25 July 2021

Science: On Relations (Quotes)

"Wit, you know, is the unexpected copulation of ideas, the discovery of some occult relation between images in appearance remote from each other." (Samuel Johnson, "The Rambler", 1750)

"[It] may be laid down as a general rule that, if the result of a long series of precise observations approximates a simple relation so closely that the remaining difference is undetectable by observation and may be attributed to the errors to which they are liable, then this relation is probably that of nature." (Pierre-Simon Laplace, "Mémoire sur les Inégalites Séculaires des Planètes et des Satellites", 1787)

"Discoveries are not generally made in the order of their scientific arrangement: their connexions and relations are made out gradually; and it is only when the fermentation of invention has subsided that the whole clears into simplicity and order. " (William Whewell, "An Elementary Treatise on Mechanics" Vol. I, 1819)

"There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations." (Auguste Comte, "The Positive Philosophy", 1830)

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action."(Carl von Clausewitz, "On War", 1832)

"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)

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

"A discovery is generally an unforeseen relation not included in theory." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"To imagine - to form an image - we must have the numerous relations of things present to the mind, and see the objects in their actual order. In this we are of course greatly aided by the mass of organised experience, which allows us rapidly to estimate the relations of gravity or affinity just as we remember that fire burns and that heated bodies expand. But be the aid great or small, and the result victorious or disastrous, the imaginative process is always the same." (George H Lewes, "The Principles of Success in Literature", 1865)

"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)

"[…] deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." (Charles S Peirce, 1885)

"The use of figures is, above all, then, for the purpose of making known certain relations between the objects that we study, and these relations are those which occupy the branch of geometry that we have called Analysis Situs [that is, topology], and which describes the relative situation of points and lines on surfaces, without consideration of their magnitude." (Henri Poincaré, "Analysis Situs", Journal de l'Ecole Polytechnique 1, 1895)

"Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth." (Charles S Peirce, "Kinds of Reasoning", cca. 1896)

"Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone." (Henri Poincaré, "Science and Hypothesis", 1901)

"The laws of nature are drawn from experience, but to express them one needs a special language: for, ordinary language is too poor and too vague to express relations so subtle, so rich, so precise. Here then is the first reason why a physicist cannot dispense with mathematics: it provides him with the one language he can speak [...]" (Henri Poincaré, "The Value of Science", 1905)

"The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things."  (Henri Poincaré, "Science and Hypothesis", 1905)

"But surely it is self-evident that every theory is merely a framework or scheme of concepts together with their necessary relations to one another, and that the basic elements can be constructed as one pleases." (Gottlob Frege, "On the Foundations of Geometry and Formal Theories of Arithmetic" , cca. 1903-1909)

"Statistics may be defined as numerical statements of facts by means of which large aggregates are analyzed, the relations of individual units to their groups are ascertained, comparisons are made between groups, and continuous records are maintained for comparative purposes." (Melvin T Copeland. "Statistical Methods" [in: Harvard Business Studies, Vol. III, Ed. by Melvin T Copeland, 1917])

"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)

"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 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."(Ludwig von Bertalanffy, "Kritische Theorie der Formbildung", 1928)

"A system is said to be coherent if every fact in the system is related every other fact in the system by relations that are not merely conjunctive. A deductive system affords a good example of a coherent system." (Lizzie S Stebbing, "A modern introduction to logic", 1930)

"To apply the category of cause and effect means to find out which parts of nature stand in this relation. Similarly, to apply the gestalt category means to find out which parts of nature belong as parts to functional wholes, to discover their position in these wholes, their degree of relative independence, and the articulation of larger wholes into sub-wholes." (Kurt Koffka, 1931)

"[…] the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well-constructed theory is in some respects undoubtedly an artistic production." (Ernest Rutherford, 1932)

"It is the function of notions in science to be useful, to be interesting, to be verifiable and to acquire value from anyone of these qualities. Scientific notions have little to gain as science from being forced into relation with that formidable abstraction, ‘general truth’." (Wilfred Trotter, [paper delivered before the Royal College of Surgeons of England] 1932)

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." (Nikola Tesla, "Radio Power Will Revolutionize the World", Modern Mechanics and Inventions, 1934)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships.  In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Albert Einstein, [Obituary for Emmy Noether], 1935)

"Matter-of-fact is an abstraction, arrived at by confining thought to purely formal relations which then masquerade as the final reality. This is why science, in its perfection, relapses into the study of differential equations. The concrete world has slipped through the meshes of the scientific net." (Alfred N Whitehead, "Modes of Thought", 1938)

"Analogies are useful for analysis in unexplored fields. By means of analogies an unfamiliar system may be compared with one that is better known. The relations and actions are more easily visualized, the mathematics more readily applied, and the analytical solutions more readily obtained in the familiar system." (Harry F Olson, "Dynamical Analogies", 1943)

"By a model we thus mean any physical or chemical system which has a similar relation-structure to that of the process it imitates. By ’relation-structure’ I do not mean some obscure non-physical entity which attends the model, but the fact that it is a physical working model which works in the same way as the process it parallels, in the aspects under consideration at any moment." (Kenneth Craik, "The Nature of Explanation", 1943)

"Given any object, relatively abstracted from its surroundings for study, the behavioristic approach consists in the examination of the output of the object and of the relations of this output to the input. By output is meant any change produced in the surroundings by the object. By input, conversely, is meant any event external to the object that modifies this object in any manner." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Teleology", Philosophy of Science 10, 1943)

"In classifying behavior the term teleology was used as synonymous with purpose controlled by feed-back. Teleology has been interpreted in the past to imply purpose and the vague concept of a ?nal cause has been often added. This concept of final causes has led to the opposition of teleology to determinism. […] teleology is not opposed to determinism, but to non-teleology. Both teleological and nonteleological systems are deterministic when the behavior considered belongs to the realm where determinism applies. The concept of teleology shares only one thing with the concept of causality: a time axis. But causality implies a one-way, relatively irreversible functional relationship, whereas teleology is concerned with behavior, not with functional relationships." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Technology", Philosophy of Science Vol. 10 (1), 1943)

"It is important to realize that it is not the one measurement, alone, but its relation to the rest of the sequence that is of interest." (William E Deming, "Statistical Adjustment of Data", 1943)

"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)

"[...] the conception of chance enters in the very first steps of scientific activity in virtue of the fact that no observation is absolutely correct. I think chance is a more fundamental conception that causality; for whether in a concrete case, a cause-effect relation holds or not can only be judged by applying the laws of chance to the observation." (Max Born, 1949)

"The act of discovery escapes logical analysis; there are no logical rules in terms of which a 'discovery machine' could be constructed that would take over the creative function of the genius. But it is not the logician’s task to account for scientific discoveries; all he can do is to analyze the relation between given facts and a theory presented to him with the claim that it explains these facts. In other words, logic is concerned with the context of justification." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"When the mathematician speaks of the existence of a 'functional relation' between two variable quantities, he means that they are connected by a simple 'formula that is to say, if we are told the value of one of the variable quantities we can find the value of the second quantity by substituting in the formula which tells us how they are related. [...] The thing to be clear about before we proceed further is that a functional relationship in mathematics means an exact and predictable relationship, with no ifs or buts about lt. It is useful in practice so long as the ifs and buts are only tiny voices which even the most ardent protagonist of proportional representation can ignore with a clear conscience." (Michael J Moroney, "Facts from Figures", 1951)

"The principle of complementarity states that no single model is possible which could provide a precise and rational analysis of the connections between these phenomena [before and after measurement]. In such a case, we are not supposed, for example, to attempt to describe in detail how future phenomena arise out of past phenomena. Instead, we should simply accept without further analysis the fact that future phenomena do in fact somehow manage to be produced, in a way that is, however, necessarily beyond the possibility of a detailed description. The only aim of a mathematical theory is then to predict the statistical relations, if any, connecting the phenomena." (David Bohm, "A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables", 1952)

"Probability is a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation." (William Feller, "An Introduction to Probability Theory and Its Applications", 1957)

"Every metaphor is the tip of a submerged model. […] Use of theoretical models resembles the use of metaphors in requiring analogical transfer of a vocabulary. Metaphor and model-making reveal new relationships; both are attempts to pour new content into old bottles." (Max Black," Models and Metaphors", 1962)

"Certain properties are necessary or sufficient conditions for other properties, and the network of causal relations thus established will make the occurrence of one property at least tend, subject to the presence of other properties, to promote or inhibit the occurrence of another. Arguments from models involve those analogies which can be used to predict the occurrence of certain properties or events, and hence the relevant relations are causal, at least in the sense of implying a tendency to co-occur." (Mary B Hesse," Models and Analogies in Science", 1963)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"Thus, there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relations or 'forces' between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general. In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for ‘systems’ in general." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)

"You cannot sum up the behavior of the whole from the isolated parts, and you have to take into account the relations between the various subordinate systems which are super-ordinated to them in order to understand the behavior of the parts." (Ludwig von Bertalanffy, "General System Theory", 1968)

"In complex systems cause and effect are often not closely related in either time or space. The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by nonlinear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive-feedback loops describing growth processes as well as negative, goal-seeking loops. In the complex system the cause of a difficulty may lie far back in time from the symptoms, or in a completely different and remote part of the system. In fact, causes are usually found, not in prior events, but in the structure and policies of the system." (Jay Wright Forrester, "Urban dynamics", 1969)

"The advantages of models are, on one hand, that they force us to present a 'complete' theory by which I mean a theory taking into account all relevant phenomena and relations and, on the other hand, the confrontation with observation, that is, reality." (Jan Tinbergen, "The Use of Models: Experience," 1969)

"Self-organization can be defined as the spontaneous creation of a globally coherent pattern out of local interactions. Because of its distributed character, this organization tends to be robust, resisting perturbations. The dynamics of a self-organizing system is typically non-linear, because of circular or feedback relations between the components. Positive feedback leads to an explosive growth, which ends when all components have been absorbed into the new configuration, leaving the system in a stable, negative feedback state. Non-linear systems have in general several stable states, and this number tends to increase (bifurcate) as an increasing input of energy pushes the system farther from its thermodynamic equilibrium." (Francis Heylighen, "The Science Of Self-Organization And Adaptivity", 1970)

"A system in one perspective is a subsystem in another. But the systems view always treats systems as integrated wholes of their subsidiary components and never as the mechanistic aggregate of parts in isolable causal relations." (Ervin László, "Introduction to Systems Philosophy", 1972)

"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)

"A system may be specified in either of two ways. In the first, which we shall call a state description, sets of abstract inputs, outputs and states are given, together with the action of the inputs on the states and the assignments of outputs to states. In the second, which we shall call a coordinate description, certain input, output and state variables are given, together with a system of dynamical equations describing the relations among the variables as functions of time. Modern mathematical system theory is formulated in terms of state descriptions, whereas the classical formulation is typically a coordinate description, for example a system of differential equations." (E S Bainbridge, "The Fundamental Duality of System Theory", 1975)

"Organization denotes those relations that must exist among the components of a system for it to be a member of a specific class. Structure denotes the components and relations that actually constitute a particular unity and make its organization real." (Humberto Maturana, "The Tree of Knowledge", 1987)

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"A semantic network or net represents knowledge as a net-like graph. An idea, event, situation or object almost always has a composite structure; this is represented in a semantic network by a corresponding structure of nodes (drawn as circles or boxes) representing conceptual units, and directed links (drawn as arrows between the nodes) representing the relations between the units." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

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

"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking - from the parts to the whole, from objects to relationships, from contents to patterns." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"[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)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Complexity is that property of a model which makes it difficult to formulate its overall behaviour in a given language, even when given reasonably complete information about its atomic components and their inter-relations." (Bruce Edmonds, "Syntactic Measures of Complexity", 1999)

"Metaphors can have profound significance because, as images or figures, they allow the mind to grasp or discover unsuspected ideal and material relationships between objects." (Giuseppe Del Re, "Cosmic Dance", 1999)

"What cognitive capabilities underlie our fundamental human achievements? Although a complete answer remains elusive, one basic component is a special kind of symbolic activity - the ability to pick out patterns, to identify recurrences of these patterns despite variation in the elements that compose them, to form concepts that abstract and reify these patterns, and to express these concepts in language. Analogy, in its most general sense, is this ability to think about relational patterns." (Keith Holyoak et al, "Introduction: The Place of Analogy in Cognition", 2001)

"Fuzzy relations are developed by allowing the relationship between elements of two or more sets to take on an infinite number of degrees of relationship between the extremes of 'completely related' and 'not related', which are the only degrees of relationship possible in crisp relations. In this sense, fuzzy relations are to crisp relations as fuzzy sets are to crisp sets; crisp sets and relations are more constrained realizations of fuzzy sets and relations."  (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)

"There exists an alternative to reductionism for studying systems. This alternative is known as holism. Holism considers systems to be more than the sum of their parts. It is of course interested in the parts and particularly the networks of relationships between the parts, but primarily in terms of how they give rise to and sustain in existence the new entity that is the whole whether it be a river system, an automobile, a philosophical system or a quality system." (Michael C Jackson, "Systems Thinking: Creative Holism for Manager", 2003)

"There is no doubt that science has made great progress by taking things apart, but it has become increasingly clear that many important questions can only be addressed by thinking more carefully about relationships between and amongst the parts. Indeed, one of the main difficulties in answering questions or solving problems - any kind of problem - is that we think the problem is in the parts, when it is really in the relationships between them." (Yaneer Bar-Yam, "Making Things Work: Solving Complex Problems in a Complex World", 2004)

"A diagram is a graphic shorthand. Though it is an ideogram, it is not necessarily an abstraction. It is a representation of something in that it is not the thing itself. In this sense, it cannot help but be embodied. It can never be free of value or meaning, even when it attempts to express relationships of formation and their processes. At the same time, a diagram is neither a structure nor an abstraction of structure." (Peter Eisenman, "Written Into the Void: Selected Writings", 1990-2004, 2007)

"Humans seem to have an inbuilt need to impose mathematical order, symmetry and cause-and-effect relationships on a natural world that often may not work in that way at all." (Michael Hanlon, "10 Questions Science Can’t Answer (Yet): A Guide to the Scientific Wilderness", 2007)

"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought." (Clifford A Pickover, "The Math Book", 2009)

"A conceptual model of an interactive application is, in summary: the structure of the application - the objects and their operations, attributes, and relation-ships; an idealized view of the how the application works – the model designers hope users will internalize; the mechanism by which users accomplish the tasks the application is intended to support." (Jeff Johnson & Austin Henderson, "Conceptual Models", 2011)

"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"Mathematical abstraction is the process of considering and manipulating op­erations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content con­nected to particular applications. […] abstraction is the process of passing from things to ideas, properties and relations, to properties of relations and relations of properties, to properties of relations between properties, etc. Being a fundamental thinking process, abstraction has two faces: a logical face and evidently a psychological aspect that is the target of cognitive sciences." (Hourya B Sinaceur,"Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)

"All the mathematical sciences are founded on the relations between physical laws and laws of numbers." (James C Maxwell)

"Human thinking depends on metaphor. We understand new and complex things in relation to the things we already know […] once you pick a metaphor it will guide your thinking." (Jonathan Haidt)

"Statistics is the science of finding relationships and actionable insights from data." (Nate Silver)

"Topology is the study of the modal relations of spatial figures and the laws of connectivity, mutual position, and ordering of points, lines, surfaces, and solids and their parts independently of measure and magnitude relations." (Johann B Listing)

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