Mental Models XLVIII

"If we wish to form a mental representation of what is going on among the molecules in calm air, we cannot do better than observe a swarm of bees, when every individual bee is flying furiously, first in one direction, and then in another, while the swarm, as a whole, either remains at rest, or sails slowly through the air." (James C Maxwell, "Molecules", Nature, 1873) 

"That phenomenon which corresponds to a thing in thought, we will call the representation, concept or mental image of a thing." (Giuseppe Veronese, "Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten in elementarer Form entwickelt", 1894)

"The image of the world around us, which we carry in our head, is just a model. Nobody in his head imagines all the world, government or country. He has only selected concepts, and relationships between them, and uses those to represent the real system." (Jay W Forrester, “Counter-Intuitive Behaviour of Social Systems”, Technological Review 73, 1971) 

"There are two subcategories of holist called irredundant holists and redundant holists. Students of both types image an entire system of facts or principles. Though an irredundant holist's image is rightly interconnected, it contains only relevant and essential constitents. In contrast, redundant holists entertain images that contain logically irrelevant or overspecific material, commonly derived from data used to 'enrich' the curriculum, and these students embed the salient facts and principles in a network of redundant items. Though logically irrelevant, the items in question are of great psychological importance to a 'redundant holist', since he uses them to access, retain and manipulate whatever he was originally required to learn." (Gordon Pask, "Learning Strategies and Individual Competence", 1972)

"Your experience in the world of physical matter flows outward from the center of your psyche. Then you perceive this experience. exterior events, circumstances and conditions are meant as a kind of living feedback. Altering the state of the psyche automatically alters the physical circumstances. There is no other valid way of changing physical events. It might help if you imagine an inner living dimension within yourself in which you create, in miniature psychic form, all the exterior conditions that you know. Simply put, you do exactly this. Your thoughts, feelings and mental pictures can be called incipient exterior events, for in one way or another, each of these is materialized into physical reality." (Jane Roberts, "The Nature of Personal Reality", 1974) 

"The psychological core of understanding consists in your having a 'working model' of the phenomenon in your mind. If you understand inflation, a mathematical proof, the way a computer works, DNA or a divorce, then you have a mental representation that serves as a model of an entity in much the same way as, say, a clock functions as a model of the earth's rotation." (Philip Johnson-Laird, "Mental Models: Towards a Cognitive Science of Language, Inference and Consciousness", 1983)

"Nothing resembles reality less than the photograph. Nothing resembles substance less than its shadow. To convey the meaning of something substantial you have to use not a shadow but a sign, not the limitation but the image. The image is a new and different reality, and of course it does not convey an impression of some object, but the mind of the subject; and that is something else again." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006) 

"Models are present in everything we do. One does not have a family or corporation in one's head. Instead, one has observations about those systems. Such observations and assumptions constitute mental models, which are then used as the basis for action. System dynamics models have little impact unless they change the way people perceive a situation. They must relate to and improve mental models if they are to fill an effective role." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"System dynamics models have little impact unless they change the way people perceive a situation. A model must help to organize information in a more understandable way. A model should link the past to the present by showing how present conditions arose, and extend the present into persuasive alternative futures under a variety of scenarios determined by policy alternatives. In other words, a system dynamics model, if it is to be effective, must communicate with and modify the prior mental models. Only people's beliefs - that is, their mental models - will determine action. Computer models must relate to and improve mental models if the computer models are to fill an effective role." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"From the mental models that people have, they can define the world around them. They can relate and share ideas with others as a way of improving their lives. […] Looking at the bigger picture, mental models not only help people in understanding how the world works, but it also draws people to comprehend their position in it." (Adam Feel, "Mental Models", 2019)

Jay W Forrester - Collected Quotes

"Negative feedback is the form normally encountered in the control of physical systems. Yet, positive feedback dominates in the growth and decline patterns of social systems." (Jay W Forrester, "Modeling the Dynamic Processes of Corporate Growth", 1964)

"Our social systems are highly nonlinear. It seems likely that such nonlinearities, coupled with the unstable tendencies caused by amplifications and time delays, create the characteristic modes of behavior […]" (Jay W Forrester, "Modeling the Dynamic Processes of Corporate Growth", 1964)

"[The engineer] must identify the significant and critical problems, but in his education, problems have been predetermined and assigned. He must develop the judgment to know what solutions to problems are possible, but in school the problems encountered are known to have answers." (Jay W Forrester, "Engineering Education and Engineering Practice in the Year 2000", 1967)

"Formulating a model of a system should start from the question 'Where is the boundary, that encompasses the smallest number of components, within which the dynamic behavior under study is generated?'" (Jay W Forrester, "Principles of Systems", 1968)

"[…] complex systems are counterintuitive. That is, they give indications that suggest corrective action which will often be ineffective or even adverse in its results." (Jay W Forrester, "Urban Dynamics", 1969)

"Nonlinear coupling allows one feedback loop to dominate the system for a time and then cause this dominance to shift to another part of the system where behavior is so different that the two seem unrelated." (Jay W. Forrester, "Urban Dynamics", 1969)

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

"The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by non‐linear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive‐feedback loops describing growth processes as well as negative, goal‐seeking loops." (Jay F Forrester, "Urban Dynamics", 1969)

"To model the dynamic behavior of a system, four hierarchies of structure should be recognized: closed boundary around the system; feedback loops as the basic structural elements within the boundary; level variables representing accumulations within the feedback loops; rate variables representing activity within the feedback loops." (Jay W Forrester, "Urban Dynamics", 1969)

"Each of us uses models constantly. Every person in his private life and in his business life instinctively uses models for decision making. The mental image of the world around you which you carry in your head is a model. […] A mental image is a model. All our decisions are taken on the basis of models." (Jay W Forrester, "Counter-Intuitive Behaviour of Social Systems", Technological Review 73, 1971)

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

"A model for simulating dynamic system behavior requires formal policy descriptions to specify how individual decisions are to be made. Flows of information are continuously converted into decisions and actions. No plea about the inadequacy of our understanding of the decision-making processes can excuse us from estimating decision-making criteria. To omit a decision point is to deny its presence - a mistake of far greater magnitude than any errors in our best estimate of the process." (Jay W Forrester, "Policies, decisions and information sources for modeling", 1994)

"First, social systems are inherently insensitive to most policy changes that people choose in an effort to alter the behavior of systems. In fact, social systems draw attention to the very points at which an attempt to intervene will fail. Human intuition develops from exposure to simple systems. In simple systems, the cause of a trouble is close in both time and space to symptoms of the trouble. If one touches a hot stove, the burn occurs here and now; the cause is obvious. However, in complex dynamic systems, causes are often far removed in both time and space from the symptoms. True causes may lie far back in time and arise from an entirely different part of the system from when and where the symptoms occur. However, the complex system can mislead in devious ways by presenting an apparent cause that meets the expectations derived from simple systems."(Jay W Forrester, "Counterintuitive Behavior of Social Systems", 1995)

"Second, social systems seem to have a few sensitive influence points through which behavior can be changed. These high-influence points are not where most people expect. Furthermore, when a high-influence policy is identified, the chances are great that a person guided by intuition and judgment will alter the system in the wrong direction." (Jay W Forrester, "Counterintuitive Behavior of Social Systems", 1995)

"System dynamics models are not derived statistically from time-series data. Instead, they are statements about system structure and the policies that guide decisions. Models contain the assumptions being made about a system. A model is only as good as the expertise which lies behind its formulation. A good computer model is distinguished from a poor one by the degree to which it captures the essence of a system that it represents. Many other kinds of mathematical models are limited because they will not accept the multiple-feedback-loop and nonlinear nature of real systems." (Jay W Forrester, "Counterintuitive Behavior of Social Systems", 1995)

"Third, social systems exhibit a conflict between short-term and long-term consequences of a policy change. A policy that produces improvement in the short run is usually one that degrades a system in the long run. Likewise, policies that produce long-run improvement may initially depress behavior of a system. This is especially treacherous. The short run is more visible and more compelling. Short-run pressures speak loudly for immediate attention. However, sequences of actions all aimed at short-run improvement can eventually burden a system with long-run depressants so severe that even heroic short-run measures no longer suffice. Many problems being faced today are the cumulative result of short-run measures taken in prior decades." (Jay W Forrester, "Counterintuitive Behavior of Social Systems", 1995)

"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W. Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"Models are present in everything we do. One does not have a family or corporation in one's head. Instead, one has observations about those systems. Such observations and assumptions constitute mental models, which are then used as the basis for action. System dynamics models have little impact unless they change the way people perceive a situation. They must relate to and improve mental models if they are to fill an effective role." (Jay W. Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"System dynamics models have little impact unless they change the way people perceive a situation. A model must help to organize information in a more understandable way. A model should link the past to the present by showing how present conditions arose, and extend the present into persuasive alternative futures under a variety of scenarios determined by policy alternatives. In other words, a system dynamics model, if it is to be effective, must communicate with and modify the prior mental models. Only people's beliefs - that is, their mental models - will determine action. Computer models must relate to and improve mental models if the computer models are to fill an effective role." (Jay W. Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

James R Thompson - Collected Quotes

"A model is generally more believable if it can predict what will happen, rather than 'explain' something that has already occurred. […] Model building is not so much the safe and cozy codification of what we are confident about as it is a means of orderly speculation." (James R Thompson, "Empirical Model Building", 1989)

"Although it is difficult to say where problem posing ends and problem solving begins, since in the ideal state there is continuous interaction between the two, model building is more concerned with the former than with the latter. Whereas problem solving can generally be approached by more or less well-defined techniques, there is seldom such order in the problem posing mode." (James R Thompson, "Empirical Model Building", 1989)

"Among the advantages of a simulation approach is principally that it enables us to eliminate time-consuming and artificial approximation theoretic activities and to spend our time in more useful pursuits. More importantly, simulation will enable us to deal with problems that are so complex in their 'closed form' manifestation that they are presently attacked only in ad hoc fashion." (James R Thompson, "Empirical Model Building", 1989)

"Empirical model building refers to a mindset that lends itself to constructing practical models useful in describing and coping with real-world situations." (James R Thompson, "Empirical Model Building", 1989)

"Whenever we axiomitize a real-world system, we always, of necessity, oversimplify. Frequently, the oversimplification will adequately describe the system for the purposes at hand. In many other cases, the oversimplification may seem deceptively close to reality, when in fact it is far wide of the mark. The best hope, of course, is the use of a model adequate to explain observation. However, when we are unable to develop an adequate model, we would generally be well advised to stick with empiricism and axiomatic imprecision." (James R Thompson, "Empirical Model Building", 1989)

Hermann G Grassmann - Collected Quotes

"Geometry can in no way be viewed […] as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I… realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry." (Hermann G Grassmann, "Ausdehnungslehre", 1844)

"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative." (Hermann G Grassmann, "Ausdehnungslehre", 1844)

"I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results." (Hermann G Grassmann, "Ausdehnungslehre", 1844)

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept." (Hermann GGrassmann, "Ausdehnungslehre", 1844) 

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"Mathematics is the science of the connection of magnitudes. Magnitude is anything that can be put equal or unequal to another thing. Two things are equal when in every assertion each may be replaced by the other."  (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"Every science aims to become a popular science. A science can only reach this goal if it also uses a popular language." (Hermann G Grassmann, 1870)

"An essential defect of previous presentations of geometry is that one usually returns to discrete numerical ratios in the treatment of similarity theory. This procedure, which at first seems simple, soon enough becomes entangled in complicated investigations concerning incommensurable magnitudes, as we have already hinted above; and the initial impression of simplicity is revenged upon problems of a purely geometrical procedure by the appearance of a set of difficult investigations of a completely heterogeneous type, which shed no light on the essence of spatial magnitudes. To be sure, one cannot eliminate the problem of measuring spatial magnitudes and expressing the results of these measurements numerically. But this problem cannot originate in geometry itself, but only arises when one, equipped on the one hand with the concept of number and on the other with spatial perceptions, applies them to that problem, and thus in a mixed branch that one can in a general sense call by the name 'theory of measurement' […] To relegate the theory of similarity, and even that of surface area, to this branch as has previously occurred (not to the form but to the substance) is to steal the essential content from what is called (pure) geometry." (Hermann G Grassmann)

Donella H Meadows - Collected Quotes

"Everything we think we know about the world is a model. Every word and every language is a model. All maps and statistics, books and databases, equations and computer programs are models. So are the ways I picture the world in my head - my mental models. None of these is or ever will be the real world. […] Our models usually have a strong congruence with the world. That is why we are such a successful species in the biosphere. Especially complex and sophisticated are the mental models we develop from direct, intimate experience of nature, people, and organizations immediately around us." (Donella H Meadows, "Limits to Growth", 1972)

"However, and conversely, our models fall far short of representing the world fully. That is why we make mistakes and why we are regularly surprised. In our heads, we can keep track of only a few variables at one time. We often draw illogical conclusions from accurate assumptions, or logical conclusions from inaccurate assumptions. Most of us, for instance, are surprised by the amount of growth an exponential process can generate. Few of us can intuit how to damp oscillations in a complex system." (Donella H Meadows, "Limits to Growth", 1972)

"Technology can relieve the symptoms of a problem without affecting the underlying causes. Faith in technology as the ultimate solution to all problems can thus divert our attention from the most fundamental problem - the problem of growth in a finite system - and prevent us from taking effective action to solve it." (Donella H Meadows, "The Limits to Growth", 1972)

"Models can easily become so complex that they are impenetrable, unexaminable, and virtually unalterable." (Donella H Meadows, "The unavoidable a priori", 1980)

"The world is a complex, interconnected, finite, ecological–social–psychological–economic system. We treat it as if it were not, as if it were divisible, separable, simple, and infinite. Our persistent, intractable global problems arise directly from this mismatch." (Donella H Meadows, "Whole Earth Models and System", 1982)

"A quantity growing exponentially toward a limit reaches that limit in a surprisingly short time." (Donella Meadows, "Thinking in systems: A Primer", 2008)

"A system is a set of things - people, cells, molecules, or whatever - interconnected in such a way that they produce their own pattern of behavior over time. […] The system, to a large extent, causes its own behavior." (Donella H Meadows, "Thinking in Systems: A Primer", 2008) 

"In fact, one of the most frustrating aspects of systems is that the purposes of subunits may add up to an overall behavior that no one wants." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)"

"In physical, exponentially growing systems, there must be at least one reinforcing loop driving growth and at least one balancing feedback loop constraining growth, because no system can grow forever in a finite environment." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"Let's face it, the universe is messy. It is nonlinear, turbulent, and chaotic. It is dynamic. It spends its time in transient behavior on its way to somewhere else, not in mathematically neat equilibria. It self-organizes and evolves. It creates diversity, not uniformity. That's what makes the world interesting, that's what makes it beautiful, and that's what makes it work." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

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

"The bounded rationality of each actor in a system may not lead to decisions that further the welfare of the system as a whole." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"[…] our mental models fail to take into account the complications of the real world - at least those ways that one can see from a systems perspective. It is a warning list. Here is where hidden snags lie. You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long-term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays. You are likely to mistreat, misdesign, or misread systems if you don’t respect their properties of resilience, self-organization, and hierarchy." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"The world is nonlinear. Trying to make it linear for our mathematical or administrative convenience is not usually a good idea even when feasible, and it is rarely feasible." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

"When there are long delays in feedback loops, some sort of foresight is essential." (Donella H Meadows, "Thinking in Systems: A Primer", 2008)

"You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays." (Donella H Meadow, "Thinking in Systems: A Primer", 2008)

Frank Wilczek - Collected Quotes

"As the idea of permanence of objects has faded, the idea of permanence of physical laws has become better established and more powerful." (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987) 

"The main problem with many nonscientific world models is the vigor with which they insist upon their rightness. Once a world model claims to be completely right, it is no longer open to any changes. […] Closed systems can be comforting, but they are limited. […] It's not the best we can do. Neither is extreme 'open-mindednesss' that slides into 'empty headedness' - the ideal that we can never really know anything."  (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987) 

"The most abstract conservation laws of physics come into their being in describing equilibrium in the most extreme conditions. They are the most rigorous conservation laws, the last to break down. The more extreme the conditions, the fewer the conserved structures. [...] In a deep sense, we understand the interior of the sun better that the interior of the earth, and the early stages of the big bang best of all."  (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987) 

"The whole idea of science is really to listen to nature, in her own language, as part of a continuing dialogue." (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987) 

"What is conserved, in modern physics, is not any particular substance or material but only much more abstract entities such as energy, momentum, and electric charge. The permanent aspects of reality are not particular materials or structures but rather the possible forms of structures and the rules for their transformation." (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987)

"In physics, your solution should convince a reasonable person. In math, you have to convince a person who's trying to make trouble. Ultimately, in physics, you're hoping to convince Nature. And I've found Nature to be pretty reasonable." (Frank Wilczek, Fantastic Realities: 49 Mind Journeys and a Trip to Stockholm, 2006) 

"An ordinary mistake is one that leads to a dead end, while a profound mistake is one that leads to progress. Anyone can make an ordinary mistake, but it takes a genius to make a profound mistake." (Frank Wilczek,"The Lightness of Being – Mass, Ether and the Unification of Forces", 2008) 

"Knowing how to calculate something is not the same as understanding it. Having a computer to calculate the origin of mass for us may be convincing, but is not satisfying. Fortunately we can understand it too." (Frank Wilczek,"The Lightness of Being – Mass, Ether and the Unification of Forces", 2008) 

"We evolved to be good at learning and using rules of thumb, not at searching for ultimate causes and making fine distinctions. Still less did we evolve to spin out long chains of calculation that connect fundamental laws to observable consequences. Computers are much better at it!" (Frank Wilczek,"The Lightness of Being – Mass, Ether and the Unification of Forces", 2008) 

"The happy coincidences between life’s requirements and nature’s choices of parameter-values might be just a series of flukes, but one could be forgiven for beginning to suspect that something deeper is at work. That suspicion is the first deep root of anthropic reasoning." (Frank Wilczek, "Multiversality", 2013) 

"Ironically, conventional quantum mechanics itself involves a vast expansion of physical reality, which may be enough to avoid Einstein Insanity. The equations of quantum dynamics allow physicists to predict the future values of the wave function, given its present value. According to the Schrödinger equation, the wave function evolves in a completely predictable way. But in practice we never have access to the full wave function, either at present or in the future, so this 'predictability' is unattainable. If the wave function provides the ultimate description of reality - a controversial issue!" (Frank Wilczek, "Einstein’s Parable of Quantum Insanity", 2015) 

Averroes - Collected Quotes

"After logic we must proceed to philosophy proper. Here too we have to learn from our predecessors, just as in mathematics and law. Thus it is wrong to forbid the study of ancient philosophy. Harm from it is accidental, like harm from taking medicine, drinking water, or studying law." (Averroes, "The Decisive Treatise", 1178)

"If teleological study of the world is philosophy, and if the Law commands such a study, then the Law commands philosophy." (Averroes, "The Decisive Treatise", 1178)

"The double meaning has been given to suit people's diverse intelligence. The apparent contradictions are meant to stimulate the learned to deeper study." (Averroes, "The Decisive Treatise", 1178)

"To master this instrument the religious thinker must make a preliminary study of logic, just as the lawyer must study legal reasoning. This is no more heretical in the one case than in the other." (Averroes, "The Decisive Treatise", 1178)

"The Law teaches that the universe was invented and created by God, and that it did not come into being by chance or by itself. (Averroes, "On the Harmony of Religions and Philosophy", 1198) 

"Knowledge is the conformity of the object and the intellect." (Averroes)

"The necessary connexion of movement and time is real and time is something the soul constructs in movement." (Averroes)

Giuseppe Peano - Collected Quotes

"Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use. (Giuseppe Peano, "I fondamenti dell’aritmetica nel Formulario del 1898" ["The Principles of Arithmetic, presented by a new method"], 1889)

"These primitive propositions […] suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. [...] All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions." (Giuseppe Peano, "I fondamenti dell’aritmetica nel Formulario del 1898" ["The Principles of Arithmetic, presented by a new method"], 1889)

"Certainly it is permitted to anyone to put forward whatever hypotheses he wishes, and to develop the logical consequences contained in those hypotheses. But in order that this work merit the name of Geometry, it is necessary that these hypotheses or postulates express the result of the more simple and elementary observations of physical figures." (Giuseppe Peano, "Sui fondamenti della geometria", 1894)

"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction. (Giuseppe Peano, "Notations de Logique Mathématique", 1894)

"Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define." (Giuseppe Peano, "Geometric Calculus", 1895)

"1. Zero is a number.
2. The successor of any number is another number.
3. There are no two numbers with the same successor.
4. Zero is not the successor of a number.
5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers." (Giuseppe Peano)

Misquoted: Jacque Hadamard on Complex Numbers

"The shortest path between two truths in the real domain passes through the complex domain." 

Probably this is one of the most known quotes on complex numbers as it easy to remember and reflects the fact that important problems in algebra, analysis, geometry, number theory and physics can be simplified by considering them into the complex plane. Even if the quote reflects pretty good the idea, the actual quote comes from Jacque Hadamard’s "An Essay on the Psychology of Invention in the Mathematical Field" published in 1945:

"It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one."

[French: "On a pu écrire depuis que la voie la plus courte et la meilleure entre deux vérités du domaine réel passe souvent par le domaine imaginaire." (Jacques Hadamard, "Essai sur la psychologie de l'invention dans le domaine mathématique", 1945)]

Here Hadamard refers to Paul Painlevé, who in his "Analyse des travaux scientifiques" published in 1900 wrote as follows:

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain."

Actually, "la voie" can be translated as "the way" as well as "the path", the latter being closer to Painlevé’s quote, to whom the metaphor can be attributed to. Painlevé is not the first who stressed this important advantage of the complex numbers over the real ones, however his metaphor captures this aspect the best. 

"At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial - or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications." (Carl F Gauss, [letter to Bessel] 1811)

"The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting." (Bernhard Riemann, "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse", 1851)

"The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments." (Thomas Hill, “The Imagination in Mathematics”, North American Review Vol. 85, 1857)

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Mathematics vs Philosophy II

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"[...] mathematics is not free to develop as it will, but is bound by certain restrictions: by conceptions derived either a posteriori from natural science, or assumed to be imposed a priori by an absolutistic philosophy." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Under the present dominance of formalism, one is tempted to paraphrase Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the of mathematics, has become empty." (Imre Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery", 1976)

"The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and account for the nature of mathematics. [...] The role of the philosophy of mathematics is to provide a systematic and absolutely secure foundation for mathematical knowledge, that is for mathematical truth." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"The controversy between those who think mathematics is discovered and those who think it is invented may run and run, like many perennial problems of philosophy. Controversies such as those between idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half thousand years. I do not expect to be able to convert those committed to the discovery view of mathematics to the inventionist view." (Paul Ernst, "Is Mathematics Discovered or Invented", 1996)

"Despite being partly familiar to all, because of these contradictory aspects, mathematics remains an enigma and a mystery at the heart of human culture. It is both the language of the everyday world of commercial life and that of an unseen and perfect virtual reality. It includes both free-ranging ethereal speculation and rock-hard certainty. How can this mystery be explained? How can it be unraveled? The philosophy of mathematics is meant to cast some light on this mystery: to explain the nature and character of mathematics. However this philosophy can be purely technical, a product of the academic love of technique expressed in the foundations of mathematics or in philosophical virtuosity. Too often the outcome of philosophical inquiry is to provide detailed answers to the how questions of mathematical certainty and existence, taking for granted the received ideology of mathematics, but with too little attention to the deeper why questions." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"The philosophy of mathematics is neither mathematics nor a subset of mathematics. It is a field of study which reflects on mathematics from the outside. It is one of a number of metatheories of mathematics, which also include the sociology, history, psychology, and anthropology of mathematics as well as mathematics education." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"Mathematics connect themselves on the one side with common life and the physical sciences; on the other side with philosophy, in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics, and the foundation of our knowledge of them." (Arthur Cayley)

"‘Tis of singular use, rightly to understand, and carefully to distinguish from hypotheses or mere suppositions, the true and certain consequences of experimental and mathematical philosophy; which do, with wonderful strength and advantage, to all such as are capable of apprehending them, confirm, establish, and vindicate against all objections, those great and fundamental truths of natural religion, which the wisdom of providence has at the same time universally implanted, in some degree, in the minds of persons even of the meanest capacities, not qualified to examine demonstrative proofs." (Samuel Clarke)

"Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason. […] Mathematical proofs are out of the reach of topical arguments, and are not to be attacked by the equivocal use of words or declamation, that make so great a part of other discourses; nay, even of controversies." (John Locke)

Mathematics vs Philosophy I

"Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason: for mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning. Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, - nay, even of controversies." (John Locke, "An Essay Concerning Human Understanding", 1690) 

"As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition." (Sir Isaac Newton, "Opticks", 1704)

"There is nothing more pleasant for man than the certainty of knowledge; whoever has once tasted of it is repelled by everything in which he perceives nothing but uncertainty. This is why the mathematicians who always deal with certain knowledge have been repelled by philosophy and other things, and have found nothing more pleasant than to spend their time with lines and letters. (Christian Wolff, 1741)

"It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

"The real finisher of our education is philosophy, but it is the office of mathematics to ward off the dangers of philosophy." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)

"In the history of mathematics, the ‘how’ always preceded the ‘why’, the technique of the subject preceded its philosophy." (Tobias Dantzig, "Number: The Language of Science", 1930)

"To create a healthy philosophy you should renounce metaphysics but be a good mathematician." (Bertrand Russell, [lecture] 1935)

"[…] all mathematical cognition has this pecularity: that it must first exhibit its concept in intuitional form. […] Without this, mathematics cannot take a single step. Its judgements are therefore always intuitional, whereas philosophy must make do with discursive judgements from mere concepts. It may illustrate its judgements by means of a visual form, but it can never derive them from such a form." (Immanuel Kant)

"In the end mathematics is but simple philosophy, and philosophy, higher mathematics in general." (Friederich von Hardenberg [Novalis])

John Stillwell - Collected Quotes

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

 

"Double periodicity is more interesting than single periodicity, because it is more varied. There is really only one periodic line, since all circles are the same up to a scale factor. However, there are infinitely many doubly periodic planes, even if we ignore scale. This is because the angle between the two periodic axes can vary, and so can the ratio of period lengths. The general picture of a doubly periodic plane is given by a lattice in the plane of complex numbers: a set of points of the form mA + nB, where A and B are nonzero complex numbers in different directions from O, and m and n run through all the integers. A and B are said to generate the lattice because it consists of all their sums and differences. […] The shape of the lattice of points mA + nB can therefore be represented by the complex number A/B. It is not hard to see that any nonzero complex number represents a lattice shape, so in some sense there is whole plane of lattice shapes. Even more interesting: the plane of lattice shapes is a periodic plane, because different numbers represent the same lattice." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"In fact, complex numbers are not much more complicated than reals, and many structures built on the complex numbers actually have simpler behavior than the corresponding structures built on the real numbers." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

 

"It is impossible for √-1 to be a real number, since its square is negative. This implies that √-1 is neither greater nor less than zero, so we cannot see √-1 on the real line. However, √-1 behaves like a number with respect to + and x. This prompts us to look elsewhere for it, and indeed we find it on another line (the imaginary axis) perpendicular to the real line." (John Stillwell, "Yearning for the impossible: the surprising truths of mathematics", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that  mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006) 

"The worst aspect of the term 'complex' - one that condemns it to eventual extinction in my opinion - is that it is also applied to structures called 'simple'. Mathematics uses the word 'simple' as a technical term for objects that cannot be 'simplified'. Prime numbers are the kind of thing that might be called 'simple' (though in their case it is not usually done) because they cannot be written as products of smaller numbers. At any rate, some of the 'simple' structures are built on the complex numbers, so mathematicians are obliged to speak of such things as 'complex simple Lie groups'. This is an embarrassment in a subject that prides itself on consistency, and surely either the word 'simple' or the word 'complex' has to go." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

 

"There are many ways to use unique prime factorization, and it is rightly regarded as a powerful idea in number theory. In fact, it is more powerful than Euclid could have imagined. There are complex numbers that behave like 'integers' and 'primes', and unique prime factorization holds for them as well. Complex integers were first used around 1770 by Euler, who found they have almost magical powers to unlock secrets of ordinary integers. For example, by using numbers of the form a + b√ -2. where a and b are integers, he was able to prove a claim of Fermat that 27 is the only cube that exceeds a square by 2. Euler's results were correct, but partly by good luck. He did not really understand complex 'primes' and their behavior." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful, knowing their geometric properties […] What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Today, most mathematicians have embraced the axiom of choice because of the order and simplicity it brings to mathematics in general. For example, the theorems that every vector space has a basis and every field has an algebraic closure hold only by virtue of the axiom of choice. Likewise, for the theorem that every sequentially continuous function is continuous. However, there are special places where the axiom of choice actually brings disorder. One is the theory of measure." (John Stillwell, "Roads to Infinity: The mathematics of truth and proof", 2010) 

On Complex Numbers XIV

"It is greatly to be lamented that this virtue of the real numbers [the ordinary integers], to be decomposable into prime factors, always the same ones [...]  does not also belong to the complex numbers [complex integers]; were this the case, the whole theory [...] could easily be brought to its conclusion. For this reason, the complex numbers we have been considering seem imperfect, and one may well ask whether one ought not to look for another kind which would preserve the analogy with the real numbers with respect to such a fundamental property." (Ernst E Kummer, 1844)

"If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -1 does not exist mathematically." (David Hilbert, [address to the International Congress of Mathematicians], 1900) 

"When we get used to playing with these complex numbers, we cease to think of a + ib as a pair of things, namely the two real numbers a and b, but we think of a+ib as an entire thing on its own, and we could use a single letter, say:, to denote the whole complex number z = a+ib. It may be checked that all the normal rules of algebra are satisfied by complex numbers. In fact, all this is a good deal more straightforward than checking everything for real numbers. […] From this point of view, it seems rather extraordinary that complex numbers were viewed with suspicion for so long, whereas the much more complicated extension from the nationals to the reals had, after ancient Greek times, been generally accepted without question." (Sir Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", 2004)

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"In fact, complex numbers are not much more complicated than reals, and many structures built on the complex numbers actually have simpler behavior than the corresponding structures built on the real numbers." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"It is impossible for √-1 to be a real number, since its square is negative. This implies that √-1 is neither greater nor less than zero, so we cannot see √-1 on the real line. However, √-1 behaves like a number with respect to + and x. This prompts us to look elsewhere for it, and indeed we find it on another line (the imaginary axis) perpendicular to the real line." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"Consider for example the complex numbers x + iy, where you of course ask what is i = √ −1 when you first encounter this mathematical construction. But that uncomfortable feeling of what this strange imaginary unit really is fades away as you get more experienced and learn that C is a field of numbers that is extremely useful, to say the least. You no longer care what kind of object i is but are satisfied only to know that i^2 = −1, which is how you calculate with i." (Andreas Rosén,"Geometric Multivector Analysis: From Grassmann to Dirac", 2019)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

"Imaginary numbers have an intuitive explanation: they 'rotate' numbers, just like negatives make a 'mirror image' of a number. […] Seeing imaginary numbers as rotations gives us a new mindset to approach problems; the 'plug and chug' formulas can make intuitive sense, even for a strange topic like complex numbers." (Kalid Azal, Math, Better Explained) 

"Zero is such a weird idea, having 'something' represent 'nothing', and it eluded the Romans. Complex numbers are similar - it’s a new way of thinking. But both zero and complex numbers make math much easier. If we never adopted strange, new number systems, we’d still be counting on our fingers." (Kalid Azal, Math,"Better Explained") 

Marcus Aurelius - Collected Quotes

"Always take the short cut; and that is the rational one. Therefore say and do everything according to soundest reason." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"And in the case of superior things like stars, we discover a kind of unity in separation. The higher we rise on the scale of being, the easier it is to discern a connection even among things separated by vast distances." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"Constantly regard the universe as one living being, having one substance and one soul; and observe how all things have reference to one perception, the perception of this one living being; and how all things act with one movement; and how all things are the cooperating causes of all things which exist; observe too the continuous spinning of the thread and the contexture of the web." (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"Either all things proceed from one intelligent source and come together as in one body, and the part ought not to find fault with what is done for the benefit of the whole; or there are only atoms, and nothing else than a mixture and dispersion. Why, then, art thou disturbed? Say to this ruling faculty, Art thou dead, art thou corrupted, art thou playing the hypocrite, art thou become a beast, dost thou herd and feed with the rest?" (Marcus Aurelius, "Meditations". cca. 121–180 AD)

 

"Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"Look to the essence of a thing, whether it be a point of doctrine, of practice, or of interpretation." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

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

 

"Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life." (Marcus Aurelius, Meditations, cca. 121–180)

 

"Observe the movements of the stars as if you were running their courses with them, and let your mind constantly dwell on the changes of the elements into each other." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"Time is a sort of river of passing events, and strong is its current; no sooner is a thing brought to sight than it is swept by and another takes its place, and this too will be swept away.“ (Marcus Aurelius, "Meditations", cca. 121–180 AD)

"The other reason is that what happens to the individual is a cause of well-being in what directs the world - of its well-being, its fulfillment, or its very existence, even. Because the whole is damaged if you cut away anything - anything at all - from its continuity and its coherence. Not only its parts, but its purposes. And that's what you're doing when you complain: hacking and destroying." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"The universal intelligence puts itself in motion for every separate effect [...] or it puts itself in motion once, and everything else comes by way of a sequence in a manner; or individual elements are the origin of all things. In a word, if there is a god, all is well; and if chance rules, do not thou be governed by it." (Marcus Aurelius, "Meditations", cca. 121–180 AD)

 

"The universe is change; our life is what our thoughts make it." (Marcus Aurelius)

Science vs Religion IV

"Modern masters of science are much impressed with the need of beginning all inquiry with a fact. The ancient masters of religion were quite equally impressed with that necessity. They began with the fact of sin - a fact as practical as potatoes. Whether or not man could be washed in miraculous waters, there was no doubt at any rate that he wanted washing." (Gilbert K Chesterton, "Orthodoxy", 1908)

"Art and Religion are, then, two roads by which men escape from circumstance to ecstasy. Between aesthetic and religious rapture there is a family alliance. Art and Religion are means to similar states of mind." (Clive Bell, "Art", 1913)

"The truths of religion are unprovable; the facts of science are unproved." (Gilbert K Chesterton, "Christian Science", 1920)

"Like the great temples of some religions, mathematics may be viewed only from the outside by those uninitiated into its mysteries. (Andrew M Gleason, "Evolution of an Active Mathematical Theory", Science Vol. 145 (3631), 1964)

"All religions, arts and sciences are branches of the same tree. All these aspirations are directed toward ennobling man’s life, lifting it from the sphere of mere physical existence and leading the individual towards freedom." (Albert Einstein, "Out of My Later Years", 1937)

"A religion old or new, that stressed the magnificence of the universe as revealed by modern science, might be able to draw forth reserves of reverence and awe hardly tapped by the conventional faiths. Sooner or later such a religion will emerge." (Carl Sagan, "Pale Blue Dot: a Vision of the Human Future in Space", 1994)

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

"Does there truly exist an insuperable contradiction between religion and science? Can religion be superseded by science? The answers to these questions have, for centuries, given rise to considerable dispute and, indeed, bitter fighting. Yet, in my own mind there can be no doubt that in both cases a dispassionate consideration can only lead to a negative answer. What complicates the solution, however, is the fact that while most people readily agree on what is meant by ‘science,’ they are likely to differ on the meaning of ‘religion’." (Albert Einstein)

"From religion comes a man's purpose; from science, his power to achieve it. Sometimes people ask if religion and science are not opposed to one another. They are: in the sense that the thumb and fingers of my hands are opposed to one another. It is an opposition by means of which anything can be grasped." (Sir William Bragg)

"He who posseses science and art, has religion; he who possesses neither science nor art, let him get religion." (Johann Wolfgang von Goethe)

Mathematics vs Religion I

"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701) 

"Like the great temples of some religions, mathematics may be viewed only from the outside by those uninitiated into its mysteries." (Andrew M Gleason, "Evolution of an Active Mathematical Theory", Science Vol. 145 (3631), 1964)
 
"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world."  (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)

"Mathematics is the life supreme. The life of the gods is mathematics. All divine messengers are mathematicians. Pure mathematics is religion. Its attainment requires a theophany." (Friederich von Hardenberg [Novalis])

"‘Tis of singular use, rightly to understand, and carefully to distinguish from hypotheses or mere suppositions, the true and certain consequences of experimental and mathematical philosophy; which do, with wonderful strength and advantage, to all such as are capable of apprehending them, confirm, establish, and vindicate against all objections, those great and fundamental truths of natural religion, which the wisdom of providence has at the same time universally implanted, in some degree, in the minds of persons even of the meanest capacities, not qualified to examine demonstrative proofs." (Samuel Clarke)

Science vs Religion III

"The essence of religion is inertia; the essence of science is change. It is the function of the one to preserve, it is the function of the other to improve. If, as in Egypt, they are firmly chained together, either science will advance, in which case the religion will be altered, or the religion will preserve its purity, and science will congeal." (Winwood Reade, "The Martyrdom of Man", 1876)

"The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet to be formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science, or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe. In the last analysis, magic, religion, and science are nothing but theories of thought." (Sir James G Frazer, "The Golden Bough", 1890)

"Both religion and natural science require a belief in God for their activities, to the former He is the starting point, and to the latter the goal of every thought process. To the former He is the foundation, to the latter, the crown of the edifice of every generalized world view." (Max Planck, "Religion and Natural Science", 1937)

"All men seek to be enlightened. Religion is but the most ancient and honorable way in which men have striven to make sense out of God's universe. Scientists seek the lawfulness of events. It is the task of Religion to fit man into this lawfulness." (Frank Herbert, "Dune", 1965)

"Both religion and science must preserve their autonomy and their distinctiveness. Religion is not founded on science nor is science an extension of religion. Each should possess its own principles, its pattern of procedures, its diversities of interpretation and its own conclusions." (Pope John Paul II, [Letter to Father George V Coyne], 1988)

"Religion is the antithesis of science; science is competent to illuminate all the deep questions of existence, and does so in a manner that makes full use of, and respects the human intellect. I see neither need nor sign of any future reconciliation." (Peter W Atkins, "Religion - The Antithesis to Science", 1997)

"As knowledge advances, science ceases to scoff at religion; and religion ceases to frown on science. The hour of mockery by the one, and of reproof by the other, is passing away. Henceforth, they will dwell together in unity and goodwill. They will mutually illustrate the wisdom, power, and grace of God. Science will adorn and enrich religion; and religion will ennoble and sanctify science." (Oliver W Holmes)

"Science is opposed to theological dogmas because science is founded on fact. To me, the universe is simply a great machine which never came into being and never will end. The human being is no exception to the natural order. Man, like the universe, is a machine." (Nikola Tesla)

"Science without religion is lame, religion without science is blind." (Albert Einstein)

"When confronted with the order and beauty of the universe and the strange coincidences of nature, it's very tempting to take the leap of faith from science into religion. I am sure many physicists want to. I only wish they would admit it." (Tony Rothman)

Science vs Religion II

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Religion and science are the two conjugated faces of phases of one and the same act of complete knowledge - the only one which can embrace the past and future of evolution so as to contemplate, measure and fulfill them." (Pierre T de Chardin, "The Phenomenon of Man", 1959)

"Science, philosophy and religion are bound to converge as they draw nearer to the whole." (Pierre T de Chardin, "The Phenomenon of Man", 1959)

"Scientists seek the lawfulness of events. It is the task of Religion to fit man into this lawfulness." (Frank Herbert, "Dune", 1965)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)

"To look at the development of physics since Newton is to observe a struggle to define the limits of science. Part of this process has been the intrusion of scientific methods and ideas into domains that have traditionally been the province of metaphysics or religion. In this conflict, Hawking’s phrase ‘to know the Mind of God’ is just one example of a border infringement. But by playing the God card, Hawking has cleverly fanned the flames of his own publicity, appealing directly to the popular allure of the scientist-as-priest."  (Peter Coles, "Hawking and the Mind of God", 2000)

"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)

"Religion claims to help us understand things about the universe, but, unlike science has no way to test or verify its claims. Both science and religion compete to understand reality, but only science has the method to verify its findings, while religion merely buttresses emotional and epistemic commitments made in advance, commitments impervious to evidence." (Jerry Coyne, 2015)

"Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature." (Freeman J Dyson)

Science vs Religion I

"Therefore science always goes abreast with the just elevation of the man, keeping step with religion and metaphysics; or, the state of science is an index of our self-knowledge." (Ralph W Emerson, "The Poet", 1844)

"A mere inference or theory must give way to a truth revealed; but a scientific truth must be maintained, however contradictory it may appear to the most cherished doctrines of religion." (David Brewster, "More Worlds Than One: The Creed of the Philosopher and the Hope of the Christian", 1856)

"Without poetry our science will appear incomplete, and most of what now passes with us for religion and philosophy will be replaced by poetry." (Matthew Arnold, "The Study of Poetry", 1880)

"Science boasts of the distance of its stars; of the terrific remoteness of the things of which it has to speak. But poetry and religion always insist upon the proximity, the almost menacing closeness of the things with which they are concerned." (Gilbert K Chesterton, 'A Glimpse of My Country', 1909)

"For a long time it has been known that the first systems of representations with which men have pictured to themselves the world and themselves were of religious origin. There is no religion that is not a cosmology at the same time that it is a speculation upon divine things. If philosophy and the sciences were born of religion, it is because religion began by taking the place of the sciences and philosophy." (Émile Durkheim, "The Elementary Forms of the Religious Life", 1912)

"The mysteries of religion are of a different order from those of science; they are parts of an arbitrary system of man’s own creation; they contradict our reason and our experience, while the mysteries of science are revealed by our reason, and transcend our experience." (John Burroughs, "Scientific Faith", The Atlantic Monthly, 1915)

"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", 1926)

"It is impossible to make a clear cut between science, religion, and art. The whole is never equal simply to the sum of its various parts." (Max Planck, "The Philosophy of Physics", 1936)

"Since science’s competence extends to observable and measurable phenomena, not to the inner being of things, and to the means, not to the ends of human life, it would be nonsense to expect that the progress of science will provide men with a new type of metaphysics, ethics, or religion."
(Jacques Maritain, "Science and Ontology", Bulletin of the Atomic Scientists Vol. 5, 1944)

"Science always goes abreast with the just elevation of the man, keeping step with religion and metaphysics; or, the state of science is an index of our self-knowledge." (Ralph W Emerson)

Walter F Buckley - Collected Quotes

"A synthetic approach where piecemeal analysis is not possible due to the intricate interrelationships of parts that cannot be treated out of context of the whole;" (Walter F Buckley, "Sociology and modern systems theory", 1967)

"A viewpoint that gets at the heart of sociology because it sees the sociocultural system in terms of information and communication nets [...]" (Walter F Buckley, "Sociology and modern systems theory", 1967)

"Adaptive system - whether on the biological, psychological, or sociocultural level - must manifest (1) some degree of 'plasticity' and 'irritability' vis-a-vis its environment such that it carries on a constant interchange with acting on and reacting to it; (2) some source or mechanism for variety, to act as a potential pool of adaptive variability to meet the problem of mapping new or more detailed variety and constraints in a changeable environment; (3) a set of selective criteria or mechanisms against which the 'variety pool' may be sifted into those variations in the organization or system that more closely map the environment and those that do not; and (4) an arrangement for preserving and/or propagating these 'successful' mappings." (Walter F Buckley," Sociology and modern systems theory", 1967)

"Basic ingredients of the decision-making focus include, then: (1) a process approach; (2) a conception of tensions as inherent in the process; and (3), a renewed concern with the role and workings of man's enlarged cortex seen as a complex adaptive subsystem operating within an interaction matrix characterized by uncertainty, conflict, and other dissociative (as well as associative) processes underlying the structuring and restructuring of the larger psychosocial system."  (Walter F Buckley," Sociology and modern systems theory", 1967)

"In essence, the process model typically views society as a complex, multifaceted, fluid interplay of widely varying degrees and intentions and intensities of association and dissociation. The "structure" is abstract construct, not something distinct from the ongoing interactive process but rather a temporary, accommodative representation of it at any one time." (Walter F Buckley," Sociology and modern systems theory", 1967)

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

"Only a modern systems approach promises to get the full complexity of the interacting phenomena - to see not only the causes acting on the phenomena under study, the possible consequences of the phenomena and the possible mutual interactions of some of these factors, but also to see the total emergent processes as a function of possible positive and/or negative feedbacks mediated by the selective decisions, or 'choices', of the individuals and groups directly involved." (Walter F Buckley, "Sociology and modern systems theory", 1967)

"That a system is open means, not simply that it engages in interchanges with the environment, but that this interchange is an essential factor underlying the system's viability, its reproductive ability or continuity, and its ability to change. [...] Openness is an essential factor underlying a system's viability, continuity, and its ability to change."  (Walter F Buckley, "Sociology and modern systems theory", 1967)

"[The equilibrium model describes systems] which, in moving to an equilibrium point, typically lose organization, and then tend to hold that minimum level within relatively narrow conditions of disturbance." (Walter F Buckley, "Sociology and modern systems theory", 1967)

"A more viable model, one much more faithful to the kind of system that society is more and more recognized to be, is in process of developing out of, or is in keeping with, the modern systems perspective (which we use loosely here to refer to general systems research, cybernetics, information and communication theory, and related fields). Society, or the sociocultural system, is not, then, principally an equilibrium system or a homeostatic system, but what we shall simply refer to as a complex adaptive system." (Walter F Buckley, "Society as a complex adaptive system", 1968)

"We have argued at some length in another place that the mechanical equilibrium model and the organismic homeostasis models of society that have underlain most modern sociological theory have outlived their usefulness." (Walter F Buckley, "Society as a complex adaptive system", 1968)

"Systems theory, in its concern for the whole and its emergent properties, ignores the components." (Walter F. Buckley, "Society: A Complex Adaptive System - Essays in Social Theory", 1998)

"The notion of system we are interested in may be described generally as a complex of elements or components directly or indirectly related in a network of interrelationships of various kinds, such that it constitutes a dynamic whole with emergent properties." (Walter F. Buckley, "Society: A Complex Adaptive System - Essays in Social Theory", 1998)

Nicomachus of Gerasa - Collected Quotes

"Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little." (Nicomachus of Gerasa,"Introductio Arithmetica", cca. 100 AD)

"There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totaling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity." (Nicomachus of Gerasa,"Introductio Arithmetica", cca. 100 AD)

"Things […] are some of them continuous […] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes', a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end […] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude […] or with multitude […] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus of Gerasa, "Introductio Arithmetica", cca. 100 AD)

"All that has by nature, with systematic method, been arranged in the universe, seems both in part and as a whole to have been determined and ordered in accordance with number, by the forethought and the mind of him that created all things; for the pattern was fixed, like a preliminary sketch, by the domination of number preexistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things: time, motion, the heavens, the stars, all sorts of revolutions." (Nicomachus of Gerasa)

"Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd." (Nicomachus of Gerasa)

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin - for motion naturally comes after rest - nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities. So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness." (Nicomachus of Gerasa)

"Since of quantity, one kind is viewed by itself, having no relation to anything else, as 'even,' 'odd,' 'perfect,' and the like, and the other is relative to something else, and is conceived of together with its relationship to another thing, like' double,', greater,' 'smaller,' 'half,' 'one and one-half times,' 'one and one-third times,' and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity: arithmetic, [with] absolute quantity; and music, [with] relative quantity." (Nicomachus of Gerasa)

"The even is that which can be divided into two equal parts without a unit intervening in the middle; and the odd is that which cannot […]" (Nicomachus of Gerasa)

"Two other sciences in the same way will accurately treat of 'size': geometry, the part that abides and is at rest, [and] astronomy, that which moves and revolves." (Nicomachus of Gerasa)

On Numbers: Perfect Numbers II

"It is the greatest that will ever be discovered, for, as they [perfect numbers] are merely curious without being useful, it is not likely that any person will attempt to find one beyond it." (Peter Barlow, "Theory of Numbers", 1811)

“The Perfect numbers are also like the virtues, few in number; whilst the other two classes are like the vices - numerous, inordinate and indefinite.” (W Wynn Westcott, “Numbers: Their Occult Power and Mystic Virtues”, 1911)

“The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. ‘Perfect numbers’ certainly never did any good, but then they never did any particular harm.” (John E Littlewood, “A Mathematician’s Miscellany”, 1953)
”Maybe some simple combination of a dozen or so primes in fact yield an odd perfect number!” (Stan Wagon, “The evidence: perfect numbers”, Mathematical Intelligencer 7(2), 1985)

”Yet, I believe the problem stands like a unconquerable fortress. For all that is known, it would be almost by luck that an odd perfect number would be found. On the other hand, nothing that has been proved is promising to show that odd perfect numbers do not exist. New ideas are required.” (Paulo Ribenboim, “The New Book of Prime Number Records”, 1996)

“We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers.” (Stanley J Bezuszka & Margaret J Kenney, ”Even perfect numbers”, Math. Teacher 90, 1997)

”Throughout both ancient and modern history the feverish hunt for perfect numbers became a religion.” (Clifford A Pickover, “Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning”, 2001)

"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)

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Resources:Wikipedia (2018) List of perfect numbers [Online] Available from: https://en.wikipedia.org/wiki/List_of_perfect_numbers