visualization

"Like a physical model, a conceptual model is an artificial system. It is however, made up of conceptual, and not physical components."

"Within sociology there have been several system theories, differing from one another in the extent to which, for example, human agency, creativity, and entrepreneurship are assumed to play a role in system formation and reformation; conflict and struggle are taken into account; power and stratification are part and parcel of the theory; structural change and transformation – and more generally, historically developments – are taken into account and explained. What the various system theories have in common is a systematic concern with complex and varied interconnections and interdependencies of social life. Complexity has been a central concept for many working in the systems perspective. The tradition is characterized to a great extent by a burning ambition and hope to provide a unifying language and conceptual framework for all the social sciences."

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it... If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’?"

"Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities."

"The term architecture is used here to describe the attributes of a system as seen by the programmer, i.e., the conceptual structure and functional behavior, as distinct from the organization of the data flow and controls, the logical design, and the physical implementation. i. Additional details concerning the architecture,"

"The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems which are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers... The eighteenth century doctrine of natural rights is a search for Euclidean axioms in politics. The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source."

"A conceptual model is a mental image of a system, its components, its interactions. It lays the foundation for more elaborate models, such as physical or numerical models. A conceptual model provides a framework in which to think about the workings of a system or about problem solving in general. An ensuing operational model can be no better than its underlying conceptualization."

"Scientists whose work has no clear, practical implications would want to make their decisions considering such things as: the relative worth of (1) more observations, (2) greater scope of his conceptual model, (3) simplicity, (4) precision of language, (5) accuracy of the probability assignment."

"Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"The Greeks... discovered mathematics and the art of deductive reasoning. Geometry, in particular, is a Greek invention, without which modern science would have been impossible."

"At a very early period the study of Geometry was regarded as a very important mental discipline, as may be shewn from the seventh book of the Republic of Plato. To his testimony may be added that of the celebrated Pascal (Œuvres, Tom. I. p. 66,) which Mr. Hallam has quoted in his History of the Literature of the Middle Ages. "Geometry," Pascal observes, "is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration." These are enumerated by him as eight in number. 1. To define nothing which cannot be expressed in clearer terms than those in which it is already expressed. 2. To leave no obscure or equivocal terms undefined. 3. To employ in the definition no terms not already known. 4. To omit nothing in the principles from which we argue, unless we are sure it is granted 5. To lay down no axiom which is not perfectly evident. 6. To demonstrate nothing which is as clear already as we can make it. 7. To prove every thing in the least doubtful, by means of self-evident axioms, or of propositions already demonstrated. 8. To substitute mentally the definition instead of the thing defined. Of these rules he says, "the first, fourth, and sixth are not absolutely necessary to avoid error, but the other five are indispensable; and though they may be found in books of logic, none but the geometers have paid any regard to them."

"The term "paradigm," from the Greek paradeigma ("pattern"), was used by Kuhn to denote a conceptual framework shared by a community of scientists and providing them with model problems and solutions"

"Various relations being established in geometry between lines constituted under given conditions, as parts of geometrical figures, if we choose to adopt the idea of expressing these lines by numerical measures, we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their geometrical relations however are absolutely general, and do not refer to any such distinction."

"A conceptual model is simply a framework or schematic to understand the interaction of workforce education and development systems with other variables in a society."

"The rule is derived inductively from experience, therefore does not have any inner necessity, is always valid only for special cases and can anytime be refuted by opposite facts. On the contrary, the law is a logical relation between conceptual constructions; it is therefore deductible from upper [übergeordnete] laws and enables the derivation of lower laws; it has as such a logical necessity in concordance with its upper premises; it is not a mere statement of probability, but has a compelling, apodictic logical value once its premises are accepted"

"Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration."

"All those who have written histories [of geometry] bring to this point their account of the development of this science. Not long after these men [pupils of Plato] came Euclid… Not much younger than these [pupils of Plato] is Euclid, who put together the Elements ,…bringing to irrefragable demonstration the things which had been only loosely proved by his predecessors. This man [must have] lived in the time of the first Ptolemy; for Archimedes, who followed closely the first [Ptolemy? book?] makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorter way to study geometry…to which he replied that there was no royal road to geometry. He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says."

"The first function of a conceptual model is relating the research to the existing body of literature. With the help of a conceptual model a researcher can indicate in what way he is looking at the phenomenon of his research."

"Conceptual models are best thought of as design-tools — a way for designers to straighten out and simplify the design and match it to the users' task-domain, thereby making it clearer to users how they should think about the application."

"In the new pattern of thought we do not assume any longer the detached observer, occurring in the idealizations of this classical type of theory, but an observer who by his indeterminable effects creates a new situation, theoretically described as a new state of the observed system. In this way every observation is a singling out of a particular factual result, here and now, from the theoretical possibilities, therefore making obvious the discontinuous aspect of physical phenomena. Nevertheless, there remains still in the new kind of theory an objective reality, inasmuch as these theories deny any possibility for the observer to influence the result of a measurement, once the experimental arrangement is chosen. Therefore particular qualities of an individual observer do not enter into the conceptual framework of the theory."

"It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. ...the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". ...If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface."

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality;"

"The conceptual model is a non-software specific description of the simulation model that is to be developed, describing the objectives, inputs, outputs, content, assumptions and simplifications of the model."

"The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other ; it is rather a word denoting a kind of conceptual construction, as, e.g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas.."

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties... of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent... and Newton, introduced transformations other than projection and section."

"I was informed by the priests at Thebes, that king Sesostris made a distribution of the territory of Egypt among all his subjects, assigning to each an equal portion of land in the form of a quadrangle, and that from these allotments he used to derive his revenue by exacting every year a certain tax. In cases however where a part of the land was washed away by the annual inundations of the Nile, the proprietor was permitted to present himself before the king, and signify what had happened. The king then used to send proper officers to examine and ascertain, by admeasurement, how much of the land had been washed away, in order that the amount of tax to be paid for the future, might be proportional to the land which remained. From this circumstance I am of opinion, that Geometry derived its origin; and from hence it was transmitted into Greece."

"The authors on may be divided into... theoretical and practical... [N]one... have combined the theory with the practice... to render the subject plain and intelligible... [T]he most valuable and scientifical are... abstruse, and the practical scarcely furnish... the rationale... The object of the ensuing treatise is to simplify the theory, yet to retain a methodical and accurate... investigation, and to exemplify this theory by... important... useful examples. ...[D]emonstrations are frequently founded on principles strictly Geometrical ...and sometimes ...by algebraical signs, particularly where the Geometrical ...would require a complicated figure, or a ...tedious process. ...[T]he algebraical mode of deduction tends greatly to simplify... yet... definitions and... elementary parts... must be acquired from Geometrical principles illustrated by diagrams; otherwise a student will never obtain a clear and satisfactory knowledge... Should any person attempt to teach the elementary principles of the science by... algebraic characters, and algebraic formulae alone, without the aid of Geometry, he would... deceive both himself and his pupils."

"The purpose of a conceptual model is to provide a vocabulary of terms and concepts that can be used to describe problems and/or solutions of design. It is not the purpose of a model to address specific problems, and even less to propose solutions for them. Drawing an analogy with linguistics, a conceptual model is analogous to a language, while design patterns are analogous to rhetorical figures, which are predefined templates of language usages, suited particularly to specific problems."

"Sometimes, however, a conceptual model is only a first step, and the second step is a mathematical representation of the conceptual model"

"Geometry has two great treasures: one is the Theorem of Phythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel."

"Geometry can in no way be viewed... as a branch of mathematics, instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry."

"The contemporary decline in interest in geometry and its gradual disappearance from school curricula... should be deplored... Geometry is the most visual of the mathematical disciplines. It is not in principle divorced from numbers, and hence neither is it divorced from algebra. Many a pupil's understanding of algebraic proofs would be considerably reinforced by... visual geometrical proofs which were the hallmark of Greek mathematics and to some extent of Arab mathematics also. ...where a geometrical proof is clear and immediate, as... with... many algebraic identities such as (a \pm b)^2 = a^2 \pm 2ab + b^2\!, the geometry should not be forgotten. The Greeks were some of the greatest teachers of all time... [and] geometric algebra was in many ways [their] greatest achievement ..."

"The geometrical spirit is not so tied to geometry that it cannot be detached from it and transported to other branches of knowledge. A work of morals or politics or criticism, perhaps even of eloquence, would be better (other things being equal) if it were done in the style of a geometer. The order, clarity, precision and exactitude which have been apparent in good books for some time might well have their source in this geometric spirit. ...Sometimes one great man gives the tone to a whole century; Descartes], to whom one might legitimately be accorded the glory of having established a new art of reasoning, was an excellent geometer."

"When we entrust the domain of values to those whose intellectual concerns are essentially centred on empirical facts, and whose conceptual frameworks are inevitably constructed around sets of empirical facts, we need not be surprised if the result is moral confusion."

"The way I have taken seems not to lead to the goal, but much rather to make the truth of geometry doubtful."

"What surprised me, which Google was part of, is that superficial search techniques over large bodies of stuff could get you what you wanted. I grew up in the AI tradition, where you have a complete conceptual model, and the information retrieval tradition, where you have complex vectors of key terms and Boolean queries. The idea that you can index billions of pages and look for a word and get what you want is quite a trick. To put it in more abstract terms, it's the power of using simple techniques over very large numbers versus doing carefully constructed systematic analysis."

"Geometry enlightens the intellect and sets one's mind right."

"[T]he system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of... practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their proper dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience... not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it from "purely axiomatic geometry.""

"One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. ... for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such an association geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not."

"A conceptual model is what in the model theory is called a set of formulas making statements about the world."

"A conceptual model is a representation of the system expertise using this formalism. An internal model is derived from the conceptual model and from a specification of the system transactions and the performance constraints."

"And the whole [is] greater than the part."

"Since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art."

"A conceptual model is one which reflects reality by placing words which are concepts into the model in the same way that the model aeroplane builder puts wings, a fuselage, and a cockpit together."

"[T]he ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact."

"There is no royal road to geometry."

"This making or imagining of models (not necessarily or usually a material model, but a conceptual model) is a recognised way of arriving at an understanding of recondite and ultra-sensual occurring say in the ether or elsewhere."

"Truly grand and powerful theories […] do not and cannot rest upon single observations. Evolution is an inference from thousands of independent sources, the only conceptual structure that can make unified sense of all this disparate information. The failure of a particular claim usually records a local error, not the bankruptcy of a central theory. […] If I mistakenly identify your father's brother as your own dad, you don't become genealogically rootless and created de novo. You still have a father; we just haven't located him properly."

"Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise."

"Historically, it was Euclidean geometry that, developed to a large extent as a votive offering to the God of Reason, opened men's eyes to the possibility of design and to the possibility of uncovering it by the pursuit of mathematics."