H Rom Harré - Collected Quotes

"A theory describes a hypothetical mechanism or hypothetical structure which stands for the unknown real structure of things and materials. The hypothetical structure is modeled on some real structure known to the scientist and his colleagues. We can speak of the hypothetical mechanism as a model of the real mechanism of nature, and as modeled on some real mechanism we know." (H Rom Harré, "Philosophical Issues and Conceptual Change", Theory Into Practice Vol. 10 (2), 1971)

"Science is the combined effort to find out what sort of behavior ensues when various conditions are fulfilled. Chemistry is the study of reactions, that is, of the behavior of different substances, combined with the effort to discover their natures in virtue of which they behave as they do." (H Rom Harré, "Philosophical Issues and Conceptual Change", Theory Into Practice Vol. 10 (2), 1971)

"An analogy is a relationship between two entities, processes, or what you will, which allows inferences to be made about one of the things, usually that about which we know least, on the basis of what we know about the other. […] The art of using analogy is to balance up what we know of the likenesses against the unlikenesses between two things, and then on the basis of this balance make an inference as to what is called the neutral analogy, that about which we do not know." (Rom Harré," The Philosophies of Science" , 1972) 

"Metaphor and simile are the characteristic tropes of scientific thought, not formal validity of argument." (Rom Harré, "Varieties of Realism", 1986)

"A model can become a symbol when its source of projection is lost or forgotten as, for instance, the bull's head became alpha, but a symbol cannot become a model." (H Rom Harré, "Modeling: Gateway to the Unknown", 2004)

"An object, real or imagined, is not a model in itself. But it functions as a model when it is viewed as being in certain relationships to other things. So the classification of models is ultimately a classification of the ways things and processes can function as models." (H Rom Harré, "Modeling: Gateway to the Unknown", 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)

"The word that has survived throughout, enlarging its meaning and in some circles losing meaning altogether, is the word 'model'. Thinking with models is likewise called 'modeling'. We make and use 'models', concrete representations of the central material entities, structures and processes of a domain into which a scientist might be enquiring. The focus shifts from discourse, talking and writing about nature, the most general theory of which is logic, to modeling, reproducing and representing nature materially, the most general theory of which is analogy." (H Rom Harré, "Modeling: Gateway to the Unknown", 2004)

Daniel Rothbart - Collected Quotes

"A narrative is similar to a model in three ways. First, narratives, like models, are conceptual constructions under the control of a story teller. Second, a narrative replicates some aspects of past experiences, recalling events that are at least temporally remote, and in most cases far away. Here's the present teller, close to the reader or listener, and there at a distance is the tale. Third, a narrative has a projective dimension. Reflection on past activity leads to planning and projection of future activity, so that the story teller anticipates encounters yet to occur. The projective aspect of narratives, and models, is essential for revealing unobserved, but observable, events." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"As the frontiers of science are continually pushed back, and the distance between experimenter and the world widens, the intelligibility of the world demands the construction and manipulation of models. Scientific discourse is often used to convey the information from well-grounded models. Scientific thinking is inescapably modeling and intimately involved with inquiry. ls, is essential for revealing unobserved, but observable, events." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"In sum, an enlightened understanding of both physical and social phenomena is possible through modeling, as various stages of inquiry. Just as real world models are inseparable from experiences of the empirical world, a narrative is thoroughly implicated in a social encounter. Actors resort to narratives as they respond to the movements of others and project possibilities for future encounters. An actor becomes a virtual witness to an idealized scene, drawing upon past encounters to construct a picture of future, hypothetical events." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"The unknown has always been alluring. Since the time of the ancients, scientists have refined techniques, apparatus, and methodologies for disclosing things and events that lie beyond the senses. Some of the greatest discoveries of science occur in areas that transcend the here and now, exposing a world that is bizarre in relation to everyday material bodies. Such discoveries reveal alien beings that challenge our capacities of imagination." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"Through modeling, scientists manipulate symbols with meanings to represent an environment with structure. Such manipulations take place to fulfill a human need, solve a problem, or create a product. When constructing a model, one works in the cognitive space of ideas. Models are used to encapsulate, highlight, replicate or represent patterns of events and the structures of things. Of course, no model provides an exact duplication of the subject matter being modeled. Details are hidden, features are skewed, and certain properties are emphasized. Models are abstract and idealized. As an abstraction, a model omits some features of the subject matter, while retaining only significant properties. As an idealization, a model depicts a subject's properties in a more perfect form." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 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)

Herbert A Simon - Collected Quotes

"All behavior involves conscious or unconscious selection of particular actions out of all those which are physically possible to the actor and to those persons over whom he exercises influence and authority." (Herbert A Simon, "Administrative Behavior: A Study of Decision-making Processes in Administrative Organization", 1947)

"Decision making processes are aimed at finding courses of action that are feasible or satisfactory in the light of multiple goals and constraints." (Herbert A Simon, "Administrative Behavior: A Study of Decision-making Processes in Administrative Organization", 1947)

"In the process of decision those alternatives are chosen which are considered to be appropriate means of reaching desired ends. Ends themselves, however, are often merely instrumental to more final objectives. We are thus led to the conception of a series, or hierarchy, of ends. Rationality has to do with the construction of means-ends chains of this kind." (Herbert A Simon, "Administrative Behavior", 1947)

"It is impossible for the behavior of a single, isolated individual to reach a high degree of rationality. The number of alternatives he must explore is so great, the information he would need to evaluate them so vast that even an approximation to objective rationality is hard to conceive. Individual choice takes place in rationality is hard to conceive. [...] Actual behavior falls short in at least three ways, of objective rationality." (Herbert A Simon, "Administrative Behavior", 1947)

"Many individuals and organization units contribute to every large decision, and the very problem of centralization and decentralization is a problem of arranging the complex system into an effective scheme." (Herbert A Simon, "Administrative Behavior: A Study of Decision-making Processes in Administrative Organization", 1947)

"Rationality requires a choice among all possible alternative behaviors. In actual behavior, only a very few of all these possible alternatives come to mind." (Herbert A Simon, "Administrative Behavior", 1947)

"Rationality requires a complete knowledge and anticipation of the consequences that will follow on each choice. In fact, knowledge of consequences is always fragmentary." (Herbert A Simon, "Administrative Behavior", 1947)

"Roughly speaking, rationality is concerned with the selection of preferred behavior alternatives in terms of some system of values, whereby the consequences of behavior can be evaluated." (Herbert A Simon, "Administrative Behavior", 1947)

"The function of knowledge in the decision-making process is to determine which consequences follow upon which of the alternative strategies. It is the task of knowledge to select from the whole class of possible consequences a more limited subclass, or even (ideally) a single set of consequences correlated with each strategy." (Herbert A Simon, "Administrative Behavior: A Study of Decision-making Processes in Administrative Organization", 1947)

"The principle of bounded rationality [is] the capacity of the human mind for formulating and solving complex problems is very small compared with the size of the problems whose solution is required for objectively rational behavior in the real world - or even for a reasonable approximation to such objective rationality." (Herbert A Simon, "Administrative Behavior", 1947)

"The first consequence of the principle of bounded rationality is that the intended rationality of an actor requires him to construct a simplified model of the real situation in order to deal with it. He behaves rationally with respect to this model, and such behavior is not even approximately optimal with respect to the real world. To predict his behavior we must understand the way in which this simplified model is constructed, and its construction will certainly be related to his psychological properties as a perceiving, thinking, and learning animal." (Herbert A Simon, "Models of Man", 1957)

"The mathematical and computing techniques for making programmed decisions replace man but they do not generally simulate him." (Herbert A Simon, "Management and Corporations 1985", 1960)

"Programs do not merely substitute brute force for human cunning. Increasingly, they imitate-and in some cases improve upon-human cunning." (Herbert A Simon, "Management and Corporations 1985", 1960)

"Roughly, by a complex system I mean one made up of a large number of parts that interact in a nonsimple way. In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole." (Herbert A Simon, "The Architecture of Complexity", Proceedings of the American Philosophical Society, Vol. 106 (6), 1962)

"Thus, the central theme that runs through my remarks is that complexity frequently takes the form of hierarchy, and that hierarchic systems have some common properties that are independent of their specific content. Hierarchy, I shall argue, is one of the central structural schemes that the architect of complexity uses." (Herbert A Simon, "The Architecture of Complexity", Proceedings of the American Philosophical Society Vol. 106 (6), 1962)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"[...] a problem of design exists when (1) there is a language for naming actions and a language for naming states of the world, (2) there is a need to find an action that will produce a specified state of the world or a specified change in the state of the world, and (3) there is no non-trivial process for translating changes in the state of the world into their corresponding actions." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"An adaptive organism is connected with its environment by two kinds of channels. Afferent channels give it information about the state of the environment; efferent channels cause action on the environment. Problem statements define solutions in terms of afferent information to the organism; the organism's task is to discover a set of efferent signals which, changing the state of the environment, will produce the appropriate afferent. But, ab initio, the mapping of efferents on afferents is entirely arbitrary; the relations can only be discovered by experiment, by acting and observing the consequences of action." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"Design problems - generating or discovering alternatives - are complex largely because they involve two spaces, an action space and a state space, that generally have completely different structures. To find a design requires mapping the former of these on the latter. For many, if not most, design problems in the real world systematic algorithms are not known that guarantee solutions with reasonable amounts of computing effort. Design uses a wide range of heuristic devices - like means-end analysis, satisficing, and the other procedures that have been outlined - that have been found by experience to enhance the efficiency of search. Much remains to be learned about the nature and effectiveness of these devices." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"Every problem-solving effort must begin with creating a representation for the problem - a problem space in which the search for the solution can take place. Of course, for most of the problems we encounter in our daily personal or professional lives, we simply retrieve from memory a representation that we have already stored and used on previous occasions. Sometimes, we have to adapt the representation a bit to the new situation, but that is usually a rather simple matter." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Natural science is knowledge about natural objects and phenomena." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Learning is any change in a system that produces a more or less permanent change in its capacity for adapting to its environment. Understanding systems, especially systems capable of understanding problems in new task domains, are learning systems." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Making discoveries belongs to the class of ill-structured problem-solving tasks that have relatively ill-defined goals." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Solving a problem simply means representing it so as to make the solution transparent." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos. […] This is the task of natural science: to show that the wonderful is not incomprehensible, to show how it can be comprehended - but not to destroy wonder. For when we have explained the wonderful, unmasked the hidden pattern, a new wonder arises at how complexity was woven out of simplicity. The aesthetics of natural science and mathematics is at one with the aesthetics of music and painting - both inhere in the discovery of a partially concealed pattern." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The more we are willing to abstract from the detail of a set of phenomena, the easier it becomes to simulate the phenomena. Moreover we do not have to know, or guess at, all the internal structure of the system but only that part of it that is crucial to the abstraction." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"[...] in an information-rich world, the wealth of information means a dearth of something else: a scarcity of whatever it is that information consumes. What information consumes is rather obvious: it consumes the attention of its recipients. Hence a wealth of information creates a poverty of attention and a need to allocate that attention efficiently among the overabundance of information sources that might consume it." (Herbert Simon, "Designing Organizations for an Information-Rich World", 1971)

"But the answers provided by the theory of games are sometimes very puzzling and ambiguous. In many situations, no single course of action dominates all the others; instead, a whole set of possible solutions are all equally consistent with the postulates of rationality." (Herbert A Simon et al, "Decision Making and Problem Solving", Interfaces Vol. 17 (5), 1987)

"[...] problem solving generally proceeds by selective search through large sets of possibilities, using rules of thumb (heuristics) to guide the search. Because the possibilities in realistic problem situations are generally multitudinous, trial-and-error search would simply not work; the search must be highly selective." (Herbert A Simon et al, "Decision Making and Problem Solving", Interfaces Vol. 17 (5), 1987)

"The way in which an uncertain possibility is presented may have a substantial effect on how people respond to it." (Herbert A Simon et al, "Decision Making and Problem Solving", Interfaces Vol. 17 (5), 1987)

Amos Tversky - Collected Quotes

"Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not 'corrected' as a chance process unfolds, they are merely diluted." (Amos Tversky & Daniel Kahneman, "Judgment Under Uncertainty: Heuristics and Biases", Science Vol. 185 (4157), 1974)

"Intuitive judgments of probability are based on a limited number of heuristics that are usually effective but sometimes lead to severe and systematic errors. Research shows, for example, that people judge the probability of a hypothesis by the degree to which it represents the evidence, with little or no regard for its prior probability. Other heuristics lead to an overestimation of the probabilities of highly available or salient events, and to overconfidence in the assessment of subjective probability distributions. These biases are not readily corrected, and they are shared by both naive and statistically sophisticated subjects." (Amos Tversky, "Assessing Uncertainty", Journal of the Royal Statistical Society B Vol. 36 (2), 1974) 

"The theory of expected utility is formulated in terms of an abstract set of consequences, that are the carriers of utilities. The axiomatic theory, by its very nature, leaves the consequences uninterpreted. Any application of the theory, of course, is based on a particular interpretation of the outcomes. Thus, the theory could be valid in one interpretation and invalid in another. The appropriateness of the interpretation, however, cannot be evaluated within the theory." (Amos Tversky, "A Critique of Expected Utility Theory: Descriptive and Normative Considerations", Erkenntnis Vol. 9 (2), 1975)

"A significant property of the value function, called loss aversion, is that the response to losses is more extreme than the response to gains. The common reluctance to accept a fair bet on the toss of a coin suggests that the displeasure of losing a sum of money exceeds the pleasure of winning the same amount. Thus the proposed value function is (i) defined on gains and losses, (ii) generally concave for gains and convex for losses, and (iii) steeper for losses than for gains." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"An essential condition for a theory of choice that claims normative status is the principle of invariance: different representations of the same choice problem should yield the same preference. That is, the preference between options should be independent of their description. Two characterizations that the decision maker, on reflection, would view as alternative descriptions of the same problem should lead to the same choice-even without the benefit of such reflection." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"Effective learning takes place only under certain conditions: it requires accurate and immediate feedback about the relation between the situational conditions and the appropriate response. The necessary feedback is often lacking for the decisions made by managers, entrepreneurs, and politicians because (i) outcomes are commonly delayed and not easily attributable to a particular action; (ii) variability in the environment degrades the reliability of the feedback, especially where outcomes of low probability are involved; (iii) there is often no information about what the outcome would have been if another decision had been taken; and (iv) most important decisions are unique and therefore provide little opportunity for learning." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"The modern theory of decision making under risk emerged from a logical analysis of games of chance rather than from a psychological analysis of risk and value. The theory was conceived as a normative model of an idealized decision maker, not as a description of the behavior of real people." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"The assumption of rationality has a favored position in economics. It is accorded all the methodological privileges of a self-evident truth, a reasonable idealization, a tautology, and a null hypothesis. Each of these interpretations either puts the hypothesis of rational action beyond question or places the burden of proof squarely on any alternative analysis of belief and choice. The advantage of the rational model is compounded because no other theory of judgment and decision can ever match it in scope, power, and simplicity." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"Theories of choice are at best approximate and incomplete. One reason for this pessimistic assessment is that choice is a constructive and contingent process. When faced with a complex problem, people employ a variety of heuristic procedures in order to simplify the representation and the evaluation of prospects. These procedures include computational shortcuts and editing operations, such as eliminating common components and discarding nonessential differences. The heuristics of choice do not readily lend themselves to formal analysis because their application depends on the formulation of the problem, the method of elicitation, and the context of choice." (Amos Tversky & Daniel Kahneman, "Advances in Prospect Theory: Cumulative Representation of Uncertainty" [in "Choices, Values, and Frames"], 2000)

"Whenever there is a simple error that most laymen fall for, there is always a slightly more sophisticated version of the same problem that experts fall for." (Amos Tversky)

On Critical Points I

"From its beginning critical point theory has been concerned with mutual relations between topology and geometric analysis, including differential geometry. Although it may have seemed to many to have been directed in its initial years toward applications of topology to analysis, one now sees that the road from topology to geometric analysis is a two-way street. Today the methods of critical point theory enter into the foundations of almost all studies of analysis or geometry 'in the large'." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)

"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Catastrophe theory is a local theory, telling us what a function looks like  in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...]  involve extrapolating these rock-solid, local results to regions that may  well be distant in time and space from the singularity." (John L Casti, "Five Golden Rules", 1995)

"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)

"The reason catastrophe theory can tell us about such abrupt changes in a system's behavior is that we usually observe a dynamical system when it's at or near its steady-state, or equilibrium, position. And under various assumptions about the nature of the system's dynamical law of motion, the set of all possible equilibrium states is simply the set of critical points of a smooth function closely related to the system dynamics. When these critical points are nondegenerate, Morse's Theorem applies. But it is exactly when they become degenerate that the system can move sharply from one equilibrium position to another. The Thorn Classification Theorem tells when such shifts will occur and what direction they will take." (John L Casti, "Five Golden Rules", 1995)

"The phenomenon of emergence takes place at critical points of instability that arise from fluctuations in the environment, amplified by feedback loops." (Fritjof Capra, "The Hidden Connections", 2002)

"This spontaneous emergence of order at critical points of instability is one of the most important concepts of the new understanding of life. It is technically known as self-organization and is often referred to simply as ‘emergence’. It has been recognized as the dynamic origin of development, learning and evolution. In other words, creativity-the generation of new forms-is a key property of all living systems. And since emergence is an integral part of the dynamics of open systems, we reach the important conclusion that open systems develop and evolve. Life constantly reaches out into novelty." (Fritjof  Capra, "The Hidden Connections", 2002)

"A commonly accepted principle of systems dynamics is that a quantitative change, beyond a critical point, results in a qualitative change. Accordingly, a difference in degree may become a difference in kind. This doesn't mean that an increased quantity of a given variable will bring a qualitative change in the variable itself. However, when the state of a system depends on a set of variables, a quantitative change in one variable beyond the inflection point will result in a change of phase in the state of the system. This change is a qualitative one, representing a whole new set of relationships among the variables involved." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"This spontaneous emergence of order at critical points of instability, which is often referred to simply as 'emergence', is one of the hallmarks of life. It has been recognized as the dynamic origin of development, learning, and evolution. In other words, creativity-the generation of new forms-is a key property of all living systems." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

H Marston Morse - Collected Quotes

"It is possible that analysis in the large may eventually reduce to topology, but not until topology has been greatly broadened. It is equally conceivable that the apparently less general situations which arise with such frequency in problems in analysis in the large may form the canonical cases about which the topology of the future can be built." (Marston Morse, "What is Analysis in the Large?", The American Mathematical Monthly Vol. 49 (6), 1942) 

"A definition is topological if it makes no use of mathematical elements other than those defined in terms of continuous deformations or transformations. Such deformations or transformations take the straightness out of planes and alter lengths and areas." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)

"Mathematicians are led to new problems not only by way of contact with the world of physical experience but also by introspective study of the methods which they have elected to use. The trend of classical analysis has been to break up the object of study into finer and finer elements without end." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)

"The concepts of an equilibrium theory are not put forward as a practical method of attaining an economic Utopia. Lack of economic data, knowledge, political and psychological understanding, are too obvious to permit this. Perhaps the greatest value which these considerations have is qualitative in nature. The implications are negative in the sense that the general mathematical theory of equilibria, points to the high a priori probability that any given state of equilibrium is unstable in character." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)

"There is no conflict between science, philosophy and theology. What conflict there may be is due to a failure of agreement as to the implications of the word 'science'." (Marston Morse, "Science in the Modern World", Mathematics Magazine Vol. 28 (4), 1955)

"This continuity of effort is particularly important in mathematics. It is needed to realize the promise of unity which modern mathematics holds. In no science does it appear truer than- in mathematics that the relatively unexplained universe of known facts can be unified by theories of a general character, built of the bricks of current techniques, if only there could rise enough men of talent with a sense of values that would hold them to their task to the very end." (Marston Morse, "Science in the Modern World", Mathematics Magazine Vol. 28 (4), 1955)

"Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse, 1959)

"From its beginning critical point theory has been concerned with mutual relations between topology and geometric analysis, including differential geometry. Although it may have seemed to many to have been directed in its initial years toward applications of topology to analysis, one now sees that the road from topology to geometric analysis is a two-way street. Today the methods of critical point theory enter into the foundations of almost all studies of analysis or geometry 'in the large'." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)

"Mathematicians are finding that the study of global analysis or differential topology requires a knowledge not only of the separate techniques of analysis, differential geometry, topology, and algebra, but also a deeper understanding of how these fields can join forces." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)

"But mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius." (Marston Morse)

"Discovery in mathematics is not a matter of logic. It is rather the result of mysterious powers which no one understands, and in which unconscious recognition of beauty must play an important part. Out of an infinity of designs, a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse)

"Mathematicians of today are perhaps too exuberant in their desire to build new logical foundations for everything. Forever the foundation and never the cathedral." (Marston Morse)

"Most convincing to me of the spiritual relations between mathematics and music, is my own very personal experience. Composing in an amateurish way, I get exactly the same elevation from a prelude that has come to me at the piano, as I do from a new idea that has come to me in mathematics." (Marston Morse)

"The creative scientist lives in a 'wildness of logic,' where reason is the handmaiden and not the master." (Marston Morse)

Philip Tetlow - Collected Quotes

"An emergent behavior or emergent property can appear when a number of simple items, entities, or agents operate in an environment, forming more complex behaviors as a collective, hence its obvious relevance to systems like the Web. The property itself, therefore, represents a new level of the system’s evolution, signifying a step change in the overall nature of a given system." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"Emergent structures are patterns not created by a single event or rule. There is nothing that commands systems to form such patterns, instead the interactions of each part to its immediate surroundings causes a complex process that leads to order. For such reasons, one might conclude that emergent structures are more than the sum of their parts because emergent order will not arise if the various parts simply coexist; the interaction of these parts is central." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"In plain English, fractal geometry is the geometry of the irregular, the geometry of nature, and, in general, fractals are characterized by infinite detail, infinite length, and the absence of smoothness or derivative." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"No investigation of complexity would be complete without a brief summary of what is often considered to be its most extreme form. Beyond the mathematical upper border of complexity lies the deceptively camouflaged notion of chaos. This is not strictly analogous to the classical interpretations of its name involving shear calamity and confusion. Instead, in mathematical or computational terms, chaos relates to much simpler notions of pattern and organization. It may be random to our native observation, certainly, but it is also far more concisely describable than complexity when inspected using modern mathematical techniques." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"The concept that all systems, no matter how complex, are merely aggregate upon aggregate of simple elemental patterns is still fundamental to the modern-day study of complex and dynamic, nonlinear systems. It is the process of combining and collecting things together that produces apparent randomness. Consequently, many interesting and complex phenomena can usefully be described as 'orderly ensemble properties' and productively understood in terms of the properties and interactions of subphenomena or elements." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"To state that the Web is different from other modern technologies is an obvious and gross oversimplification. The Web is not just different, it is different in a very specific way, and its highly connected, self-organizing complexity sets it apart from all other man-made systems. But many do not understand the very basics of complexity, let alone how these might be applicable to a modern technology or its association with a concept such as life. So it is indeed appropriate to visit this supposed black art in order to give an insight into the very nature of complexity and explain just why our current understandings are so applicable to the Web at a number of levels." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"Under certain circumstances, complex systems can demonstrate stronger types of particular correlation, some forming almost instantaneously to overwhelm their parent and transforming it into something completely and unexpectedly. This is the phenomenon we now know understand as 'emergence', the process by which complex systems transition into something that they once were not. Like complexity, emergence has a spectrum of disguises, being capable of manifesting great subtlety and power." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"When dealing with complexity and complex phenomena such as life, an inescapable problem has to be faced - namely, that by their very nature they are multifaceted, and to reduce any description down the point of even relative simplicity would involve removing much of the very essence that we are striving to capture." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"Wherever we look in our world the complex systems of nature and time seem to preserve the look of details at finer and finer scales. Fractals show a holistic hidden order behind things, a harmony in which everything affects everything else, and, above all, an endless variety of interwoven patterns. Fractal geometry allows bounded curves of infinite length, as well as closed surfaces with infinite area. It even allows curves with positive volume and arbitrarily large groups of shapes with exactly the same boundary." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

Henry P Stapp - Collected Quotes

"A long-range correlation between observables has the interesting property that the equation of motion which governs the propagation of this effect is precisely the equation of motion of a freely moving particle." (Henry P Stapp, "S-Matrix Interpretation of Quantum Theory", 1970)

 "[...] an elementary particle is not an independently existing, unanalyzable entity. It is, in essence a set of relationships that reach outward to other things."  (Henry P Stapp, "S-Matrix Interpretation of Quantum Theory", 1970)

"If the attitude of quantum mechanics is correct, in the strong sense that a description of the substructure underlying experience more complete than the one it provides is not possible, then there is no substantive physical world, in the usual sense of this term. The conclusion here is not the weak conclusion that there may not be a substantive physical world but rather that there definitely is not a substantive physical world." (Henry P Stapp, "S-Matrix Interpretation of Quantum Theory", 1970)

"[the physical world, according to quantum mechanics, is] not a structure built out oi independently existing unanalyzable entities, but lather a web of relationships between elements whose meanings arise wholly from their relationships to the whole." (Henry P Stapp, "S-Matrix Interpretation of Quantum Theory", 1970)

"Our beliefs about ourselves in relation to the world around us are the roots of our values, and our values determine not only our immediate actions, but also, over the course of time, the form of our society. Our beliefs are increasingly determined by science." (Henry P Stapp, "Mind, Matter and Quantum Mechanics", 1993)

"[…] according to the new conception, the physically described world is built not out of bits of matter, as matter was understood in the nineteenth century, but out of objective tendencies - potentialities - for certain discrete, whole actual events to occur. Each such event has both a psychologically described aspect, which is essentially an increment in knowledge, and also a physically described aspect, which is an action that abruptly changes the mathematically described set of potentialities to one that is concordant with the increase in knowledge. This coordination of the aspects of the theory that are described in physical/mathematical terms with aspects that are described in psychological terms is what makes the theory practically useful." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

"Briefly stated, the orthodox formulation of quantum theory asserts that, in order to connect adequately the mathematically described state of a physical system to human experience, there must be an abrupt intervention in the otherwise smoothly evolving mathematically described state of that system." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 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)

"The existing general descriptions of quantum theory emphasize puzzles and paradoxes in a way that tend to make non-physicists leery of using in any significant away the profound changes in our understanding of both man and nature wrought by the quantum revolution. Yet in the final analysis quantum mechanics is more understandable than classical mechanics because it is more deeply in line with our common sense ideas about our role in nature than the ‘automaton’ notion promulgated by classical physics." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

On Axioms (2010-2019)

"A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms." (Raymond S Nickerson, "Mathematical Reasoning", 2010)

"Another feature of Bourbaki is that it rejects intuition of any kind. Bourbaki books tend not to contain explanations, examples, or heuristics. One of the main messages of the present book is that we record mathematics for posterity in a strictly rigorous, axiomatic fashion. This is the mathematician’s version of the reproducible experiment with control used by physicists and biologists and chemists. But we learn mathematics, we discover mathematics, we create mathematics using intuition and trial and error. We draw pictures. Certainly, we try things and twist things around and bend things to try to make them work. Unfortunately, Bourbaki does not teach any part of this latter process." (Steven G Krantz, "The Proof is in the Pudding: The Changing Nature of Mathematical Proof", 2010)

"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C Baez, "Division Algebras and Quantum Theory", 2011)

"While mathematicians now recognize that there is some freedom in the choice of the axioms one uses, not any set of statements can serve as a set of axioms. In particular, every set of axioms must be logically consistent, which is another way of saying that it should not be possible to prove a particular statement simultaneously true and false using the given set of axioms. Also, axioms should always be logically independent - that is, no axiom should be a logical consequence of the others. A statement that is a logical consequence of some of the axioms is a theorem, not an axiom." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the 'religious' fanaticism of professional mathematicians, and their insistence on so-called rigor, that in many cases is misplaced and hypocritical, since it is based on "axioms" that are completely fictional, i. e. those that involve the so-called infinity." (Doron Zeilberger, "Doron Zeilberger's 126th Opinion", 2012)

"System meaning is informed by the circumstances and factors that surround the system. The contextual axiom's propositions are those which bound the system by providing guidance that enables an investigator to understand the set of external circumstances or factors that enable or constrain a particular system. The contextual axiom has three principles: (1) holism, (2) darkness, and (3) complementarity." (Patrick Hester & Kevin Adams," Systemic Thinking: Fundamentals for Understanding Problems and Messes", 2014)

"Mathematicians usually think not in terms of concrete realizations but in terms of rules that are given axiomatically. Mathematics is the art of arguing with some chosen logic and some chosen axioms. As such, it is simply one of the oldest games with symbols and words." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

Alfred S Posamentier - Collected Quotes

"[…] mathematics is not only to teach the algorithms and skills of mathematics - which we will agree are very important - but also to teach for understanding, with an emphasis on reasoning." (Alfred S Posamentier et al, "Exemplary Practices for Secondary Math Teachers", 2007)

"A very basic observation concerning a fundamental property of the world we live in is the existence of objects that can be distinguished from each other. For the definition of a set, it is indeed of crucial importance that things have individuality, because in order to decide whether objects belong to a particular set they must be distinguishable from objects that are not in the set. Without having made the basic experience of individuality of objects, it would be difficult to imagine or appreciate the concept of a set." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"But we also have to know that every model has its limitations. The model of natural numbers and their sums is very successful to determine the number of objects in the union of two different groups of well-distinguished objects. But as a mathematical model, the arithmetic of numbers is not generally true but only validated and confirmed for certain well-controlled situations. […] If a model makes valid predictions in many concrete cases, if it already has been applied and tested successfully in many situations, we have some right to trust in that model. By now, we believe in the model 'natural numbers and their arithmetic' and in its predictions without having to check it every time. We do not expect that the result might be wrong; hence the verification step is not needed any longer for validating the model. If the model had a flaw, it would have been eliminated already in the past." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"Mathematicians usually think not in terms of concrete realizations but in terms of rules that are given axiomatically. Mathematics is the art of arguing with some chosen logic and some chosen axioms. As such, it is simply one of the oldest games with symbols and words." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"Moreover, there is still another important observation that seems to be essential for the idea to group objects into a set: This is the human ability to recognize similarities in different objects. Usually, a collection, or group, consists of objects that somehow belong together, objects that share a common property. While a mathematical set could also be a completely arbitrary collection of unrelated objects, this is usually not what we want to count. We count coins or hours or people, but we usually do not mix these categories." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"Most mathematicians are not particularly worried by the fact that there are natural numbers so huge that they cannot be conceptualized exactly. Typically, when applying numbers to reality, approximate quantities are sufficient, and extremely large numbers would rarely be needed. In theory, the natural numbers are just a sequence whose structure is axiomatically described by the Peano axioms. As a mathematician, one typically does not care about the practical realizability of particular numbers. That every number has a unique successor is simply true by assumption; it needs no practical verification." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"People have often wondered why mathematics is able to describe many aspects of our world with high precision and accuracy. In a sense, this is not so astonishing, because from the very beginning, mathematical concepts have been formed on the basis of human experience - an experience, in turn, formed by the world surrounding us." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The act of counting is governed by five principles. They describe the conditions and prerequisites that make counting possible. We call them the 'BOCIA' principles - from the words Bijection, Ordinality, Cardinality, Invariance, and Abstraction." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The branch of philosophy of mathematics that would not accept objects or expressions that nobody can construct in any practical sense is called ultrafinitism. According to this view, not even the concept of natural numbers would be accepted without restrictions, and, of course, an ultrafinitist would refuse to talk about infinity. To most mathematicians, this view would be too extreme. Reducing mathematics to finite and not-too-large objects would restrict mathematics and its usefulness in an intolerable way." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"[…] the usefulness of mathematics is by no means limited to finite objects or to those that can be represented with a computer. Mathematical concepts depending on the idea of infinity, like real numbers and differential calculus, are useful models for certain aspects of physical reality." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015) 

On Knowledge (-1699)

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)

"Thinking is different from perceiving and is held to be in part imagination, in part judgement: we must therefore first mark off the sphere of imagination and then speak of judgement. If then imagination is that in virtue of which an image arises for us, excluding metaphorical uses of the term, is it a single faculty or disposition relative to images, in virtue of which we discriminate and are either in error or not? The faculties in virtue of which we do this are sense, opinion, knowledge, thought." (Aristotle, "De Anima", cca. 350 BC)

"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)

"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)

"We can get some idea of a whole from a part, but never knowledge or exact opinion. Special histories therefore contribute very little to the knowledge of the whole and conviction of its truth. It is only indeed by study of the interconnexion of all the particulars, their resemblances and differences, that we are enabled at least to make a general survey, and thus derive both benefit and pleasure from history." (Polybius, "The Histories", cca. 150 BC)

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular." (Proclus Lycaeus, cca 5th century)

"All knowledge or cognition possessed by creatures is limited. Infinite knowledge belongs solely to God, because of His infinite nature." (John of Salisbury, "Metalogicon", 1159)

"All things have a way of adding up together, so that one will become more proficient in any proposed branch of learning to the extent that he has mastered neighboring and related departments of knowledge." (John of Salisbury, "Metalogicon", 1159)

"In our acquisition of [scientific] knowledge, investigation is the first step, and comes before comprehension, analysis, and retention. Innate ability, although it proceeds from nature, is fostered by study and exercise. What is difficult when we first try it, becomes easier after assiduous practice, and once the rules for doing it are mastered, very easy, unless languor creeps in, through lapse of use or carelessness, and impedes our efficiency. This, in short, is how all the arts have originated: Nature, the first fundamental, begets the habit and practice of study, which proceeds to provide an art, and the latter, in turn, finally furnishes the faculty whereof we speak. Natural ability is accordingly effective. So, too, is exercise. And memory likewise, is effective, when employed by the two aforesaid. With the help of the foregoing, reason waxes strong, and produces the arts, which are proportionate to [man’s] natural talents." (John of Salisbury, "Metalogicon", 1159)

"There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured […] Of these sciences the gate and key is mathematics […] He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world." (Roger Bacon, "Opus Majus", 1267)

"There are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth unless the mind discovers it by the path of experience." (Roger Bacon, "Opus Majus", 1267)

"That faculty which perceives and recognizes the noble proportions in what is given to the senses, and in other things situated outside itself, must be ascribed to the soul. It lies very close to the faculty which supplies formal schemata to the senses, or deeper still, and thus adjacent to the purely vital power of the soul, which does not think discursively […] Now it might be asked how this faculty of the soul, which does not engage in conceptual thinking, and can therefore have no proper knowledge of harmonic relations, should be capable of recognizing what is given in the outside world. For to recognize is to compare the sense perception outside with the original pictures inside, and to judge that it conforms to them." (Johannes Kepler, "Harmonices Mundi" ["Harmony of the World"] , 1619)

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"[…] the highest probability amounts not to certainty, without which there can be no true knowledge." (John Locke, "An Essay Concerning Human Understanding", 1689)

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)

On Knowledge (1800-1824)

"Knowledge is only real and can only be set forth fully in the form of science, in the form of system." (G W Friedrich Hegel, "The Phenomenology of Mind", 1807)

"It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretched out his arms for others." (Carl F Gauss, [Letter to Farkas Bolyai] 1808)

"Thus then does the Doctrine of Knowledge, which in its substance is the realisation of the absolute Power of intelligising which has now been defined, end with the recognition of itself as a mere Schema in a Doctrine of Wisdom, although indeed a necessary and indispensable means to such a Doctrine: - a Schema, the sole aim of which is, with the knowledge thus acquired, - by which knowledge alone a Will, clear and intelligible to itself and reposing upon itself without wavering or perplexity, is possible, - to return wholly into Actual Life; - not into the Life of blind and irrational Instinct which we have laid bare in all its nothingness, but into the Divine Life which shall become visible to us." (Johann G Fichte, "Outline of the Doctrine of Knowledge", 1810)

"The most important questions of life are, for the most part, really only problems of probability. Strictly speaking one may even say that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, induction and analogy, the principal means for discovering truth, are based on probabilities, so that the entire system of human knowledge is connected with this theory." (Pierre-Simon Laplace, "Theorie Analytique des Probabilités", 1812)

"One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth - induction and analogy - are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities", 1814) 

"[...] all knowledge, and especially the weightiest knowledge of the truth, to which only a brief triumph is allotted between the two long periods in which it is condemned as paradoxical or disparaged as trivial." (Arthur Schopenhauer, "The World as Will and Representation", 1819)

"The highest knowledge can be nothing more than the shortest and clearest road to truth; all the rest is pretension, not performance, mere verbiage and grandiloquence, from which we can learn nothing." (Charles C Colton, "Lacon", 1820)

"We [...] are profiting not only by the knowledge, but also by the ignorance, not only by the discoveries, but also by the errors of our forefathers; for the march of science, like that of time, has been progressing in the darkness, no less than in the light." (Charles C Colton, "Lacon", 1820)

"The aim of every science is foresight. For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past." (Auguste Compte, "Plan des travaux scientifiques nécessaires pour réorganiser la société", 1822)

On Knowledge (1825-1849)

"It is true that of far the greater part of things, we must content ourselves with such knowledge as description may exhibit, or analogy supply; but it is true likewise, that these ideas are always incomplete, and that at least, till we have compared them with realities, we do not know them to be just. As we see more, we become possessed of more certainties, and consequently gain more principles of reasoning, and found a wider base of analogy." (Samuel Johnson, 1825)

"The first steps in the path of discovery, and the first approximate measures, are those which add most to the existing knowledge of mankind." (Charles Babbage, "Reflections on the Decline of Science in England", 1830)

"Our knowledge of circumstances has increased, but our uncertainty, instead of having diminished, has only increased. The reason of this is, that we do not gain all our experience at once, but by degrees; so our determinations continue to be assailed incessantly by fresh experience; and the mind, if we may use the expression, must always be under arms." (Carl von Clausewitz, "On War", 1832)

"Truth in itself is rarely sufficient to make men act. Hence the step is always long from cognition to volition, from knowledge to ability. The most powerful springs of action in men lie in his emotions." (Carl von Clausewitz, "On War", 1832)

"Science and knowledge are subject, in their extension and increase, to laws quite opposite to those which regulate the material world. Unlike the forces of molecular attraction, which cease at sensible distances; or that of gravity, which decreases rapidly with the increasing distance from the point of its origin; the farther we advance from the origin of our knowledge, the larger it becomes, and the greater power it bestows upon its cultivators, to add new fields to its dominions." (Charles Babbage, "On the Economy of Machinery and Manufactures", 1832)

"The peculiar character of mathematical truth is that it is necessarily and inevitably true; and one of the most important lessons which we learn from our mathematical studies is a knowledge that there are such truths." (William Whewell, "Principles of English University Education", 1838)

"[…] in order that the facts obtained by observation and experiment may be capable of being used in furtherance of our exact and solid knowledge, they must be apprehended and analysed according to some Conception which, applied for this purpose, gives distinct and definite results, such as can be steadily taken hold of and reasoned from […]" (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon their History" Vol. 2, 1840)

"But a thousand unconnected observations have no more value, as a demonstrative proof, than a single one. If we do not succeed in discovering causes by our researches, we have no right to create them by the imagination; we must not allow mere fancy to proceed beyond the bounds of our knowledge."(Justus von Liebig, "The Lancet", 1844)

"[…] there do exist among us doctrines of solid and acknowledged certainty, and truths of which the discovery has been received with universal applause. These constitute what we commonly term Sciences; and of these bodies of exact and enduring knowledge, we have within our reach so large and varied a collection, that we may examine them, and the history of their formation, with good prospect of deriving from the study such instruction as we seek." (William Whewell, "The Philosophy of the Inductive Sciences Founded upon Their History" Vol. 1, 1847)

On Knowledge (1850-1874)

"In every branch of knowledge the progress is proportional to the amount of facts on which to build, and therefore to the facility of obtaining data." (James C Maxwell, [Letter to Lewis Campbell] 1851) 

"Remember that accumulated knowledge, like accumulated capital, increases at compound interest: but it differs from the accumulation of capital in this; that the increase of knowledge produces a more rapid rate of progress, whilst the accumulation of capital leads to a lower rate of interest. Capital thus checks its own accumulation: knowledge thus accelerates its own advance. Each generation, therefore, to deserve comparison with its predecessor, is bound to add much more largely to the common stock than that which it immediately succeeds." (Charles Babbage, "The Exposition of 1851: Or the Views of Industry, Science and Government of England", 1851)

"All knowledge is profitable; profitable in its ennobling effect on the character, in the pleasure it imparts in its acquisition, as well as in the power it gives over the operations of mind and of matter. All knowledge is useful; every part of this complex system of nature is connected with every other. Nothing is isolated. The discovery of to-day, which appears unconnected with any useful process, may, in the course of a few years, become the fruitful source of a thousand inventions." (Joseph Henry, "Report of the Secretary" [Sixth Annual Report of the Board of Regents of the Smithsonian Institution for 1851], 1852)

"SYSTEM (σύστημα, σύν ἵστημιavu, to place together) - is a full and connected view of all the truths of some department of knowledge. An organized body of truth, or truths arranged under one and the same idea, which idea is as the life or soul which assimilates all those truths. No truth is altogether isolated. Every truth has relation to some other. And we should try to unite the facts of our knowledge so as to see them in their several bearings. This we do when we frame them into a system. To do so legitimately we must begin by analysis and end with synthesis. But system applies not only to our knowledge, but to the objects of our knowledge. Thus we speak of the planetary system, the muscular system, the nervous system. We believe that the order to which we would reduce our ideas has a foundation in the nature of things. And it is this belief that encourages us to reduce our knowledge of things into systematic order. The doing so is attended with many advantages. At the same time a spirit of systematizing may be carried too far. It is only in so far as it is in accordance with the order of nature that it can be useful or sound." (William Fleming, "Vocabulary of philosophy, mental, moral, and metaphysical; with quotations and references; for the use of students", 1857)

"Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained - a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas." (Chauncey Wright, "The Philosophy of Herbert Spencer", North American Review, 1865)

"One of the greatest obstacles to the free and universal movement of human knowledge is the tendency that leads different kinds of knowledge to separate into systems." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Isolated facts and experiments have in themselves no value, however great their number may be. They only become valuable in a theoretical or practical point of view when they make us acquainted with the law of a series of uniformly recurring phenomena, or, it may be, only give a negative result showing an incompleteness in our knowledge of such a law, till then held to be perfect." (Hermann von Helmholtz, "The Aim and Progress of Physical Science", 1869)

"As systematic unity is what first raises ordinary knowledge to the rank of science, that is, makes a system out of a mere aggregate of knowledge, architectonic is the doctrine of the scientific in our knowledge, and therefore necessarily forms part of the doctrine of method." (Immanuel Kant, "Critique of Pure Reason", 1871)

"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress." (William T Kelvin, "Elements of Natural Philosophy", 1873)

"Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. [...] Numerical precision is the soul of science [...]" (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

On Knowledge (1930-1939)

"The final truth about phenomena resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge is complete. We go beyond the mathematical formula at our own risk; we may find a [nonmathematical] model or picture that helps us to understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault." (James Jeans, "The Mysterious Universe", 1930)

"Scientific discovery and scientific knowledge have been achieved only by those who have gone in pursuit of them without any practical purpose whatsoever in view." (Max Planck, "Where is Science Going?", 1932)

"As facts and knowledge accumulate, the claim of the scientist to an understanding of the world in a certain sense diminishes." (Werner K Heisenberg, "Zur Geschichte der physikalischen Naturerklärung", 1933)

"The urge to knowledge is so deeply rooted in man that it can scarcely be omitted from a list of life's important needs." (Hans Reichenbach, "Atom and Cosmos: The World of Modern Physics", 1933)

"Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. Thus it may be that some part of what one knows may pertain to relations […] between the two subsystems (we shall limit ourselves to two), as follows: if a particular measurement on the first system yields this result, then for a particular measurement on the second the valid expectation statistics are such and such; but if the measurement in question on the first system should have that result, then some other expectation holds for that one the second. […] In this way, any measurement process at all or, what amounts to the same, any variable at all of the second system can be tied to the not-yet-known value of any variable at all of the first, and of course vice versa also." (Erwin Schrödinger, "The Present Situation in Quantum Mechanics", 1935)

"The laws of science are the permanent contributions to knowledge - the individual pieces that are fitted together in an attempt to form a picture of the physical universe in action. As the pieces fall into place, we often catch glimpses of emerging patterns, called theories; they set us searching for the missing pieces that will fill in the gaps and complete the patterns. These theories, these provisional interpretations of the data in hand, are mere working hypotheses, and they are treated with scant respect until they can be tested by new pieces of the puzzle." (Edwin P Whipple, "Experiment and Experience", [Commencement Address, California Institute of Technology] 1938)

"We have discovered that it is actually an aid in the search for knowledge to understand the nature of the knowledge we seek." (Arthur S Eddington, "The Philosophy of Physical Science", 1938)

"[…] reality is a system, completely ordered and fully intelligible, with which thought in its advance is more and more identifying itself. We may look at the growth of knowledge […] as an attempt by our mind to return to union with things as they are in their ordered wholeness. […] and if we take this view, our notion of truth is marked out for us. Truth is the approximation of thought to reality […] Its measure is the distance thought has travelled […] toward that intelligible system […] The degree of truth of a particular proposition is to be judged in the first instance by its coherence with experience as a whole, ultimately by its coherence with that further whole, all comprehensive and fully articulated, in which thought can come to rest." (Brand Blanshard, "The Nature of Thought" Vol. II, 1939) 

"The failure of the social sciences to think through and to integrate their several responsibilities for the common problem of relating the analysis of parts to the analysis of the whole constitutes one of the major lags crippling their utility as human tools of knowledge." (Robert S Lynd, "Knowledge of What?", 1939)

On Knowledge (1940-1949)

"It is by abstraction that one can separate movements, knowledge, and affectivity. And the analysis is, here, so far from being a real dismemberment that it is given only as probable. One can never effectively reduce an [mental] image to its elements, for the reason that an image, like all other psychic syntheses, is something more and different from the sum of its elements. […] We will always go from image to image. Comprehension is a movement which is never-ending, it is the reaction of the mind to an image by another image, to this one by another image and so on, in principle to infinity. "(Jean-Paul Sartre, "The Imaginary: A phenomenological psychology of the imagination", 1940)

"In perception, a knowledge forms itself slowly; in the [mental] image the knowledge is immediate. We see now that the image is a synthetic act which unites a concrete, nonimagined, knowledge to elements which are more actually representative. The image teaches nothing: it is organized exactly like the objects which do produce knowledge, but it is complete at the very moment of its appearance." (Jean-Paul Sartre, "The Psychology of Imagination", 1940)

"Science, in the broadest sense, is the entire body of the most accurately tested, critically established, systematized knowledge available about that part of the universe which has come under human observation. For the most part this knowledge concerns the forces impinging upon human beings in the serious business of living and thus affecting man’s adjustment to and of the physical and the social world. […] Pure science is more interested in understanding, and applied science is more interested in control […]" (Austin L Porterfield, "Creative Factors in Scientific Research", 1941)

“A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling.” (Robert Musil, “Man Without Qualities”, 1943)

"It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge."  (George Pólya, "How to solve it", 1945)

"The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach." (George Pólya, "How to solve it", 1945) 

"Our theory has some bleaker consequences. [...] What is knowledge, if we are but a part of the mechanical system of the world we seek to know? What becomes of our ceaseless effort to explain the universe we live in, if explanation is but a part of the mechanical process?" (Kenneth Craik, "The Nature of Explanation", 1943)

"Whenever a man increases the content of his mind he gains new knowledge, and this occurs each time a new relation is established between the worlds on the two sides of the sense-organs - the world of ideas in an individual mind, and the world of objects existing outside individual minds which is common to us all." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"Science usually advances by a succession of small steps, through a fog in which even the most keen-sighted explorer can seldom see more than a few paces ahead. Occasionally the fog lifts, an eminence is gained, and a wider stretch of territory can be surveyed - sometimes with startling results. A whole science may then seem to undergo a kaleidoscopic ‘rearrangement’, fragments of knowledge being found to fit together in a hitherto unsuspected manner. Sometimes the shock of readjustment may spread to other sciences; sometimes it may divert the whole current of human thought." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"The former distrust of specialization has been supplanted by its opposite, a distrust of generalization. Not only has man become a specialist in practice, he is being taught that special facts represent the highest form of knowledge." (Richard Weaver, "Ideas have Consequences", 1948)

"We all inherit a great deal of useless knowledge, and a great deal of misinformation and error (maps that were formerly thought to be accurate), so that there is always a portion of what we have been told that must be discarded. But the cultural heritage of our civilization that is transmitted to us - our socially pooled knowledge, both scientific and humane - has been valued principally because we have believed that it gives us accurate maps of experience. The analogy of verbal words to maps is an important one [...]. It should be noticed at this point, however, that there are two ways of getting false maps of the world into our heads: first, by having them given to us; second, by creating them ourselves when we misread the true maps given to us." (Samuel I Hayakawa, "Language in Thought and Action", 1949)

On Knowledge (1990-1999)

"[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge." (Howard Gardner, "The Unschooled Mind", 1991)

"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. [...] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)

"We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance." (John A Wheeler, Scientific American Vol. 267, 1992)

"Indeed, knowledge that one will be judged on some criterion of ‘creativeness’ or ‘originality’ tends to narrow the scope of what one can produce (leading to products that are then judged as relatively conventional); in contrast, the absence of an evaluations seems to liberate creativity." (Howard Gardner, "Creating Minds", 1993)

"Knowledge is theory. We should be thankful if action of management is based on theory. Knowledge has temporal spread. Information is not knowledge. The world is drowning in information but is slow in acquisition of knowledge. There is no substitute for knowledge." (William E Deming, "The New Economics for Industry, Government, Education", 1993)

"Worldviews are social constructions, and they channel the search for facts. But facts are found and knowledge progresses, however fitfully. Fact and theory are intertwined, and all great scientists understand the interaction." (Stephen J Gould, "Shields of Expectation - and Actuality", 1993)

"At the very least (there is certainly more), cybernetics implies a new philosophy about (1) what we can know, (2) about what it means for something to exist, and (3) about how to get things done. Cybernetics implies that knowledge is to be built up through effective goal-seeking processes, and perhaps not necessarily in uncovering timeless, absolute, attributes of things, irrespective of our purposes and needs." (Jeff Dooley, "Thoughts on the Question: What is Cybernetics", 1995)

"Crude complexity is ‘the length of the shortest message that will describe a system, at a given level of coarse graining, to someone at a distance, employing language, knowledge, and understanding that both parties share (and know they share) beforehand." (Murray Gell-Mann, "What is Complexity?" Complexity Vol. 1 (1), 1995)

"Humans may crave absolute certainty; they may aspire to it; they may pretend, as partisans of certain religions do, to have attained it. But the history of science - by far the most successful claim to knowledge accessible to humans - teaches that the most we can hope for is successive improvement in our understanding, learning from our mistakes, an asymptotic approach to the Universe, but with the proviso that absolute certainty will always elude us. We will always be mired in error. The most each generation can hope for is to reduce the error bars a little, and to add to the body of data to which error bars apply." (Carl Sagan, "The Demon-Haunted World: Science as a Candle in the Dark", 1995)

"The amount of understanding produced by a theory is determined by how well it meets the criteria of adequacy - testability, fruitfulness, scope, simplicity, conservatism - because these criteria indicate the extent to which a theory systematizes and unifies our knowledge." (Theodore Schick Jr., "How to Think about Weird Things: Critical Thinking for a New Age", 1995)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states." (Robert Hafner & Jim Stewart, "Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’", Science Education 79 (2), 1995) 

"Generalization is the process of matching new, unknown input data with the problem knowledge in order to obtain the best possible solution, or one close to it. Generalization means reacting properly to new situations, for example, recognizing new images, or classifying new objects and situations. Generalization can also be described as a transition from a particular object description to a general concept description. This is a major characteristic of all intelligent systems." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"In the new systems thinking, the metaphor of knowledge as a building is being replaced by that of the network. As we perceive reality as a network of relationships, our descriptions, too, form an interconnected network of concepts and models in which there are no foundations. For most scientists such a view of knowledge as a network with no firm foundations is extremely unsettling, and today it is by no means generally accepted. But as the network approach expands throughout the scientific community, the idea of knowledge as a network will undoubtedly find increasing acceptance." (Fritjof Capra," The Web of Life: a new scientific understanding of living systems", 1996)

"Discourses are ways of referring to or constructing knowledge about a particular topic of practice: a cluster (or formation) of ideas, images and practices, which provide ways of talking about, forms of knowledge and conduct associated with, a particular topic, social activity or institutional site in society. These discursive formations, as they are known, define what is and is not appropriate in our formulation of, and our practices in relation to, a particular subject or site of social activity." (Stuart Hall, "Representation: Cultural Representations and Signifying Practices", 1997)

"Data is discrimination between physical states of things (black, white, etc.) that may convey or not convey information to an agent. Whether it does so or not depends on the agent's prior stock of knowledge." (Max Boisot, "Knowledge Assets", 1998)

"The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)

"Analysis of a system reveals its structure and how it works. It provides the knowledge required to make it work efficiently and to repair it when it stops working. Its product is know-how, knowledge, not understanding. To enable a system to perform effectively we must understand it - we must be able to explain its behavior—and this requires being aware of its functions in the larger systems of which it is a part." (Russell L Ackoff, "Re-Creating the Corporation", 1999)