🪷On Mind: On Perception (1960-1969)

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

"Environments are invisible. Their ground-rules, pervasive structure, and overall patterns elude easy perception." (Marshall McLuhan, "Understanding Media", 1964)

"The direct task of our cognitive apparatus is the formation of invariants, and namely invariants of perception, thinking and motives. […] Still, in the first place, exactly these invariants are, cybernetically speaking, enabling a stabilization of the relationship between humans and their environment and between humans themselves." (Georg Klaus, "Kybernetik und Erkenntnistheorie" ["Cybernetics and Epistemology"], 1966)

"A theory has a physical meaning iff it contains interpretation assumptions that assign physical correlates to its basic concepts. These correlates (referents) need not be and in general are not perceptible. But they must be scrutable, i.e. they must show up as observable effects even if remotely, i. e. at the end of a long chain accounted for by other theories."(Mario Bunge, "Foundations of Physics", 1967)

"Basically, self-regulation requires a functional distinction between perception, decision-making, and action. This is normally achieved by a structural distinction between perceptor elements, control elements and effector elements in the system. Behaviorally, a system may be defined as a “black box” characterized by a given set or range of inputs and outputs. Adequate knowledge of any system requires both structural-functional analysis and behavioral analysis." (Charles R Decher, "The Development of Cybernetics" [in "The Social Impact of Cybernetics", 1967)

"Graphic representation constitutes one of the basic sign-systems conceived by the human mind for the purposes of storing, understanding, and communicating essential information. As a "language" for the eye, graphics benefits from the ubiquitous properties of visual perception. As a monosemic system, it forms the rational part of the world of images. […] Graphics owes its special significance to its double function as a storage mechanism and a research instrument." (Jacques Bertin, Semiology of graphics [Semiologie Graphique], 1967)

🪷On Mind: On Perception (1970-1979)

"All the model-boundedness of human encounter with the world as well as the interhuman communication is equally involved already in the modeling character of the perception process. […] The modeling character of perception forms is also not changed by the circumstance that the access to the original-side, i.e. to the properties of signal constellations from the external world, it is only always possible via the building of internal models of the external world." (Herbert Stachowiak, "Allgemeine Modelltheorie", 1973)"

"In the absolute universe all events can be regarded as absolutely deterministic, and if we can’t perceive the greater structures, it’s because our vision is faulty. If we had a real grasp of causality down to the molecular level, we wouldn’t need to rely on mathematical approximations, on statistics and probabilities, in making predictions. If our perceptions of cause and effect were only good enough, we’d be able to attain absolute knowledge of what is to come. We would make ourselves all-seeing." (Robert Silverberg, "The Stochastic Man", 1975)

"If mankind were to continue in other than the present barbarism, a new path must be found, a new civilization based on some other method than technology. Space is an illusion, and time as well. There is no such factor as either time or space. We have been blinded by our own cleverness, blinded by false perceptions of those qualities that we term eternity and infinity. There is another factor that explains it all, and once this universal factor is recognized, everything grows simple. There is no longer any mystery, no longer any wonder, no longer any doubt; for the simplicity of it all lies before us [...]" (Clifford D Simak,"A Heritage of Stars", 1977)

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

"Science sometimes improves hypothesis and sometimes disproves them. But proof would be another matter and perhaps never occurs except in the realms of totally abstract tautology. We can sometimes say that if such and such abstract suppositions or postulates are given, then such and such abstract suppositions or postulates are given, then such and such must follow absolutely. But the truth about what can be perceived or arrived at by induction from perception is something else again." (Gregory Bateson, "Mind and Nature, a Necessary Unity", 1979)

🪷On Mind: On Perception (1910-1919)

"The organized activity of the logical function draws into itself all the sensations and constructs an inner world of its own, which progressively departs from reality but yet at certain points still retains so intimate a connection with it that transitions from one to the other continually take place and we hardly notice that we are acting on a double stage - our own inner world (which, of course, we objectify as the world of sense-perception) and also an entirely different and external world." (Hans Vaihinger, "The Philosophy of 'As if': A System of the Theoretical, Practical and Religious Fictions of Mankind", 1911)

"On the other side of the subject, Geometry is an abstract rational Science which deals with the relations of objects that are no longer physical objects, although these ideal objects, points, straight lines, circles, &c., are called by the same names by which we denote their physical counterparts. At the base of this rational Science there lies a set of definitions and postulations which specify the nature of the relations between the ideal objects with which the Science deals. These postulations and definitions were suggested by our actual spatial perceptions, but they contain an element of absolute exactness which is wanting in the rough data provided by our senses. The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry." (Ernest W Hobson, "Squaring the circle", 1913)

"Would it be possible for a 'mental image' , perception or idea, to correspond to a 'physical object', if the parts of the former were not combined in the same order as the parts of the latter? […] The more fully the similarity of two mental images is 'recognized' , i.e., the more elements of both images are brought to identity in the consciousness, the greater the extent they are associated 'by similarity'." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"There is an intellectual function in us which demands unity, connection and intelligibility from any material, whether of perception or thought, that comes within its grasp; and if, as a result of special circumstances, it is unable to establish a true connection, it does not hesitate to fabricate a false one." (Sigmund Freud, 1913-1914)"

"Every great revelation is likely to have been foreshadowed in more or less imperfect forms, so as to prepare our minds and make ready the way for complete perception hereafter. It is probable that the human race is quite incompetent to receive a really great idea the first time it is offered. " (Oliver J Lodge, "Raymond, or Life and Death", 1916)

"But instinct is something which transcends knowledge. We have, undoubtedly, certain finer fibers that enable us to perceive truths when logical deduction, or any other willful effort of the brain, is futile." (Nikola Tesla, "My Inventions", 1919)

🪷On Mind: On Perception (1940-1949)

"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

"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)

"Classes and concepts may, however, also be conceived as real objects, namely classes as 'pluralities of things' or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions." (Kurt Gödel, "The Philosophy of Bertrand Russell", 1944)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

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

"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)

🪷On Mind: On Perception (1930-1939)

"Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations." (Tobias Dantzig, "Number: The Language of Science", 1930)

"Modern Physics impresses us particularly with the truth of the old doctrine which teaches that there are realities existing apart from our sense-perceptions, and that there are problems and conflicts where these realities are of greater value for us than the richest treasures of the world of experience." (Max Planck, "The Universe in the Light of Modern Physics", 1931)

"Physics is an exact Science and hence depends upon measurement, while all measurement itself requires sense-perception. Consequently all the ideas employed in Physics are derived from the world of sense-perception." (Max Planck, "The Universe in the Light of Modern Physics", 1931)

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

"Physics is an exact Science and hence depends upon measurement, while all measurement itself requires sense-perception. Consequently all the ideas employed in Physics are derived from the world of sense-perception." (Max Planck, "The Universe in the Light of Modern Physics", 1937)

"[…] mathematics is like music, freely exploring the possibilities of form. And yet, notoriously, mathematics holds true of things; hugs and permeates them far more closely than does confused and inconstant human perception; so that the dream of many exasperated critics of human error has been to assimilate all science to mathematics, so as to make knowledge safe by making it, as Locke wished, direct perception of the relations between ideas […]" (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)

"[…] science conceived as resting on mere sense-perception, with no other source of observation, is bankrupt, so far as concerns its claim to self-sufficiency. Science can find no individual enjoyment in nature: Science can find no aim in nature: Science can find no creativity in nature; it finds mere rules of succession. These negations are true of Natural Science. They are inherent in it methodology." (Alfred N Whitehead, "Modes of Thought", 1938)

🪷On Mind: On Perception (1920-1929)

"By intuition is frequently understood perception, or the knowledge of actual reality, the apprehension of something as real. […] Intuition is the undifferentiated unity of the perception of the real and of the simple image of the possible. " (Benedetto Croce, "The Essence of Æsthetic", 1921)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "The Principle of Relativity: A Collection of Original Memoirs on the, Special and General Theory of Relativity, Space and Time", Nature vol. 113, 1924)

"When a man of science speaks of his 'data', he knows very well in practice what he means. Certain experiments have been conducted, and have yielded certain observed results, which have been recorded. But when we try to define a 'datum' theoretically, the task is not altogether easy. A datum, obviously, must be a fact known by perception. But it is very difficult to arrive at a fact in which there is no element of inference, and yet it would seem improper to call something a 'datum' if it involved inferences as well as observation. This constitutes a problem [...]" (Bertrand Russell, The Analysis of Matter", 1927)

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

"Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be intuitively understood," these systems consist of nothing but 'logical relations' or 'artificial manifolds'. They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The different premises will possibly allow equally good explanations with respect to the imprecise nature of our sensory perception, because the sensory perception is, in fact, not concerned with issues of precision mathematics but of approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"When we look at a very large number of small objects that are close together the idea of continuum arises within us. […] Even if we believe to perceive a continuum in front of us, a more accurate observation will often convince us that we are only observing a dense succession of small particles." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

🪷On Mind: On Perception (1900-1909)

"All our thinking is performed upon signs of some kind or other, either imagined or actually perceived. The best thinking, especially on mathematical subjects, is done by experimenting in the imagination upon a diagram or other scheme, and it facilitates the thought to have it before one’s eyes." (Charles S Peirce, “The Principles of Mathematics”, cca. 1902)

"The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results." (Josiah W Gibbs, "Elementary Principles in Statistical Mechanics", 1902)

"The most ordinary things are to philosophy a source of insoluble puzzles. In order to explain our perceptions it constructs the concept of matter and then finds matter quite useless either for itself having or for causing perceptions in a mind. With infinite ingenuity it constructs a concept of space or time and then finds it absolutely impossible that there be objects in this space or that processes occur during this time [...] The source of this kind of logic lies in excessive confidence in the so-called laws of thought." (Ludwig E Boltzmann, "On Statistical Mechanics", 1904)

"Synergy is the principle that explains all organization and creates all structures. The products of cosmic synergy are found in all fields of phenomena. Celestial structures are worlds and world systems; chemical structures are atoms, molecules, and substances; biotic structures are protoplasm, cells, tissues, organs, and organisms. There are also psychic structures - feelings, emotions, passions, volitions, perceptions, cognitions, memory, imagination, reason, thought, and all the acts of consciousness. And then there are social structures […]. These are the products of the social forces acting under the principle of social synergy." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

"To know each other we must reach beyond the sphere of our sense perceptions." (Nikola Tesla, "Electrical World and Engineer", 1905)

"Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey - having been thus constructed - an intuitive perception of the significance of numbers being out of the question." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1908)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "Space and Time", [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

🪷On Mind: On Perception (2010-2019)

"Perception and memory are imprecise filters of information, and the way in which information is presented, that is, the frame, influences how it is received. Because too much information is difficult to deal with, people have developed shortcuts or heuristics in order to come up with reasonable decisions. Unfortunately, sometimes these heuristics lead to bias, especially when used outside their natural domains." (Lucy F Ackert & Richard Deaves, "Behavioral Finance: Psychology, Decision-Making, and Markets", 2010)

"Natural science has discovered 'chaos'. Social science has encountered 'complexity'. But chaos and complexity are not characteristics of our new reality; they are features of our perceptions and understanding. We see the world as increasingly more complex and chaotic because we use inadequate concepts to explain it. When we understand something, we no longer see it as chaotic or complex. Maybe playing the new game requires learning a new language." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Once a myth becomes established, it forms part of our mental model of the world and alters our perception, the way our brains interpret the fleeting patterns our eyes pick up." (Jeremy Wade, "River Monsters: True Stories of the Ones that Didn't Get Away", 2011)

"Although good ethical decision-making requires us carefully to take into account as much relevant information as is available to us, we have good reason to think that we commonly fall well short of this standard – either by overlooking relevant facts completely or by underestimating their significance. The mental models we employ can contribute to this problem. As we have explained, mental models frame our experiences in ways that both aid and hinder our perceptions. They enable us to focus selectively on ethically relevant matters. By their very nature, they provide incomplete perspectives, resulting in bounded awareness and bounded ethicality. Insofar as our mental modeling practices result in unwarranted partiality, or even ethical blindness, the desired reflective process is distorted. This distortion is aggravated by the fact that our mental models can have this distorting effect without our consciously realizing it. Thus, although we cannot do without mental models, they leave us all vulnerable to blindness and, insofar as we are unaware of this, self-deception." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)

"Even more important is the way complex systems seem to strike a balance between the need for order and the imperative for change. Complex systems tend to locate themselves at a place we call 'the edge of chaos'. We imagine the edge of chaos as a place where there is enough innovation to keep a living system vibrant, and enough stability to keep it from collapsing into anarchy. It is a zone of conflict and upheaval, where the old and new are constantly at war. Finding the balance point must be a delicate matter - if a living system drifts too close, it risks falling over into incoherence and dissolution; but if the system moves too far away from the edge, it becomes rigid, frozen, totalitarian. Both conditions lead to extinction. […] Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

"Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

"There is no linear additive process that, if all the parts are taken together, can be understood to create the total system that occurs at the moment of self-organization; it is not a quantity that comes into being. It is not predictable in its shape or subsequent behavior or its subsequent qualities. There is a nonlinear quality that comes into being at the moment of synchronicity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

"Our assumptions are un question ably interconnected. They are nodes with connections" (edges) to other nodes. The more foundational the assumption, the more strongly connected it is. What I’m suggesting is that our assumptions and the highly sensitive network of responses, perceptions, behaviors, thoughts, and ideas they create and interact with are a complex system. One of the most basic features of such a network is that when you move or disrupt one thing that is strongly connected, you don’t just affect that one thing, you affect all the other things that are connected to it. Hence small causes can have massive effects" (but they don’t have to, and usually don’t actually). In a system of high tension, simple questions targeting basic assumptions have the potential to transform perception in radical and unpredictable ways." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

🪷On Mind: On Perception (2000-2009)

"Bounded rationality simultaneously constrains the complexity of our cognitive maps and our ability to use them to anticipate the system dynamics. Mental models in which the world is seen as a sequence of events and in which feedback, nonlinearity, time delays, and multiple consequences are lacking lead to poor performance when these elements of dynamic complexity are present. Dysfunction in complex systems can arise from the misperception of the feedback structure of the environment. But rich mental models that capture these sources of complexity cannot be used reliably to understand the dynamics. Dysfunction in complex systems can arise from faulty mental simulation-the misperception of feedback dynamics. These two different bounds on rationality must both be overcome for effective learning to occur. Perfect mental models without a simulation capability yield little insight; a calculus for reliable inferences about dynamics yields systematically erroneous results when applied to simplistic models." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Just as dynamics arise from feedback, so too all learning depends on feedback. We make decisions that alter the real world; we gather information feedback about the real world, and using the new information we revise our understanding of the world and the decisions we make to bring our perception of the state of the system closer to our goals.( John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

"Of course, the information systems governing the feedback we receive can change as we learn. They are part of the feedback structure of our systems. Through our mental models we define constructs such as GDP or scientific research, create metrics for these ideas, and design information systems to evaluate and report them. These then condition the perceptions we form. Changes in our mental models are constrained by what we previously chose to define, measure, and attend to. Seeing is believing and believing is seeing. They feed back on one another." " (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making."(John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"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)

"The self-reinforcing feedback between expectations and perceptions has been repeatedly demonstrated […]. Sometimes the positive feedback assists learning by sharpening our ability to perceive features of the environment, as when an experienced naturalist identifies a bird in a distant bush where the novice sees only a tangled thicket. Often, however, the mutual feedback of expectations and perception blinds us to the anomalies that might challenge our mental models and lead to deep insight." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"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)

"People talk far too glibly about 'recognizing' things and then build machines that simply label patterns. There is a vast difference between recognizing patterns by labeling them correctly and knowing the objects that are perceived. Such knowledge is a happy resonance between imagination and perception, possessed neither by WISARD nor by the many neural pattern-recognition machines built over the last fifteen or so years. Something extra is required: yes, inner states are necessary, but they cannot be just any old inner states." (Igor Aleksander, "How to Build a Mind: toward machines with imagination", 2001)

"If the intensity of the material world is plotted along the horizontal axis, and the response of the human mind is on the vertical, the relation between the two is represented by the logarithmic curve. Could this rule provide a clue to the relationship between the objective measure of information, and our subjective perception of it?" (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"Knowledge is encoded in models. Models are synthetic sets of rules, and pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)

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

"Metaphor is evidence of the human ability to visualize the universe as a coherent organism. Proof of our capacity, not just to see one thing in another but to change the very nature of things. When a metaphor is accepted as fact, it enters groupthink, taking on an existence in the real world. [...] Metaphor is the default form of thought, providing many angles from which to literally 'see' the world." (Marcel Danesi, "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture", 2004)"

"The most familiar example of swarm intelligence is the human brain. Memory, perception and thought all arise out of the nett actions of billions of individual neurons. As we saw earlier, artificial neural networks" (ANNs) try to mimic this idea. Signals from the outside world enter via an input layer of neurons. These pass the signal through a series of hidden layers, until the result emerges from an output layer. Each neuron modifies the signal in some simple way. It might, for instance, convert the inputs by plugging them into a polynomial, or some other simple function. Also, the network can learn by modifying the strength of the connections between neurons in different layers." (David G Green, "The Serendipity Machine: A voyage of discovery through the unexpected world of computers", 2004)

"[...] when data is presented in certain ways, the patterns can be readily perceived. If we can understand how perception works, our knowledge can be translated into rules for displaying information. Following perception‐based rules, we can present our data in such a way that the important and informative patterns stand out. If we disobey the rules, our data will be incomprehensible or misleading." (Colin Ware, "Information Visualization: Perception for Design" 2nd Ed., 2004)

"We must begin by distinguishing between visual mental imagery and visual perception: Visual perception occurs while a stimulus is being viewed, and includes functions such as visual recognition" (i. e., registering that a stimulus is familiar) and identification" (i. e., recalling the name, context, or other information associated with the object). Two types of mechanisms are used in visual perception: ‘bottom-up’ mechanisms are driven by the input from the eyes; in contrast, ‘top-down’ mechanisms make use of stored information" (such as knowledge, belief, expectations, and goals). Visual mental imagery is a set of representations that gives rise to the experience of viewing a stimulus in the absence of appropriate sensory input. In this case, information in memory underlies the internal events that produce the experience. Unlike afterimages, mental images are relatively prolonged." (Stephen M Kosslyn, "Mental images and the brain", Cognitive Neuropsychology 22, 2005)

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

"Most dashboards fail to communicate efficiently and effectively, not because of inadequate technology" (at least not primarily), but because of poorly designed implementations. No matter how great the technology, a dashboard's success as a medium of communication is a product of design, a result of a display that speaks clearly and immediately. Dashboards can tap into the tremendous power of visual perception to communicate, but only if those who implement them understand visual perception and apply that understanding through design principles and practices that are aligned with the way people see and think." (Stephen Few, "Information Dashboard Design", 2006)

"Our inner working models, therefore, function as interpretation schemes, on the basis of which we organize our experiences. But, such schemes also distort reality in the direction of our pattern of expectations. In short: such working models organize and screen our experiences. This means that such an inner working model organizes and colours our perception of things in such a way that it can be extremely stimulating but can also sometimes slow us down considerably." (M H M de Wolf,"Freud and Mahler", 2007)

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

"Our inner working models, therefore, function as interpretation schemes, on the basis of which we organize our experiences. But, such schemes also distort reality in the direction of our pattern of expectations. In short: such working models organize and screen our experiences. This means that such an inner working model organizes and colours our perception of things in such a way that it can be extremely stimulating but can also sometimes slow us down considerably." (M H M de Wolf, "Freud and Mahler", 2007)

"[…] mathematics is not only an essential tool for science and technology, but also for humanities, in particular for art. And out of art we may say that mathematics gains one of its main reasons for developing and changing in time. Mathematics contributes to our way of conceiving and shaping the world we live in, while art develops the means to harmonize, describe, represent aesthetically - or even to transcend and transfigure - the world of our sensations and perception." (Mauro Francaviglia,"Art and Mathematics", 2008)

"Nonetheless, some hesitation persisted. After all, the very word imaginary betrays ambivalence, and suggests that in our heart of hearts we do not believe these numbers exist. On the other hand, by calling every number representable by a decimal expansion real, we are making the psychological distinction more stark. Indeed the adjective imaginary is a somewhat unfortunate one - although an intriguing name, some students’ perceptions are so colored by the word that they consequently fail to come to grips with the idea." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Perception requires imagination because the data people encounter in their lives are never complete and always equivocal." (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008) 

"Gaining awareness about how the system is built up and how it works can also help us to avoid solutions that only treat the symptoms of an underlying problem without curing the problem itself. System thinking is powerful because it helps us to see our own mental models and how these models color our perception of the world. In many cases, it is difficult for us to alter our mental models. There are always some beliefs or viewpoints that we are not willing to change, no matter what evidence is presented against it. This causes a certain resistance to new concepts. Problems can occur, however, when a rigid mental model stands in the way of a solution that might solve a problem. In such situations, adherence to mental models can be dangerous to the health of the organization." (Akhilesh Bajaj & Stanisław Wrycza, "Systems Analysis and Design for Advanced Modeling Methods: Best Practices", 2009)

"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)

"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)

"[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)

"Pattern perception" (that is, the perception of similarities in spatial or temporal configurations) has a fundamental role in playing chess [...] The two essential components in decision making in chess are recognizing patterns stored in long-term memory" (which requires an exhaustive knowledge database) and searching for a solution within the problem space. The first component uses perception and long-term memory, and the second leans mainly on the calculation of variations, which in turn has its foundations in logical reasoning." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The two essential components in decision making in chess are recognizing patterns stored in long-term memory" (which requires an exhaustive knowledge database) and searching for a solution within the problem space. The first component uses perception and long-term memory, and the second leans mainly on the calculation of variations, which in turn has its foundations in logical reasoning." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

🪷On Mind: On Perception (1990-1999)

"Systems philosophy brings forth a reorganization of ways of thinking. It creates a new worldview, a new paradigm of perception and explanation, which is manifested in integration, holistic thinking, purpose-seeking, mutual causality, and process-focused inquiry." (Béla H. Bánáthy, Systems Design of Education", 1991)

"Homeomorphism is one of the basic concepts in topology. Homeomorphism, along with the whole topology, is in a sense the basis of spatial perception. When we look at an object, we see, say, a telephone receiver or a ring-shaped roll and first of all pay attention to the geometrical shape" (although we do not concentrate on it specially) - an oblong figure thickened at the ends or a round rim with a large hole in the middle. Even if we deliberately concentrate on the shape of the object and forget about its practical application, we do not yet 'see' the essence of the shape. The point is that oblongness, roundness, etc. are metric properties of the object. The topology of the form lies 'beyond them'." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"Memory is a reenactment of perception, indistinguishable from the original act of knowing. " (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"A metaphor is not an ornament. It is an organ of perception. Through metaphors, we see the world as one thing or another." (Neil Postman,"The End of Education: Redefining the Value of School", 1996)

"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 shift of paradigms requires an expansion not only of our perceptions and ways of thinking, but also of our values. […] scientific facts emerge out of an entire constellation of human perceptions, values, and actions-in one word, out of a paradigm-from which they cannot be separated. […] Today the paradigm shift in science, at its deepest level, implies a shift from physics to the life sciences." (Fritjof Capra, "The Web of Life", 1996)

"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking—from the parts to the whole, from objects to relationships, from contents to patterns. [...] Nourishing the community means nourishing those relationships." (Fritjof Capra, The Web of Life", 1996)

"Mathematics can function as a telescope, a microscope, a sieve for sorting out the signal from the noise, a template for pattern perception, a way of seeking and validating truth. […] A knowledge of the mathematics behind our ideas can help us to fool ourselves a little less often, with less drastic consequences." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

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

"Imagining the unseeable is hard, because imagining means having an image in your mind. And how can you have a mental image of something you have never seen? Like perception itself, the models of science are embedded inextricably in the current worldview we call culture." (K C Cole, "First You Build a Cloud and Other Reflections on Physics as a Way of Life", 1999)

"In science, it is a long-standing tradition to deal with perceptions by converting them into measurements. But what is becoming increasingly evident is that, to a much greater extent than is generally recognized, conversion of perceptions into measurements is infeasible, unrealistic or counter-productive. With the vast computational power at our command, what is becoming feasible is a counter-traditional move from measurements to perceptions. […] To be able to compute with perceptions it is necessary to have a means of representing their meaning in a way that lends itself to computation." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

"Knowledge is encoded in models. Models are synthetic sets of rules, pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns. As argued by philosophers and shown by scientists, we do not have access to 'reality', only to some of its manifestations, whose regularities are used to determine rules, which when widely applicable become 'laws of nature'. These laws are constantly tested in the scientific march, and they evolve, develop and transmute as the frontier of knowledge recedes further away. " (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

"Randomness is the very stuff of life, looming large in our everyday experience. […] The fascination of randomness is that it is pervasive, providing the surprising coincidences, bizarre luck, and unexpected twists that color our perception of everyday events." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"That randomness gives rise to innovation and diversity in nature is echoed by the notion that chance is also the source of invention in the arts and everyday affairs in which naturally occurring processes are balanced between tight organization, where redundancy is paramount, and volatility, in which little order is possible. One can argue that there is a difference in kind between the unconscious, and sometimes conscious, choices made by a writer or artist in creating a string of words or musical notes and the accidental succession of events taking place in the natural world. However, it is the perception of ambiguity in a string that matters, and not the process that generated it, whether it be man-made or from nature at large." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"The abstractions of science are stereotypes, as two-dimensional and as potentially misleading as everyday stereotypes. And yet they are as necessary to the process of understanding as filtering is to the process of perception." (K C Cole, "First You Build a Cloud and Other Reflections on Physics as a Way of Life", 1999)

🪷On Mind: On Perception (1980-1989)

"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)

"A mental image occurs when a representation of the type created during the initial phases of perception is present but the stimulus is not actually being perceived; such representations preserve the perceptible properties of the stimulus and ultimately give rise to the subjective experience of perception." (Stephen M Kosslyn, "Image and Mind", 1980)

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

"That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects" (at a later stage, of marks on paper or even mental images). As a result of this experience, there is something" (a trace, an effect) in the pupil's mind which is his representation of the integers. But his representation is equivalent to mine, in the sense that we both get die same answer to any question you ask - or if we get different answers, we can compare notes and figure out what's right. We do this, not because we have been taught a set of algebraic rules, but because our mental pictures match each other." (Philip J Davis & Reuben Hersh, "The Mathematical Experience", 1981)

"Real progress in understanding nature is rarely incremental. All important advances are sudden intuitions, new principles, new ways of seeing." (Marilyn Ferguson,"The Aquarian Conspiracy: Personal and Social Transformation in the 1980s", 1980)

"That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects" (at a later stage, of marks on paper or even mental images)." (Philip J Davis & Reuben Hersh,"The Mathematical Experience", 1981)

"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)

"The thing and the perception of the thing are one amid the same thing, one and the same image, but related in one or other of two systems of reference. The thing is the image as it is in itself, as it is related to all the other images to whose action it completely submits and on which it reacts immediately. But the perception of the thing is the same image related to another special image which frames it, and which only retains a partial action from it, and only reacts to it mediately." (Gilles Deleuze, "Cinema 1: The Movement Image", 1983)

"Effect spreads its 'tentacles' not only forwards" (as a new cause giving rise to a new effect) but also backwards, to the cause which gave rise to it, thus modifying, exhausting or intensifying its force. This interaction of cause and effect is known as the principle of feedback. It operates everywhere, particularly in all self-organising systems where perception, storing, processing and use of information take place, as for example, in the organism, in a cybernetic device, and in society. The stability, control and progress of a system are inconceivable without feedback." (Alexander Spirkin, "Dialectical Materialism", 1983)

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

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline,"Mathematics and the Search for Knowledge", 1985)

"That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects" (at a later stage, of marks on paper or even mental images)." (Philip J Davis & Reuben Hersh,"The Mathematical Experience", 1985)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"Science has become a social method of inquiring into natural phenomena, making intuitive and systematic explorations of laws which are formulated by observing nature, and then rigorously testing their accuracy in the form of predictions. The results are then stored as written or mathematical records which are copied and disseminated to others, both within and beyond any given generation. As a sort of synergetic, rigorously regulated group perception, the collective enterprise of science far transcends the activity within an individual brain." (Lynn Margulis & Dorion Sagan, "Microcosmos", 1986)

"Although science literally means ‘knowledge’, the scientific attitude is concerned much more with rational perception through the mind and with testing such perceptions against actual fact, in the form of experiments and observations." (David Bohm & F David Peat, "Science, Order, and Creativity", 1987)

"[...] to acknowledge the subjectivity inherent in the interpretation of data is to recognize the central role of statistical analysis as a formal mechanism by which new evidence can be integrated with existing knowledge. Such a view of statistics as a dynamic discipline is far from the common perception of a rather dry, automatic technology for processing data." (Donald A Berry, "Statistical Analysis and the Illusion of Objectivity", American Scientist Vol. 76, 1988) 

"The common perception of science as a rational activity, in which one confronts the evidence of fact with an open mind, could not be more false. Facts assume significance only within a pre-existing intellectual structure, which may be based as much on intuition and prejudice as on reason." (Walter Gratzer, The Guardian, 1989)

"The more aware we are of our basic paradigms, maps, or assumptions, and the extent to which we have been influenced by our experience, the more we can take responsibility for those paradigms, examine them, test them against reality, listen to others and be open to their perceptions, thereby getting a larger picture and a far more objective view." (Stephen Covey, "The 7 Habits of Highly Effective People", 1989)

"The word paradigm comes from the Greek. It was originally a scientific term, and is more commonly used today to mean a model, theory, perception, assumption, or frame of reference. In the more general sense, it's the way we 'see' the world - not in terms of our visual sense of sight, but in terms of perceiving, understanding, and interpreting." (Stephen Covey, "The 7 Habits of Highly Effective People", 1989)

🪷On Mind: On Perception (1950-159)

"[…] key metaphors help determine what and how we perceive and how we think about our perceptions." (Meyer H Abrams, "The Mirror and the Lamp: Romantic Theory and the Critical Tradition", 1953)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana,"The Life of Reason, or the Phases of Human Progress", 1954)

"The suggestion is that the function of the brain and nervous system and sense organs is in the main eliminative and not productive. Each person is at each moment capable of remembering all that has ever happened to him and of perceiving everything that is happening everywhere in the universe. The function of the brain and the nervous system is to protect us from being overwhelmed and confused by this mass of largely useless and irrelevant knowledge, by shutting out most of what we should otherwise perceive or remember at any moment, and leaving only that very small and special selection which is likely to be practically useful. According to such a theory, each one of us is potentially Mind at Large." (Aldous Huxley, "The Doors of Perception", 1954)

"When, for instance, I see a symmetrical object, I feel its pleasurable quality, but do not need to assert explicitly to myself, ‘How symmetrical!’. This characteristic feature may be explained as follows. In the course of individual experience it is found generally that symmetrical objects possess exceptional and desirable qualities. Thus our own bodies are not regarded as perfectly formed unless they are symmetrical. Furthermore, the visual and tactual technique by which we perceive the symmetry of various objects is uniform, highly developed, and almost instantaneously applied. It is this technique which forms the associative 'pointer.' In consequence of it, the perception of any symmetrical object is accompanied by an intuitive aesthetic feeling of positive tone." (George D Birkhoff, "Mathematics of Aesthetics", 1956)

"So-called common sense relies on programmed nonperception, concealment, or ridicule of everything that doesn’t fit into the conventional nineteenth century vision of a world that can be explained down to the last detail." (Stanislaw Lem, "The Investigation", 1959)

🪷On Mind: On Perception (1875-1899)

"For of an image one requires some kind of sameness with the pictured object, of a statue sameness of form, of a delineation sameness of perspective projection in the visual field, of a painting also sameness of color. " (Heinrich Hertz, "The Facts in Perception", 1878)

"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [president's address] 1883)

"It is difficult to give an idea of the vast extent of modern mathematics. The word 'extent' is not the right one: I mean extent crowded with beautiful details - not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood and flower. But, as for everything else, so for a mathematical theory - beauty can be perceived but not explained." (Arthur Cayley, [address before the meeting of the British Association at Southport] 1883)

"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)

"Mathematics stands forth as that which unites, mediates between Man and Nature, inner and outer world, thought and perception, as no other subject does." (Julius Froebel, 1893)

"Experience is the collecting of what is similar in different particular perceptions." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"It is a mistake to reduce mater to the perception which we have of it, a mistake also to make of it a thing able to produce in us perceptions, but in itself is of another nature than they. Matter, in our view, is an aggregate of ‘images’. And by ‘image‘ we mean a certain existence which is more than that which the idealist calls a representation, but less than that which the realist calls a thing - an existence placed halfway between the ‘thing‘ and the ‘representation’. This conception of matter is simply that of common sense. [...] For common sense, then, the object exists in itself, and, on the other hand, the object is, in itself, pictorial, as we perceive it image it is, but a self-existing image." (Henri Bergson,"Matter and Memory", 1896)

"We call that psychical process, which is operative in the clear perception of a narrow region of the content of consciousness, attention." (Wilhelm M Wundt,"Outlines of Psychology", 1897)

🪷On Mind: On Perception (1850-1874)

"All perception of truth is the detection of an analogy [...]" (Henry D Thoreau, 1851)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"As you are aware, no perceptions obtained by the senses are merely sensations impressed on our nervous systems. A peculiar intellectual activity is required to pass from a nervous sensation to the conception of an external object, which the sensation has aroused. The sensations of our nerves of sense are mere symbols indicating certain external objects, and it is usually only after considerable practice that we acquire the power of drawing correct conclusions from our sensations respecting the corresponding objects." (Hermann von Helmholtz, "On the Physiological Causes of Harmony", 1857)

"The term 'intellect' includes all those powers by which we acquire, retain, and extend our knowledge; as perception, memory, imagination, judgment, and the like." (William Fleming, "A vocabulary of the philosophical sciences", 1857)

"Observation is so wide awake, and facts are being so rapidly added to the sum of human experience, that it appears as if the theorizer would always be in arrears, and were doomed forever to arrive at imperfect conclusion; but the power to perceive a law is equally rare in all ages of the world, and depends but little on the number of facts observed." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1862)

"If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

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

"Even if a curve is not drawn nor it is assumed to be tracked by the eye, but we have a 'perception' of it, it has in any case a limited precision and does not therefore correspond to the exact concept of a function of precision mathematics but rather to the idea of a function stripe." (Felix Klein, 1873)

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

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

🪷On Mind: On Perception (1800-1849)

"If we seek a cause wherever we perceive symmetry, it is not that we regard a symmetrical event as less possible than the others, but, since this event ought to be the effect of a regular cause or that of chance, the first of these suppositions is more probable than the second.” (Pierre-Simon de Laplace, "Philosophical Essay on Probabilities”, 1812)

"The difficulty of bringing idea and experience into relation with one another makes itself very painfully felt in all investigation of nature. The idea is independent of space and time. Research is limited in space and time. Hence in the idea simultaneous and successive features are most intimately linked, whereas these are always separated in experience; and to think of a process of nature as simultaneous and successive at once, in accordance with the idea, makes our heads spin. The understanding is unable to conceive of those sense data as jointly present which experience transmitted to it one at a time. Thus, the contradiction between ideation and perception remains forever unresolved. " (Johann Wolfgang von Goethe, "Doubt and Resignation", 1820)

"Truth travels down from the heights of philosophy to the humblest walks of life, and up from the simplest perceptions of an awakened intellect to the discoveries which almost change the face of the world. At every stage of its progress it is genial, luminous, creative." (Edward Everett, [address] 1835)

"A line, as defined by geometers, is wholly unconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. But we cannot conceive a line without breadth: we can form no mental picture of such a line: all the lines which we have in our minds are lines possessing breadth." (John S Mill, "System of Logic ratiocinative and inductive, being a connected view of the principles of evidence and the methods of scientific investigation", 1843)

"All existing things upon this earth, which have knowledge of their own existence, possess, some in one degree and some in another, the power of thought, accompanied by perception, which is the awakening of thought by the effects of external objects upon the senses." (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

🪷On Mind: On Perception (1600-1699)

"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)

"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)

"Although time, space, place, and motion are very familiar to everyone, it must be noted that these quantities are popularly conceived solely with reference to the objects of sense perception." (Isaac Newton, "The Principia: Mathematical Principles of Natural Philosophy", 1687)

"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)

“Probability is the appearance of agreement upon fallible proofs. As demonstration is the showing the agreement or disagreement of two ideas by the intervention of one or more proofs, which have a constant, immutable, and visible connexion one with another; so probability is nothing but the appearance of such an agreement or disagreement by the intervention of proofs, whose connexion is not constant and immutable, or at least is not perceived to be so, but is, or appears for the most part to be so, and is enough to induce the mind to judge the proposition to be true or false, rather than the contrary.” (John Locke, “An Essay Concerning Human Understanding”, 1689)

“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)

🪷On Mind: On Perception (1700-1799)

"Moreover, it must be confessed that perception and that which depends upon it are inexplicable on mechanical grounds, that is to say, by means of figures and motions. And supposing there were a machine, so constructed as to think, feel, and have perception, it might be conceived as increased in size, while keeping the same proportions, so that one might go into it as into a mill. That being so, we should, on examining its interior, find only parts which work one upon another, and never anything by which to explain a perception. Thus it is in a simple substance, and not in a compound or in a machine, that perception must be sought for." (Gottfried W Leibniz, "Monadology", 1714)

"[...] 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)

"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)

"Nay farther, even with relation to that succession, we cou'd only admit of those perceptions, which are immediately present to our consciousness, nor cou'd those lively images, with which the memory presents us, be ever receiv'd as true pictures of past perceptions. The memory, senses, and understanding are, therefore, all of them founded on the imagination, or the vivacity of our ideas.(David Hume, "A Treatise of Human Nature A Treatise of Human Nature", 1739)

"Nay farther, even with relation to that succession, we cou'd only admit of those perceptions, which are immediately present to our consciousness, nor cou'd those lively images, with which the memory presents us, be ever receiv'd as true pictures of past perceptions. The memory, senses, and understanding are, therefore, all of them founded on the imagination, or the vivacity of our ideas.(David Hume, "A Treatise of Human Nature", 1739)

"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)

"Show all these fanatics a little geometry, and they learn it quite easily. But, strangely enough, their minds are not thereby rectified. They perceive the truths of geometry, but it does not teach them to weigh probabilities. Their minds have set hard. They will reason in a topsy-turvy wall all their lives, and I am sorry for it." (Voltaire, "False Minds" ["Esprits faux"], A Philosophical Dictionary],1764)

"Things which do not now exist in the mind itself, can only be perceived, remembered, or imagined, by means of the ideas or images in the mind, which are the immediate objects of perception, remembrance, and imagination." (Thomas Reid, "An Inquiry into the Human Mind on the Principles", 1764)

"Psychologists have hitherto failed to realize that imagination is a necessary ingredient of perception itself." (Immanuel Kant, "Critique of Pure Reason", 1781)

🪷On Mind: On Perception (600-1599)

"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)

"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)

"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)

"[Intuitive] Understanding is consequent upon deliberation, and firmly embraces the better part. For [intuitive] understanding concerns itself with divine truths, and the relish, love, and observance of the latter constitutes true wisdom. Rather than being the [mere] product of nature, these successive steps are the result of grace. The latter, according to its own free determination, derives the various rivulets of the sciences and wisdom from the fountainhead of sense perception. Grace reveals hidden divine truths by means of those things which have been made, and by that unity which belongs to love, communicates what it has made manifest, thus uniting man to God." (John of Salisbury, "Metalogicon", 1159)

"The employment of tropes, just as the use of schemata, is the exclusive privilege of the very learned. The rules governing tropes are also very strict, so that the latitude in which they may be used is definitely limited. For the rules teach that we may not extend figures. One who is studiously imitating the authors by using metaphors and figures, must take care to avoid crude figures that are hard to interpret. What is primarily desirable in language is lucid clarity and easy comprehensibility. Therefore schemata should be used only out of necessity or for ornamentation. Speech was invented as a means of communicating mental concepts; and figures [of speech] are admitted so far as they compensate by their utility for whatever they lack in conformity to the [rules of the grammatical] art." (John of Salisbury, "Metalogicon", 1159)

"Part of the functions of the imaginative faculty is, as you well know, to retain impressions by the senses, to combine them, and chiefly to form images." (Moses Maimonides,"Guide of the Perplexed", 1190 [translated by Michael Friedländer, 1904])

"All our knowledge has its origins in our perceptions." (Leonardo da Vinci, [notebooks] 1480-1519)

🪷On Mind: On Perception (-599)

"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)

"It is obvious then, that memory belongs to that part of the soul to which imagination belongs. […] Just as the picture painted on the panel is at once a picture and a portrait, and though one and the same, is both, yet the essence of the two is not the same, and it is possible to think of it both as a picture and as a portrait, so in the same way we must regard the mental picture within us both as an object of contemplation in itself and as a mental picture of something else […]. Insofar as we consider it in relation to something else, e.g. as a likeness, it is also an aid to memory." (Aristotle,"De Memoria et Reminiscentia" [On Memory and Recollection], 4th century BC)

"Now for the thinking soul images take the place of direct perceptions; and when it asserts or denies that they are good or bad, it avoids or pursues them. Hence the soul never thinks without a mental image." (Aristotle, "De Anima" , cca. 350 BC)

"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)"

"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)

"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)

"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)

"Constantly regard the universe as one living being, having one substance and one soul; and observe how all things have reference to one perception, the perception of this one living being; and how all things act with one movement; and how all things are the cooperating causes of all things which exist; observe too the continuous spinning of the thread and the contexture of the web." (Marcus Aurelius, "Meditations". cca. 121–180 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)

"[...] so external sensation is the image of this perception of the soul, which is in its essence truer and is a contemplation of forms alone without being affected. From these forms, from which the soul alone receives its lordship over the living being, come reasonings, and opinions and noetic acts; and this is precisely where ‘we’ are." (Plotinus, "Enneads", cca. 270 AD)

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

🕸️On Graph Theory: On Neural Networks (2010-2019)

"Neural networks can model very complex patterns and decision boundaries in the data and, as such, are very powerful. In fact, they are so powerful that they can even model the noise in the training data, which is something that definitely should be avoided. One way to avoid this overfitting is by using a validation set in a similar way as with decision trees.[...] Another scheme to prevent a neural network from overfitting is weight regularization, whereby the idea is to keep the weights small in absolute sense because otherwise they may be fitting the noise in the data. This is then implemented by adding a weight size term (e.g., Euclidean norm) to the objective function of the neural network." (Bart Baesens, "Analytics in a Big Data World: The Essential Guide to Data Science and Its Applications", 2014)

"An important property of neural networks is that their weights are set randomly before learning is started, and this random initialization affects the model that is learned. That means that even when using exactly the same parameters, we can obtain very different models when using different random seeds. If the networks are large, and their complexity is chosen properly, this should not affect accuracy too much, but it is worth keeping in mind (particularly for smaller networks)." (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"Neural networks - particularly the large and powerful ones - often take a long time to train. They also require careful preprocessing of the data, as we saw here. Similarly to SVMs, they work best with 'homogeneous' data, where all the features have similar meanings. For data that has very different kinds of features, tree-based models might work better. Tuning neural network parameters is also an art unto itself. In our experiments, we barely scratched the surface of possible ways to adjust neural network models and how to train them."  (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"A neural network consists of a set of neurons that are connected together. A neuron takes a set of numeric values as input and maps them to a single output value. At its core, a neuron is simply a multi-input linear-regression function. The only significant difference between the two is that in a neuron the output of the multi-input linear-regression function is passed through another function that is called the activation function." (John D Kelleher & Brendan Tierney, "Data Science", 2018)

"Just as they did thirty years ago, machine learning programs (including those with deep neural networks) operate almost entirely in an associational mode. They are driven by a stream of observations to which they attempt to fit a function, in much the same way that a statistician tries to fit a line to a collection of points. Deep neural networks have added many more layers to the complexity of the fitted function, but raw data still drives the fitting process. They continue to improve in accuracy as more data are fitted, but they do not benefit from the 'super-evolutionary speedup'."  (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

"A neural-network algorithm is simply a statistical procedure for classifying inputs (such as numbers, words, pixels, or sound waves) so that these data can mapped into outputs. The process of training a neural-network model is advertised as machine learning, suggesting that neural networks function like the human mind, but neural networks estimate coefficients like other data-mining algorithms, by finding the values for which the model’s predictions are closest to the observed values, with no consideration of what is being modeled or whether the coefficients are sensible." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

"Deep neural networks have an input layer and an output layer. In between, are “hidden layers” that process the input data by adjusting various weights in order to make the output correspond closely to what is being predicted. [...] The mysterious part is not the fancy words, but that no one truly understands how the pattern recognition inside those hidden layers works. That’s why they’re called 'hidden'. They are an inscrutable black box - which is okay if you believe that computers are smarter than humans, but troubling otherwise." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

"Neural-network algorithms do not know what they are manipulating, do not understand their results, and have no way of knowing whether the patterns they uncover are meaningful or coincidental. Nor do the programmers who write the code know exactly how they work and whether the results should be trusted. Deep neural networks are also fragile, meaning that they are sensitive to small changes and can be fooled easily." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

"The label neural networks suggests that these algorithms replicate the neural networks in human brains that connect electrically excitable cells called neurons. They don’t. We have barely scratched the surface in trying to figure out how neurons receive, store, and process information, so we cannot conceivably mimic them with computers." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)