❄️Systems Thinking: On Stability (Quotes)

"The behavior of two individuals, consisting of effort which results in output, is considered to be determined by a satisfaction function which depends on remuneration (receiving part of the output) and on the effort expended. The total output of the two individuals is not additive, that is, together they produce in general more than separately. Each individual behaves in a way which he considers will maximize his satisfaction function. Conditions are deduced for a certain relative equilibrium and for the stability of this equilibrium, i.e., conditions under which it will not pay the individual to decrease his efforts. In the absence of such conditions ‘exploitation’ occurs which may or may not lead to total parasitism." (Anatol Rapoport, "Mathematical theory of motivation interactions of two individuals," The Bulletin of Mathematical Biophysics 9, 1947)

"[…] there are three different but interconnected conceptions to be considered in every structure, and in every structural element involved: equilibrium, resistance, and stability." (Eduardo Torroja, "Philosophy of Structure" , 1951)

"Stability is commonly thought of as desirable, for its presence enables the system to combine of flexibility and activity in performance with something of permanence. Behaviour that is goal-seeking is an example of behaviour that is stable around a state of equilibrium. Nevertheless, stability is not always good, for a system may persist in returning to some state that, for other reasons, is considered undesirable." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"As shorthand, when the phenomena are suitably simple, words such as equilibrium and stability are of great value and convenience. Nevertheless, it should be always borne in mind that they are mere shorthand, and that the phenomena will not always have the simplicity that these words presuppose." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"The static stability of a system is defined by the initial tendency to return to equilibrium conditions following some disturbance from equilibrium. […] If the object has a tendency to continue in the direction of disturbance, negative static stability or static instability exists. […] If the object subject to disturbance has neither the tendency to return nor the tendency to continue in the displacement direction, neutral static stability exists." (Hugh H Hurt, "Aerodynamics for Naval Aviators", 1960)

"While static stability is concerned with the tendency of a displaced body to return to equilibrium, dynamic stability is concerned with the resulting motion with time. If an object is disturbed from equilibrium, the time history of the resulting motion indicates the dynamic stability of the system. In general, the system will demonstrate positive dynamic stability if the amplitude of the motion decreases with time." (Hugh H Hurt, "Aerodynamics for Naval Aviators", 1960)

"A real system is subject to perturbations and it is never possible to control its initial state exactly. This raises the question of stability: under a slight perturbation will the system remain near the equilibrium state or not?" Joseph P LaSalle & Solomon Lefschetz, "Stability by Liapunov's Direct Method with Applications", 1961) 

"[...] in a state of dynamic equilibrium with their environments. If they do not maintain this equilibrium they die; if they do maintain it they show a degree of spontaneity, variability, and purposiveness of response unknown in the non-living world. This is what is meant by ‘adaptation to environment’ […] [Its] essential feature […] is stability - that is, the ability to withstand disturbances." (Kenneth Craik, 'Living organisms', “The Nature of Psychology”, 1966)

"One of the central problems studied by mankind is the problem of the succession of form. Whatever is the ultimate nature of reality (assuming that this expression has meaning). it is indisputable that our universe is not chaos. We perceive beings, objects, things to which we give names. These beings or things are forms or structures endowed with a degree of stability: they take up some part of space and last for some period of time." (René Thom, "Structural Stability and Morphogenesis", 1972)

"There seems to be a time scale in all natural processes beyond which structural stability and calculability become incompatible." (René Thom, "Structural Stability and Morphogenesis", 1972)

"The point of departure is the measurement problem, as it appears in physics; the manner in which measurements allow us to characterize subsystems; the role of such subsystems as tools in system analysis; and the relationships existing between different ways of perceiving or interacting with the same system. Our conclusions are: (1) there exists no universal family of of analytic units appropriate for the treatment of all interactions; (2) there are on the contrary many such families of analytic units, all of which are equally 'real' and entitled to be treated on the same footing; (3) the appropriate use of natural interactions can enormously extend the class of physical observables accessible to us; (4) the concept of a model must be formulated, in its most general terms, as the sharing of a subsystem by two otherwise distinct systems, capable of imposing the same dynamic on an appropriate system with which they can both interact. We establish these results through a variety of terminologies which turn out to be equivalent: stability, invariance, symmetry, homeostasis." (Robert Rosen, "Fundamentals of Measurement and Representation of Natural Systems", 1978)

"Stability theory is the study of systems under various perturbing influences. Since there are many systems, many types of influences, and many equations describing systems, this is an open-ended problem. A system is designed so that it will be stable under external influences. However, one cannot predict all external influences, nor predict the magnitude of those that occur. Consequently, we need control theory. If one is interested in stability theory, a natural result is a theory of control." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Complex systems operate under conditions far from equilibrium. Complex systems need a constant flow of energy to change, evolve and survive as complex entities. Equilibrium, symmetry and complete stability mean death. Just as the flow, of energy is necessary to fight entropy and maintain the complex structure of the system, society can only survive as a process. It is defined not by its origins or its goals, but by what it is doing." (Paul Cilliers,"Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Cybernetics is the science of effective organization, of control and communication in animals and machines. It is the art of steersmanship, of regulation and stability. The concern here is with function, not construction, in providing regular and reproducible behaviour in the presence of disturbances. Here the emphasis is on families of solutions, ways of arranging matters that can apply to all forms of systems, whatever the material or design employed. [...] This science concerns the effects of inputs on outputs, but in the sense that the output state is desired to be constant or predictable – we wish the system to maintain an equilibrium state. It is applicable mostly to complex systems and to coupled systems, and uses the concepts of feedback and transformations (mappings from input to output) to effect the desired invariance or stability in the result." (Chris Lucas, "Cybernetics and Stochastic Systems", 1999)

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

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

"Global stability of an equilibrium removes the restrictions on the initial conditions. In global asymptotic stability, solutions approach the equilibrium for all initial conditions. [...] In a study of local stability, first equilibrium solutions are identified, then linearization techniques are applied to determine the behavior of solutions near the equilibrium. If the equilibrium is stable for any set of initial conditions, then this type of stability is referred to as global stability." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)

"It is important to distinguish between local and global stability. Local stability of an equilibrium implies that solutions approach the equilibrium only if they arc initially close to it. For example, if the initial population size is very small and the zero equilibrium is stable, then extinction of the population may occur. however, it the initial population size is large, then local stability of the zero equilibrium tells nothing about population extinction. Global stability of an equilibrium is much stronger. Global stability implies that regardless of the initial population size, solutions approach the equilibrium. We state conditions for local stability and global stability of an equilibrium in the case or a scalar difference equation, where only one state is modeled such as population size. In addition, we state conditions for local stability of an equilibrium when several states are modeled by first-order difference equations or when one state is modeled by a second-order or higher-order difference equation. These latter conditions are known as the Jury conditions." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)

"Like resilience, self-organizazion is often sacrificed for purposes of short-term productivity and stability." (Donella Meadows, "Thinking in Systems: A Primer", 2008)

"There is common ground in analysing financial systems and ecosystems, especially in the need to identify conditions that dispose a system to be knocked from seeming stability into another, less happy state." (Robert M May et al, "Complex systems: Ecology for bankers" 2008)

"If a network has violently changing properties, it is most probably not very stable. How can we measure stability, if a network remains unchanged? The assessment of stability often requires a test, and this test comes in the form of a perturbation to the network. A stable network should try to restore its original status after a perturbation. However, this is not easy. Most networks are open systems and therefore undergo a continuous series of perturbations." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Network stability may be a key element in the development of multilevel, nested networks. The formation of nested networks obviously requires at least a few contacts between the bottom networks. However, evolutionary selection requires the independence and at least temporary isolation of the bottom networks themselves. Weak links are probably the only tools for solving this apparent paradox." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Among complex systems, stability is typically meta-stability, which is preserved through cycling, whilst growth and shrinkage are often components of a larger-scale, cyclic wave." (Nick Land, "Eternal Return, and After", 2011)

"But the history of large systems demonstrates that, once the hurdle of stability has been cleared, a more subtle challenge appears. It is the challenge of remaining stable when the rules change. Machines, like organizations or organisms, that fail to meet this challenge find that their previous stability is no longer of any use. The responses that once were life-saving now just make things worse. What is needed now is the capacity to re-write the procedure manual on short notice, or even (most radical change of all) to change goals." (John Gall, "The Systems Bible: The Beginner's Guide to Systems Large and Small"[Systematics 3^rd Ed.], 2011)

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

"The qualitative structure of the flow can change as parameters are varied. In particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in the dynamics are called bifurcations, and the parameter values at which they occur are called bifurcation points. Bifurcations are important scientifically - they provide models of transitions and instabilities as some control parameter is varied." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

❄️Systems Thinking: On Emergence (Quotes)

"The component parts of a vegetable or animal substance do not lose their mechanical and chemical properties as separate agents, when, by a peculiar mode of juxtaposition, they, as an aggregate whole, acquire physiological or vital properties in addition. Those bodies continue, as before, to obey mechanical and chemical laws, in so far as the operation of those laws is not counteracted by the new laws which govern them as organized beings. […] Though there are laws which, like those of chemistry and physiology, owe their existence to a breach of the principle of Composition of Causes, it does not follow that these peculiar, or, as they might be termed, heteropathic laws, are not capable of composition with one another." (John S Mill, "A System of Logic: Ratioconative and Inductive", 1843) [the heteropathic laws is synonymous with emergence]

"The higher quality emerges from the lower level of existence and has its roots therein, but it emerges therefrom, and it does not belong to that lower level, but constitutes its possessor a new order of existent with its special laws of behaviour. The existence of emergent qualities thus described is something to be noted, as some would say, under the compulsion of brute empirical fact, or, as I should prefer to say in less harsh terms, to be accepted with the ‘natural piety’ of the investigator. It admits no explanation." (Samuel Alexander, "Space, Time and Deity", 1922)

"This, however, is very speculative; the point of interest for our present enquiry is that physical reality is built up, apparently, from a few fundamental types of units whose properties determine many of the properties of the most complicated phenomena, and this seems to afford a sufficient explanation of the emergence of analogies between mechanisms and similarities of relation-structure among these combinations without the necessity of any theory of objective universals." (Kenneth Craik, "The Nature of Explanation", 1943)

"The moment of truth, the sudden emergence of a new insight, is an act of intuition. Such intuitions give the appearance of miraculous flushes, or short-circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links. (Arthur Koestler, "The Act of Creation", 1964)

"The principle that whole entities exhibit properties which are meaningful only when attributed to the whole, not to its parts - e.g. the smell of ammonia. Every model of human activity system exhibits properties as a whole entity which derive from it component activities and their structure, but cannot be reduced to them." (Peter Checkland, "Systems Thinking, Systems Practice", 1981)

"[Hierarchy is] the principle according to which entities meaningfully treated as wholes are built up of smaller entities which are themselves wholes […] and so on. In hierarchy, emergent properties denote the levels." (Peter Checkland, "Systems Thinking, Systems Practice", 1981)

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

"Catastrophes are often stimulated by the failure to feel the emergence of a domain, and so what cannot be felt in the imagination is experienced as embodied sensation in the catastrophe. (William I Thompson, "Gaia, a Way of Knowing: Political Implications of the New Biology", 1987)

"At the other far extreme, we find many systems ordered as a patchwork of parallel operations, very much as in the neural network of a brain or in a colony of ants. Action in these systems proceeds in a messy cascade of interdependent events. Instead of the discrete ticks of cause and effect that run a clock, a thousand clock springs try to simultaneously run a parallel system. Since there is no chain of command, the particular action of any single spring diffuses into the whole, making it easier for the sum of the whole to overwhelm the parts of the whole. What emerges from the collective is not a series of critical individual actions but a multitude of simultaneous actions whose collective pattern is far more important. This is the swarm model." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Clearly, complex adaptive systems have a tendency to give rise to other complex adaptive systems. […] The appearance of more and more complex forms is not a phenomenon restricted to the evolution of complex adaptive systems, although for those systems the possibility arises of a selective advantage being associated under certain circumstances with increased complexity." (Murray Gell-Mann, "What is Complexity?", Complexity Vol 1 (1), 1995)

"[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations." (Fritjof Capra, "The web of life: a new scientific understanding of living systems", 1996)

"It may not be obvious at first, but the study of emergence and model-building go hand in hand. The essence of model-building is shearing away detail to get at essential elements. A model, by concentrating on selected aspects of the world, makes possible the prediction and planning that reveal new possibilities. That is exactly the problem we face in trying to develop a scientific understanding of emergence." (John H Holland, "Emergence" , Philosophica 59, 1997)

"When the behavior of the system depends on the behavior of the parts, the complexity of the whole must involve a description of the parts, thus it is large. The smaller the parts that must be described to describe the behavior of the whole, the larger the complexity of the entire system. […] A complex system is a system formed out of many components whose behavior is emergent, that is, the behavior of the system cannot be simply inferred from the behavior of its components." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"Perhaps the greatest puzzle in self-organisation is the question of emergent properties. That is, how do features that characterise a system as a whole emerge from the interplay of the parts that go to makeup the system? […] The issue of self-organisation remains as great a challenge today as it ever has been. Though there remains much yet to learn about self-organisation, we can at least identify some of the processes involved." (David G Green, "Self-organisation in complex systems", 2000)

"The central proposition in [realistic thinking] is that human actions and interactions are processes, not systems, and the coherent patterning of those processes becomes what it becomes because of their intrinsic capacity, the intrinsic capacity of interaction and relationship, to form coherence. That emergent form is radically unpredictable, but it emerges in a controlled or patterned way because of the characteristic of relationship itself, creation and destruction in conditions at the edge of chaos." (Ralph D Stacey et al, "Complexity and Management: Fad or Radical Challenge to Systems Thinking?", 2000)

"Emergent self-organization in multi-agent systems appears to contradict the second law of thermodynamics. This paradox has been explained in terms of a coupling between the macro level that hosts self-organization (and an apparent reduction in entropy), and the micro level (where random processes greatly increase entropy). Metaphorically, the micro level serves as an entropy 'sink', permitting overall system entropy to increase while sequestering this increase from the interactions where self-organization is desired." (H Van Dyke Parunak & Sven Brueckner, "Entropy and Self-Organization in Multi-Agent Systems", Proceedings of the International Conference on Autonomous Agents, 2001)

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

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

"Emergence is not really mysterious, although it may be complex. Emergence is brought about by the interactions between the parts of a system. The galloping horse illusion depends upon the persistence of the human retina/brain combination, for instance. Elemental gases bond in combination by sharing outer electrons, thereby altering the appearance and behavior of the combination. In every case of emergence, the source is interaction between the parts - sometimes, as with the brain, very many parts - so that the phenomenon defies simple explanation." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"In particular, complexity examines how components of a system can through their dynamic interaction 'spontaneously' develop collective properties or patterns, such as colour, that do not seem implicit, or at least not implicit in the same way, within individual components.  Complexity investigates emergent properties, certain regularities of behaviour that somehow transcend the ingredients that make them up. Complexity argues against reductionism, against reducing the whole to the parts. And in so doing it transforms scientific understanding of far-from-equilibrium structures, of irreversible times and of non-Euclidean mobile spaces. It emphasizes how positive feedback loops can exacerbate initial stresses in the system and render it unable to absorb shocks to re-establish the original equilibrium. Positive feedback occurs when a change tendency is reinforced rather than dampened clown. Very strong interactions occur between the parts of such systems, with the absence of a central hierarchical structure that unambiguously' governs' and produces outcomes. These outcomes are to be seen as both uncertain and irreversible." (John Urry, "Global Complexity", 2003)

"Emergence is the phenomenon of properties, capabilities and behaviours evident in the whole system that are not exclusively ascribable to any of its parts." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"Another typical feature of theories of emergence is the layered view of nature. On this view, all things in nature belong to a certain level of existence, each according to its characteristic properties. These levels of existence constitute a hierarchy of increasing complexity that also corresponds to their order of appearance in the course of evolution." (Markus Eronen, "Emergence in the Philosophy of Mind", 2004)

"Emergence refers to the relationship between the details of a system and the larger view. Emergence does not emphasize the primary importance of the details or of the larger view; it is concerned with the relationship between the two. Specifically, emergence seeks to discover: Which details are important for the larger view, and which are not? How do collective properties arise from the properties of parts? How does behavior at a larger scale of the system arise from the detailed structure, behavior and relationships on a finer scale?" (Yaneer Bar-Yam, "Making Things Work: Solving Complex Problems in a Complex World", 2004) 

"The basic concept of complexity theory is that systems show patterns of organization without organizer (autonomous or self-organization). Simple local interactions of many mutually interacting parts can lead to emergence of complex global structures. […] Complexity originates from the tendency of large dynamical systems to organize themselves into a critical state, with avalanches or 'punctuations' of all sizes. In the critical state, events which would otherwise be uncoupled became correlated." (Jochen Fromm, "The Emergence of Complexity", 2004)

"Complexity arises when emergent system-level phenomena are characterized by patterns in time or a given state space that have neither too much nor too little form. Neither in stasis nor changing randomly, these emergent phenomena are interesting, due to the coupling of individual and global behaviours as well as the difficulties they pose for prediction. Broad patterns of system behaviour may be predictable, but the system's specific path through a space of possible states is not." (Steve Maguire et al, "Complexity Science and Organization Studies", 2006)

"The beauty of nature insists on taking its time. Everything is prepared. Nothing is rushed. The rhythm of emergence is a gradual, slow beat; always inching its way forward, change remains faithful to itself until the new unfolds in the full confidence of true arrival. Because nothing is abrupt, the beginning of spring nearly always catches us unawares. It is there before we see it; and then we can look nowhere without seeing it. (John O'Donohue, "To Bless the Space Between Us: A Book of Blessings", 2008)

[emergence:] "The process of complex pattern formation from simpler rules; emergent properties are neither properties had by any parts of the system taken in isolation nor a resultant of a mere summation of properties of parts of the system." (Ani Calinescu & Janet Efstathiou, "Measures of Network Structure", Encyclopedia of Networked and Virtual Organizations, 2008) 

"Although the potential for chaos resides in every system, chaos, when it emerges, frequently stays within the bounds of its attractor(s): No point or pattern of points is ever repeated, but some form of patterning emerges, rather than randomness. Life scientists in different areas have noticed that life seems able to balance order and chaos at a place of balance known as the edge of chaos. Observations from both nature and artificial life suggest that the edge of chaos favors evolutionary adaptation." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)

"Emergence is defined as the occurrence of new processes operating at a higher level of abstraction then is the level at which the local rules operate." (Jirí Kroc & Peter M A Sloot, "Complex Systems Modeling by Cellular Automata", Encyclopedia of Artificial Intelligence, 2009)

"If universality is one of the observed characteristics of complex dynamical systems in many fields of study, a second characteristic that flows from the study of these systems is that of emergence. As self-organizing systems go about their daily business, they are constantly exchanging matter and energy with their environment, and this allows them to remain in a state that is far from equilibrium. That allows spontaneous behavior to give rise to new patterns." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)

"The notion of emergence is used in a variety of disciplines such as evolutionary biology, the philosophy of mind and sociology, as well as in computational and complexity theory. It is associated with non-reductive naturalism, which claims that a hierarchy of levels of reality exist. While the emergent level is constituted by the underlying level, it is nevertheless autonomous from the constituting level. As a naturalistic theory, it excludes non-natural explanations such as vitalistic forces or entelechy. As non-reductive naturalism, emergence theory claims that higher-level entities cannot be explained by lower-level entities." (Martin Neumann, "An Epistemological Gap in Simulation Technologies and the Science of Society", 2011)

"System theorists know that it's easy to couple simple-to-understand systems into a ‘super system’ that's capable of displaying behavioral modes that cannot be seen in any of its constituent parts. This is the process called ‘emergence’." (John L Casti, [interview with Austin Allen], 2012)

"Every system that has existed emerged somehow, from somewhere, at some point. Complexity science emphasizes the study of how systems evolve through their disorganized parts into an organized whole." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"Things evolve to evolve. Evolutionary processes are the linchpin of change. These processes of discovery represent a complexity of simple systems that flux in perpetual tension as they teeter at the edge of chaos. This whirlwind of emergence is responsible for the spontaneous order and higher, organized complexity so noticeable in biological evolution - one–celled critters beefing up to become multicellular organisms." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

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

"Emergence is a nontrivial relationship between the properties of a system at microscopic and macroscopic scales. Macroscopic properties are called emergent when it is hard to explain them simply from microscopic properties." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

[emergence:] "A feature in a complex system that is generated through the dynamic interactions between the parts of a system at one level, and is realized at the next level of organization without intentionality or causality." (A Faye Bres, "Integral Post-Analysis of Design-Based Research of an Organizational Learning Process for Strategic Renewal of Environmental Management", Integral Theory and Transdisciplinary Action Research in Education, 2019)

🏷️Knowledge Representation: On Abstraction (Quotes)

"[Arithmetic] has a very great and elevating effect, compelling the soul to reason about abstract numbers, and rebelling against the introduction of visible or tangible objects into the argument." (Plato, "The Republic", cca 375 BC)

"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

"The exact kind of language we employ in philosophical analyses of abstract truth is one thing, and the language used in attempts to popularize the subject is another." (Marcus Tullius Cicero, "De officiis" ["On Duties"], cca.44 BC)

"It seems that all perception is but the grasping of the form of the perceived object in some manner. If, then, it is a perception of some material object, it consists in an apprehension of its form by abstracting it from matter in some way. But the kinds of abstraction are different and their degrees various. This is because, owing to matter, the material form is subject to certain states and conditions which do not belong to [the form] by itself insofar as it is this form. So sometimes the abstraction from matter is effected with all or some of these attachments, and sometimes it is complete in that the concept is abstracted from matter and from the accidents it possesses on account of the matter."(Avicenna Latinus [Ibn Sina], "Liber De anima", cca. 1014-1027)

"Sometimes a thing is perceived [via sense-perception] when it is observed; then it is imagined, when it is absent [in reality] through the representation of its form inside, Sense-perception grasps [the concept] insofar as it is buried in these accidents that cling to it because of the matter out of which it is made without abstracting it from [matter], and it grasps it only by means of a connection through position [ that exists] between its perception and its matter. It is for this reason that the form of [the thing] is not represented in the external sense when [sensation] ceases. As to the internal [faculty of] imagination, it imagines [the concept] together with these accidents, without being able to entirely abstract it from them. Still, [imagination] abstracts it from the afore-mentioned connection [through position] on which sense-perception depends, so that [imagination] represents the form [of the thing] despite the absence of the form's [outside] carrier." (Avicenna Latinus [Ibn Sina], "Pointer and Reminders", cca. 1030)

"Nor is it enough to say that the intelligible notions formed by the active intellect subsist somehow in the phantasmata (mental image), which are certainly intrinsic to us; for as we have already observed in treating the passive intellect, objects only become actually intelligible when abstracted from phantasmata; so that merely by way of the phantasmata, we cannot attribute the work of the active intellect to ourselves" (St. Thomas Aquinas, "De Anima" III, cca. 1268) [On Aristotle's phantasmata]

"Abstraction involves perceiving something, relating it to other things, grasping some common trait of those things, and conceiv­ing of the common trait as to it can be related not only to those things but also to other similar things."  (John Locke, "An Essay Concerning Human Understanding", 1689)

"But to form the idea of an object, and to form an idea simply is the same thing; the reference of the idea to an object being an extraneous denomination, of which in itself it bears no mark or character. Now as it is impossible to form an idea of an object, that is possessed of quantity and quality, and yet is possessed of no precise degree of either; it follows, that there is an equal impossibility of forming an idea, that is not limited and confined in both these particulars. Abstract ideas are therefore in themselves individual, however they may become general in their representation. The image in the mind is only that of a particular object, though the application of it in our reasoning be the same, as if it were universal." (David Hume, "Treatise of Human Nature", 1738)

"General abstract truth is the most precious of all blessings; without it, man is blind; it is the eye of reason." (Jean-Jacques Rousseau, "The Confessions of J. J. Rousseau", 1783)

"It is impossible to disassociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept a word is needed; to portray a phenomenon a concept is needed. All three mirror one and the same reality." (Antoine-Laurent Lavoisier, "Traite Elementaire de Chimie", 1789)

"Delight at having understood a very abstract and obscure system leads most people to believe in the truth of what it demonstrates." (Georg C Lichtenberg, Notebook J, 1789-1793)

"Whoever limits his exertions to the gratification of others, whether by personal exhibition, as in the case of the actor and of the mimic, or by those kinds of literary composition which are calculated for no end but to please or to entertain, renders himself, in some measure, dependent on their caprices and humours. The diversity among men, in their judgments concerning the objects of taste, is incomparably greater than in their speculative conclusions; and accordingly, a mathematician will publish to the world a geometrical demonstration, or a philosopher, a process of abstract reasoning, with a confidence very different from what a poet would feel, in communicating one of his productions even to a friend." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"Before abstraction everything is one, but one like chaos; after abstraction everything is united again, but this union is a free binding of autonomous, self-determined beings. Out of a mob a society has developed, chaos has been transformed into a manifold world." (G P Friedrich von Hardenberg [Novalis]," Blüthenstaub", 1798) 

"The expressions abstract and concrete refer not so much to the concepts themselves - for any concept is an abstract concept - as to their usage. And this usage can again have different grades; - according as one treats a concept now more, now less abstract or concrete, that is, takes away from or adds to it now more, now fewer definitions."(Immanuel Kant, "Logik: ein Handbuch zu Vorlesungen", 1800)

"Facts are the mere dross of history. It is from the abstract truth which interpenetrates them, and lies latent among them, like gold in the ore, that the mass derives its whole value: and the precious particles are generally combined with the baser in such a manner that the separation is a task of the utmost difficulty." (Thomas B Macaulay, "History", 1828)

"The domain of physics is no proper field for mathematical pastimes. The best security would be in giving a geometrical training to physicists, who need not then have recourse to mathematicians, whose tendency is to despise experimental science. By this method will that union between the abstract and the concrete be effected which will perfect the uses of mathematical, while extending the positive value of physical science. Meantime, the uses of analysis in physics is clear enough. Without it we should have no precision, and no co-ordination; and what account could we give of our study of heat, weight, light, etc.? We should have merely series of unconnected facts, in which we could foresee nothing but by constant recourse to experiment; whereas, they now have a character of rationality which fits them for purposes of prevision." (Auguste Comte, "The Positive Philosophy", 1830)

"Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space." (James J Sylvester, "A Probationary Lecture on Geometry", 1844)

"Fundamentally, as is readily seen, there exists neither force nor matter. Both are abstractions of things, such as they are, looked at from different standpoints. They complete and presuppose each other. Isolated they are meaningless." (Emil DuBois-Reymond, "Untersuchungen über tierische Elektrizität", 1848)

"Science gains from it [the pendulum] more than one can expect. With its huge dimensions, the apparatus presents qualities that one would try in vain to communicate by constructing it on a small [scale], no matter how carefully. Already the regularity of its motion promises the most conclusive results. One collects numbers that, compared with the predictions of theory, permit one to appreciate how far the true pendulum approximates or differs from the abstract system called 'the simple pendulum'." (Jean-Bernard-Léon Foucault, "Demonstration Experimentale du Movement de Rotation de la Terre", 1851)

"Beyond the little arithmetic required for the ordinary economies of life, the mass of college-bred men, unless engaged in the business of instruction or in pursuits which directly involve their application, from the time they leave their places of education, of whatever name, give up the Mathematics as a useless and hopeless abstraction." (Edward Everett, [address] 1857)

"The Mathematics, like language, (of which indeed they may be considered a species,) comprehending under that designation the whole science of number, space, form, time, and motion, as far as it can be expressed in abstract formulas, are evidently not only one of the most useful, but one of the grandest of studies." (Edward Everett, [address] 1857)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, „The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Purely mechanical phenomena do not exist […] are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it." (Ernst Mach, "The Science of Mechanics", 1883)

"The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity." (Hermann Helmholtz, "Vorträge und Reden", 1896)

"In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor." (Hermann von Helmholtz, "Vorträge und Reden", 1896)

"Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact. When the mathematician deals with facts, they become for him mere ‘hypotheses’; for with their truth he refuses to concern himself. The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality." (Charles S Peirce, "The Regenerated Logic", The Monist Vol. 7 (1), 1896)

"In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics, - provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary." (David Hilbert, "Die Theorie der algebraischen Zahlkorper", 1897)

"Our science, in contrast with others, is not founded on a single period of human history, but has accompanied the development of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in Engineering. She not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers." (Felix Klein, "Klein und Riecke: Ueber angewandte Mathematik und Physik" 1900)

"The man of science deals with questions which commonly lie outside of the range of ordinary experience, which often have no immediately discernible relation to the affairs of everyday life, and which concentrate the mind upon apparent abstractions to an extraordinary degree." (Frank W Clarke, "The Man of Science in Practical Affairs", Appletons' Popular Science Monthly Vol. XLV, 1900)

"A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought." (Paul Carus, „Friedrich Schiller: A Sketch of His Life and an Appreciation of His Poetry", 1905)

"But, once again, what the physical states as the result of an experiment is not the recital of observed facts, but the interpretation and the transposing of these facts into the ideal, abstract, symbolic world created by the theories he regards as established." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract."  (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1908)

"Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite. " (Ezra Pound, "The Spirit of Romance", 1910)

"The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i. e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet." (Felix Klein, "Lectures on Mathematics", 1911)

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. " (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)

"Even the most refined statistics are nothing but abstractions." (Walter Lippmann, "Politics, The Golden Rule and After", 1913)

"[…] science deals with but a partial aspect of reality, and there is no faintest reason for supposing that everything science ignores is less real than what it accepts. [...] Why is it that science forms a closed system? Why is it that the elements of reality it ignores never come in to disturb it? The reason is that all the terms of physics are defined in terms of one another. The abstractions with which physics begins are all it ever has to do with." (John W N Sullivan, "The Limitations of Science", 1915)

"Abstract as it is, science is but an outgrowth of life. That is what the teacher must continually keep in mind. […] Let him explain […] science is not a dead system - the excretion of a monstrous pedantism - but really one of the most vigorous and exuberant phases of human life." (George A L Sarton, "The Teaching of the History of Science", The Scientific Monthly, 1918)

"It is not surprising that the greatest mathematicians have again and again appealed to the arts in order to find some analogy to their own work. They have indeed found it in the most varied arts, in poetry, in painting, and in sculpture, although it would certainly seem that it is in music, the most abstract of all the arts, the art of number and of time, that we find the closest analogy." (Havelock Ellis, "The Dance of Life", 1923)

"Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about." (Alfred N Whitehead, "Science and the Modern World", 1925)

"Progress in truth - truth of science and truth of religion - is mainly a progress in the framing of concepts, in discarding artificial abstractions or partial metaphors, and in evolving notions which strike more deeply into the root of reality." (Alfred N Whitehead, "Religion in the Making, Truth and Criticism", 1926)

"Often a liberal antidote of experience supplies a sovereign cure for a paralyzing abstraction built upon a theory." (Benjamin N Cardozo, "The Paradoxes of Legal Science", 1928)

"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"The fundamental concepts of physical science, it is now understood, are abstractions, framed by our mind, so as to bring order to an apparent chaos of phenomena." (Sir William C Dampier, "A History of Science and its Relations with Philosophy & Religion", 1931)

"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)

"We love to discover in the cosmos the geometrical forms that exist in the depths of our consciousness. The exactitude of the proportions of our monuments and the precision of our machines express a fundamental character of our mind. Geometry does not exist in the earthly world. It has originated in ourselves. The methods of nature are never so precise as those of man. We do not find in the universe the clearness and accuracy of our thought. We attempt, therefore, to abstract from the complexity of phenomena some simple systems whose components bear to one another certain relations susceptible of being described mathematically." (Alexis Carrel, "Man the Unknown", 1935)

„[...] the abstract mathematical theory has an independent, if lonely existence of its own. But when a sufficient number of its terms are given physical definitions it becomes a part of a vital organism concerning itself at every instant with matters full of human significance. Every theorem can be given the form ‘if you do so and so, such and such will happen'." (Oswald Veblen, "Remarks on the Foundation of Geometry", Bulletin of the American Mathematical Society, Vol. 35, 1935)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T. Sanders, "Mathematics", National Mathematics Magazine, 1937)

"Sooner or later the cold plunge into pure abstraction must be taken if one is to learn to swim in mathematics and to reason as rational, thinking human beings do." (Eric T Bell, "The Handmaiden of the Sciences", 1937)

"The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes." (Eric T Bell, "Men of Mathematics", 1937)

"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)

"The first thing to realize about physics […] is its extraordinary indirectness. […] For physics is not about the real world, it is about 'abstractions' from the real world, and this is what makes it so scientific. […] Theoretical physics runs merrily along with these unreal abstractions, but its conclusions are checked, at every possible point, by experiments." (Anthony Standen, "Science is a Sacred Cow", 1950)

"In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations." (David Hilbert, "Geometry and the imagination", 1952)

"There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned. (Raymond L Wilder, Introduction to the Foundations of Mathematics, 1952)

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals – such were its abstract characteristics." (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness", 1955)

"Abstractions are wonderfully clever tools for taking things apart and for arranging things in patterns but they are very little use in putting things together and no use at all when it comes to determining what things are for." (Archibald MacLeish, "Why Do We Teach Poetry?", The Atlantic Monthly Vol. 197 (3), 1956)

"Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra." (Susanne K Langer, "Philosophy in a New Key", 1957)

"One great lesson that we can learn from its systematic absence in the work of the grand theorists is that every self-conscious thinker must at all times be aware of - and hence be able to control - the levels of abstraction on which he is working. The capacity to shuttle between levels of abstraction, with ease and with clarity, is a signal mark of the imaginative and systematic thinker." (C Wright Mills, "The Sociological Imagination", 1959)

"There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction." (Thomas Merton, "The Secular Journal of Thomas Merton", 1959)

"Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework." (Melvin Schwartz, "Principles of Electrodynamics", 1972)

"Our science, in contrast with others, is not founded on a single period of human history, but has accompanied the development of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in Engineering. She not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers." (Felix Klein, "Klein und Riecke: Ueber angewandte Mathematik und Physik" 1900)

"The man of science deals with questions which commonly lie outside of the range of ordinary experience, which often have no immediately discernible relation to the affairs of everyday life, and which concentrate the mind upon apparent abstractions to an extraordinary degree." (Frank W Clarke, "The Man of Science in Practical Affairs", Appletons' Popular Science Monthly Vol. XLV, 1900)

"A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought." (Paul Carus, „Friedrich Schiller: A Sketch of His Life and an Appreciation of His Poetry", 1905)

"But, once again, what the physical states as the result of an experiment is not the recital of observed facts, but the interpretation and the transposing of these facts into the ideal, abstract, symbolic world created by the theories he regards as established." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract."  (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1908)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics." (Eric T Bell, "The Development of Mathematics", 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for […] The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)

"I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas." (John von Neumann,  "The Mathematician", The Works of the Mind Vol. I (1), 1947)

"It is of our very nature to see the universe as a place that we can talk about. In particular, you will remember, the brain tends to compute by organizing all of its input into certain general patterns. It is natural for us, therefore, to try to make these grand abstractions, to seek for one formula, one model, one God, around which we can organize all our communication and the whole business of living." (John Z Young, "Doubt and Certainty in Science: A Biologist’s Reflections on the Brain", 1960)

"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)

"Scientists, it should already be clear, never learn concepts, laws, and theories in the abstract and by themselves. Instead, these intellectual tools are from the start encountered in a historically and pedagogically prior unit that displays them with and through their applications." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

"With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractions, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range for its applications." (Aleksandr D Aleksandrov, 1963)

"A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense..." (Hermann Bondi. "Relativity and Common Sense", 1964)

"If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details." (John G Kemeny, "Random Essays on Mathematics, Education, and Computers", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

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

"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 […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"Science uses the senses but does not enjoy them; finally buries them under theory, abstraction, mathematical generalization." (Theodore Roszak, "Where the Wasteland Ends", 1972)
 
"The beauty of physics lies in the extent which seemingly complex and unrelated phenomena can be explained and correlated through a high level of abstraction by a set of laws which are amazing in their simplicity." (Melvin Schwartz, "Principles of Electrodynamics", 1972)

"A model is an abstract description of the real world. It is a simple representation of more complex forms, processes and functions of physical phenomena and ideas." (Moshe F Rubinstein & Iris R Firstenberg, "Patterns of Problem Solving", 1975)

"The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction." (Hilary Putnam, "Mathematics, Matter, and Method", 1975)

"Every word is an abstraction or category, not a particular." (Robert Pinsky, "The Situation of Poetry - Contemporary Poetry and its Traditions", 1976)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, "The Role of Mathematics in the Rise of Science", 1981)

"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"Language is the most formless means of expression. Its capacity to describe concepts without physical or visual references carries us into an advanced state of abstraction." (Ian Wilson, "Conceptual Art", 1984)

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of mathematics is extension, generalization, and abstraction [… which] often bring increased confidence in the results of a specific application, as well as new viewpoints."  (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"In mathematics itself abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics  and that of a research subject with a highly active life of its own." (Israel N Herstein, "Abstract Algebra", 1986)

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

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

"The essence of modeling, as we see it, is that one begins with a nontrivial word problem about the world around us. We then grapple with the not always obvious problem of how it can be posed as a mathematical question. Emphasis is on the evolution of a roughly conceived idea into a more abstract but manageable form in which inessentials have been eliminated. One of the lessons learned is that there is no best model, only better ones."  (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

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

"[…] a model is the picture of the real - a short form of the whole. Hence, a model is an abstraction or simplification of a system. It is a technique by which aspects of reality can be 'artificially' represented or 'simulated' and at the same time simplified to facilitate comprehension." (Laxmi K Patnaik, "Model Building in Political Science", The Indian Journal of Political Science, Vol. 50, No. 2, 1989)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts", 1989)

"Modeling in its broadest sense is the cost-effective use of something in place of something else for some [cognitive] purpose. It allows us to use something that is simpler, safer, or cheaper than reality instead of reality for some purpose. A model represents reality for the given purpose; the model is an abstraction of reality in the sense that it cannot represent all aspects of reality. This allows us to deal with the world in a simplified manner, avoiding the complexity, danger and irreversibility of reality." (Jeff Rothenberg, "The Nature of Modeling. In: Artificial Intelligence, Simulation, and Modeling", 1989)

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 1991)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)

"Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical." (Edward Rothstein, "Emblems of Mind: The Inner Life of Music and Mathematics", 1995)

"The larger, more detailed and complex the model - the less abstract the abstraction – the smaller the number of people capable of understanding it and the longer it takes for its weaknesses and limitations to be found out." (John Adams, "Risk", 1995)

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

"Abstract concepts are largely metaphorical." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought", 1999)

"The abstractions of science are stereotypes, as two-dimensional and as potentially misleading as everyday stereotypes. And yet they are as necessary to the process of understanding as filtering is to the process of perception." (K C Cole, First You Build a Cloud and Other Reflections on Physics as a Way of Life, 1999)

"Abstraction is itself an abstract word and has no single meaning. […] Every word in our language is abstract, because it represents something else." (Eric Maisel, "The Creativity Book: A Year's Worth of Inspiration and Guidance", 2000)

"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)

"[…] we underestimate the share of randomness in about everything […]  The degree of resistance to randomness in one’s life is an abstract idea, part of its logic counterintuitive, and, to confuse matters, its realizations nonobservable." (Nassim N Taleb, "Fooled by Randomness", 2001)

"A model isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)

"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Do not be afraid of the word 'theory'. Yes, it can sound dauntingly abstract at times, and in the hands of some writers can appear to have precious little to do with the actual, visual world around us. Good theory however, is an awesome thing. [...] But unless we actually use it, it borders on the metaphysical and might as well not be used at all." (Richard Howells,  Visual Culture, 2003)

"Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verifiable predictions about the behavior of atoms, molecules and solids." (Arthur M Lesk, "Introduction to Symmetry and Group Theory for Chemists", 2004)

"Mathematics is not about abstract entities alone but is about relation of abstract entities with real entities. […] Adequacy relations between abstract and real entities provide space or opportunity where mathematical and logical thought operates parsimoniously."  (Navjyoti Singh, "Classical Indian Mathematical Thought", 2005)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 2007)

"Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities." (Tom G Palmer, Realizing Freedom: Libertarian Theory, History, and Practice, 2009)

"In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions." (Bruce Hannon and Matthias Ruth, "Dynamic Modeling of Diseases and Pests", 2009)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

"Abstract formulations of simply stated concrete ideas are often the result of efforts to create idealized models of complex systems. The models are 'idealized' in the sense that they retain only the most fundamental properties of the original systems. The vocabulary is chosen to be as inclusive as possible so that research into the model reveals facts about a wide variety of similar systems. Unfortunately, it is often the case that over time the connection between a model and the systems on which it was based is lost, and the interested reader is faced with something that looks as if it were created to be deliberately complicated - deliberately confusing - but the original intention was just the opposite. Often, the model was devised to be simpler and more transparent than any of the systems on which it was based." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Presumably, one can become a mathematical genius only if one has an outstanding capacity for forming vivid mental represen­tations of abstract mathematical concepts - mental images that soon turn into an illusion, eclipsing the human origins of mathematical objects and endowing them with the semblance of an independent existence." (Stanislas Dehaene," The Number Sense: How the Mind Creates Mathematics", 2011) 

"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand." (Peter Rowlett, "The Unplanned Impact of Mathematics", Nature Vol. 475 (7355), 2011) 

"There is no unique, global, and universal relation of identity for abstract objects. [...] Abstract objects are of different sorts and this should mean, almost by definition, that there is no global, universal identity for sorts. Each sort X is equipped with an internal relation of identity but there is no identity relation that would apply to all sorts." (Jean-Pierre Marquis," Categorical foundations of mathematics, or how to provide foundations for abstract mathematics", The Review of Symbolic Logic Vol. 6 (1), 2012) 

"Abstraction is an essential knowledge process, the process (or, to some, the alleged process) by which we form concepts. It consists in recognizing one or several common features or attributes (properties, predicates) in individ­uals, and on that basis stating a concept subsuming those common features or attributes. Concept is an idea, associated with a word expressing a prop­erty or a collection of properties inferred or derived from different samples. Subsumption is the logical technique to get generality from particulars." (Hourya B Sinaceur," Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)

"In general, when building statistical models, we must not forget that the aim is to understand something about the real world. Or predict, choose an action, make a decision, summarize evidence, and so on, but always about the real world, not an abstract mathematical world: our models are not the reality - a point well made by George Box in his oft-cited remark that "all models are wrong, but some are useful". (David Hand, "Wonderful examples, but let's not close our eyes", Statistical Science 29, 2014) 

"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)

"Models can be: formulations, abstractions, replicas, idealizations, metaphors - and combinations of these. [...] Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than (merely) formidable Delphic oracles." (Barry Mazur, "The Authority of the Incomprehensible" , 2014)

"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise." (Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])

"The theory of groups is considered the language par excellence to study symmetry in science; it provides the mathematical formalism needed to tackle symmetry in a precise way. The aim of this chapter, therefore, is to lay the foundations of abstract group theory." (Pieter Thyssen & Arnout Ceulemans, "Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table", 2017)