"As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." (Gerald Tenenbaum & Michael M France, "The Prime Numbers and Their Distribution", 2000)
"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)
"To look at the development of physics since Newton is to observe a struggle to define the limits of science. Part of this process has been the intrusion of scientific methods and ideas into domains that have traditionally been the province of metaphysics or religion. In this conflict, Hawking’s phrase ‘to know the Mind of God’ is just one example of a border infringement. But by playing the God card, Hawking has cleverly fanned the flames of his own publicity, appealing directly to the popular allure of the scientist-as-priest." (Peter Coles, "Hawking and the Mind of God", 2000)
"To say that a thing is imaginary is not to dispose of it in the realm of mind, for the imagination, or the image making faculty, is a very important part of our mental functioning. An image formed by the imagination is a reality from the point of view of psychology; it is quite true that it has no physical existence, but are we going to limit reality to that which is material? We shall be far out of our reckoning if we do, for mental images are potent things, and although they do not actually exist on the physical plane, they influence it far more than most people suspect." (Dion Fortune," Spiritualism and Occultism", 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)
"An opportunity for cybernetics to change the course of the philosophy of mind was missed when intentionality was misinterpreted as 'the providing of coded knowledge'." (Igor Aleksander, New Scientist Vol. 169, 2001)
"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?" (Stephen W Hawking, "The Universe in a Nutshell", 2001).
"Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall & Carl Pomerance, "Prime Numbers: A Computational Perspective", 2001)
"There is no ‘scientific worldview’ just as there is no uniform enterprise ‘science’- except in the minds of metaphysicians, school masters, and scientists blinded by the achievements of their own particular niche." (Paul Feyerabend, "Conquest of Abundance", 2001)
"When we look at the world around us, we find that we are not thrown into chaos and randomness but are part of a great order, a grand symphony of life. Every molecule in our body was once a part of previous bodies-living or nonliving-and will be a part of future bodies. In this sense, our body will not die but will live on, again and again, because life lives on. We share not only life's molecules but also its basic principles of organization with the rest of the living world. Arid since our mind, too, is embodied, our concepts and metaphors are embedded in the web of life together with our bodies and brains. We belong to the universe, we are at home in it, and this experience of belonging can make our lives profoundly meaningful." (Fritjof Capra, "The Hidden Connections", 2002)
"Mathematics is not just about numbers. As well as numbers, modern mathematics also looks at the relations between them. The passage from pure numerology to this new vision has derived from the realization that the most profound meaning is not in the numbers but in the relations between them. Mathematical investigation is precisely the exploration and the study of the different possible relations; some of them find a concrete and immediate application in the environment in which they are immersed, others just ‘live’ in the minds of those that conceive them." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos and Complexity: The Dynamics of Natural and Social Systems", 2003)
"The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle." (Marcus du Sautoy, "The Music of the Primes", 2003)
"I often told the fanatics of realism that there is no such thing as realism in art: it only exists in the mind of the observer. Art is a symbol, a thing conjuring up reality in our mental image. That is why I don't see any contradiction between abstract and figurative art either." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)
"Your ability to visualize is perhaps the most powerful faculty that you possess. All improvement in your life begins with an improvement in your mental pictures. You are where you are and what you are today largely because of the mental pictures that you hold in your conscious mind at the present time. As you change your mental pictures on the inside, your world on the outside will begin to change to correspond to those pictures." (Brian Tracy, "Goals!: How to Get Everything You Want - Faster Than You Ever Thought Possible", 2004)
"When a particular image appears in the mind's eye often enough it begins to connect apparently unrelated ideas leading to models and theories. […] Patterns experienced again and again become intuitions. […] Intuitive judgments are made by our use of imagery; intuition is the result of mental model building. […] The mental model used and the form of the intuition is dependent upon the question being answered." (Roger Frantz, "Two Minds", 2005)
"Nothing resembles reality less than the photograph. Nothing resembles substance less than its shadow. To convey the meaning of something substantial you have to use not a shadow but a sign, not the limitation but the image. The image is a new and different reality, and of course it does not convey an impression of some object, but the mind of the subject; and that is something else again." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)
"Why must those models be simplifications? Each model must help us to focus on only those aspects that matter in some particular context; that's what makes a map more useful to us than seeing the entire landscape that it depicts. The same applies to what we store in our minds. Consider how messy our minds would become if we filled them up with descriptions of things whose details had too little significance. So instead, we spend large parts of our lives at trying to tidy up our minds - selecting the portions we want to keep, suppressing others we'd like to forget, and refining the ones we're dissatisfied with." (Marvin Minsky, "The Emotion Machine: Commonsense thinking, artificial intelligence, and the future of the human mind", 2006)
"Your mental models shape the way you see the world. They help you to quickly make sense of the noises that filter in from outside, but they can also limit your ability to see the true picture. [...] We eventually lose all awareness that these ‘models’ are in fact internal illusions. We accept them as external reality and act on them as if they were. If they are good models, in most circumstances they more than adequately permit the mind to handle external reality. But here a danger creeps in. When the world changes in important ways, we can find ourselves with a model that is completely irrelevant to the current situation. We find ourselves wearing our street clothes when we are thrown off the deck of a ship. What we need at that point is a wet suit and lifejacket." (Colin Cook & Yoram R Wind, "The Power of Impossible Thinking: Transform the Business of Your Life and the Life of Your Business", 2006)
"But ignorance exists in the map, not in the territory. If I am ignorant about a phenomenon, that is a fact about my own state of mind, not a fact about the phenomenon itself. A phenomenon can seem mysterious to some particular person. There are no phenomena which are mysterious of themselves. To worship a phenomenon because it seems so wonderfully mysterious, is to worship your own ignorance." (Eliezer Yudkowsky, "Mysterious Answers To Mysterious Questions" 2007)
"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)
"There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)
[mental model:] "Internal representations constructed on the spot when required by demands of an external task or by a self-generated stimulus. It enables activation of relevant schemata, and allows new knowledge to be integrated. It specifies causal actions among concepts that take place within it, and it can be interacted with in the mind." (Daniel Churchill, "Mental Models" [in "Encyclopedia of Information Technology Curriculum Integration"] , 2008)
"For me mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos." (Clifford A Pickover, "The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", 2009)
"Intuition, mind’s originary act of ‘seeing’ what is given to him […]" (Hermann Weyl, "Mind and Nature", 2009)
"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre 0 Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)
"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)
"We all use mental models every day. Our minds do not contain real economic or social systems. Instead, they contain representations - models - of reality. We use these models in all aspects of decision-making. Being explicitly aware of our mental models can help us in understanding why we make the decisions we do and how we can improve our decision-making processes. If everyone’s mental models are brought to light in the context of an organization, we can begin to see where, how, and why the models diverge. This is the first step in building a shared understanding within an organization. As long as mental models remain hidden, they constitute an obstacle to building shared understanding." (Akhilesh Bajaj & Stanislaw Wrycza, "Systems Analysis and Design for Advanced Modeling Methods: Best Practices", 2009)
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