On Culture (Unsourced)

"[...] every culture in the world has had its own unique history and we can not therefore say that any culture observable in the present day world is an earlier form of any other." (Charles W Hart)

"He who cherishes the values of culture cannot fail to be a pacifist." (Albert Einstein)

"Human creative work is by excellence one of expression. The deep human desire is to be understood by other peoples. This expression which succeeds to be communicated is just what we call culture." (Grigore C Moisil)

"More and more, science has become not only increasingly necessary as a foundation for professional skill, but has come to be regarded as the most valuable instrument of culture." (Henry P Smith)

"Our models of communication [...] create what we disingenuously pretend they merely describe. As a result our science is [...] a reflexive one. We not only describe behavior; we create a particular corner of culture - culture that determines, in part, the kind of communicative world we inhabit." (James W Carey)

"Our will and testament has come about out of a vigorous conviction that a nation that does not highly esteem mathematical thought can never be capable of achieving the highest cultural goals and thereby enjoy the international respect which, in the long term, is an effective means of maintaining our position in the world, as well as asserting our right to live our own lives." (M Gustav Mittag-Leffler)

"Science with its strict analysis of the facts, its persevering search for new, more consummate truths, and its relentless struggle against discovered mistakes and prejudices - science must saturate all or technics, our culture, and everyday life." (Abram F Joffe)

"The acquiring of culture is the developing of an avid hunger for knowledge and beauty." (Jesse L Bennett [attributed])

"The highest culture is not obtained from the teacher when at school or college, so much as by our ever-diligent self-education when we become men." (Samuel Smiles)

"The mathematics of rhythm are universal. They don't belong to any particular culture." (John McLaughlin)

"The trademark of modern culture is science; if you can fake this, you’ve got it made." (Mario Bunge)

"We are caged by our cultural programming. Culture is a mass hallucination, and when you step outside the mass hallucination you see it for what it’s worth." (Terence McKenna)

On Culture (From Fiction to Science Fiction)

"The great law of culture is: Let each become all that he was created capable of being." (Thomas Carlyle, "Jean Paul Friedrich Richter" (1827)

"This history of culture will explain to us the motives, the conditions of life, and the thought of the writer or reformer." (Lev N Tolstoy, "War and Peace", 1867)

"Culture is then properly described not as having its origin in curiosity, but as having its origin in the love of perfection; it is a study of perfection." (Matthew Arnold, "Culture and Anarchy", 1869)

"Not a having and a resting, but a growing and a becoming is the character of perfection as culture conceives it." (Matthew Arnold, "Culture and Anarchy", 1869)

"The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language." (Anton Chekhov, [letter to A.S. Suvorin] 1892)

"As the traveller who has been once from home is wiser than he who has never left his own door step, so a knowledge of one other culture should sharpen our ability to scrutinise more steadily, to appreciate more lovingly, our own." (Margaret Mead, "Coming of Age in Samoa", 1928)

"The worst difficulties from which we suffer do not come from without. They come from within [...] They come from a peculiar type of brainy people, always found in our country, who if they add something to our culture, take much from its strength. Our difficulties come from the mood of unwarrantable self-abasement into which we have been cast by a powerful section of our own intellectuals." (Winston Churchill, [speech] 1933)

"There are and have been worlds and cultures without end, each nursing the proud illusion that it is unique in space and time. There have been men without number suffering from the same megalomania; men who imagined themselves unique, irreplaceable, irreproducible. There will be more [...] more plus infinity. (Alfred Bester, "The Demolished Man", 1953)

"You don't have to burn books to destroy a culture. Just get people to stop reading them." (Ray Bradbury, "Fahrenheit 451", 1953)

"No one person can change a whole culture." (Poul Anderson, "Ghetto", 1954)

"When two alien cultures meet, the stronger must transform the weaker with love or hate." (Damon Knight, "Stranger Station", 1956)

"The meeting between ignorance and knowledge, between brutality and culture - it begins in the dignity with which we treat our dead." (Frank Herbert, "Dune", 1965)

"Homo can truly be called sapiens when he practices his specialty of being unspecialized. His repeated attempts to freeze himself into an all-answering pattern or culture or ideology, or whatever he has named it, have repeatedly brought ruin. Give him the pragmatic business of making his living, and he will usually do rather well. He adapts, within broad limits." (Poul Anderson, "The Queen of Air and Darkness", 1971)

"No culture as yet has actually forgotten history because no culture has really possessed more than fragments of it." (Edgar Pangborn, "Mount Charity", 1971)

"When one culture has the big guns and the other has none, there is a certain predictability about the outcome." (Joanna Russ, ‘"When It Changed", 1972)

"Man creates culture and through culture creates himself." (Pope John Paul II, "Osservatore Romano", 1980)

"A dying culture invariably exhibits personal rudeness. Bad manners. Lack of consideration for others in minor matters. A loss of politeness, of gentle manners, is more significant than is a riot." (Robert A. Heinlein, "Friday", 1982)

"Love was always a word that covered too much territory, from loving a spouse to loving a hobby or abstract justice, and the emotion-mongers of popular entertainment portrayed it as everlasting and exclusive. In a culture under stress the truth could not be concealed by sentimental fluff. The Greenhouse people learned to appreciate love without glorifying it." (George Turner, "Drowning Towers", 1987)

"Some cultures send their young people to the desert to seek visions and guidance, searching for true thinking spawned by the openness of the place, the loneliness, the beauty of emptiness." (Pat Murphy, "Rachel in Love", 1987)

"Most people don’t listen. They use the time when someone else is speaking to think of what they’re going to say next. True Listeners have always been revered among oral cultures, and prized for their rarity value." (Terry Pratchett, "Pyramids", 1989)

"As the soil, however rich it may be, cannot be productive without culture, so the mind without cultivation can never produce good fruit." (Seneca)

"Culture is acquainting ourselves with the best that has been known and said in the world, and thus with the history of the human spirit." (Matthew Arnold)

"Culture is the arts elevated to a set of beliefs." (Thomas Wolfe)

"Culture of the mind must be subservient to the heart." (Mohandas Gandhi)

"In rhetoric, this art of omission is a chief secret of power, and, in general, it is proof of high culture, to say the greatest matters in the simplest way. Veracity first of all, and forever." (Ralph W Emerson)

"It surprises me how our culture can destroy curiosity in the most curious of all animals - human beings." (Paul Maclean)

"Language is the road of a culture. It tells you where its people come from and where they are going." (Rita Mae Brown)

"No culture can live if it attempts to be exclusive." (Mohandas Gandhi)

"Noble life demands a noble architecture of noble uses of noble men. Lack of culture means what it has always meant: ignoble civilization and therefore imminent downfall." (Frank Lloyd Wright)

On Culture (1970-1979)

"If observation be the soil, reading is the manure of intellectual culture." (Daniel Drake, "Physician to the West: Selected Writings of Daniel Drake on Science and Society", 1970)

"Surely one of the most important characteristics of a scientific, introverted, specialized, hence infinitely intellectual culture is its drive toward, and faith in, total “awareness”. Awareness of almost every conceivable factor influencing almost every conceivable situation is our characteristic panacea or cure-all. In this sense, gnosis, the total consciousness, and self-consciousness are the major goals of our secular culture. We really believe that if we know or are aware of everything, if we can understand all relevant causes and factors, we can control everything." (Langdon Gilkey, "Religion and the Scientific Future: Reflections on Myth, Science, and Theology", 1970)

"What a lost person needs is a map of the territory, with his own position marked on it so he can see where he is in relation to everything else. Literature is not only a mirror; it is also a map, a geography of the mind. Our literature is one such map, if we can learn to read it as our literature, as the product of who and where we have been. We need such a map desperately, we need to know about here, because here is where we live. For the members of a country or a culture, shared knowledge of their place, their here, is not a luxury but a necessity. Without that knowledge we will not survive." (Margaret Atwood, "Survival: A Thematic Guide to Canadian Literature", 1972)

"[…] science is not sacrosanct. The restrictions it imposes (and there are many such restrictions though it is not easy to spell them out) are not necessary in order to have general coherent and successful views about the world. There are myths, there are the dogmas of theology, there is metaphysics, and there are many other ways of constructing a worldview. It is clear that a fruitful exchange between science and such ‘nonscientific’ world-views will be in even greater need of anarchism than is science itself. Thus, anarchism is not only possible, it is necessary both for the internal progress of science and for the development of our culture as a whole." (Paul Feyerabend, "Against Method", 1975)

"The ‘culture’ of a group or class, is the peculiar and distinctive ‘way of life’ of the group or class, the meanings, values and ideas embodied in institutions, in social relations, in systems of beliefs, in mores and customs, in the uses of objects and material life. Culture is the distinctive shapes in which this material and social organization of life expresses itself. A culture includes the ‘maps of meaning’ which make things intelligible to its members. These ‘maps of meaning’ are not simply carried around in the head: they are objectivated in the patterns of social organization and relationship through which the individual becomes a ‘social individual’. Culture is the way the social relations of a group are structured and shaped: but it is also the way those shapes are experienced, understood and interpreted." (John Clark et al "Subcultures, Cultures and Class", 1975)

"The influence of modern physics goes beyond technology. It extends to the realm of thought and culture where it has led to a deep revision in man's conception of the universe and his relation to it." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"I think the most significant creative activities of our or any other human culture - legal and ethical systems, art and music, science and technology - were made possible only through the collaborative work of the left and right cerebral hemispheres. [...] We might say that human culture is the function of corpus callosum." (Carl Sagan, "The Dragons of Eden: Speculations on the Evolution of Human", 1977)

"It is hard for us today to assimilate all the new ideas that are being suggested in response to the new information we have. We must remember that our picture of the universe is based not only on our scientific knowledge but also on our culture and our philosophy. What new discoveries lie ahead no one can say. There may well be civilizations in other parts of our galaxy or in other galaxies that have already accomplished much of what lies ahead for mankind. Others may just be beginning. The universe clearly presents an unending challenge." (Necia H Apfel & J Allen Hynek, "Architecture of the Universe", 1979)

On Culture (1950-1959)

"A social system consists in a plurality of individual actors interacting with each other in a situation which has at least a physical or environmental aspect, actors who are motivated in terms of a tendency to the 'optimization of gratification' and whose relation to their situations, including each other, is defined and mediated in terms of a system of culturally structured and shared symbols." (Talcott Parsons, "The Social System", 1951)

"Culture consists of patterns, explicit and implicit, of and for behavior acquired and transmitted by symbols, constituting the distinctive achievement of human groups, including their embodiments in artifacts; the essential core of culture consists of traditional (i.e., historically derived and selected) ideas and especially their attached values; culture systems may, on the one hand, be considered as products of action, on the other as conditioning elements of further action." (Alfred L Kroeber & Clyde Kluckhohn, "Culture", 1952)

"[...] mathematics is what we make it; not by each of us acting without due regard for what constitutes mathematics in our culture, but by seeking to build up new theories in the light of the old, and to solve outstanding problems generally recognized as valuable for the progress of mathematics as we know it. Until we make it, it fails to 'exist'. And, having been made, it may at some future time even fail to be 'mathematics' any longer." (Raymond L Wilder, "Introduction to the Foundations of Mathematics", 1952)

"Rich in its past, dynamic in the present, prodigious for the future, replete with simple and yet profound ideas and methods, surely mathematics can give something to anyone’s culture." (Rudolph E Langer, "The Things I Should Have Done, I Did Not Do", The American Mathematical Monthly, Vol. 59 (7), 1952)

"The principal mathematical element in the culture, embodying the living and growing mass of modern mathematics, will be chiefly possessed by the professional mathematicians. True, certain professions, such as engineering, physics, and chemistry, which employ a great deal of mathematics, carry a sizable amount of the mathematical tradition, and in some of these, as in the case of physics and engineering research, some individuals contribute to the growth of the mathematical element in the culture. But, in the main, the mathematical element of our culture is dependent for its existence and growth on the class of those individuals known as ‘mathematicians’." (Raymond L Wilder, "Introduction to the Foundations of Mathematics", 1952)

"[...] there is a tendency to forget that all science is bound up with human culture in general, and that scientific findings, even those which at the moment appear the most advanced and esoteric and difficult to grasp, are meaningless outside their cultural context." (Erwin Schrödinger, "Are There Quantum Jumps?", The British Journal for the Philosophy of Science Vol. 3, 1952)

"'World view' differs from culture, ethos, mode of thought, and national character. It is the picture the members of a society have of the properties and characters upon their stage of action. While 'national character' refers to the way these people look to the outsider looking in on them, 'world view' refers to the way the world looks to that people looking out. Of all that is connoted by 'culture', 'world view' attends especially to the way a man, in a particular society, sees himself in relation to all else. It is the properties of existence as distinguished from and related to the self. It is, in short, a man's idea of the universe. It is that organization of ideas which answers to a man the questions: Where am I? Among what do I move? What are my relations to these things? (Robert Redfield, "The Primitive World View", 1952)

"The secret language of statistics, so appealing in a fact-minded culture, is employed to sensationalize, inflate, confuse, and oversimplify. Statistical methods and statistical terms are necessary in reporting the mass data of social and economic trends, business conditions, 'opinion' polls, the census. But without writers who use the words with honesty and understanding and readers who know what they mean, the result can only be semantic nonsense." (Darell Huff, "How to Lie with Statistics", 1954)

"Understanding mathematical logic, or the theory of relativity, is not an indispensable attribute of the cultured mind. But if one wishes to learn anything about these subjects, one must learn something. It is necessary to master the rudiments of the language, to practice a technique, to follow step by step a characteristic sequence of reasoning and to see a problem through from beginning to end." (James R Newman, "The World of Mathematics" Vol. I, 1956)

"The separation between the two cultures has been getting deeper under our eyes; there is now precious little communication between them. [...] The traditional culture [...] is, of course, mainly literary [...] the scientific culture is expansive, not restrictive." (Charles P Snow, New Statesman, 1956)

"A good many times I have been present at gatherings of people who, by the standards of traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative." (Charles P Snow, "The Two Cultures", [lecture] 1959)

"Mathematics is an aspect of culture as well as a collection of algorithms." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

On Culture (1960-1969)

"A world view is not merely a philosophical by-product of each culture, like a shadow, but the very skeleton of concrete cognitive assumptions on which the flesh of customary behavior is hung. World view, accordingly, may be expressed, more or less systematically in cosmology, philosophy, ethics, religious ritual, scientific belief, and so on, but it is implicit in almost every act. In Parsonian terms, it constitutes the set of cognitive orientations of the members of a society." (Anthony F C Wallace, "Culture and Personality", 1961)

"Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture." (George Pólya, "Mathematical Discovery", 1962)

"When technology extends one of our senses, a new translation of culture occurs as swiftly as the new technology is interiorized." (Marshall McLuhan, "The Gutenberg Galaxy", 1962)

"Science contributes to our culture in many ways, as a creative intellectual activity in its own right, as the light which has served to illuminate man’s place in the universe, and as the source of understanding of man’s own nature." (John F Kennedy, Address to the National Academy of Sciences Washington, D.C., 1963)

"Any culture is a series of related structures which comprise social forms, values, cosmology, the whole of knowledge and through which all experience is mediated." (Mary Douglas, "Purity and Danger", 1966)

"Culture, in the sense of the public, standardised values of a community, mediates the experience of individuals. It provides in advance some basic categories, a positive pattern in which ideas and values are tidily ordered. And above all, it has authority, since each is induced to assent because of the assent of others." (Mary Douglas, "Purity and Danger", 1966)

"Human action is ‘cultural’ in that meanings and intentions concerning acts are formed in terms of symbolic systems (including the codes through which they operate in patterns) that focus most generally about the universal of human societies, language." (Talcott Parsons, "Societies: Evolutionary and Comparative Perspectives" 1966)

"After all, Greek thought is expressed not only mythically, in fiction, but also directly, in theorems. The gate through which the Greek world may be discussed - and without the knowledge of which, in my opinion, one’s culture can not be deemed complete - is not necessarily Homer. Greek geometry is a wider gate, through which the eye might grasp an austere, yet essential landscape." (Dan Barbilian, 1967)

"Any understanding of social and cultural change is impossible without a knowledge of the way media works as environments." (Marshall McLuhan, "The Medium is the Massage: An inventory of effects", 1967)

"There isn’t a scientific community. It is a culture. It is a very undisciplined organization." (Isidor Isaac Rabi, "The Politics of Pure Science", 1967)

"No science of any kind can be divorced from ethical considerations [...] Science is a human learning process which arises in certain subcultures in human society and not in others, and a subculture as we seen is a group of people defined by acceptance of certain common values, that is, an ethic which permits extensive communication between them." (Kenneth E Boulding, "Economics As A Moral Science", 1969)

On Culture (1980-1989)

"Culture is the collective programming of the mind distinguishing the members of one group or category of people from others." (Geert Hofstede, "Culture's consequences: International differences in work-related values", 1980)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural." (Stephen J Gould, "The Mismeasure of Man", 1980)

"The dominant culture of a complex society is never a homogeneous structure. It is layered, reflecting different interests within the dominant class (e.g. an aristocratic versus a bourgeois outlook), containing different traces from the past (e.g. religious ideas within a largely secular culture), as well as emergent elements in the present. Subordinate cultures will not always be in open conflict with it. They may, for long periods, coexist with it, negotiate the spaces and gaps in it, make inroads into it, ‘warrening it from within’. However, though the nature of this struggle over culture can never be reduced to a simple opposition, it is crucial to replace the notion of ‘culture’ with the more concrete, historical concept of ‘cultures’; a redefinition which brings out more clearly the fact that cultures always stand in relations of domination - and subordination - to one another, are always, in some sense, in struggle with one another." (Stuart Hall et al, "Encoding, Decoding’, 1980)

"However, for most of us, science functions like myth in that we have no personal experience in the matter. We put our trust in the scientific view given us by our culture and enshrined in its myths. If asked why leaves are green, most of us would probably mutter something about “chlorophyll.” But unless we were specialists, we would simply be repeating the story of someone else’s experience." (Wallace B Clift, "Jung and Christianity", 1982)

"Culture [is] a pattern of basic assumptions invented, discovered, or developed by a given group as it learns to cope with its problems of external adaptation and internal integration that has worked well enough to be considered valid and, therefore, to be taught to new members as the correct way to perceive, think, and feel in relation to those problems." (Edgar H Schein, "Organizational Culture and Leadership", 1985)

"[…] paradigms, the core of the culture of science, are transmitted and sustained just as is culture generally: scientists accept them and become committed to them as a result of training and socialization, and the commitment is maintained by a developed system of social control." (Barry Barnes, "Thomas Kuhn", 1985)

"[...] without imagination, heightened awareness, moral sense, and some reference to the general culture, the engineering experience becomes less meaningful, less fulfilling than it should be." (Samuel C Florman, "The Civilized Engineer", 1985)

"Science develops best when its concepts and conclusions are integrated into the broader human culture and its concerns for ultimate meaning and value." (Pope John Paul II, [letter to Father George V Coyne], 1988)

"A culture may be conceived as a network of beliefs and purposes in which any string in the net pulls and is pulled by the others, thus perpetually changing the configuration of the whole. If the cultural element called morals takes on a new shape, we must ask what other strings have pulled it out of line. It cannot be one solitary string, nor even the strings nearby, for the network is three-dimensional at least." (Jacques Barzun, "The Culture We Deserve", 1989)

On Culture (1990-1999)

"Science is an integral part of culture. It’s not this foreign thing, done by an arcane priesthood. It’s one of the glories of the human intellectual tradition." (Stephen J Gould, Independent (London), 1990)

"Works of science are ways of understanding created through human effort which, like works of art, can be interrogated for what they say about ourselves and our development. By finding out about our science we find out about ourselves." (Joseph Schwartz, "The Creative Moment: How Science Made Itself Alien to Modern Culture", 1992)

"Mathematics was born and nurtured in a cultural environment. Without the perspective which the cultural background affords, a proper appreciation of the content and state of present-day mathematics is hardly possible." (Raymond L Wilder, American Mathematical Monthly, 1994)

"New knowledge is not science until it is made social. The scientific culture can be defined as new verifiable knowledge secured and distributed with fair credit meticulously given." (Edward O Wilson, "Naturalist", 1994)

"Culture in the broadest sense is a form of highly participatory activity, in which people create their societies and identities. Culture shapes individuals, drawing out and cultivating their potentialities and capacities for speech, action, and creativity." (Douglas Kellner, "Media Culture", 1995)

"Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Networks constitute the new social morphology of our societies, and the diffusion of networking logic substantially modifies the operation and outcomes in processes of production, experience, power, and culture. While the networking form of social organization has existed in other times and spaces, the new information technology paradigm provides the material basis for its pervasive expansion throughout the entire social structure." (Manuel Castells, "The Rise of the Network Society", 1996)

"Any global tradition needs to begin with a shared worldview - a culture-independent, globally accepted consensus as to how things are." (Ursula Goodenough, "The Sacred Depths of Nature", 1998)

"Despite being partly familiar to all, because of these contradictory aspects, mathematics remains an enigma and a mystery at the heart of human culture. It is both the language of the everyday world of commercial life and that of an unseen and perfect virtual reality. It includes both free-ranging ethereal speculation and rock-hard certainty. How can this mystery be explained? How can it be unraveled? The philosophy of mathematics is meant to cast some light on this mystery: to explain the nature and character of mathematics. However this philosophy can be purely technical, a product of the academic love of technique expressed in the foundations of mathematics or in philosophical virtuosity. Too often the outcome of philosophical inquiry is to provide detailed answers to the how questions of mathematical certainty and existence, taking for granted the received ideology of mathematics, but with too little attention to the deeper why questions." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"Even revolutionaries conserve; all cultures are conservative. This is so because it is a systemic phenomenon: all systems exist only as long as there is conservation of that which defines them." (Humberto M Romesin & Pille Bunnell, "Biosphere, Homosphere, and robosphere: What has that to do with Business?", Society for Organizational Learning, 1998)

"[Mathematics is] a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context." (Reuben Hersh, "What Is Mathematics, Really?", 1998)

"Cultural archetypes are the unconscious models that help us make sense of the world: they are the myths, narratives, images, symbols, and files into which we organize the data of our life experience" (Clotaire Rapaille, "Cultural Imprints", Executive Excellence Vol. 16 (10), 1999)

"Every culture has a shared pattern of thinking. It is the cement that holds a culture together, gives it unity. A culture's characteristic way of thinking is imbedded in its concept of the nature of reality, its world view. […] A change of world view not only brings about profound cultural changes, but also is responsible for what historians call a ‘change of age’. An age is a period of time in which the prevailing world view has remained relatively unchanged." (Russell L Ackoff,"Re-Creating the Corporation", 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", 1999)

On Culture (2000-2009)

"Numeracy is the ability to process, interpret and communicate numerical, quantitative, spatial, statistical, even mathematical information, in ways that are appropriate for a variety of contexts, and that will enable a typical member of the culture or subculture to participate effectively in activities that they value." (Jeff Evans, "Adults´ Mathematical Thinking and Emotion", 2000)

"The stunning variety of life forms that surround us, as well as the beliefs, practices, techniques, and behavioral forms that constitute human culture, are the product of evolutionary dynamics." (Herbert Gintis, "Game Theory Evolving", 2000)

"A scientific theory or other product is creative only if the innovation gains the acceptance of a field of experts and so transforms the culture." (Jeanne Nakamura & Mihaly Csikszentmihalyi, "Catalytic creativity: The case of Linus Pauling", 2001)

"All human knowledge - including statistics - is created  through people's actions; everything we know is shaped by our language, culture, and society. Sociologists call this the social construction of knowledge. Saying that knowledge is socially constructed does not mean that all we know is somehow fanciful, arbitrary, flawed, or wrong. For example, scientific knowledge can be remarkably accurate, so accurate that we may forget the people and social processes that produced it." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"Ecological rationality uses reason – rational reconstruction – to examine the behavior of individuals based on their experience and folk knowledge, who are ‘naïve’ in their ability to apply constructivist tools to the decisions they make; to understand the emergent order in human cultures; to discover the possible intelligence embodied in the rules, norms and institutions of our cultural and biological heritage that are created from human interactions but not by deliberate human design. People follow rules without being able to articulate them, but they can be discovered." (Vernon L Smith, "Constructivist and ecological rationality in economics", 2002)

"A meme is to thinking what a gene is to evolution. A meme is defined as any idea, behavior, or skill. Like a gene, it can replicate by transferring from one person to another by imitation: stories, fashions, inventions, recipes, songs, ways of plowing a field or throwing a baseball or making a sculpture. Like a gene, it competes with other memes, as ideas and behavior compete in a culture and between cultures." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Financial Systems", 2003)

"Just because people doing science are embedded in a particular social and cultural milieu, it doesn’t follow that science is not universal." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"Art is constructivist in nature, aimed at the deliberate refinement and elaboration of mental models and worldviews. These are the natural products of cognition itself, the outcome of the brain’s tendency to strive for the integration of perceptual and conceptual material over time. […] human culture is essentially a distributed cognitive system within which worldviews and mental models are constructed and shared by the members of a society. Artists are traditionally at the forefront of that process, and have a large influence on our worldviews and mental models." (Mark Turner, "The Artful Mind : cognitive science and the riddle of human creativity", 2006)

"Mathematics can be as effortless as humming a tune, if you know the tune. But our culture does not prepare us for appreciation of mathematics as it does for appreciation of music. Though we start hearing music very early in life, the same cannot be said of mathematics, even though the two subjects are twins. This is a shame; to know music without knowing its mathematics is like hearing a melody without its accompaniment." (Gareth Loy, "Musimathics: The Mathematical Foundations of Music" Vol. 1, 2006)

"The principle of informed responsibility: no culture is justified in creating a cultural separation of its own area; it is the responsibility of each culture to inform other cultures about their own developments and be informed about developments of other cultures." (Andrzej P. Wierzbicki et al, "Creative Space: Models of Creative Processes for the Knowledge Civilization Age", Studies in Computational Intelligence Vol. 10, 2006)

"The principle of systemic integration: whenever needed, knowledge from and about diverse cultures and disciplines might be synthesized by systemic methods, be they soft or hard, without a prior prejudice against any of them, following the principles of open and informed systemic integration." (Andrzej P. Wierzbicki et al, "Creative Space: Models of Creative Processes for the Knowledge Civilization Age", Studies in Computational Intelligence Vol. 10, 2006)

"When great learning comes up against an unsupportive organizational culture, the culture wins every time." (Marc J Rosenberg, "Beyond E-Learning", 2006)

"Beneath the problems that often seem so ‘given’ lie cultural norms and practices and ultimately whole worldviews. Our problems have contexts, backgrounds, roots. These in turn can be shifted and reconstructed. Problems can be circumvented or at least reshaped so that they arise in more manageable forms." (Anthony Weston, "How to Re-Imagine the World", 2007)

"Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems. And the culture evolves because its key concepts and results change, slowly or brutally, to be replaced by new mathematical beacons." (David Ruelle, "The Mathematician's Brain", 2007)

"Our generational perspective contributes to the mental models we hold about ourselves, the world, and the way things ‘should’ be. These beliefs create blind spots that can become our undoing as we pursue our values and seek to accomplish our goals. Likewise, they can have a powerful effect on our culture." (Deborah Gilburg,"Empowering Multigenerational Collaboration in the Workplace", The Systems Thinker Vol. 18 No. 4, 2007)

"A theory is a speculative explanation of a particular phenomenon which derives it legitimacy from conforming to the primary assumptions of the worldview of the culture in which it appears. There can be more than one theory for a particular phenomenon that conforms to a given worldview. […] A new theory may seem to trigger a change in worldview, as in this case, but logically a change in worldview must precede a change in theory, otherwise the theory will not be viable. A change in worldview will necessitate a change in all theories in all branches of study." (M G Jackson, "Transformative Learning for a New Worldview: Learning to Think Differently", 2008)

"Many mathematical game theorists dislike the dependence of an outcome on historical, cultural, or linguistic aspects of the game or on purely arbitrary devices like round numbers; they would prefer the solution be determined purely by the abstract mathematical facts about the game - the number of players, the strategies available to each, and the payoffs to each in relation to the strategy choices of all. We disagree. We think it entirely appropriate that the outcome of a game played by humans interacting in a society should depend on the social and psychological aspects of the game." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Our culture, obsessed with numbers, has given us the idea that what we can measure is more important than what we can't measure. Think about that for a minute. It means that we make quantity more important than quality." (Donella Meadows, "Thinking in Systems: A Primer", 2008)

"The brain seems to be making choices according to some stubborn timing pattern, undoubtedly influenced by both culture and gene." (John Medina, "Brain Rules", 2008)

"If we want to combat the mind viruses responsible for the decline of culture, we need to be conscious of our own programming, consciously adopting memes that take us in the direction we want to go." (Richard Brodie, "Virus of the Mind", 2009)

"If you currently believe in any concepts or subcultures or dogmas that meet these requirements, and you didn’t consciously choose to program yourself with these memes, you are infected with a mind virus." (Richard Brodie, "Virus of the Mind", 2009)

"Mathematicians seek a certain kind of beauty. Perhaps mathematical beauty is a constant - as far as the contents of mathematics are concerned - and yet the forms this beauty takes are certainly cultural. And while the history of mathematics surely is many stranded, one of its most important strands is formed by such cultural forms of mathematical beauty." (Reviel Netz, "Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic", 2009)

"The most popular and prevalent parts of our culture are the most effective at copying memes." (Richard Brodie, "Virus of the Mind", 2009)

"Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity. Why does this matter, and why does it go beyond a cultural oddity of our profession? The answer is that mathematics is deeply embedded in the reasoning used within many branches of knowledge. That reasoning often involves conjectures, assumptions, intuition. But whatever aspect has been reduced to mathematics has an absolute validity. As in other subjects search for truth, the mathematical components embedded in their search are like the boulders in the stream, providing a solid footing on which to cross from one side to the other." (James Glimm, "Reflections and Prospectives", 2009)

On Culture (2010-)

"The answers to the human problems of ecology are to be found in economy. And the answers to the problems of economy are to be found in culture and character. To fail to see this is to go on dividing the world falsely between guilty producers and innocent consumers." (Wendell Berry, "What Are People For?: Essays", 2010)

"Cultures are never merely intellectual constructs. They take form through the collective intelligence and memory, through a commonly held psychology and emotions, through spiritual and artistic communion." (Tariq Ramadan, "Islam and the Arab Awakening", 2012)

"Scientific method is the gateway into scientific discoveries that in turn prompt technological advances and cultural influences." (Hugh G Gauch Jr., "Scientific Method in Brief", 2012)

"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)

"Culture is fuzzy, easy to caricature, amenable to oversimplifications, and often used as a catchall when all other explanations fail." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"All cultures organize themselves around a story, which tells them how the world came into being - a creation myth." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Paradigm change necessarily involves a discontinuous jump. Reality is singular and each paradigm evokes its own reality. This is the reason that scientific paradigms are not changed without a great deal of conflict; the reason why deep thinking is so difficult and involves overcoming so much resistance both in the individual and in the larger culture. In fact it has been said that a scientist never really gives up the paradigm within which she has been trained." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"When a culture is founded on the principle of immediacy of experience, there is no need for numeracy. It is impossible to consume more than one thing at a time, so differentiating between 'a small amount', 'a larger amount' and 'many' is enough for survival." (The Open University, "Understanding the environment: learning and communication", 2016)

"History of mathematics is done by mathematicians as well as historians. History models mathematics as a segment of the ongoing story of human culture. Mathematicians are likely to see the past through the eyes of the present, and ask, ‘Was it important? natural? deep? surprising? elegant?’ The historian sees mathematics as a thread in the ever-growing web of human life, intimately interwoven with finance and technology, with war and peace. Today's mathematics is the culmination of all that has happened before now, yet to future viewpoints it will seem like a brief, outmoded stage of the past." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

On Knowledge (2000-2009)

"Storytelling is the art of unfolding knowledge in a way that makes each piece contribute to a larger truth." (Philip Gerard, "Writing a Book That Makes a Difference", 2000)

"There is a strong tendency today to narrow specialization. Because of the exponential growth of information, we can afford (in terms of both economics and time) preparation of specialists in extremely narrow fields, the various branches of science and engineering having their own particular realms. As the knowledge in these fields grows deeper and broader, the individual's field of expertise has necessarily become narrower. One result is that handling information has become more difficult and even ineffective." (Semyon D Savransky, "Engineering of Creativity", 2000)

"All human knowledge - including statistics - is created  through people's actions; everything we know is shaped by our language, culture, and society. Sociologists call this the social construction of knowledge. Saying that knowledge is socially constructed does not mean that all we know is somehow fanciful, arbitrary, flawed, or wrong. For example, scientific knowledge can be remarkably accurate, so accurate that we may forget the people and social processes that produced it." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"Defined from a societal standpoint, information may be seen as an entity which reduces maladjustment between system and environment. In order to survive as a thermodynamic entity, all social systems are dependent upon an information flow. This explanation is derived from the parallel between entropy and information where the latter is regarded as negative entropy (negentropy). In more common terms information is a form of processed data or facts about objects, events or persons, which are meaningful for the receiver, inasmuch as an increase in knowledge reduces uncertainty." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"Knowledge is factual when evidence supports it and we have great confidence in its accuracy. What we call 'hard fact' is information supported by  strong, convincing evidence; this means evidence that, so far as we know, we cannot deny, however we examine or test it. Facts always can be questioned, but they hold up under questioning. How did people come by this information? How did they interpret it? Are other interpretations possible? The more satisfactory the answers to such questions, the 'harder' the facts."(Joel Best, Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists, 2001)

"Knowledge maps are node-link representations in which ideas are located in nodes and connected to other related ideas through a series of labeled links. They differ from other similar representations such as mind maps, concept maps, and graphic organizers in the deliberate use of a common set of labeled links that connect ideas. Some links are domain specific (e.g., function is very useful for some topic domains...) whereas other links (e.g., part) are more broadly used. Links have arrowheads to indicate the direction of the relationship between ideas." (Angela M. O’Donnell et al, "Knowledge Maps as Scaffolds for Cognitive Processing", Educational Psychology Review Vol. 14 (1), 2002) 

"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)

"The networked world continuously refines, reinvents, and reinterprets knowledge, often in an autonomic manner." (Donald M Morris et al, "A revolution in knowledge sharing", 2003) 

"A mental model is conceived […] as a knowledge structure possessing slots that can be filled not only with empirically gained information but also with ‘default assumptions’ resulting from prior experience. These default assumptions can be substituted by updated information so that inferences based on the model can be corrected without abandoning the model as a whole. Information is assimilated to the slots of a mental model in the form of ‘frames’ which are understood here as ‘chunks’ of knowledge with a well-defined meaning anchored in a given body of shared knowledge." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", 2005)

"Evolution moves towards greater complexity, greater elegance, greater knowledge, greater intelligence, greater beauty, greater creativity, and greater levels of subtle attributes such as love. […] Of course, even the accelerating growth of evolution never achieves an infinite level, but as it explodes exponentially it certainly moves rapidly in that direction." (Ray Kurzweil, "The Singularity is Near", 2005)

“It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct.” (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"Information is just bits of data. Knowledge is putting them together. Wisdom is transcending them." (Ram Dass, "One-Liners: A Mini-Manual for a Spiritual Life (ed. Harmony", 2007)

"Science is not only the enterprise of harnessing nature to serve the practical needs of humankind. It is also part of man’s unending search for knowledge about the universe and his place within it." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

"Critical thinking is essentially a questioning, challenging approach to knowledge and perceived wisdom. It involves ideas and information from an objective position and then questioning this information in the light of our own values, attitudes and personal philosophy." Brenda Judge et al, "Critical Thinking Skills for Education Students", 2009)

"Equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge." (Robert P Crease, "The Great Equations", 2009)

"Traditional statistics is strong in devising ways of describing data and inferring distributional parameters from sample. Causal inference requires two additional ingredients: a science-friendly language for articulating causal knowledge, and a mathematical machinery for processing that knowledge, combining it with data and drawing new causal conclusions about a phenomenon."(Judea Pearl, "Causal inference in statistics: An overview", Statistics Surveys 3, 2009)

Marshall McLuhan - Collected Quotes

"A point of view can be a dangerous luxury when substituted for insight and understanding." (Marshall McLuhan, "The Gutenberg Galaxy", 1962)

"The sheer increase in the quantity of information movement favoured the visual organization of knowledge and the rise of perspective even before typography." (Marshall McLuhan, "The Gutenberg Galaxy", 1962)

"When technology extends one of our senses, a new translation of culture occurs as swiftly as the new technology is interiorized." (Marshall McLuhan, "The Gutenberg Galaxy", 1962)

"All meaning alters with acceleration, because all patterns of personal and political interdependence change with any acceleration of information." (Marshall McLuhan, "Understanding Media", 1964)

"Control over change would seem to consist in moving not with it but ahead of it. Anticipation gives the power to deflect and control force." (Marshall McLuhan, "Understanding Media", 1964)

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

"It is experience, rather than understanding, that influences behaviour." (Marshall McLuhan, "Understanding Media", 1964)

"It is the medium that shapes and controls the scale and form of human association and action." (Marshall McLuhan, "Understanding Media", 1964)

"The mark of our time is its revulsion against imposed patterns." (Marshall McLuhan, "Understanding Media", 1964)

"The 'message' of any medium or technology is the change of scale or pace or pattern that it introduces into human affairs." (Marshall McLuhan, "Understanding Media", 1964)

"The products of modern science are not in themselves good or bad; it is the way they are used that determines their value." (Marshall McLuhan, "Understanding Media", 1964)

"The specialist is one who never makes small mistakes while moving towards the grand fallacy." (Marshall McLuhan, "Understanding Media", 1964)

"Environments are not just containers, but are processes that change the content totally." (Marshall McLuhan, American Scholar Vol. 35, 1965)

"The present is always invisible because its environmental. No environment is perceptible, simply because it saturates the whole field of attention." (Marshall McLuhan, Mademoiselle Vol. 64, 1966)

"One of the effects of living with electric information is that we live habitually in a state of information overload. There's always more than you can cope with." (Marshall McLuhan, "The Best of Ideas", 1967)

"Our technology forces us to live mythically, but we continue to think fragmentarily, and on single, separate planes." (Marshall McLuhan, "The Medium is the Massage: An inventory of effects", 1967)

"Faced with information overload, we have no alternative but pattern-recognition."(Marshall McLuhan, "Counterblast", 1969)

"The future masters of technology will have to be lighthearted and intelligent. The machine easily masters the grim and the dumb." (Marshall McLuhan, "Counterblast", 1969)

"All discoveries in art and science result from an accumulation of errors." (Marshall McLuhan, "Culture Is Our Business", 1970)

"Computers can do better than ever what needn't be done at all. Making sense is still a human monopoly." (Marshall McLuhan, "Take Today: The Executive as Dropout", 1972)

"Every process pushed far enough tends to reverse or flip suddenly. Chiasmus – the reversal to process caused by increasing its speed, scope or size." (Marshall McLuhan, "Take Today: The Executive as Dropout", 1972)

"New technological environments are commonly cast in the molds of the preceding technology out of the sheer unawareness of their designers." (Marshall McLuhan, "Take Today: The Executive as Dropout", 1972)

"Paradox is the technique for seizing the conflicting aspects of any problem. Paradox coalesces or telescopes various facets of a complex process in a single instant." (Marshall McLuhan, "Take Today: The Executive as Dropout", 1972)

"The bias of each medium of communication is far more distorting than the deliberate lie." (Marshall McLuhan, Journalism quarterly Vol. 50, 1973)

"Formal logic and the logical syllogism encapsulate connectedness in reasoning." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"Metaphor has traditionally been regarded as the matrix and pattern of the figures of speech." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"The field of 'information theory' began by using the old hardware paradigm of transportation of data from point to point." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"Without an understanding of causality there can be no theory of communication. What passes as information theory today is not communication at all, but merely transportation." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"One of the effects of living with electric information is that we live habitually in a state of information overload. There's always more than you can cope with." (Marshall McLuhan, "Understanding Me: Lectures and Interviews" , 2003)

"Without an understanding of causality there can be no theory of communication. What passes as information theory today is not communication at all, but merely transportation." (Marshall McLuhan, "The Book of Probes : Marshall McLuhan", 2011) 

"As information becomes our environment, it becomes mandatory to program the environment itself as a work of art." (Marshall McLuhan)

"By simply moving information and brushing information against information, any medium whatever creates vast wealth." (Marshall McLuhan)

"Effects are perceived, whereas causes are conceived. Effects always precede causes in the actual developmental order." (Marshall McLuhan)

"The meaning of experience is typically one generation behind the experience. The content of new situations, both private and corporate, is typically the preceding situation." (Marshall McLuhan)

"When new technologies impose themselves on societies long habituated to older technologies, anxieties of all kinds result." (Marshall McLuhan)

On Symbols (1990-1999)

"When a person has learned a symbolic system well enough to use it, she has established a portable self-contained world within the mind." (Mihaly Csikszentmihalyi, "Flow", 1990)

"Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding." (Robert Kanigel, "The Man Who Knew Infinity", 1991)

"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors." (John Casti, "Reality Rules", 1992)

"Scientific claims or statements are inexact and provisional. They depend on dozens of simplifying assumptions and on a particular choice of words and symbols and on 'all other things being equal'." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"The insight at the root of artificial intelligence was that these 'bits' (manipulated by computers) could just as well stand as symbols for concepts that the machine would combine by the strict rules of logic or the looser associations of psychology." (Daniel Crevier, "AI: The tumultuous history of the search for artificial intelligence", 1993)

"Above all, words must be recognized as symbolic pointers to truth, not objective containers of truth." (John S Spong, "Resurrection: Myth or Reality?", 1994) 

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

"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"Every phenomenon on earth is symbolic, and each symbol is an open gate through which the soul, if it is ready, can enter into the inner part of the world, where you and I and day and night are all one." (Hermann Hesse, "The Fairy Tales of Hermann Hesse", 1995)

“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1995)

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 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)

"Meaning is conferred not by a one-to-one correspondence of a symbol with some external concept or object, but by the relationships between the structural components of the system itself." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

“Cultural archetypes are the unconscious models that help us make sense of the world: they are the myths, narratives, images, symbols, and files into which we organize the data of our life experience” (Clotaire Rapaille, “Cultural Imprints”, Executive Excellence Vol. 16 (10), 1999)

"In broad terms, a mental model is to be understood as a dynamic symbolic representation of external objects or events on the part of some natural or artificial cognitive system. Mental models are thought to have certain properties which make them stand out against other forms of symbolic representations." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

On Complex Numbers XV

"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813) 

"The true meaning of √-1 reveals itself vividly before my soul, but it will be very difficult to express it in words, which can give only an image suspended in the air." (Carl F Gauss, [Letter to Peter Hanson] 1825)

"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analyzing, to greater and greater abstractness and logical simplicity." (Bertrand Russell, "Introduction to Mathematical Philosophy", 1919)

"[…] the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood." (Harold S M Coxeter, "The Real Projective Plane" 3rd Ed, 1993)

"The dictum that everything that people do is 'cultural' licenses the idea that every cultural critic can meaningfully analyze even the most intricate accomplishments of art and science. [...] It is distinctly weird to listen to pronouncements on the nature of mathematics from the lips of someone who cannot tell you what a complex number is!" (Norman Levitt, "The Flight from Science and Reason", Science, 1996)

"At this stage you might be thinking that there is no justification for calling something of the form a+bi a number, even if you are prepared to countenance i = √-1 in the first place. But remember, it is not what numbers are that matters, but how they behave. Provided the complex numbers have a workable and useful (either in mathematics itself or possibly in a wider context) arithmetic, possibly forming a field, then they have as much right to be called 'numbers' as do any others." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"[…] because imaginary time is at right angles to real time, it behaves like a fourth spatial direction. It can therefore have a much richer range of possibilities than the railroad track of ordinary real time, which can only have a beginning or an end or go around in circles. It is in this imaginary sense that time has a shape." (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"To describe how quantum theory shapes time and space, it is helpful to introduce the idea of imaginary time. Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: time measured in what are called imaginary numbers. […] Imaginary numbers can then be represented as corresponding to positions on a vertical line: zero is again in the middle, positive imaginary numbers plotted upward, and negative imaginary numbers plotted downward. Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don't need a physical realization […]" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

Gregory Bateson - Collected Quotes

"Whenever we pride ourselves upon finding a newer, stricter way of thought or exposition; whenever we start insisting too hard upon ‘operationalism’ or symbolic logic or any other of these very essential systems of tramlines, we lose something of the ability to think new thoughts. And equally, of course, whenever we rebel against the sterile rigidity of formal thought and exposition and let our ideas run wild, we likewise lose. As I see it, the advances in scientific thought come from a combination of loose and strict thinking, and this combination is the most precious tool of science." (Gregory Bateson, "Culture Contact and Schismogenesis", 1935)

"In order to proceed with abstraction, the organism must be exposed to a sufficient number of events which contain the same factors. Only then is a person equipped to cope with the most frequent happenings that he may encounter." (Gregory Bateson, "Communication: The Social Matrix of Psychiatry", 1951)

"If a man achieves or suffers change in premises which are deeply embedded in his mind, he will surely find that the results of that change will ramify throughout his whole universe." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"In no system which shows mental characteristics can any part have unilateral control over the whole. In other words, the mental characteristics of the system are imminent, not in some part, but in the system as a whole." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"No organism can afford to be conscious of matters with which it could deal at unconscious levels. Broadly, we can afford to sink those sorts of knowledge which continue to be true regardless of changes in the environment, but we must maintain in an accessible place all those controls of behavior which must be modified for every instance. The economics of the system, in fact, pushes organisms toward sinking into the unconscious those generalities of relationship which remain permanently true and toward keeping within the conscious the pragmatic of particular instances." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"Still more astonishing is that world of rigorous fantasy we call mathematics." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put on paper. What is on the paper map is a representation of what was in the retinal representation of the man who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. […] Always, the process of representation will filter it out so that the mental world is only maps of maps, ad infinitum." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"What we mean by information - the elementary unit of information - is a difference which makes a difference, and it is able to make a difference because the neural pathways along which it travels and is continually transformed are themselves provided with energy. The pathways are ready to be triggered. We may even say that the question is already implicit in them." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"Information is any difference that makes a difference."(Gregory Bateson, "Mind and Nature: A necessary unity", 1979)

"Let's not pretend that mental phenomena can be mapped on to the characteristics of billiard balls." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)

"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)

"The map is not the territory, and the name is not the thing named." (Gregory Bateson, "Mind and Nature: A Necessary Unity", 1979)

"The world partly becomes - comes to be - how it is imagined." (Gregory Bateson, "Mind and Nature: A necessary unity", 1979)

"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)

"Prediction can never be absolutely valid and therefore science can never prove some generalization or even test a single descriptive statement and in that way arrive at final truth." (Gregory Bateson, Mind and Nature: A necessary unity", 1988)

J Robert Oppenheimer - Collected Quotes

"A discovery in science, or a new theory, even when it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalysed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow; it takes a vast world unchallenged and for granted. This is one reason why, however great the novelty or scope of new discovery, we neither can, nor need, rebuild the house of the mind very rapidly. This is one reason why science, for all its revolutions, is conservative. This is why we will have to accept the fact that no one of us really will ever know very much. This is why we shall have to find comfort in the fact that, taken together, we know more and more." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

 "Despite all the richness of what men have learned about the world of nature, of matter and of space, of change and of life, we carry with us today an image of the giant machine as a sign of what the objective world is really like." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

"Knowledge rests on knowledge; what is new is meaningful because it departs slightly from what was known before; this is a world of frontiers, where even the liveliest of actors or observers will be absent most of the time from most of them." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

"Often the very fact that the words of science are the same as those of our common life and tongues can be more misleading than enlightening, more frustrating to understanding than recognizably technical jargon." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

"The history of science is rich in the example of the fruitfulness of bringing two sets of techniques, two sets of ideas, developed in separate contexts for the pursuit of new truth, into touch with one another." (J Robert Oppenheimer, "Science and the Common Understanding", 1954)

"We are not today tempted to search for these keys that unlock the whole of human knowledge and man’s experience. We know that we are ignorant; we are well taught it, and the more surely and deeply we know our own job the better able we are to appreciate the full measure of our pervasive ignorance." (J. Robert Oppenheimer, "Science and the Common Understanding", 1954)

"It is proper to the role of the scientist that he not merely find new truth and communicate it to his fellows, but that he teach, that he try to bring the most honest and intelligible account of new knowledge to all who will try to learn." (J Robert Oppenheimer, "The Open Mind", 1955) 

"Science is not skepticism. It is not the practice of science to look for things to doubt." (J Robert Oppenheimer, "The Open Mind", 1955) 

"The true responsibility of a scientist, as we all know, is to the integrity and vigor of his science. And because most scientists, like all men of learning, tend in part also to be teachers, they have a responsibility for the communication of the truths they have found. This is at least a collective, if not an individual responsibility. That we should see in this any insurance that the fruits of science will be used for man’s benefit, or denied to man when they make for his distress or destruction, would be a tragic naiveté."  (J Robert Oppenheimer, "The Open Mind", 1955) 

"One thing science can do, and rarely does: it can correct the inherited views that it has by accident at another stage given to common sense, and which turn out to be not true." (J Robert Oppenheimer, "The Growth of Science and the Structure of Culture" , Daedalus, 1958) 

"A change in science, whether novelty or discovery, when properly understood, when the linguistic problem is adequately solved, will even then provide only a hunch, a starting point for looking at an area of experience other than the science in which it was nourished and born." (J Robert Oppenheimer, "The Growth of Science and the Structure of Culture", Daedalus, 1958) 

"Taken as a story of human achievement, and human blindness, the discoveries in the sciences are among the great epics." (J Robert Oppenheimer, "Reflections on the resonances of physics history" , 1972) 

"The theory of our modern technic shows that nothing is as practical as the theory." (J Robert Oppenheimer, "Reflex", 1977)

"All history teaches us that these questions that we think the pressing ones will be transmuted before they are answered, that they will be replaced by others, and that the very process of discovery will shatter the concepts that we today use to describe our puzzlement." (J Robert Oppenheimer, "Uncommon Sense", 1984)

"The greatest of the changes that science has brought is the acuity of change; the greatest novelty the extent of novelty." (J Robert Oppenheimer, "Atom and Void" , 1989)

Mental Models XV

“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) 

“Know that the figures employed by prophets are of two kinds: first, where every word which occurs in the simile represents a certain idea: and secondly, where the simile, as a whole, represents a general idea, but has a great many points which have no reference whatever to that idea: they are simply required to give to the simile its proper form and order, or better to conceal the idea: the simile is therefore continued as far as necessary, according to its literal sense.” (Moses Maimonides, “The Guide for the Perplexed”, 1190) 

“With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses.” (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

“Mental schemas are not rigid. By lasting and laborious processes the mental schemas accommodate themselves, in the long run, to the features of real situations and become progressively more fit to manage them and to solve the problems with which we are faced. Each period of mental development is characterized by a system. of basic mental schemas which determine the capacity of the child to learn, to interpret, and to use the information he gets.” (Efraim Fischbein, "Intuition and Proof", For The Leaning of Mathematics 3 (2), 1982)

“People’s views of the world, of themselves, of their own capabilities, and of the tasks that they are asked to perform, or topics they are asked to learn, depend heavily on the conceptualizations that they bring to the task. In interacting with the environment, with others, and with the artifacts of technology, people form internal, mental models of themselves and of the things with which they are interacting. These models provide predictive and explanatory power for understanding the interaction.” (Donald A Norman, “Some observations on Mental Models”, 1983)

"You must know the big ideas in the big disciplines and use them routinely – all of them, not just a few.  Most people are trained in one model – economics, for example – and try to solve all problems in one way. This is a dumb way of handling problems. […] What you need is a latticework of mental models in your head.  And, with that system, things gradually get to fit together in a way that enhances cognition.” (Charles T Munger, “Poor Charlie’s Almanack”, 2005)

“Art is constructivist in nature, aimed at the deliberate refinement and elaboration of mental models and worldviews. These are the natural products of cognition itself, the outcome of the brain’s tendency to strive for the integration of perceptual and conceptual material over time. […] human culture is essentially a distributed cognitive system within which worldviews and mental models are constructed and shared by the members of a society. Artists are traditionally at the forefront of that process, and have a large influence on our worldviews and mental models.” (Mark Turner, “The Artful Mind : cognitive science and the riddle of human creativity”, 2006)

“It is hard to navigate across one’s environment without having some ideas, however coarse, about it. Indeed, to face any situation we must know whether it is real or imaginary, profane or sacred, sensitive or insensitive to our actions, and so on. This is why even lowly organisms develop, if not worldviews, at least rough sensory maps of their immediate environment – as noted by ethologists from the start. But it is generally assumed that only humans can build conceptual models of their environments. And, except for some philosophers, humans distinguish maps from the territories they represent.” (Mario Bunge, “Matter and Mind: A Philosophical Inquiry”, 2010)

“Also known as worldview, mental model, or mind-set, our perspective of the world is based on the sum total of our knowledge and experiences. It defines us, shaping our thoughts and actions because it represents the way we see ourselves and situations, how we judge the relative importance of things, and how we establish a meaningful relationship with everything around us.” (Navi Radjou, Prasad Kaipa, “From Smart to Wise: Acting and Leading with Wisdom”, 2013) 

On Worldviews (2000-2009)

“One measure of the depth of a physical theory is the extent to which it poses serious challenges to aspects of our worldview that had previously seemed immutable.” (Brian Greene, “The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest, for the Ultimate Theory”, 2000)

"When we acquire a language we don’t simply learn how to use the correct words, grammar and conventions for speaking appropriately in context, we also acquire a ‘world view’: an implicit set of assumptions and presuppositions regarding how to understand the world, who and what we are within it, and everything else that is entailed in categorising our experience." (Michael Forrester," Psychology of the Image", 2000)

“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)

"It is not so much that particular languages evolve and then cause us to see the world in a given way, but that language and worldview develop side by side to the point where language becomes so ingrained that it constantly supports a specific way of seeing and structuring the world. In the end it becomes difficult to see the world in any other light."  (F David Peat, "From Certainty to Uncertainty", 2002)

“Art is constructivist in nature, aimed at the deliberate refinement and elaboration of mental models and worldviews. These are the natural products of cognition itself, the outcome of the brain’s tendency to strive for the integration of perceptual and conceptual material over time. […] human culture is essentially a distributed cognitive system within which worldviews and mental models are constructed and shared by the members of a society. Artists are traditionally at the forefront of that process, and have a large influence on our worldviews and mental models.” (Mark Turner, “The Artful Mind : cognitive science and the riddle of human creativity”, 2006) 

“Mathematics is more than a tool and language for science. It is also an end in itself, and as such, it has, over the centuries, affected our worldview in its own right.” (Stephen Hawking, “God Created the Integers”, 2007)

“Mathematics provides a good part of the cultural context for the worlds of science and technology. Much of that context lies not only in the explicit mathematics that is used, but also in the assumptions and worldview that mathematics brings along with it.” (William Byers, “How Mathematicians Think”, 2007)

“Beneath the problems that often seem so ‘given’ lie cultural norms and practices and ultimately whole worldviews. Our problems have contexts, backgrounds, roots. These in turn can be shifted and reconstructed. Problems can be circumvented or at least reshaped so that they arise in more manageable forms.” (Anthony Weston, “How to Re-Imagine the World”, 2007)

“Systemic problems trace back in the end to worldviews. But worldviews themselves are in flux and flow. Our most creative opportunity of all may be to reshape those worldviews themselves. New ideas can change everything.” (Anthony Weston, “How to Re-Imagine the World”, 2007)

“One of the things cognitive science teaches us is that when people define their very identity by a worldview, or a narrative, or a mode of thought, they are unlikely to change-for the simple reason that it is physically part of their brain, and so many other aspects of their brain structure would also have to change; that change is highly unlikely.” (George Lakoff, “The Political Mind: A Cognitive Scientist's Guide to Your Brain and Its Politics”, 2008)

“A worldview must be coherent, logical and adequate. Coherence means that the fundamental ideas constituting the worldview must be seen as proceeding from a single, unifying, overarching concept. A logical worldview means simply that the various ideas constituting it should not be contradictory. Adequate means that it is capable of explaining, logically and coherently, every element of contemporary experience.” (M. G. Jackson, “Transformative Learning for a New Worldview: Learning to Think Differently”, 2008)

“A theory is a speculative explanation of a particular phenomenon which derives it legitimacy from conforming to the primary assumptions of the worldview of the culture in which it appears. There can be more than one theory for a particular phenomenon that conforms to a given worldview. […]  A new theory may seem to trigger a change in worldview, as in this case, but logically a change in worldview must precede a change in theory, otherwise the theory will not be viable. A change in worldview will necessitate a change in all theories in all branches of study.” (M G Jackson, “Transformative Learning for a New Worldview: Learning to Think Differently”, 2008)

“Great stories agree with our worldview. The best stories don't teach people anything new. Instead the best stories agree with what the audience already believes and makes the members of the audience feel smart and secure when reminded how right they were in the thirst place.” (Seth Godin, “All Marketers are Liars”, 2009)

Mental Models XII

"It has become increasingly evident in recent times, however, that nature works on a different plan. Her fundamental laws do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies." (Paul Dirac, "The Principles of Quantum Mechanics", 1935)

"The solution of the difficulty is that the two mental pictures which experiment lead us to form - the one of the particles, the other of the waves - are both incomplete and have only the validity of analogies which are accurate only in limiting cases." (Werner Heisenberg, "The Physical Principles of the Quantum Theory", 1949)

"We must not leap to the fatalistic conclusion that we are stuck with the conceptual scheme that we grew up in. We can change it, bit by bit, plank by plank, though meanwhile there is nothing to carry us along but the evolving conceptual scheme itself. The philosopher's task was well compared by Neurath to that of a mariner who must rebuild his ship on the open sea." (Willard v. O Quine, "The Journal of Philosophy", 1950)

"A conceptual scheme is never discarded merely because of a few stubborn facts with which it cannot be reconciled; a conceptual scheme is either modified or replaced by a better one, never abandoned with nothing left to take its place." (James B Conant, "Science and Common Sense", 1951)

"Our acceptance of an ontology is, I think, similar in principle to our acceptance of a scientific theory, say a system of physics; we adopt, at least insofar as we are reasonable, the simplest conceptual scheme into which the disordered fragments of raw experience can be fitted and arranged." (Willard van Orman Quine, "From a Logical Point of View", 1953)

"Mental pictures offer us an opportunity to practice new traits and attitudes, which otherwise we could not do. This is possible because again - your nervous system cannot tell the difference between an actual experience and one that is vividly imagined." (Maxwell Maltz, "Psycho-Cybernetics", 1960)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience." (Northrop Frye, "The Educated Imagination", 1963)

"Information is recorded in vast interconnecting networks. Each idea or image has hundreds, perhaps thousands, of associations and is connected to numerous other points in the mental network." (Peter Russell, "The Brain Book: Know Your Own Mind and How to Use it", 1979)

"Perceptual interpretive processes are applied to mental images in much the same way that they are applied to actual physical objects. In this sense, imagined objects can be 'interpreted' much like physical objects." (Ronald A. Finke, "Creative Imagery", 1990)

"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)