"Abstraction is itself an abstract word and has no single meaning. […] Every word in our language is abstract, because it represents something else." (Eric Maisel, "The Creativity Book: A Year's Worth of Inspiration and Guidance", 2000)
"In the language of mental models, such past experience provided the default assumptions necessary to fill the gaps in the emerging and necessarily incomplete framework of a relativistic theory of gravitation. It was precisely the nature of these default assumptions that allowed them to be discarded again in the light of novel information - provided, for instance, by the further elaboration of the mathematical formalism - without, however, having to abandon the underlying mental models which could thus continue to function as heuristic orientations." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", [in "The Universe of General Relativity"] 2000)
"A symbol is a mental representation regarding the internal reality referring to its object by a convention and produced by the conscious interpretation of a sign. In contrast to signals, symbols may be used every time if the receiver has the corresponding representation. Symbols also relate to feelings and thus give access not only to information but also to the communicator’s motivational and emotional state. The use of symbols makes it possible for the organism using it to evoke in the receiver the same response it evokes in himself. To communicate with symbols is to use a language." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 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)
"Concept maps have long provided visual languages widely used in many different disciplines and application domains. Abstractly, they are sorted graphs visually represented as nodes having a type, name and content, some of which are linked by arcs. Concretely, they are structured diagrams having discipline- and domain-specific interpretations for their user communities, and, sometimes, formally defining computer data structures. Concept maps have been used for a wide range of purposes and it would be useful to make such usage available over the World Wide Web." (Brian R Gaines, "WebMap: Concept Mapping on the Web", 2001)
"Statistical mechanics is the science of predicting the observable properties of a many-body system by studying the statistics of the behaviour of its individual constituents, be they atoms, molecules, photons etc. It provides the link between macroscopic and microscopic states. […] classical thermodynamics. This is a subject dealing with the very large. It describes the world that we all see in our daily lives, knows nothing about atoms and molecules and other very small particles, but instead treats the universe as if it were made up of large-scale continua. […] quantum mechanics. This is the other end of the spectrum from thermodynamics; it deals with the very small. It recognises that the universe is made up of particles: atoms, electrons, protons and so on. One of the key features of quantum mechanics, however, is that particle behaviour is not precisely determined (if it were, it would be possible to compute, at least in principle, all past and future behaviour of particles, such as might be expected in a classical view). Instead, the behaviour is described through the language of probabilities." (A Mike Glazer & Justin S Wark, "Statistical Mechanics: A survival guide", 2001)
"Systems science, with such an ambition and with its basic Systems Theory, provides a general language with which to tie together various areas in interdisciplinary communication. As such it automatically strives towards a universal science, i.e. to join together the many splintered disciplines with a 'law of laws', applicable to them all and integrating all scientific knowledge." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)
"Those who violate the rules of a language do not enter new territory; they leave the domain of meaningful discourse. Even facts in these circumstances dissolve, because they are shaped by the language and subjected to its limitations." (Paul K Feyerabend,"Conquest of Abundance", 2001)
"What cognitive capabilities underlie our fundamental human achievements? Although a complete answer remains elusive, one basic component is a special kind of symbolic activity - the ability to pick out patterns, to identify recurrences of these patterns despite variation in the elements that compose them, to form concepts that abstract and reify these patterns, and to express these concepts in language. Analogy, in its most general sense, is this ability to think about relational patterns." (Keith Holyoak et al,"Introduction: The Place of Analogy in Cognition", 2001)
"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)
"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. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music." (Northrop Frye,"The Educated Imagination", 2002)
"The claim that scientific models are metaphors is tied to the fact that often an analogy is exploited to construct a model about a phenomenon. [...] Scientific models appear to be, contrary to past research traditions, as central in scientific practice for describing and communicating aspects of the empirical world as metaphors are in ordinary language." (Daniela M Bailer-Jones," Models, Metaphors and Analogies", 2002)
"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)
"Quantum theory forces us to see the limits of our abilities to make images, to create metaphors, and push language to its ends. As we struggle to gaze into the limits of nature we dimly begin to discern something hidden in the dark shadows. That something consists of ourselves, our minds, our language, our intellect, and our imagination, all of which have been stretched to their limits." (F David Peat, "From Certainty to Uncertainty", 2002)
"A person thinking in the nonverbal mode is actually thinking with the meaning of the language in the form of mental pictures of the concepts and ideas it contains. Nonverbal thought doesn't require literacy. An illiterate person can communicate without knowing what the symbols look like. [...] Literacy, then, is established as the person learns how the symbols look and becomes able to recognize them as representing certain things or concepts." (Ronald D Davis & Eldon M Braun, "The Gift of Learning", 2003)
"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. [… Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)
"Contrary to popular belief, mathematics is not a universal language. Rather, mathematics is based on a strict set of definitions and rules that have been instated and to which meaning has been given." (Christopher Tremblay,"Mathematics for Game Developers", 2004)
"Graphical design notations have been with us for a while [...] their primary value is in communication and understanding. A good diagram can often help communicate ideas about a design, particularly when you want to avoid a lot of details. Diagrams can also help you understand either a software system or a business process. As part of a team trying to figure out something, diagrams both help understanding and communicate that understanding throughout a team. Although they aren't, at least yet, a replacement for textual programming languages, they are a helpful assistant." (Martin Fowler," UML Distilled: A Brief Guide to the Standard Object Modeling", 2004)
"Mathematicians have evolved a systematic way of thinking about symmetries that is fairly easy to grasp at the outset and a lot of fun to play with. This almost magical subject is known as group theory. […] Group theory is the mathematical language of symmetry, and it is so important that it seems to play a fundamental role in the very structure of nature. It governs the forces we see and is believed to be the organizing principle underlying all of the dynamics of elementary particles. Indeed, in modem physics the concept of symmetry serves as perhaps the most crucial concept of all. Symmetry principles are now known to dictate the basic laws of physics, to control the structure and dynamics of matter, and to define the fundamental forces in nature. Nature, at its most fundamental level, is defined by symmetry." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Models, analogies and metaphors are closely related, though not identical tools for rational thought. […] A model for something, be it thing or process, can be described in the language of simile as a thing or process analogous to that of which it is a model. […] The model offers us nothing by way of explanation, and no existential hypotheses, but it does provide, in the system of metaphors, a picturesque terminology. Many metaphors are indeed just this, the terminological debris of a dead model." (H Rom Harré, "Modeling: Gateway to the Unknown", 2004)
"Probability is a mathematical language for quantifying uncertainty." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)
"The essence of metaphor is understanding and experiencing one kind of thing in terms of another. […] Metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature." (George Lakoff and Mark Johnson, Metaphors We Live By, 1980)
"Theoretical physics borrows from mathematics (or, if there's none to borrow, they invent new mathematics) in order to create a mathematical roadmap of things that can happen in the real world, in nature. It strives to explain all of the many different phenomena observed in the universe, perhaps ultimately seeking one elegant and economical logical system. However, physicists usually settle for lesser triumphs, in which many physical systems with common and comprehensible behaviors are successfully described. This description is always created in the abstract language of mathematics." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"We have come, in our time, to systematize our understanding of the rules of nature. We say that these rules are the laws of physics. The language of the laws of nature is mathematics. We acknowledge that our understanding of the laws is still incomplete, yet we know how to proceed to enlarge our understanding by means of the 'scientific method' - a logical process of observation and reason that distills the empirically true statements we can make about nature." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"What was impossible, inconceivable, and incoherent based on literal vocabulary becomes possible, conceivable, and coherent through metaphoric redescription. Combinations of terms that were incoherent, in relation to the conventional rules of meaning, become meaningful. Metaphoric description arises from a momentary suspension of the rules for literal vocabulary. The semantics of a metaphor convey an alternative realm of conceptual possibilities, through a new set of possible attributes. Of course, not all scientific language is metaphoric. But when unexpected empirical findings raise serious doubts about a familiar scientific theory, a satisfactory resolution occur through the use of metaphoric vocabulary." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)
"If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language, whose beauty eventually - in retrospect - compensates for all the drudgery." (Volker Runde, "A Taste of Topology", 2005)
"A symbol is an object, act, or event that conveys meaning to others. Symbols can be considered a rich, non-verbal language that vibrantly conveys the organization’s important values concerning how people relate to one another and interact with the environment." (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)
"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"Quantum theory can be described as a new kind of language to be used in a dialogue between us and the systems we study with our instruments. [...] It tells us nothing about what the world would be like in our absence." (Lee Smolin, "The Trouble with Physics: The Rise of String Theory, The Fall of a Science and What Comes Next", 2006)
"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)
"A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"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)
"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain "recipes" for calculating things." (Ian Stewart, "Why Beauty Is Truth", 2007)
"Art and music make manifest, by bringing into conscious awareness, that which has previously been felt only tentatively and internally. Art, in its widest sense, is a form of play that lies at the origin of all making, of language, and of the mind's awareness of its place within the world. Art, in all its forms, makes manifest the spiritual dimension of the cosmos, and expresses our relationship to the natural world. This may have been the cause of that natural light which first illuminated the preconscious minds of early hominids." (F David Peat, "Pathways of Chance", 2007)
"Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception." (David Ruelle, "The Mathematician's Brain", 2007)
"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)
"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)
"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)
"Our best way of summarizing and communicating knowledge tends to be through language. And when mathematics became formalized, it did so essentially by emulating the symbolic structure of traditional human natural language." (Stephen Wolfram, Some Modern Perspectives on the Quest for Ultimate Knowledge, 2007)
"Programming is the ability to talk to the computer in a language it can understand and using grammar and syntax that it can follow to get it to perform useful tasks for you." (Adrian Kingsley-Hughes & Kathie Kingsley-Hughes, "Beginning Programming", 2007)
"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain ‘recipes’ for calculating things." (Ian Stewart, "Why Beauty Is Truth", 2007)
"Mathematics is not an inevitable body of knowledge. Understanding it and doing it requires a consciousness of the ‘rules’ and the awareness that they are rules or conventions. Such awareness is particularly needed at the early stages where we often act as if there is nothing to be surprised about." (Bill Barton, "The Language of Mathematics: Telling Mathematical Tales", 2008)
"A modeling language is usually based on some kind of computational model, such as a state machine, data flow, or data structure. The choice of this model, or a combination of many, depends on the modeling target. Most of us make this choice implicitly without further thinking: some systems call for capturing dynamics and thus we apply for example state machines, whereas other systems may be better specified by focusing on their static structures using feature diagrams or component diagrams. For these reasons a variety of modeling languages are available." (Steven Kelly & Juha-Pekka Tolvanen, "Domain-specific Modeling", 2008)
"A perspective is a map from reality to an internal language such that each distinct object, situation, problem, or event gets mapped to a unique word." (Scott E Page, "The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools and Societies", 2008)
"When the words are used without mental image or concrete objects, we label them as metaphor. […] While concepts are being internalised, language is not only appropriated but metaphorised." (Lynne Cameron, "Metaphor in the construction of a learning environment", 2008)
"Why, though, is symmetry so pervasive in nature? It is not just a matter of aesthetics. Just as it is for me and mathematics, symmetry in nature is about language. It provides a way for animals and plants to convey a multitude of messages, from genetic superiority to nutritional information. Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)
"Mathematics is not a language, it's an adventure." (Paul Lockhart,"A Mathematician's Lament", 2009)
"Fuzzy set theory [...] is primarily concerned with quantifying and reasoning using natural language in which words can have ambiguous meanings. It is widely used in a variety of fields because of its simplicity and similarity to human reasoning." (Tzung-Pei Hong et al, "Genetic-Fuzzy Data Mining Techniques", 2009)
"In our modern era, God and mathematics are usually placed in totally separate arenas of human thought. But [...] this has not always been the case, and even today many mathematicians find the exploration of mathematics akin to a spiritual journey. The line between religion and mathematics becomes indistinct. In the past, the intertwining of religion and mathematics has produced useful results and spurred new areas of scientific thought. […] In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics attempt to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable truths and then often try to apply these truths to the real world. Some atheists claim another similarity: mathematics and religion are the most powerful evidence of the inventive genius of the human race. Of course, there are also many differences between mathematics and religion." (Clifford A Pickover, "The Loom of God: Tapestries of Mathematics and Mysticism", 2009)
"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)
"Symbolic reasoning falls short not only in modeling low level behaviors but is also difficult to ground "into real world interactions and to scale upon dynamic environments […] This has lead many […] to abandon symbolic systems […] and […] focus on parallel distributed, entirely sub-symbolic approaches […] well suited for many learning and control tasks, but difficult to apply [in] areas such as reasoning and language." (Joscha Bach, "Principles of Synthetic Intelligence PSI: An Architecture of Motivated Cognition", 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)
"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)
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