On Art VII: Knowledge

"The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter." (Augustus De Morgan, "Calculus of Functions" Encyclopaedia of Pure Mathematics, 1847)

"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", 1930)

"The diagrams and circles aid the understanding by making it easy to visualize the elements of a given argument. They have considerable mnemonic value […] They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"There is beauty in discovery. There is mathematics in music, a kinship of science and poetry in the description of nature, and exquisite form in a molecule. Attempts to place different disciplines in different camps are revealed as artificial in the face of the unity of knowledge. All illiterate men are sustained by the philosopher, the historian, the political analyst, the economist, the scientist, the poet, the artisan, and the musician." (Glenn T Seaborg, 1958)

"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)

"There is one great difficulty with a good hypothesis. When it is completed and rounded, the corners smooth and the content cohesive and coherent, it is likely to become a thing in itself, a work of art. It is then like a finished sonnet or a painting completed. One hates to disturb it. Even if subsequent information should shoot a hole in it, one hates to tear it down because it once was beautiful and whole."  (John Steinbeck, "Sea of Cortez", 1982)

"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)

"It is ironic but true: the one reality science cannot reduce is the only reality we will ever know. This is why we need art. By expressing our actual experience, the artist reminds us that our science is incomplete, that no map of matter will ever explain the immateriality of our consciousness." (Jonah Lehrer, "Proust Was a Neuroscientist", 2011)

Scientific Experience VI: Knowledge

"Without experience nothing can be sufficiently known. For there are two modes of acquiring knowledge, namely, by reasoning and by experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain [...] unless the mind discovers it by the method of experience." (Roger Bacon, "Opus Majus", 1267)

"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)

"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact." (Albert Einstein, 1930)

"There is no field of experience which cannot, in principle, be brought under some form of scientific law, and no type of speculative knowledge about the world which it is, in principle, beyond the power of science to give." (Alfred J Ayer, "Language, Truth and Logic", 1936)

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

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

"Models constitute a framework or a skeleton and the flesh and blood will have to be added by a lot of common sense and knowledge of details."(Jan Tinbergen, "The Use of Models: Experience," 1969)

"Concepts form the basis for any science. These are ideas, usually somewhat vague (especially when first encountered), which often defy really adequate definition. The meaning of a new concept can seldom be grasped from reading a one-paragraph discussion. There must be time to become accustomed to the concept, to investigate it with prior knowledge, and to associate it with personal experience. Inability to work with details of a new subject can often be traced to inadequate understanding of its basic concepts." (William C Reynolds & Harry C Perkins, "Engineering Thermodynamics", 1977)

"The evolutionary vision is agnostic in regard to systems in the universe of greater complexity than those of which human beings have clear knowledge. It recognizes aesthetic, moral, and religious ideas and experiences as a species, in this case of mental structures or of images, which clearly interacts with other species in the world's great' ecosystem." (Kenneth Boulding," Ecodynamics: A New Theory of Societal Evolution", 1978)

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

On Series XIV: Knowledge

"Analysis is a method where one assumes that which is sought, and from this, through a series of implications, arrives at something which is agreed upon on the basis of synthesis; because in analysis, one assumes that which is sought to be known, proved, or constructed, and examines what this is a consequence of and from what this latter follows, so that by backtracking we end up with something that is already known or is part of the starting points of the theory; we call such a method analysis; it is, in a sense, a solution in reversed direction. In synthesis we work in the opposite direction: we assume the last result of the analysis to be true. Then we put the causes from analysis in their natural order, as consequences, and by putting these together we obtain the proof or the construction of that which is sought. We call this synthesis." (Pappus of Alexandria, cca. 4th century BC)

"In the series of things those which follow are always aptly fitted to those which have gone before; for this series is not like a mere enumeration of disjointed things, which has only a necessary sequence, but it is a rational connection: and as all existing things are arranged together harmoniously, so the things which come into existence exhibit no mere succession, but a certain wonderful relationship." (Marcus Aurelius, "Meditations", cca. 180 AD)

"With the synthesis of every new concept in the aggregation of coordinate characteristics the extensive or complex distinctness is increased; with the further analysis of concepts in the series of subordinate characteristics the intensive or deep distinctness is increased. The latter kind of distinctness, as it necessarily serves the thoroughness and conclusiveness of cognition, is therefore mainly the business of philosophy and is carried farthest especially in metaphysical investigations." (Immanuel Kant, "Logic", 1800)

"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)

"The steps to scientific as well as other knowledge consist in a series of logical fictions which are as legitimate as they are indispensable in the operations of thought, but whose relations to the phenomena whereof they are the partial and not unfrequently merely symbolical representations must never be lost sight of." (John Stallo, "The Concepts and Theories of Modern Physics", 1884) 

"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line [...]" (Vladimir I Lenin, "On the Question of Dialectics", 1915)

"Knowledge is not a series of self-consistent theories that converges toward an ideal view; it is rather an ever increasing ocean of mutually incompatible (and perhaps even incommensurable) alternatives, each single theory, each fairy tale, each myth that is part of the collection forcing the others into greater articulation and all of them contributing, via this process of competition, to the development of our consciousness." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

On Structure: Structure in Knowledge (2000-2009)

"[...] information feedback about the real world not only alters our decisions within the context of existing frames and decision rules but also feeds back to alter our mental models. As our mental models change we change the structure of our systems, creating different decision rules and new strategies. The same information, processed and interpreted by a different decision rule, now yields a different decision. Altering the structure of our systems then alters their patterns of behavior. The development of systems thinking is a double-loop learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view and then redesign our policies and institutions accordingly." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

"Deep change in mental models, or double-loop learning, arises when evidence not only alters our decisions within the context of existing frames, but also feeds back to alter our mental models. As our mental models change, we change the structure of our systems, creating different decision rules and new strategies. The same information, interpreted by a different model, now yields a different decision. Systems thinking is an iterative learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view, reinventing our policies and institutions accordingly." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 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)

"Representations of real or imaginary structure in the human mind enabling orientation as well as goal orientated actions and movements" (Ralf Wagner, "Customizing Multimedia with Multi-Trees" [in "Encyclopedia of Multimedia Technology and Networking" 2nd Ed.], 2009)

On Structure: Structure in Knowledge (1990-1999)

"A mental model is a knowledge structure that incorporates both declarative knowledge (e.g., device models) and procedural knowledge (e.g., procedures for determining distributions of voltages within a circuit), and a control structure that determines how the procedural and declarative knowledge are used in solving problems (e.g., mentally simulating the behavior of a circuit)." (Barbara Y White & John R Frederiksen, "Causal Model Progressions as a Foundation for Intelligent Learning Environments", Artificial Intelligence 42, 1990)

"The essential idea of semantic networks is that the graph-theoretic structure of relations and abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The great organizing principle of thought is abstraction. By assigning particular things to abstract categories we are able to dispense with irrelevant detail and yet instantly draw copious conclusions about a thing due to its membership in various categories. Semantic networks specify the structure of interrelated abstract categories and use this structure to draw conclusions." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 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)

"All systems evolve, although the rates of evolution may vary over time both between and within systems. The rate of evolution is a function of both the inherent stability of the system and changing environmental circumstances. But no system can be stabilized forever. For the universe as a whole, an isolated system, time’s arrow points toward greater and greater breakdown, leading to complete molecular chaos, maximum entropy, and heat death. For open systems, including the living systems that are of major interest to us and that interchange matter and energy with their external environments, time’s arrow points to evolution toward greater and greater complexity. Thus, the universe consists of islands of increasing order in a sea of decreasing order. Open systems evolve and maintain structure by exporting entropy to their external environments." (L Douglas Kiel, "Chaos Theory in the Social Sciences: Foundations and Applications", 1996)

"Ideas about organization are always based on implicit images or metaphors that persuade us to see, understand, and manage situations in a particular way. Metaphors create insight. But they also distort. They have strengths. But they also have limitations. In creating ways of seeing, they create ways of not seeing. There can be no single theory or metaphor that gives an all-purpose point of view, and there can be no simple 'correct theory' for structuring everything we do." (Gareth Morgan, "Imaginization", 1997)

"[Schemata are] knowledge structures that represent objects or events and provide default assumptions about their characteristics, relationships, and entailments under conditions of incomplete information." (Paul J DiMaggio, "Culture and Cognition", Annual Review of Sociology No. 23, 1997)

"[A mental model] is a relatively enduring and accessible, but limited, internal conceptual representation of an external system (historical, existing, or projected) [italics in original] whose structure is analogous to the perceived structure of that system." (James K Doyle & David N Ford, "Mental models concepts revisited: Some clarifications and a reply to Lane", System Dynamics Review 15 (4), 1999)

"[…] philosophical theories are structured by conceptual metaphors that constrain which inferences can be drawn within that philosophical theory. The (typically unconscious) conceptual metaphors that are constitutive of a philosophical theory have the causal effect of constraining how you can reason within that philosophical framework." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought", 1999)

"What it means for a mental model to be a structural analog is that it embodies a representation of the spatial and temporal relations among, and the causal structures connecting the events and entities depicted and whatever other information that is relevant to the problem-solving talks. […] The essential points are that a mental model can be nonlinguistic in form and the mental mechanisms are such that they can satisfy the model-building and simulative constraints necessary for the activity of mental modeling." (Nancy J Nersessian, "Model-based reasoning in conceptual change", 1999)

On Structure: Structure in Knowledge (1980-1989)

"A schema, then is a data structure for representing the generic concepts stored in memory. There are schemata representing our knowledge about all concepts; those underlying objects, situations, events, sequences of events, actions and sequences of actions. A schema contains, as part of its specification, the network of interrelations that is believed to normally hold among the constituents of the concept in question. A schema theory embodies a prototype theory of meaning. That is, inasmuch as a schema underlying a concept stored in memory corresponds to the meaning of that concept, meanings are encoded in terms of the typical or normal situations or events that instantiate that concept." (David E Rumelhart, "Schemata: The building blocks of cognition", 1980)

"These organizational processes result in our perceptions being structured into units corresponding to objects and properties of objects. It is these larger units that may be stored and later assembled into images that are experienced as quasi-pictorial, spatial entities resembling those evoked during perception itself [...] It is erroneous to equate image representations with mental photographs, since this would overlook the fact that images are composed from highly processed perceptual encodings." (Stephen Kosslyn, "Image and Mind", 1980)

"Facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world's data. Theories are structures of ideas that explain and interpret facts. Facts do not go away while scientists debate rival theories for explaining them." (Stephen J Gould "Evolution as Fact and Theory", 1981)

"At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs." (Philip Johnson-Laird, "Mental Models" 1983)

"Myth is the system of basic metaphors, images, and stories that in-forms the perceptions, memories, and aspirations of a people; provides the rationale for its institutions, rituals and power structure; and gives a map of the purpose and stages of life." (Sam Keen, "The Passionate Life", 1983)

"A mental model is a cognitive construct that describes a person's understanding of a particular content domain in the world." (John Sown, "Conceptual Structures: Information Processing in Mind and Machine", 1984)

"Curiously, the unexpected complexity that has been discovered in nature has not led to a slowdown in the progress of science, but on the contrary to the emergence of new conceptual structures that now appear as essential to our understanding of the physical world - the world that includes us. (Isabelle Stengers, "Order Out of Chaos", 1984)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to motor mechanisms' of the knowledge." (John F Sowa, "Conceptual Structures", 1984)

"The basic idea is that schemata are data structures for representing the generic concepts stored in memory. There are schemata for generalized concepts underlying objects, situations, events, sequences of events, actions, and sequences of actions. Roughly, schemata are like models of the outside world. To process information with the use of a schema is to determine which model best fits the incoming information. Ultimately, consistent configurations of schemata are discovered which, in concert, offer the best account for the input. This configuration of schemata together constitutes the interpretation of the input." (David E Rumelhart, Paul Smolensky, James L McClelland & Geoffrey E Hinton, "Schemata and sequential thought processes in PDP models", 1986)

"A mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding." (Mark Johnson, "The Body in the Mind", 1987)

"The mapping from linguistic inputs to mental models is not a one-one mapping. So semantic properties of sentences may not be recoverable from a mental model. Reading or listening is typically for content not for form. People want to know what is being said to them, not how it is being said. [...] A mental model is a representation of the content of a text that need bear no resemblance to any of the text's linguistic representations. Its structure is similar to the situation described by the text." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"[…] a mental model is a mapping from a domain into a mental representation which contains the main characteristics of the domain; a model can be ‘run’ to generate explanations and expectations with respect to potential states. Mental models have been proposed in particular as the kind of knowledge structures that people use to understand a specific domain […]" (Helmut Jungermann, Holger Schütz & Manfred Thuering, "Mental models in risk assessment: Informing people about drugs", Risk Analysis 8 (1), 1988)

"Model is used as a theory. It becomes theory when the purpose of building a model is to understand the mechanisms involved in the developmental process. Hence as theory, model does not carve up or change the world, but it explains how change takes place and in what way or manner. This leads to build change in the structures." (Laxmi K Patnaik, "Model Building in Political Science", The Indian Journal of Political Science Vol. 50 (2), 1989)

On Structure: Structure in Knowledge (1970-1979)

"In general, one might define a complex of semantic components connected by logical constants as a concept. The dictionary of a language is then a system of concepts in which a phonological form and certain syntactic and morphological characteristics are assigned to each concept. This system of concepts is structured by several types of relations. It is supplemented, furthermore, by redundancy or implicational rules […] representing general properties of the whole system of concepts. […] At least a relevant part of these general rules is not bound to particular languages, but represents presumably universal structures of natural languages. They are not learned, but are rather a part of the human ability to acquire an arbitrary natural language." (Manfred Bierwisch, "Semantics", 1970)

"Mental models are fuzzy, incomplete, and imprecisely stated. Furthermore, within a single individual, mental models change with time, even during the flow of a single conversation. The human mind assembles a few relationships to fit the context of a discussion. As debate shifts, so do the mental models. Even when only a single topic is being discussed, each participant in a conversation employs a different mental model to interpret the subject. Fundamental assumptions differ but are never brought into the open. […] A mental model may be correct in structure and assumptions but, even so, the human mind - either individually or as a group consensus - is apt to draw the wrong implications for the future." (Jay W Forrester, "Counterintuitive Behaviour of Social Systems", Technology Review, 1971)

"The essential functions of the mind consist in understanding and in inventing, in other words, in building up structures by structuring reality." (Jean Piaget, 1971)

"A person is changed by the contingencies of reinforcement under which he behaves; he does not store the contingencies. In particular, he does not store copies of the stimuli which have played a part in the contingencies. There are no 'iconic representations' in his mind; there are no 'data structures stored in his memory'; he has no 'cognitive map' of the world in which he has lived. He has simply been changed in such a way that stimuli now control particular kinds of perceptual behavior." (Burrhus F Skinner, "About behaviorism", 1974)

"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful' . In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed." (Michael Dummett, "The philosophical basis of intuitionistic logic", 1975)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"Owing to his lack of knowledge, the ordinary man cannot attempt to resolve conflicting theories of conflicting advice into a single organized structure. He is likely to assume the information available to him is on the order of what we might think of as a few pieces of an enormous jigsaw puzzle. If a given piece fails to fit, it is not because it is fraudulent; more likely the contradictions and inconsistencies within his information are due to his lack of understanding and to the fact that he possesses only a few pieces of the puzzle. Differing statements about the nature of things […] are to be collected eagerly and be made a part of the individual's collection of puzzle pieces. Ultimately, after many lifetimes, the pieces will fit together and the individual will attain clear and certain knowledge." (Alan R Beals, "Strategies of Resort to Curers in South India" [contributed in Charles M. Leslie (ed.), "Asian Medical Systems: A Comparative Study", 1976]) 

"The cognitive map is not a picture or image which 'looks like' what it represents; rather, it is an information structure from which map-like images can be reconstructed and from which behaviour dependent upon place information can be generated." (John O'Keefe & Lynn Nadel, "The Hippocampus as a Cognitive Map", 1978)

"The evolutionary vision is agnostic in regard to systems in the universe of greater complexity than those of which human beings have clear knowledge. It recognizes aesthetic, moral, and religious ideas and experiences as a species, in this case of mental structures or of images, which clearly interacts with other species in the world's great' ecosystem." (Kenneth Boulding," Ecodynamics: A New Theory of Societal Evolution", 1978)

"The use of metaphor is one of many devices available to the scientific community to accomplish the task of accommodation of language to the causal structure of the world." (Richard Boyd, "Metaphor and theory change: what is ‘metaphor’ a metaphor for?", 1979)

On Structure: Structure in Knowledge (1960-1969)

"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one’s craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful." (Jerome S Bruner, "On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)

"For Science in its totality, the ultimate goal is the creation of a monistic system in which - on the symbolic level and in terms of the inferred components of invisibility and intangibly fine structure - the world’s enormous multiplicity is reduced to something like unity, and the endless successions of unique events of a great many different kinds get tidied and simplified into a single rational order. Whether this goal will ever be reached remains to be seen. Meanwhile we have the various sciences, each with its own system coordinating concepts, its own criterion of explanation." (Aldous Huxley, "Literature and Science", 1963)

"This other world is the so-called physical world image; it is merely an intellectual structure. To a certain extent it is arbitrary. It is a kind of model or idealization created in order to avoid the inaccuracy inherent in every measurement and to facilitate exact definition." (Max Planck, "The Philosophy of Physics", 1963)

"[...] 'information' is not a substance or concrete entity but rather a relationship between sets or ensembles of structured variety." (Walter F Buckley, "Sociology and modern systems theory", 1967)

"It [knowledge] is clearly related to information, which we can now measure; and an economist especially is tempted to regard knowledge as a kind of capital structure, corresponding to information as an income flow. Knowledge, that is to say, is some kind of improbable structure or stock made up essentially of patterns - that is, improbable arrangements, and the more improbable the arrangements, we might suppose, the more knowledge there is." (Kenneth E Boulding, "Beyond Economics: Essays on Society", 1968)

"Knowing reality means constructing systems of transformations that correspond, more or less adequately, to reality. They are more or less isomorphic to transformations of reality. The transformational structures of which knowledge consists are not copies of the transformations in reality; they are simply possible isomorphic models among which experience can enable us to choose. Knowledge, then, is a system of transformations that become progressively adequate." (Jean Piaget, "Genetic Epistemology", 1968)

"The idea of knowledge as an improbable structure is still a good place to start. Knowledge, however, has a dimension which goes beyond that of mere information or improbability. This is a dimension of significance which is very hard to reduce to quantitative form. Two knowledge structures might be equally improbable but one might be much more significant than the other." (Kenneth E Boulding, "Beyond Economics: Essays on Society", 1968)

"Modern science is characterized by its ever-increasing specialization, necessitated by the enormous amount of data, the complexity of techniques and of theoretical structures within every field. Thus science is split into innumerable disciplines continually generating new subdisciplines. In consequence, the physicist, the biologist, the psychologist and the social scientist are, so to speak, encapusulated in their private universes, and it is difficult to get word from one cocoon to the other." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Visual thinking calls, more broadly, for the ability to see visual shapes as images of the patterns of forces that underlie our existence - the functioning of minds, of bodies or machines, the structure of societies or ideas." (Rudolf Arnheim, "Visual Thinking", 1969)

On Networks (1970-1979)

"Nature is a network of happenings that do not unroll like a red carpet into time, but are intertwined between every part of the world; and we are among those parts. In this nexus, we cannot reach certainty because it is not there to be reached; it goes with the wrong model, and the certain answers ironically are the wrong answers. Certainty is a demand that is made by philosophers who contemplate the world from outside; and scientific knowledge is knowledge for action, not contemplation. There is no God’s eye view of nature, in relativity, or in any science: only a man’s eye view." (Jacob Bronowski, "The Identity of Man", 1972)

"In the province of the mind, what one believes to be true is true or becomes true, within certain limits to be found experientially and experimentally. These limits are further beliefs to be transcended. In the mind, there are no limit. […] In the province of connected minds, what the network believes to be true, either is true or becomes true within certain limits to be found experientially and experimentally. These limits are further beliefs to be transcended. In the network's mind there are no limits." (John C Lilly, "The Human Biocomputer", 1974)

"As with any graphic, networks are used in order to discover pertinent troups of to inform others of the groups and structures discovered. It is a good means of displaying structures, However, it ceases to be a means of discovery when the elements are numerous. The figure rapidly becomes complex, illegible and untransformable." (Jacques Bertin, "Graphics and graphic information processing", 1977)

"An autopoietic system is organized (defined as a unity) as a network of processes of production (transformation and destruction) of components that produces the components that: (a) through their interactions and transformations continuously regenerate and realize the network of processes (relations) that produce them and, (b) constitute it (the machine) as a concrete unity in the space in which they exist by specifying the topological domain of its realization as such a network." (Francisco Varela, "Principles of Biological Autonomy", 1979)

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

On Knowledge (1900-1929)

"A system is not so important as a method. A system is of significance because it brings order and clearness into our knowledge, but he who hopes by its help to reach something more, he who thinks to extend his knowledge by means of a system is self-deceived." (Harald Høffding, "A history of modern philosophy", 1900)

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments." (David Hilbert, "Mathematical Problems", Bulletin American Mathematical Society Vol. 8, 1901-1902)

"Man's determination not to be deceived is precisely the origin of the problem of knowledge. The question is always and only this: to learn to know and to grasp reality in the midst of a thousand causes of error which tend to vitiate our observation." (Federigo Enriques, "Problems of Science", 1906)

"If the fresh facts which come to our knowledge all fit themselves into the scheme, then our hypothesis may gradually become a solution." (Arthur C Doyle, "The Adventure of Wisteria Lodge", 1908)

"Knowledge is the distilled essence of our intuitions, corroborated by experience." (Elbert Hubbard, "A Thousand & One Epigrams, 1911)

"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)

"The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another." (Henri Poincaré, 1913)

"By intuition is frequently understood perception, or the knowledge of actual reality, the apprehension of something as real. […] Intuition is the undifferentiated unity of the perception of the real and of the simple image of the possible. " (Benedetto Croce, "The Essence of Æsthetic", 1921)

"Observed facts must be built up, woven together, ordered, arranged, systematized into conclusions and theories by reflection and reason, if they are to have full bearing on life and the universe. Knowledge is the accumulation of facts. Wisdom is the establishment of relations. And just because the latter process is delicate and perilous, it is all the more delightful." (Gamaliel Bradford, "Darwin", 1926)

"Hypothesis, however, is an inference based on knowledge which is insufficient to prove its high probability." (Frederick L Barry, "The Scientific Habit of Thought", 1927) 

"With fuller knowledge we should sweep away the references to probability and substitute the exact facts." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

John M Ziman - Collected Quotes

"Many philosophers have now sadly come to the conclusion that there is no ultimate procedure which will wring the last drops of uncertainty from what scientists call their knowledge." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

"Although the best and most famous scientific discoveries seem to open whole new windows of the mind, a typical scientific paper has never pretended to be more than another piece in a large jig-saw not significant in itself but as an element in a grander scheme. This technique, of soliciting many modest contributions to the vast store of human knowledge, has been the secret of western science since the seventeenth century, for it achieves a corporate collective power that is far greater than any one individual can exert. Primary scientific papers are not meant to be final statements of indisputable truths; each is merely a tiny tentative step forward, through the jungle of ignorance." (John M Zimer, Vol. 224, 1969)

"It is not enough to observe, experiment, theorize, calculate and communicate; we must also argue, criticize, debate, expound, summarize, and otherwise transform the information that we have obtained individually into reliable, well established, public knowledge." (John M Ziman, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)

"The sooner we all face up to the fact that theory and practice are indissoluble, and that there is no contradiction between the qualities of usefulness and beauty, the better." (John M Ziman, "Growth and Spread of Science", Nature Vol. 221 (5180), 1969)

"A significant fraction of the ordinary scientific literature in any field is concerned with essentially irrational theories put forward by a few well-established scholars who have lost touch with reality." (John M Ziman, "Some Pathologies of the Scientific Life", Nature Vol. 227, 1970)

"The communication of modern science to the ordinary citizen, necessary, important, desirable as it is, cannot be considered an easy task. The prime obstacle is lack of education. [...] There is also the difficulty of making scientific discoveries interesting and exciting without completely degrading them intellectually. [...] It is a weakness of modern science that the scientist shrinks from this sort of publicity, and thus gives an impression of arrogant mystagoguery." (John M Ziman,"The Force of Knowledge: The Scientific Dimension of Society", 1976)

"Physics defines itself as the science devoted to discovering, developing and refining those aspects of reality that are amenable to mathematical analysis." (John M Ziman, "Reliable Knowledge: An Exploration of the Grounds for Belief in Science", 1978)

"The most astonishing achievements of science, intellectually and practically, have been in physics, which many people take to be the ideal type of scientific knowledge. In fact, physics is a very special type of science, in which the subject matter is deliberately chosen so as to be amenable to quantitative analysis." (John M Ziman, "Reliable Knowledge: An Exploration of the Grounds for Belief in Science", 1978)

"'Disorder' is not mere chaos; it implies defective order." (John M Ziman, "Models of Disorder", 1979)

"A philosopher is a person who knows less and less about more and more, until he knows nothing about everything. […] A scientist is a person who knows more and more about less and less, until he knows everything about nothing." (John M Ziman, "Knowing Everything about Nothing: Specialization and Change in Scientific Careers", 1987)

"Any research organization requires generous measures of the following: (1) Social space for personal initiative and creativity; (2) Time for ideas to grow to maturity; (3) Openness to debate and criticism; (4) Hospitality towards novelty; and (5) Respect for specialized expertise." (John M Ziman, "Prometheus Bound", 1994)

"Theoretical physicists are like pure mathematicians, in that they are often interested in the hypothetical behaviour of entirely imaginary objects, such as parallel universes, or particles traveling faster than light, whose actual existence is not being seriously proposed at all." (John M Ziman, "Real Science: What it Is, and what it Means", 2000)

Out of Context: Knowledge is ... (Definitions)

"Knowledge is made by oblivion, and to purchase a clear and warrantable body of truth, we must forget and part with much we know." (Sir Thomas Browne, "Pseudodoxia Epidemica", 1646)

"Those who have not imbibed the prejudices of philosophers, are easily convinced that natural knowledge is to be founded on experiment and observation." (Colin Maclaurin, "An Account of Sir Isaac Newton’s Philosophical Discoveries", 1748)

"Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts." (Immanuel Kant, "Critique of Pure Reason", 1781)

"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)

"One may even say, strictly speaking, that almost all our knowledge is only probable [...]" (Pierre-Simon Laplace, "Philosophical Essay on Probabilities", 1814)

"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." (Joseph Henry, "Report of the Secretary" [Sixth Annual Report of the Board of Regents of the Smithsonian Institution for 1851], 1852)

"The easiest and surest way of acquiring facts is to learn them in groups, in systems, and systematized knowledge is science." (Oliver W Holmes, "Medical Essays", 1883)

"Knowledge is the distilled essence of our intuitions, corroborated by experience." (Elbert Hubbard, "A Thousand & One Epigrams, 1911)

"Observed facts must be built up, woven together, ordered, arranged, systematized into conclusions and theories by reflection and reason, if they are to have full bearing on life and the universe. Knowledge is the accumulation of facts. Wisdom is the establishment of relations. And just because the latter process is delicate and perilous, it is all the more delightful." (Gamaliel Bradford, "Darwin", 1926)

"Empirical knowledge is deficient at practically all levels." (Kenneth E. Boulding, "General Systems Theory: The Skeleton of Science", 1956)

"Scientific knowledge is not created solely by the piecemeal mining of discrete facts by uniformly accurate and reliable individual scientific investigations." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

"Knowledge is encoded in models." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)

"Knowledge is acquired by the network from its environment through a learning process, and interneuron connection strengths (synaptic weighs) are used to store the acquired knowledge." (Larbi Esmahi et al, "Adaptive Neuro-Fuzzy Systems", 2009)

"[...] the real test of knowledge is not truth, but utility." (Yuval N Harari, "Sapiens: A brief history of humankind", 2017)

Out of Context: On Probability (Definitions)

"Probability is the very guide of life." (Marcus Tullius Cicero, "De Natura Deorum" ["On the Nature of the Gods"], 45 BC) [attributed] 

"Probability is a degree of possibility." (Gottfried W Leibniz, "On estimating the uncertain", 1676)

"Probability is the appearance of agreement upon fallible proofs." (John Locke, "An Essay Concerning Human Understanding", Book IV, 1689)

"Probability is likeliness to be true, the very notation of the word signifying such a proposition, for which there be arguments or proofs to make it pass, or be received for true." (John Locke, "An Essay Concerning Human Understanding", Book IV, 1689)

"Probability is a degree of certainty and it differs from certainty as a part from a whole." (Jacob Bernoulli, "Ars Conjectandi", 1713)

"[…] to us, probability is the very guide of life." (Joseph Butler, "The Analogy of Religion", 1736) 

"Probability is relative in part to this ignorance, and in part to our knowledge." (Pierre-Simon Laplace, "Mémoire sur les Approximations des Formules qui sont Fonctions de Très Grands Nombres", 1783) 

"Probability is expectation founded upon partial knowledge." (George Boole, "The Laws of Thought", 1854)

"Probability is, so far as measurement is concerned, closely analogous to similarity." (John M Keynes, "A Treatise on Probability", 1921)

"The Theory of Probability is concerned with that part which we obtain by argument, and it treats of the different degrees in which the results so obtained are conclusive or inconclusive." (John M Keynes, "A Treatise on Probability", 1921)

"Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means." (Bertrand Russell, 1929)

"Probability is truth in some degree […]" (Errol E Harris, "Hypothesis and Perception: The Roots of Scientific Method", 1970)

"Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs." (Bruno de Finetti, "Theory of Probability", 1974)

"The logic of certainty furnishes us with the range of possibility (and the possible has no gradations); probability is an additional notion that one applies within the range of possibility, thus giving rise to graduations (‘more or less’ probable) that are meaningless in the logic of uncertainty."  (Bruno de Finetti, "Theory of Probability", 1974)

"Many modern philosophers claim that probability is relation between an hypothesis and the evidence for it." (Ian Hacking, "The Emergence of Probability", 1975)

"The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable." (Benoit Mandelbrot, "The Fractal Geometry of Nature", 1977)

"Probability is the mathematics of uncertainty." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"Probabilities are summaries of knowledge that is left behind when information is transferred to a higher level of abstraction." (Judea Pearl, Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference, 1988)

"Probability is the branch of mathematics that describes randomness." (David S Moore, "Uncertainty", 1990)

"[...] probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive." (Nassim N Taleb, "Fooled by Randomness", 2001)

"Probability is a mathematical language for quantifying uncertainty." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"Probability is a liberal art; it is a child of skepticism, not a tool for people with calculators on their belts to satisfy their desire to produce fancy calculations and certainties." (Nassim N Taleb, “The Black Swan”, 2007)

"We have to be aware that probabilities are relative to a level of observation, and that what is most probable at one level is not necessarily so at another."(Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"Probability is the science of uncertainty. It provides precise mathematical rules for understanding and analyzing our own ignorance." (Michael J Evans & Jeffrey S Rosenthal, "Probability and Statistics: The Science of Uncertainty", 2009)

Edward R Dougherty - Collected Quotes

"An advantage of a deterministic theory is that, assuming sufficient knowledge, there is no uncertainty in the evolution of the state of the system. In practice, measurements are not perfectly precise, so there is always uncertainty as to the value of any variable." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"Foretelling the future is the crux. A model may fit existing data, but the model must incorporate mathematical machinery that makes it predictive across time to be scientifically valid." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016) 

"Limitations on experimentation can result in limitations on the complexity or details of a theory. To be validated, a theory cannot exceed the experimentalist’s ability to conceive and perform appropriate experiments. With the uncertainty theory, modern physics appears to have brought us beyond the situation where limitations on observation result only from insufficient experimental apparatus to a point where limitations are unsurpassable in principle." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"In the classical deterministic scenario, a model consists of a few variables and physical constants. The relational structure of the model is conceptualized by the scientist via intuition gained from thinking about the physical world. Intuition means that the scientist has some mental construct regarding the interactions beyond positing a skeletal mathematical system he believes is sufficiently rich to capture the interactions and then depending upon data to infer the relational structure and estimate a large number of parameters." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016) 

"Parameter estimation is a basic aspect of model construction and historically it has been assumed that data are sufficient to estimate the parameters, for instance, correlations that are part of the model; however, when the number of parameters is too large for the amount of data, accurate parameter estimation becomes impossible. The result is model uncertainty." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"[…] people attempt to use highly flexible mathematical structures with large numbers of parameters that can be adjusted to fit the data, the result often being models that fit the data well but lack structural representation of the phenomena and thus are not predictive outside the range of the data. The situation is exacerbated by uncertainty regarding model parameters on account of insufficient data relative to model complexity, which in fact means uncertainty regarding the models themselves. More importantly from the standpoint of epistemology, the amount of available data is often miniscule in comparison to the amount needed for validation. The desire for knowledge has far outstripped experimental/observational capability. We are starved for data." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"Scientific knowledge is worldly knowledge in the sense that it points into the future by making predictions about events that have yet to take place. Scientific knowledge is contingent, always awaiting the possibility of its invalidation. Its truth or falsity lies in the verity of its predictions and, since these predictions depend upon the outcomes of experiments, ultimately the validity of scientific knowledge is relative to the methodology of verification." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"The foundations of a discipline are inseparable from the rules of its game, without which there is no discipline, just idle talk. The foundations of science reside in its epistemology, meaning that they lie in the mathematical formulation of knowledge, structured experimentation, and statistical characterization of validity. Rules impose limitations. These may be unpleasant, but they arise from the need to link ideas in the mind to natural phenomena. The mature scientist must overcome the desire for intuitive understanding and certainty, and must live with stringent limitations and radical uncertainty." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"The model (conceptual system) is a creation of the imagination, in accordance with the rules of the game. The manner of this creation is not part of the scientific theory. The classical manner is that the scientist combines an appreciation of the problem with reflections upon relevant phenomena and, based on mathematical knowledge, creates a model." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016) 

On Algorithms I

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

"An algorithm must be seen to be believed, and the best way to learn what an algorithm is all about is to try it." (Donald E Knuth, The Art of Computer Programming Vol. I, 1968)

"Scientific laws give algorithms, or procedures, for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"Algorithmic complexity theory and nonlinear dynamics together establish the fact that determinism reigns only over a quite finite domain; outside this small haven of order lies a largely uncharted, vast wasteland of chaos." (Joseph Ford, "Progress in Chaotic Dynamics: Essays in Honor of Joseph Ford's 60th Birthday", 1988)

"On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting - the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings." (John D Barrow, New Theories of Everything", 1991)

"Algorithms are a set of procedures to generate the answer to a problem." (Stuart Kauffman, "At Home in the Universe: The Search for Laws of Complexity", 1995)

"Let us regard a proof of an assertion as a purely mechanical procedure using precise rules of inference starting with a few unassailable axioms. This means that an algorithm can be devised for testing the validity of an alleged proof simply by checking the successive steps of the argument; the rules of inference constitute an algorithm for generating all the statements that can be deduced in a finite number of steps from the axioms." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)

"An algorithm is a simple rule, or elementary task, that is repeated over and over again. In this way algorithms can produce structures of astounding complexity." (F David Peat, "From Certainty to Uncertainty", 2002)

"Many people have strong intuitions about whether they would rather have a vital decision about them made by algorithms or humans. Some people are touchingly impressed by the capabilities of the algorithms; others have far too much faith in human judgment. The truth is that sometimes the algorithms will do better than the humans, and sometimes they won’t. If we want to avoid the problems and unlock the promise of big data, we’re going to need to assess the performance of the algorithms on a case-by-case basis. All too often, this is much harder than it should be. […] So the problem is not the algorithms, or the big datasets. The problem is a lack of scrutiny, transparency, and debate." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

On Algorithms II

"The vast majority of information that we have on most processes tends to be nonnumeric and nonalgorithmic. Most of the information is fuzzy and linguistic in form." (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 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)

"Swarm Intelligence can be defined more precisely as: Any attempt to design algorithms or distributed problem-solving methods inspired by the collective behavior of the social insect colonies or other animal societies. The main properties of such systems are flexibility, robustness, decentralization and self-organization." ("Swarm Intelligence in Data Mining", Ed. Ajith Abraham et al, 2006)

"The burgeoning field of computer science has shifted our view of the physical world from that of a collection of interacting material particles to one of a seething network of information. In this way of looking at nature, the laws of physics are a form of software, or algorithm, while the material world - the hardware - plays the role of a gigantic computer." (Paul C W Davies, "Laying Down the Laws", New Scientist, 2007)

"An algorithm refers to a successive and finite procedure by which it is possible to solve a certain problem. Algorithms are the operational base for most computer programs. They consist of a series of instructions that, thanks to programmers’ prior knowledge about the essential characteristics of a problem that must be solved, allow a step-by-step path to the solution." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Programming is a science dressed up as art, because most of us don’t understand the physics of software and it’s rarely, if ever, taught. The physics of software is not algorithms, data structures, languages, and abstractions. These are just tools we make, use, and throw away. The real physics of software is the physics of people. Specifically, it’s about our limitations when it comes to complexity and our desire to work together to solve large problems in pieces. This is the science of programming: make building blocks that people can understand and use easily, and people will work together to solve the very largest problems." (Pieter Hintjens, "ZeroMQ: Messaging for Many Applications", 2012)

"These nature-inspired algorithms gradually became more and more attractive and popular among the evolutionary computation research community, and together they were named swarm intelligence, which became the little brother of the major four evolutionary computation algorithms." (Yuhui Shi, "Emerging Research on Swarm Intelligence and Algorithm Optimization", Information Science Reference, 2014)

"Again, classical statistics only summarizes data, so it does not provide even a language for asking [a counterfactual] question. Causal inference provides a notation and, more importantly, offers a solution. As with predicting the effect of interventions [...], in many cases we can emulate human retrospective thinking with an algorithm that takes what we know about the observed world and produces an answer about the counterfactual world." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

"An algorithm, meanwhile, is a step-by-step recipe for performing a series of actions, and in most cases 'algorithm' means simply 'computer program'." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"Big data is revolutionizing the world around us, and it is easy to feel alienated by tales of computers handing down decisions made in ways we don’t understand. I think we’re right to be concerned. Modern data analytics can produce some miraculous results, but big data is often less trustworthy than small data. Small data can typically be scrutinized; big data tends to be locked away in the vaults of Silicon Valley. The simple statistical tools used to analyze small datasets are usually easy to check; pattern-recognizing algorithms can all too easily be mysterious and commercially sensitive black boxes." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"Each of us is sweating data, and those data are being mopped up and wrung out into oceans of information. Algorithms and large datasets are being used for everything from finding us love to deciding whether, if we are accused of a crime, we go to prison before the trial or are instead allowed to post bail. We all need to understand what these data are and how they can be exploited." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

Out of Context: On Science (Definitions)

"Science is, I believe, nothing but trained and organized common sense [...]" (Thomas H Huxley, "On the Educational Value of the Natural History Sciences", 1854)

"Science is organized knowledge; and before knowledge can be organized, some of it must first be possessed." (Herbert Spencer, "The Art of Education", The North British Review, 1854)

"In its broadest sense science is organised knowledge, and its methods consist of the observation and classification of the phenomena of which we become conscious through our senses, and the investigation of the causes of which these are the effects." (Richard Strachey, Nature Vol. 12 [E], 1875)

"Science is the observation of things possible, whether present or past; prescience is the knowledge of things which may come to pass, though but slowly." (Leonardo da Vinci, "The Notebooks of Leonardo da Vinci", 1883)

"Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts." (Paul Carus, "Philosophy as a Science", 1909)

"Science is the only truth and it is the great lie. It knows nothing, and people think it knows everything. It is misrepresented. People think that science is electricity, automobilism, and dirigible balloons. It is something very different. It is life devouring itself. It is the sensibility transformed into intelligence. It is the need to know stifling the need to live. It is the genius of knowledge vivisecting the vital genius." (Rémy de Gourmont, "Art and Science", cca. 1905-1909)

"Science is frankly empirical in method and aim; it seeks to discover the laws of concrete being and becoming, and to formulate these in the simplest terms, which are either immediate data of experience or verifiably derived therefrom." (J Arthur Thomson, "The System of Animate Nature" Vol. 1, 1920)

"Science is simply setting out on a fishing expedition to see whether it cannot find some procedure which it can call measurement of space and some procedure which it can call the measurement of time, and something which it can call a system of forces, and something which it can call masses." (Alfred N Whitehead, "The Concept of Nature", 1920)

"Science is a magnificent force, but it is not a teacher of morals." (William J Bryan, "Undelivered Trial Summation Scopes Trial", 1925)

"Science is either an important statement of systematic theory correlating observations of a common world or is the daydream of a solitary intelligence with a taste for the daydream of publication." (Alfred N Whitehead, "Process and Reality: An Essay in Cosmology", 1929)

"Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear." (Paul Valéry, "Moralités" ["Morality"], 1932)

"Science is the attempt to discover, by means of observation, and reasoning based upon it, first, particular facts about the world, and then laws connecting facts with one another and (in fortunate cases) making it possible to predict future occurrences." (Bertrand Russell, "Religion and Science, Grounds of Conflict", 1935)

"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." (Austin L Porterfield, "Creative Factors in Scientific Research", 1941)

"Physics is not about the real world, it is about ‘abstractions’ from the real world, and this is what makes it so scientific." (Anthony Standen, "Science is a Sacred Cow", 1950)

"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)

"Science is the creation of concepts and their exploration in the facts." (Jacob Bronowski, "Science and Human Values", 1956)

"Science is the reduction of the bewildering diversity of unique events to manageable uniformity within one of a number of symbol systems [...]" (Aldous L Huxley, "Essay", Daedalus, 1962)

"Science is what scientists do. Science is knowledge, a body of information about the external world. Science is the ability to predict. Science is power, it is engineering. Science explains, or gives causes and reasons." (John Bremer "What Is Science?" [in "Notes on the Nature of Science"], 1962)

"Science is the combined effort to find out what sort of behavior ensues when various conditions are fulfilled." (H Rom Harré, "Philosophical Issues and Conceptual Change", Theory Into Practice Vol. 10 (2), 1971)

"Science is a product of the human mind, but this product is as objective as a cathedral."  (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"Science is systematic organisation of knowledge about the universe on the basis of explanatory hypotheses which are genuinely testable." (Francisco J Ayala, "Studies in the Philosophy of Biology: Reduction and Related Problems", 1974)

"Within a Metaphysics of Quality, science is a set of static intellectual patterns describing this reality, but the patterns are not the reality they describe." (Robert M Pirsig, "Zen and the Art of Motorcycle Maintenance", 1974)

"Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation." (Stephen J Gould, "Ever Since Darwin", 1977)

"Engineering or Technology is the making of things that did not previously exist, whereas science is the discovering of things that have long existed." (David Billington, "The Tower and the Bridge: The New Art of Structural Engineering", 1983)

"Science is human experience systematically extended (by intent, methodology and instrumentation) for the purpose of learning more about the natural world and for the critical empirical testing and possible falsification of all ideas about the natural world." (Robert E Kofahl, Correctly Redefining Distorted Science: A Most Essential Task", Creation Research Society Quarterly Vol. 23, 1986)

"Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"Science is more than a mere attempt to describe nature as accurately as possible. Frequently the real message is well hidden, and a law that gives a poor approximation to nature has more significance than one which works fairly well but is poisoned at the root." (Robert H March, "Physics for Poets", 1996)

"Science is like photographing a series of close-ups with your back to the sun. No matter which way you move, your shadow always falls across the scene you photograph. No matter what you do, you can never efface yourself from the photographed scene." (F David Peat, "From Certainty to Uncertainty", 2002)

"Science is that story our society tells itself about the cosmos." (F David Peat, "From Certainty to Uncertainty", 2002)

"Science is the art of the appropriate approximation." (Byron K Jennings, "On the Nature of Science", Physics in Canada Vol. 63 (1), 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)

"Science is about discovery, but it is also about communication. An idea can hardly be said to exist if you do not awaken that same idea in someone else." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Science isn’t about being right. It is about convincing others of the correctness of an idea through a methodology all will accept using data everyone can trust. New ideas take time to be accepted because they compete with others that have already passed the test." (Tom Koch, "Commentary: Nobody loves a critic: Edmund A Parkes and John Snow’s cholera", International Journal of Epidemiology Vol. 42 (6), 2013)

"Science, at its core, is simply a method of practical logic that tests hypotheses against experience." (John M Greer, "After Progress: Reason and Religion at the End of the Industrial Age", 2015)

"Science is an endeavor to understand and describe the real world out there to (at best) alleviate and enrich human existence." (Achim Zielesny, "From Curve Fitting to Machine Learning" 2^nd Ed., 2016)

"Science is a game - but a game with reality, a game with sharpened knives." (Erwin Schrödinger)

"Science is not a technique or a body of knowledge, though it uses both. It is rather an attitude of inquiry, or observation and reasoning, with respect to the world. It can be developed, not by memorizing facts or juggling formulas to get an answer, but only by actual practice of scientific observation and reasoning." (Karl T Compton)

"Science is the knowledge of the many, orderly and methodically-arranged, so as to become comprehended by one." (Sir John Herschel)

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)