🪷On Mind: 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)

"[…] 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)

"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)

"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)