"Mathematics is an exploratory science that seeks to understand every kind of pattern - patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns." (Lynn A Steen, "The Future of Mathematics Education", 1990)
"Hard though the scientists of mental imagery try, they cannot get around the fact that the representations they deal with are like pictures. […] The methods have to assume, and the experiments continually corroborate, that having imagery is somehow like perceptual seeing, and that it is somehow like seeing pictures. […] The minimal reason for this assumption is that people do naturally talk of seeing pictures before their mind’s eye." (Eva T H Brann,"The World of Imagination", 1991)
"There is one qualitative aspect of reality that sticks out from all others in both profundity and mystery. It is the consistent success of mathematics as a description of the workings of reality and the ability of the human mind to discover and invent mathematical truths." (John D Barrow, "Theories of Everything: The quest for ultimate explanation", 1991)
"[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost." (Mark Kac, "Mathematics: Tensions", 1992)
"For a musician, visualization is the process of picturing in our minds eye what we hear in our mind's ear. Visualization is something we all do. In fact, putting a visual form before the mind's eye or forming a mental image is something that precedes most things that we do." (Jerry Bergonzi, "Melodic Structures", 1992)
"[…] number theory […] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato’s shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 5 July, 1993)
"Intuition is the art, peculiar to the human mind, of working out the correct answer from data that is, in itself, incomplete or even, perhaps, misleading." (Isaac Asimov, "Forward the Foundation", 1993)
"It is astonishingly beautiful and interesting, how thought is absent when you have an insight. Thought cannot have an insight. It is only when the mind is not operating mechanically in the structure of thought that you have an insight. Having had an insight, thought draws a conclusion from that insight. And then thought acts and thought is mechanical. So I have to find out whether having an insight into myself, which means into the world, and not drawing a conclusion from it is possible. If I draw a conclusion, I act on an idea, on an image, on a symbol, which is the structure of thought, and so I am constantly preventing myself from having insight, from understanding things as they are." (Jiddu Krishnamurti," On Mind and Thought", 1993)
"The voyage of discovery into our own solar system has taken us from clockwork precision into chaos and complexity. This still unfinished journey has not been easy, characterized as it is by twists, turns, and surprises that mirror the intricacies of the human mind at work on a profound puzzle. Much remains a mystery. We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty." (Ivars Peterson, "Newton’s Clock", 1993)
"World view, a concept borrowed from cultural anthropology, refers to the culturally dependent, generally subconscious, fundamental organization of the mind. This conceptual organization manifests itself as a set of presuppositions that predispose one to feel, think, and act in predictable patterns." (Kenneth G Tobin, "The practice of constructivism in science education", 1993)
"Mental models are the images, assumptions, and stories which we carry in our minds of ourselves, other people, institutions, and every aspect of the world. Like a pane of glass framing and subtly distorting our vision, mental models determine what we see. Human beings cannot navigate through the complex environments of our world without cognitive ‘mental maps’; and all of these mental maps, by definition, are flawed in some way." (Peter M Senge et al, "The Fifth Discipline Fieldbook: Strategies and Tools for Building a Learning Organization", 1994)
"Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations." (Anna Sierpinska, "Understanding in Mathematics", 1994)
"People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like something else (association); and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. Personally, I put a lot of effort into ‘listening’ to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations." (William P Thurston, "On proof and progress in mathematics", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)
"It is the great irony of life that a mindless act repeated in sequence can only lead to greater depths of absurdity, while a mindless act performed in parallel by a swarm of individuals can, under the proper conditions, lead to all that we find interesting." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)
"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)
"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & ?K Nishijima , "Broken Symmetry: Selected Papers Of Y Nambu", 1995)
"The leading edge of growth of intelligence is at the cultural and societal level. It is like a mind that is struggling to wake up. This is necessary because the most difficult problems we face are now collective ones. They are caused by complex global interactions and are beyond the scope of individuals to understand and solve. Individual mind, with its isolated viewpoints and narrow interests, is no longer enough." (Jeff Wright, "Basic Beliefs", [email] 1995)
"The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility, and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities." (George Sarton, The American Mathematical Monthly, Vol. 102, No. 4, 1995)
"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)
"In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world." (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)
"Reality contains not only evidence, but also the means (such as our minds, and our artefacts) of understanding it. There are mathematical symbols in physical reality. The fact that it is we who put them there does not make them any less physical." (David Deutsch, "The Fabric of Reality", 1997)
"Suppose the reasoning centers of the brain can get their hands on the mechanisms that plop shapes into the array and that read their locations out of it. Those reasoning demons can exploit the geometry of the array as a surrogate for keeping certain logical constraints in mind. Wealth, like location on a line, is transitive: if A is richer than B, and B is richer than C, then A is richer than C. By using location in an image to symbolize wealth, the thinker takes advantage of the transitivity of location built into the array, and does not have to enter it into a chain of deductive steps. The problem becomes a matter of plop down and look up. It is a fine example of how the form of a mental representation determines what is easy or hard to think." (Steven Pinker, "How the Mind Works", 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)
"Mathematics is a product - a discovery - of the human mind. It enables us to see the incredible, simple, elegant, beautiful, ordered structure that lies beneath the universe we live in. It is one of the greatest creations of mankind - if it is not indeed the greatest." (Keith Devlin, "Life By the Numbers", 1998)
"A collective mental map functions first of all as a shared memory. Various discoveries by members of the collective are registered and stored in this memory, so that the information will remain available for as long as necessary. The storage capacity of this memory is in general much larger than the capacities of the memories of the individual participants. This is because the shared memory can potentially be inscribed over the whole of the physical surroundings, instead of being limited to a single, spatially localized nervous system. Thus, a collective mental map differs from cultural knowledge, such as the knowledge of a language or a religion, which is shared among different individuals in a cultural group but is limited by the amount of knowledge a single individual can bear in mind." (Francis Heylighen, "Collective Intelligence and its Implementation on the Web", 1999)
"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)
"Metaphors can have profound signi?cance because, as images or ?gures, they allow the mind to grasp or discover unsuspected ideal and material relationships between objects." (Giuseppe Del Re, "Cosmic Dance", 1999)
"Partial models, imperfect as they may be, are the only means developed by science for understanding the universe. This statement does not imply an attitude of defeatism but the recognition that the main tool of science is the human mind and that the human mind is finite." (Nancy Cartwright, "The Dappled World: A Study of the Boundaries of Science", 1999)
"The spirit of mathematics and the essence of its beauty is remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful mathematical vista with one another, but there is no camera that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails." (Silvio Levy, "The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)
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