🦋Science: On Errors (Quotes)

"The probable is something which lies midway between truth and error" (Christian Thomasius, "Institutes of Divine Jurisprudence", 1688)

"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 errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning." (Thomas Hobbes, "The Moral and Political Works of Thomas Hobbes of Malmesbury", 1750)

"Men are often led into errors by the love of simplicity, which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"The orbits of certainties touch one another; but in the interstices there is room enough for error to go forth and prevail." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833)

"Nothing hurts a new truth more than an old error." (Johann Wolfgang von Goethe, "Sprüche in Prosa", 1840)

"Every detection of what is false directs us towards what is true: every trial exhausts some tempting form of error. Not only so; but scarcely any attempt is entirely a failure; scarcely any theory, the result of steady thought, is altogether false; no tempting form of error is without some latent charm derived from truth." (William Whewell, "Lectures on the History of Moral Philosophy in England", 1852)

"[…] ideas may be both novel and important, and yet, if they are incorrect - if they lack the very essential support of incontrovertible fact, they are unworthy of credence. Without this, a theory may be both beautiful and grand, but must be as evanescent as it is beautiful, and as unsubstantial as it is grand." (George Brewster, "A New Philosophy of Matter", 1858)

"When a power of nature, invisible and impalpable, is the subject of scientific inquiry, it is necessary, if we would comprehend its essence and properties, to study its manifestations and effects. For this purpose simple observation is insufficient, since error always lies on the surface, whilst truth must be sought in deeper regions." (Justus von Liebig," Familiar Letters on Chemistry", 1859)

"As in the experimental sciences, truth cannot be distinguished from error as long as firm principles have not been established through the rigorous observation of facts." (Louis Pasteur, "Étude sur la maladie des vers à soie", 1870)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)

"Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent." (William S Jevons, "The Principles of Science", 1887)

"One is almost tempted to assert that quite apart from its intellectual mission, theory is the most practical thing conceivable, the quintessence of practice as it were, since the precision of its conclusions cannot be reached by any routine of estimating or trial and error; although given the hidden ways of theory, this will hold only for those who walk them with complete confidence." (Ludwig E Boltzmann, "On the Significance of Theories", 1890)

"[…] to kill an error is as good a service as, and sometimes even better than, the establishing of a new truth or fact." (Charles R Darwin, "More Letters of Charles Darwin", Vol 2, 1903)

"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)

"The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, ‘Seek simplicity and distrust it’." (Alfred N Whitehead, "The Concept of Nature", 1919)

"One grievous error in interpreting approximations is to allow only good approximations." (Preston C Hammer, "Mind Pollution", Cybernetics, Vol. 14, 1971)

"A mature science, with respect to the matter of errors in variables, is not one that measures its variables without error, for this is impossible. It is, rather, a science which properly manages its errors, controlling their magnitudes and correctly calculating their implications for substantive conclusions." (Otis D Duncan, "Introduction to Structural Equation Models", 1975)

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

"Repeated observations of the same phenomenon do not always produce the same results, due to random noise or error. Sampling errors result when our observations capture unrepresentative circumstances, like measuring rush hour traffic on weekends as well as during the work week. Measurement errors reflect the limits of precision inherent in any sensing device. The notion of signal to noise ratio captures the degree to which a series of observations reflects a quantity of interest as opposed to data variance. As data scientists, we care about changes in the signal instead of the noise, and such variance often makes this problem surprisingly difficult." (Steven S Skiena, "The Data Science Design Manual", 2017)

🦋Science: On Definitions (Quotes)

"Rules for Demonstrations. I. Not to undertake to demonstrate any thing that is so evident of itself that nothing can be given that is clearer to prove it. II. To prove all propositions at all obscure, and to employ in their proof only very evident maxims or propositions already admitted or demonstrated. III. To always mentally substitute definitions in the place of things defined, in order not to be misled by the ambiguity of terms which have been restricted by definitions." (Blaise Pascal, "Pensées", 1670)

"Rules necessary for definitions. Not to leave any terms at all obscure or ambiguous without definition; Not to employ in definitions any but terms perfectly known or already explained. […] A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the art of persuading." (Blaise Pascal, "Pensées", 1670)

"The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning." (Thomas Hobbes, "The Moral and Political Works of Thomas Hobbes of Malmesbury", 1750)

"Definitions might be good if we did not employ words in making them." (Jean-Jacques Rousseau, "Emile, or, Treatise on Education", 1762)

"A definition is nothing else but an explication of the meaning of a word, by words whose meaning is already known. Hence it is evident that every word cannot be defined; for the definition must consist of words; and there could be no definition, if there were not words previously understood without definition." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"There is nothing more difficult than a good definition, for it is scarcely possible to express, in a few words, the abstracted view of an infinite variety of facts." (Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher", 1830)

"It is the essence of a scientific definition to be causative, not by introduction of imaginary somewhats, natural or supernatural, under the name of causes, but by announcing the law of action in the particular case, in subordination to the common law of which all the phenomena are modifications or results." (Samuel T Coleridge, "Hints Towards the Formation of a More Comprehensive Theory of Life, The Nature of Life", 1847)

"The dimmed outlines of phenomenal things all merge into one another unless we put on the focusing-glass of theory, and screw it up sometimes to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world." (James C Maxwell, "Are There Real Analogies in Nature?", 1856)

"Questions of Definition are of the very highest importance in Philosophy, and they need to be watched accordingly." (George Campbell, "A Fourth State of Matter, Nature", 1880)

"We say it is ‘explanation’ but it is only in ‘description’ that we are in advance of the older stages of knowledge and science. We describe better we explain just as little as our predecessors." (Friedrich W Nietzsche, "The Joyful Wisdom", 1887)

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

"A definition is the enclosing a wilderness of idea within a wall of words." (Samuel Butler, "The Note-Books of Samuel Butler", 1912)

"In every new and growing science there are many working hypotheses that never attain to any sort of reality. On the other hand, in the old and abstract sciences of mathematics, where it is hard to tell how much is mere definition or convention, the problem of reality is not so much doubtful as it is meaningless." (Gilbert N Lewis, "The Anatomy of Science", 1926)

"If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions. [...] Hence the question of whether axioms are a priori becomes pointless since they are arbitrary." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"A Weltanschauung [worldview] is an intellectual construction which solves all the problems of our existence uniformly on the basis of one overriding hypothesis, which, accordingly, leaves no question unanswered and in which everything that interests us finds its fixed place [...] the worldview of science already departs noticeably from our definition. It is true that it too assumes the uniformity of the explanation of the universe; but it does so only as a programme, the fulfillment of which is relegated to the future." Sigmund Freud, "New introductory lectures on psycho-analysis", 1932)

"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1937)

"The view is often defended that sciences should be built up on clear and sharply defined basal concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them." (Sigmund Freud, "Collected Papers", 1950)

"Being built on concepts, hypotheses, and experiments, laws are no more accurate or trustworthy than the wording of the definitions and the accuracy and extent of the supporting experiments." (Gerald Holton, "Introduction to Concepts and Theories in Physical Science", 1952)

"The word 'definition' has come to have a dangerously reassuring sound, owing no doubt to its frequent occurrence in logical and mathematical writings." (Willard van Orman Quine, "From a Logical Point of View", 1953)

"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)

"The exact verbal definition of qualitative concepts is more often the province of philosophy than of physical science." (Ronnie Bell, "The Proton in Chemistry", 1959)

"In order that a mathematical science of any importance may be founded upon conventional definitions, the entities created by them must have properties which bear some affinity to the properties of existing things." (John H C Whitehead, A Treatise on Universal Algebra, with Applications, 1960)

"The view is often defended that sciences should be built up on clear and sharply defined basal concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them." (Sigmund Freud, "General Psychological Theory", 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)

"If all the theories pertinent to systems engineering could be discussed within a common framework by means of a standard set of nomenclature and definitions, many separate courses might not be required." (A Wayne Wymore, "A Mathematical Theory of Systems Engineering", 1967)

"Definitions are the guardians of rationality, the first line of defense against the chaos of mental disintegration. (Ayn Rand, The Romantic Manifesto, 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)

"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)

"When terms [...] evolve and change definition with time; and when the social reality which terms are intended to organize and render intelligible is also seen to be in flux, capturing the truth in a net of words becomes a matter of intuition and style more than of any scientific method that can be replicated by others and made to achieve the same result every time someone asks the same question, or undertakes the same operations." (William H McNeill, "Discrepancies among the Social Sciences", 1981)

"Definitions are temporary verbalizations of concepts, and concepts - particularly difficult concepts - are usually revised repeatedly as our knowledge and understanding grows." (Ernst Mayr, "The Growth of Biological Thought", 1982)

"The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"A full definition of an object must include the whole of human experience, both as a criterion of truth and a practical indicator of its connection with human wants." (Vladimir Lenin)

"Every physical concept must be given a definition such that one can in principle describe, in virtue of this definition, whether or not it applies in each particular case." (Albert Einstein)

"Fundamental definitions do not arise at the start but at the end of the exploration, because in order to define a thing you must know what it is and what it is good for." (Hans Freudenthal)

"We begin to reason from sensible objects, and definition is the end and epilogue of science. It is not the beginning of our knowing, but only of our teaching." (Tommaso Campanella)

🔖Knowledge Representation: Models (Part I: In Search of a Definition)

Introduction
 Imagine you are talking with another person, you ramble about models here and there, and suddenly in the middle of the conversation a question falls like thunder, freezing the discussion for a few seconds: “you are talking about models, what is a model?”. The word become so common nowadays that such a question might seem a menace for the conversation. You might remember then some of the dictionary definitions you heard so often, though they pull with them other terms that need further clarification. Some well-known approximate quotations on models might spring into your thinking process, though they don’t seem of any help now. The seconds pass and the time of silence looks like an eternity, while your interlocutor stares at you waiting for an answer. Unless you are prepared to give a definition on interlocutor’s understanding, you’ll more likely be in the chase of a such definition. Words and ideas succeed fast, the silence grows deeper, your interlocutor becomes more anxious, and is waiting for an answer.

What is a Model?

The easiest way to tackle such concepts is to start with a common representative example that can be used as scaffolding for a definition. If few people heard of “role models”, more likely almost all people having a TV, PC or similar device have heard of “top models” or “fashion models”. The “fashion model” as well the “role model” are idealized types of people, the first type considering mainly the appearance characteristics that makes a person attractive for presenting fashion, while the second considers mainly the characteristics that other people would want to copy. Both categories represent some idealizations in which some characteristics, qualities or features are evidenced.

When one speaks about fashion models one can easily come with some names of such models who were en vogue over the years. On the other side maybe for many will be more difficult to come with names of people who can be considered as role models, unless they had in their life people who functioned as role models. A “role model” seems to be more abstract as concept than the “fashion model”, as the physical qualities are easier to grasps then the inner qualities. There can be defined qualities like success, confidence, hardworking or sociability that are attributed to both roles. In fact the qualities of the two concepts can intertwine to the degree that for some people a fashion model and role model may be defined by the same qualities, or the differences are so small that they can be ignored.

Fashion mode vs role model

Based on the two examples we can construct now a definition. Let’s consider a model as an idealization of a concrete or abstract object, in which some characteristics, qualities or features are considered as important and definitory.

The definition seems to be concise, clear, objective and precise, as much such a definition can be, so it looks like a good definition one can work with. The question is whether the praxis might invalidate it, where the gap between it and other definitions from technical literature is so big that makes this definition unusable. At least the following definitions seem to accord:

“A model is a simplification of reality intended to promote understanding.” [1]
“Models are replicas or representations of particular aspects and segments of the real world.” [2]
“A model is a representation of reality intended for some definite purpose .”[3].
“A model is an imitation of reality.” [4]
“A model is a representation of some subject matter.” [5]

From my point of view all these definitions aren’t good definitions because by attempting to be too general they lack precision. Is it any representation of some subject, of reality or part of it a model? Probably not! This raises another question: what isn’t a model? Unfortunately the answer isn’t that simple. Not any representation of something can be considered a model. Is it for example a cubistic representation of an object a model? The answer is again no! The model needs to keep certain characteristics of the object intended to represent, though more about it in other posts.

 Here are a few other alternative definitions for models:
”A model is a physical, mathematical, or logical representation of a system entity, phenomenon, or process.” [6]
“model: the artificial object which explains all the empirical facts under consideration” [7]

Models in Context

Depending on the context in which is used, the concept of model is more elaborated and adapted as needed. One can talk about business, conceptual/thinking, mental, mathematical, statistical, scientific, mechanistic, behavioral, structural, social, political, logical, data or diagrammatic models. In fact models are the cornerstone of all sciences and pseudo-sciences, the foundation on which our understanding and knowledge is based upon. From the inner to the outer world, from simple to complex activities, from mathematics to religion, we use models on a daily basis. We think in models, or at least we use models in our thinking even if we are not aware of it.

 From the literature reviewed over the years it can be observed that most of the definitions converge within the same boundaries, the difference residing mainly in the vocabulary used – idealization and characteristics being replaced by the family of words from the below table:

idealization	characteristics
simplification miniature representation imitation likeness description approximation abstraction reduction	qualities features attributes properties variables dimensions parameters

In various sources the denomination of model is substituted with synonyms that make better appeal of people’s understanding: pattern, example, emulation, analogy, visualization, system, replica, mold, mock-up, sketch, prototype, wireframe, schema, blueprint, (visual) aid, image (also picture, depiction), map, theory, principle, approximation, representation, allegory, metaphor, archetype, etc. All these can be considered upon context models, even if they might refer to totally different things.

Conclusion:

Even if our thinking became more complex, the average individual juggling with increasingly abstract concepts, it seems that we fail to define basic concepts like the one of model. Starting from two examples I attempted to provide a good definition one can work with. I tried also to provide a mechanism that can be used to extrapolate definitions for similar concepts.

References:
[1] SystemsThinking.org (2004) Models, by Gene Bellinger  http://www.systems-thinking.org/simulation/model.htm
[2] Data Modeling Fundamentals: A Practical Guide for IT Professionals, by Paulraj Ponniah, John Wiley & Sons, ISBN: 978-0-471-79049-5, 2007
[3] Tools for Thinking: Modelling in Management Science, 3rd Ed., by Michael Pidd, John Wiley & Sons, ISBN-13: 978-0470721421, 2009
[4] Process Systems Engineering: Process Modelling and Model Analysis Vol. IV, by Ian T. Cameron, Katalin M Hangos, Academic Press, ISBN-13: 978-0121569310, 2001
[5] Workflow Modeling: Tools for Process Improvement and Applications Development, 2nd Ed, by Alec Sharp Patrick McDermott. Artech House, ISBN-13: 978-1-59693-192-3, 2009
[6] Systems Engineering Fundamentals. Defense Acquisition University Press, 2001
[7] The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, by Alain Badiou, re.press, ISBN-13: 978-0980305234, 2007

 

🏷️Knowledge Representation: On Definitions (Quotes)

"The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning." (Thomas Hobbes, "The Moral and Political Works of Thomas Hobbes of Malmesbury", 1750)

"Definitions might be good if we did not employ words in making them." (Jean-Jacques Rousseau, "Emile, or, Treatise on Education", 1762)

"A definition is nothing else but an explication of the meaning of a word, by words whose meaning is already known. Hence it is evident that every word cannot be defined; for the definition must consist of words; and there could be no definition, if there were not words previously understood without definition." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"There is nothing more difficult than a good definition, for it is scarcely possible to express, in a few words, the abstracted view of an infinite variety of facts." (Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher" , 1830)

"It is the essence of a scientific definition to be causative, not by introduction of imaginary somewhats, natural or supernatural, under the name of causes, but by announcing the law of action in the particular case, in subordination to the common law of which all the phenomena are modifications or results." (Samuel T Coleridge, "Hints Towards the Formation of a More Comprehensive Theory of Life, The Nature of Life", 1847)

"The dimmed outlines of phenomenal things all merge into one another unless we put on the focusing-glass of theory, and screw it up sometimes to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world." (James C Maxwell, "Are There Real Analogies in Nature?", 1856)

"Questions of Definition are of the very highest importance in Philosophy, and they need to be watched accordingly." (George Campbell, "A Fourth State of Matter, Nature", 1880)

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

"A definition is the enclosing a wilderness of idea within a wall of words." (Samuel Butler, "The Note-Books of Samuel Butler", 1912)

"In every new and growing science there are many working hypotheses that never attain to any sort of reality. On the other hand, in the old and abstract sciences of mathematics, where it is hard to tell how much is mere definition or convention, the problem of reality is not so much doubtful as it is meaningless." (Gilbert N Lewis, "The Anatomy of Science", 1926)

"Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought. (Frank P Ramsey, 'Last Papers: Philosophy', 1929)

"A Weltanschauung [worldview] is an intellectual construction which solves all the problems of our existence uniformly on the basis of one overriding hypothesis, which, accordingly, leaves no question unanswered and in which everything that interests us finds its fixed place [...] the worldview of science already departs noticeably from our definition. It is true that it too assumes the uniformity of the explanation of the universe; but it does so only as a programme, the fulfillment of which is relegated to the future." Sigmund Freud, "New introductory lectures on psycho-analysis", 1932)

"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1937)

"The view is often defended that sciences should be built up on clear and sharply defined basal concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them." (Sigmund Freud, "Collected Papers", 1950)

"Being built on concepts, hypotheses, and experiments, laws are no more accurate or trustworthy than the wording of the definitions and the accuracy and extent of the supporting experiments." (Gerald Holton, "Introduction to Concepts and Theories in Physical Science", 1952)

"The word 'definition' has come to have a dangerously reassuring sound, owing no doubt to its frequent occurrence in logical and mathematical writings." (Willard van Orman Quine, From a Logical Point of View, 1953)

"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)

"The view is often defended that sciences should be built up on clear and sharply defined basal concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them." (Sigmund Freud, "General Psychological Theory", 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)

"Definitions are the guardians of rationality, the first line of defense against the chaos of mental disintegration. (Ayn Rand, The Romantic Manifesto, 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)

"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)

"When terms [...] evolve and change definition with time; and when the social reality which terms are intended to organize and render intelligible is also seen to be in flux, capturing the truth in a net of words becomes a matter of intuition and style more than of any scientific method that can be replicated by others and made to achieve the same result every time someone asks the same question, or undertakes the same operations." (William H McNeill, "Discrepancies among the Social Sciences", 1981)

"Definitions are temporary verbalizations of concepts, and concepts - particularly difficult concepts - are usually revised repeatedly as our knowledge and understanding grows." (Ernst Mayr, "The Growth of Biological Thought", 1982) 

"A full definition of an object must include the whole of human experience, both as a criterion of truth and a practical indicator of its connection with human wants." (Vladimir Lenin) 

"Fundamental definitions do not arise at the start but at the end of the exploration, because in order to define a thing you must know what it is and what it is good for." (Hans Freudenthal)

"We begin to reason from sensible objects, and definition is the end and epilogue of science. It is not the beginning of our knowing, but only of our teaching." (Tommaso Campanella)

🔖Knowledge Representation: Knowledge Maps (Part V: Propositions)

   Unfortunately the richness of natural language goes above the capabilities of the current representational tools, for example the temporal, spatial, causal and conditional associations, or the associations between whole propositions are not so easy to represent in many of the K-maps available. Given the rules of syntax, in natural language a proposition is formed typically at minimum of a triple, while a typical proposition could be expressed as one or more such triples. On the other side, a chunk of meaning could be represented as one, two or more triples that span within one or more propositions, same as from a whole text only 2-3 triples could be relevant. As chunks could spread along several propositions, it is necessary to identify the meaningful triples and recombine them in more complex constructs, such a construct being the knowlet.

   Because has been discussed in several occasions about chunks of meaning, some examples are necessary in order to understand what is meant by this construct. The simplest chunk of meaning is the concept itself, for example “collective”, “intelligence”, “collective intelligence” and “harnessing collective intelligence” are such chunks. The next logical type of chunk of meaning is a simple triple formed actually from three concepts, for example “harnessing collective intelligence is a web 2.0 principle” is based on “harnessing collective intelligence”, “is a” and “web 2.0 principle”. A minimum of 4 concepts could result in two triples, for example “web as platform and harnessing collective intelligence are web 2.0 principles”, the two triples could be split in different propositions. By adding one more concept we arrive to three triples, and the logic could be followed, resulting several types of patterns.

     Is the latest example still a chunk of meaning? Yes, it is because they share together the same object within the same proposition, enclosed in a definition-like clause. Definitions are typically examples of such unitary chunks of meaning, for example let’s consider the definition of K-mapping given by Dr. Ann Hylton:

    "Knowledge Mapping is the process of surveying, assessing and linking the information, knowledge, competencies and proficiencies held by individuals and groups within an organization." [1]

     Proposition’s structure could be represented in a K-map resembling a Concept Map as follows:

  Knowledge Mapping definition - Concept Map

     The representational language depends from person to person and from one KM to another, for example the same proposition could be rewritten as follows:

Knowledge Mapping definition - Knowledge Map

  Here’s another example following a similar structure, this time based on  D. Hyerle’s definition for K-mapping:

     Knowledge Mapping is “a rich synthesis of thinking processes, mental strategies, techniques and technologies, and knowledge that enables humans to investigate unknowns, show patterns of information, and then use the map to express, build, and assess new knowledge” [2].

 
   Which KM is better is a matter of personal taste, in theory the radial/hierarchical structures are easier to understand and navigate than the networked structures, though the later category of structures is closer to the structure of knowledge, the way concepts are structured. On the other side it could be problematic to represent the two definitions in the same K-map. Here’s an attempt based on propositions’ structure, respectively by restructuring concepts and associations:

 

   Each of the two approaches comes with its advantages and disadvantages, the first K-map being closer to initial definitions’ formulation, highlighting the concepts found closer to the subject, but more difficult to represent, while the second, more condensed, closer to the representation of knowledge as triples, but keeping less from the initial structure. The maps are not necessarily representative, and neither too elaborate, they represent just the raw representation of two definitions. It would be useful to include multiple definitions in the same K-map, attempting to represent all the attributes of a concept and most representative associations. The downside of such K-maps is that they are more complex and all the consequences deriving from it.

References:
[1] M. Jafari, P. Akhavan, A. Bourouni, R. H. Amiri, (2009). A Framework For The Selection Of Knowledge Mapping Techniques. Journal of Knowledge Management Practice, Vol. 10, No. 1, [Online] Available from: http://www.tlainc.com/articl180.htm (Accessed:16 October 2010)
[2] Hyerle, D. (2008). Thinking Maps®: A Visual Language for Learning. In: Thinking Maps®: A Visual Language for Learning, ISBN: 978-1-84800-149-7. [Online] Available from: http://www.springerlink.com/content/x57121720731381j/ (Accessed: 23 June 2009)