"We invoke the imagination and the intervals that it furnishes, since the form itself is without motion or genesis, indivisible and free of all underlying matter, though the elements latent in the form are produced distinctly and individually on the screen of imagination. What projects the images is the understanding; the source of what is projected is the form in the understanding; and what they are projected in is this 'passive nous' that unfolds in revolution about the partlessness of genuine Nous." (Proclus Lycaeus, "A Commentary on the First Book of Euclid’s Elements", cca 5th century)
"Since it seems that there is nothing outside and separate in existence from sensible spatial magnitudes, the objects of thought are in the sensible forms, viz. both the abstract objects and all the states and affections of sensible things. Hence no one can learn or understand anything in the absence of sense, and when the mind is actively aware of anything it is necessarily aware of it along with an image; for images are like sensuous contents except in that they contain no matter. Imagination is different from assertion and denial; for what is true or false involves a synthesis of thoughts. In what will the primary thoughts differ from images? Must we not say that neither these nor even our other thoughts are images, though they necessarily involve them?" (Aristotle, "De Anima", cca. 350 BC)
"[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)
"There are three kinds of intelligence: one kind understands things for itself, the other appreciates what others can understand, the third understands neither for itself nor through others. This first kind is excellent, the second good, and the third kind useless." (Niccolò Machiavelli, "The Prince", 1532)
"The human understanding resembles not a dry light, but admits a tincture of the will and passions, which generate their own system accordingly; for man always believes more readily that which he prefers." (Sir Francis Bacon, "Novum Organum", 1620)
"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)
"If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty." (Edmund Burke, "Essay on the Sublime and Beautiful", 1756)
"The analysis of concepts is for the understanding nothing more than what the magnifying glass is for sight." (Moses Mendelssohn, 1763)
"This formal and pure condition of sensibility to which the employment of the concept of understanding is restricted, we shall entitle the schema of the concept. The procedure of understanding in these schemata we shall entitle the schematism of pure understanding.
The schema is in itself always a product of imagination. Since, however, the synthesis of imagination aims at no special intuition, but only at unity in the determination of sensibility, the schema has to be distinguished from the image." (Immanuel Kant," Critique of Pure Reason", 1781)
“This schematism of our understanding, in its application to appearances and their mere form, is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze.” (Immanuel Kant, “Critique of Pure Reason”, 1781)
"Thoughts without content are empty, intuitions without concepts are blind. The understanding can intuit nothing, the senses can think nothing. Only through their unison can knowledge arise." (Immanuel Kant, "Critique of Pure Reason", 1781)
"As facts and knowledge accumulate, the claim of the scientist to an understanding of the world in a certain sense diminishes." (Werner K Heisenberg, "Zur Geschichte der physikalischen Naturerklärung", 1933)
"We have discovered that it is actually an aid in the search for knowledge to understand the nature of the knowledge we seek." (Arthur S Eddington, "The Philosophy of Physical Science", 1938)
"[…] as a model of a complex system becomes more complete, it becomes less understandable. Alternatively, as a model grows more realistic, it also becomes just as difficult to understand as the real world processes it represents." (Jay M Dutton & William H Starbuck," Computer simulation models of human behavior: A history of an intellectual technology", IEEE Transactions on Systems, 1971)
"The aim of the model is of course not to reproduce reality in all its complexity. It is rather to capture in a vivid, often formal, way what is essential to understanding some aspect of its structure or behavior." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation" , 1976)
"New metaphors are capable of creating new understandings and, therefore, new realities. This should be obvious in the case of poetic metaphor, where language is the medium through which new conceptual metaphors are created." (George Lakoff and Mark Johnson, "Metaphors We Live By", 1980)
"The essence of metaphor is understanding and experiencing one kind of thing in terms of another. […] Metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature." (George Lakoff & Mark Johnson, "Metaphors We Live By", 1980)
"After all of this it is a miracle that our models describe anything at all successfully. In fact, they describe many things well: we observe what they have predicted, and we understand what we observe. However, this last act of observation and understanding always eludes physical description." (Yuri I Manin, "Mathematics and Physics", 1981)
"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)
"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding. " (Mark Johnson, "The Body in the Mind", 1987)
"The model and the theory it represents must be accepted, at least temporarily, or rejected, depending on the agreement or disagreement between observed data and the hypotheses and implications of the model. When neither the hypotheses nor the implications of a theory can be confronted with the real world, that theory is devoid of any scientific interest. Mere logical, even mathematical, deduction remains worthless in terms of the understanding of reality if it is not closely linked to that reality." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)
"Wisdom is your perspective on life, your sense of balance, your understanding of how the various parts and principles apply and relate to each other." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)
"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)
"[...] cognitive maps can be seen as a picture or visual aid in comprehending the mappers' understanding of particular, and selective, elements of the thoughts (rather than thinking) of an individual, group or organization. They may also be seen as a representation that is amenable to analysis by both the mapper and others." (Colin Eden, "One the nature of cognitive maps", Journal of Management Studies 29 (3), 1992)
"From a very early age, we form concepts. Each concept is a particular idea or understanding we have about our world. These concepts allow us to make sense of and reason about the things in our world. These things to which our concepts apply are called objects." (James Martin, 1993)
"[For] us to be able to speak and understand novel sentences, we have to store in our heads not just the words of our language but also the patterns of sentences possible in our language. These patterns, in turn, describe not just patterns of words but also patterns of patterns. Linguists refer to these patterns as the rules of language stored in memory; they refer to the complete collection of rules as the mental grammar of the language, or grammar for short." (Ray Jackendoff, "Patterns in the Mind", 1994)
"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […] The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)
"We all depend on models to interpret our everyday experiences. We interpret what we see in terms of mental models constructed on past experience and education. They are constructs that we use to understand the pattern of our experiences." (David Bartholomew, "What is Statistics?", 1995)
"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)
"Models of the real world are not always easy to formulate because of the richness, variety, and ambiguity that exists in the real world or because of our ambiguous understanding of it." (George Dantzig & Mukund N Thapa, Linear Programming, Vol I, 1997)
"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)
"Simple observation generally gets us nowhere. It is the creative imagination that increases our understanding by finding connections between apparently unrelated phenomena, and forming logical, consistent theories to explain them. And if a theory turns out to be wrong, as many do, all is not lost. The struggle to create an imaginative, correct picture of reality frequently tells us where to go next, even when science has temporarily followed the wrong path." (Richard Morris, "The Universe, the Eleventh Dimension, and Everything: What We Know and How We Know It", 1999)
"A mental model is a representation of some domain or situation that supports understanding, reasoning, and prediction. Mental models permit reasoning about situations not directly experienced. They allow people to mentally simulate the behavior of a system. Many mental models are based on generalizations and analogies from experience." (D Gentner, "Psychology of Mental Models" [in "International Encyclopedia of the Social & Behavioral Sciences"], 2001)
"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)
"The understanding of a thing begins and ends with some conceptual model of it. The model is the better, the more accurate, and inclusive. But even rough models can be used to guide - or misguide - research." (Bunge A Mario, "Philosophy in Crisis: The Need for Reconstruction", 2001)
"We reach wisdom when we achieve a deep understanding of acquired knowledge, when we not only 'get it', but when new information blends with prior experience so completely that it makes us better at knowing what to do in other situations, even if they are only loosely related to the information from which our original knowledge came. Just as not all the information we absorb leads to knowledge, not all of the knowledge we acquire leads to wisdom." (Alberto Cairo, "The Functional Art", 2011)
"Understanding reduces the complexity of data by collapsing the dimensionality of information to a lower set of known variables. s revolutions, be they tiny or vast, technological or social."
"Understanding transcends context, since the different contexts collapse according to their previously unknown similarity, which the principle contains. That is what understanding does. And you actually feel it in your brain when it happens. Your 'cognitive load' decreases, your level of stress and anxiety decrease, and your emotional state improves." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)
"[…] many-model thinking produces wisdom through a diverse ensemble of logical frames. The various models accentuate different causal forces. Their insights and implications overlap and interweave. By engaging many models as frames, we develop nuanced, deep understandings." (Scott E Page, "The Model Thinker", 2018)
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