🏅Nobel Lectures (2010-2019)

"Like writing, reading is a protest against the insufficiencies of life. When we look in fiction for what is missing in life, we are saying, with no need to say it or even to know it, that life as it is does not satisfy our thirst for the absolute – the foundation of the human condition – and should be better. We invent fictions in order to live somehow the many lives we would like to lead when we barely have one at our disposal." (Mario V Llosa, [Nobel lecture] 2010)

"Part of a meaningful quantitative analysis is to look at models and try to figure out their deficiencies and the ways in which they can be improved. A more subtle challenge for statistical methods is to explore systematically potential modeling errors in order to assess the quality of the model predictions. This kind of uncertainty about the adequacy of a model or model family is not only relevant for econometricians outside the model but potentially also for agents inside the models." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"Uncertainty, generally conceived, is not often embraced in public discussions of economic policy. When uncertainty includes incomplete knowledge of dynamic responses, we might well be led away from arguments that 'complicated problems require complicated solutions'. When complexity, even formulated probabilistically, is not fully understood by policy makers, perhaps it is the simpler policies that are more prudent. This could well apply to the design of monetary policy, environmental policy and financial market oversight. Enriching our toolkit to address formally such challenges will improve the guidance that economists give when applying models to policy analysis." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"Using random processes in our models allows economists to capture the variability of time series data, but it also poses challenges to model builders. As model builders, we must understand the uncertainty from two different perspectives. Consider first that of the econometrician, standing outside an economic model, who must assess its congruence with reality, inclusive of its random perturbations. An econometrician’s role is to choose among different parameters that together describe a family of possible models to best mimic measured real world time series and to test the implications of these models. I refer to this as outside uncertainty. Second, agents inside our model, be it consumers, entrepreneurs, or policy makers, must also confront uncertainty as they make decisions. I refer to this as inside uncertainty, as it pertains to the decision-makers within the model. What do these agents know? From what information can they learn? With how much confidence do they forecast the future? The modeler’s choice regarding insiders’ perspectives on an uncertain future can have significant consequences for each model’s equilibrium outcomes." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"When confronted with multiple models, I find it revealing to pose the resulting uncertainty as a two-stage lottery. For the purposes of my discussion, there is no reason to distinguish unknown models from unknown parameters of a given model. I will view each parameter configuration as a distinct model. Thus a model, inclusive of its parameter values, assigns probabilities to all events or outcomes within the model’s domain. The probabilities are often expressed by shocks with known distributions and outcomes are functions of these shocks. This assignment of probabilities is what I will call risk. By contrast there may be many such potential models. Consider a two-stage lottery where in stage one we select a model and in stage two we draw an outcome using the model probabilities. Call stage one model ambiguity and stage two risk that is internal to a model." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"When there is a reference to a decision problem, an analysis with multiple priors can deduce bounds on the expected utility consequences of alternative decisions, and more generally a mapping from alternative priors into alternative expected outcomes." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"Why is it fruitful to consider model misspecification? In economics and as in other disciplines, models are intended to be revealing simplifications, and thus deliberately are not exact characterizations of reality; it is therefore specious to criticize economic models merely for being wrong. The important criticisms are whether our models are wrong in having missed something essential to the questions under consideration." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

🏅Nobel Lectures (1960-1969)

"Physics does not endeavour to explain nature. In fact, the great success of physics is due to a restriction of its objectives: it only endeavours to explain the regularities in the behavior of objects." (Eugene P Wigner, "Events, Laws of Nature, and Invariance Principles", [Nobel Lecture], 1963)

"We have ceased to expect from physics an explanation of all events, even of the gross structure of the universe, and we aim only at the discovery of the laws of nature, that is the regularities, of the events." (Eugene P Wigner, "Events, Laws of Nature, and Invariance Principles", [Nobel Lecture], 1963)

"I think the problem is not to find the best or most efficient method to proceed to a discovery, but to find any method at all." (Richard P Feynman, "The Development of the Space-Time View of Quantum Electrodynamics", [Nobel lecture] 1965)

"Imagination reaches out repeatedly trying to achieve some higher level of understanding, until suddenly I find myself momentarily alone before one new corner of nature’s pattern of beauty and true majesty revealed." (Richard P Feynman, "The Development of the Space-Time View of Quantum Electrodynamics", [Nobel lecture] 1965)

"Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing." (Richard P Feynman, "The Development of the Space-Time View of Quantum Electrodynamics", [Nobel lecture] 1965)

"The chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction - a direction obvious from an unfashionable view of field theory - who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unfashionable point of view; one that he may have to invent for himself." (Richard P Feynman, "The Development of the Space-Time View of Quantum Electrodynamics", [Nobel lecture] 1965)

"I have often had cause to feel that my hands are cleverer than my head. That is a crude way of characterizing the dialectics of experimentation. When it is going well, it is like a quiet conversation with Nature. One asks a question and gets an answer; then one asks the next question, and gets the next answer. An experiment is a device to make Nature speak intelligibly. After that one has only to listen." (George Wald, "The Molecular Basis of Visual Excitation", [Nobel Lecture] 1967)

"Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop." (Chen Ning Yang, [Nobel lecture] 1967)

"Experimental physicists [...] walk a narrow path with pitfalls on either side. If we spend all our time developing equipment, we risk the appellation of 'plumber', and if we merely use the tools developed by others, we risk the censure of our peers for being parasitic." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

"Most of us who become experimental physicists do so for two reasons; we love the tools of physics because to us they have intrinsic beauty, and we dream of finding new secrets of nature as important and as exciting as those uncovered by our scientific heroes." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

"How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature? Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really refl ects the interconnectedness of Nature’s laws at different levels." (Murray Gell-Mann,"[Nobel speech] 1969) 

🏅Nobel Lectures (1970-1979)

"As long as economic theory still works on a purely qualitative basis without attempting to measure the numerical importance of the various factors, practically any 'conclusion' can be drawn and defended." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)

"A work of art bears within itself its own verification: conceptions which are devised or stretched do not stand being portrayed in images, they all come crashing down, appear sickly and pale, convince no one. But those works of art which have scooped up the truth and presented it to us as a living force - they take hold of us, compel us, and nobody ever, not even in ages to come, will appear to refute them." (Aleksandr Solzhenitsyn, [Nobel lecture], 1970)

"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)

"Science is a field which grows continuously with ever expanding frontiers. Further, it is truly international in scope. Any particular advance has been preceded by the contributions of those from many lands who have set firm foundations for further developments. The Nobel awards should be regarded as giving recognition to this general scientific progress as well as to the individuals involved. Further, science is a collaborative effort. The combined results of several people working together is often much more effective than could be that of an individual scientist working alone." (John Bardeen, [Nobel lecture] 1972) 

"For a scientist, it is a unique experience to live through a period in which his field of endeavor comes to bloom - to be witness to those rare moments when the dawn of understanding fi nally descends upon what appeared to be confusion only a while ago - to listen to the sound of) darkness crumbling." (George E Palade, [Nobel lecture] 1974

"Life, this anti-entropy, ceaselessly reloaded with energy, is a climbing force, toward order amidst chaos, toward light, among the darkness of the indefinite, toward the mystic dream of Love, between the fire which devours itself and the silence of the Cold. Such a Nature does not accept abdication, nor skepticism." (Albert Claude, [Nobel lecture] 1974)

"A solution of newly appearing economic problems, and in particular those connected with the scientific-technical revolution often cannot be based on existing methods but needs new ideas and approaches. Such one is the problem of the protection of nature. The problem of economic valuation of technical innovations efficiency and rates of their spreading cannot be solved only by the long-term estimation of direct outcomes and results without accounting peculiarities of new industrial technology, its total contribution to technical progress." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)

"In infinite space many civilizations are bound to exist, among them societies that may be wiser and more 'successful' than ours. I support the cosmological hypothesis which states that the development of the universe is repeated in its basic characteristics an infi nite number of times. [...] Yet this should not minimize our sacred endeavors in this world of ours, where, like faint glimmers in the dark, we have emerged for a moment from the nothingness of dark unconscious into material existence." (Andrei Sakharov, "Peace, Progress, Human Rights", [Nobel lecture] 1975)

"In our time mathematics has penetrated into economics so solidly, widely and variously, and the chosen theme is connected with such a variety of facts and problems that it brings us to cite the words of Kozma Prutkov which are very popular in our country: 'One can not embrace the unembraceable'. The appropriateness of this wise sentence is not diminished by the fact that the great thinker is only a pen-name." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)

"The hard thing in a model realization is to receive and often to construct necessary data which in many cases have considerable errors and sometimes are completely absent, since none needed them previously. Difficulties of principle lie in the future prediction data and in the estimation of industry development variants." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)

"The treatment of the economy as a single system, to be controlled toward a consistent goal, allowed the efficient systematization of enormous information material, its deep analysis for valid decision-making. It is interesting that many inferences remain valid even in cases when this consistent goal could not be formulated, either for the reason that it was not quite clear or for the reason that it was made up of multiple goals, each of which to be taken into account." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)

"The accounting methods based on mathematical models, the use of computers for computations and information data processing make up only one part of the control mechanism, another part is the control structure." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)

"Cosmology is a science which has only a few observable facts to work with." (Robert W Wilson, [Nobel lecture] 1978)

"[...] scientific thought and its creation is the common and shared heritage of mankind." (Abdus Salam, "Gauge Unifi cation of Fundamental Forces", [Nobel lecture] 1979)

"Tapestries are made by many artisans working together. The contributions of separate workers cannot be discerned in the completed work, and the loose and false threads have been covered over. So it is in our picture of particle physics." (Sheldon L Glashow, "Towards a Unified Theory - Threads in a Tapestry", [Nobel lecture] 1979)

"The confusion of the past [in particle physics] is now replaced by a simple and elegant synthesis. [This] standard theory may survive as a part of the ultimate theory, or it may turn out to be fundamentally wrong. In either case, it will have been an important way-station, and the next theory will have to be better." (Sheldon L Glashow, "Towards a Unified Theory - Threads in a Tapestry", [Nobel lecture] 1979)

🏅Nobel Lectures (1980-1999)

"At any one time there is a natural tendency among physicists to believe that we already know the essential ingredients of a comprehensive theory. But each time a new frontier of observation is broached we inevitably discover new phenomena which force us to modify substantially our previous conceptions. I believe this process to be unending, that the delights and challenges of unexpected discovery will continue always." (Val Logsdon, [Nobel lecture] 1980)

"Earlier contentions that the right hemisphere is not even conscious largely gave way by the mid seventies to an intermediate position conceding that the mute hemisphere may be conscious at some lower elemental levels, but claiming that it lacks the higher, reflective, self-conscious kind of inner awareness that is special to the human mind and is needed, so it is said, to qualify the right conscious system as a 'self' or 'person'. Self awareness in particular is reported, on the basis of mirror tests mainly, to be a predominantly human attribute and is rated by developmental as well as by evolutionary standards to be a highly advanced phase of conscious awareness." (Roger W Sperry, [Nobel lecture], 1981)

"There are almost unlimited possibilities for making discoveries and to uncover the unknown. It is in the nature of the discovery that it can not be planned or programmed. On the contrary it consists of surprises and appears many times in the most unexpected places. However, the basis of the discovery is imagination, careful reasoning and experimentation where the use of knowledge created by those who came before is an important component." (Bengt I Samuelsson, [Nobel speech] 1982) 

"A black hole partitions the three-dimensional space into two regions: an inner region which is bounded by a smooth two-dimensional surface called the event horizon; and an outer region, external to the event horizon, which is asymptotically flat; and it is required (as a part of the definition) that no point in the inner region can communicate with any point of the outer region. This incommunicability is guaranteed by the impossibility of any light signal, originating in the inner region, crossing the event horizon. The requirement of asymptotic flatness of the outer region is equivalent to the requirement that the black hole is isolated in space and that far from the event horizon the space-time approaches the customary space-time of terrestrial physics." (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability", [Nobel lecture] 1983)

"It is the great glory of the quest for human knowledge that, while making some small contribution to that quest, we can also continue to learn and to take pleasure in learning." (William A Fowler, [Nobel lecture] 1983)

"The mathematical theory of black holes is a subject of immense complexity; but its study has convinced me of the basic truth of the ancient mottoes 'The simple is the seal of the true' and 'beauty is the splendor of truth.'" (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability", [Nobel lecture] 1983)

"Turning to the physical properties of the black holes, we can study them best by examining their reaction to external perturbations such as the incidence of waves of different sorts. Such studies reveal an analytic richness of the Kerr space-time which one could hardly have expected. This is not the occasion to elaborate on these technical matters. Let it suffice to say that contrary to every prior expectation, all the standard equations of mathematical physics can be solved exactly in the Kerr space-time. And the solutions predict a variety and range of physical phenomena which black holes must exhibit in their interaction with the world outside." (Subrahmanyan Chandrasekhar, "On Stars, Their Evolution, and Their Stability", [Nobel lecture] 1983)

"To apply tools of science, physicians must learn to think like scientists. They must acquire technical ability, taste in evaluating experiments, and a sense of creative adventure." (Michael S Brown, [Nobel  lecture] 1985)

"All science is based on models, and every scientific model comprises three distinct stages: statement of well-defined hypotheses; deduction of all the consequences of these hypotheses, and nothing but these consequences; confrontation of these consequences with observed data." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Nobel lecture] 1988

"However, mathematics is not and cannot be anything more than a tool, and all my work rests on the conviction that, in its use, the only two really fruitful stages in the scientific approach are, firstly, a thorough examination of the initial hypotheses; and secondly, a discussion of the meaning and empirical relevance of the results obtained. What remains is but tautological calculation, which is of interest only to the mathematician, and the mathematical rigour of the reasoning can never justify a theory based on postulates if these postulates do not correspond to the true nature of the observed phenomena." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Nobel lecture] 1988) 

"The use of even the most sophisticated forms of mathematics can never be considered as a guarantee of quality. Mathematics is, and can only be, a means of expression and reasoning. The real substance on which the economist works remains economic and social. Indeed, one must avoid the development of a complex mathematical apparatus whenever it is not strictly indispensable. Genuine progress never consists in a purely formal exposition, but always in the discovery of the guiding ideas which underlie any proof. It is these basic ideas which must be explicitly stated and discussed." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Nobel lecture] 1988)

"The submission to observed or experimental data is the golden rule which dominates any scientific discipline. Any theory whatever, if it is not verified by empirical evidence, has no scientific value and should be rejected. This is true, for example, of the contemporary theories of general economic equilibrium." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Nobel lecture] 1988)

"Life, like the first blooming, emerges tantalizing to the curious: Interlocked questions arising from the mysterious encompassing matters quite serious." (Martin Rodbell, [Nobel lecture] 1994) 

"Experimental science is a craft and an art, and part of the art is knowing when to end a fruitless experiment. There is a danger of becoming obsessed with a fruitless experiment even if it goes nowhere." (Martin L Perl, [lecture] 1995)

"The chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction - a direction obvious from an unfashionable view of field theory / who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unfashionable point of view; one that he may have to invent for himself." (Robert Curl, [Nobel lecture] 1996)

"I realized that nature is filled with a limitless number of wonderful things which have causes and reasons like anything else but nonetheless cannot be forseen but must be discovered, for their subtlety and complexity transcends the present state of science." (Robert B Laughlin, [Nobel lecture] 1998)

"Real understanding of a thing comes from taking it apart oneself, not reading about it in a book or hearing about it in a classroom. To this day I always insist on working out a problem from the beginning without reading up on it first, a habit that sometimes gets me into trouble but just as often helps me see things my predecessors have missed" (Robert B Laughlin, [Nobel lecture] 1998)

"I have no idea whether the properties of the universe as we know it are fundamental or emergent, but I believe that the mere possibility of the latter should give the string theorists pause, for it would imply that more than one set of microscopic equations is consistent with experiment - so that we are blind to these equations until better experiments are designed - and also that the true nature of the microscopic equations is irrelevant to our world." (Robert B Laughlin, "Fractional quantization", Reviews of Modern Physics vol. 71 (4), [Nobel lecture] 1999)

🏅Nobel Lectures (2000-2009)

"Ultimately, progress in applications is not deterministic, but opportunistic, exploiting for new applications whatever new science and technology happen to be coming along." (Herbert Kroemer, "Quasi-Electric Fields and Band Offsets: Teaching Electrons New Tricks" [Nobel lecture] 2000) 

"Scientists tell us that the world of nature is so small and interdependent that a butterfly flapping its wings in the Amazon rainforest can generate a violent storm on the other side of the earth. This principle is known as the 'Butterfly Effect". Today, we realize, perhaps more than ever, that the world of human activity also has its own "Butterfly Effect'" (Kofi Annan, [Nobel lecture] 2001)

"The need for general scientific understanding by the public has never been larger, and the penalty for scientific illiteracy never harsher… Lack of scientific fundamentals causes people to make foolish decisions about issues such as the toxicity of chemicals, the efficacy of medicines, the changes in the global climate." (Peter Agre, [Nobel lecture] 2003)

"The people who do make big discoveries are the ones who somehow manage to free themselves from conventional ways of thinking and to see the subject from a new perspective." (Anthony James, [Nobel lecture] 2003)

"In many real-world situations, cooperation may be easier to sustain in a long-term relationship than in a single encounter. Analyses of short-run games are, thus, often too restrictive. Robert Aumann was the first to conduct a full-fledged formal analysis of so-called infinitely repeated games. His research identified exactly what outcomes can be upheld over time in long-run relations." (Robert Auman, [Nobel lecture] 2003) 

"[...] the boundaries between chemistry, biology, physics and medicine are rapidly disappearing. Only a comprehensive understanding of the chemical and physical processes in our cells and organ systems will yield the insights needed to develop rational approaches to the prevention and treatment of disease." (Avram Hershko, [Nobel lecture] 2004)

“The universe has been expanding since the big bang, thus early on it was hot and dense. To trace the history of the universe we must understand the dynamics that operates when the universe was hot and particles were very energetic. Before the standard model we could not go back further than 200,000 years after the big bang. Today, especially since QCD simplifies at high energy, we can extrapolate to very eary times, where nucleons melt and quarks and gluons are liberated to form a quark-gluon plasma." (David Gross "The Discovery of Asymptotic Freedom and the Emergence of QCD", [Nobel lecture] 2004)

"In many real-world situations, cooperation may be easier to sustain in a long-term relationship than in a single encounter. Analyses of short-run games are, thus, often too restrictive. Robert Aumann was the first to conduct a full-fledged formal analysis of so-called infinitely repeated games. His research identified exactly what outcomes can be upheld over time in long-run relations." (Robert Aumann, [Nobel lecture] 2005)

"I think game theory creates ideas that are important in solving and approaching conflict in general." (Robert Aumann, [Nobel lecture] 2005)

"Humans have a more complex motivational structure and more capability to solve social dilemmas than posited in earlier rational-choice theory. Designing institutions to force (or nudge) entirely self-interested individuals to achieve better outcomes has been the major goal posited by policy analysts for governments to accomplish for much of the past half century. Extensive empirical research leads me to argue that instead, a core goal of public policy should be to facilitate the development of institutions that bring out the best in humans."  (Elinor Ostrom, [Nobel lecture] 2009) 

"We should continue to use simple models where they capture enough of the core underlying structure and incentives that they usefully predict outcomes. When the world we are trying to explain and improve, however, is not well described by a simple model, we must continue to improve our frameworks and theories so as to be able to understand complexity and not simply reject it." (Elinor Ostrom, [Nobel lecture] 2009)

🪷On Mind: On Reasoning (1960-1869)

 "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)

"[...] observation is the mind's support in reasoning [...]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"An anticipative idea or an hypothesis is, then, the necessary starting point for all experimental reasoning. Without it, we could not make any investigation at all nor learn anything; we could only pile up sterile observations. If we experiment without a preconceived idea, we should move at random […]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Considered by itself, the experimental method is nothing but reasoning by whose help we methodically submit our ideas to experience, - the experience of facts." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Every process of what has been called Universal Geometry - the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them - is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics." (John S Mill, "An Examination of Sir William Hamilton’s Philosophy", 1865)

"If the facts used as a basis of reasoning are ill-established or erroneous, everything will crumble or be falsified; and it is thus that errors in scientific theories most often originate in errors of fact." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"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)

"Speaking concretely, when we say ‘making experiments or making observations’, we mean that we devote ourselves to investigation and to research, that we make attempts and trials in order to gain facts from which the mind, through reasoning, may draw knowledge or instruction. Speaking in the abstract, when we say, ‘relying on observation and gaining experience’, we mean that observation is the mind’s support in reasoning, and experience the mind’s support in deciding, or still better, the fruit of exact reasoning applied to the interpretation of facts. Observation, then, is what shows facts; experiment is what teaches about facts and gives experience in relation to anything." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"The degree in which each mind habitually substitutes signs for images will be, CETERIS PARIBUS [with other conditions remaining the same], the degree in which it is liable to error. This is not contradicted by the fact that mathematical, astronomical, and physical reasonings may, when complex, be carried on more successfully by the employment of signs; because in these cases the signs themselves accurately represent the abstractness of the relations. Such sciences deal only with relations, and not with objects; hence greater simplification ensures greater accuracy. But no sooner do we quit this sphere of abstractions to enter that of concrete things, than the use of symbols becomes a source of weakness. Vigorous and effective minds habitually deal with concrete images." (George H Lewes, "The Principles of Success in Literature", 1865)

"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)

🪷On Mind: On Reasoning (1880-1889)

"[…] it must be noticed that these diagrams do not naturally harmonize with the propositions of ordinary life or ordinary logic. […] The great bulk of the propositions which we commonly meet with are founded, and rightly founded, on an imperfect knowledge of the actual mutual relations of the implied classes to one another. […] one very marked characteristic about these circular diagrams is that they forbid the natural expression of such uncertainty, and are therefore only directly applicable to a very small number of such propositions as we commonly meet with." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"Symbolical reasoning may be said to have pretty much the same relation to ordinary reasoning that machine-labour has to manual labour. In the case of machine labour we see some ingeniously contrived arrangement of wheels, levers, &c., producing with speed and facility results which the hands of man without such aid could only accomplish slowly and with difficulty, or which they would be utterly powerless to accomplish at all. In the case of symbolical reasoning we find in an analogous manner some regular system of rules and formulae, easy to retain in the memory from their general symmetry and interdependence, economizing or superseding the labour of the brain, and enabling any ordinary mind to obtain by simple mechanical processes results which would be beyond the reach of the strongest intellect if left entirely to its own resources." (Hugh MacColl, Symbolical reasoning. Mind 5" (17), 1880)

"We endeavour to employ only symmetrical figures, such as should not only be an aid to reasoning, through the sense of sight, but should also be to some extent elegant in themselves." (John Venn, "Symbolic Logic", 1881)

"[…] in the Law of Errors we are concerned only with the objective quantities about which mathematical reasoning is ordinarily exercised; whereas in the Method of Least Squares, as in the moral sciences, we are concerned with a psychical quantity - the greatest possible quantity of advantage." (Francis Y Edgeworth, "The method of least squares", 1883)

"Our reasoning appears to become more accurate as our ignorance becomes more complete; that when we have embarked upon chaos we seem to drop down into a cosmos." (Francis Y Edgeworth, "The Philosophy of Chance", Mind Vol. 9, 1884) 

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

"I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream, - not any particular existence, and yet not general. At that moment we are contemplating an icon." (Charles S Peirce, "On The Algebra of Logic : A Contribution to the Philosophy of Notation" in The American Journal of Mathematics 7, 1885)

"Observations and statistics agree in being quantities grouped about a Mean; they differ, in that the Mean of observations is real, of statistics is fictitious. The mean of observations is a cause, as it were the source from which diverging errors emanate. The mean of statistics is a description, a representative quantity put for a whole group, the best representative of the group, that quantity which, if we must in practice put one quantity for many, minimizes the error unavoidably attending such practice. Thus measurements by the reduction of which we ascertain a real time, number, distance are observations. Returns of prices, exports and imports, legitimate and illegitimate marriages or births and so forth, the averages of which constitute the premises of practical reasoning, are statistics. In short, observations are different copies of one original; statistics are different originals affording one ‘generic portrait’. Different measurements of the same man are observations; but measurements of different men, grouped round l’homme moyen, are prima facie at least statistics." (Francis Y Edgeworth, 1885)

"There is no more remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with, and justify, the conclusions to which mankind have been led, not by reasoning, but by instinct and experience, both of the individual and of the race. At the same time it has corrected, extended, and invested them with a definiteness and precision of which these crude, though sound, appreciations of common sense were till then devoid." (Morgan W Crofton, "Probability", Encyclopaedia Britannica 9th Ed,, 1885)

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

"There are three aspects of thinking as more or less complete stages of it. These are, conception, judgment, and reasoning. They are not to be considered three distinct acts; not even three successive stages. No one of them could occur without each of the others." (John Dewey, "Psychology", 1887) 

🪷On Mind: On Reasoning (1870-1879)

 "Ideas are substitutions which require a secondary process when what is symbolized by them is translated into the images and experiences it replaces; and this secondary process is frequently not performed at all, generally only performed to a very small extent. Let anyone closely examine what has passed in his mind when he has constructed a chain of reasoning, and he will be surprised at the fewness and faintness of the images which have accompanied the ideas." (George H Lewes "Problems of Life and Mind", 1873)

“Nor do I know any study which can compete with mathematics in general in furnishing matter for severe and continued thought. Metaphysical problems may be even more difficult; but then they are far less definite, and, as they rarely lead to any precise conclusion, we miss the power of checking our own operations, and of discovering whether we are thinking and reasoning or merely fancying and dreaming." (Isaac Todhunter,"Conflict of Studies and Other Essays", 1873)

"With Algebra we enter a new sphere, that of symbolical quantities; here letters are symbols of any values we please; all we deal with in them is the relations of equality which the letters symbolise. Although the values are changeable, jet, once assigned, they must remain fixed throughout the operation. Illogical reasoning, in philosophic as in ordinary minds, is not due to any irregularity in the normal operation, but to a departure &om the values assigned." (George H Lewes "Problems of Life and Mind", 1873)

"By deductive reasoning and calculation, we must endeavor to anticipate such new phenomena, especially those of a singular and exceptional nature, as would necessarily happen if the hypothesis be true." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874) 

"Deduction is certain and infallible, in the sense that each step in deductive reasoning will lead us to some result, as certain as the law itself. But it does not follow that deduction will lead the reasoner to every result of a law or combination of laws." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"It must be the ground of all reasoning and inference that what is true of one thing will be true of its equivalent, and that under carefully ascertained conditions Nature repeats herself." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. [...] Numerical precision is the soul of science [...]" (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"There is nothing physical to be learned a priori. We have no right whatever to ascertain a single physical truth without seeking for it physically, unless it be a necessary consequence of other truths already acquired by experiment, in which case mathematical reasoning is alone requisite." (Peter G Tait, "Lectures on Some Recent Advances in Physical Science, With a Special, Lecture on Force", 1876)

"The very genius of the common geometry as a method of reasoning - a system of investigation - is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn" (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds." (Edward Olney, "Mathematics", The Cyclopedia of Education, 1877

 

🪷On Mind: On Reasoning (2010-2019)

"Infinite reasoning is likewise essential for graph theory. The field had its origins in topology, and it is still relevant there, but it has expanded extraordinarily far in other directions. Graph theory today is exploring the boundaries of finite provability first exposed by Gödel’s incompleteness theorem." (John Stillwell, "Mathematics and Its History", 2010)

"Mathematics is a model of exact reasoning, the most precise branch of human knowledge." (Paul Hartal, "Mathematics and Reality", 2010)

"This is always the case in analogical reasoning: Relations between two dissimilar domains never map completely to one another. In fact, it is often the salient similarities between the base and target domains that provoke thought and increase the usefulness of an analogy as a problem-solving tool." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Venn diagramming, it turns out, is a very effective technique for performing syllogistic reasoning. Its chief advantage (over the Euler graph in particular as we noted earlier) is the ability to incrementally add knowledge to the diagram. While an Euler graph has visual power in terms of representing the relations between sets very intuitively, it is impossible to combine more than one piece of information onto a Euler graph. A Venn diagram, on the other hand, easily lends itself to the representation of partial knowledge and can be manipulated to add successively more knowledge to the diagram. This means that when our knowledge of the relations between sets increases, we simply put in more symbols and shadings into the appropriate compartments of the Venn diagram. Thus we are able to accumulate knowledge in a Venn diagram. This capability turns out to be a powerful feature, one that endows Venn diagrams with a more dynamic quality that is sorely lacking in the Euler system." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"[...] a model is a tool for taking decisions and any decision taken is the result of a process of reasoning that takes place within the limits of the human mind. So, models have eventually to be understood in such a way that at least some layer of the process of simulation is comprehensible by the human mind. Otherwise, we may find ourselves acting on the basis of models that we don’t understand, or no model at all." (Ugo Bardi, "The Limits to Growth Revisited", 2011)

"Graphs are gimmicks, substituting fancy displays for careful analysis and rigorous reasoning. It is basically a trade-off: the snazzier your display, the more you can get away with a crappy underlying analysis. Conversely, a good analysis does not need a fancy graph to sell itself. The best quantitative research has an underlying clarity and a substantive importance whose results are best presented in a sober, serious tabular display. And the best quantitative researchers trust their peers enough to present their estimates and standard errors directly, with no tricks, for all to see and evaluate." (Andrew Gelman et al, "Why Tables Are Really Much Better Than Graphs", Journal of Computational and Graphical Statistics, Vol. 20(1), 2011)

"It turns out that our knowledge is always too incomplete and our visual data is too noisy and cluttered to be interpreted by deduction. In this situation, the method of reasoning needed to parse a real-world scene must be statistical, not deductive. To implement this form of reasoning, our knowledge of the world must be encoded in a probabilistic form, known as an a priori probability distribution." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity [...] (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"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) 

What  brings  us  mathematical  knowledge?  The  carriers  of mathematical knowledge  are  proofs, more  generally  arguments  and constructions,  as embedded  in  larger  contexts. Mathematicians  and teachers  of higher mathematics  know  this,  but  it  should  be  said. Issues  about  competence  and intuition  can be  raised as  well  as  factors  of knowledge  involving  the general  dissemination  of analogical  or inductive  reasoning  or the  specific conveyance  of methods,  approaches  or ways  of thinking.  But  in the  end, what  can be  directly  conveyed  as  knowledge  are  proofs." (Akihiro Kanamori, "Mathematical  Knowledge:  Motley and  Complexity of Proof",  Annals  of  the  Japan Association  for  Philosophy  of  Science Vol. 21, 2013)

"In natural language, even the most carefully chosen words drag along concealed meanings that have the power to manipulate reasoning. [...] Symbols of mathematics too sometimes have concealed meanings, but their purpose is to bring along pure thought. It is possible to learn what a mathematical symbol stands for by context. We learn the meanings of mathematical symbols mostly from their definitions: Mostly, because in formal mathematics not everyone easily grasps definitions that are not linked to the familiar properties of experience." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Deductive and inductive reasoning schemas essentially regulate inferences. They tell us what kinds of inferences are valid and what kinds are invalid. […] Dialectical reasoning isn’t formal or deductive and usually doesn’t deal in abstractions. It’s concerned with reaching true and useful conclusions rather than valid conclusions. In fact, conclusions based on dialectical reasoning can actually be opposed to those based on formal logic." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Mathematics is both abstract and concrete, revealing much of the mental experiment, working with unobserved abstractions and objects, and the current scientific progress depended on the ability to operate precisely with abstractions and force of reasoning; […]" (Octavian Stanasila, Metabolism of Mathematics and Computer Science No. 8, 2015)

"Science is often described as a 'seamless web'. What’s meant by that is that the facts, methods, theories, and rules of inference discovered in one field can be helpful for other fields. And philosophy and logic can affect reasoning in literally every field of science.(Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Since we cannot completely eliminate uncertainty, we need to model it. In real life when we are faced with uncertainty, we use plausible reasoning. We adjust our belief about something, based on the occurrence or nonoccurrence of something else." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

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

"In any analysis of any part of the world, it is mandatory to institute a general reasoning in which the whole - the Absolute - is also included. This is what science scrupulously avoids. Science is all about the parts, and ignoring the whole. Science is non-holistic, which is why it cannot arrive at a grand unified, final theory of everything. From the whole you can get to every part, because the whole defines the parts. If you start with the parts, as science does, you can never get to the whole because the parts are necessarily defined piecemeal, heuristically and with no regard to the whole, since the whole is unknown. A bottom-up approach can never work. Only top-down approaches have any chance of working. Empiricists are always parts people and bottom-up people. Rationalists are holistic and top-down. These are opposite worldviews. The PSR is an explanatory, top-down principle. Randomness is a non-explanatory, bottom-up speculation." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"When the scientific question of interest involves retrospective thinking, we call on another type of expression unique to causal reasoning called a counterfactual. […] Counterfactuals are the building blocks of moral behavior as well as scientific thought. The ability to reflect on one’s past actions and envision alternative scenarios is the basis of free will and social responsibility. The algorithmization of counterfactuals invites thinking machines to benefit from this ability and participate in this" (until now) uniquely human way of thinking about the world. " (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

🪷On Mind: On Reasoning (2000-2009)

"Arithmetic and number theory study patterns of number and counting. Geometry studies patterns of shape. Calculus allows us to handle patterns of motion. Logic studies patterns of reasoning. Probability theory deals with patterns of chance. Topology studies patterns of closeness and position." (Keith Devlin, "The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip", 2000)

"Yet, there is a perfectly good definition of the subject, or rather of the concept of a combinatorial argument. Combinatorics is the area of mathematics that is concerned with, relates to, employs, or studies combinatorial arguments. So what is a combinatorial argument? [...] A combinatorial argument is the one that consists predominately of ingenuity or detailed reasoning rather than knowledge of existing mathematics. This is in contrast with knowledge-based argument, which relies heavily on piecing together known results." (Daniel J Kleitman, "On the future of combinatorics", 2000) 

"A mental model is a representation of some domain or situation that supports understanding, reasoning, and prediction. Mental models permit reasoning about situations not directly experienced. They allow people to mentally simulate the behavior of a system. Many mental models are based on generalizations and analogies from experience." (D Gentner, "Psychology of Mental Models" [in "International Encyclopedia of the Social & Behavioral Sciences"], 2001)

"A mental model is a representation of some domain or situation that supports understanding, reasoning, and prediction." (D Gentner, "Psychology of Mental Models" [in "International Encyclopedia of the Social & Behavioral Sciences"], 2001)

"External representations are not just passive inputs to an active mind. They can do part of the reasoning or calculation, making relevant properties of the same information more accessible." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"In chaos theory this 'butterfly effect' highlights the extreme sensitivity of nonlinear systems at their bifurcation points. There the slightest perturbation can push them into chaos, or into some quite different form of ordered behavior. Because we can never have total information or work to an infinite number of decimal places, there will always be a tiny level of uncertainty that can magnify to the point where it begins to dominate the system. It is for this reason that chaos theory reminds us that uncertainty can always subvert our attempts to encompass the cosmos with our schemes and mathematical reasoning." (F David Peat, "From Certainty to Uncertainty", 2002)

"We build models to increase productivity, under the justified assumption that it's cheaper to manipulate the model than the real thing. Models then enable cheaper exploration and reasoning about some universe of discourse. One important application of models is to understand a real, abstract, or hypothetical problem domain that a computer system will reflect. This is done by abstraction, classification, and generalization of subject-matter entities into an appropriate set of classes and their behavior." (Stephen J Mellor, "Executable UML: A Foundation for Model-Driven Architecture", 2002) 

"Both induction and deduction, reasoning from the particular and the general, and back again from the universal to the specific, form the essence to scientific thinking." (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Art and Visual Perception: A Psychology of the Creative Eye", 2004)

"In mathematics a good diagram is an invaluable aid to clear reasoning, whereas a bad one can seriously mislead. But it is not often that a diagram is good enough to suggest fresh avenues of inquiry altogether." (Anthony W F Edwards, "Cogwheels of the mind: The story of Venn diagrams", 2004)

"The process of constructing a map thus allows you to reflect on your own thinking, or that of someone else. Mapping helps you become conscious of your reasoning and to understand clearly what another person is thinking. In turn, that consciousness and understanding can help you reaffirm what you think or else change your mind. Shared understanding and mutual agreement also become possible." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"This covert mechanism would be the source of what we call intuition, the mysterious mechanism by which we arrive at the solution of a problem without reasoning toward it." (Antonio Damasio,"Descartes' Error", 2005)

"What a wealth of insight Euler’s formula reveals and what delicacy and precision of reasoning it exhibits. It provides a definition of complex exponentiation: It is a definition of complex exponentiation, but the definition proceeds in the most natural way, like a trained singer’s breath. It closes the complex circle once again by guaranteeing that in taking complex numbers to complex powers the mathematician always returns with complex numbers. It justifies the method of infinite series and sums. And it exposes that profound and unsuspected connection between exponential and trigonometric functions; with Euler’s formula the very distinction between trigonometric and exponential functions acquires the shimmer of a desert illusion." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"With our heads spinning in the world of coincidence and chaos, we nevertheless must make decisions and take steps into the minefield of our future. To avoid explosive missteps, we rely on data and statistical reasoning to inform our thinking." (Michael Starbird, "Coincidences, Chaos, and All That Math Jazz", 2005) 

“It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"Some number patterns, like even and odd numbers, lie on the surface. But the more you learn about numbers, both experimentally and theoretically, the more you discover patterns that are not so obvious. […] After a hidden pattern is exposed, it can be used to find more hidden patterns. At the end of a long chain of patterned reasoning, you can get to very difficult theorems, exploring facts about numbers that you otherwise would not know were true." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"According to mental model theory, human reasoning relies on the construction of integrated mental representations of the information that is given in the reasoning problem's premises. These integrated representations are the mental models. A mental model is a mental representation that captures what is common to all the different ways in which the premises can be interpreted. It represents in small scale" how "reality" could be— according to what is stated in the premises of a reasoning problem. Mental models, though, must not be confused with images. A mental model often forms the basis of one or more visual images, but some of them represent situations that cannot be visualized. Instead, mental models are often likened to diagrams since, as with diagrams, their structure is analogous to the structure of the states of affairs they represent." (Carsten Held et al, "Mental Models and the Mind", 2006)

"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)

"It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"Logic is the study of methods and principles of reasoning, where reasoning means obtaining new propositions from existing propositions. In classical logic, propositions are required to be either true or false; that is, the truth value of a proposition is either 0 or 1. Fuzzy logic generalizes classical two-value logic by allowing the truth values of a proposition to be any numbers in [0, 1]. This generalization allows us to perform fuzzy reasoning, also called approximate reasoning; that is, deducing imprecise conclusions" (fuzzy propositions) from a collection of imprecise premises" (fuzzy propositions). In this section, we first introduce some basic concepts and principles in classical logic and then study their generalizations to fuzzy logic." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006) 

"Mental models abide by the principle of parsimony: They represent only possibilities compatible with the premises, and they represent clauses in the premises only when they hold in a possibility. Fully explicit models represent clauses when they do not hold too. The advantage of mental models over fully explicit models is that they contain less information, and so they are easier to work with. But they can lead reasoners astray. The occurrence of these systematic and compelling fallacies is shocking. The model theory predicts them, and they are a 'litmus' test for mental models, because no other current theory predicts them. They have so far resisted explanation by theories of reasoning based on formal rules of inference, because these theories rely on valid rules." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"Metaphorizing is a manner of thinking, not a property of thinking. It is a capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is" (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects." (Ivan Mladenov, "Conceptualizing Metaphors: On Charles Peirce’s marginalia", 2006)

“When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze." (William P Thurston, [preface to John H Hubbard's"Teichmüller Theory and Applications", Vol. 1, 2006])

"[…] mathematics is not only to teach the algorithms and skills of mathematics - which we will agree are very important - but also to teach for understanding, with an emphasis on reasoning." (Alfred S Posamentier et al, "Exemplary Practices for Secondary Math Teachers", 2007)

"The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an elementary" convincing argument." (Yuri I Manin, "Mathematics as Metaphor: Selected Essays of Yuri I. Manin", 2007)

"The most serious problem in applied ethics, or at least in business ethics, is not that we frame experiences; it is not that these mental models are incomplete, sometimes biased, and surely parochial. The larger problem is that most of us either individually or as managers do not realize that we are framing, disregarding data, ignoring counterevidence, or not taking into account other points of view." (Patricia H Werhane "A Place for Philosophers in Applied Ethics and the Role of Moral Reasoning in Moral Imagination", Business Ethics Quarterly 16" (3), 2007)

"There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements […]." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Fuzzy logic is an application area of fuzzy set theory dealing with uncertainty in reasoning. It utilizes concepts, principles, and methods developed within fuzzy set theory for formulating various forms of sound approximate reasoning. Fuzzy logic allows for set membership values to range" (inclusively) between 0 and 1, and in its linguistic form, imprecise concepts like 'slightly', 'quite' and 'very'. Specifically, it allows partial membership in a set." (Larbi Esmahi et al,  Adaptive Neuro-Fuzzy Systems, 2009)

"Fuzzy set theory [...] is primarily concerned with quantifying and reasoning using natural language in which words can have ambiguous meanings. It is widely used in a variety of fields because of its simplicity and similarity to human reasoning." (Tzung-Pei Hong et al, "Genetic-Fuzzy Data Mining Techniques", 2009)

"Granular computing is a general computation theory for using granules such as subsets, classes, objects, clusters, and elements of a universe to build an efficient computational model for complex applications with huge amounts of data, information, and knowledge. Granulation of an object a leads to a collection of granules, with a granule being a clump of points" (objects) drawn together by indiscernibility, similarity, proximity, or functionality. In human reasoning and concept formulation, the granules and the values of their attributes are fuzzy rather than crisp. In this perspective, fuzzy information granulation may be viewed as a mode of generalization, which can be applied to any concept, method, or theory." (Salvatore Greco et al, "Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach", 2009)

"In essence, logic is concerned with formalization of reasoning. Correspondently, fuzzy logic is concerned with formalization of fuzzy reasoning, with the understanding that precise reasoning is a special case of fuzzy reasoning." (Lofti A Zadeh, "Fuzzy Logic", 2009) 

"Mathematicians seek a certain kind of beauty. Perhaps mathematical beauty is a constant - as far as the contents of mathematics are concerned - and yet the forms this beauty takes are certainly cultural. And while the history of mathematics surely is many stranded, one of its most important strands is formed by such cultural forms of mathematical beauty." (Reviel Netz,"Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic", 2009)

"Pattern perception" (that is, the perception of similarities in spatial or temporal configurations) has a fundamental role in playing chess [...] The two essential components in decision making in chess are recognizing patterns stored in long-term memory" (which requires an exhaustive knowledge database) and searching for a solution within the problem space. The first component uses perception and long-term memory, and the second leans mainly on the calculation of variations, which in turn has its foundations in logical reasoning." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Philosophers have sometimes made a distinction between analytic and synthetic truths. Analytic truths are not verified by observation; true analytic statements are tautologies and are true by virtue of the definitions of their terms and their logical structure. Synthetic truths relate to the material world; the truth of synthetic statements depends on their correspondence to how physical reality works. Mathematics, according to this distinction, deals exclusively with analytic truths. Its statements are all tautologies and are" (analytically) true by virtue of their adherence to formal rules of construction." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"Symbolic reasoning falls short not only in modeling low level behaviors but is also difficult to ground into real world interactions and to scale upon dynamic environments […] This has lead many […] to abandon symbolic systems […] and […]  focus on parallel distributed, entirely sub-symbolic approaches […] well suited for many learning and control tasks, but difficult to apply [in] areas such as reasoning and language." (Joscha Bach, "Principles of Synthetic Intelligence PSI: An Architecture of Motivated Cognition", 2009)

"The characterization of mathematics as a deductive discipline is accurate but incomplete. It represents the finished and polished consequences of the work of mathematicians, but it does not adequately represent the doing of mathematics. It describes theorem proofs but not theorem proving. Moreover, the history of mathematics is not the emotionless chronology of inventions of evermore esoteric formalisms that some people imagine it to be. It has its full share of color, mystery, and intrigue." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

"The two essential components in decision making in chess are recognizing patterns stored in long-term memory" (which requires an exhaustive knowledge database) and searching for a solution within the problem space. The first component uses perception and long-term memory, and the second leans mainly on the calculation of variations, which in turn has its foundations in logical reasoning." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity. Why does this matter, and why does it go beyond a cultural oddity of our profession? The answer is that mathematics is deeply embedded in the reasoning used within many branches of knowledge. That reasoning often involves conjectures, assumptions, intuition. But whatever aspect has been reduced to mathematics has an absolute validity. As in other subjects search for truth, the mathematical components embedded in their search are like the boulders in the stream, providing a solid footing on which to cross from one side to the other." (James Glimm, Reflections and Prospectives", 2009)