite, the other infinite, 
reality is blocked by form and image." (Jalaluddin Rumi, "Masnavi-ye Ma ‘navi" Vol. I ["Spritual Verses"], 1262-1264)

"There is nothing in us which we must attribute to our soul except our thoughts. These are of two principal kinds, some being actions of the soul and others its passions. Those I call its actions are all our volitions […] the various perceptions or modes of knowledge present in us may be called its passions in the general sense, for it is often not our soul which makes them such as they are, and the soul always receives them from the things that are represented by them." (René Descartes, "Les passions de l’âme" ["Passions of the Soul"], 1649)

"Modern mathematics, that most astounding of intellectual creations, has projected the mind's eye through infinite time and the mind's hand into boundless space." (Nicholas M Butler, "What Knowledge is of Most Worth?", 1895)

"Thus the essence of memory is not constituted by the image, but by having immediately before the mind an object which is recognised as past." (Bertrand Russell, "Theory of Knowledge" , 1913)

"The paradox of reality is that no image is as compelling as the one which exists only in the mind's eye." (Shana Alexander, "Talking Woman", 1976)

"[…] new insights fail to get put into practice because they conflict with deeply held internal images of how the world works [...] images that limit us to familiar ways of thinking and acting. That is why the discipline of managing mental models - surfacing, testing, and improving our internal pictures of how the world works - promises to be a major breakthrough for learning organizations." (Peter Senge, "The Fifth Discipline: The Art and Practice of the Learning Organization", 1990)

"I believe that philosophy is part of literature, and not the reverse. Writing is not possible without images. Yet, images don't have to be descriptive; they can be concepts […]. Concepts are mental images." (Paul Virilio, [interview with Louise Wilson] 1994)

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

"Presumably, one can become a mathematical genius only if one has an outstanding capacity for forming vivid mental represen­tations of abstract mathematical concepts - mental images that soon turn into an illusion, eclipsing the human origins of mathematical objects and endowing them with the semblance of an independent existence." (Stanislas Dehaene," The Number Sense: How the Mind Creates Mathematics", 2011) 

"We call it 'explanation', but it is 'description' which distinguishes us from earlier stages of knowledge and science. We describe better - we explain just as little who came before us [...] We operate with nothing but things which do not exist, with lines, planes, bodies, atoms, divisible time, divisible space - how should explanation even be possible when we first make everything into an image, into our image!" (Friedrich W Nietzsche)

On Abstraction (-1699)

"[Arithmetic] has a very great and elevating effect, compelling the soul to reason about abstract numbers, and rebelling against the introduction of visible or tangible objects into the argument." (Plato, "The Republic", cca 375 BC)

"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

"The exact kind of language we employ in philosophical analyses of abstract truth is one thing, and the language used in attempts to popularize the subject is another." (Marcus Tullius Cicero, "De officiis" ["On Duties"], cca.44 BC)

"It seems that all perception is but the grasping of the form of the perceived object in some manner. If, then, it is a perception of some material object, it consists in an apprehension of its form by abstracting it from matter in some way. But the kinds of abstraction are different and their degrees various. This is because, owing to matter, the material form is subject to certain states and conditions which do not belong to [the form] by itself insofar as it is this form. So sometimes the abstraction from matter is effected with all or some of these attachments, and sometimes it is complete in that the concept is abstracted from matter and from the accidents it possesses on account of the matter."(Avicenna Latinus [Ibn Sina], "Liber De anima", cca. 1014-1027)

"Sometimes a thing is perceived [via sense-perception] when it is observed; then it is imagined, when it is absent [in reality] through the representation of its form inside, Sense-perception grasps [the concept] insofar as it is buried in these accidents that cling to it because of the matter out of which it is made without abstracting it from [matter], and it grasps it only by means of a connection through position [ that exists] between its perception and its matter. It is for this reason that the form of [the thing] is not represented in the external sense when [sensation] ceases. As to the internal [faculty of] imagination, it imagines [the concept] together with these accidents, without being able to entirely abstract it from them. Still, [imagination] abstracts it from the afore-mentioned connection [through position] on which sense-perception depends, so that [imagination] represents the form [of the thing] despite the absence of the form's [outside] carrier." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"If one, with the scientist, studies the works of nature, which are made up of elements or matter and form, his reasoning is dependent on the data provided by sense-experience. And if one, with the mathematician, abstracts figures or calculates numerically, he must, in order to gain assent, accurately adduce many examples of both differentiated plurality and quantitative extension. The like holds true of the philosopher, whose domain is [abstract] reasoning, and who is the client of both the scientist and the mathematician. For the philosopher, too, begins with those things which are based on the evidence of the senses and contribute to the knowledge of immaterial intelligibles." (John of Salisbury, "Metalogicon", 1159)

"It is true that mathematics, which deals theoretically with abstractions, and in its subtle analysis separates things that are united in nature, treats matter and form apart from one another, so that the nature of what is composite may be more accurately and definitely understood. Still, the one cannot exist apart from the other, as [in this case] either matter would be without form, or form would lack a subject and hence be ineffective." (John of Salisbury, "Metalogicon", 1159)

"Nor is it enough to say that the intelligible notions formed by the active intellect subsist somehow in the phantasmata (mental image), which are certainly intrinsic to us; for as we have already observed in treating the passive intellect, objects only become actually intelligible when abstracted from phantasmata; so that merely by way of the phantasmata, we cannot attribute the work of the active intellect to ourselves" (St. Thomas Aquinas, "De Anima" III, cca. 1268) [On Aristotle's phantasmata]

"Abstraction involves perceiving something, relating it to other things, grasping some common trait of those things, and conceiv­ing of the common trait as to it can be related not only to those things but also to other similar things."  (John Locke, "An Essay Concerning Human Understanding", 1689)

On Spacetime (1800-1849)

"Genius and science have burst the limits of space, and few observations, explained by just reasoning, have unveiled the mechanism of the universe. Would it not also be glorious for man to burst the limits of time, and, by a few observations, to ascertain the history of this world, and the series of events which preceded the birth of the human race?" (Georges Cuvier, "Essays on the Theory of the Earth", 1822)

"History in general is therefore the development of Spirit in Time, as Nature is the development of the Idea is Space." (Georg W F Hegel, "Lectures on the Philosophy of History", 1837)

"Yet time and space are but inverse measures of the force of the soul. The spirit sports with time." (Ralph W Emerson, "Essays", 1841)

"Great distance in either time or space has wonderful power to lull and render quiescent the human mind." (Abraham Lincoln, [An Address Delivered by Abraham Lincoln], 1842)

"Poetry is the universal art of the spirit which has become free in itself and which is not tied down for its realization to external sensuous material; instead, it launches out exclusively in the inner space and the inner time of ideas and feelings." (G W Friedrich Hegel, "Introduction to Aesthetics", 1842)

"Language has time as its element; all other media have space as their element." (Søren Kierkegaard, "Either/Or: A Fragment of Life", 1843)

Mental Models XXXVI

"Conscious apprehension seems to exist […] as happens in a mirror-image when the smooth and bright surface is peaceful." (Plotinus, "Enneads", cca. 270 AD)

"In the same way as regards the soul, when that kind of thing in us which mirrors the images of thought and intellect is undisturbed, we see them and know them in a way parallel to sense-perception, along with the prior knowledge that it is intellect and thought that are active. But when this is broken because the harmony of the body is upset, thought and intellect operate without an image, and then intellectual activity takes place without a mind-picture." (Plotinus, "Enneads", cca. 270 AD)

"The noetic act is without parts and has not, so to speak, come out into the open, but remains unobserved within, but the verbal expression unfolds its content and brings it out of the noetic act into the image making power, and so shows the noetic act as if in a mirror, and this is how there is conscious apprehension and persistence and memory of it." (Plotinus, "Enneads", cca. 270 AD)

"All the perceptions of the human mind resolve themselves into two distinct kinds, which I shall call impressions and ideas. The difference betwixt these consists in the degrees of force and liveliness, with which they strike upon the mind, and make their way into our thought or consciousness. Those perceptions, which enter with most force and violence, we may name impressions; and under this name I comprehend all our sensations, passions and emotions, as they make their first appearance in the soul. By ideas I mean the faint images of these in thinking and reasoning. I believe it will not be very necessary to employ many words in explaining this distinction." (David Hume, "A Treatise of Human Nature", 1738)

"You cannot crown the edifice by this abstraction. The scientific imagination, which is here authoritative, demands as the origin and cause of a series of ether waves a particle of vibrating matter quite as definite, though it may be excessively minute, as that which gives origin to a musical sound. Such a particle we name an atom or a molecule. I think the imagination when focused so as to give definition without penumbral haze, is sure to realise this image at last." (John Tyndall, "The Scientific Use of the Imagination", 1870) 

"The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"Images tell us nothing, either right or wrong, about the external world. […] It is just because forming images is a voluntary activity that it does not instruct us about the external world. […] When we form an image of something we are not observing. The coming and going of the pictures is not something that happens to us. We are not surprised by these pictures, saying ‘Look!’"  (Ludwig Wittgenstein, "Zettel", 1967)

“The power of images consists largely in the fact that they integrate different types of knowledge and experience.” (David Gooding, "Creative Rationality", 1996)

"When we talk of seeing an image, either in front of us or visualised with closed eyes, we invoke a range of metaphors and ideas which highlight the relationship between perception and imagery. For those of us with unimpaired vision, to see with ‘the mind’s eye’ conjures up a picture of perception where there is not a great deal of difference between an external or internal image." (Michael Forrester," Psychology of the Image", 2000) 

"[mental] images are sui generis, and should be added as a third great category of intentionality to the twin pillars of perception and cognition […] Neither is it at all obvious that images necessarily carry a [conceptual] thought component […] Images are not just minor variations on perception and thought, of negligible theoretical interest; they are a robust mental category in need of independent investigation." (Colin McGinn, Mindsight, 2009)

Avicenna Latinus - Collected Quotes

"The faculty which grasps such concepts acquires intelligible forms from sense-perception by force of an inborn disposition, so that forms, which are in the form-bearing faculty and the memorizing faculty, are made present to [the rational soul] with the assistance of the imaginative and estimative [faculties]." (Avicenna Latinus [Ibn Sina], "A Compendium on the Soul", cca. 996-997)

"When knowledge reaches the specific natures and what is accidental to them, inquiry stops and is not followed by the fleeting knowledge of individuals to which our souls are not at all inclined." (Avicenna Latinus [Ibn Sina], "The Physics of The Healing", cca. 1014)

"a [thing’s] likeness often appears and seems to those who perceive it as if the image itself were speaking, as if they heard the words that they held and read." (Avicenna Latinus [Ibn Sina], "Liber de Anima", cca. 1014-1027)

"It seems that all perception is but the grasping of the form of the perceived object in some manner. If, then, it is a perception of some material object, it consists in an apprehension of its form by abstracting it from matter in some way. But the kinds of abstraction are different and their degrees various. This is because, owing to matter, the material form is subject to certain states and conditions which do not belong to [the form] by itself insofar as it is this form. So sometimes the abstraction from matter is effected with all or some of these attachments, and sometimes it is complete in that the concept is abstracted from matter and from the accidents it possesses on account of the matter."(Avicenna Latinus [Ibn Sina], "Liber De anima", cca. 1014-1027)

"The animal faculties assist the rational soul in various ways, one of them being that sense-perception brings to it particulars, from which four things result in [the rational soul]: One of them is that the mind extracts single universals from the particulars, by abstracting their concepts from matter and the appendages of matter and its accidents, by considering what is common in it and what different, and what in its existence is essential and what accidental. From this the principles of conceptualization come about [in] the soul: and this with the help of its employing imagination and estimation." (Avicenna Latinus [Ibn Sina], "Liber De Anima", cca. 1014-1027)

"When the intellectual faculty considers the particulars which are [stored] in imagination and the light of the above-mentioned active intellect shines upon them in us, then the [particulars] are transformed into something abstracted from matter and from the [material] attachments and get imprinted in the rational soul, but not in the sense that the particulars themselves are transferred from imagination to our intellect, nor in the sense that the concept buried in [material] attachments - which in itself and with regard to its essence is abstract - produces a copy of itself, but in the sense that looking at the particulars disposes the soul for something abstracted to flow upon it from the active intellect. (Avicenna Latinus [Ibn Sina], "Liber De Anima", cca. 1014-1027)

"Sometimes a thing is perceived [via sense-perception] when it is observed; then it is imagined, when it is absent [in reality] through the representation of its form inside, Sense-perception grasps [the concept] insofar as it is buried in these accidents that cling to it because of the matter out of which it is made without abstracting it from [matter], and it grasps it only by means of a connection through position [ that exists] between its perception and its matter. It is for this reason that the form of [the thing] is not represented in the external sense when [sensation] ceases. As to the internal [faculty of] imagination, it imagines [the concept] together with these accidents, without being able to entirely abstract it from them. Still, [imagination] abstracts it from the afore-mentioned connection [through position] on which sense-perception depends, so that [imagination] represents the form [of the thing] despite the absence of the form's [outside] carrier." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"The multiplicity of the soul's occupations with sense-perceptible imaginable forms and connotational images, which are in the form-bearing and the remembering [faculties respectively], with the help of the estimative and cogitative faculty, makes the soul obtain a disposition for the reception of abstractions of them [i.e., of the imaginable forms and images] from the separate substance through some kind of relationship between the two. Observation and inspection of the issue verify this. These occupations [with imaginable forms and images] are those which give [the soul] a perfect disposition that is specific for [the reception of] each individual form, though an intellectual concept may [also] provide this specific [disposition] for [the reception of] another intellectual concept." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"Absence of understanding does not warrant absence of existence." (Avicenna Latinus [Ibn Sina]) 

"Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned." (Avicenna Latinus [Ibn Sina])

"Cultivate the self with learning in order to progress and leave all else; for knowledge is an abode of all things. 

The self is like glass, the knowledge, like a lamp, and the wisdom of God, like oil. 

When your self is illuminated - you are alive, and when there is darkness - you are dead." (Avicenna Latinus [Ibn Sina])

"Friendship is always a sweet responsibility, never an opportunity." (Avicenna Latinus [Ibn Sina])

"Further, this power which conceives these ideas does at times gain from sense forms mental, imaginative, and innate in (instinctive to) itself; and in such a case it does this in that it lays before itself the forms that are in the conceiving power and in the remembering (preserving) power, by employing the imaginative and the conjecturing power, and then contemplates them, and finds them to have participated in some forms and to have differed in some other forms; and finds some amongst the forms that are in these powers to be essential and others to be accidental." (Avicenna Latinus [Ibn Sina])

"It is evident that everything which does not exist at first and then exists, is determined by something other than itself." (Avicenna Latinus [Ibn Sina])

"Medicine deals with the states of health and disease in the human body. It is a truism of philosophy that a complete knowledge of a thing can only be obtained by elucidating its causes and antecedents, provided, of course, such causes exist. In medicine it is, therefore, necessary that causes of both health and disease should be determined." (Avicenna Latinus [Ibn Sina]) 

"Now it is established in the sciences that no knowledge is acquired save through the study of its causes and beginnings, if it has had causes and beginnings; nor completed except by knowledge of its accidents and accompanying essentials." (Avicenna Latinus [Ibn Sina]) 

"Pain is a sensation produced by something contrary to the course of nature and this sensation is set up by one of two circumstances: either a very sudden change of the temperament (or the bad effect of a contrary temperament) or a solution of continuity." (Avicenna Latinus [Ibn Sina]) 

"The different sorts of madness are innumerable." (Avicenna Latinus [Ibn Sina]) 

 

"The knowledge of anything, since all things have causes, is not acquired or complete unless it is known by its causes." (Avicenna Latinus [Ibn Sina]) 

"The mind (Understanding, Reason) is in fact and deed wholly and solely nothing else than the forms of mentally-grasped things, if these be arrayed in the very mind potentially, and through it they are brought out to effective action; and hence it is said that the mind is in fact and deed at once both understanding and understood. Amongst the properties of the understanding power is this, that it unifies the many and multiplies the one through analysis and synthesis. As to multiplication, it is such as the analysis of one man into essence, body, nourishment-getting, animal, speaking (rational). As to unification of the many, it is such as the composition (synthesis) of this one man out of essence, body, animal, speaking (rational) into one notion which is mankind (human being)."(Avicenna Latinus [Ibn Sina])

"The theory of medicine, therefore, presents what is useful in thought, but does not indicate how it is to be applied in practice - the mode of operation of these principles. The theory, when mastered, gives us a certain kind of knowledge. Thus we say, for example, there are three forms of fevers and nine constitutions. The practice of medicine is not the work which the physician carries out, but is that branch of medical knowledge which, when acquired, enables one to form an opinion upon which to base the proper plan of treatment." (Avicenna Latinus [Ibn Sina]) 

On Discovery (1700-1799)

"Wit, you know, is the unexpected copulation of ideas, the discovery of some occult relation between images in appearance remote from each other." (Samuel Johnson, "The Rambler", 1750)

"[…] chance, that is, an infinite number of events, with respect to which our ignorance will not permit us to perceive their causes, and the chain that connects them together. Now, this chance has a greater share in our education than is imagined. It is this that places certain objects before us and, in consequence of this, occasions more happy ideas, and sometimes leads us to the greatest discoveries […]" (Claude Adrien Helvetius, "On Mind", 1751)

"If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty." (Edmund Burke, "Essay on the Sublime and Beautiful", 1756)

"To endeavor at discovering the connections that subsist in nature, is no way inconsistent with prudence; but it is downright folly to push these researches too far; as it is the lot only of superior Beings to see the dependence of events, from one end to the other, of the chain which supports them." (Pierre Louis Maupertuis, "An Essay Towards a History of the Principal Comets Since 1742", 1769)

"Cultivate that kind of knowledge which enables us to discover for ourselves in case of need that which others have to read or be told of." (Georg C Lichtenberg, Notebook D, 1773-1775)

"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)

"This schematism of our understanding, in its application to appearances and their mere form, is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze." (Immanuel Kant, "Critique of Pure Reason", 1781)

"A good method of discovery is to imagine certain members of a system removed and then see how what is left would behave: for example, where would we be if iron were absent from the world: this is an old example." (Georg C Lichtenberg, Notebook J, 1789-1793)

"Every science has for its basis a system of principles as fixed and unalterable as those by which the universe is regulated and governed. Man cannot make principles; he can only discover them." (Thomas Paine, "The Age of Reason", 1794)

Mental Models XVII

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

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

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

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

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

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

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

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

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

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

On Intuition (until 1699)

"Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience."(Roger Bacon, "Opus Majus", cca. 1267) 

"The highest endeavor of the mind, and the highest virtue, it to understand things by intuition." (Baruch Spinoza, "The Road to Inner Freedom: The Ethics" , 1667)

“Education produces natural intuitions, and natural intuitions are erased by education.” (Blaise Pascal, “Pensées”, 1670)

 “Those rational instincts, the connate principles engraven in the human soul, though they are truths acquirable and deducible by rational consequence and argumentation, yet seem to be inscribed in the very crasis and texture of the soul, antecedent to any acquisition by industry or the exercise of the discursive faculty in man.” (Sir Matthew Hale, “The Primitive Origination of Mankind: Considered and Examined According to the Light of Nature”, 1677)

“Sometimes the mind perceives the agreement or disagreement of two ideas immediately by themselves, without the intervention of any other; and this, I think, we may call intuitive knowledge. [...] Intuitive knowledge needs no probation, nor can have any, this being the highest of all human certainty.” (John Locke, “An Essay Concerning Human Understanding”, 1689)

“Truth is the object of our understanding, as good is of the will.” (John Dryden, [Preface to (C A du Fresnoy, “The Art of Painting”, 1668)] 1695)

On Metaphysics (1700-1799)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (Gottfried W Leibniz,  "Monadology", 1714)

“By the very nature of poetry it is impossible for everyone to be at the same time a sublime poet and a sublime metaphysician, for metaphysics abstracts the mind from the senses, and the poetic faculty must submerge the whole mind in the senses. Metaphysics soars up to universals, and the poetic faculty must plunge deep into particulars.” (Giambattista Vico, "The New Science”, 1725) 

“The topics of ontology, or metaphysic, are cause, effect, action, passion, identity, opposition, subject, adjunct, and sign.” (Isaac Watts, “Logic, or The right use of reason, in the inquiry after truth”, 1725)

“One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.” (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

“Thus, metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role.” (Jean d'Alembert, "Discours Preliminaire de L'Encyclopedie", 1751)

“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)

"That metaphysics has hitherto remained in so vacillating a state of uncertainty and contradiction, is only to be attributed to the fact, that this great problem, and perhaps even the difference between analytical and synthetical judgements, did not sooner suggest itself to philosophers. Upon the solution of this problem, or upon sufficient proof of the impossibility of synthetical knowledge a priori, depends the existence or downfall of metaphysics. (Immanuel Kant, "Critique of Pure Reason" , 1781)

“Metaphysical truths can only be established by producing effects from corresponding causes; and though we may confront such demonstrative evidence with the immutable laws of mathematical decision, we must be sensible that there will still remain some pretense for doubt; thus the basis of that knowledge, which on these principles we have been long labouring to accomplish, will become an endless toil, an endless force for controversy: and having the passions and the prejudices of mankind to combat, which mathematical certainty can alone effectually suppress, we must content ourselves only with making converts of those who have minds sufficiently expansive without the shackles of Euclid, and the vanity of displaying their own learning and pedantry.” (James Douglas, “A Dissertation on the Antiquity of the Earth”, 1785)

“All that can be said upon the number and nature of elements is, in my opinion, confined to discussions entirely of a metaphysical nature. The subject only furnishes us with indefinite problems, which may be solved in a thousand different ways, not one of which, in all probability, is consistent with nature.” (Antoine-Laurent Lavoisier, “Elements of Chemistry”, 1790)

“Nothing universal can be rationally affirmed on any moral or any political subject. Pure metaphysical abstraction does not belong to these matters. The lines of morality are not like the ideal lines of mathematics. They are broad and deep as well as long. They admit of exceptions; they demand modifications. These exceptions and modifications are not made by the process of logic, but by the rules of prudence. Prudence is not only the first in rank of the virtues political and moral, but she is the director, regulator, the standard of them all. Metaphysics cannot live without definition; but prudence is cautious how she defines.” (Edmund Burke, “Appeal from the New to the Old Whigs”, 1791)

Mental Models VI

“The intellectual capacity thinks the forms in the phantasmata (mental images) […] And for the following reason, as without having perceptual awareness no one could either learn or understand anything, so when one engages in intellectual activity one must at that time do so by means of a phantasma. For, phantasmata are just as perceptual states (aisthemata) are [in actual external perception] but without matter.” (Aristotle, “De Anima” III, cca. 350 BC)

“Nor is it enough to say that the intelligible notions formed by the active intellect subsist somehow in the phantasmata (mental image), which are certainly intrinsic to us; for as we have already observed in treating the passive intellect, objects only become actually intelligible when abstracted from phantasmata; so that merely by way of the phantasmata, we cannot attribute the work of the active intellect to ourselves” (St. Thomas Aquinas, “De Anima” III, cca. 1268) [On Aristotle's phantasmata]

“From all this I am beginning to have a rather better understanding of what I am. But it still appears - and I cannot stop thinking this - that the corporeal things of which images are formed in my thought, and which the senses investigate, are known with much more distinctness than this puzzling 'I' which cannot be pictured in the imagination.” (René Descartes, “Meditations” II, 1641)

“For after the object is removed, or the eye shut, we still retain an image of the thing seen, though more obscure than when we see it. And this is it the Latins call imagination, from the image made in seeing, and apply the same, though improperly, to all the other senses. But the Greeks call it fancy, which signifies appearance, and is as proper to one sense as to another. IMAGINATION, therefore, is nothing but decaying sense; and is found in men and many other living creatures, as well sleeping as waking.” (Thomas Hobbes, “Leviathan: The Matter, Form and Power of a Commonwealth  Ecclesiastical and Civil”, 1651)

“It is also only by virtue of the continual action of God upon us that we have in our soul the ideas of all things; that is to say, since every effect expresses its cause, the essence of our soul is a certain expression, imitation or image of the divine essence, thought, and will and of all the ideas which are comprised in God.“ (Gottfried W Leibniz, “Discourse on Metaphysics”, 1686)

“As in a block of marble all possible figures are potentially contained in it, and can be drawn out of it by the movement or by the action of the chisel, so in the same way all intelligible figures are potentially in intelligible extension and are discovered in it according to the different ways in which this extension is represented to the mind, as a consequence of the general laws which God has established according to which he continuously acts in us. “ (Nicolas Malebranche , “Dialogues On Metaphysics And Religion”, 1688)

“But to form the idea of an object, and to form an idea simply is the same thing; the reference of the idea to an object being an extraneous denomination, of which in itself it bears no mark or character. Now as it is impossible to form an idea of an object, that is possessed of quantity and quality, and yet is possessed of no precise degree of either; it follows, that there is an equal impossibility of forming an idea, that is not limited and confined in both these particulars. Abstract ideas are therefore in themselves individual, however they may become general in their representation. The image in the mind is only that of a particular object, though the application of it in our reasoning be the same, as if it were universal.” (David Hume, “Treatise of Human Nature”, 1738)

“While all that we have is a relation of phenomena, a mental image, as such, in juxtaposition with or soldered to a sensation, we can not as yet have assertion or denial, a truth or a falsehood. We have mere reality, which is, but does not stand for anything, and which exists, but by no possibility could be true. […] the image is not a symbol or idea. It is itself a fact, or else the facts eject it. The real, as it appears to us in perception, connects the ideal suggestion with itself, or simply expels it from the world of reality. […] you possess explicit symbols all of which are universal and on the other side you have a mind which consists of mere individual impressions and images, grouped by the laws of a mechanical attraction.” (Francis H Bradley, "Principles of Logic", 1883)

“Memory-images, it is true, cannot be directly aroused through external sense impressions, but follow them after a longer or shorter interval. Still, it is obvious that their attributes, and especially their relation to the primary ideas through direct impressions, can be most accurately be learned, not by waiting for their chance arrival, but by using such memory-ideas as may be aroused in a systematic, experimental way, through immediately preceding impressions.” (Wilhelm M Wundt, “Outlines of Psychology”, 1897)

“In perception, a knowledge forms itself slowly; in the [mental] image the knowledge is immediate. We see now that the image is a synthetic act which unites a concrete, nonimagined, knowledge to elements which are more actually representative. The image teaches nothing: it is organized exactly like the objects which do produce knowledge, but it is complete at the very moment of its appearance. […] Thus, the object presents itself in the image as having to be apprehended in a multiplicity of synthetic acts. Due to this fact, and because its content retains a sensible opacity, like a phantom, because it does not involve either essences or generating laws but only an irrational quality, it gives the impression of being an object of observation: from this point of view the image appears to be more like a perception than a concept.” (Jean-Paul Sartre, “The Psychology of Imagination”, 1940)

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

The Arithmetic behind Numbers

"[Arithmetic] has a very great and elevating effect, compelling the soul to reason about abstract numbers, and rebelling against the introduction of visible or tangible objects into the argument." (Plato)

 “Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.” (James J Sylvester, “A Probationary Lecture on Geometry”, 1844)

"The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods; the gods are there, behind the wall, at play with numbers." (Le Corbusier)

“I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetical act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding […]” (Richard Dedekind, “On Continuity and Irrational Numbers”, 1872)

“Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.” (Bertrand Russell, “The Principles of Mathematics”, 1903)

“Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them.” (Henri Poincaré, “Annual Report of the Board of Regents of the Smithsonian Institution”, 1909)

“The way to enable a student to apprehend the instrumental value of arithmetic is not to lecture him on the benefit it will be to him in some remote and uncertain future, but to let him discover that success in something he is interested in doing depends on ability to use numbers.” (John Dewey, “Democracy and Education: An Introduction to the Philosophy of Education”, 1916)

“In other words, without a theory, a plan, the mere mechanical manipulation of the numbers in a problem does not necessarily make sense just because you are using Arithmetic!” (Lillian R Lieber, “The Education of T.C. MITS”, 1944)

“For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.” (David Hilbert, “Anschauliche Geometrie”, 1932)

“Arithmetic, then, means dealing logically with certain facts that we know, about numbers, with a view to arriving at knowledge which as yet we do not possess.” (Anonymous)

On Numbers: The Power of Numbers II

“All things which can be known have number; for it is not possible that without number anything can be either conceived or known.” (Philolaus)

“Whenever a man can get hold of numbers, they are invaluable: if correct, they assist in informing his own mind, but they are still more useful in deluding the minds of others. Numbers are the masters of the weak, but the slaves of the strong.” (Charles Babbage, “Passages From the Life of a Philosopher”, 1864)

“Although he may not always recognize his bondage, modern man lives under a tyranny of numbers.” (Nicholas Eberstadt)

“Words and numbers are of equal value, for, in the cloak of knowledge, one is warp and the other woof. It is no more important to count the sands than it is to name the stars.” (Norton Juster, “The Phantom Tollbooth”, 1989)

“Numbers, in fact, are the atoms of the universe, combining with everything else.” (Calvin C Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers”, 1996)

“There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations.” (Auguste Comte)

“Numbers have undoubted powers to beguile and benumb, but critics must probe behind numbers to the character of arguments and the biases that motivate them.” (Stephen Jay Gould, “An Urchin in the Storm: Essays About Books and Ideas”, 1987)

“[…] a single number has more genuine and permanent value than an expensive library full of hypotheses.” (Robert Mayer, [Letter to Griesinger], 1844)

“Numbers have souls, and you can’t help but get involved with them in a personal way.” (Paul Auster, “The Music of Chance”, 1990)

“You can be moved to tears by numbers - provided they are encoded and decoded fast enough.” (Richard Dawkins, “River Out of Eden: A Darwinian View of Life”, 1995)

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On Numbers: The Power of Numbers I

"The qualities of number appear to lead to the apprehension of truth." (Plato)

"Wherever there is number, there is beauty." (Diadochus Proclus)

"All was numbers." (Pythagoras)

"Number is the ruler of forms and ideas, and the cause of gods and demons." (Pythagoras)

"Number was the substance of all things." (Pythagoras)

"Number rules the universe." (Pythagoras)

"Take from all things their number and all shall perish." (Saint Isidore of Seville)

"The God that reigns in Olympus is Number Eternal." (Carl Gustav Jacobi)

"All mathematical forms have a primary subsistence in the soul so that prior to the sensible she contains self-motive numbers." (Thomas Taylor)

"Numbers are the sources of form and energy in the world. They are dynamic and active even among themselves […] almost human in their capacity for mutual influence." (Theon of Symyma)

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On Mind: Mathematics upon Mind

"This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth." (Proclus Lycaeus, cca 5th century)

 “If a man’s wit be wandering, let him study mathematics; for in demonstrations, if his wit be called away never so little, he must begin again.” (Francis Bacon)

"Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition. This it does in two ways: 1st, by accustoming us to examine, and not to take thigs upon trust. 2nd By giving us a clear and extensive knowledge of the system of the world […]." (Dr. John Arbuthnot, “Usefulness of Mathematical Learning”, 1745)

“Mathematics is the science that yields the best opportunity to observe the working of the mind. Its study is the best training of our abilities as it develops both the power and the precision of our thinking. Mathematics is valuable on account of the number and variety of its applications. And it is equally valuable in another respect: By cultivating it, we acquire the habit of a method of reasoning which can be applied afterwards to the study of any subject and can guide us in life's great and little problems.” (Marquis de Condorcet)

"Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality." (Richard Courant & Herbert Robbins, “What Is Mathematics?”, 1941)

"For me mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos." (Clifford A Pickover, “The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics”, 2009)

"I shall not attempt to prove that mathematics is useful. I will admit it and so save myself the trouble that here is a great and respected discipline where all is impossible yet much is useful. The usefulness largely flows from the impossibility. Mathematical concepts have been simplified and generalized until they describe an imaginative world no part of which could possibly exist outside men’s minds." (Billy E Goetz, “The Usefulness of the Impossible”, 1963)

“Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics.” (Johann F Herbart)

"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline, “Mathematics and the Physical World”, 1959)

“Mathematical inquiry lifts the human mind into closer proximity with the divine than is attainable through any other medium.” (Hermann Weyl)

"Mathematics is a spirit of rationality. It is this spirit that challenges, simulates, invigorates and drives human minds to exercise themselves to the fullest. It is this spirit that seeks to influence decisively the physical, normal and social life of man, that seeks to answer the problems posed by our very existence, that strives to understand and control nature and that exerts itself to explore and establish the deepest and utmost implications of knowledge already obtained." (Morris Kline)